ABSTRACT 
 
 
 
 
Title of Dissertation: ONLINE INVENTORY REPLENISHMENT 
AND FLEET ROUTING DECISIONS 
UNDER REAL-TIME INFORMATION   
  
 Ricardo Giesen, Ph.D., 2007 
Dissertation Directed By: Professor Hani S. Mahmassani, Civil and 
Environmental Engineering Department 
 
 
Logistics managers rely on increasingly sophisticated technologies to track 
demand and associated inventories, allowing rapid response to meet anticipated 
demand, avoid shortfalls while minimizing transportation and inventory carrying 
costs.  This ability to respond gives rise to complex decision problems, characterized 
by combinatorial underlying problems under progressively unfolding demand.  Real-
time information also increases the ability to coordinate effectively inventory 
management and transportation service. 
The advantages of coordinating inventory replenishment with vehicle routing 
decisions have long been recognized, giving rise to the inventory routing problem, 
which arises in the context of vendor-managed inventories.  These typify an emerging 
class of collaborative logistics arrangements facilitated by information and 
communication technologies.  The ability to coordinate inventory with routing 
decisions in real-time adds an important dimension to the problem.  While fleet 
management decisions under real-time information have been studied extensively, 
coupling these with inventory replenishment decisions in real-time remains in the 
  
early stages of conceptualization and development.  The main objective of this 
dissertation is to examine effectiveness of policies for managing inventories taking 
into consideration the interaction between inventory replenishment, retailer 
sequencing and transportation cost. 
A major motivation for the online inventory routing problem is the presence 
of uncertainty about future consumption rates at different facilities.  The possibility of 
updating plans on a continuous basis, based on real-time information about demand 
realizations makes possible decisions to modify the set and/or the sequence of 
subsequent facilities to be visited, diverting a truck from its current destination to visit 
a different facility, and adjusting amounts to be delivered to subsequent customers in 
the route. 
This dissertation proposes two decomposition approaches, in which a 
simplified version of either the inventory-control or the routing side is solved first, 
and then that solution is used as a soft constraint when solving the other side.  For 
each approach, different operational polices are proposed, reflecting different degrees 
of sophistication in terms of technology and optimization capabilities.  These 
operational policies are based on a rolling-horizon framework, wherein new plans are 
repeatedly generated, based on updated information.  Finally, the performance of 
proposed strategies is simulated and the impacts of using sophisticated real-time 
strategies are discussed. 
 
  
 
 
 
 
 
 
 
 
 
ONLINE INVENTORY REPLENISHMENT AND FLEET ROUTING 
DECISIONS UNDER REAL-TIME INFORMATION    
 
 
 
By 
 
 
Ricardo Giesen 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2007 
 
 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Hani S. Mahmassani, Chair 
Professor Patrick Jaillet 
Professor Ali Haghani 
Professor Bruce L. Golden 
Associate Professor Elise Miller-Hooks 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Ricardo Giesen 
2007 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ii 
 
Dedication 
 
 
 
 
 
 
To my parents, 
 and my family: Macarena, Tom?s & Amelia 
 
 iii 
 
Acknowledgements 
I am deeply indebted to my advisors professors Hani S. Mahmassani and 
Patrick Jaillet.  Their encouragement, assistance, patience, and perseverance played a 
fundamental role throughout my doctoral studies, and made possible the completion 
of this dissertation.  I am really grateful that they had invited me to continue working 
with them after they left the University of Texas.  The possibility to complete my 
studies at the University of Maryland and to be a visiting student at MIT, gave me a 
unique and great learning opportunity.  It has been a privilege to work with two 
mentors that I admire and respect so greatly.  Their mentorship and commitment to 
excellence will inspire me throughout my academic career. 
I would like to thank the remaining members of my dissertation committee 
professors Bruce L. Golden, Ali Haghani and Elise Miller-Hooks for their interest in 
this work and valuable comments on the proposal and defense of this dissertation.  It 
has been a real honor to have such a distinguish committee. 
I must acknowledge my friend Miguel Figliozzi for valuable comments and 
discussion on early stages of this research.  I thank my friends in the transportation 
programs at UT and UMD.  There are many but I would like to mention Yongjin 
Kim, Jordan Ludders, Pam Murray-Tuitte, Rachel Gossen, Xuesong Zhou, Xiao Qin, 
Roger Chen, Hayssam Sbayti, Jason Lu, Sevgi Erdogan, Yeonjoo Min, Stacy 
Eisenman, Jing Dong, Ivan Damnjanovic, Arild Vold, Rahul Nair, April Kuo, 
Somnuk Gnamchai, Masoud Hamedi, Felipe Targa, Evangelos Kaisar, Dan Ilescu, 
and Aaron Kosuki. 
 
 iv 
 
Appreciation also goes to Rebecca A. Weaver-Gill for her collaboration in 
administrative matters; Forest Will for his great job editing this dissertation. 
 I am also grateful to all my friends and colleagues at the Transportation and 
Logistics Engineering Program at the Catholic University of Chile for introducing me 
to transportation engineering, encouraging me to pursue graduate studies, and 
welcoming back after graduation. 
Finally, I would like to thank all my extended family.  First and foremost, I 
am deeply grateful to my wife Macarena who stood behind me in this incredible 
adventure across the U.S.A.  This dissertation might have not been possible without 
her endless love and encouragement.  I am thankful to the memory of my dad who 
early on thought me the value of giving my best and persevering.  Thanks are also 
extended to Renato and Loreto who welcomed us in Austin, my mom and mother-in-
law for their support, visits and much needed baby-sitting, Gloria for hosting me in 
Rockville during the final stage, and my brother for his support and friendship. 
 
 v 
 
Table of Contents 
 
 
Dedication..................................................................................................................... ii 
Acknowledgements......................................................................................................iii 
Table of Contents.......................................................................................................... v 
Lists of Tables............................................................................................................. vii 
Lists of Figures ..........................................................................................................viii 
Chapter 1: Introduction................................................................................................. 1 
1.1. Motivation..................................................................................................... 1 
1.1.1. The Importance of Logistics and Distribution Systems.............................. 1 
1.1.2. The Coordination of Logistic Operations ................................................... 2 
1.1.3. Technologies that Enable Operations with Real-Time Information........... 3 
1.1.4. Other Motivations ....................................................................................... 7 
1.2. Problem Statement........................................................................................ 8 
1.3. Research Context and Scope....................................................................... 10 
1.4. Research Objectives.................................................................................... 13 
1.5. Main Contributions ..................................................................................... 15 
1.6. Dissertation Structure.................................................................................. 16 
Chapter 2: Background Review and Previous Research............................................. 19 
2.1. Previous Work on Single-Item Inventory Models for a Single Facility ..... 19 
2.1.1. Inventory Costs ......................................................................................... 19 
2.1.2. EOQ Model for Constant Demand Rate ................................................... 21 
2.1.3. Models with a Stationary Stochastic Demand Rate.................................. 22 
2.2. Background in Routing and Scheduling Problems ..................................... 27 
2.2.1. Classification of Fleet Routing and Scheduling Problems ....................... 28 
2.2.2. Inventory Routing Problems (IRPs).......................................................... 30 
2.2.3. Real-Time Fleet Operations...................................................................... 40 
2.3. Background in Real-Time Combinatorial Optimization............................. 43 
Chapter 3: Problem Definition and General Approach............................................... 45 
3.1. Problem Context and Main Assumptions ................................................... 45 
3.2. Problem Formulation .................................................................................. 48 
3.2.1. Preliminaries and Problem Parameters ..................................................... 48 
3.2.2. Decision Variables.................................................................................... 51 
3.2.3. Main Constraints....................................................................................... 52 
3.2.4. Objective Function.................................................................................... 54 
3.3. General Approach ....................................................................................... 56 
3.3.1. Simulation Framework.............................................................................. 59 
3.3.2. Benchmark Policies .................................................................................. 60 
3.3.3. Performance Measures.............................................................................. 61 
Chapter 4: Formulation and Design of Optimization Based Control Strategies for 
Fixed Inventory Target Levels.................................................................................... 64 
4.1. Off-Line Optimization Problem Formulation............................................. 64 
4.1.1. Preliminaries ............................................................................................. 64 
4.1.2. Optimization of Refilling Levels .............................................................. 66 
 
 vi 
 
4.1.3. Mathematical Formulation of the Problem............................................... 68 
4.2. Optimization Based Real-Time Strategies.................................................. 74 
4.2.1. Replan at Tour Completions (RTC) Strategy ........................................... 75 
4.2.2. Replan at Deliver Epochs (RDE) Strategy................................................ 75 
4.2.3. Replan at Deliver Epochs with possible en-route diversions (RDE+div) 
Strategy ............................................................................................................... 76 
Chapter 5: Formulation and Design of Fixed-Tour Based Control Strategies............ 77 
5.1. Fixed Tour at Regular Intervals (FTRI) Strategy ....................................... 80 
5.1.1. Optimization of Refilling Levels for FTRI Strategy................................. 80 
5.1.2. Procedure for Obtaining FTRI-Strategy Tours......................................... 85 
5.2. Fixed Tour Updating Intervals (FTUI) Strategy......................................... 85 
5.2.1. Update of Truck Idle-Times on FTUI Strategy ........................................ 86 
5.2.2. Procedure Used to Implement Updating Interval Strategies..................... 89 
5.3. Fixed Tour Skipping Retailer (FTSR) Strategy.......................................... 90 
5.3.1. Skip Decision on FTSR Strategy .............................................................. 90 
5.3.2. Procedure Used to Implement FTSR Strategy.......................................... 93 
Chapter 6: Simulation Experiments............................................................................ 94 
6.1. Simulation Scenarios .................................................................................. 94 
6.1.1. Set of Fixed Parameters ............................................................................ 95 
6.1.2. Scenarios with Steady-State Demand Patterns ......................................... 98 
6.1.3. Scenarios with Unpredicted Changes in Demand Patterns..................... 100 
6.2. Simulation Results .................................................................................... 101 
6.3. Analysis of Results ................................................................................... 114 
6.3.1. Analysis of the Product Inventory Holding Costs Impact ...................... 115 
6.3.2. Analysis of Changes in Transportation Costs......................................... 116 
6.3.3. Analysis of Tour Length ......................................................................... 118 
6.3.4. Analysis of Demand-Variability Impact................................................. 120 
6.3.5. Analysis of Demand-Disruptions Scenarios ........................................... 122 
6.4. Summary of Main Results ........................................................................ 124 
Chapter 7: Conclusions............................................................................................. 126 
7.1. Summary of Contributions and Findings.................................................. 126 
7.2. Future Research and Extensions ............................................................... 129 
Appendix A: Notation............................................................................................... 133 
Appendix B: Glossary............................................................................................... 135 
Appendix C: Inventory Reorder Level Parameters................................................... 136 
Appendix D: Detailed Results .................................................................................. 148 
Bibliography ............................................................................................................. 174 
 
 
 
 vii 
 
Lists of Tables  
 
Table 2- 1: Classification of Fleet Routing and Scheduling Problems....................... 28 
Table 2- 2: Previous Research in Inventory-Routing Problems (IRPs)...................... 33 
 
Table 6- 1: Simulation Scenarios................................................................................ 99 
 
Table C- 1: Inventory Target Levels for Set of Parameters 1................................... 136 
Table C- 2: Inventory Target Levels for Set of Parameters 2................................... 137 
Table C- 3: Inventory Target Levels for Set of Parameters 3................................... 138 
Table C- 4: Inventory Target Levels for Set of Parameters 4................................... 139 
Table C- 5: Inventory Target Levels for Set of Parameters 5................................... 140 
Table C- 6: Inventory Target Levels for Set of Parameters 6................................... 141 
Table C- 7: Inventory Target Levels for Set of Parameters 7................................... 142 
Table C- 8: Inventory Target Levels for Set of Parameters 8................................... 143 
Table C- 9: Inventory Target Levels for Set of Parameters 9................................... 144 
Table C- 10: Inventory Target Levels for Set of Parameters 10............................... 145 
Table C- 11: Inventory Target Levels for Set of Parameters 11............................... 146 
Table C- 12: Inventory Target Levels for Set of Parameters 12............................... 147 
 
Table D- 1: Simulation Results: Set of Parameters 1 ............................................... 148 
Table D- 2: Simulation Results: Set of Parameters 2 ............................................... 150 
Table D- 3: Simulation Results: Set of Parameters 3 ............................................... 152 
Table D- 4: Simulation Results: Set of Parameters 4 ............................................... 154 
Table D- 5: Simulation Results: Set of Parameters 5 ............................................... 156 
Table D- 6:  Simulation Results: Set of Parameters 6 .............................................. 158 
Table D- 7: Simulation Results: Set of Parameters 7 ............................................... 160 
Table D- 8: Simulation Results: Set of Parameters 8 ............................................... 162 
Table D- 9: Simulation Results: Set of Parameters 9 ............................................... 164 
Table D- 10: Simulation Results: Set of Parameters 10 ........................................... 166 
Table D- 11: Simulation Results: Set of Parameters 11 ........................................... 168 
Table D- 12: Simulation Results: Set of Parameters 12 ........................................... 170 
Table D- 13: Simulation Results: Set of Parameters 1, under Twice Demand Arrivals 
at Retailer 4 and Without Inventory Target Update.................................................. 172 
Table D- 14: Simulation Results: Set of Parameters 7, under Twice Demand Arrivals 
at Retailer 4 and Without Inventory Target Update.................................................. 173 
 
 viii 
 
 
Lists of Figures 
Figure 2- 1: (s, S) Policy under Periodic Review........................................................ 24 
 
Figure 5-1: Expected Weekly Costs for FTRI as Function of L
I
................................ 83 
Figure 5-2: Expected Weekly Costs for FTRI as Function of L
I
................................ 84 
Figure 5-3: Expected Weekly Costs for FTRI as Function of L
I
................................ 84 
Figure 5-4: Next Tour Expected Costs per Unit of Time as a Function of t
0
............. 88 
Figure 5-5: Next Tour Expected Costs per Unit of Time as a Function of t
0
............. 88 
 
Figure 6- 1: Location of facilities for Case 0 (Symmetric case)................................. 96 
Figure 6- 2: Location of facilities for Case 1.............................................................. 96 
Figure 6- 3: Location of facilities for Case 2.............................................................. 97 
Figure 6- 4: Location of facilities for Case 3.............................................................. 97 
Figure 6- 5: Results for the Set of Parameters 1 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 102 
Figure 6- 6: Results for the Set of Parameters 2 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 103 
Figure 6- 7: Results for the Set of Parameters 3 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 104 
Figure 6- 8: Results for the Set of Parameters 4 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 105 
Figure 6- 9: Results for the Set of Parameters 5 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 106 
Figure 6- 10: Results for the Set of Parameters 6 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 107 
Figure 6- 11: Results for the Set of Parameters 7 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 108 
Figure 6- 12: Results for the Set of Parameters 8 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 109 
Figure 6- 13: Results for the Set of Parameters 9 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 110 
Figure 6- 14: Results for the Set of Parameters 10 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 111 
Figure 6- 15: Results for the Set of Parameters 11 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 112 
Figure 6- 16: Results for the Set of Parameters 12 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases...................................................... 113 
Figure 6- 17: Impact of Inventory Holding Cost ...................................................... 116 
Figure 6- 18: Impact of Increments in Transportation Costs in Scenarios with High 
Inventory Holding Cost, Set of Parameters 1, 2 and 3.............................................. 117 
Figure 6- 19: Impact of Increments in Transportation Costs in Scenarios with Low 
Inventory Holding Cost, Set of Parameters 7, 8 and 9.............................................. 117 
Figure 6- 20: Average Number of Visits per Tour and Average Tour Length......... 119 
 
 ix 
 
Figure 6- 21: Impact of Increments in Demand Variability in Scenarios with High 
Inventory-Holding Costs, Set of Parameters 1, 4, 5 and 6........................................ 121 
Figure 6- 22: Impact of Increments in Demand Variability in Scenarios with Low 
Inventory-Holding Costs, Set of Parameters 7, 10, 11 and 12.................................. 121 
Figure 6- 23: Average Weekly Costs with 95% C.I. for the mean, under Twice 
Demand Arrivals at Retailer 4 and Without Inventory Target Update,..... for the Set of 
Parameters 1.............................................................................................................. 123 
Figure 6- 24: Average Weekly Costs with 95% C.I. for the mean, under Twice 
Demand Arrivals at Retailer 4 and Without Inventory Target Update, for the Set of 
Parameters 7.............................................................................................................. 123 
 
 
 1 
 
Chapter 1: Introduction 
 
1.1. Motivation 
 This research studies real-time distribution strategies and their associated 
benefits for a two-level distribution system, from one depot to N retailers, wherein 
vehicle delivery routes can be updated using real-time information about current 
inventory levels and the status of the vehicle.  This section discusses the need to 
conduct research in the field of logistics and distribution systems, especially in the 
area of real-time operations. 
 This research is motivated by three main considerations: (i) the importance of 
logistics and distribution systems in the national and local economy, (ii) the current 
trend to coordinate logistic operations, and (iii) the opportunities offered by current 
information and communication technologies (ICT) to operate and control a system in 
real-time.  Those considerations are discussed in the following subsections. 
1.1.1. The Importance of Logistics and Distribution Systems 
 Logistics and distribution systems are critical components of any modern 
economy wherein most products are consumed away from their production points.  In 
the United States, during 2003, logistics activities accounted for $936 billion, 
equivalent to 8.5 percent of the nominal gross demographic product (GDP).  The two 
main components of logistic costs are transportation and inventory costs, accounting 
for 63 percent ($593 billion) and 24 percent ($222 billion) respectively.  Among 
transportation modes, the trucking industry represents 81 percent ($ 482 billion) of 
 
 2 
 
the total transportation expenses, which is more than half of the total logistic costs 
(Council_of_Logistic_Management, 2004).  Furthermore, at the level of individual 
firms is estimated that distribution costs represent between 10 and 30 percent of a 
product's sales price (La Londe, 1994, Sharp and Goetschalckx, 1999, Ghiani et al., 
2004).  Therefore, given the large amount of resources involved, even slight 
improvements in logistic operations could have a significant impact on the overall 
economy.  That explains, in part, why in the last two decades some of the largest and 
most successful companies, such as Dell and Wal-Mart, have transformed the role of 
logistic operations to that of a strategic weapon rather than just a support function to 
coordinate the movement and storage of products required to satisfy consumers? 
demands (Ball, 2002). 
1.1.2. The Coordination of Logistic Operations 
The second significant motivation for this research is the current trend to 
coordinate logistical functions, such as inventory control and transportation, in order 
to improve the efficiency of supply chains, taking advantage of different logistical 
operations synergies (Thomas and Griffin, 1996).  The coordination of inventory and 
transportation operations is particularly relevant when customers are part of the same 
company or when vendor-managed inventory (VMI) strategies are employed 
(Campbell et al., 1998).  Under VMI agreements the supplier, who could be a 
manufacturer or a distributor, takes control of buyers? inventory levels, ensuring that 
adequate service levels are maintained.  Companies that have successfully 
implemented VMI agreements include Dell (Kapuscinski et al., 2004), Wall-Mart, 
Procter and Gamble, and Campbell Soup (Buzzell and Ortmeyer, 1995, Fisher, 1997, 
 
 3 
 
Lee, 2004, Mishra and Raghunathan, 2004).  Some examples of distribution systems 
where inventory and routing decisions had been integrated are fuel oil delivery to gas 
stations, industrial gas distribution, beer and soft drink distribution, vending machine 
replenishment, cash replenishment to automatic teller machines (ATMs), and 
supermarket product replenishment.  
1.1.3. Technologies that Enable Operations with Real-Time Information 
 The third significant motivation for this research are the opportunities offered 
by ongoing developments in information and communication technologies (ICT), 
which allow sharing information between different stages in a supply chain at 
progressively reduced costs (Rabah and Mahmassani, 2002).  ICT developments can 
be divided into three groups:  
a) communication and tracking of devices that allow automating the way 
information is input to computer systems and transmitted between them;  
b) commercial vehicle operations (CVO) technologies that allow the control 
of a fleet of vehicles on a real-time basis; and 
c) software and decision-support systems (DSS) that provide data processing 
capabilities at a particular facility.  
These systems increase the speed and accuracy at which data is entered, 
gathered, and communicated, and they provide better real-time visibility about 
inventory levels throughout the distribution system and better control over a fleet of 
vehicles on a real-time basis. 
 
 4 
 
a) Communication and Tracking Devices 
 Communication and tracking devices can be divided into two groups: data 
transmission between computer systems and physical transaction tracking. 
 The development of protocols and associated company standards to transmit 
business transaction data between computer systems?such as Electronic Data 
Interchange (EDI), and more recently eXchange Markup Language (XML) to 
communicate using the Internet (see www.rosettanet.org)?have facilitated 
information and data exchange between different computer systems, thereby avoiding 
paperwork.  
 In terms of physical transactions tracking, bar coding systems (see Masters 
and La Londe, 1994), and more recently, radio frequency identification (RFID) 
transponders and readers, which do not require the manual scanning of products, are 
instrumental in improving the speed and accuracy at which transactions and 
movements of products are recorded and updated in computer systems.  The interest 
on RFID is rapidly increasing after the Department of Defense (DoD) and Wal-Mart 
mandated that all cases and pallets entering their systems must have RFID 
transponders after 2004 (Datta, 2003).  However, because of privacy concerns, RFID 
is unlikely to soon replace bar coding at the final consumer level (Blanchard, 2003). 
b) CVO Technologies 
 At the level of commercial vehicle operations (CVO), ICT developments that 
enable real-time operations include automatic vehicle identification (AVI), two-way 
communication systems, automatic vehicle location (AVL), and other related 
technologies, such as on board computers (OBC), and navigation devices (Regan et 
al., 1995). 
 
 5 
 
i) AVI systems are basically applications of RFID technologies to CVO, 
consisting of adding a transponder to each tractor, which allow recognition 
it when passing through a reader.  Some applications of AVI include 
electronic toll systems, weigh-in motion systems, and RFID at terminal 
gates for tracking tractor arrivals and departures.  
ii) Two-way communications systems allow transferring voice and data 
between the dispatch center and drivers in real-time.  Available two-way 
systems vary from VHF radios, cellular phones to satellite 
communications, and differ in kinds of communication permitted (voice 
and/or data), range of operation, and cost. 
iii) AVL systems are used to map vehicle positions in real-time.  The 
technology leader in this market is Global Positioning System (GPS) 
receivers, which can compute current position and speed within meters of 
accuracy by sending signals to four out of twenty-four GPS satellites and 
triangulating.  Some leading GPS receivers suppliers include Novatel, 
Garmin, Navman, and Magellan. 
iv) Among other related technologies, the current trend in the industry is to 
integrate on-board computers (OBCs), using generic PDAs, with AVL 
systems.  Some integrated systems on the market include OmniTracs from 
Qualcomm, MobileMax from Aether Systems, VMX 8700 from Data Ltd 
Inc, Mobius TTS from Cadec Inc, g2x system from PeopleNet, i58sr and 
i88s from Motorola, and iPAQ PDA from Compaq. 
 
 6 
 
c) Software and Decision-Support Systems 
 Among software and decision-support systems (DSS), the main development 
was the Enterprise Resource Planning (ERP) system.  ERP systems use a centralized 
database to collect, manage, and share organizational information across business 
functions.  As Rutner et al. (2003) state ?ERP is becoming a widely accepted 
computerized process for handling data in American corporations with over 92% of 
companies using or in the process of implementing? it.  ERP systems vary in terms of 
sophistication from a simple transactional database to multi-component decision-
support systems (DSS), also known as ERP-II.  Among the main ERP components 
are production scheduling, material requirements planning (MRP), financial 
management, inventory management, demand planning, transportation management, 
and human resource management.  Market leading ERP vendors include SAP, Baan, 
PepleSoft, J.D. Edwards, and Oracle. 
 In addition to components included in ERP systems, there had been, during 
the last decade, an increasing interest in developing DSS for specific operational 
purposes, such as supply chain planning (SCP) systems, warehouse management 
systems (WMS), transportation management systems (TMS), and advance planning 
and scheduling (APS).  However, notwithstanding their names, in most cases they 
lack true optimization capabilities and rely on simple heuristics to obtain feasible 
solutions (Fleischmann and Meyr, 2003, Simchi-Levi et al., 2003, p. 317).  Some 
leading providers of such systems are Manugistic, i2, and Manhattan Associates.  
 Another important development is the spatial Geographic Information System 
(GIS) database, which allows presenting and manipulating geographically referenced 
data.  GIS capabilities have been implemented in many graphical user interfaces 
 
 7 
 
(GUI) used by DSSs.  Some leading GIS providers include ESRI and Caliper 
Corporation. 
In summary, all of the above-described ICT developments provide access to 
real-time information on the current state of the system? i.e. inventory levels at each 
facility and status of the fleet?which allows managers to make online decisions on a 
continuing basis to improve routing plans.  Those developments give managers new 
opportunities to react faster to changes in predicted demand patterns or traffic 
conditions, and adjust plans accordingly.  However, the operational decisions are 
complex, since the underlying problems are combinatorial and unfold in real-time, 
precluding the evaluation of all possible alternatives by the decision maker.  
Moreover, the stochastic nature of such systems implies that information about the 
state of the system is gradually revealed and cannot be accurately predicted in 
advance.  Therefore, in order to take maximum advantage of the extensive quantities 
of real-time information made available by ICTs, supply-chain managers need to use 
information effectively.  That requires the development of models and algorithms that 
can exploit the full potential of real-time information for distribution-logistic 
operations. 
1.1.4. Other Motivations 
 Finally, as it will be discussed in Chapter 2, previous research on real-time 
fleet management has focus on how to serve load demands for transportation services 
that are exogenous to the system, in the context of dynamic vehicle-routing problems 
(Gendreau et al., 1999, Larsen et al., 2002) and pick up and delivery problems (Regan 
et al., 1995, Regan et al., 1996a, Yang et al., 2004, Kim et al., 2002a, Kim et al., 
 
 8 
 
2004).  In this research, routing decisions are coordinated with inventory control.  In 
that fashion, it is expected that monitoring inventory levels would allow improving 
the forecast and coordination of transportation activities, giving the operator the 
option to visit a facility earlier than needed to take advantage of transportation 
savings.  That could be particularly useful when demand is highly variable and/or 
unpredictable, which is normally the case when final consumers are separated by 
many echelons from the echelon considered, or as a consequence of the phenomenon 
known as the bullwhip effect (Lee et al., 1997, Fine, 1998, Chen et al., 2000). 
Having established the main motivations for this research, then, the next 
section presents the specific problem studied. 
1.2. Problem Statement 
 The focus of this research is on formulating inventory-routing problems 
(IRPs) in a stochastic dynamic environment with real-time information about current 
inventory levels, as well as delivery vehicle locations and status. 
The specific distribution system considered is a two-level supply chain, in 
which a set of geographically dispersed facilities facing stochastic demands have to 
be repeatedly replenished from a central warehouse (or depot) over a long period of 
time.  The facilities to be replenished could represent final customers, retailers who 
serve demand from final customers, or distribution centers from which a set of 
additional facilities are replenished.  In this system, products are transported from the 
depot to the set of retailers by a vehicle with limited capacity, the plans for which can 
be updated with real-time information about the state of the system, thanks to modern 
information and communication capabilities.  This problem is designated as the 
 
 9 
 
online inventory routing problem (OIRP) under real-time information.  The OIRP is 
formulated and solved considering inventory allocation and transportation decisions 
together.  As such, the OIRP considers the trade-off among transportation, inventory 
holding, and stock-out costs. 
In this research?contrary to the common view in real-time fleet management 
problems where load demands are exogenous to the system?decisions to replenish 
inventory, by how much, in what sequence, and by which vehicle, are conducted in an 
integrated real-time decision framework.  In addition a central-planer approach to the 
problem is assumed.  That is, the system is assumed to be operated and controlled by 
a central decision maker who seeks to move inventories in the system in such a way 
as to maximize total expected profit in the long-run for the complete system.  
Moreover, the central decision maker operates with real-time information about the 
complete state of the system. 
Key features of this OIRP are the presence of uncertainty about future 
consumption rates at different facilities and the possibility of updating plans based on 
accurate real-time information about the complete state of the system; i.e., accurate 
real-time knowledge of all local inventory levels, and location of and remaining load 
in each truck.  That contrast with deterministic environments, in which decisions can 
be made with perfect hindsight, thus real-time operational capabilities would not 
modify the nature of the problem.  The possibility of updating plans on a continuous 
basis, based on real-time information about demand realizations makes possible some 
additional decisions to update truck-route plans, such as modifying the set and/or the 
sequence of subsequent customers to be visited; diverting a truck from its current 
 
 10 
 
destination to visit a different facility; and adjusting amounts to be delivered to 
subsequent customers in the route. 
Such a new operational environment could enable more efficient use of 
existing resources and increase system reliability.  However, the design of efficient 
strategies to operate the system can be extremely difficult.  On one hand, the 
dispatcher faces a fleet-routing and scheduling problem?which is combinatorial?to 
obtain new operational plans.  Since, even simplified static and deterministic versions 
of the inventory-routing problem are computationally hard (Bertazzi et al., 2007), a 
trade off between quality of solutions and speed should be considered in the search of 
new plans.  On the other hand, given that plans can be modified at any time, based on 
new information, the events and circumstances under which a plan update would be 
beneficial should be specified. 
1.3. Research Context and Scope 
The broad context for this research is product distribution and logistics 
operations in which a set of facilities need to be repeatedly refilled from a single 
facility with the same product over time.  The problem studied entails the 
management of a fleet of trucks that moves the product from the depot to the set of 
retailers, combining deliveries to different facilities in the same route.  This type of 
fleet operation is known as less than truck load (LTL), since a vehicle could transport 
loads for different facilities simultaneously.  In this research while the vehicle is 
enroute reallocation of loads among retailers is considered.  However, this 
dissertation does not address distribution operations with transshipments, that is, it is 
 
 11 
 
assumed that after products have been delivered to a particular facility, they cannot be 
reclaimed and reassigned to a different facility.  
Distribution systems can be operated either centrally (i.e., vertically 
integrated) or decentrally, in which different agents control different parts of the 
chain.  In this research the distribution system is assumed to be controlled and 
managed by a central agent who seeks the best performance for the complete system; 
therefore, this research does not consider coordination mechanisms to achieve system 
optimal decision in a decentralized supply chain.  Accordingly, pricing and incentive 
mechanisms that could align the strategic decisions in a supply chain are out of the 
scope of this work.  In addition, a single product is considered in the analysis.  Hence, 
decisions related to the mix of products transported, where a supplier provides 
complement and substitute products, are not studied. 
Generally, logistics decisions are classified according to the planning horizon 
involved, from longer- to shorter-term into strategic, tactical, and operational (see, for 
example Ghiani et al., 2004, Simchi-Levi et al., 2003).  This research deals with real-
time operational decisions.  In particular, the possibilities opened by decisions during 
en-route distribution operations are studied.  It is assumed that upper hierarchical 
(strategic and tactical) decisions about the system configuration are given, e.g., the set 
of facilities to be refilled from a particular distribution center, and the characteristics 
of the fleet of vehicles assigned to serve those facilities are not directly considered.  
Moreover, a single-vehicle approach to the problem is assumed.  Hence, strategies 
that could split deliveries to a particular facility from more than one depot or truck are 
precluded and left for future research. 
 
 12 
 
In this research, demand processes at different facilities are assumed to be the 
only source of uncertainty, even though, in real-world applications, particularly in 
urban areas, there are also uncertainties of traffic conditions that could lead to 
significantly varying travel times as a consequence of network congestion.  In this 
research, travel time between facilities is assumed to be fixed and known.  Moreover, 
time associated with loading and unloading operations is not considered; thus, the 
only source of delay in vehicle routes is travel time between facilities.  In addition, is 
assumed that demands are known in probability distribution, and that these demand 
processes at retailers cannot be affected by the central decision maker. 
This work investigates scenarios wherein plans could be continuously 
updated, based on accurate real-time information about fleet status and inventory 
levels at each facility.  In those scenarios, in which the distribution system could be 
monitored and controlled on a real-time basis, the main issues studied are: when and 
how to update distribution plans, based on real-time information.  The scenarios are 
compared with operations wherein some or all these information and communication 
technologies are not available. 
Another important assumption in this research is that daily and weekly cycle 
operation characteristics are not taken into account, that is the system is assumed to 
be operating continuously, without interruption.  Moreover, labor-related constraints 
are not considered.  In short, it is assumed that the vehicle and all facilities are always 
in operation; i.e., deliveries can be scheduled at any time, with neither time windows 
for particular facilities nor restrictions on the number of hours that a driver can 
 
 13 
 
operate a vehicle.  Therefore, delivery routes are constrained only by the vehicle?s 
capacity to transport products. 
The main focus of this research is on problem formulation and design of real-
time strategies.  The implementation and analysis of the proposed strategies in large 
size problems is out of the scope of this research and is left as a future endeavor. 
1.4. Research Objectives 
The main objectives of this research are to:  
1. formulate and analyze the online inventory routing problem (OIRP), 
taking into account explicitly real-time information about fleet status and 
inventory levels at different facilities; 
2. develop operational-control strategies to operate a distribution system in 
which transportation operations and inventory control are coordinated?
the strategies should be tailored to different degrees of availability of real-
time information associated to different scenarios in terms of ICT 
installed; 
3. evaluate the benefits of the proposed real-time operational strategies and 
the value of using real-time information and sophisticated optimization 
techniques in a centrally operated distribution system, establishing the 
characteristics of distribution systems for which real-time operational 
capabilities would be more beneficial, in terms of demand variability, 
location and distance between facilities, and the relationship between 
inventory holdings, stock outs, and transportation costs. 
 
 14 
 
In order to address those main objectives, the following specific tasks are 
considered: 
1. Formulate the OIRP under real-time information about inventory levels at 
different facilities and fleet status (location and load remaining in the 
vehicle).  This formulation should take into account the possibility of 
modifying delivery plans at any time, based on accurate real-time 
information about the state of the system. 
2. Develop dynamic operational-control strategies or policies to operate a 
distribution system in which transportation and inventory control decisions 
are centralized.  These strategies determine when and how to update 
operational plans for scenarios with different degree of real-time 
information. 
3. Formulate local off-line problems and heuristics used to update 
distribution plans for different operational-control strategies. 
4. Propose a methodology to evaluate the performance of the developed 
dynamic decision strategies. 
5. Develop a simulation framework to analyze distribution operations from a 
central facility to a set of retailers facing stochastic demands under real-
time information. 
6. Identify evaluation benchmarks in a dynamic environment for one-to-
many distribution systems. 
7. Evaluate the competitive performance of different strategies under 
different degrees of sophistication in terms of the ICT used to operate the 
 
 15 
 
system, particularly in the degree that current plans can be updated, 
quantifying the possible benefits of operations under real-time 
information. 
8. Study the characteristics of distribution-system that could most benefit by 
implementing sophisticated strategies using real-time information?in 
terms of: i) the location and distance between facilities; ii) the variability 
in facilities? demands; iii) the relationship among lost sales, inventory 
holding, and transportation costs; iv) the presence of disruption in demand 
patterns; and v) other parameters, such as the ratio between the capacity of 
the truck and that of the facilities. 
1.5. Main Contributions 
A primary contribution of this dissertation is to incorporate the processes that 
generate demands for transportation services in the study of real-time fleet operations.  
The specific contributions of this research are related with the main task presented, 
and include:  
a) the formulation of the online inventory routing problem (OIRP), taking 
into account explicitly real-time information about fleet status and 
inventory levels at different facilities; 
b) the development and design of operational-control strategies to operate a 
distribution system in which transportation operations and inventory 
control are coordinated tailored to different degrees of availability of real-
time information associated to different scenarios in terms of ICT 
installed; 
 
 16 
 
c) the formulation of local off-line problems and heuristics used to update 
distribution plans for different operational-control strategies; 
d) the development of a methodology and a simulation framework to analyze 
and evaluate the performance of dynamic decision strategies in the context 
of one-to-many distribution systems; 
e) the identification of evaluation benchmarks in a dynamic environment for 
distribution operations from a central facility to a set of retailers facing 
stochastic demands; 
f) improve the understanding of the characteristics of distribution-system 
that could most benefit by implementing sophisticated control-strategies 
using real-time information, and portray the main expected benefits 
associated with those strategies.  
1.6. Dissertation Structure 
This dissertation is organized as follows. After this introductory chapter, 
which presents the main motivations for research in this area introduces the specific 
problem studied, and states the research scope, main objectives and expected 
contributions, chapter 2 presents a review of related research in the literature.  This 
background chapter is divided into three parts.  First, a review of single item 
inventory models with particular attention to results used in this research is presented.  
Second, routing and scheduling problems are classified, and the problem studied in 
this research is place in the context of previous research on inventory routing 
problems (IRPs) and real-time fleet operations.  Third, a summary of real-time 
combinatorial optimization approaches is presented.  
 
 17 
 
Chapter 3 formulates the specific problem studied in this research. In this 
chapter the problem context and main assumptions are stated.  In addition sources of 
complexity are explained and the two approaches being used to deal with this 
problem are introduced. 
Chapter 4 presents the formulation and design of optimization based 
strategies, in which the inventory control side of the problem is solved a priori and its 
results are used as target levels when plans are updated.  In these strategies an off-line 
optimization problem is formulated and employed to update routing and inventory 
allocation plans.  It presents different control strategies based on different degrees of 
real-time information availability for controlling the system.  This chapter formulates 
a local off-line problem, which is used to update distribution plans in all optimization 
based strategies.  Also, it presents an optimization framework for adjusting policy 
parameters for each strategy. 
Chapter 5 is dedicated to the formulation and design of fixed-tour based 
control strategies. In this case the routing side of the problem is solved a priori.  
These strategies are based on a priori set of routes to refill retailers with recourse 
actions depending on different degrees of real-time information capabilities for 
controlling the system.  This chapter presents the rationale and characteristics of these 
strategies.  In addition, it offers an analysis and optimization of policy parameters for 
each case. 
Chapter 6 presents different experiments designed to evaluate and compare 
the set of proposed real-time policies.  It describes the set of scenarios used, including 
 
 18 
 
scenarios with steady-state demand processes, and scenarios with sudden changes in 
demand patterns.  Finally, it presents and discusses experimental results. 
To conclude, the last chapter presents a summary of the main contributions, 
findings and results.  In addition, it presents a list of possible extensions and 
directions for future research. 
 
 
 19 
 
Chapter 2: Background Review and Previous 
Research 
 
This chapter reviews previous research that relates to the problem studied in 
this research.  This review is divided into three parts. Section 1 reviews previous 
work on single-item inventory control for a single facility.  Section 2 presents a 
classification of vehicle-routing literature, and the main contributions in inventory-
routing and real-time fleet operations are categorized and described.  Section 3 
presents a summary of real-time combinatorial optimization approaches. 
2.1. Previous Work on Single-Item Inventory Models for a Single Facility 
This section reviews the main results of research on single-item inventory 
models for a single facility used in our research. First, main sources of inventory costs 
are examined. Then the classic Economic Order Quantity (EOQ) model for 
deterministic demand is presented. Finally, periodic and continuous review models 
with stationary stochastic demands are reviewed. 
2.1.1. Inventory Costs  
Before reviewing material on minimizing inventory costs, the main costs 
associated with inventory are discussed.  In general, inventory costs can be divided 
into three categories: ordering or procurement costs; inventory holding costs; and 
inventory shortage costs. 
Ordering and procurement costs are associated with purchase, transport, and 
handling of products to a particular facility, and they include fixed costs for each 
 
 20 
 
order and variable costs per unit of product.  Despite order size, there are fixed costs 
or setup costs per order.  Fixed costs are explained by economies of scale in 
production and by consolidation of products for transportation and handling. 
Inventory holding costs are related to products or material stored per unit of 
time. Those costs include opportunity costs of capital immobilized in inventory and in 
warehousing.  Among warehousing costs are insurance of items, taxes, rent for 
warehouse space, maintenance, and handling costs.  In addition, there are 
obsolescence costs in the case of perishable and seasonal goods.  Obsolescence costs 
are not discussed in this review, which assumes a constant value of products 
distributed. 
Shortage or stock-out costs are incurred when demands cannot be met. 
Shortage could also result in lost sales or backorders.  When demands could be 
satisfied by a competitor, shortage could lead to lost-sales costs, which include profit 
lost from not selling the product, and could have a negative impact on future demands 
because of lost of consumer goodwill.  On the other hand, when items are difficult to 
substitute, stock-outs may entail delayed demand satisfaction with associated 
backorder costs.  In some instances, when products supplied are raw material for 
other production processes, stock-outs may lead to disruption of the entire production 
line. 
Since shortage costs are hard to quantify, some inventory models service use 
levels of order fulfillment instead.  Two common service-level performance measures 
used are percentage of demand fulfilled from on-hand inventory, also known as fill 
rate, and percentage of time with shortages.  In those models orders are placed so that 
 
 21 
 
the expected service levels satisfy a specific target value.  However, given that any 
service level used has an implicit shortage-cost value, and that our objective is to 
study the impact of different distribution strategies on the system performance 
without imposing restriction on cost trade-offs between different alternatives, service-
levels approaches are not used in this research and, consequently, they are not 
reviewed in this section. 
During the past century and particularly since the Second World War, 
inventory management and the trade-offs between different sources of inventory costs 
have been extensively studied. Good recent reviews of inventory-control literature 
can be found in Graves et al. (1993), Axs?ter (2000), and Zipkin (2000). The next 
subsections review the main results for single-facility inventory systems used in our 
research.  
2.1.2. EOQ Model for Constant Demand Rate  
In the context of steady-state deterministic demand in a single facility, Harris 
(1913) introduced a simple model, known as the Economic Order Quantity (EOQ), to 
study the trade-off between inventory-holding and order costs. The EOQ model 
assumes that (i) demand is constant at rate ?  per unit of time, (ii) shortages are 
prohibited, (iii) orders are delivered complete and instantaneously with zero lead 
time, (iv) costs are constant and no discount rate of money is considered, (v) order 
costs are composed of a fixed part? K, per order and a variable part, c, per item 
ordered, and (vi) inventory-holding costs are accrued at a rate h per unit of time.  
Based on those assumptions is relatively easy to show that the optimal policy is to 
order a batch of size 
*
Q , also known EOQ, when the inventory level reaches zero. 
 
 22 
 
 
*
2QKh?=??  (2.1) 
Even though that is a simple model, its results are very robust with respect to 
demand rate and cost parameters (see for example Lee and Nahmias, 1993).  
Moreover, the third assumption could be relaxed to include deterministic lead times, 
in which case orders should be placed so that they arrive when the inventory level 
reaches zero. 
The next subsection reviews the main results relevant to our research, among 
models with stationary stochastic demands. 
2.1.3. Models with a Stationary Stochastic Demand Rate 
 One way to classify stochastic inventory models is in relation to their review 
process, i.e. when and how often are inventory levels reviewed and decisions made to 
place orders.  Using that criterion, inventory models can be classified as either 
periodic-review or continuous-review models.  The next subsections present the main 
results from the literature used in our research. 
2.1.3.1. Periodic-Review Models 
 In periodic-review models, inventory level is known at the beginning of each 
period and orders can be made only at those epochs. 
 The most basic model in this group is the newsvendor or newsboy model in 
which the number of periods considered is only one.  In that model, before a 
stochastic demand of size D is realized, a decision to stock y should be made.  If h is 
the overage cost per unit of remaining inventory at end of the period, p is the penalty 
cost per unit of unsatisfied demand, and c is the cost of each unit ordered, then for a 
given demand ?  the total cost at the end of the period is 
 
 23 
 
()()
()Gy cy hy p y??
++
=+ ? + ? , which is a convex function on y, where 
() { }
max 0,x x
+
= .  It could be shown that the optimal stocking decision should satisfy 
the optimality conditions given by (see for example Lee and Nahmias, 1993): 
 
()
*
Pr
p
DS
hp
?=
+
 (2.2) 
 Where p/(h+p) is known as the critical ratio.  Then the optimal policy for a 
newsvendor problem is to order up to 
*
S  whenever the initial inventory level is below 
*
S , otherwise do not place an order.  This is also known as ?order up to? policy. 
 The model could be extended to include fixed-order costs, K, which are 
accrued only when an order is placed.  In this case the optimal policy will place 
orders only when the initial inventory level is below a threshold 
*
sS< , given by the 
solution of 
*
() ( )Gs GS K=+, i.e. orders are placed only when the expected benefits 
are higher than K.  This policy is known as (s, S) policy, and can be stated: whenever 
the current inventory level is below the reorder point s, an order is placed to bring the 
inventory level to S; otherwise, do not place an order.  
 When multiple periods are considered, Scarf (1960) presented a finite-horizon 
model with fixed ordering costs and backlogging, and showed that an (s, S) policy, 
which might have different parameters at each period, is optimal when the value 
function is K-convex.  For infinite-horizon problems, the optimality of the stationary 
(s, S) policy was shown by Iglehart (1963).  In multi-period models, the concept of 
inventory position, defined as the sum of inventory on hand plus inventory in transit 
(already ordered), minus backorders, is normally used in the definition of inventory 
 
 24 
 
policies instead of initial inventory level.  Figure 2.1 presents an (s, S) policy with a 
review period of size T.  
 
Figure 2- 1: (s, S) Policy under Periodic Review 
 
2.1.3.2. Continuous-Review Models 
 In continuous-review models, inventory is always assessed and orders can be 
placed at any time.  In particular, inventory replenishment decisions can be made as 
soon as new demands are served.  The optimality results of (s, S) policies have been 
extended to the continuous review case (Beckmann, 1961, Zheng, 1991). 
 In order to analyze continuous-review inventory systems, a demand process 
should be specified.  Following the notation in Lee and Nahmias (1993), a demand 
Inventory Level 
s 
 
S 
T T 
Inventory Position
Time
       LT           LT   
Inventory 
 
 25 
 
process can be described by the probability distributions of demand interarrival times 
and demand size at each demand arrival.  In general is assumed that demand 
interarrival times are IID random variables with a finite mean 1/? .  In addition, 
demand distribution at each demand epoch is assumed to have a probability mass 
function (pmf) ()? ?  and cumulative density function (cdf) ()? ?  with finite mean ? .  
In the backlogging case, when demand follows a compound Poisson process, i.e. 
interarrival times are Poisson distributed and arrivals are a Poisson process, it can be 
demonstrated that the steady state distribution of the inventory position (IP) is: 
 
1
Pr( )
S
kj
js
IPk m m
=+
==
?
        , for 1,  2,  ...,  1j ss S= ++ ? (2.3) 
where 
1
()
S
jk
kj
mmkj?
=+
=??
?
 is the average number of visits to IP = j during a 
replenishment cycle.  Without loss of generality is assumed that (0) 0? = , otherwise 
the demand process can be replaced by an equivalent process with (1 (0))?? ?=??
null
 
and () ()(1 (0))jj? ??=??null  for j > 0.  In addition, it is required that not all demand 
sizes be a multiple of some integer larger than one.  Based on expression (2.3), the 
steady state distribution of the inventory level (IL) could be computed as: 
 
{}max 1,
Pr( ) Pr( ) Pr( ( ) )
S
jsk
ILk IP j DLT jk
=+
== =? =?
?
    , for kS?  (2.4) 
where D(LT) is the total demand during a deterministic lead time of length LT.  In 
case of unit demand sizes, i.e. demand according to a Poisson process, it can be 
shown that in a steady state the inventory position is uniformly distributed in (s+1, 
s+2, ?, S).  That result can be extended to different IID interarrival time distributions, 
as long as the demands are unitary (Richards, 1975). 
 
 26 
 
 In scenarios with lost sales, results are harder to obtain, because when 
shortages occur, lost demands do not change the inventory position.  However, results 
can be obtained using a standard simplifying assumption that the number of 
outstanding orders can be at most one, i.e. ( )sQ Ss< =?.  Because of that 
simplification, inventory level and inventory position are always the same before an 
order is placed, which can be used as a renewal epoch.  In that case, if the total 
demand during a time unit is approximated, using a normal distribution with mean ?  
and standard deviation ? , then demand during a deterministic lead time, D(LT) ~ 
2
(, )N ? ?nullnull  with probability density function (pdf) 
()
()
DLT
f ? , where LT? ?=?null  and 
LT? ?=?null .  Based in these simplifications the expected cost per unit of time can be 
expressed as: 
 
(Cost per Cycle)
(Cycle Length)
E
EC
E
=
()()
()
2
1
() ()
()
Q
h EsDLT pEsDLT
QEsDLT
?
+ ?
?
??
?+ ? +? ?
??
=
??
+?
??
 (2.5) 
where: 
 ()
( ) ( )
() 1 (
xs
s
Es DLT dx G
? ?
??
?
?
?
??
?=??=?
?
nullnull
nullnull
null  (2.6) 
 ()()()
()
() () ()
s
E s DLT E s DLT E s DLT s G
?
?
??
+?
?
?=?+?=?+?
null
null
nullnull  (2.7) 
Therefore: 
 
( ) ( )
()
2
Q ss
s
hQ s G p G
EC
QG
? ?
??
?
?
?? ? ?
?
??
?
??
?+?+? +???
??
=
??
+?
??
nullnull
nullnull
null
null
nullnull null
null
 (2.8) 
where ()G ?  is the loss function, which gives the expected number of units of demand 
lost as a function of the initial inventory level when demand is distributed normal 
 
 27 
 
standard, i.e. () ( )()
y
Gy x y xdx?
?
=?
?
, where 
( )
? ?  is the pdf of normal standard 
distribution. 
As mentioned by Axs?ter (2000, pp.73), expression (2.8) can be further 
simplified by neglecting the second term in the denominator and the same term in the 
first term in the numerator.  In addition, the parameters should satisfy (2)phQ? >? , 
otherwise, it would be not profitable to operate the system. 
The optimization of the parameters of an (s, S) policy can be done using 
iterative procedures (see Axs?ter, 2000), or using an efficient optimization procedure 
developed by Zheng and Federgruen (1991). 
This section reviewed the most important results on inventory models for 
single item and single facility used in this research.  The next section reviews the 
main literature in fleet routing and scheduling, particularly previous work on 
inventory-routing problems. 
2.2. Background in Routing and Scheduling Problems 
 This section reviews previous work in routing and scheduling of vehicles 
relevant to this research.  First, a classification of research on routing and scheduling 
problems is offered.  The second subsection reviews and discusses previous research 
in inventory-routing problems (IRPs), i.e. vehicle routing in which inventory 
replenishment decisions are combined.  The third subsection presents previous 
research in real-time fleet operations. 
 
 28 
 
2.2.1. Classification of Fleet Routing and Scheduling Problems 
 Fleet routing and scheduling problems have received extensive and fruitful 
attention since the late nineteen fifties.  Broad overview can be found in Bodin et 
al.(1983), Christofides (1985), Golden and Assad (1988), Fisher (1995), and more 
recently Toth and Vigo (2002).  Problems found therein can be classified according to 
their main characteristics, as shown in Table 2.1 (Bodin et al., 1983, Assad, 1988, 
Psaraftis, 1988). 
Table 2- 1: Classification of Fleet Routing and Scheduling Problems 
Characteristics Options 
1. Fleet Size 
Single vs. Multiple vehicles 
Fixed vs. Variable fleet size 
2. Fleet Type 
Homogeneous vs. Heterogeneous vehicle types 
Single vs. Multiple Compartments 
3. Vehicle Terminals Single vs. Multiple terminals (or depots)  
4. Nature of Demands 
Deterministic (known) vs. Stochastic demands 
Partial satisfaction of demand allowed vs. not allowed 
Customers with different priorities vs. same priorities 
5. Location of Demands At nodes vs. On arcs (or mixed) 
6. Information on Parameters 
   (demand and travel times) 
Deterministic parameters vs. Uncertain (Stochastic) parameters 
All data known in advance (static problems) vs. Real-time 
inflow of data (dynamic problems) 
7. Underlying network 
Undirected vs. directed (or mixed) 
Euclidean vs. No-euclidean distances 
Deterministic vs. Stochastic travel times 
8. Route Restrictions 
Vehicle maximum capacity vs. vehicles with unlimited capacity 
Max route length (or time) vs. not imposed (unlimited) 
Max number of customer per route vs. not imposed (unlimited) 
Loading restrictions/equipments vs. unrestricted 
Vehicle type/site dependencies 
9. Operations 
Pure pickups or pure deliveries vs. mixed (pickups and 
deliveries) 
Split deliveries allowed vs. Split deliveries disallowed 
Truckloads (TL) vs. Less than truckloads (LTL) 
Single commodities vs. Multiple commodities 
10. Costs 
Variable or routing costs (per distance) 
Fixed operating costs (per vehicle in the fleet) 
Opportunity (penalty) costs associated to unserved demands 
11. Objectives 
Minimize total routing cost 
Minimize sum of fixed and variable costs 
Minimize number of vehicle required 
Maximize utility function based on service or convenience 
Maximize utility function based on customers priorities 
  
 
 29 
 
Using the above classification scheme, the online inventory-routing problem 
(OIRP) studied in our research can be categorized in terms of the vehicle fleet used as 
a single vehicle with a single terminal (depot).  The demand locations are known and 
occur at facilities that are represented by nodes of the underlying network.  Travel 
times among those nodes are deterministic and known.  Moreover, they are 
proportional to the distance traveled, according to a Euclidean metric.  The demands 
at those nodes are stochastic and dynamically revealed in a real-time fashion.  In 
addition, those facilities should be repeatedly refilled over time, in contrast with 
problems where immediate visits do not have a direct impact on future visits.  
Moreover, facilities might have different priorities, i.e., inventory holding and 
shortage costs can be different at different facilities.  In terms of route restriction, 
truck capacity is the only route constraint considered.  The type of operations 
considered is pure deliveries of a single commodity, in which visits to different 
facilities, can be combined in the same truck tour with the possibility of using truck 
loads (TL) and/or less than truck loads (LTL), and splitting deliveries to a particular 
customer into different tours.  Finally, in term of objectives considered, an 
operational planning perspective is taken, in which long term decisions?such as fleet 
size, set of customers to be served from a particular depot, and assignment of vehicles 
to a set of customers?are considered as given.  Therefore, at this operational level, 
the cost objectives to minimize are transportation costs proportional to travel distance, 
and inventory holding and shortage costs, which can be different among facilities. 
 In the next subsection the main references for inventory-routing problems 
(IRPs) studied in the literature are described and categorized. 
 
 30 
 
2.2.2. Inventory Routing Problems (IRPs) 
 In general, inventory-routing problems (IRPs) are long-term dynamic control 
problems.  Those types of problems are very difficult to formulate.  In large practical 
cases, which are common in real-world applications, IRPs are almost impossible to 
solve to optimality, even with accurate data.  Therefore, most approaches that deal 
with real-world problems address them in a short-term planning horizon, where the 
long-term effects are included, using some approximation, but some of the more 
complex features of IRPs are ignored. 
 Even though previous research on IRPs share some common elements, most 
of the problems presented in the literature address systems having different 
characteristics.  A detailed review of the works is presented in Baita et al. (1998), 
Campbell et al. (1998), Campbell and Savelsbergh (2002), and Kleywegt et al. 
(2002).  In Table 2.2, previous research is classified according to the following 
specific characteristics: 
a) time horizon considered, which can be single period or multiple periods 
?either finite or infinite number of periods? with either discrete or 
continuous time; 
b) demands, which can be deterministic or stochastic.  In the deterministic 
case, demand can be constant or time-varying.  In the stochastic case, 
demands can be either stationary or non-stationary; 
c) objective can be profit maximization, or minimization of costs, which can 
include transportation costs (
Tr
C ), inventory holding costs (
IH
C ),inventory 
stock-out costs (
ISO
C ),and crew-associated costs (C
Crew
); 
 
 31 
 
d) fleet size, which can be limited to a fixed number of vehicles (single or 
multiple vehicles), or variable numbers of vehicles; 
e) route constraints, which can be related to vehicle capacity and/or 
maximum length of route, either an upper bound in route distance or time; 
f) number of visits per vehicle tour, which can be to a single facility or to 
multiple facilities; 
g) routes can be fixed-static, variable-static, or dynamic.  In fixed-static, 
facilities are always visited in the same sequence.  In variable-static, a new 
route is obtained according to the state of the system and then 
implemented without en-route modifications.  In dynamic, en-route 
modifications are allowed, either changing the sequence of facilities to be 
visited, and/or the amount to be refilled at each facility in the tour; 
h) information about inventory levels at each facility used to set up plans can 
be on-line accurate information about the state of the system, or forecasted 
information based on expected consumption since the previous visit to 
each facility.  Some models assume systems with forecasted information 
about inventory levels that also receive on-line information about stock 
outs.  Those models are denoted Forecasted & SO.  That distinction is 
relevant in stochastic demand models, since in deterministic demand 
models, the state of the system is known at any time; and, 
i) plan updates considered, which can be event driven, time driven, or mixed 
(event and time) driven.  Among time driven updates the most common 
are rolling horizon (RH) approaches, in which plan updates can take place 
 
 32 
 
at each period or at regular intervals of time.  In some cases a plan update 
is imputed, since the authors did not discuss how their formulations should 
be implemented. 
 After classification, some formulations and solution methods that are 
representative of the approaches proposed to deal with IRPs are highlighted. 
 
 
 33 
 
Table 2- 2: Previous Research in Inventory-Routing Problems (IRPs) 
 
Reference Time Horizon Demands Objective to Min 
Fleet 
size 
Route 
Constraint 
No. 
Visits 
Routes Information Plan Update(*) 
Bell et al.(1983) Finite Discrete Constant Deterministic - Profit Limited Veh. Cap. Multiple Var.-Static N/A Periodically (RH) 
Federgruen & Zipkin (1984) Single Period Stochastic 
Tr
C +
IH
C +
ISO
C  Limited Veh. Cap. Multiple Var.-Static Initial State N/A 
Golden et al. (1984) Finite Discrete Non-Stationary Stochastic 
Tr
C +
ISO
C +CCrew Unlimited Veh Cap-Time Multiple Var.-Static Forecast & SO Periodically (RH) &SO 
Dror & Ball (1987) Finite Discrete Non-Stationary Stochastic 
Tr
C +
ISO
C  Limited Veh. Cap. Multiple Var.-Static Forecast Periodically (RH) & SO 
Chien et al. (1989) Single Period Deterministic -Rev.+
Tr
C +
ISO
C  Limited Veh. Cap. Multiple Var.-Static N/A N/A 
Anily and Federgruen (1990) Infinite Discrete Constant Deterministic 
Tr
C +
IH
C  Unlimited Veh. Cap. Multiple Fixed-Static N/A N/A 
Viswanathan & Mathur 
(1997) 
Infinite Discrete Constant Deterministic 
Tr
C +
IH
C  Unlimited No & Veh Cap Multiple Fixed-Static N/A N/A 
Bard et al. (1998a; 1998b) Finite Discrete Stationary Stochastic 
Tr
C +
ISO
C  Limited Veh Cap-Time Multiple Var.-Static Forecast & SO Periodically (RH) & SO 
Reiman et al. (1999) Infinite Continuous Stationary Stochastic  
Tr
C +
IH
C +
ISO
C  Single Veh. Cap. 
Single and 
Multiple 
Fixed-Static On-Line Continuous 
Rabah & Mahmassani 
(2002) 
Finite Discrete Stationary Stochastic 
Tr
C +
IH
C +
ISO
C  Single Veh. Cap. Multiple Var.-Static On-Line Only at SO 
Bertazzi et al 2002(2002) Finite Discrete Varying Deterministic  
Tr
C +
IH
C  Single Veh Cap Multiple Var.-Static N/A N/A 
Kleywegt et al. (2002) Infinite Discrete  Stationary Stochastic -Rev.+
Tr
C +
IH
C +
ISO
C  Limited Veh. Cap. Single Var.-Static On-Line Each Period 
Campbell & Savelsbergh 
(2004) 
Finite Discrete Constant Deterministic 
Tr
C  Limited Veh Cap-Time Multiple Var.-Static N/A Periodically (RH) 
Kleywegt et al. (2004) Infinite Discrete  Stationary Stochastic -Rev.+
Tr
C +
IH
C +
ISO
C  Limited Veh. Cap. At most 2 Var.-Static On-Line Each Period 
Adelman (2004) Infinite Discrete Stationary Stochastic 
Tr
C +
IH
C +
ISO
C  Limited Veh. Cap. Multiple Var - Static On-Line Each Period 
Aghezzaf et al. (2006) Infinite Continuous Constant Deterministic 
Tr
C +
IH
C +CCrew Limited Veh. Cap. Multiple Fixed-Static N/A N/A 
Savelsbergh and Song 
(2007) 
Finite Continuous Constant Deterministic 
Tr
C  Limited Veh Cap Multiple Var.-Static Initial State Periodically (RH) 
*: (RH= Rolling-horizon) 
 
 
 34 
 
 Bell et al. (1983) present a successful implementation of a decision-support 
system for the distribution of industrial gases, in which vehicle routes are designed 
solving a deterministic vehicle-routing problem (VRP) based on forecasted 
consumptions at each facility. 
 Federgruen and Zipkin (1984) formulate an IRP for a single day as a non-
linear integer program.  Their model assumes that the demand at the customers? site is 
stochastic, the depot has limited capacity, and cost functions are non-linear.  That is a 
distinctive feature of the model, since depots with limited capacity are not normally 
included in other models.  The non-linear cost function includes transportation costs, 
inventory holding costs, and shortage costs for each customer.  The objective is to 
construct delivery routes that minimize the total cost incurred on the day under 
consideration.  The solution approach starts with a feasible solution.  Then, the 
procedure iteratively exchanges customers among routes.  The main drawback of that 
formulation is: the consequences of today?s decisions on following days are not 
considered. 
 Golden et al. (1984) also formulate a single day problem with the objective 
being to minimize total costs.  The objective is sought while maintaining an adequate 
level of inventory, with a pre-specified target value, at each customer.  To solve the 
problem, they use a heuristic that selects the set of customers to be visited based on 
an urgency measure for each customer.  The urgency is determined by the ratio of the 
inventory level to inventory capacity.  Customers who are below a set target level are 
scheduled to be visited.  The solution is obtained by iteratively incorporating clients 
 
 35 
 
to a traveling salesman problem (TSP).  Clients are incorporated until the vehicle 
transportation capacity is reached and no more clients need to be visited. 
 Even though the pre-specified target inventory levels for each customer is 
intended to take into account the long-term effect of short-term decisions, the main 
limitation of those two pioneer works is the limited planning horizon.  In the nineteen 
nineties, as a result of increasing computer capabilities, new approaches to overcome 
that drawback were developed.  Those approaches deal with long-term problems, and 
are described below. 
 Campbell and Savelsbergh (2004) propose a two-step integer programming 
solution approach based on a rolling-horizon framework.  Even though they assume 
deterministic consumption rates at customers? sites, they consider safety stocks to 
handle the stochastic nature of demand.  Additionally, they assume that an unlimited 
amount of product is available at the depot and they do not incorporate the inventory 
holding costs either at the depot or at retailers? sites. 
 Their problem is formulated as a two-phase Integer Program (IP).  In Phase I, 
they determine when to deliver and how much to supply on each visit to each 
customer.  In Phase II, they solve the following problems.  First, they determine 
delivery routes for each day, which involves solving a vehicle-routing problem with 
time windows (VRPTW).  Second, they construct vehicles routes and schedules for 
two consecutive days.  The solution of Phase II is constrained: the quantities 
delivered to each customer should be equal to or greater than the solution provided by 
the first phase.  Finally, they solve Phase I for one month and, using this solution, 
they solve Phase II for only the first two days. 
 
 36 
 
 This problem is combinatorial, because there are a large number of possible 
delivery routes, and the problem needs to be solved for a long planning horizon.  In 
order to make this IP computationally tractable, they make additional assumptions.  
One of them is to consider only a small, but good, set of delivery routes.  Such 
clusters are normally selected as a prepossessing step.  Another assumption is that 
time periods towards the end of the planning horizon are aggregated.  An additional 
assumption to reduce the number of integer variables is to relax integrality restrictions 
on the variables representing weekly decisions.  Moreover, they also reduce the set of 
customers to those that require a delivery in the very short term, i.e. the next few 
days; customers with large impact on efficiency of the schedule, either with high 
demand or be being very distant from the depot; and customers that, though not 
require an immediate delivery, are near or in the same cluster as the first two types. 
 Bard et al. (1998a, 1998b) and Jaillet et al. (2002) study a similar two-phase 
approach.  The main differences are: customer inventory levels are not continuously 
reviewed, and satellite facilities are considered.  Satellite facilities are additional 
depots where vehicles can be reloaded, avoiding the necessity to return to the central 
depot.  Another difference is that the objective function combines two criteria.  First, 
the marginal costs associated with visiting customers on a different day than the 
optimal day between deliveries.  And second, the minimization of daily transportation 
cost.  
 The first criterion, incremental cost, is calculated using the approximation 
presented in Jaillet et al. (2002).  This is computed by first obtaining the optimal 
number of days between deliveries for each client.  Then the cost of serving each 
 
 37 
 
client on a day different than the optimal day is computed as an incremental cost.  
That value is calculated, assuming that future deliveries are maintained according to 
the optimal interval.  
 As a second criterion in the objective function, they take into account routing 
customers within a given day.  That is, this criterion attempts to minimize daily 
transportation cost. 
 The problem is solved in two steps, using a rolling-horizon framework.  First, 
one-year optimal delivery days are computed for each customer.  Second, an 
assignment problem is solved for the first two weeks.  The second problem is solved 
considering transportation costs and incremental costs associated with visiting 
customers on days different from the one obtained in the first step.  Finally, they 
implement the solution for the first week only. 
 Kleywegt et al. (2002, 2004) formulate a general IRP as a discrete time 
Markov decision process (MDP).  They make six basic assumptions: (i) that 
inventories at customers? sites can be measured once a day at no additional cost; (ii) 
that unsatisfied demands are lost, i.e., not backlogged; (iii) that inventory holding 
costs are incurred at customers? sites for each unit of inventory per day; (iv) that the 
supplier obtains revenues every time he/she dispatches products to a retailer 
proportional to the quantity delivered; (v) that the depot has an unlimited supply of 
the product and its inventory holding costs are not considered; and (vi) that the 
supplier knows the cost associated with each decision before hand (i.e., there are no 
uncertainties with respect to transportation costs associated with each possible policy 
 
 38 
 
and to inventory holding costs at each customer site).  Moreover, the supplier knows 
the values associated with shortage penalties. 
 The objective function is to maximize the discounted sum of net benefits over 
an infinite horizon.  At each stage, net benefits take into account: the revenues 
obtained from the quantities delivered to each customer, the transportation costs of 
product from depot to customers, the inventory holding costs at each customer?s site, 
and the expected costs associated with expected shortage penalties. 
 The Markov Decision process (MDP) associated with this problem is 
extremely hard to solve when there are more than four customers and a limited 
number of vehicles.  To overcome the problem, the authors develop an approximation 
technique for the optimal value function.  The approximation is based on 
decomposition per customer.  The decomposition is easily computed given the 
smaller state space.  Then an approximate value function is obtained by optimally 
assigning the fleet capacity by solving a non-linear knapsack problem. 
 The authors show near-to-optimality results for small problem-instances and 
better solutions than other approximate policies in larger problem-instances.  The 
small problem-instances considered have up to five customers and have demands 
taking less than 10 discrete values per customer; the larger ones, up to 60 consumers 
and 30 vehicles but only up to two levels of demand per consumer. 
 Adelman (2004) also formulate a similar stochastic inventory-routing problem 
as a discrete time Markov decision process (MDP).  The main difference between his 
work and Kleywegt et al. (2002, 2004) is that he presents a math-programming 
approach which uses dual prices of linear program relaxations to approximate the 
 
 39 
 
value function, instead of using a simulation-based approach.  As in most 
approximate DP approaches, only instances with small state spaces can be solved.  In 
addition, in his problem setting, lost sales are considered, and no constraints are 
explicitly imposed on the number of facilities visited per tour.  Adelman presented 
results in which his approach outperforms Kleywegt et al.?s direct shipping policy.  
 These DP formulations are very interesting from a formulation standpoint.  
However, it is extremely difficult, normally impossible, to solve even for small 
problem-instances.  The DP method ?provides more benefit if the available 
information about the future is more accurate? (Kleywegt et al., 2002, p. 115).  In 
reality, information about the future is not very accurate, since information about 
demand distributions is not exact.  Therefore, in real-world applications the benefits 
of DP approaches are expected to be lower than those presented in simulations, where 
demand follows exact, known demand distributions. 
 The main difference between the problems found in the literature and the one 
addressed in this research is that all previous works have considered static vehicle 
routes, i.e. vehicle routes are not modified after they started until they are completed.  
However, with modern information and communication technologies, it would be 
possible to establish mid-route communication with the drivers to modify their plans 
(Regan et al., 1995, Regan et al., 1996a). 
 To summarize thus far, the main decisions addressed in IRPs have been 
established, the extension under real-time information has been introduced, and the 
main previous primary research in the area has been reviewed.  The following section 
 
 40 
 
analyzes previous research on fleet operations under real-time information?research 
that is relevant to the IRP addressed in our research. 
2.2.3. Real-Time Fleet Operations  
 Real-time techniques are important in a context where information about the 
state of the system is gradually revealed during the operation and cannot be 
accurately predicted in advance.  That area of research for fleet management is 
relatively new, Psaraftis (1988) points out that by the end of the nineties not much 
had been published on real-time vehicle- routing problems.  For recent surveys on 
dynamic vehicle-routing problems and related routing problems see Psaraftis (1995), 
Powell et al.(1995), Bertsimas and Simchi-Levi (1996), and Powell (2003). 
 The main two approaches followed to deal with operations under real-time 
information have been rolling-horizon methods and stochastic methods to address an 
infinite-horizon system under steady-state conditions. 
 The first approach uses a rolling-horizon framework (see for example 
Winston, 1994), in which a new problem-instance is solved as new information 
become available.  But instead of implementing the solution for the complete 
planning horizon, the solution for only the first part of the planning period is 
implemented, and the process is repeated.  During the time a new solution is 
computed the vehicles continue moving and new events could unfold; so, there is a 
trade-off between the time required to obtain a new solution and the quality of the 
solution (Ichoua et al., 2000).  Moreover, given that the problems are NP-Hard, 
optimal solutions would lead to long computation times, which would make them 
 
 41 
 
impractical for real-size problems.  Hence, normally fast heuristics have been 
implemented that take advantage of local operations, such as insertion.  
 One interesting implementation of the rolling-horizon approach to the general 
dynamic VRP with a time window is proposed by Gendreau et al. (1999) and 
extended by Ichoua et al. (2000) to include diversions.  They propose a general 
heuristic strategy in which a tabu search procedure is continuously running, trying to 
improve the current solution, and new requests are handled with a faster local-search 
heuristic for inserting new demands.  That strategy allows them to take acceptance or 
rejection decisions in a fixed amount of time.  One of the interesting features of the 
implementation is the time projection used to update the state of the system.  It is 
used to correctly reflect the initial conditions on the problem to be solved when a new 
demand is known.  Instead of considering the actual state of the system at time t in the 
insertion procedure, they project it to a time (t+ t? ) where t?  is the time required in 
the optimization procedure. 
 In the context truckload (TL) pick up and delivery problems, Regan et al. 
(1995, 1996b) propose and investigate various local rules for the dynamic assignment 
of vehicles to loads under real-time information.  The rules are easy to implement and 
fast to execute, but they could be improved, using formal optimization techniques.  
Yang et al. (1998, 2004) extend that work to consider re-optimization real-time 
policies; the main drawback of this approach is the computation time required, which 
limits the applicability of the approach to limited-size problems.  To overcome those 
difficulties, Kim et al. (2002b) consider a two-phase optimization approach: in a first 
step, new demands are inserted if they are feasible to the truck with minimum 
 
 42 
 
insertion cost; then, in a second step, re-assignments of loads between different trucks 
are considered, but restricted to a subset of vehicles to maintain computation times 
stable. 
 Another application relevant to our research is the dynamic allocation of 
inventories for a fixed delivery route presented by Kumar et al. (1995).  They 
compare static-allocation policies?where the replenishment quantities at each 
retailer are determined simultaneously for all retailers?and dynamic-allocations 
policies?where replenishment quantities are determined sequentially, upon arrival of 
the delivery vehicle at each retailer, on the basis of the inventory level at subsequent 
retailers in the fixed route.  They show that even under the ?dynamic-allocation 
assumption??where the dynamic-allocation problem at each retailer is relaxed, 
allowing negative replenishment quantities?dynamic policies yield lower expected 
cost per replenishment and allocation cycle than static policies. 
 A second, more ambitious approach is the use of stochastic methods, in 
which, instead of reacting to new information, the future is forecast.  Among 
stochastic methods there are two main categories: Stochastic Programming and 
Markov Decision Processes.  The main literature in this area can be found in Powell 
et al. (1995), Powell (2003), and Gans and Van Ryzin (1999).  Unfortunately, those 
approaches have computation time that grows exponentially with the size of the 
problem, making them more suitable for a priori plans than for real-time re-
optimization. 
 
 43 
 
2.3. Background in Real-Time Combinatorial Optimization 
 One of the characteristics of operations in real time is that information about 
problems needing to be solved by decision makers is dynamically revealed.  That 
contrasts with traditional static optimization, in which it is normally assumed that all 
relevant data to solve a problem-instance is known in advance.  In real-time 
operations decisions should be made without complete information about future 
outcomes, and since those outcomes are not known in advance, they could only be 
considered in a probabilistic sense at any decision epoch.  In addition, plans or 
policies can be updated with online information about the state of the system.  For 
that reason, the implementation of an operational-control strategy should establish (i) 
when, i.e. what events should trigger plan updates, and (ii) how to update plans.  
 In terms of plan-update epoch decisions, the most common operational 
strategies are (i) event-driven strategies, in which plan-updates are triggered 
whenever the state of the system satisfies certain criteria, (ii) time-driven strategies, in 
which plans are updated at regular time intervals, for example periodic review 
strategies on inventory control, and (iii) mixed strategies, in which event- and time-
driven strategies are considered together. 
 With respect to how plans are updated, planning decisions can be classified as 
(i) reactive, by which the previous plan is locally modified to accommodate recent 
events, (ii) incremental, by which the previous plan is more than slightly modified, 
and (iii) deliberative (or re-plan), by which a completely new plan is built from 
scratch; this is normally performed when the state of the system significantly deviates 
from its forecast (Seguin et al., 1997, Gr?tschel et al., 2001b).  In general, the 
 
 44 
 
recommended type of planning decision depends on a trade-off between the benefits 
of fast reaction to unpredicted events and the quality of the resulting solutions.  
Normally, the more time spent on evaluating alternatives, the better the plan selected.  
 In order to evaluate and compare different real-time operational strategies, 
there have been two main approaches proposed in the literature: competitive analysis 
and discrete-events simulation. 
 Competitive analysis is a form of worst-case analysis, in which the evaluation 
of each decision is based on the worst-possible sequence of events resulting from that 
decision.  The main limitation of that approach is that, in many cases, results are 
unduly pessimistic.  Even though some modifications have been presented to 
overcome that limitation, it is still complicated to obtain meaningful results for 
combinatorial problems.  In addition, competitive analysis does not take into account 
real-time requirements of real-world systems in which the trade-off between solution 
quality and speed is a relevant issue.  A detailed overview of competitive analysis and 
extensions is presented in Gr?tschel et al. (2001b). 
 The second approach to evaluating and comparing real-time strategies is 
discrete-events simulation, in which the operation of the system is mimicked under 
different operational strategies.  Those experiments are conducted for different 
realizations of the same stream of events over long periods of time, and statistics 
about the performance of the system using different criteria are gathered (see for 
example Law and Kelton, 2000).  The main advantages of simulation experiments 
are: they provide results for analytically intractable systems, and they provide a full 
range of statistics about system performance.
 
 45 
 
Chapter 3: Problem Definition and General Approach 
 
This chapter provides a detailed formulation of the online inventory routing 
problem (OIRP) studied in this research and describes the general approach used to 
solve the problem.  Section 1 states the problem context and specific assumptions.  
Section 2 introduces the main notation used and formulates the OIRP as a real-time 
combinatorial optimization problem.  Section 3 presents the major sources of 
complexity, and the general approach used to deal with the OIRP. 
3.1. Problem Context and Main Assumptions 
 The general characteristics of the problem studied were introduced in Chapter 
1.  This section presents a specific definition of the OIRP and the main assumptions 
related to thereto.  
In the OIRP a two-echelon distribution system for a single product from one 
to many facilities is considered.  The system is composed of a single-vehicle fleet 
with limited capacity, a single depot that keeps an infinite supply at no cost, where 
the vehicle is reloaded, and a set of N retailers which face independent and stochastic 
demand processes and which need to be repeatedly refilled over time.  The vehicle 
moves products from the depot to the retailers, and can consolidate loads to different 
facilities on the same route.  In addition, the vehicle can reallocate loads among 
retailers while en route, but transshipments are not allowed.  Consequently, after 
products have been delivered to a particular facility, they cannot be reclaimed and 
reassigned to a different facility. 
 
 46 
 
This research assumes that demands are the only source of uncertainty.  
Variability in travel times because of incidents, congestion in the network, or possible 
vehicle breakdowns are not considered.  Moreover, it is assumed that loading and 
unloading time is negligible.  Yet, real-time operational capabilities might be also 
provide benefits in those circumstances, allowing the operator of the system to 
respond faster to possible contingencies. 
The system is assumed to be operating continuously, without interruption, that 
is, daily and weekly cycle operation characteristics are not taken into account.  
Moreover, labor-related constraints are not considered.  In short, it is assumed that the 
vehicle and all facilities are always in operation; i.e., deliveries can be scheduled at 
any time, with neither time windows for particular facilities nor restrictions on the 
number of hours that a driver can operate a vehicle.  Therefore, delivery routes are 
constrained only by the vehicle?s capacity to transport products. 
The system is operated by a central decision maker, whose objective is to 
move inventories in the system so as to maximize profit in the long-run.  It is 
assumed that the demand processes at retailers cannot be affected by the decision 
maker decisions; that is, short-term pricing incentives are not considered.  Therefore, 
the problem is equivalent to minimizing the expected total operating cost per unit of 
time.  
The operating costs considered consist of transportation, inventory holding, 
and lost-sales penalty costs.  Transportation costs are assumed to be only proportional 
to the total distance traveled by the vehicle.  That is consistent with a hierarchical 
decision-making perspective, in which strategic and tactical decisions, such as the 
 
 47 
 
fleet size for a given day, are fixed and given.  Accordingly, in the short-term 
operational problem studied here, fixed-fleet costs are considered to be sunk costs; 
hence, the relevant decision become how much to use those resources.  In addition, 
transportation costs per unit of distance will not depend on the amount of load 
transported by the vehicle, which is a common assumption in the vehicle-routing 
literature (Christofides, 1985, Golden and Assad, 1988, Toth and Vigo, 2002).  In 
relation to inventory costs, each retailer i accrues inventory-holding cost, h
i
, per unit 
of inventory on hand per unit of time, and the demand during stock-out is lost with an 
associated penalty cost, 
i
p , for each unit of demand lost per retailer i.  Those costs 
parameters are considered to be known and fixed for the planning horizon. 
 The central decision maker operates the system with real-time information 
about the complete state of the system.  In other words, the central decision maker has 
accurate real-time knowledge of all local inventory levels, the location of the vehicle, 
and the load remaining in the vehicle.  The decision maker also has real-time two-
way communication with the truck driver and can update truck plans at any time.  
However, if the vehicle is traveling when an update decision is made, a time lag is 
imposed before the new plan can be implemented.  
 The main decisions available to the decision maker are related to truck plans 
and are defined by i) the sequence of facilities to be visited, ii) departure time from 
the depot, which is the only place where the truck can be idle, and iii) the amounts to 
be delivered (or picked up, in the case of the depot) to subsequent facilities on the 
route.  Therefore, the main decision alternatives at a given plan update epoch are: 
 
 48 
 
a) modify the set and/or the sequence of subsequent customers on the 
planned routes, 
b) divert a truck from its current destination to a different facility, if the truck 
is traveling, 
c) adjust amounts to be delivered to subsequent facilities on the route, or 
d) change the amount of time spent at the depot. 
 Even though the idealized problem described here represents a simplified 
version of real-world logistic-distribution problems, in which some issues are 
deliberately ignored, its analysis can provides relevant insights about how to use real-
time information and control capabilities in distribution operations, and the associated 
benefits therefrom. 
3.2. Problem Formulation 
 This section formally presents the online inventory-routing problem (OIRP) 
being investigated.  First, the main notation and parameters used to describe the 
problem are introduced.  Second, the main variables and additional notation used to 
describe the OIRP are presented.  Third, main constraints to be satisfied in the 
operation of the system are formally stated.  Finally, the objective function of the 
OIRP is specified.  
3.2.1. Preliminaries and Problem Parameters 
 In order to present the OIRP, the following general notation is used to 
describe the elements of the system.  The set of retailers is designated as ?, 
{ }1, 2,..., ,...,iN?= , and the set of all facilities (depot and retailers) as 
0
? , 
 
 49 
 
{ }
0
0???? .  Those N+1 facilities are denoted by sub-index i=0,1, 2, ?, N (sub-
index 0 is for the depot) and are located in a bounded subset in the Euclidean space.  
Those locations are denoted l(i) for 
0
i?? .  The function ( ),d ? ?  gives the Euclidean 
distance between two facilities or between a facility and the vehicle location.  Each 
retailer i has a maximum capacity to store inventory, 
i
? , measured in the units of the 
single product considered.  In addition, the vehicle has limited capacity, ? , measured 
in the same units, and its assumed to travel at constant speed according to the 
Euclidean metrics.  Without loss of generality, the vehicle speed is assumed to be 
one. 
 Each retailer i serves an independent demand process.  In general, it is 
assumed that each facility serves a compound Poisson demand process, in which 
customer arrivals to retailers follow Poisson processes, and customers? demand sizes 
are independent discrete random variables.  Demand processes have associated arrival 
rates ()
i
t?  for retailer i at time t, and associated probability mass function (pmf) 
()
j
i
t? , for the probability that a customer arriving at time t to retailer i has a demand 
size equal to j.  In addition, unless otherwise noted, customer demand sizes are 
assumed to be Poisson distributed with mean ( )
i
t? .  Thus the expected demand per 
unit of time at retailer i at time t, 
( )
i
t? , can be calculated as 
( )()()
iii
ttt???=?.  In 
these demand processes, arrivals times and demand sizes are denoted 
,im
?  and 
,im
?  
respectively, for the m
th
 customer arrival to facility i.  Thus, the total number of 
customer arrivals to retailer i that have occurred by time t is 
 
 50 
 
() { }
,
max 0 :
iim
At m t?=??, and the total demand at customer i until time t is 
()
()
,
1
i
At
iim
m
Dt ?
=
=
?
, including satisfied and lost demands. 
 The state of the system at time t, ( )X t , can be described by the following 
parameters: (i) inventory levels at time t, 
( )
1
( ) ( ),..., ( ),...., ( )
iN
ttt t?? ? ?= , where ( )
i
t?  is 
the inventory level at facility i at time t, (ii) location of the truck at time t, ()tnull , and 
(iii) load remaining in the truck at time t, ( )t? .  Hence, the state of the system at time 
t can be expressed as: 
 
( ) ( ) ( ) ( )
X tttt??=? ?
? ?
null  (3.1) 
 The decision maker can update plans at any epoch t based on ()X t  and past 
events, but without knowledge of future events.  Plan updates are implemented 
immediately unless the vehicle is moving, in which case a time lag?between the 
epoch when a decision to update a plan is made and the plan is implemented?is 
considered.  This is modeled using a time projection, which takes into account the 
time from the moment the decision to update the current plan is made until the new 
plan begins to be executed.  Hence, instead of considering the actual state of the 
system at time t in the solution procedure, the state of the system is projected to a 
time ()tt+? , ()X tt+?
null
, assuming expected consumption rates and truck current 
speed and destination, where t?  is the projection time, which includes any solution 
procedure used to update plans and the time required for the driver to modify his 
current destination. 
 
 51 
 
 In the OIRP, there are three sets of cost parameters: (i) transportation cost per 
unit of distance traveled by the vehicle, TC, (ii) inventory holding costs at each 
retailer i, 
i
h  for retailer i, and (iii) penalty associated with each unit of demand lost 
during stock-out, 
i
p  is the at retailer i . 
3.2.2. Decision Variables 
In this OIRP system, the only decisions available to the decision maker are 
related to truck plans, and can be summarized as, when, and how truck plans are 
updated.  As mentioned, a plan or policy, ? , can be specified by the sequence of 
facilities to be visited, 
[ ]
12 L
ss s=s ? , amounts to be delivered, 
[ ]
12 L
qq q=q ? , and arrival times, 
[ ]
12 L
tt t=t ? , to each one of those 
facilities, in which L  is the length of the planning horizon in terms of number of 
visits programmed.  Thus, a plan or policy can be written as: 
 
[ ]
? = sqt (3.2) 
In addition, since the state of the system is continuously monitored and plans can be 
updated at any time, plan update epochs are also decision variables.  The sequence of 
update epochs are denoted 
[ ]
12
uu=u ? , in which 
n
u  is the time of the n
th
 plan 
update satisfying 
1
0
nn
uu
+
>? for all n.  Let 
( ) { }
max 0 :
n
Ut n u t= ?? be the 
number of plan updates up to epoch t.  Then 
()
()
Ut
t? ??  is the current plan at epoch t, 
and 
()
()
Ut
ut u?  is the time of the last plan update.  Accordingly, the set of update 
epochs, 
{ }
12
,,,()uu ut? , and associated policies { }
12
,,,()t?? ?? , give the complete 
history of vehicle deliveries until epoch t.  
 
 52 
 
The following additional notation is introduced, before introducing the main 
constraints to be satisfied in the OIRP.  Let ( )
n
H ?  be the number of facilities visited 
while following the n
th
 policy, i.e., ( ) { }
1
max : ( )
ninn
Hitu??
+
=? which satisfies 
() ()
nn
HL? ?? , and let 
()
()Ht?  be the number of facilities already visited under the 
current plan at epoch t.  Let ( )
i
B t  be the total number of visits to facility i that have 
occurred by epoch t, which can be expressed as: 
 
()
()()
11
() ( )
n
HUt
ijn
nj
Bt i s
?
?
==
??
1      , for 
0
i?? , and all t (3.3) 
where 
()?1  is an indicator function that takes the value one if the argument is true and 
zero otherwise.  In addition, let 
,ik
?  and 
,ik
q  denote the k
th
 arrival time and quantity 
refilled at facility i, respectively. 
 
{ }
,1,1,1
():() ( );() ; ()
ik jn in in jn ik jn ik
tBukBus i t?? ????
+?+
=<?=<< (3.4) 
 
{ }
():() ( );() ; ()
ik jn in in jn ik jn ik
qq BukBus i t??
+?+
=<?=<< (3.5) 
Thus, the total amount of products refilled to retailer i until time t is 
()
()
,
1
i
Bt
iik
k
Qt q
=
=
?
. 
3.2.3. Main Constraints  
The main constraints that must be satisfied in this OIRP are related to the 
dynamics of the system and could be stated as follows: 
a) Inventory levels at each retailer are always non-negatives and less than 
their capacity: 
 ( )0
ii
t? ???     , for i??, and all t (3.6) 
 
 53 
 
b) Inventory levels at retailers decrease with consumptions and increase with 
deliveries: 
 
( ){ }
,,,
0
lim ( ) max 0, ( )
i im i im im
?
?? ? ?? ?
?
+= ?      , for i??, and all m (3.7) 
 
,,,
0
lim ( ) ( )
i ik i ik ik
q
?
? ????
?
+= +      , for i??, and all k (3.8) 
c) The load remaining in the truck is always non negative: 
 () 0t? ?      , for all t (3.9) 
d) The amount delivered to a retailer i is not greater than the load remaining 
in the vehicle at that delivery epoch, and the load remaining in the vehicle 
after the delivery is decreased by the quantity delivered. 
 
,,
()
ik ik
q ? ??      , for i??, and all (3.10) 
 
,,,
0
lim ( ) ( )
ik ik ik
q
?
? ????
?
+= ?     , for i??, and all (3.11) 
e) The total amount delivered in a tour does not exceed its capacity.  
 
0, 1
0,
()
,
()
ik
ik
B
ir
rB i
q
?
?
+
=??
??
??
     , for all k (3.12) 
f) The location of the truck is modified whenever the truck is not idle, and 
the truck moves toward the next facility at unit speed, so that arrival times 
should satisfy:  
 
( )
()
11
() () (),(
, for  1, 2, , 1 , and all 
h n hn h n hn
nn
ttdss
hH
?? ??
? ?
++
??
=??
 (3.13) 
If the decision space is restricted to send vehicle to the next facility and 
vehicle can only be idle at the depot, then this constraint should be 
satisfied with equality whenever 
1
()0
hn
s ?
+
>  and ( ) 0
hn
s ? > .  In addition, 
 
 54 
 
for this case, the vehicle location at the beginning of the n
th
 plan can be 
written as: 
 
111
1
(( ) (1 )( ( ) , if 1
()
( ( ))                                     ,  otherwise
Hn H n
n
Hn
ls ls
u
ls
?? ? ? ?
?
?+?
?
?+?? ??
=
?
?
null  (3.14) 
()
11 1 11
,  where ( ) ( ( ), ( ))
HnnHnHn
tudss?? ? ?
+? ? +?
=? , and the location of 
the vehicle at time t: 
 
()() ()( )
()()
1
() (1 ) () , if 1
()
( )                                      , otherwise
HH
H
ls t ls t
t
ls t
?? ? ? ?
?
+
? ?+?? ?
?
=
?
?
?
null  (3.15) 
( )()( ) ( )( )
11
, where () () () , ()
HHH
ttutdstst?? ? ?
++
=? .  
 Finally, the set of policies until epoch t, 
{ }
12
,,,()t?? ?? , that satisfy all these 
constraints for a given stream of demand realization, 
{ }
12
1,1 1,2 1, ( ) 2,1 2,2 2, ( ) ,1 ,2 , ( )
() , , , , , , , , , , , ,
N
At A t N N NA t
t ?? ? ?? ? ? ? ?=? ???? , is denoted ()t? . 
3.2.4. Objective Function  
 The objective of the central decision maker is to move the inventories in the 
system so as to minimize the expected total operating cost, composed of 
transportation, inventory holding, and lost sales costs.  Using the notation introduced 
in the previous subsection, those three components can be written as: 
a) Total transportation costs until epoch t, TTC(t) = 
 ()( )( )
()()
11
12
( ), ( ) ( ), ( ) ( ( )), ( ( ))
n
HUt
nn hnhn H
nh
TC d u s d s s d s t u t
?
????
?
==
??
??
??
?+ +
??
??
??
??
??
nullnull(3.16) 
b) Total inventory holding costs until epoch t, TIHC(t) = 
 
 55 
 
 
()
()
()
()
()
{}
()
(){}
,
,,
,1
,
,,
,
()
,
,1 ,1
0
max ( ),
,,,1
()2
max ,
iik
i
iik iik
iik i
iik
iik ik
iA
Bt
ii ik
iA iA
ik
AAt
iim im im
mA
ht
?
??
?
?
?? ? ?
?? ? ?
?? ? ?
+
++
?=
?
=+
????
????
??
?? +
??
??
?? ?
???+
?? ?
?? ?
?? ?
????
????
??
??
?
?
?
?
 (3.17) 
where: 
 
()
()
( )
(){}
,1
,
,1 , , ,
max 0,
iim
iim
B
i im ik i im im
kB
q
?
?
?? ?? ?
+
+
=
=+ ?
?
 (3.18) 
 
()
()
( )
()
{ }
,,
,
,,
max 0,
iik iik
iik i
iA iA??
?? ?? ?=? (3.19) 
c) Total lost sales penalty costs until epoch t, TLSC(t) = 
 
{}
()
,1
,
max ( ), ( )
()
,,1 ,1
1(1
max 0,
iik i
i
iik
AAt
Bt
iimik
kmA
pq
?
?
???
+
??
==
??
??
?? ?
??
???
?? ?
??
??
??
 (3.20) 
 
 Therefore the OIRP objective function can be written as: 
 
{}
() () (){}
1, , ( )
,
1
min lim
nn
nUt
ut
TTC t TIHC t TLSC t
t
?
=
??
?++
?
 (3.21) 
where the set of update epochs and policies are restricted so as to use information 
about current state of the system and past events without knowledge of the future, and 
{ }
1, , ( )
,()
nn
nUt
ut?
=
??
?
. 
Thus far, the formal definition of the problem studied in this dissertation has 
been introduced.  The next section discusses major sources of complexity, the general 
approach that is being used to deal with the OIRP, and the main limitations. 
 
 56 
 
3.3. General Approach 
 As previously stated, the OIRP does not seem to be tractable.  Among the 
main difficulties in solving this control problem are: 
a) Simplified static and deterministic versions of the problem are NP-Hard, 
i.e., given the complete stream of future demands at each retailer, the 
associated inventory-routing problem, to optimally schedule deliveries to 
retailers, is very difficult to solve.  In addition to the sequencing 
complexity of the problem, in the scheduling of visits to facilities is 
difficult to correctly capture the effect of short-term decisions on long-
term costs, since deliveries depend on the time and amount reloaded in the 
previous visit to that facility.  For that reason, an optimal solution to the 
problem would require a long-term planning horizon; therefore, even for 
small problem-instances, it is unlikely that the problem could be solved to 
even near optimality in a reasonable time.  That precludes the use of a 
complete static and deterministic IRP formulation for re-planning 
purposes in real-time operations. 
b) In addition to the combinatorial challenge of the static version of the 
problem, demands are dynamic and stochastic, and decisions can be 
updated at any time.  In fact?in contrast with other real-time fleet 
operation problems in which requests to the system are clearly decision 
epochs?in the OIRP, final customer-demand epochs occur so often that it 
would be infeasible to adjust plans at each one of them.  Thus, update 
 
 57 
 
epochs are not clearly defined, and obtaining the best update epoch is a not 
trivial undertaking.  
c) Because retailer deliveries can be combined on the same route, optimal 
policies to serve each retailer depend not only upon that retailer?s 
inventory level, but also upon the state of the complete system.  In fact, 
transportation costs to service a particular facility are not fixed, but depend 
upon the set of facilities served on the same route (Campbell et al., 1998).  
Moreover, since a single vehicle serves all retailers, the lead time to 
replenish a particular retailer might be affected by congestion, in terms of 
the number of additional deliveries that are scheduled before that visit. 
d) The advances in real-time online combinatorial optimization neither 
provide tools to solve problems, such as the OIRP, to optimality nor give 
clear guidance on how to exploit online information in its operation 
(Gr?tschel et al., 2001b, Gr?tschel et al., 2001a). 
e) Finally, as in most real-time combinatorial problems, there is a trade-off 
between the quality of a new plan and the response time at update epochs. 
 Those difficulties prevent solving the problem or finding an optimal policy 
directly from the formulation presented in the previous section.  Instead, two 
approaches are proposed wherein either the inventory control side or the routing side 
of the problem are solved first.  Those formulations take into account only a 
simplified version of the other side of the problem, but allow formulating 
optimization problems for that other side, in which those a priori solutions are used 
as soft-constraints.  In the first approach, inventory reorder parameters are established 
 
 58 
 
for each facility and then used as target levels on a routing problem used to update 
plans.  In the second case, the sequence in which facilities are visited is fixed, and 
then inventory allocation decisions are taken respecting that sequence. 
On each approach, different operational policies are proposed, tailored for 
different degrees of sophistication in terms of ICT.  Those operational policies are 
based on a rolling-horizon framework, wherein new operational plans are repeatedly 
generated, based on updated information about the state of the system, and they are 
implemented until the next update epoch is reached.  In that scheme, operational 
strategies are defined by when and how plans are updated.  
In terms of plan update epochs, three different operational-strategy cases are 
analyzed, based on different degrees of ICT capabilities considered.  They can be 
ordered in terms of decreasing ICT requirements as: i) truck routes can be 
continuously updated, allowing for en-route diversions, ii) truck routes can be 
updated only at facilities (en-route diversion not being allowed), and iii) truck routes 
cannot be updated after the truck leaves the depot, i.e. truck plans can be updated only 
upon tour completions.  In all cases, full information about the state of the system at 
plan update epochs is assumed. 
In terms of optimization capabilities, two cases are considered for obtaining 
new plans: i) simple rules, and ii) specialized software that allows solving 
combinatorial problems on a real-time basis.  In both cases, operational strategies 
should establish what rules and/or mathematical-program formulations will be used to 
obtain new plans.  This research proposes mixed-integer programming (MIP) 
problem formulation for new plan generations. 
 
 59 
 
Thus, different operational policies could be implemented, based on how often 
the off-line problem is solved and/or how many steps of the current solution are 
implemented before solving a new problem-instance with updated information about 
the state of the system. 
 In order to evaluate and compare real-time strategies, between the two 
approaches discussed in section 2.3 (competitive analysis and discrete event-
simulation)?given the difficulties, which were previously discussed in point ?a,? 
related to obtaining analytical solutions for the deterministic IRP?competitive 
analysis approaches are discarded in our analysis.  So, evaluation and comparison of 
proposed policies are conducted using discrete-event simulation experiments.  
3.3.1. Simulation Framework 
 In discrete-event simulation, the state of the system is traced as the events that 
modify its status unfold.  At each one of these events, the state of the system is 
updated and the simulation clock is advanced to the next event until the end of the 
simulation (Law and Kelton, 2000).  Those events can be divided into two categories: 
stochastic and deterministic.  In the OIRP, stochastic events include demand at 
retailers? sites; and deterministic events?which, in our case, are policy related, since 
only deterministic policies are studied?include truck arrival time at a facility with 
associated delivery; truck diversion; tour-completion time with associated 
replenishment; and tour beginning time.  
 Simulation experiments enable the analysis of the system in a replicable and 
controlled environment, in which different policies can be fairly compared under 
similar conditions.  Different policies are compared, based on an identical stream of 
 
 60 
 
stochastic events, which are generated using the same set of random number seed 
generators (see, for example, Law and Kelton, 2000).  Thus, results of simulation 
experiments reflect operation of the system under steady-state conditions for the 
particular set of parameters studied, in which statistics about the specific state of the 
system can be derived.  
 In this manner, simulation experiments allow comparison of the performance 
among proposed policies in a wide range of scenarios related to different 
combinations of problem parameters, such as demand rates, demand uncertainties, the 
existence of disruptive demand patterns, trade-off among cost parameters, vehicle and 
facilities capacities, geographic location of facilities, etc. 
3.3.2. Benchmark Policies 
 An additional difficulty in evaluating real-time fleet operational strategies is 
that there are not settled benchmarks to compare proposed policies.  As discussed by 
Kim (2003), ?detailed specifications of the problem have a significant impact on the 
performance of a policy.? Since the OIRP has not been studied before, two 
benchmark policies are introduced and developed.  
 The first benchmark policy, BENCH1, emulates what can be achieved 
operating the system in a decentralized manner with agents following optimal 
policies.  In BENCH1, each retailer manages his own inventory, placing orders to a 
central supplier who, once a day, schedules deliveries for previous-day orders.  In this 
case, on one hand, based on the orders received at the end of each day, the supplier 
creates routes solving a vehicle-routing problem (VRP).  Then the VRP solution is 
implemented to make deliveries for her customers.  On the other hand, each retailer 
 
 61 
 
will follow an optimal continuous-review policy to control his inventory, which in 
this case, as discussed in subsection 2.1.3, corresponds to an (s, S) policy.  That is, 
each retailer will place an order of size S-s immediately if his inventory level is below 
s.  The optimal parameters for an (s, S) policy can be obtained using Zheng and 
Federgruen algorithm (1991) or by exhaustive search over the feasible region.  
The second benchmark policy, most-urgent-next (MUN) is based on a simple 
greedy decision rule.  In MUN at each delivery epoch the vehicle is send next to refill 
the retailer closest to run out of inventory.  To select the next retailer to be refilled, 
inventory levels are inspected and based on average consumption rates, the time at 
which each retailer would run-out of inventory if not visited is calculated.  If the 
vehicle had enough load remaining to refill the selected retailer, it would be send 
directly to that location, otherwise it would go first to be refilled at the depot and then 
to that retailer.  In that policy, each retailer i is refilled up to a pre-specified target 
level, 
i
S , or up to capacity 
i
? .  MUN implementation assumes that inventory levels 
are monitored and that routes are created so that the next delivery is decided upon 
refilling a retailer.  
In these benchmark policies, operational control parameters, such as reorder 
levels, are adjusted based on the specific parameters of each scenario.  The details 
will be presented in Chapter 6, as part of the simulation experiments. 
3.3.3. Performance Measures 
 As stated in the objective function of the OIRP, the main performance index 
used to evaluate any proposed real-time strategy is the minimization of expected total 
costs per unit of time (week or day), which include expected inventory holding, 
 
 62 
 
stock-outs, and transportation costs.  Those indicators contain common performance 
measures used in distribution operation, such as fill rates and fleet utilization.  In the 
OIRP, fill rates, which measure the proportion of demand served from inventory on 
hand, are equivalent to stock-outs.  Also, fleet utilization, which measures the 
percentage of time the vehicle is not idle, can be obtained directly from transportation 
costs, as the ratio between transportation costs per period and the cost of running the 
vehicle continuously during that period.  In addition, other important performance 
measures, not included in the objective function, that are considered in our analysis 
include: 
a) Variability of total costs per unit of time, which is measured as the 
standard deviation of weekly total costs, and can also be decomposed in its 
components.  This is an indicator of reliability and consistency in the costs 
of operating the system using a particular strategy.  In general, managers 
prefer strategies with high consistency, to avoid the additional burden of 
explaining bad outcomes and dealing with disruption operations. 
b) Route lengths in terms of number of retailers visited and distance traveled.  
Even though the proposed models do not consider any restrictions on tour 
length, it is interesting to analyze whether the application of a particular 
strategy would lead to unreasonably long routes. 
So far, the online inventory routing problem (OIRP) has been formulated, 
previous research that relates to it has been reviewed, and the general approach used 
to study this problem has been presented.  In the next chapters, proposed real-time 
strategies are presented and then evaluated.  The proposed real-time strategies are 
 
 63 
 
classified according to the technology used to update plans in (i) optimization based 
strategies, and (ii) simple-heuristic based strategies.  In the first group, an off-line 
optimization problem is formulated and employed to update plans, and in the second 
group, simple-heuristic rules are used to update plans.  In this manner, the first two 
tasks will be presented together in two chapters, one dedicated to optimization based 
strategies, and other devoted to simple-heuristic based strategies.  Then, the last task, 
evaluation of proposed real-time strategies and analysis of benefits for different 
distribution-system scenarios, is presented in one chapter describing simulation 
experiments and results. 
 
 
 
 64 
 
Chapter 4: Formulation and Design of Optimization 
Based Control Strategies for Fixed Inventory Target 
Levels 
 
This chapter presents the formulation and design of optimization based 
strategies, in which the inventory control side of the problem is solved first without 
considering joint replenishment to different facilities.  First, a local off-line problem 
is formulated which is then used in different real-time control strategies to update 
plans.  Second, it presents different control strategies based on different degrees of 
real-time information availability for controlling the system.  Also, it presents an 
optimization framework for adjusting policy parameters for each strategy. 
4.1. Off-Line Optimization Problem Formulation  
 This section presents the local off-line problem used on optimization based 
control strategies to update plans.  First, the general approach to locally update plan is 
presented.  Second, optimization of refilling levels method is presented.  Finally, a 
mathematical formulation of the routing problem used to update delivery plans is 
presented.  
4.1.1. Preliminaries  
One of the main difficulties in formulating this problem is to be able to 
capture the effect of short-term decisions on long-run costs.  If the customers were 
visited in isolation of each other using direct deliveries from the depot, served by 
independent vehicles, the optimal policy for each customer could be computed.  In 
this case, a well known result on inventory control for single items inventory systems 
 
 65 
 
with stochastic consumption rates, constant replenishment lead times, and standard 
cost assumptions is the optimality of (s, S) policies, see for example Axs?ter (2000) 
and Zipkin (2000).  In an (s, S) policy each time the inventory position (inventory on 
hand plus on order minus backorders) is below s a delivery is scheduled to send a 
quantity equal to S minus the inventory position, so the inventory position becomes 
equal to S. 
However, since only one truck is serving all customers and customer 
deliveries can be combined on the same route, transportation (delivery) costs are not 
fixed.  Indeed, they would depend on the set of customers that are served together on 
the same route.  Then the optimal policy to serve each customer would depend not 
only on its inventory level, but also on the complete state of the system. 
To deal with this problem, optimal refilling levels for each facility are first 
specified, assuming that there is no pattern of deliveries.  These levels are then kept 
as targets to refill up to, on each delivery, when plans are generated.  However, there 
are only penalties associated with violating them as they are not included as hard 
constraints in the off-line routing problem.  The off-line routing problem generates a 
plan that stipulates for each customer the next delivery time and quantity to refill, 
based on reorder quantities and on the current state of the system, i.e. inventory levels 
at each facility, and location and load remaining on the truck.  This plan is obtained 
by minimizing the sum of transportation costs, and expected lost sales penalty (LSP) 
costs, subject to visiting all customers once during the planning horizon. 
 
 66 
 
In the next subsection, the method used to compute reorder quantities for each 
facility is presented, followed by the mathematical formulation of the off-line routing 
problem. 
4.1.2. Optimization of Refilling Levels  
To compute the target refilling levels for each facility, the sum of expected 
average cost for all facilities is minimized, assuming that there is no pattern of 
deliveries and the truck visits only one customer per route.  That is the truck goes 
back to the depot after refilling a customer and from there it would go to the next 
customer if it were necessary.  Since the truck visits only one customer per route, 
transportation costs associated with serving a particular customer are fixed.  Even 
though customers are visited in isolation, given that there is a single truck to serve all 
of them, the possibility of waiting for service due to the truck serving other facilities 
is incorporated.  Additionally, in contrast with traditional inventory systems where 
quantities are fixed after orders are made, in the system of interest quantities can be 
updated after arriving to a customer.  
Then, using a policy that places orders when the inventory level is s and refills 
up to level S, the expected average cost (AC) at steady state at each facility could be 
calculated using the renewal reward theorem (see for example Wolff, 1989).  In 
equation (4.2) the right hand side is only an approximation because the impact of 
expected stock-out time during the cycle is neglected from the cycle length and 
holding costs, (see Axs?ter, 2000 pp. 65)  
 
E[cost  per cycle]
E[cycle  length]
i
AC =  (4.1) 
 
 67 
 
 
 
1
()( )
2
ii iii
i i i i ii i i ii i i i
i
ii
i
ii
i
i
Ss sL
FTC h L s L S s L p L G
L
AC
Ss
L
?
???
?
?
?
??
????
??
++?+?++??
??
????
??
??
??
?
???
+
??
??
(4.2) 
 
where FTC
i
 is the fixed transportation cost associated to serve retailer i (in isolation); 
)(xG  is the loss function that gives the expected number of lost sales at the end of a 
period with demand distributed N(0,1) given that the initial inventory level is x, see 
(Axs?ter, 2000); and L
i
= T + d(0, i) + W
i
 , is the sum of the review period, T , and the 
total lead time.  The length of the review period, T, is the time between plan updates 
and depends on the policy implemented.  The lead time is composed of travel time 
from the depot to retailer i, d(0, i), and the expected waiting time for service for 
customer i, W
i
.  This expected waiting time could be expressed, as a function of 
reorder quantities, using the following recursive expression 
W
i
 =
() ( )Pr{cust.  is in service} (0, ) Pr{cust.  is waiting for service} 2 (0, )
ji
j
j
j dj j dj
?
?
?
??
??
?+ ?
??
??
?
nullnullnullnullnullnullnullnullnullnullnull nullnullnullnullnullnullnullnullnullnullnullnullnullnullnull
(4.3) 
 
2(0,)
j
jj
j
j
dj
Ss
L
?
?
=
???
+
??
??
 (4.4) 
, and  
 
j
j
jj
j
j
W
Ss
L
?
?
=
???
+
??
??
 (4.5) 
where ? and ? are the proportion of the cycle in which a retailer is being served and 
waits for service, respectively.  To evaluate the waiting times for all facilities, given a 
 
 68 
 
vector of reorder quantities, a bisection procedure is used iteratively until the waiting 
times for all facilities are consistent. 
Finally, to obtain the optimal reorder quantities, the sum of average cost for 
all facilities is minimized, subject to L
i
= T + d(0, i) + W
i
; equations (4.3), (4.4), and 
(4.5); and 
()
2(0,) ( ) 1
iii
i
di Ss???<
?
 .  The first four are definitional constraints, and the 
fourth implies that truck utilization rate should be less than 100%.  This problem is 
solved using a steepest decent numerical procedure, in which at each step the gradient 
is evaluated numerically.  Then, the solutions found in this step are used as input 
parameters every time the off-line routing problem presented in the next subsection is 
called. 
4.1.3. Mathematical Formulation of the Problem  
 In this off-line routing problem, the current inventory levels at all facilities are 
considered as given, and the load remaining and the distance to all facilities of the 
truck.  When the truck is at the depot the load remaining is equal to the truck 
capacity.  Additional input parameters in this formulation are transportation cost TC 
[$/hr]; inventory holding cost h
i
 [$/unit-day]; lost sales cost p
i
 [$/unit]; and order up 
to level S
i
 [units]. 
It is assumed that the central decision maker would try to follow the optimal 
reorder up to S policy for each customer.  However, since patterns of deliveries are 
not considered, he/she would deviate from that policy to take advantage of 
transportation savings.  In order to measure the impact on transportation and 
inventory cost of deviating from the reorder up to S policy, incremental inventory 
costs (IIC) for each facility are computed.  These IIC are calculated as a one time 
 
 69 
 
deviation from the reorder up to S policy, assuming that after this deviation the 
optimal policy is resumed.  These IIC can be expressed as the sum of expected 
incremental transportation costs (ITC), and expected lost sales penalty costs (LSP).  
Notice that the impacts on holding costs are only considered through the specification 
of reorder up to levels.  To compute ITC, first notice that, if each retailer is 
considered in isolation, for a given consumption rate ? , the minimum transportation 
costs are achieved when deliveries arrive when the inventory level is zero and the 
quantity delivered is S.  In this case transportation costs per unit of time are 
()/FTC S?? .  Then ITC, associated with scheduling a delivery of size q units at time 
t after the current time, given that the current inventory level is ? , could be expressed 
as 
 
() ()
(,/) 1
q
S
FTC FTC q q q
ITC q t FTC FTC
qS S
??
??
?
??
??
???? ??
??
??
=? =?? =??
???? ?? ? ?
??
?? ????
??
 (4.6) 
where 0( )qS t???? ?+?, and S is the optimal reorder up to level.  On the other hand, 
expected lost sale penalty (LSP) costs, associated with scheduling a delivery at time t 
after the current time, given that the current inventory level is ? , could be computed, 
approximating the distribution of total demand during t as 
2
N( , )tt? ? , as: 
 
()
()()
Dt
LSP (t / ) p u f u du
?
??
?
=? ?
?
 (4.7) 
 (/)LSP t ? ()
t
t
tt
pt v vdvptG
??
?
? ???
???
??
?
?
????
?? ??
=? ? =? ?
??
?? ??
?? ??
??
?
 (4.8) 
 Based on these IIC an off-line problem could be formulated similarly to a 
vehicle routing problem (VRP), where the next visit to each customer is scheduled 
 
 70 
 
based on his or her current inventory level, but adding in the objective function the 
IIC.  This static off-line problem is formulated as minimizing the sum of IIC for all 
retailers and total transportation costs for the next delivery, subject to visiting all 
customers once during the planning period (next week) and inventory levels not 
exceeding order up to levels, S, for each retailer. 
 Thus, the variables of this problem are: 
r
i
q
 : Quantity to be delivered to retailer i by the truck in its r
th
 tour, where 
tours are numbered from 0 (0 is the current tour). 
?
?
?
=
                                                                                                       Otherwise.       ,0
 tour. itsin  truck  by the facility after y immediatel  visitedis facility  If        ,1
th
rij
x
r
ij
 
?
?
?
=
                                                           Otherwise.       ,0
. tour itsin  truck  by the  served is facility  If        ,1
th
ri
y
r
i
 
t
i
 : Arrival time to retailer i. ( i??). 
r
t
0
: Arrival time to the depot by the truck in its r
th
 tour. 
0
0
t  is the truck 
arrival time to the depot in its current tour. 
 In addition, the parameters of this model are: 
i
?  : Retailer i current inventory level. 
i
?  : Retailer i capacity to store inventory. 
?  : Truck capacity. 
?  :  Load remaining in the truck, which is equal to ?  when the truck is at 
the depot. 
TC  : Transportation cost per unit of distance traveled by the truck.  This is 
measured in [$/hr], since the truck moves at constant speed. 
h
i
 : Retailer i inventory holding cost [$/unit-day]. 
 
 71 
 
p
i
 : Retailer i lost sales cost per unit of demand not satisfied [$/unit]. 
S
i
 : Retailer i order up to level [units]. 
s
i
 : Retailer i reorder level [units]. 
null: Facility where the truck is currently located.  
{ }0,1, 2,..., , 1NN? +null , it 
is equal to N+1 when truck is en-route.  In this case, a dummy node, 
N+1, is created at the projected position. 
ij
d
: Distance from facility i to facility j.  Notice that when the truck is en-
route distance from the dummy node N+1 to all facilities should be 
included. 
{ }0,1, 2,..., R??  : Set of tours (routes) for the truck in the planning horizon, 
where R is the maximum number of tours not considering the current 
tour (r=0).  Thus, r??. 
?  : Length of planning horizon (maximum number of hours of operation). 
 The mixed integer programming (MIP) formulation is presented below: 
 Min. 
00
:
(/) ( ,/)
rr r
i i i i i i i i ij ij
iir iji
LSP t ITC q t y TC d x??
?? ?? ?? ?? ?? ? ??
??
+?+??
??
??
??? ??
 (4.9) 
 
00
:
  
r
riii i
iii ii i
r
i
ii
r
ij ij
ijji r
tq
ptG FTCy FTC
S
t
TC d x
??
?
?
?? ?? ?? ?? ??
?? ?? ? ??
??
??
???
??
=?? + ?? ?+??
?? ??
??
??
??
??
??
??
??
????
?? ?
 (4.10) 
 Subject to: 
 
0
:
1
r
ij
jjir
x
?? ? ??
=
??
 , for i?? (4.11) 
 
0
1
r
i
i
x
??
=
?
 , for r =1, 2,?, R (4.12) 
 
 72 
 
 
00
01,0
1
iN
i
xx
+
??
+=
?
  (4.13) 
 
0
0
:
1
j
jj
x
?? ?
=
? null
null
  (4.14) 
 
{ }
00
00
::
1
ij ji
jji jji
xxi
?? ? ?? ?
=+=
??
null     , if 1N? +null ; for i?? (4.15) 
 
00
000
1,
::
ij ji N i
jji jji
x xx
+
?? ? ?? ?
=+
??
        , if 1N= +null , for i?? (4.16) 
 
00
::
rr
ij ji
jji jji
x x
?? ? ?? ?
=
??
 , for 
0
i?? ; r =1, 2,?, R (4.17) 
 
1
10
00
1
N
ji
ji
x x
+
?? =
?
??
  (4.18) 
 
(1)
00
rr
ji
ji
xx
?
?? ??
?
??
 , for r = 2, 3,?, R  (4.19) 
 
00
:
iij
iii
x x
?? ?? ?
?
??null
 , for j?? (4.20) 
 
0
:
rr
iij
iii
x x
?? ?? ?
?
??
 , for j??; r =1, 2,?, R (4.21) 
 
( )
1
r
j iij ij
ttdM x?+ ? ?
 , for i??; j??; r?? (4.22) 
 
( )
1
r
j iij ij
ttdM x?+ + ?
 , for i??; j??; r?? (4.23) 
 
()
0
1
j jj
tdM x?? ?
nullnull
 , for j?? (4.24) 
 
()
0
1
j jj
tdM x?+ ?
nullnull
 , if 0?null , for j?? (4.25) 
 
()
(1)
00 0
1
rr
jj j
ttd M x
+
?+ ? ?
 , for j??, r?? (4.26) 
 
0=
null
t
 , if 1N? +null  (4.27) 
 
( )
0
0,10,1
0
0
1
++
???
NN
xMdt
 , if null=N+1 (4.28) 
 
( )
0
0,10,1
0
0
1
++
?+?
NN
xMdt
 , if null=N+1 (4.29) 
 
 73 
 
 
( )
00 0
1
rr
ii i
ttdM x?+ ? ?
 , for i??; r?? (4.30) 
 
( )
00 0
1
rr
ii i
ttd M x?+ + ?
 , for i??; r?? (4.31) 
 
(1)
00
rr
tt
?
?  , for r =1, 2,?, R (4.32) 
 
0
R
t ??
  (4.33) 
 
0
:
rr
ij i
jji
x y
?? ?
=
?
 , for i??; r?? (4.34) 
 
0
rr
ii
qy????
 , for i??; r?? (4.35) 
 
r
i
i
q
??
??
?
 , for r = 1,2,?, R (4.36) 
 
0
i
i
q ?
??
?
?
  (4.37) 
 
r
ii ii i ii ii
r
s tqS t? ???
??
?+ ? ? ?+
?
, for :ii???null (4.38) 
 
0
min{( ), }sqS? ??????
nullnull null nullnull
 , if 1N? +null  (4.39) 
 {} {}
1,0 and 1,0 ??
r
i
r
ij
yx
 , for 
0
i?? ; 
0
j?? ; r?? (4.40) 
 {}
 1,0
0
,1
?
+ jN
x
 , if null=N+1, for 
0
j??  (4.41) 
 The objective function (4.9)-(4.10) minimizes the sum of the total expected 
lost sale penalties at each facility for the next scheduled visit, and the total 
transportation costs. 
 All tours start at the depot with a full truck, with the exception of the current 
tour, r=0, in which the truck starts at any facility or en-route (at node N+1), and the 
truck might not be full (its current load is ? ). 
 Constraint (4.11) ensures that the next visit for each retailer is programmed.  
Equations (4.12) and (4.13) ensure that the truck returns to the depot in all its tours.  
 
 74 
 
Constraint (4.14) dictates that the truck should leave from its current location.  
Constraints (4.15)-(4.16)-(4.17) give continuity of flow ensuring that the number of 
arrivals equals the number of departures at each node.  Constraints (4.18)-(4.19) 
ensure that subsequent routes could be traveled only if the previous route is 
completed.  Constraints (4.20)-(4.21) ensure that current route leave the initial node 
and subsequent routes leave the depot to visit retailers.  Constraints (4.22) through 
(4.32) ensure that the arrival times at each facility are consistent with travel times 
between them and the initial conditions, where M is a big number.  Constraint (4.33) 
dictates that the last route should be completed before the end of the planning horizon 
? .  Constraint (4.34) relates facilities served on each route with its links.  
Constraints (4.35) guarantee that only customers visited from a particular route could 
receive deliveries from it.  Constraints (4.36)-(4.37) ensure that the truck capacities 
are not exceeded and that the quantity delivered cannot exceed the load remaining in 
the vehicle.  Constraints (4.38)-(4.39) guarantee that inventory levels should not 
exceed the order up to level, S
i
 and inventory level should be greater than s
i
 after 
refilling; however an exception is allowed at the current facility if the load remaining 
in the truck is insufficient (4.39). 
 As mentioned, the purpose of this formulation is to update truck plans making 
use of updated information about the state of the system.  The next section describes 
three strategies that solve this formulation in a rolling horizon framework 
4.2. Optimization Based Real-Time Strategies  
The off-line problem described in the previous section will be used to 
determine how to update truck routes and inventory allocations in a rolling horizon 
 
 75 
 
framework.  Different policies could be implemented based on how often the off-line 
problem is solved and/or how many steps of the current solution are implemented 
before solving a new instance with updated information about the state of the system.  
Three policies were implemented using the off-line IRP presented in the previous 
section.  These are Replan at Tour Completions (RTC), Replan at Delivery Epochs 
(RDE), and Replan at Delivery Epochs with possible en-route diversions (RDE+div), 
which differ in how often the off-line problem is solved.  These three policies are 
presented in what follows, ordered in terms of increasing ICT requirement. 
4.2.1. Replan at Tour Completions (RTC) Strategy  
In Replan at Tour Completions (RTC), the off-line IRP is solved each time the 
truck returns to the depot, i.e. completes a tour, and only the first route of the current 
solution is implemented.  In this policy, the review period, T, used to compute the 
optimal refilling levels, is obtained as the expected distance on a tour over the set of 
retailers. 
4.2.2. Replan at Deliver Epochs (RDE) Strategy  
In Replan at Delivery Epochs (RDE), the off-line problem is called at delivery 
epochs.  Each time a truck arrives to a facility, either a retailer or the depot (delivery 
epoch), an off-line IRP is solved and the solution implemented until the next delivery 
epoch.  That is the amount specified by the solution is delivered at the current facility, 
and the truck is sent to the next facility specified by the solution.  In this policy, the 
review period, T, used to compute the optimal refilling levels, is obtained as the 
expected distance between two retailers.  
 
 76 
 
4.2.3. Replan at Deliver Epochs with possible en-route diversions (RDE+div) 
Strategy  
In Replan at Delivery Epochs with possible en-route diversions (RDE+div), 
plans are updated at delivery epochs, as in RDE, but in addition plans are updated 
when demand disruptions occur.  In this case, inventory levels are continuously 
monitored while the vehicle is traveling; whenever a facility?s consumption since the 
last plan update is below or above 3 standard deviations from its expected demand, 
the current plan is updated.  To update the plan, the state of the system, i.e. the 
location of the truck, and inventory levels assuming expected consumption rates, is 
first projected.  Then based on the projected state of the system, an off-line routing 
problem is solved and the next step implemented.  In this strategy, the truck could be 
diverted if in the new plan the next facility to be visited differs from the current 
destination.  
In order to solve the off-line IRP formulation used in these strategies, the first 
term in equation is piecewise linearized, so that small instances can be efficiently 
solved using CPLEX 10.0 with default settings.  This problem can be solved in a few 
seconds for most instances with less than six facilities and few minutes for instance 
with less than nine facilities.  The design of heuristics to solve larger size instances is 
beyond the scope of this dissertation, and is left as a future extension.  
 
 
 
 77 
 
Chapter 5: Formulation and Design of Fixed-Tour 
Based Control Strategies 
 
This chapter states the rationale and characteristics of a set of control 
strategies wherein facilities are visited in a predetermined sequence, i.e. a set of a 
priori routes is used.  As a part of fixed-tour strategies, recourse actions are 
introduced to illustrate what can be achieved with different degrees of real-time 
information available for controlling the system.  In addition, an analysis and 
optimization of policy parameters is presented. 
One of the main difficulties in implementing the set of policies presented in 
the previous chapter is that they require solving difficult combinatorial problems at 
decision epochs.  In addition, even though metaheuristics can be developed to solve 
the proposed formulation, there is no proof that better strategies can obtained by 
allowing for complete flexibility at decision epochs or solving off-line problems with 
longer planning horizons.  Moreover, in the proposed formulation, when inventory 
target levels are obtained, it is assumed that each facility is visited independent of the 
others.  Possibly, if joint replenishment efforts were introduced, when inventory 
target levels were established, better performance of the system could be achieved. 
This chapter deals with the formulation and design of fixed-tour based control 
strategies in order to better understand the impact of: i) restricting the set of feasible 
decisions at plan-update epochs, particularly restricting the sequence in which 
facilities are visited; and ii) coordinating visits to facilities that are close to each 
other. 
 
 78 
 
 Fixed-tour strategies discussed in this chapter are based on a common 
distribution strategy known as milk-runs, in which retailers are refilled, using fixed 
routes (see, for example, Chopra and Meindl, 2003, pp. 243-244). 
Among the anticipated advantages of using fixed-tour strategies are the 
following:  
i) Fixed-tour policies are simpler to implement in the field, since they permit 
drivers to know in advance what routes they will be driving.  For that 
reason, they might be preferred, even when their expected performance 
might be inferior to a more sophisticate policy. 
ii) A better formulation of the inventory-control side of the problem can be 
obtained when the sequence in which facilities are visited is fixed.  In 
particular, when the time between replenishments to a particular retailer is 
constant, inventory replenishment levels can be reduced.  
iii) If transportation costs were predominant in the total cost function, then 
using effective tours might be more effective than having the flexibility of 
repeatedly changing them during execution.  Thus, tours with unnecessary 
zigzags are avoided. 
iv) The scheduling of visits to near-by facilities is always coordinated to 
reduce transportation costs per visit. 
However, restricting the flexibility to attend retailers who are close to stock-
out will, presumably, increase stock-out penalties.  That could be particularly costly 
when demand variability is high or demand forecasts are wrong.  
 
 79 
 
The most simple of such policies is illustrated by the case in which no real-
time communication capabilities are present.  In that case, a fixed-tour at regular-
intervals (FTRI) can be implemented.  In FTRI a fixed delivery tour is implemented 
without updating plans, even when that might be profitable.  However, when real-
time communication capabilities are available, recourse action can be introduced to 
react to deviations from projected consumption patterns. 
There are several ways to improve upon FTRI, when real-time information 
about inventory levels is available.  In this research two of them that are studied: 
? Update the intervals between delivery tours, based on updated information 
about inventory levels at retailers.  Then, only the ability to monitor retailer 
inventory levels from the depot is required, since en-route vehicle plans would 
not be modified. 
? Skip retailers whenever the expected total cost savings are greater than the 
expected increment of stock-out penalty costs.  Skipping strategies can be 
implemented with or without vehicle-communication capabilities, as long as 
inventory levels can be centrally monitored.  In the first case, skip decisions 
can be made en-route, based on updated information about inventory levels; 
and in the second case, only before leaving the depot.  In this research, only 
en-route skipping decisions are studied. 
 In the following sections those strategies are discussed, explaining how 
decision parameters are obtained for each case. 
 
 80 
 
5.1. Fixed Tour at Regular Intervals (FTRI) Strategy  
In FTRI, facilities are visited following an a priori sequence, in which each 
retailer i is refilled to a pre-specified target level, 
i
S , or up to capacity 
i
? .  FTRI 
implementation assumes that inventory levels are not monitored and that routes are 
created so that deliveries do not exceed vehicle capacity, i.e. route failures are not 
permitted.  One way to ensure that the vehicle does not run out product in route is to 
impose that the sum of target levels of retailers visited on the route is less than the 
vehicle capacity. 
5.1.1. Optimization of Refilling Levels for FTRI Strategy  
For an FTRI strategy, the expected cost per unit of time can be evaluated 
analytically, assuming that the total demand during a tour interval, at each retailer i, is 
normally distributed with mean '
iiI
L? ?= , and standard deviation '
iiI
L??= , 
where L
I
  is the time interval between successive tours, which should be not smaller 
than the tour length, L
T
, i.e. L
I 
?L
T
 .  In that case, the expected cost per unit of time 
can be expressed as: 
 
()
()
()
() ()
2
'
1
''
'
1
2' '
'
1
2' '
'
1
''
()
              ( )
              ( )
              ( )
i
ii
ii
i
iii
ii
iii
ii i
i
i
Ii
S
x
T
FTRI i i i i
i
I
S
xx
i
Sx
x
S
i
px
iiL
L
EC TC h S x dx
L
hdx
xS dx
?
??
?
??
?
??
?
??
?
?
?
?
?
??
??
?
??
??
?
?
??
?
??
=? + ? ?
??
??
+?
+?
+??
?
?
?
?
?
?
i
i
S
i
?
??
?
?
 (5.1) 
 In equation (5.1) the first term represents the total transportation costs per unit 
of time, which is the product of transportation costs per unit of time and the fraction 
 
 81 
 
of time that the vehicle is being used.  The second term represents the inventory 
holding costs associated with the remaining inventory from the previous visit to each 
facility, also known as safety stock.  The third and fourth terms account for inventory 
holding costs of products consumed during a cycle at each facility.  Those two terms 
can be approximated, replacing S
2
 by x
2
 in the fourth term, obtaining 
'
2
i
i
i
h
?
??
?
?
.  
That gives an upper bound to the expected total costs?which, when the probability 
of stock-outs are very low, is a good approximation, since in that case the fourth term 
is insignificant, compared with the remaining terms.  On the other hand, a lower 
bound can be obtained ignoring those two terms.  That lower bound would be tight 
only for very long tour intervals, in which demand realizations would be orders of 
magnitude higher than S.  In general, the expected costs should be closer to the upper 
bound, since in this research scenario with high lost-sale penalties are relevant.  
Finally, the last term represents the lost-sale penalties per unit of time at each facility.  
Thus, grouping terms, equation (5.1) can be approximated with these upper and lower 
bound expressions: 
 
()
(){}
() ()
'
2
'
1
''
'
             ( )
i
ii
Iii
i
T
FTRI i i i
i
I
px
iii iL
S
i
L
EC TC h S
L
hxS dx
?
?
??
?
?
??
?
?
??
??
?? + ? ? +
??
??
++ ?
?
?
?
 (5.2) 
 
()()
() ()
'
2
'
'
             '
i
iii
Ii
T
FTRI i i
i
I
pS
iiL
i
L
EC TC h S
L
hG
?
?
?
?
??
?
??
??
?? + ??
??
??
++??
?
?
 (5.3) 
 
 82 
 
 
()
() ()
'
'
'
             '
iii
Ii
T
FTRI i i i
i
I
pS
iiL
i
L
EC TC h S
L
hG
?
?
?
?
??
?
??
??
?? + ??
??
??
++??
?
?
 (5.4) 
, where ( )G ?  is the loss function, which gives the expected number of units of 
demand lost as a function of the initial inventory level, when demand is normal 
standard distributed, i.e. 
[]
() ( )() () 1 ()
y
Gy x y xdx y y y??
?
?? =???
?
, where ( )? ?  
and ( )?? are the pdf and cdf of the normal standard distribution respectively.  
For a given tour interval, L
I
, the refilling levels for each facility that minimize 
(5.1) can be obtained from the first order condition, since that function is convex.  
Lower and upper bounds are constructed using bounds on the second term in (5.5).  In 
that manner, bounds are obtained for the optimal refilling levels, which are presented 
in (5.7). 
 
()
() ()()
''
11
'' '
1
ii i
ii I
i
xp S
FTRI
iii ixL
S
i
EC
hhS dx h
S
??
?? ?
?
?
??
?
? ?
=+ + + ? ?
? ?
?
?
 (5.5) 
 
()() () ()
iii ii
Ii I
pS pS
FTRI
ii iLL
i
EC
hh h
S
????
?
?? ??
++ ? ?? ?+ ? ?
?? ??
?
 (5.6) 
 
( ) ( )
1* 1
'' ''
iiI i
I
phL p
ii i iipphL
S?? ??
???
+
+? ??+?  (5.7) 
Those target levels are the solution to the newsvendor problem presented in 
(2.2), which should not be a surprise, since in FTRI each retailer is visited in every 
tour and refilled up to S.  In scenarios where inventory-holding costs are insignificant 
with respect to lost sale costs, i.e. hL<<p, target levels should be as high as possible.  
Then, depending on the problem parameters, either full truck loads or filling up to 
facilities? capacities would be the best strategy. 
 
 83 
 
To obtain the best tour frequencies, the upper bound on optimal refilling 
levels presented in (5.7) is used to evaluate expected total costs (5.2).  Thus, 
evaluating that expression for different values of L
I
, the best tour frequency can be 
obtained.  In Figures 5-1 to 5-3, the expected weekly costs as a function of tour 
intervals, for scenarios including seven retailers with the same demand parameters, 
are presented.  As shown, scenarios with higher inventory holding costs and higher 
demand variability tend to be more sensitive to tour intervals.  In particular, when 
retailer capacities to hold inventory are binding (see Figure 5-3) deviation from 
optimal tour frequency has a greater effect on expected costs.  
 
Expected Weekly Cost as a Function of Tour Intervals,
? = 50 [items/day]; ? = 10 [items/day]; p =100[$/unit]; TC =100 [$/hr]; 
-
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
50,000
8.6
5
11
.
1
0
13
.
5
5
16
.
0
0
18
.
4
5
20
.
9
0
23
.
3
5
25
.
8
0
28
.
2
5
30
.
7
0
33
.
1
5
35
.
6
0
38
.
0
5
40
.
5
0
42
.
9
5
45
.
4
0
47
.
8
5
50
.
3
0
52
.
7
5
55
.
2
0
57
.
6
5
60
.
1
0
62
.
5
5
65
.
0
0
67
.
4
5
69
.
9
0
72
.
3
5
74
.
8
0
77
.
2
5
79
.
7
0
Tour Interval (L
I
)
[$
/
w
e
e
k] EC (h=10 [$/day])
EC (h=5 [$/day])
EC (h=1 [$/day])
 
Figure 5-1: Expected Weekly Costs for FTRI as Function of L
I
 
 
 
 84 
 
Expected Weekly Cost as a Function of Tour Intervals,
? = 50 [items/day]; ? = 25 [items/day]; p =100[$/unit]; TC =100 [$/hr]; 
-
10,000
20,000
30,000
40,000
50,000
60,000
8.6
5
11
.
1
0
13
.
5
5
16
.
0
0
18
.
4
5
20
.
9
0
23
.
3
5
25
.
8
0
28
.
2
5
30
.
7
0
33
.
1
5
35
.
6
0
38
.
0
5
40
.
5
0
42
.
9
5
45
.
4
0
47
.
8
5
50
.
3
0
52
.
7
5
55
.
2
0
57
.
6
5
60
.
1
0
62
.
5
5
65
.
0
0
67
.
4
5
69
.
9
0
72
.
3
5
74
.
8
0
77
.
2
5
79
.
7
0
Tour Interval (L
I
)
[
$
/week
]
EC (h=10 [$/day])
EC (h=5 [$/day])
EC (h=1 [$/day])
 
Figure 5-2: Expected Weekly Costs for FTRI as Function of L
I
 
 
Expected Weekly Cost as a Function of Tour Intervals,
? = 50 [items/day]; ? = 25 [items/day]; p =100[$/unit]; TC  =100 [$/hr]; ?=100 [items/day] 
-
10,000
20,000
30,000
40,000
50,000
60,000
10
.
0
0
11
.
0
5
12
.
1
0
13
.
1
5
14
.
2
0
15
.
2
5
16
.
3
0
17
.
3
5
18
.
4
0
19
.
4
5
20
.
5
0
21
.
5
5
22
.
6
0
23
.
6
5
24
.
7
0
25
.
7
5
26
.
8
0
27
.
8
5
28
.
9
0
29
.
9
5
31
.
0
0
32
.
0
5
33
.
1
0
34
.
1
5
35
.
2
0
36
.
2
5
37
.
3
0
38
.
3
5
39
.
4
0
Tour Interval (L
I
)
[
$
/week
]
EC (h=10 [$/day])
EC (h=5 [$/day])
EC (h=1 [$/day])
 
Figure 5-3: Expected Weekly Costs for FTRI as Function of L
I
 
 
 
 85 
 
5.1.2. Procedure for Obtaining FTRI-Strategy Tours 
To construct the a priori tours used in FTRI, a simple route first-cluster 
second heuristic is implemented(Beasley, 1983).  In this heuristic, a priori routes are 
constructed, following a four-step procedure:  
i) obtain an optimal or close-to-optimal solution to the traveling-salesperson 
problem (TSP) over the set of facilities;  
ii) assuming that the tour is traveled uninterruptedly (i.e., the time interval 
between visits to a particular retailer is equal to the tour travel time), 
obtain optimal refilling levels for the retailers in that tour, using the 
solution to the newsvendor problem presented in subsection 5.1.1;  
iii) create different routes, dividing the tour into segments, using the Optimal 
Partitioning heuristic introduced by Beasley (1983); and 
iv) optimize the time that the vehicle is idle at the depot between tours, using 
a bisection search method, in which the effect of different idle times is 
assessed numerically, evaluating the expected cost of the system per unit 
of time, as presented in subsection 5.1.1. 
5.2. Fixed Tour Updating Intervals (FTUI) Strategy 
When inventory levels at different facilities can be monitored in real-time, the 
tour frequency can be updated either to avoid stock-out penalties at some facilities or 
to save on transportation costs when consumption rates have been less than expected.  
This possibility of updating tour intervals is particularly attractive when demand 
variability is high, capacity constraints at retailer sites to store inventory are binding, 
or demand parameters are not well known. 
 
 86 
 
In this strategy, whenever the vehicle is at the depot, a decision on waiting an 
additional time or departing immediately has to be continuously weighed until vehicle 
departure.  How to evaluate the trade-off between expected benefits and costs and 
how to make that decision are covered in the next subsection. 
5.2.1. Update of Truck Idle-Times on FTUI Strategy 
To evaluate the effect of waiting-additional-time decisions, the total expected 
cost per unit of time until completion of the next tour is computed.  In that 
computation, inventory costs associated with a particular retailer are accounted only 
until the next visit to that facility.  That approach facilitates the analysis, since, for 
each facility, the time at which it is refilled is a renewal epoch for its demand process.  
That assumes that after a facility had been refilled, the probability of stock-out before 
tour completion is null, which is a reasonable assumption when lost-sale penalties are 
high, because in that case target levels are high enough to avoid the occurrence of 
stock-outs at the beginning of the cycle.  
The expected costs per unit of time until completion of the next tour, ECNT, 
can be computed as a function of the additional time spend at the depot, t
0
, given the 
current inventory level at each retailer, 
i
? , as follows: 
 
()
( )
()
00
00 00
2
00
00
0
0
()
11
00 2
() (
00
()
1
00 2
() (
00
(/)
1
                 + ( )
()
1
                 + ( )
()
i
ii i
ii ii
ii
i
ii i
T
T
xtd
iii
td td
i
i
xtd
iix
td
i
TC L
ECNT t
tL
htd x x
td
htd
td
?
?
??
??
??
?
??
?
?+
++??
??
?+
+
???
=
??
+
??
+? ?
+
+?
+
?
?
null
nullnull
null
null
null
null
null
null
( )
()
( )
00
00
00 00
)
()
1
() ()
00
1
                 +
()
ii
ii i
ii iii
i
td
i
xtd
iii
td td
i
i
dx
px x
td
?
?
???
??
?
+
??
?
?+
++
??
?
+
?
?
?
?
null
null
nullnull
null
 (5.8) 
 
 87 
 
, where 
01,
1
i
ikk
k
dd
?
=
=
?
null
 is the time since the vehicle left the depot until retailer i is 
visited.  Applying a procedure similar to the one used to obtain (5.2) upper and lower 
bounds for expression (5.8) can be constructed, as shown in equations (5.9) and 
(5.10) respectively.  
 
()
()
( )
00
00
00
1
02
0
()
00
()
(/) + ( )
                 + ( )
iii
i
ii
T
ii i i
i
T
ptd
ii i
td
td
i
TC L
ECNT t h t d
tL
htdG
??
?
???
?
??
?+
+
+
??
???
???+
??
+
??
++?
?
?
null
null
null
null
null
 (5.9) 
 
()
()
( )
00
00
00
0
0
()
00
()
(/) + ( )
                 + ( )
iii
i
ii
T
ii i i
i
T
ptd
ii i
td
td
i
TC L
ECNT t h t d
tL
htdG
??
?
???
?
??
?+
+
+
??
???
???+
??
+
??
++?
?
?
null
null
null
null
null
 (5.10) 
Figures 5-4 and 5-5 present plots of the expressions for scenarios with high 
and low inventory holding costs, where ECNT
u
(t
0
) and ECNT
l
(t
0
) are the upper and 
lower bounds presented in (5.9)-(5.10) respectively.  In addition, each term is plotted: 
TC(t
0
) is the first term in both expressions, EIHCNT
u
(t
0
) or EIHCNT
l
(t
0
) is the 
second term in (5.9)-(5.10) respectively, and ELSPNT corresponds to the third term.  
As could be anticipated, both bounds are very tight when inventory holding costs are 
low. 
 
 88 
 
Next Tour Expected Costs
?=2.5 [items/hr]; ?
2
=0.5 [items/hr]
2
; p =100[$/unit]; TC  =100 [$/hr]; h =0.5 [$/hr] 
0.0
200.0
400.0
600.0
800.0
1000.0
1200.0
0
.0
0
0.
70
1
.4
0
2.
10
2.
80
3.
50
4.
20
4.
90
5.
60
6.
30
7.
00
7.
7
0
8.
40
9.
1
0
9.
80
1
0
.
50
11
.
20
1
1
.
90
12
.6
0
1
3
.
30
14
.0
0
1
4
.
70
15
.4
0
1
6
.
10
16
.8
0
1
7
.
50
18
.2
0
1
8
.
90
19
.6
0
Additional Time Idle
[$/h
r
]
TC(t0)
EIHCNTu(t0)
EIHCNTl(t0)
ELSPNT(t0)
ECNTu(t0)
ECNTl(t0)
 
Figure 5-4: Next Tour Expected Costs per Unit of Time as a Function of t
0 
 
 
Next Tour Expected Costs
?=2.5 [items/hr]; ?
2
=0.5 [items/hr]
2
; p =100[$/unit]; TC  =100 [$/hr]; h =0.05 [$/hr] 
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
0
.0
0
0.
70
1
.4
0
2.
10
2.
80
3.
50
4.
20
4.
90
5.
60
6.
30
7.
00
7.
7
0
8.
40
9.
1
0
9.
80
1
0
.
50
11
.
20
1
1
.
90
12
.6
0
1
3
.
30
14
.0
0
1
4
.
70
15
.4
0
1
6
.
10
16
.8
0
1
7
.
50
18
.2
0
1
8
.
90
19
.6
0
Additional Time Idle
[$/h
r
]
TC(t0)
EIHCNTu(t0)
EIHCNTl(t0)
ELSPNT(t0)
ECNTu(t0)
ECNTl(t0)
 
Figure 5-5: Next Tour Expected Costs per Unit of Time as a Function of t
0
 
 
 
 89 
 
Expressions (5.9) and (5.10) prevent obtaining optimal conditions 
analytically, and their minima do not coincide.  In fact, it is easy to verify that the 
departure time that minimizes the upper bound function (5.9) is lower than the one 
that minimizes the lower bound function (5.10).  In addition, the next-tour-expected 
cost function is asymmetric, and?for the set of parameters of interest?the increment 
in expected costs is lower when deviating the same amount from the optimal 
departure time to an earlier epoch than to a later one.  For that reason, the upper 
bound function is used to decide whether to wait additional time at the depot. 
5.2.2. Procedure Used to Implement Updating Interval Strategies 
In order to implement updating-tour-interval decisions, the derivative of 
equation (5.9) is evaluated numerically, since the optimal departure time cannot be 
obtained analytically from that expression.  Equation (5.11) presents the marginal 
increment in expected cost of the next tour.  
 
()
()
( )
00
00
00
0 1
22
0
0
()
00
()
0
(/)
-
                 + ( )
iii
i
ii
u T
ii
i
T
ptd
ii i
td
td
i
ECNT t TC L
h
t
tL
htdG
t
??
?
?
?
?
??
?+
+
+
??
??
? ??
=???
?
+
??
?
++?
?
?
?
null
null
null
null
 (5.11) 
The expression is evaluated repeatedly at small constant intervals of time, ?t, 
to decide when to leave the depot.  Since expression (5.9) is quasi-convex, whenever 
(5.11) is positive during the interval, immediate departure is recommended.  For that 
reason, to make a decision is necessary only to evaluate the equation at the end of the 
next time interval.  Therefore, when updating-interval strategies are implemented, 
every time the vehicle is at the depot, expression (5.11) is repeatedly evaluated with 
 
 90 
 
updated inventory levels at t
0
= ?t , where ?t is the incremental time that the vehicle 
would wait at the depot at each decision epoch until departure. 
5.3. Fixed Tour Skipping Retailer (FTSR) Strategy 
One of the main disadvantages of previously proposed fixed-tour strategies is 
that whenever replenishments are executed, all the facilities in the tour are visited, 
even though some facilities might have high inventory levels, which are probably 
adequate to last until the next delivery.  Thus, savings could be achieved if those 
facilities were skipped in the current tour. 
The decision to skip a particular retailer can be made before leaving the depot 
or enroute.  If the decision about the set of retailers to skip is made at the beginning of 
the tour, the probability of skipping retailers near the end of the tour might be low 
compared to those near the beginning.  In that case, tour orientation, i.e. the direction 
in which the tour is traveled, might become a relevant question.  In order to simplify 
the analysis, skipping facilities in the tour is considered only in scenarios in which 
vehicles are equipped with two-ways communications capability.  In that case, the 
decision to skip a particular retailer can be made up to the time of departure from the 
previous facility, when additional information about demand realization at facility 
will be available. 
5.3.1. Skip Decision on FTSR Strategy 
It is assumed that the decision to skip the next retailer is made without 
considering the inventory levels at the remaining facilities on the tour and, 
furthermore, assuming that they will be visited.  That approach is taken to avoid 
 
 91 
 
addressing the combinatorial problem of choosing which set of the remaining retailers 
to visit, given the current conditions.  That is, when making the decision to skip or 
visit the i
th
 facility, it is assumed that facilities (i+1)
th
, (i+2)
th
, etc. will be visited.  In 
that context, the additional savings and costs of skipping retailer i when the vehicle is 
at retailer null are computed as follows. 
The expected incremental cost of skipping (EICS) retailer i, and the expected 
incremental cost savings of skipping (EISS) retailer i can be computed, given that the 
vehicle is at facility null and the current inventory level at i.  They are calculated in 
equations (5.12) and (5.13), respectively. 
 
( ) ( ) ( )
,,
(/) ( )/ / /
ii I i i i i I i
EICS LSP L d LSP d LSP L S???=+? ?
nullnull
null  (5.12) 
 
()
()
( ) ()
()
()
()( )
( ) ()
,
2
,,
,
2
,,
,,1,1
1
1
2
1
(/)
()
         
ii
ii iI
ii iIiIi
Ii
ii iI
ii iIiIi
ii i ii i
x
xd yL
Iii
dLLd
yx
Lx
Iii I iy
x
xd yL
dLLd
i
EISS TC d d d
LS x dxdy
LS x L x y
dxdy
h
??
? ?
???
?
?
? ?
???
?
???
??
??
++
?
? ?
?
=?? =??
?
=??
? ?
?
=? + ?
?+
??
?+ ? ? +?
??
+
?
??
+?
??
null
nullnull
null
nullnull
nullnull
null
()( ) ()
()()
( ) ()
() ( ) ()
22
,
2
,,
,
2
,,
2
,
2
,,
()
11
2
11
2
1
2
ii
i
i
iiii iI
ii iIiIi
i
i
ii iI
ii iIiIi
i
iiiiI
ii iIiIi
S
yx
xdSx yL
I yy
dLLd
yS x
S
xd yL
Ii
dLLd
yx
xdSyL
I y
dLLd
x
d
L y S y dxdy
xdy
?
?
???
???
? ?
???
?
? ?
???
?
??
??
??
=?
?
???
?
==??
?
? ?
?
=?? =
? ?
?
=
+?
++?
+
??
??
??
null
nullnull
null
nullnull
null
nullnull
ii
yS
??
=
??
??
??
??
??
??
(5.13) 
The EICS is related to the additional expected lost sales during the next tour 
cycle if facility i is skipped on the current tour.  The EISS is the sum of the 
transportation costs saved by skipping i on the current tour, and inventory holding 
savings at i during the next tour.  In equation (5.13), the last two terms are relevant 
 
 92 
 
only when the probability of a stock-out is not negligible if facility i is not visited on 
the current tour.  If that were the case, for scenarios with relevant parameters, facility 
i should not be skipped, and EICS should be higher than EISS.  A lower bound on the 
EISS can be constructed as follows: 
 
()
()
()( )
( ) ()
()
,
,
,
2
,,
,,1,1
0
1
2
(/)
    (
iiiiiI
iI i i
ii
ii iI
ii iIiIi
Ii
I
ii i ii i
dSL
ii
Ld
S
xd yL
I
x
dLLd
yx
Lx
iIy
L
EISS TC d d d
S
L
dxdy
hLxy
???
??
?
? ?
???
?
?
?
??
?
++
??
? ?
?
=?? =??
>
?
<
?? + ?
??
?? ?
??
+
??
??
??
+?? ? +?
??
??
null
null
null
nullnull
nullnull
null
nullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnull
nullnullnullnullnull
( ) ()
( ) ()
,
2
,,
22
,
2
,,
1
()
()
11
2
0
)
ii
ii iI
ii iIiIi
i
ii
i
iiii iI
ii iIiIi
i
S
xd yL
i
dLLd
yxx
S
xdSx yL
I yy
dLLd
yS x
dxdy
Ldxd
?
? ?
???
?
?
?
???
???
??
??
? ?
?
=? =??
<?
?
???
?
==??
>
??
??
?
??
??
?
?
?
??
?
??
+?
?
??
?
?
?
??
??
??
null
nullnull
null
nullnull
nullnullnullnullnull
nullnullnullnullnull
()()
()( )
{}
,
,
1
,,1,1 2
     
iiiiiI
iI i i
dSL
iii i i I ii
Ld
TC d d d h L S
???
??
?
??
++
?
?
?
?
?
?
?
?
??
?? + ? +? ?? ?
??
??
null
null
nullnull
(5.14) 
Thus, the decision to skip a customer is based on a threshold policy, i.e. 
whenever a retailer?s current inventory level is above a certain level, it is skipped; 
otherwise it is visited. Those threshold-levels depend on the current facility nullat 
which the truck is located. They are computed as the critical levels that change the 
sign of the following expression. 
 
() ()()
( ) ( )
{ }
()()()
,
,
1
,,1,1 2
,,
/
              ( ) / / /
iiiiiI
iI i i
dSL
ii i ii i i I i i
Ld
Iii ii Ii
Skip TC d d d h L S
LSP L d LSP d LSP L S
???
??
??
??
??
++
? ?
=? + ? +? ?? ?
? ?
? ?
?++ +
null
null
nullnull
nullnull
null
(5.15) 
If that expression was positive, facility i should be skipped, otherwise it 
should be visited.  
 
 93 
 
5.3.2. Procedure Used to Implement FTSR Strategy  
FTSR strategy is implemented using the same tours and reorder levels 
obtained for FTRI and FTUI strategies, and the procedure presented in 5.2.2 for 
updating tour intervals.  In addition, the possibility of skipping a facility is always 
evaluated before departing to it based on expression (5.15).  Thus, FTSR only differs 
from FTUI in that skipping decisions are considered before departing to a facility. 
 
To compare all proposed operational strategies in this and the previous 
chapter, simulation experiments were designed and run, as described in the next 
chapter.  
 
 
 94 
 
Chapter 6: Simulation Experiments 
 
This chapter documents experiments designed to evaluate and compare 
proposed real-time policies.  It describes the set of scenarios, including those with 
steady-state demand processes and those with sudden changes in demand patterns.  
Finally, it presents and discusses experiment results. 
 Simulation runs were performed with all eight real-time strategies presented in 
previous chapters; namely: i) the two benchmark policies, BENCH1 and MUN, ii) the 
three re-optimization strategies, RTC, RDE and RDE+div, presented in Chapter 4, 
and iii) the three fixed-tour strategies, FTRI, FTUI and FTSR, presented in Chapter 5.  
In addition, RDE+FT_S strategy was considered.  RDE+FT_S is the same RDE 
strategy, but implemented using inventory target levels obtained for the FTRI 
strategy. 
6.1. Simulation Scenarios  
 This section presents the main elements and defining parameters of the 
simulated scenarios.  First, the set of fixed parameters used in all simulation is 
introduced.  Second, the set of parameters considered in scenarios with steady-state 
demand processes is presented.  Third, inventory reorder levels are obtained for each 
combination of strategy and scenario simulated.  Finally, experiments with demand 
disruption at one facility are presented. 
Because of limited computational resources, and since each simulation run 
takes hours of computer time?even days for some re-optimization strategies?the 
strategies were not tested under all possible parameters.  For each strategy, 
 
 95 
 
simulations were performed for four cases of facility layouts, and 12 sets of 
parameters, representing typical cost settings, probabilistic scenarios, and constraints. 
6.1.1. Set of Fixed Parameters 
Distances between facilities are Euclidean and are measured in units of time 
[hours], because it is assumed, without loss of generality, that the vehicle moves at 
unit speed.  All facilities are located in a square region, with side length of four hours, 
with the depot in the center of the square region, i.e. the depot is locate at (2, 2).  In 
all cases simulated, seven retailers and one depot are considered.  In Case 0, retailers 
are symmetrically distributed around the depot at 1.2 hours apart, and in Cases 1 to 3 
they are randomly distributed in the region.  Figures 6.1 through 6.4 show the 
locations of facilities for each case.  In those figures the TSP tour visiting all facilities 
is drawn.  The distances of those TSP tours are 8.65, 12.23, 8.38, and 8.69 hours, 
respectively.  Facilities were renumbered so that their numbers coincide with their 
positions in the TSP tour. 
In addition to the location of facilities for each case studied, the following set 
of parameters is considered as fixed: the vehicle capacity, ?  = 400 [units]; the length 
of the planning horizon used on the off-line problem for re-optimization strategies, ?  
= 100 [hrs] which is also assumed to be the amount of working hours per week; lost 
sales penalty costs, 
i
p  = 100 [$/unit] for all retailers; and the fixed transportation 
costs used to obtain refilling levels for re-optimization strategies, TCdFTC
ii
??=
0
2 , 
which is computed as twice the cost of a tour from the depot considering only that 
retailer. 
 
 
 96 
 
Location of Facilities for Case 0
0
1
2
3
4
1234
[hrs]
[hrs
]
Depot
Retailers 0
3
4
5
2
1
6
7
 
Figure 6- 1: Location of facilities for Case 0 (Symmetric case) 
 
Location of Facilities for Case 1
0
1
2
3
4
1234
[hrs]
[hrs]
Depot
Retailers 1
3
4
5
2
1
6
7
 
Figure 6- 2: Location of facilities for Case 1 
 
 97 
 
Location of Facilities for Case 2
0
1
2
3
4
1234
[hrs]
[hrs
]
Depot
Retailers 2
3
4
5
2
1
6
7
 
Figure 6- 3: Location of facilities for Case 2 
 
Location of Facilities for Case 3
0
1
2
3
4
01234
[hrs]
[hr
s
]
Depot
Retailers 3
3
4
5
2
1
6
7
 
Figure 6- 4: Location of facilities for Case 3 
 
 98 
 
6.1.2. Scenarios with Steady-State Demand Patterns  
 Two sets of scenarios were studied: i) products with high inventory-holding 
costs and no capacity constraints at retailers? sites, and ii) products with low 
inventory-holding costs and capacity constraints at retailers? sites.  Scenarios with 
low inventory holding costs and no capacity constraints at retailers were not 
considered, since for those scenarios the best policy would be full-truck load 
deliveries. 
 For each set of scenarios, a base case was considered.  The set of Parameters 
No. 1 is the base case for high inventory-holding costs scenarios, in which TC=100 
[$/hr], 
i
h  = 10 [$/unit-day] for all retailers (inventory holding cost), and demand 
parameters are the same for all retailers, and equal to ?
i
=50 [arrivals/Day] and ?
i
 = 1 
[units], for all i.  This demand process can be approximated as N(50,10
2
) for daily 
periods.  For low inventory-holding cost and limited capacity at retailers? sites 
scenarios, the set of Parameters No. 7 is the base case, in which 
i
h  = 1 [$/unit-day], 
?
i
= 100[units] for all retailers, and the remaining parameters are the same as in the set 
of Parameters No. 1. 
 In order to analyze the impact of transportation costs and demand variability 
on the proposed policies, ten additional scenarios, in which those parameters vary, 
were studied.  Table 6-1 shows the parameters for those remaining scenarios.  In 
scenarios with high (low) inventory holding costs and without (with) capacity 
constraint at retailers? sites, set of Parameters No. 2 and No. 3 (No. 8 and No. 9) 
permits studying the effects of changes in transportation costs, and set of Parameters 
 
 99 
 
No. 4 to No. 6 (No. 10 to No. 12) permits studying the effects of increments on 
demand variability. 
Table 6- 1: Simulation Scenarios 
Parameter 
Set 
TC 
[$/hr] 
h 
[$/day] 
? 
[units] 
? 
[arrivals/day]
? 
[units] 
Approx. 
N(?,? 
2
) 
1 100 10 ? 50 1 N(50,10
2
) 
2 33 10 ? 50 1 N(50,10
2
) 
3 300 10 ? 50 1 N(50,10
2
) 
4 100 10 ? 10.5 4.8 N(50,17
2
) 
5 100 10 ? 4.35 11.5 N(50,25
2
) 
6 100 10 ? 2.4 20.8 N(50,33
2
) 
7 100 1 100 50 1 N(50,10
2
) 
8 33 1 100 50 1 N(50,10
2
) 
9 300 1 100 50 1 N(50,10
2
) 
10 100 1 100 10.5 4.8 N(50,17
2
) 
11 100 1 100 4.35 11.5 N(50,25
2
) 
12 100 1 100 2.4 20.8 N(50,33
2
) 
 
 In general, inventory-holding and lost-sale costs have important differences, 
depending on the nature of the product distributed.  For that reason?and taking into 
account that this research deals with general distribution systems?the scenarios 
studied consider important variations of ratios between transportation and inventory 
costs.  Thus, in the experimental design, the magnitude of each cost parameter is less 
relevant than the relationship among them. 
 In Appendix C, reorder parameters for each set of experiments, are presented.  
The reorder parameters of the (s, S) used in BENCH1 were obtained using Zheng and 
 
 100 
 
Federgruen algorithm (1991).  RTC and RDE reorder parameters were obtained using 
the procedure described in subsection 4.1.2.  The only difference between RTC and 
RDE is the review period considered.  In RTC the expected TSP length was used, and 
in RDE the expected distance between two facilities was used.  For MUN strategy, 
RDE refilling up to levels, S, were used.  Refilling levels for FTRI strategy were 
obtained using the procedure presented in subsection 5.1.1.  The remaining fixed-tour 
strategies?FTUI and FTSR?used the same S parameters.  The last columns of the 
tables in Appendix C, show the inventory target parameters used in the RDE+FT_S 
experiments.  In those experiments the RDE strategy was executed with refilling up to 
levels obtained for FTRI. 
6.1.3. Scenarios with Unpredicted Changes in Demand Patterns  
 In all previous scenarios was assumed that demand-process parameters could 
be precisely estimated; however, this is hardly ever true, particularly for products 
with a short life-cycle.  Thus, the purpose of the experiments was to investigate the 
performance of different strategies under disruption demand patterns at a particular 
facility.  In distribution systems when such disruptions are observed, it is difficult for 
the decision maker to update inventory target parameters, since in the short term those 
changes can be attributed to deviation in normal consumption patterns. 
 Two experiments were performed with unpredicted changes in demand 
patterns for the set of Parameters No. 1 and No. 7.  These experiments were only 
performed for the Case 0, symmetric location of facilities, in order to isolate the effect 
of the demand disruption with respect to the location of facilities.  For these 
 
 101 
 
experiments the arrival rate of customers to retailer 4 was doubled without updating 
demand parameters on each strategy. 
6.2. Simulation Results  
For every combination of strategy and set of parameters studied, simulations 
were carried for 30 replication runs of 100 hours (1 week) each, and all four Cases of 
facility layouts.  For each set of parameters, different strategies were simulated with 
common random numbers, and the same initial conditions.  The initial conditions for 
the first replication were the same for all strategies in the same scenario, starting with 
the vehicle at the depot, and initial inventory levels presented in Appendix C.  The 
effect of initial conditions is only relevant up to the first visit to each facility, which is 
small compared to the length of each run to have significant effects.  Moreover, those 
initial conditions were only used in the first replication, and results suggest that 
transient-state effects are negligible. 
 For each simulation run, the following measures of performance were 
examined: average transportation costs; average inventory holding costs; average lost 
sales penalty costs; and average tour length.  For each one of those measures, interval 
estimates were obtained.  Those results are presented in Appendix D in Tables D-1 
through D-12.  Based on those results, 95% confidence intervals for the average total 
cost per week were computed, and presented in Figures 6-5 through 6-16.  In 
addition, the results for the two scenarios with unpredicted changes in demand 
patterns are presented in Tables D-13 and D-14, and Figures 6-23 and 6-24. 
 
 
 102 
 
Average Weekly Cost (95% C. I.) Set of Parameters 1
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
0123
Case
[$
/
w
e
e
k]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
RDE_FT_S
 
Average Weekly Cost Set of Parameters 1
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR RDE+FT_S
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 5: Results for the Set of Parameters 1 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 103 
 
Average Weekly Cost (95% C. I.) Set of Parameters 2
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
0123
Case
[$
/we
e
k]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
RDE_FT_S
 
Average Cost Set of Parameters 2
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR RDE+FT_S
Strategy
[$/
w
eek]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 6: Results for the Set of Parameters 2 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 
 104 
 
Average Weekly Cost (95% C. I.) Set of Parameters 3
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
34,000
36,000
38,000
40,000
42,000
44,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
RDE_FT_S
Average Cost Set of Parameters 3
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
34,000
36,000
38,000
40,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR RDE+FT_S
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 7: Results for the Set of Parameters 3 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 105 
 
Average Weekly Cost (95% C. I.) Set of Parameters 4
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
34,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
RDE+FT_S
 
Average Cost Set of Parameters 4
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR RDE+FT_S
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 8: Results for the Set of Parameters 4 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 106 
 
Average Weekly Cost (95% C. I.) Set of Parameters 5
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
34,000
36,000
38,000
40,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
RDE_FT_S
 
Average Cost Set of Parameters 5
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
34,000
36,000
38,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR RDE+FT_S
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 9: Results for the Set of Parameters 5 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 107 
 
Average Weekly Cost (95% C. I.) Set of Parameters 6
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
34,000
36,000
38,000
40,000
42,000
44,000
46,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
RDE+FT_S
 
Average Cost Set of Parameters 6
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
34,000
36,000
38,000
40,000
42,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR RDE+FT_S
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 10: Results for the Set of Parameters 6 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 108 
 
Average Weekly Cost (95% C. I.) Set of Parameters 7
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
10,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
 
Average Cost Set of Parameters 7
-
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 11: Results for the Set of Parameters 7 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 109 
 
Average Weekly Cost (95% C. I.) Set of Parameters 8
0
1,000
2,000
3,000
4,000
5,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
 
Average Cost Set of Parameters 8
-
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
4,500
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR
Strategy
[$
/w
e
ek]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 12: Results for the Set of Parameters 8 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 110 
 
Average Weekly Cost (95% C. I.) Set of Parameters 9
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
 
Average Cost Set of Parameters 9
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 13: Results for the Set of Parameters 9 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 111 
 
Average Weekly Cost (95% C. I.) Set of Parameters 10
0
2,000
4,000
6,000
8,000
10,000
12,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
 
Average Cost Set of Parameters 10
-
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 14: Results for the Set of Parameters 10 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 112 
 
Average Weekly Cost (95% C. I.) Set of Parameters 11
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0123
Case
[$/we
e
k
]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
 
Average Cost Set of Parameters 11
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 15: Results for the Set of Parameters 11 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 113 
 
Average Weekly Cost (95% C. I.) Set of Parameters 12
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
0123
Case
[$
/w
e
ek]
BENCH1
MUN
RTC
RDE
RDE+div
FTRI
FTUI
FTSR
 
Average Cost Set of Parameters 12
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR
Strategy
[$/week
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 16: Results for the Set of Parameters 12 with 95% C.I. for the mean, and 
Average Cost for each Strategy among all cases 
 
 
 114 
 
6.3. Analysis of Results  
Simulations were carried only under typical?and in some cases promising?
system parameters, and run under idealized probabilistic distributions.  Thus, the 
results presented are valid only for the range of values studied.  Nevertheless, the 
simulated scenarios were intended to replicate real-world applications and the results 
are adequate to serve as general guidelines for them. 
For all sets of parameters considered, the developed online strategies 
systematically outperformed benchmark strategies.  The best proposed strategies 
achieved reductions in average total costs of approximately 30% and 15% compared 
against benchmark policies BENCH1 and MUN, respectively.  The average cost 
improvements were computed as the average of: 
 
()Avg. Total Cost of Strategy Avg. Total Cost of BENCH
100
Avg. Total Cost of BENCH
%
?
?  (6.1) 
for all sets of parameters and cases considered.  Moreover, the optimal decentralized 
benchmark policy, BENCH1, was systematically outperformed by centralized 
strategies.  That can be explained in part by the fact that BENCH1 tends to carry 
more inventories to be protected from longer lead times.  In addition, all proposed 
strategies achieved less variability in average costs than the BENCH1 strategy. 
In general, re-optimization strategies with appropriate inventory target levels 
had the lowest average costs.  However, in many scenarios with moderate demand 
variability, there were no significant differences, in terms of average total cost, 
between the best re-optimization and the best fixed-tour strategies. 
Among re-optimization strategies, those that update plans at delivery epochs, 
RDE and RDE+div, were the best strategies for the set of parameters considered.  The 
 
 115 
 
possibility of diversions?either en-route or when the vehicle is idle at the depot?
improves system performance in scenarios with low inventory-holding costs and 
high-demand variability.  However, further research is needed to identify scenarios in 
which en-route diversion could be beneficial, since in RDE vehicle idle time at the 
depot is set upon arrival and not updated, even when that might be profitable.  The 
benefits of re-planning at delivery epochs tend to be higher in cases where there are 
clusters of facilities close to each other and/or near to the depot, such as in Case 2. 
Among fixed-tour strategies, the possibility of updating tour intervals offered 
benefits of up to 10% in scenarios with low inventory-holding costs and high demand 
variability.  Moreover, for scenarios studied, the possibility of skipping retailers in the 
route produced small benefits in scenarios with high demand variability and small 
tour intervals.  Otherwise, that possibility did not produce significant benefits, since 
in those scenarios the probabilities of skipping were too small.  Among the cases of 
location of facilities, the possibility of skipping presented higher benefits in Case 2, 
wherein one facility had a high insertion cost in the tour. 
6.3.1. Analysis of the Product Inventory Holding Costs Impact 
The experiments were carried out for two sets of scenarios: i) products with 
high inventory-holding costs, and ii) products with low inventory-holding costs and 
capacity constraints at retailers? sites.  As shown in Figure 6-17, a comparison of the 
two sets of scenarios illustrates that re-planning strategies tended to increase their 
benefits vs. benchmark policies when they were applied in scenarios with low 
inventory-holding costs, whereas fixed-tour strategies tended to have similar benefits. 
 
 116 
 
Impact of Inventory Holding Costs
-35.0%
-30.0%
-25.0%
-20.0%
-15.0%
-10.0%
-5.0%
0.0%
5.0%
RT
C
 v
s
.
 
BE
NC
H1
RT
C v
s
.
 
M
U
N
R
D
E
 vs
.
 BE
NC
H1
RD
E
 vs.
 
M
U
N
RD
E+di
v vs.
 BE
NC
H1
RD
E
+
d
i
v vs
.
 M
U
N
F
T
R
I
 v
s
.
 
BE
NC
H1
F
T
RI
 vs.
 
MU
N
F
T
S
R
 v
s
.
 
BE
NC
H1
FT
S
R
 vs.
 
MU
N
Av
er
a
g
e Cos
t
 
D
i
ffer
e
nc
e
Set of Scenarios with high h
Set of Scenarios with Low h and k
 
Figure 6- 17: Impact of Inventory Holding Cost 
6.3.2. Analysis of Changes in Transportation Costs   
In Figures 6-18 and 6-19, reductions in average total costs for different re-
planning and fixed-tour strategies vs. the two benchmark policies are illustrated.  In 
scenarios with high inventory-holding costs, the benefits of the proposed strategies 
tend to decrease (increase) as transportation costs increase, when compared with 
BENCH1 (MUN).  That can be explained mainly by the fact that in BENCH1 there 
are longer lead-times, so facilities need to carry more inventory.  Thus, the more 
relevant the inventory-holding costs, the worse the performance of BENCH1.  In 
scenarios with low inventory-holding costs and retailer capacities, the benefit of the 
proposed strategies tends to increase, as transportation costs increase when compared 
to any of the benchmark policies.  Thus, with the exception of BENCH1, for 
scenarios with high inventory-holding costs, the benefits of the proposed policies tend 
 
 117 
 
to increase as a function of the proportion of transportation costs in the total cost 
function. 
Impact of Increments in Transportation Costs in Scenarios with High Inventory Holding Costs
-45.0%
-40.0%
-35.0%
-30.0%
-25.0%
-20.0%
-15.0%
-10.0%
-5.0%
0.0%
5.0%
RTC vs.
BENCH1
RTC vs.
MUN
RDE vs.
BENCH1
RDE vs.
MUN
RDE+div
vs.
BENCH1
RDE+div
vs. MUN
FTRI vs.
BENCH1
FTRI vs.
MUN
FTSR vs.
BENCH1
FTSR vs.
MUN
A
ver
age C
o
st
 
D
i
f
f
e
r
e
n
c
e
TC=33 [$/hr]
TC=100[$/hr]
TC=300 [$/hr]
 
Figure 6- 18: Impact of Increments in Transportation Costs in Scenarios with High 
Inventory Holding Cost, Set of Parameters 1, 2 and 3 
 
Impact of Increments in Transportation Costs in Scenarios with Low Inventory Holding Costs 
and Capacity Constraints at Retailers' Sites 
-35.0%
-30.0%
-25.0%
-20.0%
-15.0%
-10.0%
-5.0%
0.0%
RTC vs.
BENCH1
RTC vs.
MUN
RDE vs.
BENCH1
RDE vs.
MUN
RDE+div vs.
BENCH1
RDE+div vs.
MUN
FTRI vs.
BENCH1
FTRI vs.
MUN
FTSR vs.
BENCH1
FTSR vs.
MUN
A
ver
age C
o
st
 D
i
f
f
e
r
e
nce
TC=33 [$/hr]
TC=100 [$/hr]
TC=300 [$/hr]
 
Figure 6- 19: Impact of Increments in Transportation Costs in Scenarios with Low 
Inventory Holding Cost, Set of Parameters 7, 8 and 9 
 
 118 
 
Among re-planning strategies when transportation costs are more significant 
in the total costs, re-planning at delivery epochs is less beneficial, compared with re-
planning only at tour completions.  A comparison between RDE and RTC shows that 
RDE reduced by approximately 11%, 8%, and 4% the average total costs in scenarios 
with high inventory-holding costs, set of parameters 2 (TC=33), 1 (TC=100), and 3 
(TC=300), respectively.  In scenarios with low inventory-holding costs, the 
differences are less dramatic and remain relatively constant (around 2.5%) with 
respect to changes in transportation costs.  Those values were computed in the same 
manner as Eq.  (6.1), but replacing BENCH with RTC, for all sets of cases 
considered.  Those differences could be explained mainly by the fact that in RTC 
higher inventory levels are maintained, therefore when inventory-holding costs are 
predominant in the total cost function, differences between RDE and RTC are higher. 
6.3.3. Analysis of Tour Length  
In Figure 6-20, the average number of visits per tour and the average tour 
length (in hours) are presented for each strategy.  MUN was the only strategy with 
average tour lengths longer than 10 hours, because in that strategy the vehicle did not 
return to the depot until it was empty and might have delivered small quantities to 
facilities that were not near stock-out. 
 
 119 
 
Number of Visits per Tour and Average Tour Length
in Scenarios with High Inventory Holding Costs
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
14.00
15.00
16.00
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR RDE+FT_S
Strategy
N
o
. Vis
i
ts / 
h
r
s.
No. of Visits per Tour Tour Length [hr]
 
Number of Visits per Tour and Average Tour Length
in Scenarios with Low Inventory Holding Costs and Capacity Constraints at Retailers' Sites
0.00
2.00
4.00
6.00
8.00
10.00
12.00
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR
Strategy
N
o
. V
i
si
t
s
 
/
 h
r
s
.
No. of Visits per Tour Tour Length [hr]
 
Figure 6- 20: Average Number of Visits per Tour and Average Tour Length 
 
 
 120 
 
For scenarios with high inventory-holding costs, fixed-tour and RDE+FT_S 
(RDE with fixed-tour inventory-target levels) strategies tend to have longer routes 
and more visits than re-planning strategies with original inventory target levels, which 
is consistent with lower reorder quantities. It is interesting to observe that in those 
scenarios, fixed-tour and RDE+FT_S strategies produce higher vehicle-utilization 
rates.  At the operational level studied in this research, higher utilization rates were 
good as long as they reduced total costs.  However, when the system is designed, 
higher utilization rates might imply a larger fleet size.  Hence, at a strategic-decision 
level, the trade-off between operational costs and fleet size fixed-costs should be 
taken into account. 
In contrast, for scenarios with low inventory-holding costs, fixed-tour and re-
planning strategies present similar average numbers of visits per tour and tour 
lengths, possibly because, in those scenarios, the quantities delivered in all proposed 
strategies are similar. 
6.3.4. Analysis of Demand-Variability Impact 
As shown in Figure 6-21, in scenarios with high inventory-holding costs, as 
demand variability increases, the benefits of the proposed strategies tend to decrease, 
compared to benchmark strategies. Conversely, in scenarios with low inventory-
holding costs and capacity constraints, as demand variability increases, the benefits of 
the proposed strategies tend to increase vs. BENCH1 and decrease vs. MUN, as 
illustrated in Figure 6-22.  As expected, MUN begins to be competitive in scenarios 
with very high demand variability. 
 
 121 
 
Impact of Increments in Demand Variability in Scenarios with High Inventory Holding Costs
-35.0%
-30.0%
-25.0%
-20.0%
-15.0%
-10.0%
-5.0%
0.0%
5.0%
10.0%
15.0%
RTC vs.
BENCH1
RTC vs.
MUN
RDE vs.
BENCH1
RDE vs.
MUN
RDE+div vs.
BENCH1
RDE+div vs.
MUN
FTRI vs.
BENCH1
FTRI vs.
MUN
FTSR vs.
BENCH1
FTSR vs.
MUN
D
i
ffer
e
nc
e
Low Demand Variability
Medium Demand Variability
High Demand Variability
Very High Demand Variability
 
Figure 6- 21: Impact of Increments in Demand Variability in Scenarios with High 
Inventory-Holding Costs, Set of Parameters 1, 4, 5 and 6 
 
Impact of Increments in Demand Variability in Scenarios with Low Inventory Holding Costs 
and Capacity Constraints at Retailers' Sites
-60.0%
-50.0%
-40.0%
-30.0%
-20.0%
-10.0%
0.0%
10.0%
20.0%
RTC vs.
BENCH1
RTC vs.
MUN
RDE vs.
BENCH1
RDE vs.
MUN
RDE+div vs.
BENCH1
RDE+div vs.
MUN
FTRI vs.
BENCH1
FTRI vs.
MUN
FTSR vs.
BENCH1
FTSR vs.
MUN
A
v
er
a
g
e
 Co
st
 D
i
ff
e
r
en
ce
Low Demand Variability
Medium Demand Variability
High Demand Variability
Very High Demand Variability
 
Figure 6- 22: Impact of Increments in Demand Variability in Scenarios with Low 
Inventory-Holding Costs, Set of Parameters 7, 10, 11 and 12 
 
 122 
 
In scenarios with increased demand variability, i.e. those with set of 
Parameters No. 5, No. 6, No. 11, and No. 12, the advantage of re-planning strategies 
at delivery epochs, compared with re-planning only at tour completions, tends to be 
slightly higher than in scenarios with less demand variability. 
Among fixed-tour strategies, in scenarios with low inventory-holding costs, 
the capability to update intervals offers increased benefits, compared to FTRI, as 
demand variability increases.  In scenarios with high inventory-holding costs, the 
benefits of updating intervals, compared to regular intervals, remain relatively stable, 
since the reduction in lost sales tends to be compensated for by increments in holding 
costs. 
The possibility of diversions?either en-route or when the vehicle is idle at the 
depot?improves system performance in scenarios with low inventory-holding costs 
and high demand variability. However, further research is needed to identify 
scenarios in which en-route diversion would be beneficial. 
6.3.5. Analysis of Demand-Disruptions Scenarios 
Finally, two scenarios for the set of Parameters No. 1 and No. 7, in which 
demand disruption occurs at a particular facility are studied.  For those scenarios, the 
arrival rate of customers to Retailer 4 was doubled without updating the demand 
parameters on each strategy.  The experiments were performed only for Case 0 
(symmetric location of facilities) in order to isolate the impact of the demand 
disruption with its location.  
 
 123 
 
Average Weekly Costs (95% C.I.) Symmetric Case under Demand Disruption at Retailer 4
Set of Parameters 1
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
34,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR RDE+FT_S
Strategy
[$/
w
eek
]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 23: Average Weekly Costs with 95% C.I. for the mean, under Twice Demand 
Arrivals at Retailer 4 and Without Inventory Target Update, for the Set of Parameters 1 
Average Weekly Costs (95% C.I.) Symmetric Case under Demand Disruption at Retailer 4
Set of Parameters 7
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
22,000
BENCH1 MUN RTC RDE RDE+div FTRI FTUI FTSR
Strategy
[$/
w
eek]
Inv. Holding Cost
Lost Sales Cost
Transp. Cost
 
Figure 6- 24: Average Weekly Costs with 95% C.I. for the mean, under Twice Demand 
Arrivals at Retailer 4 and Without Inventory Target Update, for the Set of Parameters 7 
 
 124 
 
 
As illustrated in Figure 6-23 and 6-24, the average total costs decomposed in 
transportation, lost sales, and inventory-holding costs for each policy.  Also, 95% 
confidence intervals for the mean are presented under brackets for each component. 
Re-planning strategies, particularly RDE+div and RDE+FT_S, systematically 
outperform fixed-tour strategies.  That could be explained in part by the additional 
flexibility to react to demand disruptions in re-planning strategies; for example, it is 
possible to return to a facility already visited. 
The MUN strategy had the best performance in scenarios with low inventory-
holding costs and was among the best in scenarios with high inventory-holding costs.  
That is not surprising, since MUN is a greedy rule that cares well for the facility 
closest to running out of inventory.  Nevertheless, when the parameters used in re-
planning or fixed-tour strategies are accurate, MUN is systematically outperformed. 
 
6.4. Summary of Main Results  
The following is a summary of the main results reported in this chapter. 
? The optimal decentralized benchmark policy, BENCH1, is systematically 
outperformed by centralized strategies, probably because BENCH1 tends to 
carry more inventory, thereby being protected from longer lead times. 
? The developed online strategies systematically outperform benchmark 
strategies.  The best proposed strategies achieved average-total-costs 
reductions of approximately 30% and 15%, respectively, compared with 
benchmark policies BENCH1 and MUN. 
 
 125 
 
? Strategies that included complete re-planning at delivery epochs, with 
appropriate inventory-control parameters, had the best performance of the 
studied strategies for the set of parameters considered.  
? Among re-planning strategies, those that updated plans at delivery epochs 
(RDE and RDE+div) were the best for the set of parameters considered.  The 
possibility of diversions?either en-route or when the vehicle was idle at the 
depot?improve system performance in scenarios with low inventory-holding 
costs and high demand variability. However, further research is needed to 
identify scenarios in which en-route diversion would be beneficial. 
? Among fixed-tour strategies, the capability to update tour intervals provided 
benefits of up to 10% in scenarios with low inventory-holding costs and high 
demand variability.  Moreover, the possibility of skipping retailers on the 
route did not produce significant benefits in the scenarios studied.  
? In scenarios with high inventory-holding costs and moderate demand 
variability, fixed-tour strategies performed among the best of those studied.  
In general, fixed-tour strategies are a very good benchmark when evaluating 
the implementation of more sophisticated real-time control strategies. 
 
 
 126 
 
Chapter 7: Conclusions 
 
This last chapter summarizes this dissertation?s main contributions, findings, 
and conclusions.  In addition, it lists possible extensions and directions for future 
research. 
7.1. Summary of Contributions and Findings  
 In this section the main objectives of this dissertation, which were outlined in 
Section 1.4, are reexamined, presenting related findings and conclusions.   
The first objective of this research was to formulate and state the online 
inventory routing problem (OIRP) in a manner that explicitly took into account real-
time information about fleet status and inventory levels at different facilities. That 
objective was achieved by providing a formal definition of the OIRP and discussing 
various problem features. 
The second main objectives were the development and design of operational-
control strategies, and the formulation of local off-line problems and heuristics used 
to update distribution plans within distribution systems, wherein transportation 
operations and inventory control operations were coordinated.  Those objectives were 
accomplished by proposing two decomposition approaches in which a simplified 
version of either the inventory-control side or the routing side of the problem is 
solved first, and then that solution was used as a soft constraint when solving the 
other side.  In the first approach, inventory-reorder parameters were first established 
for each facility, then used as target levels on a developed routing problem used to 
update plans.  The main contributions in that approach were the formulation of a 
 
 127 
 
mathematical programming model for the short-term off-line IRP and the 
development of an inventory-control model that explicitly recognized queueing 
effects on multiple-orders.  A key contribution in that MIP model was the 
development of an objective function that recognizes the operational trade-offs 
involved in online distribution decisions.  In the second approach, the routing side of 
the problem was solved a priori, establishing fixed-tours to re-supply retailers; then, 
inventory allocation decisions were taken respecting that sequence.  An important 
contribution of the second approach was the analytical derivation of expected costs 
for fixed tours at regular interval strategies.  For both approaches, different rolling-
horizon strategies were developed, tailored to different degrees of availability of real-
time information associated with different scenarios in terms of the ICT installed. 
The two final objectives in this research were: to improve the understanding 
of the relationship among problem parameters of a distribution system that could 
most benefit by implementing sophisticated control strategies; and to estimate the 
major benefits associated with implementing proposed real-time strategies.  This last 
objective was accomplished primarily by simulation experiments for different 
scenarios.  In order to fulfill that objective, a simulation framework was developed to 
analyze and evaluate the performance of the proposed dynamic-decision strategies for 
scenarios with: i) different facilities layout, ii) different relationship among cost 
parameters, and iii) different demand variabilities.  In addition, two evaluation 
benchmarks for the OIRP were identified.  The first was a decentralized system in 
which each agent followed optimal policies; namely, that each retailer applied an 
optimal single-echelon inventory-control policy, and that the vendor scheduled 
 
 128 
 
deliveries solving a VRP.  The second benchmark took into consideration a 
centralized system with real-time communication capabilities in which a simple 
greedy rule was used to schedule the next delivery.  Under that policy, at each 
delivery epoch, the vehicle is sent next to re-supply the retailer nearest to running out 
of inventory. 
The main findings of those experiments can be summarized as follows:  
? The developed online strategies systematically outperform benchmark 
strategies.  Moreover, those strategies are able to reduce stock-out penalties 
better than the benchmarks considered. The best proposed strategies achieve 
reductions in average total costs of approximately 30% and 15% compared 
against benchmark policies BENCH1 and MUN respectively. 
? Strategies that use complete re-planning at delivery epochs, with appropriate 
inventory-control parameters, have the best performance of the studied 
strategies for the set of parameters considered.  
? Among re-planning strategies, those that update plans at delivery epochs, 
RDE and RDE+div, were the best strategies for the set of parameters 
considered.  The possibility of diversions?either en-route or when the vehicle 
is idle at the depot?improve system performance in scenarios with low 
inventory-holding costs and high demand variability. However, further 
research is needed to identify scenarios in which en-route diversion would be 
beneficial. 
? Among fixed-tour strategies, the possibility of updating tour intervals offers 
benefits of up to 10% in scenarios with low inventory-holding costs and high 
 
 129 
 
demand variability.  Moreover, the possibility of skipping retailers in the route 
did not produce significant benefits for scenarios studied.  
? In scenarios with high inventory-holding costs and moderate demand 
variability, fixed-tour strategies perform among the best studied strategies.  In 
general, fixed-tour strategies are a very good benchmark when evaluating the 
implementation of more sophisticated real-time control strategies.  
7.2. Future Research and Extensions  
 This dissertation presents the first study of the OIRP.  Given the scope of this 
research, there are many promising extensions appropriate for attention in future 
research.  The first set of extensions relate to topics that are either direct extensions of 
the problem studied or study aspects for which inconclusive answers were found.  
? The results obtained in this dissertation provide a limited understanding of the 
benefits associated with en-route truck diversions.  A more comprehensive 
understanding would require comparing strategies with diversions against 
similar strategies without diversions but including updates in idle time at the 
depot.  Moreover, for scenarios with low stock-out probabilities, an analysis 
might require the design of different experiments in which those low 
probability events are sampled, using importance-sampling methods, for 
example. 
? The improvement of the formulation of the inventory-control side of the 
problem in re-optimization strategies in order to incorporate synergies 
associated with serving clusters of retailers together. 
 
 130 
 
? The development of hybrid strategies?those that integrate fixed-tour 
deliveries for normal conditions and the plan re-optimization approach when 
demand disruptions occur?would follow some promising avenues.  In 
general, incorporating the possibility of updating idle time at the depot in re-
optimization strategies is needed to better understand the advantages of re-
routing vs. fixed-tour strategies.  
The second set of extensions relate to topics that were beyond the scope of 
this dissertation. Such extensions can be divided into three groups: i) demand 
processes and types of operations considered, ii) types of uncertainties considered; 
and iii) design of distribution system for online operations.  In terms of demand 
processes and types of operations allowed, the following can be considered: 
a) The analysis of the performance of the proposed strategies could be 
extended to incorporate priorities for some facilities and scenarios with 
asymmetric demand patterns.   
b) In this research a single product was considered in the analysis.  However, 
most distribution systems deal with multiple products.  An important 
extension might incorporate decisions related to the mix of products 
transported. 
c) This research is assumed that demand processes at retailers could not be 
affected by the central decision maker.  An extension to this research 
might deal with scenarios in which distribution decisions are combined 
with real-time pricing to, for example avoid stock-outs at particular 
facilities.  Further refinements in that direction might consider scenarios in 
 
 131 
 
which a supplier provides complement and substitute products and 
combine inventory and pricing decisions. 
d) Incorporating daily- and weekly-cycle operational characteristics in the 
analysis, such as changes in demand patterns during the day and week, and 
end-of-cycle constraints, would be logical extensions.  Moreover, 
including labor-related constraints in the vehicle routes, such as the 
number of hours that a driver can operate a vehicle, and time windows for 
particular facilities, would be worthy extensions. 
e) Another possible extension would be to incorporate transshipment 
operations between retailers, since this dissertation assumed that products 
that had been delivered to one facility could not be reclaimed and 
reassigned to another facility. 
In this research the only source of uncertainty were demand processes at 
different facilities.  However, in real-world inventory-routing systems, additional 
activities are affected by uncertainties. 
a) Traffic conditions are an important source of uncertainties, particularly in 
urban areas, as they can cause significantly varying travel times as a 
consequence of network congestion.  Thus, extending this research to 
account for the uncertainty of travel times would seem to be a worthy 
extension. 
b) In addition, time required for loading and unloading, which was not 
considered in this dissertation, is also affected by uncertainties.  That is 
particularly relevant in maritime operations, in which ports are affected by 
 
 132 
 
weather and congestion which can significantly affect the amount of time 
spent on loading and unloading vehicles. 
Finally, in terms of the design of distribution systems for online operations, 
there are topics that would extend this research, such as the following: 
a) The implementation and analysis of the proposed strategies in large-scale 
inventory-routing systems.  In particular, the development of solution 
approaches for re-planning strategies in scenarios with large number of 
retailers.  Since for the proposed re-optimization formulation, with current 
technology, solution times grow exponentially with problem size, the 
implementation of re-planning strategies in large size instances probably 
would require the development of fast heuristics or metaheuristics. 
b) The design of large-scale inventory-routing systems composed of multiple 
depots, retailers, and vehicles, to be operated with real-time ICT 
capabilities?in particular, the allocation of vehicles and retailers to 
depots.  One example might be the analysis of strategies that could divide 
deliveries to a particular facility among more than one depot or truck. 
c) The analysis of fleet-sizing decisions for real-time operation under 
different cost parameters. 
d) The analysis of coordination mechanisms to achieve close-to-optimal 
system decisions in a decentralized inventory-routing system, such as 
pricing incentives that could align the strategic decisions of each player in 
a decentralized distribution system.  
 
 
 133 
 
Appendix A: Notation 
 
?: 
{ }
1,2,..., ,...,iN?=  set of retailers  
0
? , 
{ }
0
0????  set of all facilities (depot and retailers). Those N+1 facilities are 
denoted by sub-index i=0,1, 2, ?, N (sub-index 0 is for the depot) 
l(i) locations for 
0
i?? . 
(),d ??
function: gives the Euclidean distance between two facilities or between a 
facility and the vehicle location 
i
? :retailer i maximum capacity to store inventory 
? : vehicle limited capacity. 
()
i
t?  Customers? arrival rates for retailer i at time t 
()
j
i
t? : probability that a customer arriving at time t to retailer i has a demand size 
equal to j. 
()
i
t? . mean of customer demand sizes when they are assumed to be Poisson 
distributed. 
()
i
t? : the expected demand per unit of time at retailer i at time t, which can be 
calculated as 
() () ( )
iii
ttt???=?. 
,im
?  :Arrivals times for the m
th
 customer arrival to facility i. 
,im
?  Demand sizes for the m
th
 customer arrival to facility i. 
() { }
,
max 0 :
iim
A tm t?=??: Total number of customer arrivals to retailer i that have 
occurred by time t  
()
()
,
1
i
At
iim
m
Dt ?
=
=
?
: Total demand at customer i until time t 
()X t : state of the system at time t. ( ) ( ) ( ) ( )X tttt??=? ?
? ?
null . 
()
i
t?  : inventory level at facility i at time t. 
1
( ) ( ),..., ( ),...., ( )
iN
ttt t?? ? ?= : vector of inventory levels at time t 
()tnull : Location of the truck at time t 
()t? : Load remaining in the truck at time t,. 
() () () ()X tttt??=??
??
null  
TC: transportation cost per unit of distance traveled by the vehicle  
i
h : Inventory holding costs at each retailer i 
i
p : Penalty associated with each unit of demand lost during stock-out at retailer i 
? : a plan or policy, which can be specified by 
[ ]
? = sqt 
[ ]
12 L
ss s=s ? : Sequence of facilities to be visited. 
[ ]
12 L
qq q=q ? : Amounts to be delivered. 
 
 134 
 
[ ]
12 L
tt t=t ? , arrival times to each one of those facilities, in which L  is the 
length of the planning horizon in terms of number of visits programmed. 
[ ]
12
uu=u ? : The sequence of update epochs in which 
n
u  is the time of the n
th
 
plan update satisfying 
1
0
nn
uu
+
>? for all n.  
() { }max 0 :
n
Ut n u t=?? : number of plan updates up to epoch t.  
()
()
Ut
t? ??  is the current plan at epoch t 
()
()
Ut
ut u?  is the time of the last plan update 
 
Tr
C : Transportation costs  
IH
C : Inventory holding costs. 
ISO
C : Inventory stock-out costs 
C
Crew
 : Crew-associated costs  
 
 
(s, S) policy, and can be stated: whenever the current inventory level is below the 
reorder point s, an order is placed to bring the inventory level to S; otherwise, do not 
place an order.  
 
 
 135 
 
Appendix B: Glossary 
 
APS advance planning and scheduling 
ATMs: automatic teller machines  
AVI automatic vehicle identification  
AVL automatic vehicle location  
CVO: commercial vehicle operations. 
DP: Dynamic Programming 
DSS: decision-support systems 
EDI: Electronic Data Interchange 
EOQ Economic Order Quantity  
ERP: Enterprise Resource Planning 
ERP-II: multi-component decision-support systems  
GDP: gross demographic product  
GIS Geographic Information System  
GPS Global Positioning System 
ICT: information and communication technologies  
IID: independent and identically distributed 
IP: Integer Program  
IRPs inventory-routing problems 
LTL: less than truck load 
MDP: Markov decision process  
MRP: material requirements planning 
OBC on board computers 
OIRP online inventory routing problem 
PDA: personal digital assistant 
RFID: radio frequency identification  
RH: rolling horizon  
SCP supply chain planning systems 
SO: stock-outs 
TL: truck loads  
TMS transportation management systems  
TSP: traveling salesman problem  
VMI: vendor-managed inventory  
VRP: vehicle-routing problem  
VRPTW: vehicle-routing problem with time windows 
WMS warehouse management systems  
XML: eXchange Markup Language 
 
 136 
 
Appendix C: Inventory Reorder Level Parameters  
 
 
Table C- 1: Inventory Target Levels for Set of Parameters 1 
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 9.3 62.3 53.5 39.6 30.1 19.9 43.5 43.1 16.4 26.7
2 35.7 62.3 53.5 39.6 30.1 19.9 43.5 43.1 16.4 26.7
3 22.0 62.3 53.5 39.6 30.1 19.9 43.5 43.1 16.4 26.7
4 4.4 62.3 53.5 39.6 30.1 19.9 43.5 43.1 16.4 26.7
5 40.5 62.3 53.5 39.6 30.1 19.9 43.5 43.1 16.4 26.7
6 5.3 62.3 53.5 39.6 30.1 19.9 43.5 43.1 16.4 26.7
0 
7 30.1 62.3 53.5 39.6 30.1 19.9 43.5 43.1 16.4 26.7
1 30.8 62.8 49.2 46.5 24.8 26.8 37.3 50.1 14.5 35.6
2 47.6 61.2 63.8 46.1 39.9 27.2 54.0 50.1 21.4 28.7
3 18.7 60.1 76.3 46.4 53.6 28.0 68.4 50.1 28.1 22.0
4 9.5 61.7 58.9 46.1 34.8 27.0 48.4 50.1 19.0 31.1
5 22.4 61.1 65.1 46.1 41.3 27.3 55.5 50.1 22.0 28.1
6 12.1 61.5 60.9 46.1 36.8 27.1 50.6 50.1 19.9 30.2
1 
7 38.7 61.1 64.8 46.1 41.0 27.3 55.0 50.1 21.8 28.3
1 9.4 66.0 26.4 40.4 8.2 18.8 15.3 42.5 5.9 36.6
2 10.9 62.4 52.7 38.5 29.7 18.8 42.6 42.5 16.0 26.5
3 36.4 61.3 62.6 38.6 39.7 19.5 53.3 42.5 20.8 21.8
4 48.2 61.7 59.0 38.5 36.0 19.2 49.4 42.5 19.0 23.5
5 12.1 63.2 45.7 38.7 23.0 18.5 35.3 42.5 13.0 29.5
6 58.5 60.7 69.2 38.9 46.6 20.0 60.5 42.5 24.2 18.3
2 
7 2.1 66.0 26.7 40.4 8.4 18.8 15.7 42.5 6.0 36.5
1 32.4 63.3 44.3 41.9 21.0 21.8 33.0 43.2 12.4 30.8
2 18.9 60.9 67.0 41.6 43.7 22.8 57.8 43.2 23.0 20.2
3 0.8 62.5 51.6 41.5 27.9 21.9 40.8 43.2 15.5 27.7
4 18.9 63.8 40.8 42.2 17.9 21.8 29.3 43.2 11.0 32.2
5 7.0 61.4 61.7 41.5 38.2 22.4 51.9 43.2 20.3 22.9
6 30.3 61.5 60.8 41.5 37.2 22.4 51.0 43.2 19.9 23.3
3 
7 28.6 61.1 65.5 41.6 42.1 22.7 56.1 43.2 22.2 21.0
 
 
 137 
 
 
Table C- 2: Inventory Target Levels for Set of Parameters 2  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 6.5 65.0 32.5 45.1 18.6 26.0 30.5 33.0 16.4 16.6
2 25.0 65.0 32.5 45.1 18.6 26.0 30.5 33.0 16.4 16.6
3 15.4 65.0 32.5 45.1 18.6 26.0 30.5 33.0 16.4 16.6
4 3.1 65.0 32.5 45.1 18.6 26.0 30.5 33.0 16.4 16.6
5 28.4 65.0 32.5 45.1 18.6 26.0 30.5 33.0 16.4 16.6
6 3.7 65.0 32.5 45.1 18.6 26.0 30.5 33.0 16.4 16.6
0 
7 21.2 65.0 32.5 45.1 18.6 26.0 30.5 33.0 16.4 16.6
1 21.9 65.4 30.1 53.3 17.5 35.0 26.6 42.9 14.5 28.4
2 36.0 64.1 38.5 52.6 28.4 34.7 40.9 42.9 21.4 21.5
3 14.7 63.2 45.6 52.5 40.0 35.2 53.7 42.9 28.1 14.8
4 7.1 64.5 35.7 52.8 24.3 34.7 36.0 42.9 19.0 23.9
5 17.0 64.0 39.2 52.5 29.6 34.8 42.2 42.9 22.0 20.9
6 9.1 64.3 36.8 52.7 25.9 34.7 37.9 42.9 19.9 23.0
1 
7 29.4 64.0 39.0 52.6 29.3 34.8 41.8 42.9 21.8 21.0
1 5.3 68.1 17.0 44.7 6.1 24.9 8.6 32.3 5.9 26.4
2 7.5 65.0 32.1 43.2 17.2 23.9 29.1 32.3 16.0 16.2
3 25.8 64.2 37.8 43.1 24.3 24.3 37.7 32.3 20.8 11.5
4 33.7 64.5 35.7 43.1 21.6 24.1 34.5 32.3 19.0 13.3
5 8.0 65.7 28.1 43.5 13.1 23.8 23.3 32.3 13.0 19.3
6 42.3 63.7 41.6 43.2 29.5 24.7 43.7 32.3 24.2 8.1
2 
7 1.2 68.0 17.2 44.6 6.2 24.9 8.9 32.3 6.0 26.3
1 22.0 65.9 27.3 48.0 13.2 28.6 22.4 33.1 12.4 20.7
2 14.2 63.9 40.3 47.2 29.3 28.8 43.3 33.1 23.0 10.1
3 0.6 65.2 31.4 47.6 17.3 28.5 28.9 33.1 15.5 17.6
4 12.6 66.2 25.3 48.4 11.1 28.8 19.5 33.1 11.0 22.1
5 5.1 64.3 37.3 47.2 25.0 28.6 38.3 33.1 20.3 12.8
6 22.3 64.3 36.8 47.2 24.1 28.6 37.4 33.1 19.9 13.3
3 
7 21.3 64.0 39.4 47.2 27.9 28.8 41.8 33.1 22.2 10.9
 
 
 138 
 
 
Table C- 3: Inventory Target Levels for Set of Parameters 3  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 16.6 59.1 89.4 33.2 64.5 14.3 77.7 74.5 16.4 58.2
2 63.8 59.1 89.4 33.2 64.5 14.3 77.7 74.5 16.4 58.2
3 39.3 59.1 89.4 33.2 64.5 14.3 77.7 74.5 16.4 58.2
4 7.8 59.1 89.4 33.2 64.5 14.3 77.7 74.5 16.4 58.2
5 72.4 59.1 89.4 33.2 64.5 14.3 77.7 74.5 16.4 58.2
6 9.5 59.1 89.4 33.2 64.5 14.3 77.7 74.5 16.4 58.2
0 
7 53.9 59.1 89.4 33.2 64.5 14.3 77.7 74.5 16.4 58.2
1 57.5 59.6 82.1 35.9 55.5 16.8 69.7 84.8 14.5 70.3
2 83.4 57.8 107.3 36.1 79.9 17.7 94.6 84.8 21.4 63.4
3 31.7 56.4 128.8 36.8 100.7 18.9 115.8 84.8 28.1 56.7
4 16.9 58.4 98.9 35.9 71.8 17.3 86.3 84.8 19.0 65.8
5 39.1 57.6 109.5 36.1 82.0 17.8 96.8 84.8 22.0 62.7
6 21.5 58.1 102.2 36.0 75.0 17.5 89.6 84.8 19.9 64.9
1 
7 67.7 57.7 108.9 36.1 81.4 17.8 96.2 84.8 21.8 62.9
1 19.9 63.6 42.6 34.6 20.2 13.4 32.3 73.4 5.9 67.5
2 19.8 59.2 88.0 33.0 63.4 14.0 77.1 73.4 16.0 57.4
3 63.9 57.9 105.2 33.2 80.1 14.8 93.7 73.4 20.8 52.6
4 85.4 58.4 98.9 33.1 74.0 14.5 87.4 73.4 19.0 54.4
5 22.3 60.1 76.0 33.0 51.8 13.6 65.0 73.4 13.0 60.4
6 102.0 57.1 116.6 33.6 91.2 15.5 105.5 73.4 24.2 49.2
2 
7 4.3 63.5 43.1 34.6 20.7 13.4 32.8 73.4 6.0 67.4
1 61.2 60.3 73.6 34.1 48.7 14.6 62.2 73.4 12.4 61.0
2 32.9 57.4 112.7 34.5 86.4 16.2 100.6 73.4 23.0 50.4
3 1.5 59.3 86.1 34.0 60.7 15.0 74.5 73.4 15.5 57.9
4 36.4 60.9 67.6 34.3 42.9 14.5 56.3 73.4 11.0 62.4
5 12.3 58.0 103.7 34.2 77.7 15.7 91.7 73.4 20.3 53.1
6 53.7 58.1 102.1 34.2 76.2 15.7 90.2 73.4 19.9 53.5
3 
7 49.9 57.6 110.1 34.4 83.9 16.1 98.1 73.4 22.2 51.2
 
 
 139 
 
 
Table C- 4: Inventory Target Levels for Set of Parameters 4  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 10.4 70.1 56.9 45.4 35.8 22.9 48.5 48.7 23.7 25.1
2 39.8 70.1 56.9 45.4 35.8 22.9 48.5 48.7 23.7 25.1
3 24.5 70.1 56.9 45.4 35.8 22.9 48.5 48.7 23.7 25.1
4 4.9 70.1 56.9 45.4 35.8 22.9 48.5 48.7 23.7 25.1
5 45.2 70.1 56.9 45.4 35.8 22.9 48.5 48.7 23.7 25.1
6 5.9 70.1 56.9 45.4 35.8 22.9 48.5 48.7 23.7 25.1
0 
7 33.6 70.1 56.9 45.4 35.8 22.9 48.5 48.7 23.7 25.1
1 36.0 70.8 52.7 51.8 31.2 29.0 43.6 56.0 21.2 34.8
2 54.3 68.4 67.2 50.6 47.0 29.0 61.6 56.0 30.2 25.8
3 21.2 66.5 79.6 50.4 61.6 29.5 77.6 56.0 38.7 17.2
4 10.9 69.2 62.3 50.9 41.6 28.9 55.6 56.0 27.0 28.9
5 25.6 68.2 68.5 50.6 48.5 29.0 63.3 56.0 31.0 24.9
6 13.9 68.8 64.3 50.8 43.8 28.9 58.0 56.0 28.3 27.7
1 
7 44.2 68.2 68.1 50.6 48.1 29.0 62.8 56.0 30.8 25.2
1 11.2 76.0 30.0 48.2 12.2 22.9 18.2 48.1 9.2 39.0
2 12.1 70.2 56.1 45.0 34.1 22.1 47.1 48.1 23.2 24.9
3 40.1 68.6 66.0 44.6 44.5 22.6 58.7 48.1 29.4 18.7
4 53.2 69.1 62.4 44.7 40.7 22.4 54.5 48.1 27.1 21.0
5 13.5 71.5 49.2 45.5 27.3 22.0 39.3 48.1 19.1 29.0
6 64.5 67.6 72.6 44.5 51.7 23.0 66.7 48.1 33.8 14.3
2 
7 2.4 75.9 30.3 48.1 12.4 22.9 18.6 48.1 9.3 38.8
1 37.1 71.8 47.8 48.0 26.7 24.8 37.7 48.8 18.4 30.4
2 21.1 67.9 70.4 46.6 50.6 25.3 64.6 48.8 32.3 16.5
3 0.9 70.4 55.0 47.2 33.9 24.8 46.1 48.8 22.5 26.3
4 21.9 72.5 44.3 48.4 23.5 25.0 33.8 48.8 16.4 32.4
5 7.8 68.7 65.1 46.7 44.8 25.1 58.2 48.8 28.8 20.0
6 34.0 68.8 64.2 46.7 43.8 25.0 57.1 48.8 28.2 20.6
3 
7 31.9 68.1 68.9 46.6 48.9 25.3 62.8 48.8 31.3 17.5
 
 
 140 
 
 
Table C- 5: Inventory Target Levels for Set of Parameters 5  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 11.1 78.2 61.1 52.0 40.7 26.7 52.1 48.7 32.0 23.2
2 42.7 78.2 61.1 52.0 40.7 26.7 52.1 48.7 32.0 23.2
3 26.3 78.2 61.1 52.0 40.7 26.7 52.1 48.7 32.0 23.2
4 5.2 78.2 61.1 52.0 40.7 26.7 52.1 48.7 32.0 23.2
5 48.5 78.2 61.1 52.0 40.7 26.7 52.1 48.7 32.0 23.2
6 6.4 78.2 61.1 52.0 40.7 26.7 52.1 48.7 32.0 23.2
0 
7 36.1 78.2 61.1 52.0 40.7 26.7 52.1 48.7 32.0 23.2
1 39.7 79.2 56.9 57.4 37.6 32.4 48.2 56.0 28.7 36.0
2 59.3 75.8 71.3 55.5 54.3 32.0 67.2 56.0 40.2 24.5
3 23.1 73.2 83.7 54.7 69.8 32.1 84.4 56.0 50.9 13.8
4 11.9 76.9 66.5 56.0 48.6 32.0 60.8 56.0 36.3 28.4
5 27.9 75.5 72.6 55.4 55.9 32.0 69.0 56.0 41.3 23.4
6 15.2 76.4 68.4 55.8 50.9 32.0 63.3 56.0 37.8 26.9
1 
7 48.2 75.6 72.3 55.5 55.5 32.0 68.5 56.0 41.0 23.7
1 12.6 86.0 34.7 55.6 17.2 27.8 20.4 48.1 12.9 41.6
2 12.9 78.4 60.3 51.2 40.5 26.1 50.3 48.1 31.4 23.1
3 42.7 76.1 70.1 50.3 51.5 26.3 62.6 48.1 39.2 15.3
4 56.7 76.9 66.5 50.6 47.5 26.2 58.1 48.1 36.3 18.2
5 14.4 80.1 53.4 52.0 33.3 26.1 42.1 48.1 26.1 28.4
6 68.8 74.6 76.7 50.0 59.3 26.6 71.1 48.1 44.8 9.7
2 
7 2.7 85.9 35.0 55.6 17.5 27.8 20.8 48.1 13.1 41.4
1 40.4 80.5 52.1 54.9 31.5 28.9 41.0 48.8 25.1 30.1
2 22.8 75.1 74.5 52.3 56.3 28.7 69.5 48.8 42.9 12.4
3 1.0 78.6 59.2 53.8 39.0 28.6 49.9 48.8 30.5 24.7
4 23.9 81.5 48.6 55.5 28.2 29.1 37.0 48.8 22.6 32.6
5 8.4 76.3 69.2 52.7 50.2 28.6 62.7 48.8 38.5 16.7
6 36.6 76.5 68.4 52.8 49.2 28.6 61.6 48.8 37.8 17.4
3 
7 34.4 75.4 73.0 52.4 54.6 28.7 67.6 48.8 41.6 13.6
 
 
 141 
 
 
Table C- 6: Inventory Target Levels for Set of Parameters 6  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 12.0 85.4 65.6 58.3 44.8 30.0 56.4 48.7 40.3 21.1
2 46.3 85.4 65.6 58.3 44.8 30.0 56.4 48.7 40.3 21.1
3 28.5 85.4 65.6 58.3 44.8 30.0 56.4 48.7 40.3 21.1
4 5.7 85.4 65.6 58.3 44.8 30.0 56.4 48.7 40.3 21.1
5 52.6 85.4 65.6 58.3 44.8 30.0 56.4 48.7 40.3 21.1
6 6.9 85.4 65.6 58.3 44.8 30.0 56.4 48.7 40.3 21.1
0 
7 39.1 85.4 65.6 58.3 44.8 30.0 56.4 48.7 40.3 21.1
1 43.7 86.7 61.4 63.5 42.3 35.3 53.0 56.0 36.3 38.2
2 64.6 82.4 75.7 60.8 59.4 34.4 73.3 56.0 50.3 24.3
3 25.1 79.2 87.9 59.3 75.5 34.2 91.9 56.0 63.1 11.5
4 13.0 83.8 70.9 61.6 53.5 34.6 66.4 56.0 45.5 29.1
5 30.4 82.1 76.9 60.7 61.0 34.3 75.2 56.0 51.6 23.0
6 16.6 83.2 72.8 61.3 55.8 34.5 69.2 56.0 47.4 27.2
1 
7 52.6 82.2 76.6 60.7 60.6 34.4 74.7 56.0 51.2 23.4
1 14.1 94.7 39.9 63.1 20.9 32.1 22.9 48.1 16.6 44.2
2 13.9 85.6 64.8 57.6 44.5 29.5 54.2 48.1 39.5 21.3
3 46.0 82.8 74.5 56.2 55.8 29.5 67.4 48.1 49.1 11.8
4 61.1 83.8 70.9 56.7 51.6 29.4 62.5 48.1 45.6 15.3
5 15.6 87.8 58.0 58.8 37.3 29.8 45.5 48.1 33.1 27.7
6 74.1 81.0 81.0 55.6 63.8 29.6 76.6 48.1 55.8 5.1
2 
7 3.1 94.5 40.2 63.0 21.2 32.0 23.4 48.1 16.9 44.0
1 44.2 88.2 56.7 61.1 36.3 32.3 45.0 48.8 31.9 29.5
2 24.7 81.6 78.8 57.5 62.0 31.4 75.4 48.8 53.5 8.0
3 1.1 86.0 63.7 59.7 44.0 31.8 54.3 48.8 38.5 22.9
4 26.3 89.4 53.3 61.9 33.0 32.6 40.7 48.8 28.8 32.6
5 9.1 83.0 73.6 58.1 55.6 31.5 68.0 48.8 48.2 13.2
6 39.7 83.3 72.7 58.2 54.6 31.5 66.7 48.8 47.4 14.1
3 
7 37.3 82.0 77.3 57.7 60.2 31.5 73.3 48.8 52.0 9.5
 
 
 142 
 
 
Table C- 7: Inventory Target Levels for Set of Parameters 7  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 17.5 63.6 36.4 38.4 61.6 18.1 81.9 48.7 16.4 83.6
2 67.2 63.6 36.4 38.4 61.6 18.1 81.9 48.7 16.4 83.6
3 41.4 63.6 36.4 38.4 61.6 18.1 81.9 48.7 16.4 83.6
4 8.2 63.6 36.4 38.4 61.6 18.1 81.9 48.7 16.4 83.6
5 76.3 63.6 36.4 38.4 61.6 18.1 81.9 48.7 16.4 83.6
6 10.0 63.6 36.4 38.4 61.6 18.1 81.9 48.7 16.4 83.6
0 
7 56.8 63.6 36.4 38.4 61.6 18.1 81.9 48.7 16.4 83.6
1 62.8 64.5 35.5 44.4 55.6 23.9 76.1 56.0 14.5 85.5
2 66.5 61.6 38.4 44.4 55.6 24.6 75.4 56.0 21.4 78.6
3 20.4 59.3 40.7 44.8 55.2 25.5 74.5 56.0 28.1 71.9
4 14.8 62.6 37.4 44.4 55.6 24.3 75.7 56.0 19.0 81.0
5 30.4 61.4 38.6 44.5 55.5 24.7 75.3 56.0 22.0 78.0
6 18.1 62.2 37.8 44.4 55.6 24.4 75.6 56.0 19.9 80.1
1 
7 53.0 61.5 38.5 44.5 55.5 24.7 75.3 56.0 21.8 78.2
1 38.0 70.1 29.9 39.2 48.7 16.5 61.5 48.1 5.9 94.1
2 21.0 63.8 36.2 38.5 61.5 18.2 81.8 48.1 16.0 84.0
3 55.2 61.9 38.1 38.9 61.1 19.1 80.9 48.1 20.8 79.2
4 79.3 62.6 37.4 38.7 61.3 18.8 81.2 48.1 19.0 81.0
5 28.2 65.3 34.7 38.4 61.6 17.7 82.3 48.1 13.0 87.0
6 77.6 60.6 39.4 39.2 60.8 19.7 80.3 48.1 24.2 75.8
2 
7 8.1 70.0 30.0 39.2 49.7 16.5 62.1 48.1 6.0 94.0
1 79.5 65.6 34.4 40.0 60.0 19.2 80.8 48.8 12.4 87.6
2 25.9 61.0 39.0 40.5 59.5 20.9 79.1 48.8 23.0 77.0
3 1.6 64.0 36.0 40.1 59.9 19.7 80.3 48.8 15.5 84.5
4 52.4 66.3 33.7 40.0 60.0 19.0 81.0 48.8 11.0 89.0
5 10.7 62.0 38.0 40.3 59.7 20.4 79.6 48.8 20.3 79.7
6 47.4 62.2 37.8 40.3 59.7 20.4 79.6 48.8 19.9 80.1
3 
7 40.3 61.3 38.7 40.4 59.6 20.8 79.2 48.8 22.2 77.8
 
 
 143 
 
 
Table C- 8: Inventory Target Levels for Set of Parameters 8  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 17.2 68.4 31.6 40.7 59.3 19.4 80.6 94.9 14.5 80.4
2 66.1 68.4 31.6 40.7 59.3 19.4 80.6 94.9 21.4 73.6
3 40.7 68.4 31.6 40.7 59.3 19.4 80.6 94.9 28.1 66.9
4 8.1 68.4 31.6 40.7 59.3 19.4 80.6 94.9 19.0 76.0
5 75.1 68.4 31.6 40.7 59.3 19.4 80.6 94.9 22.0 72.9
6 9.8 68.4 31.6 40.7 59.3 19.4 80.6 94.9 19.9 75.0
0 
7 55.9 68.4 31.6 40.7 59.3 19.4 80.6 94.9 21.8 73.1
1 51.9 69.0 31.0 47.4 52.6 26.0 62.9 82.3 5.9 76.5
2 64.2 66.9 33.1 48.0 52.0 27.2 72.8 82.3 16.0 66.3
3 19.6 65.3 34.7 48.7 51.3 28.4 71.6 82.3 20.8 61.6
4 14.3 67.6 32.4 47.8 52.2 26.8 73.2 82.3 19.0 63.4
5 29.4 66.7 33.3 48.1 51.9 27.3 72.7 82.3 13.0 69.3
6 17.5 67.3 32.7 47.9 52.1 27.0 73.0 82.3 24.2 58.2
1 
7 51.2 66.8 33.2 48.1 51.9 27.3 72.7 82.3 6.0 76.4
1 20.9 73.3 26.7 41.7 20.2 17.7 33.9 82.4 12.4 69.9
2 20.6 68.5 31.5 41.0 59.0 19.8 80.2 82.4 23.0 59.4
3 54.0 67.1 32.9 41.7 58.3 21.0 79.0 82.4 15.5 66.8
4 77.6 67.6 32.4 41.4 58.6 20.5 79.5 82.4 11.0 71.3
5 23.2 69.6 30.4 41.1 53.5 18.8 67.6 82.4 20.3 62.1
6 75.6 66.2 33.8 42.2 57.8 21.8 78.2 82.4 19.9 62.5
2 
7 4.5 73.2 26.8 41.7 20.8 17.7 34.4 82.4 22.2 60.2
1 63.0 69.8 30.2 42.9 50.0 20.5 64.1 83.6 16.4 67.2
2 25.2 66.5 33.5 43.6 56.4 23.1 76.9 83.6 16.4 67.2
3 1.5 68.7 31.3 42.6 57.4 21.2 75.4 83.6 16.4 67.2
4 37.8 70.4 29.6 42.9 44.0 20.2 58.4 83.6 16.4 67.2
5 10.4 67.2 32.8 43.3 56.7 22.5 77.5 83.6 16.4 67.2
6 46.2 67.3 32.7 43.2 56.8 22.4 77.6 83.6 16.4 67.2
3 
7 39.2 66.7 33.3 43.5 56.5 22.9 77.1 83.6 16.4 67.2
 
 
 144 
 
 
Table C- 9: Inventory Target Levels for Set of Parameters 9  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 17.9 56.9 43.1 35.0 65.0 16.0 84.0 100.0 14.5 85.5
2 68.9 56.9 43.1 35.0 65.0 16.0 84.0 100.0 21.4 78.6
3 42.4 56.9 43.1 35.0 65.0 16.0 84.0 100.0 28.1 71.9
4 8.4 56.9 43.1 35.0 65.0 16.0 84.0 100.0 19.0 81.0
5 78.2 56.9 43.1 35.0 65.0 16.0 84.0 100.0 22.0 78.0
6 10.2 56.9 43.1 35.0 65.0 16.0 84.0 100.0 19.9 80.1
0 
7 58.2 56.9 43.1 35.0 65.0 16.0 84.0 100.0 21.8 78.2
1 65.1 58.3 41.7 40.2 59.8 21.1 78.9 100.0 5.9 94.1
2 69.4 53.4 46.6 39.6 60.4 21.3 78.7 100.0 16.0 84.0
3 21.4 48.4 51.6 39.4 60.6 21.7 78.3 100.0 20.8 79.2
4 15.4 55.1 44.9 39.8 60.2 21.2 78.8 100.0 19.0 81.0
5 31.8 52.9 47.1 39.6 60.4 21.4 78.6 100.0 13.0 87.0
6 18.9 54.4 45.6 39.7 60.3 21.3 78.7 100.0 24.2 75.8
1 
7 55.4 53.1 46.9 39.6 60.4 21.4 78.6 100.0 6.0 94.0
1 52.0 65.9 34.1 36.8 63.2 15.7 84.3 100.0 12.4 87.6
2 21.5 57.2 42.8 35.2 64.8 16.1 83.9 100.0 23.0 77.0
3 56.9 53.8 46.2 35.1 64.9 16.6 83.4 100.0 15.5 84.5
4 81.6 55.1 44.9 35.1 64.9 16.4 83.6 100.0 11.0 89.0
5 28.9 59.4 40.6 35.4 64.6 15.9 84.1 100.0 20.3 79.7
6 80.2 51.4 48.6 35.1 64.9 17.1 82.9 100.0 19.9 80.1
2 
7 11.0 65.8 34.2 36.8 63.2 15.7 84.3 100.0 22.2 77.8
1 81.4 59.8 40.2 36.8 63.2 17.3 82.7 100.0 16.4 83.6
2 26.8 52.3 47.7 36.3 63.7 18.1 81.9 100.0 16.4 83.6
3 1.7 57.5 42.5 36.5 63.5 17.5 82.5 100.0 16.4 83.6
4 53.6 60.9 39.1 37.1 62.9 17.2 82.8 100.0 16.4 83.6
5 11.0 54.1 45.9 36.3 63.7 17.9 82.1 100.0 16.4 83.6
6 48.9 54.5 45.5 36.3 63.7 17.8 82.2 100.0 16.4 83.6
3 
7 41.7 52.8 47.2 36.3 63.7 18.1 81.9 100.0 16.4 83.6
 
 
 145 
 
 
Table C- 10: Inventory Target Levels for Set of Parameters 10  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 16.3 69.4 30.6 47.3 52.7 23.7 76.3 100.0 21.2 78.8
2 62.6 69.4 30.6 47.3 52.7 23.7 76.3 100.0 30.2 69.8
3 38.6 69.4 30.6 47.3 52.7 23.7 76.3 100.0 38.7 61.3
4 7.7 69.4 30.6 47.3 52.7 23.7 76.3 100.0 27.0 73.0
5 71.1 69.4 30.6 47.3 52.7 23.7 76.3 100.0 31.0 69.0
6 9.3 69.4 30.6 47.3 52.7 23.7 76.3 100.0 28.3 71.7
0 
7 52.9 69.4 30.6 47.3 52.7 23.7 76.3 100.0 30.8 69.2
1 65.1 70.8 29.2 56.0 44.0 21.1 78.9 100.0 9.2 90.8
2 69.4 66.2 33.8 54.6 45.4 21.3 78.7 100.0 23.2 76.8
3 21.4 62.4 37.6 53.8 46.2 21.7 78.3 100.0 29.4 70.6
4 15.4 67.7 32.3 55.0 45.0 21.2 78.8 100.0 27.1 72.9
5 31.8 65.8 34.2 54.5 45.5 21.4 78.6 100.0 19.1 80.9
6 18.9 67.1 32.9 54.8 45.2 21.3 78.7 100.0 33.8 66.2
1 
7 55.4 65.9 34.1 54.5 45.5 21.4 78.6 100.0 9.3 90.7
1 40.5 79.6 20.4 49.6 50.4 22.9 65.6 100.0 18.4 81.6
2 19.5 69.7 30.3 47.4 52.6 23.9 76.1 100.0 32.3 67.7
3 51.6 66.5 33.5 47.0 53.0 24.5 75.5 100.0 22.5 77.5
4 74.0 67.7 32.3 47.1 52.9 24.2 75.8 100.0 16.4 83.6
5 26.2 72.0 28.0 47.8 52.2 23.5 76.5 100.0 28.8 71.2
6 72.6 64.5 35.5 46.9 53.1 25.0 75.0 100.0 28.2 71.8
2 
7 8.7 79.5 20.5 49.6 50.4 22.9 66.5 100.0 31.3 68.7
1 73.1 72.5 27.5 50.1 49.9 25.7 74.3 100.0 23.7 76.3
2 24.0 65.2 34.8 48.9 51.1 26.6 73.4 100.0 23.7 76.3
3 1.5 70.1 29.9 49.6 50.4 25.9 74.1 100.0 23.7 76.3
4 48.1 73.8 26.2 50.4 49.6 25.6 74.4 100.0 23.7 76.3
5 9.9 66.8 33.2 49.0 51.0 26.3 73.7 100.0 23.7 76.3
6 43.9 67.1 32.9 49.1 50.9 26.3 73.7 100.0 23.7 76.3
3 
7 37.4 65.7 34.3 48.9 51.1 26.5 73.5 100.0 23.7 76.3
 
 
 146 
 
 
Table C- 11: Inventory Target Levels for Set of Parameters 11  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 14.9 71.2 28.8 56.8 43.2 30.1 69.9 100.0 28.7 71.3
2 57.3 71.2 28.8 56.8 43.2 30.1 69.9 100.0 40.2 59.8
3 35.3 71.2 28.8 56.8 43.2 30.1 69.9 100.0 50.9 49.1
4 7.0 71.2 28.8 56.8 43.2 30.1 69.9 100.0 36.3 63.7
5 65.1 71.2 28.8 56.8 43.2 30.1 69.9 100.0 41.3 58.7
6 8.5 71.2 28.8 56.8 43.2 30.1 69.9 100.0 37.8 62.2
0 
7 48.5 71.2 28.8 56.8 43.2 30.1 69.9 100.0 41.0 59.0
1 48.8 72.9 27.1 68.1 31.9 40.9 59.1 100.0 12.9 87.1
2 53.4 67.0 33.0 64.4 35.6 39.4 60.6 100.0 31.4 68.6
3 16.8 62.0 38.0 61.9 38.1 38.5 61.5 100.0 39.2 60.8
4 11.8 69.0 31.0 65.5 34.5 39.8 60.2 100.0 36.3 63.7
5 24.5 66.5 33.5 64.2 35.8 39.3 60.7 100.0 26.1 73.9
6 14.5 68.2 31.8 65.1 34.9 39.7 60.3 100.0 44.8 55.2
1 
7 42.7 66.7 33.3 64.2 35.8 39.3 60.7 100.0 13.1 86.9
1 39.3 82.4 17.6 62.2 37.8 30.8 63.7 100.0 25.1 74.9
2 17.9 71.5 28.5 56.9 43.1 30.3 69.7 100.0 42.9 57.1
3 47.4 67.5 32.5 55.6 44.4 30.5 69.5 100.0 30.5 69.5
4 67.9 69.0 31.0 56.0 44.0 30.4 69.6 100.0 22.6 77.4
5 23.9 74.3 25.7 57.9 42.1 30.3 69.7 100.0 38.5 61.5
6 67.0 64.9 35.1 55.0 45.0 30.8 69.2 100.0 37.8 62.2
2 
7 8.5 82.3 17.7 62.3 37.7 30.8 65.0 100.0 41.6 58.4
1 65.5 74.9 25.1 61.2 38.8 33.4 66.6 100.0 32.0 68.0
2 21.9 65.8 34.2 57.5 42.5 33.0 67.0 100.0 32.0 68.0
3 1.4 72.0 28.0 59.8 40.2 33.2 66.8 100.0 32.0 68.0
4 43.0 76.4 23.6 62.0 38.0 33.6 66.4 100.0 32.0 68.0
5 9.0 67.9 32.1 58.2 41.8 33.0 67.0 100.0 32.0 68.0
6 39.9 68.2 31.8 58.3 41.7 33.0 67.0 100.0 32.0 68.0
3 
7 34.1 66.4 33.6 57.7 42.3 33.0 67.0 100.0 32.0 68.0
 
 
 147 
 
 
Table C- 12: Inventory Target Levels for Set of Parameters 12  
BENCH1 RTC RDE FTRI RDE+FT_S 
Case Retailer ? (0) 
s S-s s S-s s S-s S s S-s 
1 13.6 68.4 31.6 64.8 35.2 36.3 63.7 100.0 40.3 59.7
2 52.3 68.4 31.6 64.8 35.2 36.3 63.7 100.0 40.3 59.7
3 32.2 68.4 31.6 64.8 35.2 36.3 63.7 100.0 40.3 59.7
4 6.4 68.4 31.6 64.8 35.2 36.3 63.7 100.0 40.3 59.7
5 59.4 68.4 31.6 64.8 35.2 36.3 63.7 100.0 40.3 59.7
6 7.8 68.4 31.6 64.8 35.2 36.3 63.7 100.0 40.3 59.7
0 
7 44.2 68.4 31.6 64.8 35.2 36.3 63.7 100.0 40.3 59.7
1 42.4 70.1 29.9 73.9 26.1 48.6 51.4 100.0 36.3 63.7
2 48.3 63.9 36.1 68.0 32.0 45.3 54.7 100.0 50.3 49.7
3 15.6 58.2 41.8 63.5 36.5 42.8 57.2 100.0 63.1 36.9
4 10.5 66.0 34.0 69.8 30.2 46.3 53.7 100.0 45.5 54.5
5 22.2 63.3 36.7 67.5 32.5 45.0 55.0 100.0 51.6 48.4
6 13.0 65.2 34.8 69.1 30.9 45.9 54.1 100.0 47.4 52.6
1 
7 38.7 63.5 36.5 67.6 32.4 45.1 54.9 100.0 51.2 48.8
1 37.6 78.6 21.4 74.0 26.0 39.0 61.0 100.0 16.6 83.4
2 16.2 68.7 31.3 65.1 34.9 36.7 63.3 100.0 39.5 60.5
3 43.5 64.4 35.6 62.8 37.2 36.3 63.7 100.0 49.1 50.9
4 62.1 66.0 34.0 63.6 36.4 36.4 63.6 100.0 45.6 54.4
5 21.6 71.6 28.4 67.0 33.0 37.1 62.9 100.0 33.1 66.9
6 61.8 61.5 38.5 61.5 38.5 36.1 63.9 100.0 55.8 44.2
2 
7 8.0 78.5 21.5 73.8 26.2 39.0 61.0 100.0 16.9 83.1
1 58.3 72.1 27.9 70.7 29.3 40.7 59.3 100.0 31.9 68.1
2 20.1 62.5 37.5 63.9 36.1 38.6 61.4 100.0 53.5 46.5
3 1.2 69.2 30.8 68.2 31.8 39.9 60.1 100.0 38.5 61.5
4 38.1 73.5 26.5 71.9 28.1 41.2 58.8 100.0 28.8 71.2
5 8.2 64.8 35.2 65.3 34.7 39.0 61.0 100.0 48.2 51.8
6 36.3 65.2 34.8 65.5 34.5 39.1 60.9 100.0 47.4 52.6
3 
7 31.2 63.2 36.8 64.3 35.7 38.7 61.3 100.0 52.0 48.0
 
 
 148 
 
Appendix D: Detailed Results  
 
 
Table D- 1: Simulation Results: Set of Parameters 1 
TC= 100 [$/hr], h
i
= 50[$/week], ?
i
= 50[arrivals/day], ?
i
= 1 [units] for all i 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 22,569                 2,041                   24,610                 3,428                   28,038                 
St Dev 862                      1,474                   1,244                   164                      1,240                   
CV 0.04                     0.72                     0.05                     0.05                     0.04                     
Mean 12,843                 1,901                   14,744                 6,415                   21,159                
St Dev 539                      1,798                   1,572                   494                      1,633                   
CV 0.04                     0.95                     0.11                     0.08                     0.08                     
Mean 16,221                 167                      16,388                 4,621                   21,008                
St Dev 646                      278                      591                      241                      561                      
CV 0.04                     1.67                     0.04                     0.05                     0.03                     
Mean 13,997                 312                      14,308                 4,943                   19,252                
St Dev 227                      387                      486                      258                      543                      
CV 0.02                     1.24                     0.03                     0.05                     0.03                     
Mean 14,158                 228                      14,386                 4,980                   19,366                
St Dev 234                      304                      307                      248                      397                      
CV 0.02                     1.33                     0.02                     0.05                     0.02                     
Mean 9,703                   1,211                   10,914                 7,032                   17,946                
St Dev 203                      895                      833                      72                        825                      
CV 0.02                     0.74                     0.08                     0.01                     0.05                     
Mean 9,884                   499                      10,382                 7,205                   17,587                
St Dev 174                      519                      500                      238                      561                      
CV 0.02                     1.04                     0.05                     0.03                     0.03                     
Mean 9,884                   499                      10,382                 7,205                   17,587                
St Dev 174                      519                      500                      238                      561                      
CV 0.02                     1.04                     0.05                     0.03                     0.03                     
Mean 9,679                   561                      10,239                 7,233                   17,472                
St Dev 157                      509                      501                      192                      507                      
CV 0.02                     0.91                     0.05                     0.03                     0.03                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 22,812                 2,928                   25,739                 4,363                   30,103                 
St Dev 841                      2,302                   1,852                   432                      1,963                   
CV 0.04                     0.79                     0.07                     0.10                     0.07                     
Mean 16,361                 1,763                   18,124                 7,437                   25,561                
St Dev 725                      1,672                   1,393                   545                      1,541                   
CV 0.04                     0.95                     0.08                     0.07                     0.06                     
Mean 18,895                 321                      19,215                 5,308                   24,523                
St Dev 703                      498                      886                      330                      911                      
CV 0.04                     1.55                     0.05                     0.06                     0.04                     
Mean 17,287                 307                      17,594                 5,536                   23,130                
St Dev 316                      410                      498                      317                      602                      
CV 0.02                     1.34                     0.03                     0.06                     0.03                     
Mean 17,238                 239                      17,477                 5,606                   23,083                
St Dev 327                      271                      417                      354                      592                      
CV 0.02                     1.13                     0.02                     0.06                     0.03                     
Mean 10,953                 1,606                   12,558                 8,197                   20,756                
St Dev 249                      1,191                   1,136                   117                      1,139                   
CV 0.02                     0.74                     0.09                     0.01                     0.05                     
Mean 11,301                 786                      12,087                 8,605                   20,693                
St Dev 221                      821                      816                      458                      1,004                   
CV 0.02                     1.04                     0.07                     0.05                     0.05                     
Mean 11,301                 786                      12,087                 8,605                   20,693                
St Dev 221                      821                      816                      458                      1,004                   
CV 0.02                     1.04                     0.07                     0.05                     0.05                     
Mean 11,033                 824                      11,857                 8,607                   20,464                
St Dev 159                      769                      753                      215                      779                      
CV 0.01                     0.93                     0.06                     0.03                     0.04                     
Strategy
Case 0 (Symmetric case)
BENCH1
RDE
RDE+div
FTRI
RTC
RDE+FT_S
FTUI
FTRI
RDE+FT_S
MUN
BENCH1
Strategy
Case 1
RTC
FTUI
FTSR
RDE
FTSR
MUN
RDE+div
 
 
 149 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 22,721                 1,432                   24,153                 3,041                   27,194                 
St Dev 683                      1,691                   1,576                   216                      1,665                   
CV 0.03                     1.18                     0.07                     0.07                     0.06                     
Mean 12,725                 612                      13,337                 7,366                   20,703                
St Dev 381                      1,202                   1,104                   525                      1,372                   
CV 0.03                     1.97                     0.08                     0.07                     0.07                     
Mean 14,903                 660                      15,563                 4,367                   19,930                
St Dev 362                      755                      693                      264                      701                      
CV 0.02                     1.14                     0.04                     0.06                     0.04                     
Mean 12,647                 680                      13,327                 4,922                   18,249                
St Dev 250                      574                      636                      230                      640                      
CV 0.02                     0.84                     0.05                     0.05                     0.04                     
Mean 12,658                 443                      13,101                 4,919                   18,019                
St Dev 252                      592                      667                      235                      739                      
CV 0.02                     1.34                     0.05                     0.05                     0.04                     
Mean 9,549                   1,746                   11,295                 6,928                   18,223                
St Dev 188                      1,117                   1,039                   121                      1,031                   
CV 0.02                     0.64                     0.09                     0.02                     0.06                     
Mean 9,732                   873                      10,604                 7,152                   17,756                
St Dev 153                      749                      760                      284                      882                      
CV 0.02                     0.86                     0.07                     0.04                     0.05                     
Mean 9,721                   877                      10,597                 7,111                   17,708                
St Dev 148                      743                      746                      270                      843                      
CV 0.02                     0.85                     0.07                     0.04                     0.05                     
Mean 9,645                   669                      10,314                 7,247                   17,561                
St Dev 121                      400                      401                      266                      557                      
CV 0.01                     0.60                     0.04                     0.04                     0.03                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 23,221                 1,821                   25,041                 3,355                   28,396                 
St Dev 901                      1,620                   1,410                   279                      1,488                   
CV 0.04                     0.89                     0.06                     0.08                     0.05                     
Mean 14,696                 1,046                   15,742                 6,650                   22,392                
St Dev 616                      1,234                   981                      555                      1,244                   
CV 0.04                     1.18                     0.06                     0.08                     0.06                     
Mean 16,938                 534                      17,473                 4,466                   21,939                
St Dev 528                      667                      855                      282                      851                      
CV 0.03                     1.25                     0.05                     0.06                     0.04                     
Mean 14,960                 257                      15,217                 4,727                   19,943                
St Dev 337                      576                      535                      222                      627                      
CV 0.02                     2.24                     0.04                     0.05                     0.03                     
Mean 15,020                 212                      15,232                 4,805                   20,037                
St Dev 308                      326                      410                      198                      478                      
CV 0.02                     1.54                     0.03                     0.04                     0.02                     
Mean 9,696                   1,405                   11,100                 7,041                   18,142                
St Dev 221                      961                      943                      69                        976                      
CV 0.02                     0.68                     0.08                     0.01                     0.05                     
Mean 9,920                   839                      10,759                 7,295                   18,054                
St Dev 160                      747                      768                      328                      803                      
CV 0.02                     0.89                     0.07                     0.04                     0.04                     
Mean 9,915                   926                      10,841                 7,290                   18,131                
St Dev 149                      761                      795                      329                      833                      
CV 0.02                     0.82                     0.07                     0.05                     0.05                     
Mean 9,679                   932                      10,611                 7,498                   18,108                
St Dev 175                      765                      729                      259                      673                      
CV 0.02                     0.82                     0.07                     0.03                     0.04                     
30 replication with common random numbers
Results in [$/week]
RDE
MUN
FTUI
FTRI
RDE+div
FTSR
RDE+FT_S
BENCH1
Strategy
FTRI
RDE+FT_S
RDE
Case 2
FTUI
FTSR
BENCH1
Strategy
Case 3
RTC
RTC
MUN
RDE+div
 
 
 
 150 
 
 
Table D- 2: Simulation Results: Set of Parameters 2 
TC= 33 [$/hr], h
i
= 50[$/week], ?
i
= 50[arrivals/day], ?
i
= 1 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 21,344                 181                      21,525                 1,408                   22,932                 
St Dev 461                      502                      544                      38                        536                      
CV 0.02                     2.78                     0.03                     0.03                     0.02                     
Mean 11,904                 1,748                   13,651                 2,476                   16,128                
St Dev 489                      1,579                   1,290                   186                      1,278                   
CV 0.04                     0.90                     0.09                     0.08                     0.08                     
Mean 14,906                 315                      15,220                 1,689                   16,910                
St Dev 477                      670                      775                      72                        770                      
CV 0.03                     2.13                     0.05                     0.04                     0.05                     
Mean 12,649                 187                      12,837                 1,890                   14,727                
St Dev 231                      282                      281                      72                        301                      
CV 0.02                     1.50                     0.02                     0.04                     0.02                     
Mean 12,635                 236                      12,871                 1,901                   14,772                
St Dev 248                      313                      312                      84                        293                      
CV 0.02                     1.33                     0.02                     0.04                     0.02                     
Mean 7,802                   1,141                   8,943                   3,299                   12,242                
St Dev 160                      726                      714                      18                        715                      
CV 0.02                     0.64                     0.08                     0.01                     0.06                     
Mean 7,736                   1,141                   8,877                   3,257                   12,134                
St Dev 138                      723                      684                      32                        689                      
CV 0.02                     0.63                     0.08                     0.01                     0.06                     
Mean 7,736                   1,141                   8,877                   3,257                   12,134                
St Dev 138                      723                      684                      32                        689                      
CV 0.02                     0.63                     0.08                     0.01                     0.06                     
Mean 7,809                   627                      8,436                   3,242                   11,678                
St Dev 141                      595                      581                      29                        592                      
CV 0.02                     0.95                     0.07                     0.01                     0.05                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 20,641                 1,886                   22,527                 1,920                   24,447                 
St Dev 699                      1,950                   1,512                   72                        1,535                   
CV 0.03                     1.03                     0.07                     0.04                     0.06                     
Mean 15,338                 2,274                   17,612                 2,632                   20,244                
St Dev 802                      2,434                   2,076                   189                      2,081                   
CV 0.05                     1.07                     0.12                     0.07                     0.10                     
Mean 17,971                 432                      18,402                 1,873                   20,275                
St Dev 703                      801                      869                      119                      891                      
CV 0.04                     1.86                     0.05                     0.06                     0.04                     
Mean 16,246                 282                      16,527                 1,980                   18,507                
St Dev 313                      330                      412                      124                      399                      
CV 0.02                     1.17                     0.02                     0.06                     0.02                     
Mean 16,212                 243                      16,455                 1,987                   18,442                
St Dev 297                      367                      415                      91                        403                      
CV 0.02                     1.51                     0.03                     0.05                     0.02                     
Mean 9,652                   1,545                   11,196                 3,300                   14,496                
St Dev 225                      1,245                   1,184                   26                        1,193                   
CV 0.02                     0.81                     0.11                     0.01                     0.08                     
Mean 9,580                   1,717                   11,297                 3,268                   14,565                
St Dev 211                      1,067                   1,061                   32                        1,059                   
CV 0.02                     0.62                     0.09                     0.01                     0.07                     
Mean 9,573                   1,940                   11,513                 3,265                   14,778                
St Dev 217                      1,582                   1,512                   42                        1,528                   
CV 0.02                     0.82                     0.13                     0.01                     0.10                     
Strategy
Case 0 (Symmetric case)
RDE
RDE+div
Case 1
BENCH1
FTUI
FTSR
BENCH1
RTC
FTUI
FTSR
RTC
FTRI
RDE+div
FTRI
RDE+FT_S
Strategy
MUN
MUN
RDE
 
 
 151 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 20,559                 787                      21,345                 1,300                   22,646                 
St Dev 418                      859                      835                      55                        848                      
CV 0.02                     1.09                     0.04                     0.04                     0.04                     
Mean 11,087                 985                      12,072                 2,576                   14,648                
St Dev 405                      1,006                   873                      177                      864                      
CV 0.04                     1.02                     0.07                     0.07                     0.06                     
Mean 13,720                 490                      14,210                 1,635                   15,845                
St Dev 348                      632                      671                      81                        707                      
CV 0.03                     1.29                     0.05                     0.05                     0.04                     
Mean 11,243                 518                      11,761                 1,903                   13,664                
St Dev 203                      611                      619                      81                        639                      
CV 0.02                     1.18                     0.05                     0.04                     0.05                     
Mean 11,275                 386                      11,661                 1,952                   13,613                
St Dev 193                      411                      439                      109                      452                      
CV 0.02                     1.07                     0.04                     0.06                     0.03                     
Mean 7,610                   1,847                   9,457                   3,298                   12,755                
St Dev 124                      1,140                   1,106                   25                        1,098                   
CV 0.02                     0.62                     0.12                     0.01                     0.09                     
Mean 7,551                   1,740                   9,291                   3,236                   12,526                
St Dev 116                      1,086                   1,071                   51                        1,071                   
CV 0.02                     0.62                     0.12                     0.02                     0.09                     
Mean 7,573                   1,713                   9,286                   3,244                   12,530                
St Dev 110                      1,042                   1,048                   43                        1,054                   
CV 0.01                     0.61                     0.11                     0.01                     0.08                     
Mean 7,554                   1,418                   8,973                   3,246                   12,219                
St Dev 125                      1,024                   972                      35                        965                      
CV 0.02                     0.72                     0.11                     0.01                     0.08                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 21,317                 493                      21,811                 1,389                   23,199                 
St Dev 482                      745                      843                      43                        842                      
CV 0.02                     1.51                     0.04                     0.03                     0.04                     
Mean 13,287                 909                      14,195                 2,316                   16,511                
St Dev 504                      1,452                   1,146                   181                      1,151                   
CV 0.04                     1.60                     0.08                     0.08                     0.07                     
Mean 15,735                 247                      15,981                 1,604                   17,585                
St Dev 464                      480                      666                      72                        697                      
CV 0.03                     1.95                     0.04                     0.04                     0.04                     
Mean 13,682                 363                      14,045                 1,751                   15,795                
St Dev 209                      501                      516                      75                        512                      
CV 0.02                     1.38                     0.04                     0.04                     0.03                     
Mean 13,766                 238                      14,004                 1,790                   15,794                
St Dev 205                      374                      410                      90                        426                      
CV 0.01                     1.57                     0.03                     0.05                     0.03                     
Mean 7,768                   1,382                   9,150                   3,300                   12,450                
St Dev 155                      1,026                   1,012                   18                        1,007                   
CV 0.02                     0.74                     0.11                     0.01                     0.08                     
Mean 7,725                   1,709                   9,433                   3,248                   12,682                
St Dev 115                      1,067                   1,032                   44                        1,050                   
CV 0.01                     0.62                     0.11                     0.01                     0.08                     
Mean 7,733                   1,462                   9,195                   3,256                   12,451                
St Dev 115                      1,071                   1,037                   46                        1,053                   
CV 0.01                     0.73                     0.11                     0.01                     0.08                     
30 replication with common random numbers
Results in [$/week]
Case 3
FTUI
RTC
RDE
RDE+div
RDE+div
RTC
MUN
Strategy
BENCH1
RDE
FTUI
FTSR
Strategy
BENCH1
FTRI
FTSR
MUN
FTRI
RDE+FT_S
Case 2
 
 
 152 
 
 
Table D- 3: Simulation Results: Set of Parameters 3 
TC= 300 [$/hr], h
i
= 50[$/week], ?
i
= 50[arrivals/day], ?
i
= 1 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 29,895                 488                      30,383                 8,441                   38,825                 
St Dev 642                      731                      761                      688                      1,156                   
CV 0.02                     1.50                     0.03                     0.08                     0.03                     
Mean 19,248                 829                      20,077                 15,671                 35,748                
St Dev 707                      1,386                   1,114                   1,459                   2,103                   
CV 0.04                     1.67                     0.06                     0.09                     0.06                     
Mean 20,780                 368                      21,147                 9,666                   30,814                
St Dev 628                      577                      743                      749                      1,050                   
CV 0.03                     1.57                     0.04                     0.08                     0.03                     
Mean 19,086                 448                      19,534                 10,027                 29,562                
St Dev 470                      547                      517                      682                      931                      
CV 0.02                     1.22                     0.03                     0.07                     0.03                     
Mean 19,167                 452                      19,619                 10,063                 29,682                
St Dev 437                      483                      538                      709                      914                      
CV 0.02                     1.07                     0.03                     0.07                     0.03                     
Mean 15,416                 1,549                   16,965                 12,329                 29,293                
St Dev 415                      990                      941                      385                      1,067                   
CV 0.03                     0.64                     0.06                     0.03                     0.04                     
Mean 15,565                 709                      16,274                 12,506                 28,780                
St Dev 365                      518                      639                      831                      1,226                   
CV 0.02                     0.73                     0.04                     0.07                     0.04                     
Mean 15,565                 709                      16,274                 12,506                 28,780                
St Dev 365                      518                      639                      831                      1,226                   
CV 0.02                     0.73                     0.04                     0.07                     0.04                     
Mean 15,828                 702                      16,530                 11,829                 28,359                
St Dev 391                      766                      771                      732                      1,155                   
CV 0.02                     1.09                     0.05                     0.06                     0.04                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 30,680                 1,703                   32,383                 10,581                 42,965                 
St Dev 980                      1,742                   1,813                   845                      2,226                   
CV 0.03                     1.02                     0.06                     0.08                     0.05                     
Mean 21,104                 1,529                   22,633                 16,244                 38,877                
St Dev 1,063                   1,710                   1,426                   1,885                   2,254                   
CV 0.05                     1.12                     0.06                     0.12                     0.06                     
Mean 24,095                 609                      24,705                 11,183                 35,888                
St Dev 874                      934                      1,132                   897                      1,555                   
CV 0.04                     1.53                     0.05                     0.08                     0.04                     
Mean 22,833                 529                      23,362                 11,523                 34,886                
St Dev 536                      624                      800                      1,062                   1,364                   
CV 0.02                     1.18                     0.03                     0.09                     0.04                     
Mean 22,800                 505                      23,305                 11,578                 34,883                
St Dev 738                      682                      966                      1,012                   1,448                   
CV 0.03                     1.35                     0.04                     0.09                     0.04                     
Mean 17,195                 2,197                   19,392                 14,106                 33,499                
St Dev 505                      1,321                   1,280                   1,302                   1,966                   
CV 0.03                     0.60                     0.07                     0.09                     0.06                     
Mean 17,715                 725                      18,440                 14,691                 33,130                
St Dev 686                      606                      934                      1,198                   1,705                   
CV 0.04                     0.84                     0.05                     0.08                     0.05                     
Mean 17,715                 725                      18,440                 14,691                 33,130                
St Dev 686                      606                      934                      1,198                   1,705                   
CV 0.04                     0.84                     0.05                     0.08                     0.05                     
Mean 17,336                 1,014                   18,350                 14,356                 32,706                
St Dev 405                      1,077                   1,026                   839                      1,269                   
CV 0.02                     1.06                     0.06                     0.06                     0.04                     
Strategy
Case 0 (Symmetric case)
FTRI
RDE+FT_S
RDE
RDE+div
FTUI
FTSR
RDE+FT_S
BENCH1
Case 1
Strategy
BENCH1
RDE
RDE+div
FTRI
RTC
MUN
RTC
MUN
FTUI
FTSR
 
 
 153 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 28,522                 727                      29,250                 7,470                   36,719                 
St Dev 661                      1,167                   1,231                   716                      1,562                   
CV 0.02                     1.60                     0.04                     0.10                     0.04                     
Mean 16,594                 4,798                   21,393                 15,627                 37,020                
St Dev 781                      3,346                   2,759                   2,391                   2,722                   
CV 0.05                     0.70                     0.13                     0.15                     0.07                     
Mean 19,246                 771                      20,017                 8,978                   28,995                
St Dev 655                      1,172                   1,023                   796                      1,247                   
CV 0.03                     1.52                     0.05                     0.09                     0.04                     
Mean 17,627                 553                      18,180                 9,578                   27,758                
St Dev 428                      765                      888                      819                      1,218                   
CV 0.02                     1.38                     0.05                     0.09                     0.04                     
Mean 17,561                 469                      18,031                 9,566                   27,597                
St Dev 460                      417                      537                      663                      869                      
CV 0.03                     0.89                     0.03                     0.07                     0.03                     
Mean 15,204                 2,134                   17,338                 12,178                 29,516                
St Dev 340                      1,184                   976                      530                      1,163                   
CV 0.02                     0.55                     0.06                     0.04                     0.04                     
Mean 15,473                 809                      16,282                 12,565                 28,846                
St Dev 351                      911                      1,054                   787                      1,620                   
CV 0.02                     1.13                     0.06                     0.06                     0.06                     
Mean 15,459                 718                      16,177                 12,521                 28,698                
St Dev 390                      896                      1,044                   846                      1,621                   
CV 0.03                     1.25                     0.06                     0.07                     0.06                     
Mean 15,373                 604                      15,977                 11,189                 27,167                
St Dev 270                      554                      549                      627                      691                      
CV 0.02                     0.92                     0.03                     0.06                     0.03                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 29,530                 1,009                   30,539                 8,368                   38,907                 
St Dev 905                      1,053                   1,118                   745                      1,399                   
CV 0.03                     1.04                     0.04                     0.09                     0.04                     
Mean 20,035                 449                      20,484                 14,654                 35,138                
St Dev 818                      555                      949                      1,645                   2,134                   
CV 0.04                     1.24                     0.05                     0.11                     0.06                     
Mean 22,267                 372                      22,639                 9,184                   31,823                
St Dev 636                      444                      689                      821                      1,207                   
CV 0.03                     1.19                     0.03                     0.09                     0.04                     
Mean 20,555                 561                      21,116                 9,563                   30,678                
St Dev 416                      806                      822                      832                      1,286                   
CV 0.02                     1.44                     0.04                     0.09                     0.04                     
Mean 20,594                 573                      21,167                 9,585                   30,751                
St Dev 533                      592                      673                      803                      1,089                   
CV 0.03                     1.03                     0.03                     0.08                     0.04                     
Mean 15,115                 1,938                   17,053                 12,309                 29,362                
St Dev 356                      1,619                   1,498                   482                      1,548                   
CV 0.02                     0.84                     0.09                     0.04                     0.05                     
Mean 15,431                 963                      16,394                 12,672                 29,067                
St Dev 358                      640                      696                      866                      1,217                   
CV 0.02                     0.66                     0.04                     0.07                     0.04                     
Mean 15,431                 963                      16,394                 12,672                 29,067                
St Dev 358                      640                      696                      866                      1,217                   
CV 0.02                     0.66                     0.04                     0.07                     0.04                     
Mean 15,248                 1,030                   16,279                 12,018                 28,296                
St Dev 389                      935                      1,039                   649                      1,143                   
CV 0.03                     0.91                     0.06                     0.05                     0.04                     
30 replication with common random numbers
Results in [$/week]
Case 3
RDE+FT_S
RDE+div
RTC
RDE
MUN
Case 2
Strategy
BENCH1
FTRI
FTUI
FTSR
Strategy
BENCH1
RTC
RDE+div
RDE+FT_S
FTUI
FTSR
MUN
FTRI
RDE
 
 
 
 154 
 
 
Table D- 4: Simulation Results: Set of Parameters 4 
TC= 100 [$/hr], h
i
= 50[$/week], ?
i
= 10.5 [arrivals/day], ?
i
= 4.8 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 26,668                 1,924                   28,592                 3,293                   31,885                 
St Dev 833                      1,779                   1,870                   238                      1,922                   
CV 0.03                     0.92                     0.07                     0.07                     0.06                     
Mean 16,535                 923                      17,458                 6,903                   24,360                
St Dev 531                      996                      1,118                   514                      1,339                   
CV 0.03                     1.08                     0.06                     0.07                     0.05                     
Mean 20,424                 863                      21,287                 4,757                   26,044                
St Dev 636                      989                      1,109                   331                      1,169                   
CV 0.03                     1.15                     0.05                     0.07                     0.04                     
Mean 17,050                 1,018                   18,068                 5,197                   23,265                
St Dev 355                      855                      798                      303                      820                      
CV 0.02                     0.84                     0.04                     0.06                     0.04                     
Mean 16,934                 727                      17,660                 5,326                   22,987                
St Dev 436                      804                      769                      333                      831                      
CV 0.03                     1.11                     0.04                     0.06                     0.04                     
Mean 12,128                 3,471                   15,599                 7,684                   23,283                
St Dev 332                      2,209                   2,134                   71                        2,138                   
CV 0.03                     0.64                     0.14                     0.01                     0.09                     
Mean 12,348                 2,139                   14,488                 8,030                   22,518                
St Dev 197                      1,472                   1,484                   478                      1,569                   
CV 0.02                     0.69                     0.10                     0.06                     0.07                     
Mean 12,328                 2,104                   14,432                 7,909                   22,341                
St Dev 218                      1,820                   1,779                   400                      1,756                   
CV 0.02                     0.87                     0.12                     0.05                     0.08                     
Mean 12,169                 1,533                   13,702                 7,919                   21,621                
St Dev 161                      1,112                   1,124                   380                      1,013                   
CV 0.01                     0.73                     0.08                     0.05                     0.05                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 26,615                 2,797                   29,412                 4,326                   33,738                 
St Dev 1,055                   2,524                   2,131                   401                      2,200                   
CV 0.04                     0.90                     0.07                     0.09                     0.07                     
Mean 19,530                 1,282                   20,812                 7,222                   28,034                
St Dev 957                      1,953                   1,806                   543                      1,972                   
CV 0.05                     1.52                     0.09                     0.08                     0.07                     
Mean 23,530                 524                      24,054                 5,388                   29,442                
St Dev 908                      839                      1,185                   317                      1,225                   
CV 0.04                     1.60                     0.05                     0.06                     0.04                     
Mean 20,716                 649                      21,365                 5,867                   27,231                
St Dev 430                      690                      701                      480                      882                      
CV 0.02                     1.06                     0.03                     0.08                     0.03                     
Mean 20,810                 307                      21,118                 5,872                   26,990                
St Dev 428                      555                      611                      431                      686                      
CV 0.02                     1.81                     0.03                     0.07                     0.03                     
Mean 13,594                 3,147                   16,741                 8,857                   25,597                
St Dev 317                      1,654                   1,596                   85                        1,562                   
CV 0.02                     0.53                     0.10                     0.01                     0.06                     
Mean 13,717                 2,377                   16,094                 9,038                   25,132                
St Dev 249                      1,286                   1,181                   324                      1,182                   
CV 0.02                     0.54                     0.07                     0.04                     0.05                     
Mean 13,730                 2,077                   15,807                 8,976                   24,783                
St Dev 239                      1,440                   1,402                   353                      1,418                   
CV 0.02                     0.69                     0.09                     0.04                     0.06                     
Mean 13,618                 1,499                   15,117                 9,270                   24,387                
St Dev 284                      1,073                   1,156                   240                      1,173                   
CV 0.02                     0.72                     0.08                     0.03                     0.05                     
Strategy
Case 0 (Symmetric case)
RDE+FT_S
FTRI
BENCH1
RDE
RDE+div
FTRI
Case 1
RDE+FT_S
Strategy
BENCH1
MUN
RTC
FTUI
FTSR
RDE
RDE+div
MUN
RTC
FTUI
FTSR
 
 
 155 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 26,427                 1,817                   28,244                 2,953                   31,196                 
St Dev 1,117                   1,751                   1,602                   254                      1,662                   
CV 0.04                     0.96                     0.06                     0.09                     0.05                     
Mean 15,341                 1,651                   16,992                 7,352                   24,343                
St Dev 540                      1,621                   1,570                   563                      1,590                   
CV 0.04                     0.98                     0.09                     0.08                     0.07                     
Mean 19,364                 545                      19,909                 4,340                   24,249                
St Dev 751                      596                      982                      287                      989                      
CV 0.04                     1.09                     0.05                     0.07                     0.04                     
Mean 15,385                 699                      16,084                 4,976                   21,060                
St Dev 273                      641                      617                      311                      673                      
CV 0.02                     0.92                     0.04                     0.06                     0.03                     
Mean 15,458                 812                      16,270                 5,125                   21,395                
St Dev 269                      784                      806                      310                      817                      
CV 0.02                     0.96                     0.05                     0.06                     0.04                     
Mean 12,047                 2,750                   14,797                 7,586                   22,383                
St Dev 263                      1,477                   1,427                   65                        1,437                   
CV 0.02                     0.54                     0.10                     0.01                     0.06                     
Mean 12,114                 1,679                   13,793                 7,670                   21,462                
St Dev 214                      1,108                   1,153                   545                      1,139                   
CV 0.02                     0.66                     0.08                     0.07                     0.05                     
Mean 12,205                 1,386                   13,591                 7,474                   21,064                
St Dev 234                      1,054                   1,089                   497                      1,089                   
CV 0.02                     0.76                     0.08                     0.07                     0.05                     
Mean 12,062                 1,240                   13,302                 7,800                   21,102                
St Dev 172                      1,200                   1,221                   378                      1,287                   
CV 0.01                     0.97                     0.09                     0.05                     0.06                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 27,080                 2,313                   29,393                 3,318                   32,711                 
St Dev 849                      2,482                   2,267                   255                      2,309                   
CV 0.03                     1.07                     0.08                     0.08                     0.07                     
Mean 17,761                 860                      18,621                 6,730                   25,351                
St Dev 682                      965                      1,089                   550                      1,238                   
CV 0.04                     1.12                     0.06                     0.08                     0.05                     
Mean 21,753                 1,165                   22,918                 4,513                   27,431                
St Dev 943                      1,582                   1,557                   294                      1,565                   
CV 0.04                     1.36                     0.07                     0.07                     0.06                     
Mean 18,207                 911                      19,118                 4,893                   24,011                
St Dev 439                      1,205                   1,020                   305                      1,031                   
CV 0.02                     1.32                     0.05                     0.06                     0.04                     
Mean 18,213                 578                      18,791                 4,949                   23,740                
St Dev 510                      672                      627                      322                      764                      
CV 0.03                     1.16                     0.03                     0.07                     0.03                     
Mean 12,217                 3,094                   15,311                 7,701                   23,012                
St Dev 288                      2,051                   2,019                   78                        2,033                   
CV 0.02                     0.66                     0.13                     0.01                     0.09                     
Mean 12,343                 1,825                   14,168                 7,948                   22,116                
St Dev 230                      1,194                   1,265                   397                      1,304                   
CV 0.02                     0.65                     0.09                     0.05                     0.06                     
Mean 12,327                 1,670                   13,997                 7,806                   21,803                
St Dev 206                      1,241                   1,255                   317                      1,298                   
CV 0.02                     0.74                     0.09                     0.04                     0.06                     
Mean 12,117                 1,467                   13,584                 8,262                   21,846                
St Dev 213                      997                      890                      349                      941                      
CV 0.02                     0.68                     0.07                     0.04                     0.04                     
30 replication with common random numbers
Results in [$/week]
Strategy
Case 3
RDE+div
Strategy
BENCH1
RDE
MUN
RTC
Case 2
FTRI
RDE+FT_S
BENCH1
FTUI
FTSR
RDE
RDE+div
FTRI
RDE+FT_S
RTC
FTSR
MUN
FTUI
 
 
 156 
 
 
Table D- 5: Simulation Results: Set of Parameters 5 
TC= 100 [$/hr], h
i
= 50[$/week], ?
i
= 4.35 [arrivals/day], ?
i
= 11.5 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 30,862                 2,273                   33,135                 3,105                   36,241                 
St Dev 1,292                   2,761                   2,590                   243                      2,615                   
CV 0.04                     1.21                     0.08                     0.08                     0.07                     
Mean 19,999                 1,509                   21,508                 7,251                   28,759                
St Dev 1,043                   1,693                   1,803                   598                      2,105                   
CV 0.05                     1.12                     0.08                     0.08                     0.07                     
Mean 25,127                 2,488                   27,615                 4,981                   32,596                
St Dev 989                      2,867                   2,857                   437                      2,860                   
CV 0.04                     1.15                     0.10                     0.09                     0.09                     
Mean 19,988                 1,895                   21,883                 5,358                   27,241                
St Dev 419                      1,540                   1,473                   436                      1,615                   
CV 0.02                     0.81                     0.07                     0.08                     0.06                     
Mean 20,272                 1,383                   21,654                 5,558                   27,212                
St Dev 485                      1,386                   1,370                   381                      1,499                   
CV 0.02                     1.00                     0.06                     0.07                     0.06                     
Mean 14,743                 5,912                   20,655                 8,473                   29,128                
St Dev 399                      3,563                   3,391                   44                        3,402                   
CV 0.03                     0.60                     0.16                     0.01                     0.12                     
Mean 14,877                 5,311                   20,188                 8,462                   28,650                
St Dev 335                      3,080                   3,135                   504                      3,199                   
CV 0.02                     0.58                     0.16                     0.06                     0.11                     
Mean 14,911                 4,769                   19,680                 8,287                   27,968                
St Dev 333                      2,890                   2,917                   521                      3,016                   
CV 0.02                     0.61                     0.15                     0.06                     0.11                     
Mean 14,923                 2,847                   17,770                 8,240                   26,010                
St Dev 264                      2,001                   2,041                   419                      2,038                   
CV 0.02                     0.70                     0.11                     0.05                     0.08                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 30,842                 4,103                   34,945                 4,083                   39,028                 
St Dev 1,354                   4,971                   4,560                   375                      4,586                   
CV 0.04                     1.21                     0.13                     0.09                     0.12                     
Mean 23,346                 1,259                   24,606                 7,219                   31,825                
St Dev 1,117                   1,514                   1,478                   853                      2,139                   
CV 0.05                     1.20                     0.06                     0.12                     0.07                     
Mean 28,961                 1,647                   30,607                 5,616                   36,223                
St Dev 1,605                   1,640                   2,242                   472                      2,197                   
CV 0.06                     1.00                     0.07                     0.08                     0.06                     
Mean 24,254                 1,248                   25,502                 6,386                   31,888                
St Dev 637                      1,519                   1,367                   571                      1,487                   
CV 0.03                     1.22                     0.05                     0.09                     0.05                     
Mean 24,592                 959                      25,551                 6,327                   31,878                
St Dev 709                      1,350                   1,375                   584                      1,455                   
CV 0.03                     1.41                     0.05                     0.09                     0.05                     
Mean 16,810                 4,568                   21,378                 9,969                   31,347                
St Dev 548                      3,081                   3,014                   74                        3,015                   
CV 0.03                     0.67                     0.14                     0.01                     0.10                     
Mean 16,581                 5,129                   21,711                 9,422                   31,132                
St Dev 399                      3,255                   3,249                   455                      3,300                   
CV 0.02                     0.63                     0.15                     0.05                     0.11                     
Mean 16,700                 5,142                   21,842                 9,205                   31,047                
St Dev 412                      3,053                   3,027                   512                      3,263                   
CV 0.02                     0.59                     0.14                     0.06                     0.11                     
Mean 16,871                 2,762                   19,633                 9,441                   29,074                
St Dev 363                      2,495                   2,453                   289                      2,558                   
CV 0.02                     0.90                     0.12                     0.03                     0.09                     
FTUI
FTSR
MUN
RDE+FT_S
RDE+div
FTRI
RTC
RDE
Strategy
Case 0 (Symmetric case)
BENCH1
RDE
RTC
FTSR
MUN
RDE+div
Case 1
FTRI
RDE+FT_S
FTUI
Strategy
BENCH1
 
 
 157 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 30,787                 2,473                   33,260                 2,919                   36,179                 
St Dev 1,061                   2,254                   2,035                   274                      2,165                   
CV 0.03                     0.91                     0.06                     0.09                     0.06                     
Mean 18,397                 2,570                   20,967                 7,416                   28,384                
St Dev 599                      2,245                   2,082                   770                      2,188                   
CV 0.03                     0.87                     0.10                     0.10                     0.08                     
Mean 24,345                 1,885                   26,229                 4,363                   30,593                
St Dev 963                      2,152                   2,193                   482                      2,229                   
CV 0.04                     1.14                     0.08                     0.11                     0.07                     
Mean 18,311                 2,872                   21,182                 5,361                   26,543                
St Dev 418                      2,646                   2,545                   501                      2,645                   
CV 0.02                     0.92                     0.12                     0.09                     0.10                     
Mean 18,589                 1,596                   20,184                 5,312                   25,496                
St Dev 344                      1,633                   1,741                   501                      1,848                   
CV 0.02                     1.02                     0.09                     0.09                     0.07                     
Mean 14,752                 4,408                   19,160                 8,381                   27,541                
St Dev 350                      3,131                   2,988                   0                          2,988                   
CV 0.02                     0.71                     0.16                     0.00                     0.11                     
Mean 14,665                 3,653                   18,317                 8,136                   26,453                
St Dev 244                      2,408                   2,311                   597                      2,446                   
CV 0.02                     0.66                     0.13                     0.07                     0.09                     
Mean 14,742                 3,923                   18,665                 7,352                   26,017                
St Dev 247                      2,908                   2,907                   578                      2,944                   
CV 0.02                     0.74                     0.16                     0.08                     0.11                     
Mean 14,738                 2,968                   17,706                 8,081                   25,787                
St Dev 248                      1,729                   1,624                   478                      1,679                   
CV 0.02                     0.58                     0.09                     0.06                     0.07                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 31,057                 3,478                   34,536                 3,098                   37,634                 
St Dev 1,391                   3,501                   3,377                   252                      3,371                   
CV 0.04                     1.01                     0.10                     0.08                     0.09                     
Mean 21,483                 999                      22,482                 6,897                   29,378                
St Dev 1,187                   1,158                   1,457                   848                      2,098                   
CV 0.06                     1.16                     0.06                     0.12                     0.07                     
Mean 26,585                 1,774                   28,359                 4,703                   33,062                
St Dev 1,204                   2,158                   2,348                   303                      2,343                   
CV 0.05                     1.22                     0.08                     0.06                     0.07                     
Mean 21,582                 1,767                   23,348                 5,591                   28,940                
St Dev 494                      1,567                   1,615                   551                      1,651                   
CV 0.02                     0.89                     0.07                     0.10                     0.06                     
Mean 21,587                 905                      22,492                 5,639                   28,131                
St Dev 525                      1,315                   1,209                   544                      1,384                   
CV 0.02                     1.45                     0.05                     0.10                     0.05                     
Mean 14,928                 4,988                   19,916                 8,495                   28,411                
St Dev 375                      3,258                   3,147                   64                        3,152                   
CV 0.03                     0.65                     0.16                     0.01                     0.11                     
Mean 14,920                 3,592                   18,512                 8,431                   26,943                
St Dev 261                      2,479                   2,423                   590                      2,671                   
CV 0.02                     0.69                     0.13                     0.07                     0.10                     
Mean 14,987                 3,168                   18,155                 7,986                   26,141                
St Dev 263                      2,473                   2,491                   558                      2,569                   
CV 0.02                     0.78                     0.14                     0.07                     0.10                     
Mean 14,879                 3,145                   18,025                 8,984                   27,009                
St Dev 365                      2,486                   2,367                   387                      2,371                   
CV 0.02                     0.79                     0.13                     0.04                     0.09                     
30 replication with common random numbers
Results in [$/week]
Strategy
Case 3
RTC
FTUI
FTSR
FTRI
Case 2
BENCH1
RDE
RDE+div
MUN
RDE+FT_S
RDE+FT_S
BENCH1
Strategy
FTUI
FTSR
MUN
FTRI
RTC
RDE
RDE+div
 
 
 158 
 
 
Table D- 6:  Simulation Results: Set of Parameters 6 
TC= 100 [$/hr], h
i
= 50[$/week], ?
i
= 2.4 [arrivals/day], ?
i
= 20.8 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 34,425                 4,670                   39,096                 3,044                   42,140                 
St Dev 1,691                   4,308                   3,834                   325                      3,758                   
CV 0.05                     0.92                     0.10                     0.11                     0.09                     
Mean 23,539                 1,954                   25,493                 7,302                   32,795                
St Dev 1,197                   1,728                   1,783                   881                      1,941                   
CV 0.05                     0.88                     0.07                     0.12                     0.06                     
Mean 29,653                 4,261                   33,914                 5,281                   39,195                
St Dev 1,614                   3,799                   3,574                   523                      3,630                   
CV 0.05                     0.89                     0.11                     0.10                     0.09                     
Mean 23,151                 4,040                   27,190                 5,717                   32,907                
St Dev 633                      2,860                   2,841                   479                      2,775                   
CV 0.03                     0.71                     0.10                     0.08                     0.08                     
Mean 23,482                 2,630                   26,112                 5,793                   31,906                
St Dev 524                      2,704                   2,699                   417                      2,749                   
CV 0.02                     1.03                     0.10                     0.07                     0.09                     
Mean 17,395                 10,358                 27,753                 9,330                   37,083                
St Dev 547                      4,903                   4,723                   49                        4,719                   
CV 0.03                     0.47                     0.17                     0.01                     0.13                     
Mean 17,314                 9,936                   27,250                 8,888                   36,137                
St Dev 381                      4,878                   4,781                   489                      4,822                   
CV 0.02                     0.49                     0.18                     0.05                     0.13                     
Mean 17,550                 8,015                   25,565                 8,568                   34,133                
St Dev 480                      4,378                   4,287                   565                      4,244                   
CV 0.03                     0.55                     0.17                     0.07                     0.12                     
Mean 17,574                 6,129                   23,703                 8,433                   32,136                
St Dev 371                      4,033                   4,017                   390                      3,976                   
CV 0.02                     0.66                     0.17                     0.05                     0.12                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 34,452                 5,503                   39,955                 4,050                   44,005                 
St Dev 1,673                   4,789                   4,108                   414                      4,211                   
CV 0.05                     0.87                     0.10                     0.10                     0.10                     
Mean 26,848                 2,233                   29,081                 7,225                   36,306                
St Dev 1,119                   2,763                   2,382                   776                      2,527                   
CV 0.04                     1.24                     0.08                     0.11                     0.07                     
Mean 33,758                 2,504                   36,262                 5,824                   42,086                
St Dev 1,463                   3,145                   3,157                   479                      3,215                   
CV 0.04                     1.26                     0.09                     0.08                     0.08                     
Mean 27,965                 3,227                   31,192                 7,002                   38,194                
St Dev 841                      2,572                   2,350                   560                      2,517                   
CV 0.03                     0.80                     0.08                     0.08                     0.07                     
Mean 28,194                 1,324                   29,518                 6,785                   36,303                
St Dev 833                      1,997                   2,050                   720                      2,134                   
CV 0.03                     1.51                     0.07                     0.11                     0.06                     
Mean 20,375                 7,116                   27,491                 9,998                   37,490                
St Dev 669                      3,654                   3,461                   127                      3,494                   
CV 0.03                     0.51                     0.13                     0.01                     0.09                     
Mean 20,072                 7,259                   27,332                 9,303                   36,635                
St Dev 648                      4,173                   4,141                   388                      4,145                   
CV 0.03                     0.57                     0.15                     0.04                     0.11                     
Mean 20,161                 6,841                   27,002                 8,939                   35,941                
St Dev 725                      4,156                   4,052                   560                      4,053                   
CV 0.04                     0.61                     0.15                     0.06                     0.11                     
Mean 20,618                 4,295                   24,912                 9,440                   34,352                
St Dev 455                      3,310                   3,323                   249                      3,375                   
CV 0.02                     0.77                     0.13                     0.03                     0.10                     
Strategy
Case 0 (Symmetric case)
BENCH1
RDE
RTC
MUN
RDE+div
FTRI
RDE+div
RTC
Strategy
FTUI
FTSR
RDE+FT_S
MUN
Case 1
BENCH1
RDE
FTRI
RDE+FT_S
FTUI
FTSR
 
 
 159 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 34,610                 3,952                   38,563                 2,734                   41,297                 
St Dev 1,464                   2,876                   2,592                   331                      2,660                   
CV 0.04                     0.73                     0.07                     0.12                     0.06                     
Mean 21,356                 3,562                   24,918                 7,523                   32,441                
St Dev 841                      2,225                   1,930                   857                      1,944                   
CV 0.04                     0.62                     0.08                     0.11                     0.06                     
Mean 28,890                 1,965                   30,855                 4,419                   35,274                
St Dev 1,271                   2,334                   2,813                   498                      2,829                   
CV 0.04                     1.19                     0.09                     0.11                     0.08                     
Mean 21,304                 4,612                   25,916                 5,247                   31,163                
St Dev 587                      3,785                   3,640                   548                      3,844                   
CV 0.03                     0.82                     0.14                     0.10                     0.12                     
Mean 21,667                 2,631                   24,298                 5,800                   30,098                
St Dev 453                      2,452                   2,386                   646                      2,430                   
CV 0.02                     0.93                     0.10                     0.11                     0.08                     
Mean 17,423                 6,490                   23,913                 9,197                   33,110                
St Dev 540                      3,432                   3,233                   56                        3,225                   
CV 0.03                     0.53                     0.14                     0.01                     0.10                     
Mean 17,232                 7,102                   24,334                 8,290                   32,624                
St Dev 368                      3,794                   3,665                   685                      3,839                   
CV 0.02                     0.53                     0.15                     0.08                     0.12                     
Mean 17,452                 6,159                   23,612                 7,353                   30,964                
St Dev 388                      3,829                   3,662                   573                      3,612                   
CV 0.02                     0.62                     0.16                     0.08                     0.12                     
Mean 17,508                 5,770                   23,279                 8,242                   31,521                
St Dev 337                      2,931                   2,820                   430                      2,877                   
CV 0.02                     0.51                     0.12                     0.05                     0.09                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 35,055                 3,169                   38,224                 3,119                   41,343                 
St Dev 1,409                   2,945                   3,135                   290                      3,112                   
CV 0.04                     0.93                     0.08                     0.09                     0.08                     
Mean 25,234                 1,656                   26,891                 7,029                   33,920                
St Dev 1,309                   1,979                   1,978                   789                      2,347                   
CV 0.05                     1.19                     0.07                     0.11                     0.07                     
Mean 31,198                 2,420                   33,618                 4,926                   38,544                
St Dev 1,359                   2,973                   2,462                   423                      2,527                   
CV 0.04                     1.23                     0.07                     0.09                     0.07                     
Mean 24,729                 4,071                   28,800                 6,065                   34,865                
St Dev 762                      3,254                   2,833                   634                      2,964                   
CV 0.03                     0.80                     0.10                     0.10                     0.09                     
Mean 25,215                 2,420                   27,635                 6,213                   33,848                
St Dev 609                      2,029                   1,893                   642                      1,937                   
CV 0.02                     0.84                     0.07                     0.10                     0.06                     
Mean 17,498                 7,497                   24,995                 9,374                   34,369                
St Dev 509                      3,449                   3,249                   44                        3,262                   
CV 0.03                     0.46                     0.13                     0.00                     0.09                     
Mean 17,321                 7,267                   24,588                 8,688                   33,276                
St Dev 309                      3,887                   3,871                   717                      4,118                   
CV 0.02                     0.53                     0.16                     0.08                     0.12                     
Mean 17,648                 5,909                   23,557                 7,886                   31,443                
St Dev 309                      3,664                   3,672                   807                      3,922                   
CV 0.02                     0.62                     0.16                     0.10                     0.12                     
Mean 17,465                 4,871                   22,336                 9,068                   31,404                
St Dev 388                      2,871                   2,712                   408                      2,787                   
CV 0.02                     0.59                     0.12                     0.05                     0.09                     
30 replication with common random numbers
Results in [$/week]
FTUI
RDE+FT_S
Strategy
Strategy
Case 2
Case 3
BENCH1
RDE
MUN
RDE+FT_S
FTUI
FTSR
BENCH1
FTSR
MUN
RDE
RDE+div
FTRI
RTC
FTRI
RDE+div
RTC
 
 
 160 
 
 
Table D- 7: Simulation Results: Set of Parameters 7 
TC= 100 [$/hr], h
i
= 5[$/week], ?
i
= 50[arrivals/day], ?
i
= 1 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,178                   440                      2,618                   4,158                   6,776                   
St Dev 41                        605                      594                      173                      588                      
CV 0.02                     1.38                     0.23                     0.04                     0.09                     
Mean 2,066                   762                      2,828                   4,648                   7,476                  
St Dev 75                        1,357                   1,325                   484                      1,355                   
CV 0.04                     1.78                     0.47                     0.10                     0.18                     
Mean 2,153                   183                      2,336                   3,215                   5,551                  
St Dev 58                        350                      351                      172                      414                      
CV 0.03                     1.92                     0.15                     0.05                     0.07                     
Mean 2,098                   220                      2,318                   3,134                   5,452                  
St Dev 46                        246                      232                      211                      328                      
CV 0.02                     1.12                     0.10                     0.07                     0.06                     
Mean 2,104                   93                        2,197                   3,183                   5,380                  
St Dev 42                        227                      233                      177                      309                      
CV 0.02                     2.44                     0.11                     0.06                     0.06                     
Mean 2,226                   119                      2,345                   3,402                   5,747                  
St Dev 54                        350                      349                      383                      592                      
CV 0.02                     2.94                     0.15                     0.11                     0.10                     
Mean 2,118                   159                      2,277                   3,135                   5,412                  
St Dev 47                        337                      334                      243                      447                      
CV 0.02                     2.12                     0.15                     0.08                     0.08                     
Mean 2,118                   159                      2,277                   3,135                   5,412                  
St Dev 47                        337                      334                      243                      447                      
CV 0.02                     2.12                     0.15                     0.08                     0.08                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,004                   2,128                   4,132                   5,872                   10,004                 
St Dev 67                        1,633                   1,590                   196                      1,643                   
CV 0.03                     0.77                     0.38                     0.03                     0.16                     
Mean 2,024                   1,299                   3,323                   6,397                   9,720                  
St Dev 87                        1,892                   1,845                   680                      1,800                   
CV 0.04                     1.46                     0.56                     0.11                     0.19                     
Mean 2,135                   177                      2,311                   4,360                   6,671                  
St Dev 74                        341                      341                      306                      548                      
CV 0.03                     1.93                     0.15                     0.07                     0.08                     
Mean 2,098                   307                      2,405                   4,208                   6,613                  
St Dev 44                        479                      467                      334                      606                      
CV 0.02                     1.56                     0.19                     0.08                     0.09                     
Mean 2,087                   184                      2,271                   4,286                   6,557                  
St Dev 45                        292                      286                      339                      508                      
CV 0.02                     1.59                     0.13                     0.08                     0.08                     
Mean 2,202                   226                      2,429                   4,525                   6,953                  
St Dev 68                        512                      521                      423                      763                      
CV 0.03                     2.26                     0.21                     0.09                     0.11                     
Mean 2,132                   273                      2,405                   4,303                   6,708                  
St Dev 66                        398                      413                      390                      654                      
CV 0.03                     1.46                     0.17                     0.09                     0.10                     
Mean 2,132                   273                      2,405                   4,303                   6,708                  
St Dev 66                        398                      413                      390                      654                      
CV 0.03                     1.46                     0.17                     0.09                     0.10                     
FTUI
FTSR
RTC
BENCH1
MUN
Strategy
Case 1
RDE
RDE+div
FTRI
FTUI
FTSR
BENCH1
RDE
RDE+div
FTRI
MUN
Case 0 (Symmetric case)
Strategy
RTC
 
 
 161 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,196                   603                      2,799                   3,931                   6,731                   
St Dev 56                        781                      768                      200                      753                      
CV 0.03                     1.30                     0.27                     0.05                     0.11                     
Mean 1,991                   429                      2,420                   4,729                   7,149                  
St Dev 63                        648                      644                      490                      813                      
CV 0.03                     1.51                     0.27                     0.10                     0.11                     
Mean 2,056                   444                      2,500                   3,059                   5,559                  
St Dev 61                        476                      442                      249                      472                      
CV 0.03                     1.07                     0.18                     0.08                     0.08                     
Mean 1,955                   323                      2,278                   3,062                   5,339                  
St Dev 41                        650                      647                      196                      718                      
CV 0.02                     2.02                     0.28                     0.06                     0.13                     
Mean 1,945                   175                      2,119                   3,071                   5,191                  
St Dev 42                        266                      273                      236                      394                      
CV 0.02                     1.53                     0.13                     0.08                     0.08                     
Mean 2,201                   320                      2,521                   3,287                   5,808                  
St Dev 45                        571                      567                      365                      746                      
CV 0.02                     1.79                     0.23                     0.11                     0.13                     
Mean 2,104                   420                      2,524                   3,061                   5,584                  
St Dev 55                        472                      458                      277                      563                      
CV 0.03                     1.13                     0.18                     0.09                     0.10                     
Mean 2,104                   420                      2,524                   3,061                   5,584                  
St Dev 55                        472                      458                      277                      563                      
CV 0.03                     1.13                     0.18                     0.09                     0.10                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,165                   1,108                   3,273                   4,215                   7,489                   
St Dev 60                        1,241                   1,214                   205                      1,271                   
CV 0.03                     1.12                     0.37                     0.05                     0.17                     
Mean 2,083                   424                      2,507                   5,073                   7,579                  
St Dev 77                        647                      616                      526                      757                      
CV 0.04                     1.53                     0.25                     0.10                     0.10                     
Mean 2,151                   264                      2,416                   3,185                   5,600                  
St Dev 61                        488                      494                      252                      637                      
CV 0.03                     1.85                     0.20                     0.08                     0.11                     
Mean 2,115                   188                      2,303                   3,156                   5,458                  
St Dev 38                        311                      305                      225                      407                      
CV 0.02                     1.66                     0.13                     0.07                     0.07                     
Mean 2,114                   105                      2,218                   3,205                   5,424                  
St Dev 48                        227                      232                      249                      314                      
CV 0.02                     2.17                     0.10                     0.08                     0.06                     
Mean 2,212                   302                      2,514                   3,348                   5,862                  
St Dev 67                        512                      528                      360                      748                      
CV 0.03                     1.69                     0.21                     0.11                     0.13                     
Mean 2,107                   332                      2,440                   3,118                   5,558                  
St Dev 60                        432                      445                      256                      590                      
CV 0.03                     1.30                     0.18                     0.08                     0.11                     
Mean 2,107                   332                      2,440                   3,118                   5,558                  
St Dev 60                        432                      445                      256                      590                      
CV 0.03                     1.30                     0.18                     0.08                     0.11                     
30 replication with common random numbers
Results in [$/week]
RTC
MUN
BENCH1
FTUI
FTSR
FTRI
RDE
RDE+div
Strategy
Case 3
FTRI
RTC
RDE+div
FTUI
FTSR
RDE
Case 2
MUN
BENCH1
Strategy
 
 
 162 
 
 
Table D- 8: Simulation Results: Set of Parameters 8 
TC= 33 [$/hr], h
i
= 5[$/week], ?
i
= 50[arrivals/day], ?
i
= 1 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,228                   107                      2,335                   1,411                   3,746                   
St Dev 46                        333                      324                      34                        330                      
CV 0.02                     3.11                     0.14                     0.02                     0.09                     
Mean 2,086                   364                      2,450                   1,575                   4,025                  
St Dev 97                        619                      576                      158                      612                      
CV 0.05                     1.70                     0.24                     0.10                     0.15                     
Mean 2,153                   126                      2,279                   1,062                   3,341                  
St Dev 53                        315                      292                      89                        286                      
CV 0.02                     2.50                     0.13                     0.08                     0.09                     
Mean 2,106                   133                      2,240                   1,049                   3,288                  
St Dev 37                        324                      317                      68                        322                      
CV 0.02                     2.43                     0.14                     0.06                     0.10                     
Mean 2,109                   93                        2,202                   1,053                   3,255                  
St Dev 36                        168                      172                      77                        202                      
CV 0.02                     1.80                     0.08                     0.07                     0.06                     
Mean 1,897                   146                      2,043                   1,401                   3,444                  
St Dev 47                        430                      415                      81                        439                      
CV 0.02                     2.94                     0.20                     0.06                     0.13                     
Mean 1,839                   96                        1,936                   1,322                   3,258                  
St Dev 38                        270                      269                      90                        287                      
CV 0.02                     2.80                     0.14                     0.07                     0.09                     
Mean 1,839                   96                        1,936                   1,322                   3,258                  
St Dev 38                        270                      269                      90                        287                      
CV 0.02                     2.80                     0.14                     0.07                     0.09                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,082                   727                      2,809                   1,989                   4,799                   
St Dev 59                        859                      839                      48                        842                      
CV 0.03                     1.18                     0.30                     0.02                     0.18                     
Mean 2,028                   860                      2,888                   2,142                   5,030                  
St Dev 80                        1,184                   1,142                   160                      1,184                   
CV 0.04                     1.38                     0.40                     0.07                     0.24                     
Mean 2,148                   322                      2,470                   1,440                   3,910                  
St Dev 58                        439                      458                      109                      461                      
CV 0.03                     1.36                     0.19                     0.08                     0.12                     
Mean 2,064                   358                      2,423                   1,427                   3,850                  
St Dev 51                        549                      541                      112                      577                      
CV 0.02                     1.53                     0.22                     0.08                     0.15                     
Mean 2,068                   213                      2,281                   1,448                   3,729                  
St Dev 50                        338                      340                      96                        373                      
CV 0.02                     1.59                     0.15                     0.07                     0.10                     
Mean 2,103                   300                      2,403                   1,596                   3,999                  
St Dev 64                        484                      474                      117                      511                      
CV 0.03                     1.61                     0.20                     0.07                     0.13                     
Mean 2,075                   91                        2,166                   1,562                   3,728                  
St Dev 52                        249                      257                      157                      305                      
CV 0.03                     2.74                     0.12                     0.10                     0.08                     
Mean 2,075                   91                        2,166                   1,562                   3,728                  
St Dev 52                        249                      257                      157                      305                      
CV 0.03                     2.74                     0.12                     0.10                     0.08                     
MUN
RTC
FTUI
FTSR
MUN
RDE+div
FTRI
RDE
RDE+div
FTRI
RTC
FTUI
FTSR
BENCH1
Case 1
Strategy
RDE
BENCH1
Strategy
Case 0 (Symmetric case)
 
 
 163 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,231                   308                      2,539                   1,359                   3,897                   
St Dev 40                        593                      591                      42                        588                      
CV 0.02                     1.92                     0.23                     0.03                     0.15                     
Mean 1,845                   90                        1,934                   1,797                   3,731                  
St Dev 49                        184                      173                      174                      259                      
CV 0.03                     2.05                     0.09                     0.10                     0.07                     
Mean 1,874                   377                      2,251                   1,053                   3,304                  
St Dev 38                        635                      615                      71                        621                      
CV 0.02                     1.68                     0.27                     0.07                     0.19                     
Mean 1,745                   421                      2,166                   1,092                   3,258                  
St Dev 37                        447                      440                      70                        435                      
CV 0.02                     1.06                     0.20                     0.06                     0.13                     
Mean 1,750                   236                      1,986                   1,089                   3,075                  
St Dev 48                        363                      347                      88                        360                      
CV 0.03                     1.54                     0.17                     0.08                     0.12                     
Mean 1,860                   367                      2,227                   1,382                   3,609                  
St Dev 40                        527                      510                      90                        514                      
CV 0.02                     1.44                     0.23                     0.07                     0.14                     
Mean 1,802                   190                      1,992                   1,308                   3,300                  
St Dev 48                        411                      409                      76                        426                      
CV 0.03                     2.17                     0.21                     0.06                     0.13                     
Mean 1,802                   190                      1,992                   1,308                   3,300                  
St Dev 48                        411                      409                      76                        426                      
CV 0.03                     2.17                     0.21                     0.06                     0.13                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,215                   194                      2,409                   1,417                   3,826                   
St Dev 51                        534                      542                      35                        532                      
CV 0.02                     2.76                     0.22                     0.02                     0.14                     
Mean 2,027                   259                      2,286                   1,753                   4,039                  
St Dev 83                        458                      419                      158                      429                      
CV 0.04                     1.77                     0.18                     0.09                     0.11                     
Mean 2,101                   280                      2,381                   1,079                   3,460                  
St Dev 57                        501                      496                      74                        526                      
CV 0.03                     1.79                     0.21                     0.07                     0.15                     
Mean 2,007                   170                      2,177                   1,099                   3,276                  
St Dev 43                        273                      277                      70                        299                      
CV 0.02                     1.61                     0.13                     0.06                     0.09                     
Mean 2,009                   98                        2,107                   1,121                   3,228                  
St Dev 40                        228                      230                      71                        247                      
CV 0.02                     2.33                     0.11                     0.06                     0.08                     
Mean 1,861                   310                      2,172                   1,397                   3,569                  
St Dev 43                        507                      495                      78                        503                      
CV 0.02                     1.63                     0.23                     0.06                     0.14                     
Mean 1,818                   109                      1,927                   1,343                   3,270                  
St Dev 49                        227                      230                      97                        260                      
CV 0.03                     2.08                     0.12                     0.07                     0.08                     
Mean 1,818                   109                      1,927                   1,343                   3,270                  
St Dev 49                        227                      230                      97                        260                      
CV 0.03                     2.08                     0.12                     0.07                     0.08                     
30 replication with common random numbers
Results in [$/week]
MUN
RTC
MUN
Strategy
FTRI
FTSR
FTUI
RDE
RDE+div
BENCH1
FTRI
RTC
RDE+div
FTUI
FTSR
Strategy
BENCH1
RDE
Case 2
Case 3
 
 
 164 
 
 
Table D- 9: Simulation Results: Set of Parameters 9 
TC= 300 [$/hr], h
i
= 5[$/week], ?
i
= 50[arrivals/day], ?
i
= 1 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,007                   3,317                   5,324                   11,593                 16,917                 
St Dev 63                        2,482                   2,445                   550                      2,587                   
CV 0.03                     0.75                     0.46                     0.05                     0.15                     
Mean 2,077                   426                      2,503                   14,269                 16,772                
St Dev 95                        442                      427                      1,532                   1,648                   
CV 0.05                     1.04                     0.17                     0.11                     0.10                     
Mean 2,132                   172                      2,304                   9,507                   11,811                
St Dev 63                        279                      283                      776                      826                      
CV 0.03                     1.62                     0.12                     0.08                     0.07                     
Mean 2,096                   192                      2,288                   9,358                   11,646                
St Dev 36                        301                      290                      491                      642                      
CV 0.02                     1.57                     0.13                     0.05                     0.06                     
Mean 2,109                   140                      2,249                   9,466                   11,715                
St Dev 44                        251                      251                      597                      621                      
CV 0.02                     1.80                     0.11                     0.06                     0.05                     
Mean 2,226                   118                      2,344                   10,205                 12,549                
St Dev 54                        344                      343                      1,150                   1,296                   
CV 0.02                     2.91                     0.15                     0.11                     0.10                     
Mean 2,051                   578                      2,629                   9,048                   11,677                
St Dev 57                        626                      648                      722                      1,101                   
CV 0.03                     1.08                     0.25                     0.08                     0.09                     
Mean 2,051                   578                      2,629                   9,048                   11,677                
St Dev 57                        626                      648                      722                      1,101                   
CV 0.03                     1.08                     0.25                     0.08                     0.09                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 1,799                   7,725                   9,524                   15,720                 25,244                 
St Dev 62                        2,998                   2,977                   1,000                   3,075                   
CV 0.03                     0.39                     0.31                     0.06                     0.12                     
Mean 2,009                   929                      2,937                   18,889                 21,826                
St Dev 67                        1,205                   1,168                   1,384                   1,469                   
CV 0.03                     1.30                     0.40                     0.07                     0.07                     
Mean 2,060                   1,001                   3,060                   12,338                 15,399                
St Dev 62                        837                      852                      853                      1,310                   
CV 0.03                     0.84                     0.28                     0.07                     0.09                     
Mean 2,029                   873                      2,902                   12,072                 14,974                
St Dev 39                        745                      741                      945                      1,097                   
CV 0.02                     0.85                     0.26                     0.08                     0.07                     
Mean 2,031                   493                      2,524                   12,172                 14,696                
St Dev 47                        635                      638                      783                      1,046                   
CV 0.02                     1.29                     0.25                     0.06                     0.07                     
Mean 2,200                   220                      2,420                   13,574                 15,994                
St Dev 67                        475                      486                      1,270                   1,483                   
CV 0.03                     2.16                     0.20                     0.09                     0.09                     
Mean 2,061                   614                      2,675                   12,243                 14,918                
St Dev 59                        670                      662                      973                      1,176                   
CV 0.03                     1.09                     0.25                     0.08                     0.08                     
Mean 2,061                   614                      2,675                   12,243                 14,918                
St Dev 59                        670                      662                      973                      1,176                   
CV 0.03                     1.09                     0.25                     0.08                     0.08                     
RTC
MUN
FTUI
FTSR
BENCH1
BENCH1
RDE
RDE+div
FTRI
RTC
MUN
Case 1
Strategy
RDE
RDE+div
FTUI
FTSR
FTRI
Strategy
Case 0 (Symmetric case)
 
 
 165 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,056                   3,174                   5,229                   10,616                 15,845                 
St Dev 69                        2,274                   2,233                   748                      2,359                   
CV 0.03                     0.72                     0.43                     0.07                     0.15                     
Mean 2,022                   2,323                   4,345                   12,921                 17,266                
St Dev 101                      2,007                   1,966                   1,470                   2,024                   
CV 0.05                     0.86                     0.45                     0.11                     0.12                     
Mean 2,089                   685                      2,774                   8,874                   11,647                
St Dev 61                        703                      678                      753                      1,110                   
CV 0.03                     1.03                     0.24                     0.08                     0.10                     
Mean 2,058                   430                      2,488                   8,747                   11,234                
St Dev 46                        563                      549                      619                      871                      
CV 0.02                     1.31                     0.22                     0.07                     0.08                     
Mean 2,065                   380                      2,445                   8,881                   11,326                
St Dev 55                        1,100                   1,085                   662                      1,236                   
CV 0.03                     2.89                     0.44                     0.07                     0.11                     
Mean 2,201                   413                      2,614                   9,861                   12,475                
St Dev 51                        696                      684                      1,096                   1,419                   
CV 0.02                     1.68                     0.26                     0.11                     0.11                     
Mean 2,060                   720                      2,780                   8,910                   11,690                
St Dev 63                        779                      768                      666                      1,010                   
CV 0.03                     1.08                     0.28                     0.07                     0.09                     
Mean 2,060                   743                      2,804                   8,881                   11,684                
St Dev 58                        761                      753                      559                      933                      
CV 0.03                     1.02                     0.27                     0.06                     0.08                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 1,991                   3,514                   5,505                   11,643                 17,148                 
St Dev 66                        2,453                   2,412                   667                      2,464                   
CV 0.03                     0.70                     0.44                     0.06                     0.14                     
Mean 2,050                   880                      2,930                   14,527                 17,457                
St Dev 89                        1,169                   1,116                   1,470                   2,068                   
CV 0.04                     1.33                     0.38                     0.10                     0.12                     
Mean 2,096                   721                      2,816                   9,178                   11,995                
St Dev 62                        807                      831                      689                      1,194                   
CV 0.03                     1.12                     0.30                     0.08                     0.10                     
Mean 2,072                   617                      2,688                   9,036                   11,724                
St Dev 48                        637                      632                      733                      1,041                   
CV 0.02                     1.03                     0.23                     0.08                     0.09                     
Mean 2,056                   493                      2,550                   9,121                   11,670                
St Dev 46                        569                      570                      570                      821                      
CV 0.02                     1.15                     0.22                     0.06                     0.07                     
Mean 2,198                   384                      2,582                   10,044                 12,626                
St Dev 55                        629                      624                      1,079                   1,387                   
CV 0.03                     1.64                     0.24                     0.11                     0.11                     
Mean 2,064                   631                      2,695                   9,059                   11,755                
St Dev 61                        644                      640                      571                      810                      
CV 0.03                     1.02                     0.24                     0.06                     0.07                     
Mean 2,064                   631                      2,695                   9,059                   11,755                
St Dev 61                        644                      640                      571                      810                      
CV 0.03                     1.02                     0.24                     0.06                     0.07                     
30 replication with common random numbers
Results in [$/week]
RTC
FTUI
FTSR
MUN
FTRI
RDE
RDE+div
Strategy
BENCH1
FTRI
Strategy
BENCH1
Case 2
RDE
FTUI
FTSR
MUN
RTC
RDE+div
Case 3
 
 
 166 
 
 
Table D- 10: Simulation Results: Set of Parameters 10 
TC= 100 [$/hr], h
i
= 5[$/week], ?
i
= 10.5 [arrivals/day], ?
i
= 4.8 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,173                   2,082                   4,255                   4,119                   8,374                   
St Dev 58                        2,259                   2,229                   200                      2,281                   
CV 0.03                     1.09                     0.52                     0.05                     0.27                     
Mean 2,227                   808                      3,035                   5,007                   8,042                  
St Dev 107                      914                      915                      600                      1,130                   
CV 0.05                     1.13                     0.30                     0.12                     0.14                     
Mean 2,395                   658                      3,054                   3,782                   6,836                  
St Dev 94                        786                      768                      237                      801                      
CV 0.04                     1.19                     0.25                     0.06                     0.12                     
Mean 2,273                   652                      2,925                   3,603                   6,528                  
St Dev 57                        1,264                   1,250                   297                      1,307                   
CV 0.02                     1.94                     0.43                     0.08                     0.20                     
Mean 2,281                   602                      2,883                   3,620                   6,503                  
St Dev 52                        877                      859                      321                      913                      
CV 0.02                     1.46                     0.30                     0.09                     0.14                     
Mean 2,464                   526                      2,990                   4,302                   7,292                  
St Dev 77                        843                      823                      266                      878                      
CV 0.03                     1.60                     0.28                     0.06                     0.12                     
Mean 2,339                   736                      3,075                   3,758                   6,833                  
St Dev 66                        879                      868                      331                      997                      
CV 0.03                     1.20                     0.28                     0.09                     0.15                     
Mean 2,335                   489                      2,824                   3,751                   6,574                  
St Dev 69                        448                      452                      372                      607                      
CV 0.03                     0.92                     0.16                     0.10                     0.09                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,000                   3,985                   5,986                   5,721                   11,707                 
St Dev 68                        2,999                   2,958                   257                      2,971                   
CV 0.03                     0.75                     0.49                     0.04                     0.25                     
Mean 2,183                   1,023                   3,206                   6,661                   9,867                  
St Dev 106                      1,241                   1,199                   571                      1,162                   
CV 0.05                     1.21                     0.37                     0.09                     0.12                     
Mean 2,419                   726                      3,145                   5,181                   8,326                  
St Dev 114                      901                      917                      269                      953                      
CV 0.05                     1.24                     0.29                     0.05                     0.11                     
Mean 2,273                   634                      2,907                   5,212                   8,119                  
St Dev 52                        906                      900                      553                      1,074                   
CV 0.02                     1.43                     0.31                     0.11                     0.13                     
Mean 2,292                   637                      2,929                   5,080                   8,009                  
St Dev 49                        932                      927                      357                      993                      
CV 0.02                     1.46                     0.32                     0.07                     0.12                     
Mean 2,484                   525                      3,009                   5,722                   8,731                  
St Dev 50                        744                      726                      265                      784                      
CV 0.02                     1.42                     0.24                     0.05                     0.09                     
Mean 2,337                   572                      2,909                   5,013                   7,922                  
St Dev 63                        670                      654                      497                      817                      
CV 0.03                     1.17                     0.22                     0.10                     0.10                     
Mean 2,341                   529                      2,870                   5,040                   7,910                  
St Dev 60                        670                      660                      532                      889                      
CV 0.03                     1.27                     0.23                     0.11                     0.11                     
MUN
Case 1
MUN
RTC
FTUI
FTSR
RDE
RDE+div
Strategy
BENCH1
RTC
FTUI
FTSR
FTRI
BENCH1
RDE
RDE+div
FTRI
Case 0 (Symmetric case)
Strategy
 
 
 167 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,199                   2,081                   4,280                   3,842                   8,122                   
St Dev 75                        1,643                   1,630                   196                      1,613                   
CV 0.03                     0.79                     0.38                     0.05                     0.20                     
Mean 2,161                   541                      2,702                   4,971                   7,674                  
St Dev 110                      707                      663                      487                      926                      
CV 0.05                     1.31                     0.25                     0.10                     0.12                     
Mean 2,408                   466                      2,874                   3,459                   6,334                  
St Dev 110                      578                      591                      271                      645                      
CV 0.05                     1.24                     0.21                     0.08                     0.10                     
Mean 2,190                   690                      2,881                   3,586                   6,466                  
St Dev 43                        946                      949                      484                      1,120                   
CV 0.02                     1.37                     0.33                     0.13                     0.17                     
Mean 2,195                   364                      2,559                   3,431                   5,990                  
St Dev 54                        520                      504                      284                      596                      
CV 0.02                     1.43                     0.20                     0.08                     0.10                     
Mean 2,475                   464                      2,939                   4,156                   7,095                  
St Dev 65                        522                      524                      268                      533                      
CV 0.03                     1.12                     0.18                     0.06                     0.08                     
Mean 2,305                   341                      2,646                   3,587                   6,234                  
St Dev 75                        598                      598                      406                      759                      
CV 0.03                     1.75                     0.23                     0.11                     0.12                     
Mean 2,314                   421                      2,735                   3,587                   6,322                  
St Dev 67                        593                      592                      385                      733                      
CV 0.03                     1.41                     0.22                     0.11                     0.12                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,159                   2,412                   4,571                   4,126                   8,697                   
St Dev 72                        2,049                   2,017                   225                      2,051                   
CV 0.03                     0.85                     0.44                     0.05                     0.24                     
Mean 2,233                   570                      2,804                   5,155                   7,959                  
St Dev 101                      737                      729                      533                      836                      
CV 0.05                     1.29                     0.26                     0.10                     0.11                     
Mean 2,442                   710                      3,152                   3,805                   6,957                  
St Dev 108                      939                      919                      288                      910                      
CV 0.04                     1.32                     0.29                     0.08                     0.13                     
Mean 2,297                   468                      2,765                   3,653                   6,418                  
St Dev 50                        758                      746                      264                      790                      
CV 0.02                     1.62                     0.27                     0.07                     0.12                     
Mean 2,296                   558                      2,854                   3,744                   6,598                  
St Dev 45                        864                      868                      298                      899                      
CV 0.02                     1.55                     0.30                     0.08                     0.14                     
Mean 2,483                   592                      3,075                   4,201                   7,276                  
St Dev 81                        1,196                   1,159                   237                      1,167                   
CV 0.03                     2.02                     0.38                     0.06                     0.16                     
Mean 2,340                   898                      3,238                   3,676                   6,914                  
St Dev 74                        1,038                   1,030                   370                      1,131                   
CV 0.03                     1.16                     0.32                     0.10                     0.16                     
Mean 2,342                   882                      3,223                   3,666                   6,889                  
St Dev 69                        1,027                   1,010                   357                      1,107                   
CV 0.03                     1.16                     0.31                     0.10                     0.16                     
30 replication with common random numbers
Results in [$/week]
FTSR
MUN
FTUI
RTC
FTUI
FTSR
FTRI
RDE
RDE+div
FTRI
BENCH1
Case 2
Strategy
BENCH1
RDE
MUN
RTC
RDE+div
Strategy
Case 3
 
 
 168 
 
 
Table D- 11: Simulation Results: Set of Parameters 11 
TC= 100 [$/hr], h
i
= 5[$/week], ?
i
= 4.35 [arrivals/day], ?
i
= 11.5 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,155                   7,635                   9,790                   4,017                   13,807                 
St Dev 112                      5,935                   5,883                   297                      5,990                   
CV 0.05                     0.78                     0.60                     0.07                     0.43                     
Mean 2,432                   985                      3,417                   5,626                   9,042                  
St Dev 123                      1,166                   1,159                   560                      1,302                   
CV 0.05                     1.18                     0.34                     0.10                     0.14                     
Mean 2,704                   1,200                   3,903                   4,612                   8,515                  
St Dev 139                      1,811                   1,743                   360                      1,969                   
CV 0.05                     1.51                     0.45                     0.08                     0.23                     
Mean 2,438                   2,003                   4,441                   4,102                   8,543                  
St Dev 59                        1,963                   1,958                   359                      2,036                   
CV 0.02                     0.98                     0.44                     0.09                     0.24                     
Mean 2,467                   1,476                   3,943                   4,199                   8,142                  
St Dev 57                        1,778                   1,758                   355                      1,965                   
CV 0.02                     1.20                     0.45                     0.08                     0.24                     
Mean 2,705                   1,108                   3,814                   5,670                   9,483                  
St Dev 84                        1,945                   1,899                   236                      1,999                   
CV 0.03                     1.76                     0.50                     0.04                     0.21                     
Mean 2,553                   1,365                   3,918                   4,717                   8,635                  
St Dev 70                        1,156                   1,164                   411                      1,286                   
CV 0.03                     0.85                     0.30                     0.09                     0.15                     
Mean 2,548                   1,812                   4,360                   4,600                   8,959                  
St Dev 82                        1,789                   1,778                   339                      1,711                   
CV 0.03                     0.99                     0.41                     0.07                     0.19                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 1,997                   11,111                 13,108                 5,463                   18,571                 
St Dev 112                      5,915                   5,856                   334                      5,932                   
CV 0.06                     0.53                     0.45                     0.06                     0.32                     
Mean 2,365                   1,828                   4,193                   7,101                   11,294                
St Dev 115                      2,188                   2,141                   679                      2,086                   
CV 0.05                     1.20                     0.51                     0.10                     0.18                     
Mean 2,703                   1,454                   4,157                   6,321                   10,479                
St Dev 134                      1,850                   1,793                   417                      1,933                   
CV 0.05                     1.27                     0.43                     0.07                     0.18                     
Mean 2,462                   1,752                   4,214                   6,097                   10,311                
St Dev 66                        2,302                   2,289                   596                      2,526                   
CV 0.03                     1.31                     0.54                     0.10                     0.24                     
Mean 2,488                   921                      3,409                   6,006                   9,414                  
St Dev 64                        1,253                   1,225                   521                      1,543                   
CV 0.03                     1.36                     0.36                     0.09                     0.16                     
Mean 2,719                   897                      3,616                   7,541                   11,157                
St Dev 64                        1,247                   1,228                   173                      1,277                   
CV 0.02                     1.39                     0.34                     0.02                     0.11                     
Mean 2,540                   2,104                   4,644                   6,033                   10,677                
St Dev 60                        2,686                   2,697                   629                      2,890                   
CV 0.02                     1.28                     0.58                     0.10                     0.27                     
Mean 2,514                   2,067                   4,582                   5,778                   10,359                
St Dev 77                        2,309                   2,318                   617                      2,340                   
CV 0.03                     1.12                     0.51                     0.11                     0.23                     
FTRI
RTC
RDE
Strategy
BENCH1
RDE+div
Strategy
Case 0 (Symmetric case)
BENCH1
RDE
FTUI
FTSR
MUN
RDE+div
FTRI
Case 1
RTC
FTUI
FTSR
MUN
 
 
 169 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,199                   6,968                   9,167                   3,689                   12,857                 
St Dev 84                        4,807                   4,759                   282                      4,894                   
CV 0.04                     0.69                     0.52                     0.08                     0.38                     
Mean 2,406                   1,310                   3,716                   5,258                   8,975                  
St Dev 133                      2,227                   2,155                   756                      2,089                   
CV 0.06                     1.70                     0.58                     0.14                     0.23                     
Mean 2,728                   1,732                   4,460                   4,306                   8,766                  
St Dev 131                      1,873                   1,841                   374                      1,947                   
CV 0.05                     1.08                     0.41                     0.09                     0.22                     
Mean 2,398                   1,974                   4,372                   4,129                   8,501                  
St Dev 58                        1,550                   1,561                   620                      1,679                   
CV 0.02                     0.79                     0.36                     0.15                     0.20                     
Mean 2,436                   653                      3,088                   4,073                   7,161                  
St Dev 61                        940                      944                      577                      1,060                   
CV 0.03                     1.44                     0.31                     0.14                     0.15                     
Mean 2,726                   908                      3,634                   5,477                   9,111                  
St Dev 70                        1,499                   1,470                   245                      1,524                   
CV 0.03                     1.65                     0.40                     0.04                     0.17                     
Mean 2,533                   1,565                   4,098                   4,350                   8,448                  
St Dev 72                        2,134                   2,135                   526                      2,182                   
CV 0.03                     1.36                     0.52                     0.12                     0.26                     
Mean 2,513                   1,865                   4,378                   4,073                   8,451                  
St Dev 98                        2,315                   2,302                   474                      2,400                   
CV 0.04                     1.24                     0.53                     0.12                     0.28                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,144                   7,516                   9,659                   3,960                   13,620                 
St Dev 88                        4,136                   4,083                   197                      4,123                   
CV 0.04                     0.55                     0.42                     0.05                     0.30                     
Mean 2,415                   946                      3,361                   5,691                   9,052                  
St Dev 85                        1,372                   1,337                   654                      1,289                   
CV 0.04                     1.45                     0.40                     0.11                     0.14                     
Mean 2,715                   1,144                   3,859                   4,678                   8,537                  
St Dev 140                      1,398                   1,353                   398                      1,444                   
CV 0.05                     1.22                     0.35                     0.09                     0.17                     
Mean 2,459                   1,823                   4,282                   4,458                   8,740                  
St Dev 71                        2,178                   2,142                   530                      2,402                   
CV 0.03                     1.19                     0.50                     0.12                     0.27                     
Mean 2,502                   784                      3,287                   4,562                   7,849                  
St Dev 74                        1,138                   1,130                   376                      1,162                   
CV 0.03                     1.45                     0.34                     0.08                     0.15                     
Mean 2,718                   859                      3,576                   5,517                   9,093                  
St Dev 64                        1,161                   1,136                   246                      1,199                   
CV 0.02                     1.35                     0.32                     0.04                     0.13                     
Mean 2,532                   1,715                   4,247                   4,464                   8,711                  
St Dev 69                        1,943                   1,930                   524                      1,901                   
CV 0.03                     1.13                     0.45                     0.12                     0.22                     
Mean 2,523                   1,832                   4,355                   4,259                   8,613                  
St Dev 62                        2,066                   2,064                   356                      2,040                   
CV 0.02                     1.13                     0.47                     0.08                     0.24                     
30 replication with common random numbers
Results in [$/week]
FTUI
FTSR
MUN
FTRI
RTC
RDE
RDE+div
BENCH1
Case 2
BENCH1
RDE
RDE+div
Strategy
Case 3
RTC
FTUI
FTSR
Strategy
MUN
FTRI
 
 
 170 
 
 
Table D- 12: Simulation Results: Set of Parameters 12 
TC= 100 [$/hr], h
i
= 5[$/week], ?
i
= 2.4 [arrivals/day], ?
i
= 20.8 [units] for all i 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,137                   18,784                 20,921                 3,758                   24,679                 
St Dev 145                      8,613                   8,512                   363                      8,639                   
CV 0.07                     0.46                     0.41                     0.10                     0.35                     
Mean 2,673                   1,174                   3,847                   6,441                   10,288                
St Dev 118                      1,487                   1,460                   671                      1,474                   
CV 0.04                     1.27                     0.38                     0.10                     0.14                     
Mean 2,906                   4,250                   7,156                   5,436                   12,592                
St Dev 149                      4,277                   4,239                   471                      4,401                   
CV 0.05                     1.01                     0.59                     0.09                     0.35                     
Mean 2,626                   3,741                   6,367                   4,641                   11,008                
St Dev 82                        3,575                   3,563                   385                      3,475                   
CV 0.03                     0.96                     0.56                     0.08                     0.32                     
Mean 2,660                   2,214                   4,874                   4,779                   9,653                  
St Dev 66                        2,021                   2,034                   517                      2,128                   
CV 0.02                     0.91                     0.42                     0.11                     0.22                     
Mean 2,898                   2,904                   5,802                   7,515                   13,317                
St Dev 85                        3,275                   3,257                   174                      3,233                   
CV 0.03                     1.13                     0.56                     0.02                     0.24                     
Mean 2,728                   3,364                   6,092                   4,518                   10,610                
St Dev 80                        3,075                   3,090                   544                      3,212                   
CV 0.03                     0.91                     0.51                     0.12                     0.30                     
Mean 2,686                   2,874                   5,560                   5,117                   10,677                
St Dev 84                        2,860                   2,845                   553                      2,925                   
CV 0.03                     1.00                     0.51                     0.11                     0.27                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,017                   17,912                 19,928                 5,131                   25,059                 
St Dev 127                      7,455                   7,373                   349                      7,482                   
CV 0.06                     0.42                     0.37                     0.07                     0.30                     
Mean 2,571                   2,220                   4,791                   7,824                   12,615                
St Dev 127                      2,385                   2,345                   733                      2,368                   
CV 0.05                     1.07                     0.49                     0.09                     0.19                     
Mean 2,866                   3,121                   5,986                   7,362                   13,349                
St Dev 129                      3,558                   3,512                   560                      3,604                   
CV 0.05                     1.14                     0.59                     0.08                     0.27                     
Mean 2,652                   1,920                   4,572                   7,253                   11,826                
St Dev 68                        1,667                   1,654                   605                      1,739                   
CV 0.03                     0.87                     0.36                     0.08                     0.15                     
Mean 2,660                   1,946                   4,606                   7,451                   12,057                
St Dev 71                        1,832                   1,814                   543                      1,889                   
CV 0.03                     0.94                     0.39                     0.07                     0.16                     
Mean 2,910                   1,214                   4,124                   9,990                   14,115                
St Dev 72                        1,697                   1,684                   148                      1,699                   
CV 0.02                     1.40                     0.41                     0.01                     0.12                     
Mean 2,706                   2,934                   5,640                   7,170                   12,810                
St Dev 73                        2,656                   2,641                   815                      2,677                   
CV 0.03                     0.91                     0.47                     0.11                     0.21                     
Mean 2,689                   2,668                   5,357                   6,583                   11,940                
St Dev 65                        2,926                   2,921                   662                      2,915                   
CV 0.02                     1.10                     0.55                     0.10                     0.24                     
FTUI
FTSR
Case 1
BENCH1
RDE
RDE+div
RTC
Strategy
FTRI
RDE+div
FTRI
FTUI
FTSR
MUN
Strategy
Case 0 (Symmetric case)
BENCH1
RDE
RTC
MUN
 
 
 171 
 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,232                   13,941                 16,173                 3,398                   19,571                 
St Dev 122                      8,261                   8,195                   303                      8,219                   
CV 0.05                     0.59                     0.51                     0.09                     0.42                     
Mean 2,617                   2,381                   4,997                   5,639                   10,636                
St Dev 132                      2,314                   2,324                   711                      2,297                   
CV 0.05                     0.97                     0.47                     0.13                     0.22                     
Mean 2,926                   3,228                   6,154                   4,816                   10,970                
St Dev 130                      3,391                   3,354                   486                      3,479                   
CV 0.04                     1.05                     0.54                     0.10                     0.32                     
Mean 2,616                   2,712                   5,328                   4,683                   10,011                
St Dev 74                        2,434                   2,426                   759                      2,389                   
CV 0.03                     0.90                     0.46                     0.16                     0.24                     
Mean 2,669                   1,805                   4,474                   4,667                   9,142                  
St Dev 78                        1,884                   1,861                   604                      2,051                   
CV 0.03                     1.04                     0.42                     0.13                     0.22                     
Mean 2,924                   2,105                   5,029                   7,260                   12,289                
St Dev 75                        2,361                   2,338                   177                      2,327                   
CV 0.03                     1.12                     0.46                     0.02                     0.19                     
Mean 2,718                   3,925                   6,642                   5,219                   11,861                
St Dev 78                        3,435                   3,410                   658                      3,598                   
CV 0.03                     0.88                     0.51                     0.13                     0.30                     
Mean 2,697                   3,488                   6,185                   4,428                   10,613                
St Dev 73                        2,680                   2,667                   463                      2,686                   
CV 0.03                     0.77                     0.43                     0.10                     0.25                     
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,168                   14,369                 16,537                 3,766                   20,303                 
St Dev 97                        7,092                   7,049                   289                      7,109                   
CV 0.04                     0.49                     0.43                     0.08                     0.35                     
Mean 2,711                   1,360                   4,071                   6,624                   10,695                
St Dev 136                      1,965                   1,967                   792                      2,176                   
CV 0.05                     1.44                     0.48                     0.12                     0.20                     
Mean 2,917                   2,907                   5,824                   5,410                   11,235                
St Dev 121                      4,036                   4,015                   386                      4,119                   
CV 0.04                     1.39                     0.69                     0.07                     0.37                     
Mean 2,668                   2,066                   4,734                   5,480                   10,214                
St Dev 78                        2,432                   2,423                   650                      2,565                   
CV 0.03                     1.18                     0.51                     0.12                     0.25                     
Mean 2,710                   1,175                   3,886                   5,405                   9,291                  
St Dev 69                        1,753                   1,732                   744                      2,099                   
CV 0.03                     1.49                     0.45                     0.14                     0.23                     
Mean 2,922                   1,223                   4,144                   7,299                   11,443                
St Dev 61                        1,630                   1,610                   136                      1,601                   
CV 0.02                     1.33                     0.39                     0.02                     0.14                     
Mean 2,721                   2,469                   5,190                   5,416                   10,606                
St Dev 92                        2,897                   2,890                   571                      2,905                   
CV 0.03                     1.17                     0.56                     0.11                     0.27                     
Mean 2,728                   2,346                   5,074                   4,819                   9,892                  
St Dev 69                        2,195                   2,221                   514                      2,403                   
CV 0.03                     0.94                     0.44                     0.11                     0.24                     
30 replication with common random numbers
Results in [$/week]
FTRI
FTUI
FTSR
MUN
Case 3
BENCH1
RDE
RDE+div
RTC
BENCH1
RDE
RDE+div
FTRI
RTC
Strategy
Case 2
FTUI
FTSR
MUN
Strategy
 
 
 
 172 
 
 
Table D- 13: Simulation Results: Set of Parameters 1, under Twice Demand 
Arrivals at Retailer 4 and Without Inventory Target Update 
TC= 100 [$/hr], h
i
= 50[$/week], ?
i
= 50[arrivals/day], ?
i
= 1 [units] for i={1,2,3,5,6,7} 
, ?
4
= 100[arrivals/day], ?
4
= 1 [units] 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 22,099                 5,370                   27,470                 3,615                   31,085                 
St Dev 3,358                   2,535                   3,445                   139                      3,438                   
CV 0.15                     0.47                     0.13                     0.04                     0.11                     
Mean 13,594                 2,216                   15,810                 7,845                   23,655                
St Dev 3,336                   1,225                   3,197                   463                      3,332                   
CV 0.25                     0.55                     0.20                     0.06                     0.14                     
Mean 15,196                 7,751                   22,946                 5,226                   28,172                
St Dev 2,308                   2,670                   3,133                   247                      3,097                   
CV 0.15                     0.34                     0.14                     0.05                     0.11                     
Mean 13,592                 5,602                   19,194                 5,384                   24,578                
St Dev 1,380                   2,540                   2,960                   268                      2,947                   
CV 0.10                     0.45                     0.15                     0.05                     0.12                     
Mean 13,673                 3,203                   16,876                 5,416                   22,291                
St Dev 980                      1,549                   1,600                   209                      1,627                   
CV 0.07                     0.48                     0.09                     0.04                     0.07                     
Mean 9,094                   17,124                 26,217                 7,032                   33,249                
St Dev 679                      4,090                   3,862                   72                        3,865                   
CV 0.07                     0.24                     0.15                     0.01                     0.12                     
Mean 10,180                 8,900                   19,079                 8,613                   27,692                
St Dev 567                      2,839                   3,033                   298                      3,168                   
CV 0.06                     0.32                     0.16                     0.03                     0.11                     
Mean 10,179                 8,876                   19,055                 8,614                   27,669                
St Dev 564                      2,888                   3,063                   296                      3,193                   
CV 0.06                     0.33                     0.16                     0.03                     0.12                     
30 replication with common random numbers
Results in [$/week]
FTUI
FTSR
MUN
BENCH1
RDE
RDE+div
FTRI
RTC
Strategy
Case 0 (Symmetric case)
 
 
 
 
 173 
 
 
Table D- 14: Simulation Results: Set of Parameters 7, under Twice Demand 
Arrivals at Retailer 4 and Without Inventory Target Update 
TC= 100 [$/hr], h
i
= 5[$/week], ?
i
= 50[arrivals/day], ?
i
= 1 [units] for i={1,2,3,5,6,7} 
, ?
4
= 100[arrivals/day], ?
4
= 1 [units] 
Inv. Holding Cost Lost Sales Cost Total Inv. Cost Transp. Cost Total Cost
Mean 2,055                   7,787                   9,842                   4,336                   14,177                 
St Dev 237                      3,156                   3,099                   177                      3,125                   
CV 0.12                     0.41                     0.31                     0.04                     0.22                     
Mean 2,197                   905                      3,102                   5,445                   8,547                  
St Dev 314                      669                      787                      601                      934                      
CV 0.14                     0.74                     0.25                     0.11                     0.11                     
Mean 2,065                   8,581                   10,646                 3,602                   14,247                
St Dev 157                      3,229                   3,212                   221                      3,211                   
CV 0.08                     0.38                     0.30                     0.06                     0.23                     
Mean 2,052                   5,937                   7,989                   3,522                   11,511                
St Dev 185                      2,621                   2,533                   211                      2,535                   
CV 0.09                     0.44                     0.32                     0.06                     0.22                     
Mean 2,036                   3,291                   5,327                   3,453                   8,780                  
St Dev 142                      2,271                   2,272                   226                      2,269                   
CV 0.07                     0.69                     0.43                     0.07                     0.26                     
Mean 2,097                   17,119                 19,216                 3,402                   22,618                
St Dev 295                      3,987                   4,002                   383                      4,115                   
CV 0.14                     0.23                     0.21                     0.11                     0.18                     
Mean 2,443                   3,666                   6,109                   4,802                   10,912                
St Dev 293                      1,588                   1,644                   344                      1,843                   
CV 0.12                     0.43                     0.27                     0.07                     0.17                     
Mean 2,444                   3,666                   6,110                   4,802                   10,913                
St Dev 327                      1,588                   1,654                   344                      1,855                   
CV 0.13                     0.43                     0.27                     0.07                     0.17                     
30 replication with common random numbers
Results in [$/week]
RTC
Case 0 (Symmetric case)
FTUI
FTSR
RDE+div
FTRI
BENCH1
RDE
MUN
Strategy
 
 
 
 
 
 174 
 
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