ABSTRACT 
 
 
Title of Dissertation: PARAMETER SENSITIVITY MEASURES FOR 
SINGLE OBJECTIVE, MULTI-OBJECTIVE, AND 
FEASIBILITY ROBUST DESIGN OPTIMIZATION  
  Subroto Gunawan, Doctor of Philosophy, 2004 
 
Dissertation directed by: Shapour Azarm, Professor 
  Department of Mechanical Engineering 
   
Uncontrollable variations are unavoidable in engineering design. If ignored, such 
variations can seriously deteriorate performance of an optimum design. Robust 
optimization is an approach that optimizes performance of a design and at the same time 
reduces its sensitivity to variations. The literature reports on numerous robust 
optimization techniques. In general, these techniques have three main shortcomings: (i) 
they presume probability distributions for parameter variations, which might be invalid, 
(ii) they limit parameter variations to a small (linear) range, and (iii) they use gradient 
information of objective/constraint functions. These shortcomings severely restrict 
applications of the techniques reported in the literature.  
The objective of this dissertation is to present a robust optimization method that 
addresses all of the above-mentioned shortcomings. In addition to being efficient, the 
robust optimization method of this dissertation is applicable to both single and multi-
objective optimization problems.  
There are two steps in our robust optimization method. In the first step, the method 
measures robustness for a design alternative. The robustness measure is developed based 
on a concept that associated with each design alternative there is a sensitivity region in 
parameter variation space that determines how much variation a design alternative can 
absorb. The larger the size of this region, the more robust the design. The size of the 
sensitivity region is estimated by a hyper-sphere, using a worst-case approach. The radius 
of this hyper-sphere is obtained by solving an inner optimization problem. By comparing 
this radius to an actual range of parameter variations, it is determined whether or not a 
design alternative is robust. This comparison is added, in the second step, as an additional 
constraint to the original optimization problem. An optimization technique is then used to 
solve this problem and find a robust optimum design solution.  
As a demonstration, the robust optimization method is applied to numerous 
numerical and engineering examples. The results obtained are numerically analyzed and 
compared to nominal optimum designs, and to optimum designs obtained by a few well-
known methods from the literature. The comparison study verifies that the solutions 
obtained by our method are indeed robust, and that the method is efficient.  
 
 
 
 
 
 
PARAMETER SENSITIVITY MEASURES FOR  
SINGLE OBJECTIVE, MULTI-OBJECTIVE, AND FEASIBILITY  
ROBUST DESIGN OPTIMIZATION 
 
 
 
by 
 
Subroto Gunawan 
 
 
 
 
 
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 
2004 
 
 
 
 
 
 
 
 Advisory Committee:  
 
Professor Shapour Azarm, Advisor/Chair 
Professor Amr Baz 
Professor Roberto Celi 
Assistant Professor Steven A. Gabriel 
Associate Professor Jeffrey Herrmann 
Associate Professor Peter Sandborn 
     
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
 
Subroto Gunawan 
 
2004 
 
 
ACKNOWLEDGEMENTS 
 
 
First of all, I would like to thank my advisor, Dr. Shapour Azarm, for his guidance 
and support throughout my graduate study. I also would like to thank the committee 
members, Dr. Baz, Dr. Celi, Dr. Gabriel, Dr. Herrmann, and Dr. Sandborn, for their 
inputs and comments. Secondly, I would like to thank our sponsor, Dr. Ng from ONR, 
whose generous support made this research possible. Special thanks to Mr. Art Boyars 
from NSWC Indian Head who in numerous occasions helped and provided us with the 
information necessary to formulate the payload optimization problem. His comments and 
suggestions throughout the research have been invaluable as well. Thanks also to Dr. 
Magrab who provided support during the first year of my graduate study. The research 
work presented in this thesis was also supported in part by the National Science 
Foundation through Grant DMI 0200029. Such support does not constitute an 
endorsement by the funding agency of the opinions expressed in the paper. 
Thirdly, I want to thank my colleagues, past and present, from the Design Decision 
Support Laboratory: Mr. Kumar Maddulapalli, Mr. Babak Besharati, Mr. Genzi Li, Dr. 
Ali Farhang-Mehr, Dr. Jin Wu, and others who have provided much valuable inputs and 
suggestions. I also want to thank my parents and my family for their continuous support 
during my study. Very special thanks to my girlfriend, Agnes Susianti, whose love and 
understanding helped me carry on during difficult times. Lastly, I wish to thank all my 
friends, Koswara, Franky, Andre, Steven, Ronald, and many others, for their support and 
encouragements.  
 ii
 
TABLE OF CONTENTS 
 
 
 
List of Figures.................................................................................................................... vi 
 
List of Tables ..................................................................................................................... ix 
 
Nomenclature...................................................................................................................... x 
 
CHAPTER 1: INTRODUCTION....................................................................................... 1 
1.1. MOTIVATION AND OBJECTIVE............................................................................ 1 
1.2. RESEARCH COMPONENTS..................................................................................... 4 
1.2.1. Research Component 1: Single Objective Robust Optimization.................. 4 
1.2.2. Research Component 2: Multi-Objective Robust Optimization................... 5 
1.2.3. Research Component 3: Feasibility Robust Optimization............................ 5 
1.3. ASSUMPTIONS.......................................................................................................... 6 
1.4. ORGANIZATION OF DISSERTATION ................................................................... 7 
 
CHAPTER 2: DEFINITIONS AND PREVIOUS WORKS............................................... 9 
2.1. INTRODUCTION ....................................................................................................... 9 
2.2. DEFINITIONS AND TERMINOLOGIES.................................................................. 9 
2.3. OVERVIEW OF PREVIOUS WORK ...................................................................... 13 
2.3.1. Single Objective Robust Optimization ....................................................... 16 
2.3.2. Multi-Objective Robust Optimization ........................................................ 18 
2.3.3. Feasibility Robust Optimization ................................................................. 19 
 
CHAPTER 3: SINGLE OBJECTIVE ROBUST OPTIMIZATION ................................ 21 
3.1. INTRODUCTION ..................................................................................................... 21 
3.2. TWO-SIDED SENSITIVITY MEASURE................................................................ 22 
3.2.1. Sensitivity Set ............................................................................................. 22 
3.2.2. Sensitivity Region....................................................................................... 25 
3.2.3. Directional Sensitivity ................................................................................ 32 
3.2.4. Worst Case Sensitivity Region ................................................................... 36 
3.2.5. Normalization ............................................................................................. 44 
3.3. ROBUST OPTIMIZATION ...................................................................................... 49 
3.3.1. Robustness Index ........................................................................................ 49 
3.3.2. Constrained Robustness Approach ............................................................. 51 
3.4. COMPARISON STUDY........................................................................................... 54 
3.4.1. Wine-Bottle Function.................................................................................. 55 
3.4.2. Design of a Three-Bar Truss....................................................................... 58 
3.4.3. Design of A Welded Beam ......................................................................... 64 
3.4.4. Design of a Compression Spring ................................................................ 72 
3.5. SUMMARY............................................................................................................... 77 
 
 
 iii
 
CHAPTER 4: MULTI-OBJECTIVE ROBUST OPTIMIZATION ................................. 80 
4.1. INTRODUCTION ..................................................................................................... 80 
4.2. BASIC CONCEPTS .................................................................................................. 81 
4.2.1. Multi-Objective Robustness........................................................................ 81 
4.2.2. Multi-Objective Robust Optimality ............................................................ 83 
4.3. TWO-SIDED SENSITIVITY OF MULTIPLE FUNCTIONS.................................. 85 
4.3.1. Generalized Sensitivity Set and Sensitivity Region.................................... 85 
4.3.2. Generalized Worst Case Sensitivity Region............................................... 88 
4.3.3. Normalization ............................................................................................. 91 
4.4. ROBUST OPTIMIZATION ...................................................................................... 94 
4.5. COMPARISON STUDY........................................................................................... 96 
4.5.1. Numerical Example .................................................................................... 97 
4.5.2. Design of a Vibrating Platform................................................................. 103 
4.5.3. Design of a Speed Reducer....................................................................... 111 
4.5.4. Design of a Power Electronic Module...................................................... 117 
4.6. SUMMARY............................................................................................................. 126 
 
CHAPTER 5: FEASIBILITY ROBUST OPTIMIZATION........................................... 128 
5.1. INTRODUCTION ................................................................................................... 128 
5.2. ONE-SIDED SENSITIVITY MEASURE............................................................... 129 
5.2.1. Single Constraint....................................................................................... 129 
5.2.2. Multiple Constraints.................................................................................. 134 
5.3. ROBUST OPTIMIZATION .................................................................................... 139 
5.3.1. Determination of Increment Limit............................................................ 139 
5.3.2. Normalization and Feasibility Robustness Index ..................................... 141 
5.4. COMPARISON STUDY......................................................................................... 143 
5.4.1. Numerical Example .................................................................................. 143 
5.4.2. Design of an Explosive Actuated Cylinder............................................... 150 
5.4.3. Design of a Belleville Spring.................................................................... 157 
5.4.4. Design of a Control Valve Actuator Linkage ........................................... 163 
5.5. SUMMARY............................................................................................................. 172 
 
CHAPTER 6: DISCUSSIONS ....................................................................................... 173 
6.1. INTRODUCTION ................................................................................................... 173 
6.2. OBJECTIVE AND FEASIBILITY ROBUST OPTIMIZATION........................... 173 
6.2.1. Design of a Payload for an Undersea Autonomous Vehicle (UAV) ........ 178 
6.3. ONE-SIDED SENSITIVITY FOR OBJECTIVE ROBUSTNESS ......................... 185 
6.4. ASYMMETRICAL TWO-SIDED SENSITIVITY MEASURE............................. 187 
6.5. ASYMMETRICAL PARAMETER VARIATIONS............................................... 188 
6.6. ROBUSTNESS INDEX VS. ROBUSTNESS PROBABILITY ............................. 190 
6.7. SUMMARY............................................................................................................. 195 
 
CHAPTER 7: CONCLUSIONS ..................................................................................... 197 
7.1. CONCLUDING REMARKS................................................................................... 197 
7.1.1. Verification ............................................................................................... 198 
7.1.2. Computational Efficiency ......................................................................... 200 
 iv
 
7.1.3. Advantages and Disadvantages................................................................. 202 
7.2. CONTRIBUTIONS ................................................................................................. 205 
7.3. FUTURE RESEARCH DIRECTIONS ................................................................... 206 
 
REFERENCES ............................................................................................................... 209 
 
 v
 
LIST OF FIGURES 
 
 
Figure 1.1: Organization of dissertation. 
 
Figure 3.1: Sensitivity region of a design alternative. 
Figure 3.2: Geometric condition for a boundary point. 
Figure 3.3: One-dimensional SR example of a discontinuous function. 
Figure 3.4: One-dimensional example of equality inner point condition. 
Figure 3.5: Boundedness of a sensitivity region. 
Figure 3.6: Example of a condition that causes SR to be unbounded. 
Figure 3.7: Example of a disjointed sensitivity region. 
Figure 3.8: (a) Sensitivity region of the piston pin. (b) Surface plot of ?W(??,?r). 
Figure 3.9: Example of a directional sensitivity. 
Figure 3.10: SR of the aluminum and stainless steel pins. 
Figure 3.11: Most and least sensitive directions of a SR. 
Figure 3.12: An example where best-case direction has no meaning. 
Figure 3.13: Worst Case Sensitivity Region. 
Figure 3.14: SR and WCSR of the (a) steel pin, and (b) aluminum pin. 
Figure 3.15: Mathematical WCSR of the (a) steel pin, and (b) aluminum pin. 
Figure 3.16: Equal-scaled SR and WCSR of (a) steel and (b) aluminum pins. 
Figure 3.17: Normalized SR and WCSR of (a) steel pin, and (b) aluminum pin. 
Figure 3.18: Normalized ?p
0
 ranges. 
Figure 3.19: Comparison between ? and ?p
0
. 
Figure 3.20: Effect of adding robustness constraint. 
Figure 3.21: Surface plot of the wine-bottle function. 
Figure 3.22: Distributions of robust optima on a contour plot.  
Figure 3.23: Distributions of robust optima on a cross-section plot. 
Figure 3.24: A three-bar truss. 
Figure 3.25: Sensitivity analysis of the three-bar truss optima. 
Figure 3.26: SR and WCSR of the nominal optimum. 
Figure 3.27: SR and WCSR of the robust optimum. 
Figure 3.28: SR and WCSR of the mean optimum. 
Figure 3.29: SR and WCSR of the mean+std optimum. 
Figure 3.30: A welded beam assembly. 
Figure 3.31: SR and WCSR of the optima: (a) nominal, (b) robust. 
Figure 3.32: Sensitivity analysis of the welded beam optima. 
Figure 3.33: Sensitivity analysis of the welded beam optima. 
Figure 3.34: A compression spring. 
Figure 3.35: Sensitivity analysis of the nominal optimum. 
Figure 3.36: Sensitivity analysis of the robust optimum. 
Figure 3.37: Sensitivity analysis of the gradient optimum. 
 
Figure 4.1: Graphical illustration of a multi objective robustness. 
Figure 4.2: Comparison between point and set robustness. 
Figure 4.3: Comparison between nominal and robust Pareto set. 
 vi
 
Figure 4.4: Graphical definition of the generalized SR. 
Figure 4.5: (a) Generalized SR, and (b) its corresponding ?f
0
 ranges. 
Figure 4.6: Generalized WCSR. 
Figure 4.7: Graphical interpretation of the simplified constraint. 
Figure 4.8: A piston-pin-arm assembly. 
Figure 4.9: Overall SR and WCSR of the stainless steel pin. 
Figure 4.10: Constrained MORO approach. 
Figure 4.11: Nominal and robust Pareto set of the numerical example. 
Figure 4.12: Behavior of robust Pareto set as constraint is relaxed. 
Figure 4.13: Robust solutions using Monte Carlo simulations. 
Figure 4.14: Sensitivity analysis of the robust 1 design. 
Figure 4.15: Sensitivity analysis of the robust 2 design. 
Figure 4.16: Sensitivity analysis of the nominal 1 design. 
Figure 4.17: Sensitivity analysis of the nominal 2 design. 
Figure 4.18: Sensitivity analysis of the nominal 3 design. 
Figure 4.19: A pinned-pinned vibrating platform. 
Figure 4.20: Pareto sets of the vibrating platform example. 
Figure 4.21: Sensitivity of: (a) nominal, (b) robust, and (c) probabilistic designs. 
Figure 4.22: SR and WCSR of: (a) nominal, (b) robust, and (c) probabilistic designs. 
Figure 4.23: A speed reducer. 
Figure 4.24: Nominal and robust Pareto solutions of the speed reducer problem. 
Figure 4.25: Sensitivity analysis of the robust-1 (left) and robust-2 (right) design. 
Figure 4.26: Sensitivity analysis of the nominal-1 (left) and nominal-2 (right) design. 
Figure 4.27(a): SR and WCSR of the nominal-1 design. 
Figure 4.27(b): SR and WCSR of the robust-1 design. 
Figure 4.28: A power electronic module. 
Figure 4.29: Nominal and robust Pareto solutions of the power electronic problem. 
Figure 4.30: Sensitivity analysis of the nominal and robust designs. 
 
Figure 5.1: Feasibility Sensitivity Region for the inequality constraint. 
Figure 5.2: Feasibility WCSR for the inequality constraint. 
Figure 5.3: Overall FSR and FWCSR for the two constraints. 
Figure 5.4: Graphical derivation of the simplified constraint. 
Figure 5.5: Objective and constraint contours of the numerical example. 
Figure 5.6: Sensitivity analysis of the optimum designs. 
Figure 5.7: Plot of ?x
0
 vs. Prob of the FWCSR optima. 
Figure 5.8: Plot of ?x
0
 vs. Prob of the gradient and moment matching optima. 
Figure 5.9: Plot of ?x
0
 vs. f* of the four robust optimization methods. 
Figure 5.10: An explosive actuated cylinder. 
Figure 5.11: Sensitivity analysis of the nominal optimum. 
Figure 5.12: Sensitivity analysis of the probabilistic optima. 
Figure 5.13: Sensitivity analysis of the robust FWCSR optima. 
Figure 5.14: A Belleville spring. 
Figure 5.15: Plot of f vs. ?
g
 of the Belleville spring. 
Figure 5.16: Sensitivity analysis of the nominal and min-max optima. 
Figure 5.17: Sensitivity analysis of the robust (0.3) and robust (0.6) optima. 
 vii
 
Figure 5.18: Sensitivity analysis of the robust (0.8) and robust (1.0) optima. 
Figure 5.19: A control valve actuator linkage. 
Figure 5.20: Forces acting on the linkage. 
Figure 5.21: Trade-off frontier of the actuator linkage. 
Figure 5.22: FSR and FWCSR of nominal and robust optima. 
 
Figure 6.1: Intersection of SR and FSR. 
Figure 6.2: Pareto sets of the payload problem. 
Figure 6.3: Sensitivity analysis of the nominal optimum payload. 
Figure 6.4: Sensitivity analysis of the robust optimum payload. 
Figure 6.5: Sensitivity analysis of the probabilistic optimum payload. 
Figure 6.6: Comparison between the estimate and actual P
s
. 
 viii
 
LIST OF TABLES 
 
 
Table 3.1: Optimum designs of the three-bar truss. 
Table 3.2: Optimum designs of the welded beam. 
Table 3.3: Constraints of the robust optimum design. 
Table 3.4: Optimum designs of the compression spring. 
 
Table 4.1: Material properties. 
Table 4.2: Sensitivity of the constraints. 
Table 4.3: Nominal and robust designs to be analyzed. 
 
Table 5.1: Optimum designs of the numerical example. 
Table 5.2: Optimum designs of the explosive actuated cylinder. 
Table 5.3: Constraints of the optimum designs. 
Table 5.4: Probability of constraint satisfaction of the optima. 
Table 5.5: Optimum designs of the Belleville spring. 
Table 5.6: Constraint values of the Belleville spring optima. 
Table 5.7: Optimum designs of the control valve actuator linkage. 
Table 5.8: Sensitivity analysis of the optima. 
Table 5.9: Constraint values of the nominal and robust optima. 
 
Table 6.1: Objective and constraint values of the optima. 
Table 6.2: Design variables of the optima. 
Table 6.3: Probability values for uniform distribution. 
Table 6.4: Probability values for normal distribution. 
 
Table 7.1: Summary of sensitivity analysis results. 
Table 7.2: Summary of average number of function evaluations. 
Table 7.3: Summary of computational time. 
 ix
 
NOMENCLATURE 
 
 
 
FSR  Feasibility Sensitivity Region 
FWCSR Feasibility Worst Case Sensitivity Region 
GA  Genetic Algorithm 
MC  Monte Carlo simulation 
MOGA Multi Objective Genetic Algorithm 
MORO Multi Objective Robust Optimization 
SR  Sensitivity Region 
WCSR  Worst Case Sensitivity Region 
?f  variations in objectives 
?f
0
  maximum acceptable variations in the objectives 
?g  variations in constraints 
?g
0
  maximum acceptable increment in the constraints 
?p  variations in design parameters 
?p
0
  known ranges of parameter variations 
?
?
0
p   lower bound of known parameter variation ranges 
+
?
0
p   upper bound of known parameter variation ranges 
p?   normalized design parameters 
f  design objectives 
g  inequality design constraints 
G  number of design parameters 
h  equality design constraints 
 x
 
+?/
?   asymmetric robustness index 
+?/
0
?   desired level of asymmetric objective robustness 
+
0
?   desired level of one-sided objective robustness  
+
?   one-sided robustness index  
?  robustness index 
?
0
  desired level of objective robustness specified by designer 
?
g
  feasibility robustness index  
?
g,0
  desired level of feasibility robustness specified by designer 
?
min
   overall robustness index  
J  number of inequality constraints 
K  number of equality constraints 
M  number of design objectives 
N  number of design variables 
p  design parameters 
p
0
  nominal value of design parameters (values used in the optimization) 
P
s
  probability of constraint satisfaction 
f
R   normalized WCSR radius 
+?/
R
f
  radius of asymmetrical WCSR 
+
f
R   radius of one-sided WCSR 
R  radius of the worst-case estimate of the SR and FSR intersection 
R
f
  radius of WCSR 
R
g
  radius of FWCSR 
 xi
 
S
f
  sensitivity set  
S
g
  feasibility sensitivity set 
x  design variables 
x
0
  a particular instance of design variables 
 xii
 
CHAPTER 1 
INTRODUCTION 
 
1.1. MOTIVATION AND OBJECTIVE 
Back in the late 1970?s, there was a tile manufacturer in Japan called Ina Tile 
Company. One day the company discovered that an uneven temperature profile of its 
kilns was causing unacceptable variations in the size of its manufactured tiles. An 
obvious way to solve the problem was to modify the kilns by adding thermocouples and 
temperature controllers to monitor and correct the malfunction. However, this 
modification would have been very expensive. Instead, the company chose to make an 
inexpensive modification to their tile design to reduce the sensitivity of the manufactured 
tiles to temperature variations. Using statistically designed experiments, they found that 
increasing the lime content of their clay-mix from 1% to 5% reduced the variations in 
their tile size by a factor of 10 (Leon et al., 1987).  
Uncontrollable variations and noises are unavoidable in engineering design. 
Temperature variations, deviation of material properties from specifications, and 
dimensional tolerances of a design are just a few examples of uncontrollable parameter 
variations. When designing a system, these variations cannot and should not be ignored 
because they can seriously affect the performance of a design. As in the Ina Tile 
Company example above, one way to counter the effects of these variations is to try to 
reduce or eliminate the parameter variations themselves. However, this approach is 
usually very difficult to undertake and/or expensive to implement. Furthermore, it is quite 
possible that such variations will re-appear some other time in the future. A better 
 1
 
approach is to try to reduce the sensitivity of the design to the variations so that 
deteriorations caused by these variations are kept within an acceptable level.  
Dr. Genichi Taguchi from Japan is commonly credited for introducing the idea of 
reducing the sensitivity of a design, a process he called parameter design. Since then, this 
?least-sensitive design? idea has been developed much further, and later the term ?robust 
design? was coined to refer to a design alternative that is insensitive to parameter 
variations. With the introduction of design optimization into system design, it was not 
long before the idea of a robust and optimum design surfaced, and the concept of robust 
optimization became popular among researchers in the field. Following conventional 
terminologies, Parkinson et al. (1993) later introduced the term ?objective robustness? 
and ?feasibility robustness? to refer to robustness with respect to objective and constraint 
functions in an optimization problem, respectively. 
Many robust optimization methods have been developed in the literature, as will be 
reviewed in detail in Chapter 2. However, the applicability of these methods is limited to 
optimization problems with small variations, and continuous and/or differentiable 
objective and constraint functions. In addition, these methods typically presume a certain 
form of probability distribution function of the uncertain parameters, and are applicable 
for single objective optimization problems only. The computational cost of these methods 
also often limits their application to relatively simple optimization problems.  
Practically, real world optimization problems rarely exhibit the properties mentioned 
above. The functions involved in real world optimization problems are typically non-
differentiable. The parameter variations of interest are very often large, beyond the 
validity of gradient estimation. Probability distribution of the uncertain parameters is also 
 2
 
generally not known, or is difficult and expensive to estimate accurately. Many problems 
have multiple objectives that need to be considered simultaneously. In our case study for 
example, the design of a payload for an Undersea Autonomous Vehicle (Chapter 6), the 
objective and constraint functions are discontinuous, the parameter variations are large, 
and the probability distribution of parameter variations is unknown. This problem also 
has multiple objectives, instead of just one objective, for which we want to find the 
robust optimum solutions. To make matter worse, in reality it is computationally 
expensive to compute the functions involved in the problem, so computational efficiency 
of the method used is important.   
The overall objective of this dissertation is to develop an efficient robust 
optimization method for both single- and multi-objective design optimization problems to 
obtain optimum designs that are robust with respect to both objectives and constraints, 
without having to: (1) presume a probability distribution of the parameter variations, (2) 
limit parameter variations to a small (linear) range, and (3) use the gradient information 
of objective/constraint functions.  
Before we continue, it is important to provide a distinction between the concept of 
robust optimization and Post Optimality Sensitivity Analysis (POSA) (e.g., Fiacco, 
1983). Both concepts deal with the uncertainties and variabilities that exist in an 
optimization problem. However, POSA is a posteriori approach where it determines the 
sensitivity and stability of the solution due to variability after the optimization process is 
completed. The goal of POSA is to provide information to the designer regarding the 
behavior of the optimum solution and the active constraints if some of the parameters 
vary. It is hoped that based on this information, the designer can then take appropriate 
 3
 
measures to maintain the performance of the optimum design. It is essentially a passive 
approach to optimization under uncertainty. In contrast, robust optimization is an active 
approach in dealing with uncertainty whereas the sensitivity of the design is considered 
during the optimization process. The goal of robust optimization is not to inform the 
designers of how to guard the optimum design against variations, but rather to reduce the 
sensitivity of the optimum design obtained so that there is little need for the designer to 
devise corrective measures when the variabilities exist.  
 
1.2. RESEARCH COMPONENTS 
To achieve the overall objective, we developed a step-by-step approach for the 
research in this dissertation. We first developed three research components for different 
types of robustness. These research components are: (1) single objective robustness, (2) 
multi-objective robustness, and (3) feasibility robustness. Next, we combined these 
different types of robustness to obtain a robust optimization method that accounts for 
both objective and feasibility robustness. In the next three sub-sections, an overview and 
objective of each research component is given. 
 
1.2.1. Research Component 1: Single Objective Robust Optimization 
The first research component is concerned with variations in the objective value of 
an optimum design due to uncontrollable variations in the parameters. This so-called 
?objective robustness? of an optimum design is important because if its objective value 
changes significantly, then performance of the design can degrade so much that it may be 
deemed unsatisfactory. Objective robustness of an optimum design is especially 
 4
 
important if the design is part of a larger system because deviations in the design?s 
performance could affect the rest of the system.  
The objective of the first research component is to develop a novel robust 
optimization method for single objective optimization problems that can obtain an 
optimum design solution that is robust in terms of the objective function.  
 
1.2.2. Research Component 2: Multi-Objective Robust Optimization 
The second research component is concerned with performance variations of an 
optimum design when there are multiple objectives involved. Similar to the first research 
component, here we also look into the changes in the objective values of a design due to 
variations in the parameters. However, since there are now multiple objectives, we have 
to examine the performance variation of the design with respect to each objective, and 
then based on it, determine the overall robustness of the design.  
The objective of the second research component is twofold. First, since the notion of 
a design that is optimum and robust for multiple objectives has not yet been defined in the 
literature, this research component aims to introduce and develop the concept of ?multi-
objective robustness? and ?multi-objective robust optimality? of a design alternative. 
Second, this research component seeks to develop a novel method for robust optimization 
of a design in multi-objective optimization problems.  
 
1.2.3. Research Component 3: Feasibility Robust Optimization 
The third research component is concerned with the feasibility of an optimum design 
due to uncontrollable variations in parameters. Typically, an optimum solution to an 
 5
 
engineering optimization problem is a boundary optimum, i.e., at the optimum at least 
one of the constraints is active (Papalambros and Wilde, 2000). Because of this, if some 
of the problem?s parameters vary, the optimum design may no longer be feasible. A 
design that is always feasible even if there are parameter variations is called ?feasibly 
robust,? and the method to obtain a feasibly robust solution is called ?feasibility robust 
optimization? method.  
The objective of the third research component is to develop a novel and efficient 
feasibility robust optimization method to obtain an optimum design that is always feasible 
regardless of parameter variations. 
 
1.3. ASSUMPTIONS 
In developing our robust optimization method, we make the following assumptions: 
? For objective robust optimization, we assume that there exists a trade-off between 
objective values of a design, and its robustness. If such a trade-off does not exist, 
then an optimum design is also a robust design, and there is no need to conduct 
robust optimization. 
? For feasibility robust optimization, we assume that the optimum is on (or near) the 
boundary of the feasible domain. If the optimum is well inside the feasible 
domain, then most likely that optimum is already feasibly robust, and we do not 
need to conduct robust optimization.  
? We assume that the range of parameter variations is known a priori, and that they 
are symmetric. (Robust optimization with asymmetric parameter variations will 
be discussed briefly in Chapter 6.) 
 6
 
1.4. ORGANIZATION OF DISSERTATION 
The rest of the dissertation is organized as follows. Chapter 2 gives the definitions of 
concepts and terminologies used throughout the dissertation, as well as a comprehensive 
review of related previous work in the literature. We develop the method for objective 
robust optimization of a single objective optimization problem in Chapter 3 (Research 
Component 1), and extend it to multi-objective problems in Chapter 4 (Research 
Component 2). In Chapter 5, we develop a method for feasibility robust optimization 
(Research Component 3). Chapter 6 presents our combined objective and feasibility 
robust optimization method. To demonstrate the applications of our method, several 
numerical and engineering examples are given in Chapters 3 through 6. Chapter 7 
concludes the dissertation with some remarks as well as a discussion on the contributions 
of the dissertation and potential future research directions.  
After reading this chapter and the next, we recommend that the readers continue with 
Chapter 3 first because it contains the bulk of our research results that will become the 
foundation of Chapters 4, 5 and 6. Chapters 4 and 5 may be read separately; however, 
Chapter 6 should be read after Chapters 3, 4, and 5.  
Figure 1.1 shows the organization and flow of information in this dissertation.  
 
 7
 
CHAPTER 1
Motivation and overall objective; 
research components; assumptions
CHAPTER 2
Definitions and terminologies; 
review of related works
CHAPTER 3
Single objective robust 
optimization
CHAPTER 4
Multi-objective robust 
optimization
CHAPTER 5
Feasibility robust optimization
CHAPTER 6
Combined objective and feasibility 
robust optimization
CHAPTER 7
Conclusions; contributions; future 
research directions
 
Figure 1.1: Organization of dissertation. 
 8
 
CHAPTER 2 
DEFINITIONS AND PREVIOUS WORK 
 
2.1. INTRODUCTION 
In this chapter, we provide several definitions and terminologies that will be used 
throughout the dissertation. In addition, we also give a comprehensive review of previous 
work in the literature related to single objective, multi-objective, and feasibility robust 
optimization concepts and methods. 
 
2.2. DEFINITIONS AND TERMINOLOGIES 
 A general single objective optimization problem can be formulated as shown in 
Eq. (2.1) below. 
K,...,1;0),(h
J,...,1;0),(g:subject to
),(minimize
==
=?
k
j
f
k
j
p
p
p
x
x
x
x
   (2.1) 
Here, f is the objective function to be optimized, x = [x
1
,?,x
N
]
t
 is the design variable 
vector (with superscript ?t? referring to the transpose of the row vector), and p = 
[p
1
,?,p
G
]
t
 is the set of parameters. For practical reasons, x and p are restricted to real 
values. The problem has J inequality constraints, g
j
, j = 1,...,J, and K equality constraints, 
h
k
, k = 1,?,K.  
In this dissertation, the notation p represents problem factors that have variability, 
including design variables. Following this notation, if there are variations in some of the 
design variables, then a subset of x belongs to p, i.e., these are the design factors that we 
control during the optimization of f but these factors have variability. Some researchers 
 9
 
prefer to differentiate between variations in x and variations in p, the so-called type-1 and 
type-2 variations (Chen et al., 1996; Kalsi et al., 2001). For simplicity, we do not make 
that distinction. Also, the parameters value p = p
0
 that we use to optimize a design is 
called the nominal parameters value.  
Because of noise and uncertainty, parameter values vary by some amount: ?p = 
(?p
1
,?,?p
G
)
t
, and in turn these variations affect the objective and constraint values of a 
design.  
The goal of objective robust optimization is to obtain a design variable vector x
R
 
whose objective value f(x
R
,p) is not only minimum but also remains within an acceptable 
bound when p varies. In other words, the objective value of the design is insensitive to 
variations in p.  
The goal of feasibility robust optimization is to obtain a design variable vector x
R
 
whose inequality constraint values g
j
(x
R
,p), j=1,..,J, are always feasible regardless of ?p 
variations. Feasibility robust optimization is concerned only with inequality constraints 
(i.e., equality constraints are not considered in feasibility robust optimization). This is 
because equality constraints are ?hard? to consider, i.e., unless the ?p variations are such 
that h
k
(x,p+?p) = 0 for all k=1,?,K, there is no way to guarantee that these equality 
constraints will always be satisfied.  
When there is more than one objective function to minimize, the problem becomes a 
multi-objective optimization problem as shown in Eq. (2.2).  
[ ]
K,...,1;0),(h
J,...,1;0),(g:subject to
,...,,),(minimize
M21
==
=?
=
k
j
fff
k
j
p
p
p
x
x
xf
x
       (2.2) 
 10
 
The formulation in Eq. (2.2) is the same as that of a single objective optimization 
problem, except that now we have M > 1 objectives to minimize simultaneously. Here, 
we assume that the number of objectives is finite (i.e., M < ?). We also assume that at 
least two of the objectives are conflicting, i.e., for a given design, as we decrease 
(improve) the value of one objective, the value of at least one other objective increases 
(worsens). Because of this trade-off among the objectives, there is generally a set of 
optimum solutions to the problem in Eq. (2.2). This set is called a Pareto set (see 
definitions below), and the design solutions in the set are called Pareto designs. The 
Pareto set is a trade-off set meaning that there is no design in the set that dominates or is 
better than the other designs in the set. Extensive reviews of multi-objective optimization 
concepts and methods are given by Miettinen (1999), and in evolutionary multi-objective 
optimization by Deb (2001) and Coello Coello et al. (2002). 
When there are variations in design parameters (i.e., ?p), some or all of the M 
objectives will be affected. The goal of multi-objective robust optimization is to obtain a 
design variable vector x
R
 whose objective values f(x
R
,p) are Pareto optimum, and at the 
same time are insensitive to these ?p variations for all objectives.  
Next, we provide several definitions and terminologies used in this dissertation. 
Objective space (f-space): An M-dimensional space in which the coordinate axes are the 
objective values. 
Parameter variation space (?p-space): A G-dimensional space in which the coordinate 
axes are the parameter variation (?p) values. 
Normalized parameter variation space ( p? -space): A G-dimensional ?p-space where all 
the entries are normalized by the known ranges of parameter variations (?p
0
), i.e., 
 11
 
,0
p
p
p
i
i
i
?
?
=?
, for all i=1,?,G. 
Maximum acceptable performance variation (?f
0
 = [?f
1,0
,?,?f
M,0
]
t
): The maximum 
acceptable change in the objective function values f. These values are determined by 
the designer. The maximum acceptable change may also be given as a percentage 
value.  
Inferiority, non-inferiority, and dominance: In multi-objective minimization, a feasible 
design point x
a
 is said to be inferior with respect to (w.r.t.) another feasible design 
point x
b
 if f
i
(x
b
) ? f
i
(x
a
) for all i=1,?,M, with strict inequality for at least one i. 
Correspondingly, the design point x
b
 is said to dominate x
a
. If x
a
 neither dominates 
nor is inferior to x
b
, then x
a
 and x
b
 are said to be non-inferior w.r.t. each other.  
Inferior, non-inferior, and dominant regions: In multi-objective minimization, the inferior 
region of a design x
a
 is defined to be the region in the objective space where the 
design points are dominated by x
a
. Similarly, regions in the objective space where 
the design points in the regions are non-inferior and dominate x
a
 are called the non-
inferior and dominant regions of x
a
, respectively. 
Trade-off set: A set of design points is a trade-off set if all points in the set are non-
inferior with respect to each other (although there might be design points outside the 
set that dominate the points in the set).  
Pareto optimality, Pareto set, and Pareto frontier: In multi-objective optimization, a 
feasible design point x
p
 is Pareto optimum if it is not inferior w.r.t. any other feasible 
design point. The set of all Pareto optimum points is called the Pareto set. The plot of 
the Pareto set in the objective space is called the Pareto frontier.  
 12
 
Nominal Pareto set: Pareto set of a multi-objective optimization problem without 
robustness consideration. 
Robust Pareto set: A trade-off set whose elements are both multi-objectively robust and 
Pareto optimum (see Chapter 4 for further explanations). 
Vector operator ?: Let a and b be nx1 vectors. We define the operation: c = a ? b, where 
c is a nx1 vector whose elements are:  c = (a
1
b
1
, a
2
b
2
,?, a
n
b
n
)
t
.  
Ternary vector operator ??? : Let a, b, and c be nx1 vectors. We define the operation: d = 
, where d is a nx1 vector whose i-th element is d  for all 
i=1,?,n. 
?? cba ,,
?
?
?
>
?
=
0a if,ca
0a if,ba
iii
iii
i
 
2.3. OVERVIEW OF PREVIOUS WORK 
Most of the robust optimization methods in the literature are developed to account 
for both objective and feasibility robustness of an optimum design. However, some 
methods are developed to account for objective robustness only, while others are for 
feasibility robustness only. There are also some related works that developed methods for 
robustness measurement of a design only, without optimization. 
In general, these methods can be categorized into three main groups: (i) experiment-
based methods, (ii) deterministic methods, and (iii) probabilistic methods. Experiment-
based methods are those methods that perform local sampling around the nominal value 
of a design to probe its behavior under parameter variations. Taguchi?s orthogonal array 
(Taguchi and Phadke, 1984; Kackar, 1985; Phadke, 1989) and simple random sampling 
(Branke, 1998, 2001) are examples of experiment-based methods. Deterministic methods 
 13
 
are those methods that do not use statistical measures in calculating a design?s 
robustness; rather, they use some deterministic measures. Gradient minimization 
(Belegundu and Zhang, 1992), worst-case analysis (Parkinson et al., 1993), and the mini-
max method (Hirokawa and Fujita, 2002) are examples of deterministic methods. 
Probabilistic methods are those methods that use statistical measures, such as mean and 
variance, to calculate a design?s robustness. These methods often use a Taylor series 
expansion to estimate these statistics. Examples of probabilistic methods include Yu and 
Ishii?s variation pattern method (1994, 1998), reliability index method (Tu et al., 1999; 
Youn et al., 2003), and most probable point method (Du and Chen, 2000).  
In addition to these three main groups, there are other more specialized robust 
optimization methods such as fuzzy robustness method (Otto and Antonsson, 1993; 
Arakawa and Yamakawa, 1998), tolerance maximization method (Balling et al., 1998), 
and Zhu and Ting?s performance sensitivity distribution method (2001). There are also 
methods that combine several of the methods from the above three main groups.  
Experiment-based methods are generally simple and straightforward, but their 
computational efforts grow rapidly, and eventually become impractical for problems with 
many parameters because they are essentially based on exhaustive permutations of all 
possible parameter variations. The computational cost of these methods is even more 
prohibitive as the number of discrete variation levels to be analyzed becomes large. Even 
for methods that use only partial permutations (Branke, 2001) the computational cost is 
still very high. Because of this computational issue, often these methods require 
preliminary experiments to eliminate those parameters that are statistically insignificant, 
 14
 
and to determine the levels of variations to be analyzed. These preliminary experiments 
are often costly and difficult.  
Many deterministic methods (Belegundu and Zhang, 1992; Parkinson et al., 1993) 
try to reduce the computational effort by using gradient information to estimate a 
design?s robustness. Estimating the gradient of a function is indeed computationally more 
efficient than exhaustive permutations. However, since these methods need gradient 
information, obviously they are only applicable to optimization problems whose 
functions are differentiable. These methods cannot solve robust optimization problems 
having non-smooth objective and/or constraint functions (e.g., a step function). Besides, 
as parameter variations grow large (beyond the range in which linear approximation is 
valid), gradient estimation will cease to be valid. 
Some deterministic methods use worst-case analysis to calculate a design?s 
robustness (Badhrinath and Rao, 1994; Hirokawa and Fujita, 2002). These methods are 
also computationally more efficient than experiment-based methods. However, the results 
obtained are typically conservative because they use the worst possible instance of a 
design?s performance as its robustness measure.  
Probabilistic methods (Yu and Ishii, 1994, 1998; Tu et al., 1999) extend the 
experiment-based methods by calculating probability information of a design based on a 
probability distribution of the parameters. For objective robustness, they calculate the 
mean and variance of a design?s performance. For feasibility robustness, they calculate 
the probability of constraint satisfaction for a design. Clearly, these methods require that 
the probability distribution of parameters is known a priori (which often is not the case), 
and the results obtained from these methods are dependent on the validity of the assumed 
 15
 
distribution. The computational cost of these methods is also more prohibitive than 
experiment-based methods because they need to calculate probability information. Even 
for those probabilistic methods that claim to be efficient (Du and Chen, 2000), the 
number of function evaluations needed is still very high. 
Almost all of the robust optimization methods in the literature are only applicable to 
single objective optimization problems. It is widely acknowledged, however, that an 
engineering design problem generally has multiple conflicting objectives. Very few 
papers address the issue of multiple objectives (sometimes also called multi-criteria): Rao 
(1984), Pignatiello (1993), and Ramakrishnan and Rao (1996). However, these methods 
essentially convert a multi-objective robust optimization problem into a single-objective 
one by aggregating the performance variations, and do not take into account trade-offs 
among the solutions. To our knowledge, there is no reported work in the literature that 
has formulated and defined the concept of a robust and multi-objectively optimum design 
as will be developed in this dissertation.  
A more detail discussion of related works in the literature is given in the next three 
sections. For clarity, we divide our discussions according to our research components: (i) 
single objective robust optimization, (ii) multi-objective robust optimization, and (iii) 
feasibility robust optimization.  
 
2.3.1. Single Objective Robust Optimization 
One of the earliest works in objective robustness is the parameter design method of 
Taguchi (Taguchi, 1978; Taguchi and Phadke, 1984; Kackar, 1985; Phadke, 1989). This 
method is an experiment-based method that uses full-factorial experiments to obtain the 
 16
 
responses of a design, calculates the mean and variance of the responses, and then based 
on these values, minimizes a quantity called signal-to-noise (S/N) ratio. Taguchi?s 
method has been widely used to obtain a robust design, e.g., Pignatiello and Ramberg 
(1985), Wang et al. (1999), Hwang et al. (2001). However, it also has received much 
criticism for its use of the S/N ratio because it could result in a design with very low 
performance or very high variance. Leon et al. (1987) proposed an alternative to S/N 
ratio, and developed a robustness measure called PerMIA (Performance Measure 
Independent of Adjustment). They showed that PerMIA is a quality loss measure that 
Taguchi originally proposed in his parameter design concept, but did not use in his 
formulation. They also showed that PerMIA is a more reliable measure than S/N ratio in 
terms of solution quality, and that for certain special cases, PerMIA simplifies to the S/N 
ratio. The use of PerMIA in factorial experiments is also proposed and discussed in 
Pignatiello and Ramberg (1987) and Box (1988).  
In a different approach to Taguchi?s experiment-based method, some methods 
calculate a design?s robustness deterministically. Many methods (e.g., Belegundu and 
Zhang, 1992) use the gradient of the objective function as a robustness measure, and 
minimize a weighted sum (or some other combinations) of the objective and gradient 
values. However, Badhrinath and Rao (1994) showed that in general this weighted-sum 
approach is not reliable because it could lead to a local maximum solution. Instead they 
proposed a worst-case approach where they minimize the maximum objective value 
within the given parameter range. Parkinson (1998, 2000) and Hirokawa and Fujita 
(2002) also developed methods based on this worst-case min-max strategy. However, this 
min-max approach is often computationally extensive because it calculates the maximum 
 17
 
objective value every time a new feasible solution is obtained. To reduce computation 
cost, Sundaresan et al. (1992, 1993) proposed to simply calculate objective values at the 
?corners? of parameter range, and used their average as a robustness measure. 
Balling et al. (1986) proposed an interesting deterministic method for robust 
optimization. Unlike other methods, their method works ?backward? in that they first 
specify an acceptable variation in objective value, and then maximize the parameter 
range. Zhu and Ting (2001) also proposed a similar ?backward? approach where they 
define relative robustness of a design based on the relationship between variation in the 
objective value and the parameter range corresponding to this variation. 
A large portion of the literature uses probabilistic measures to determine robustness, 
most commonly the mean and variance of objective value. The simplest of these methods 
is the sampling approach where the mean and variance values are estimated by local 
sampling around the nominal objective value: Tsutsui and Gosh (1997), Branke (1998, 
2001), Tsutsui (1999). Many probabilistic methods estimate the mean and variance using 
a Taylor series expansion, and then minimize a weighted sum of the two values: Yu and 
Ishii (1994, 1998), Du and Chen (2000), Jung and Lee (2002). Many other methods 
propose to treat mean and variance as two conflicting objectives, and use multi-objective 
optimization to simultaneously optimize them: Chen et al. (1996), Simpsons et al. (1997), 
Chen and Yuan (1999), Chen et al. (2000), Kalsi et al. (2001).  
 
2.3.2. Multi-Objective Robust Optimization 
 Only a few methods in the literature are applicable to multi-objective robust 
optimization problems, and most of them are generalizations of Taguchi?s loss function 
 18
 
(i.e., the S/N ratio). The first work to extend the loss function to multiple responses is by 
Pignatiello (1993) for the case of nominal-the-best. Tsui (1999) later expanded the work 
to include smaller-the-better and larger-the-better cases. Not wanting to use the S/N ratio 
in their robust design, Elsayed and Chen (1993) developed a quantity called PerMQ 
(Performance Measure on Quality), a generalization of the PerMIA measure of Leon et 
al. (1987) for single-response problems.  
Other non-Taguchi based methods have been developed as well: Rao (1984), 
Ramakrishnan and Rao (1996), Lee and Lee (2001), Messac and Yahaya (2002), 
Shelokar et al. (2002), but they all use different approaches and are not focused on a 
certain approach.   
None of the methods mentioned above take into account the trade-off that exists 
among the multiple objectives. A few researchers attempt to include Pareto dominance 
when searching for a robust solution (Kunjur and Krishnamurty, 1997; Fernandez et al., 
2001). However, a true multi-objective robust optimization method that accounts for both 
multi-response robustness and Pareto optimality, as presented in this dissertation, has not 
yet been developed.  
 
2.3.3. Feasibility Robust Optimization 
Many deterministic methods have been developed for feasibility robust optimization, 
the simplest being the use of a safety factor. Several methods propose a worst-case 
approach where they constrain the largest constraint variation instead of the original 
constraint value: Parkinson et al. (1993), Yu and Ishii (1994, 1998), Hirokawa and Fujita 
(2002). Other methods propose non-mathematical ?procedures? to guarantee a design?s 
 19
 
robustness: ?closest feasible point? method (Balling et al., 1986), ?corner space? method 
(Sundaresan et al., 1992, 1993). Yet some other methods use fuzzy logic rules to 
determine if a design is feasibly robust: Arakawa and Yamakawa (1998), Rao and Cao 
(2002).  
The most popular of all feasibility robustness approaches is the probabilistic 
approach. Some methods calculate the statistical variance of the constraint value using a 
Taylor series expansion, and then add this variance to the original constraint, a so-called 
?moment matching? approach: Wu et al. (1990), Parkinson et al. (1993), Chen et al. 
(1996), and Ramakrishnan and Rao (1996). Some other methods replace the original 
constraints with a constraint on the probability of constraint satisfaction of the optimum 
design. This ?chance-constrained? approach has been well developed for linear problems 
(Charnes and Cooper, 1959; Charnes and Cooper, 1963), as well as non-linear ones (Du 
and Chen, 2000; Jung and Lee, 2002).  
The chance-constrained approach has been much expanded and developed further 
using reliability analysis techniques, where a design?s probability of constraint 
satisfaction is calculated via a so-called ?reliability index?: Chandu and Grandhi (1995), 
Choi et al. (2001), Choi and Youn (2002), Youn et al. (2003). Tu et al. (1999) generalize 
this approach to two approaches: the conventional Reliability Index Approach (RIA) and 
the Performance Measure Approach (PMA), and show that in most cases PMA is better 
than RIA. Du and Chen (2000) later propose to perform a local sampling around the so-
called ?most probable point? to get a more accurate estimate of probability of constraint 
satisfaction, as well as to improve computational efficiency.  
 20
 
CHAPTER 3 
SINGLE OBJECTIVE ROBUST OPTIMIZATION 
 
3.1. INTRODUCTION 
When optimizing a design or a system, it is important to ensure that its performance 
at the optimum does not change significantly if some parameters vary uncontrollably. In 
designing a racecar, for example, it is often desirable that its weight be minimized. At the 
same time, it is also necessary to guarantee that a change in its weight (due to a change in 
some uncontrollable parameters) is limited. If the weight of the vehicle increases 
significantly, the vehicle speed may decrease considerably. If the weight decreases 
significantly, the lift that the vehicle experiences might result in lost traction. This so-
called performance robustness (or objective robustness) of a design is especially critical if 
the design is part of a larger system. A robotic arm for a cutting tool, for instance, is often 
optimized for maximum movement speed. However, once an optimum speed is decided, 
it must be maintained so that it is compatible with other parts of the robotic system that 
are designed for that speed.  
The purpose of this chapter is to present a robust optimization method that can obtain 
a design solution that is objectively robust for a single objective optimization problem. A 
design is objectively robust if the variation in its objective function (i.e., variation in its 
performance) is small, within an acceptable range specified by the designer.  
We begin this chapter by presenting a method to measure the sensitivity (or inversely 
the robustness) of a design using a sensitivity region concept. We then present an 
approach to use such a measure to obtain a robust optimum design. Several numerical 
 21
 
and engineering comparison studies are given in this chapter to demonstrate the 
applications of the method. Note that in this chapter we consider objective robustness 
only, and assume that the robust optimum obtained is feasible when parameter variations 
occur. We will present our feasibility robustness approach later on in Chapter 5. 
 
3.2. TWO-SIDED SENSITIVITY MEASURE 
We start by developing a method to measure the two-sided sensitivity of a design. 
This sensitivity measure is developed based on the notion that for each design alternative, 
there is a region in ?p-space that can be used to evaluate that design?s sensitivity. This 
measure is a ?two-sided? measure because we limit both the increase and decrease of the 
design performance (unlike feasibility robustness in Chapter 5, which is ?one-sided?). 
 
3.2.1. Sensitivity Set 
Let x
0
 be a design alternative whose sensitivity we want to measure, and let p
0
 be the 
nominal parameter values for which the objective value of that design is defined, i.e., 
f(x
0
,p
0
). If a subset of x belongs to p, then the p
0
 values of this subset are its x
0
 values. 
Also let ?f
0
?0 be the maximum acceptable changes for the design performance as 
determined by the designer. For this design and given ?f
0
, there is a set of ?p values such 
that the changes in f(x
0
,p
0
) due to these ?p?s are less than or equal to ?f
0
. This set is 
called the ?sensitivity set? (S
f
) of the design, and mathematically it is defined by 
Eq. (3.1). (We use the square of ?f to account for negative values.) 
[ ] [ ]{ }
),(),()(:where
)(:R)(
0000
2
0
2G
0
pppp
ppS
xx
x
fff
ff
f
??+=??
??????=
   (3.1) 
 22
 
Let us take a moment to discuss the importance of this set. What exactly is a 
sensitivity set? As shown in Eq. (3.1), S
f
 is a set of ?p?s that can be allowed to happen if 
we want variation in f(x
0
,p
0
) to be within ?f
0
. So, a sensitivity set is essentially a 
collection of parameter changes ?p that a design can ?absorb? before it violates the 
acceptable performance variation limit ?f
0
. Clearly, a design that can absorb a large 
amount of ?p is less sensitive (or more robust) than a design that can absorb only a small 
amount. This observation implies that S
f
 is an indicator of a design?s sensitivity. As the 
number of elements in the set S
f
 increases, the design can allow more changes in p. This 
in turn brings up two key observations: 
1. Given two designs, the design with a larger S
f
 is less sensitive (more robust) to 
changes in p than the design with a smaller S
f
 
2. If we can control ?p such that it is always a member of S
f
, then we can guarantee 
that the ?f
0
 limit will always be satisfied. 
Provided f(x
0
,p
0
) is defined and thus exists, S
f
 is guaranteed to exist and is always 
unique. The existence of S
f
 is easy to see. Because f is defined for the pair (x
0
,p
0
), then 
the smallest set possible is S
f
 = {0}. An empty S
f
 implies that f(x
0
,p
0
) does not exist; a 
contradiction to our assumption. The uniqueness of S
f
 is also straightforward. Suppose x
0
 
has J non-unique sensitivity sets that satisfy Eq. (3.1): S
f,1
,?,S
f,J
. Then the unions of all 
these sets must necessarily also satisfy Eq. (3.1), and this superset becomes the unique 
sensitivity set of x
0
: S
f
 = {S
f,1
 ??? S
f,J
 }. 
Let us demonstrate with an example how to use the sensitivity set of a design to 
measure its robustness.  
 
 23
 
Example 3.1 
Consider a cylindrical piston pin made out of stainless steel (density ? = 8.0 gr.cm
-3
) 
whose height and radius are: h = 5 cm and r = 2 cm, respectively. The nominal weight of 
the pin is W = 502.6 gr, but suppose due to variations in ? and r, the actual value of the 
pin?s weight varies. If we want the weight to vary by at most ?W
0
 = 10 gr, determine the 
?? and ?r values that can be allowed to occur. 
Solution 
The weight of the pin is given by W = ??r
2
h. When there are variations ?? and ?r, 
the weight becomes W? = (?+??)?(r+?r)
2
h. Setting the square of the weight difference 
(with and without the variations): ?W
2
 = (W??W)
2
, to be less than (?W
0
)
2
, we obtain a 
quadratic inequality: 
 0r?
)? h(
W
t)? r2(t
42
2
2
022
?
?
?
?
?
?
?
?
?
?
?
??     (3.2) 
where: t = uv
2
, u = (?+??), and v = (r+?r). Solving the inequality in Eq. (3.2) and using 
backward substitution, we obtain the sensitivity set of the pin design: 
?
?
?
?
?
?
?
?
?
?
?
?+
?
+
????
?+
?
?
??= ?
)rr(
? h
W
? r
??
)rr(
? h
W
? r
:)r,?(
2
0
2
2
0
2
f
S              (3.3) 
As long as the condition in Eq. (3.3) is satisfied, the pin weight will always remain within 
?10 gr. For a simple verification, if (??,?r) = (-0.4, 0.05), then the pin weight is 501.7 gr 
and the condition in Eq. (3.3) is satisfied. If (??,?r) = (1.0, -0.02), then the pin weight is 
554.2 gr, and the condition in Eq. (3.3) is violated. If (??,?r) = (0.495, -0.04), the 
 24
 
condition is actively (as an equality) satisfied, and the pin weight is 512.6 gr (exactly 
equal to: nominal weight plus ?W
0
).  
The reader might have readily observed that there is an apparent relationship 
between strict satisfaction of Eq. (3.3) and the amount of the ?W
0
 limit being used up. 
This observation is not a coincidence. In fact, it is an important property that later will 
become the basis for all of our robust optimization methods. ? 
 
3.2.2. Sensitivity Region 
Because we are dealing with continuous ?p, mathematically the size of S
f
 is infinite, 
and as such we cannot explicitly use it as a measure of a design?s robustness. However, if 
we plot S
f
 in the ?p-space, we obtain a region surrounding the origin that we call the 
?sensitivity region? (SR) of the design. The size of this region explicitly corresponds to 
the size of S
f
, and therefore the size of a SR is also a measure of a design?s sensitivity: the 
larger the region, the less sensitive (more robust) the design. Figure 3.1 shows a typical 
example of a SR (for a two-parameter function). 
1
p?
2
p?
Point A
Point B
Point C
0
[][]
2
0
2
C
)( ff ?>?? p
[][]
2
0
2
A
)( ff ?=?? p
[][
2
0
2
B
)( ff ?<?? p ]
 
Figure 3.1: Sensitivity region of a design alternative. 
Since SR is simply a plot of S
f
, the conditions for a ?p point to be inside, outside, or 
on the boundary of a SR can be derived from Eq. (3.1). It is obvious that if a point is 
 25
 
outside a SR (e.g., point C), then Eq. (3.1) is not satisfied, i.e., [?f (?p)]
2
 > [?f
0
]
2
. What 
about points inside and on the boundary of a SR? It is not obvious from Eq. (3.1) what 
the conditions for these points are. As it turns out, a point inside a SR (e.g., point B) 
satisfies [?f (?p)]
2
 < [?f
0
]
2
, while a point on the boundary of a SR satisfies Eq. (3.1) with 
an equality [?f (?p)]
2
 = [?f
0
]
2
.  
Let us discuss the boundary point condition first. The argument for this condition is 
as follows. Suppose a point is on the boundary of a SR but Eq. (3.1) is satisfied as an 
inequality. Then the inequality implies that ?p can change by an infinitesimal amount in 
any direction, and the condition [?f (?p)]
2
 ? [?f
0
]
2
 is still satisfied. But geometrically, if 
the point is on the boundary of a SR, then there is at least one direction along which the 
change in ?p, no matter how small, will push the point to the outside of SR, i.e., 
[?f (?p)]
2
 > [?f
0
]
2
 (Figure 3.2). This is a contradiction, so we conclude that if a point is 
on the boundary of a SR, then Eq. (3.1) must necessarily be satisfied with an equality. 
1
p?
2
p?
0
Point A
cone of 
out-of-bound
directions
sensitivity
region
 
Figure 3.2: Geometric condition for a boundary point. 
It should be noted that the above argument is valid only if f(x,p) is continuous with 
respect to p (but not necessarily differentiable). If f(x,p) (hence ?f (?p)) is discontinuous, 
the condition [?f (?p)]
2
 = [?f
0
]
2
 is still valid as long as the discontinuity does not occur 
 26
 
on the SR boundary (i.e., as long as it is an inner discontinuity). However, even if the 
discontinuity does occur on the boundary, the condition can still be valid provided the 
discontinuity is such that [?f (?p)]
2
 = [?f
0
]
2
 as the discontinuity is approached from the 
left or from the right (Figure 3.3). Most engineering functions are continuous, so we 
rarely have to deal with this situation (recall that we are not dealing with discrete ?p). 
However, if the function of interest happens to be discontinuous, then we assume that the 
discontinuity observes either one of the two conditions described above. 
p?
f?
0
inner discontinuity
0
f??
0
f?+
sensitivity region
[][]
2
0
2
)p( ff ?=??
is valid
boundary 
discontinuity
[][]
2
0
2
)p( ff ?=??
is valid
boundary 
discontinuity
[][
2
0
2
)p( ff ?=?? ]
is not valid
 
Figure 3.3: One-dimensional SR example of a discontinuous function. 
The argument for the condition of a point inside a SR is similar to that of a boundary 
point condition, but there is a small complication that must be addressed. Strictly 
speaking, the inner point condition should be [?f (?p)]
2
 ? [?f
0
]
2
 because it is possible that 
the equality is satisfied by inner points also (as shown in Figure 3.4). However, since we 
are using the size of SR to measure the robustness of a design, we are interested primarily 
in the SR boundary. So, to avoid computational problems it is necessary to make a clear 
distinction between inner and boundary point conditions. To make this distinction, we 
modify the given ?f
0
 to , where ?
f
ff ?
?
00
+?=?
f
 is an infinitesimal positive number. 
 27
 
Using , the inner point condition becomes [  and the boundary point 
condition becomes [ . 
0
?
f?
2
0
2
]
?
[)]( ff ?<?? p
f
2
0
2
]
?
[)]( ff ?=?? p
?
0
inner point
[][]
2
0
2
)p f?=(f ??
2
)]p
2
)]p
p?
 
0
f??
0
f?+
sensitivity region
inner points
[][]
2
0
2
)p( ff ?=??
Figure 3.4: One-dimensional example of equality inner point condition. 
Mathematically, the boundaries defined by [  and 
 are slightly different. However, since ?
2
0
2
]
?
[)]( ff ?=?? p
2
[)]( ff ?<?? p
2
0
][([ ff ?=??
2
0
]
?
[([ ff ?=??
f
 is infinitesimal, practically 
they are the same. For the rest of our discussions, we will use [ and 
 for inner and boundary point conditions, respectively. For simplicity, 
we will use the notation ?f
2
0
]
?
0
 to refer to . 
0
?
f?
The existence and uniqueness of a SR follows directly from the existence and 
uniqueness of S
f
. One property of SR that should be noted is that it always encloses the 
origin of the ?p-space because the set {0} is always a member of S
f
. This property will 
become important later on. It should also be noted that a SR does not necessarily have to 
be fully bounded although we showed a fully bounded SR example in Figure 3.1 and 3.2. 
By fully bounded we mean that in the G-dimensional ?p-space, the SR boundary is a 
closed hyper-surface. However, we do assume that SR is bounded along at least one ?p-
direction, not necessarily the coordinate axis (Figure 3.5). 
 28
 
1
p?
2
p?
0
closed
curve
(a) Fully bounded SR
1
p?
2
p?
0
bounded
direction
(b) Not fully bounded SR
 
Figure 3.5: Boundedness of a sensitivity region. 
The assumption for the SR?s one-directional boundedness is necessary because if the 
SR is completely unbounded, then it implies that the design is already inherently robust, 
and hence there is no need to calculate its robustness. Two conditions can cause a SR to 
be unbounded. First, the objective function f(x
0
,p
0
) is independent of ?p, i.e., 
f(x
0
,p
0
) = f(x
0
,p
0
+?p), ??p. In turn, this means that the objective value of the design x
0
 is 
unaffected by changes in p (i.e., the hyper-surface of f(x
0
,p
0
) with respect to p is flat). 
Second, ?f
0
 is larger than the maximum possible ?f (e.g., the sinus function in 
Figure 3.6). This also implies that x
0
 is practically unaffected by ?p; at least for the given 
?f
0
. If either condition holds, then any design alternative is already inherently robust, and 
there is no need to search for the robust optimum design. 
-6 -4 -2 0 2 4
-1
-0.5
0.5
1
6
p?
)psin(?=?f
0
f?+
0
f??
max(?f) < ?f
0
1.5
-1.5
 
Figure 3.6: Example of a condition that causes SR to be unbounded. 
 29
 
 Note also that a SR does not have to be a single connected region. It can be a 
collection of several disjointed regions in which one of them encloses the origin 
(Figure 3.7). Each of the disjointed regions can be either fully or partially bounded. 
1
p?
2
p?
0
encloses
origin
 
Figure 3.7: Example of a disjointed sensitivity region. 
A SR has a geometric significance as well. If we look into the hyper-surface of the 
function ?f(?p), then SR is simply the collection of points sandwiched between two 
parallel ??f
0
 and +?f
0
 hyper-planes, projected onto the ?p hyper-plane (as shown in 
Figure 3.3 and 3.4 for one dimension). In other words, SR is the region between the 
contours of ?f(?p) for values ?f = ??f
0
 and ?f = +?f
0
. Since contours of a function 
always exist and are unique, this observation provides yet another justification for the 
existence and uniqueness of SR. In addition, it also provides an explanation as to why it 
is possible for a SR to be not fully bounded or even be disjointed.  
 
Example 3.2 
Let us revisit our piston pin design example again. In Example 3.1 we derived the 
sensitivity set of the design when its height and radius are h = 5 cm and r = 2 cm, 
respectively. Given ?W
0
 = 10 gr, what is the sensitivity region of this design?  
 30
 
Solution 
Since we already have S
f
, we can directly use Eq. 3.3 to plot the SR of this design as 
shown in Figure 3.8(a) below. 
-0.4
-0.2
0.4
-0.04 -0.02 0 0.02 0.04
0.2
r?
??
SR
?
)rr(
? h
W
? r
?
2
02
?
?+
?
+
=?
?
)rr(
? h
W
? r
?
2
02
?
?+
?
?
=?
(a) (b)
??
r?
W?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
 
Figure 3.8: (a) Sensitivity region of the piston pin. (b) Surface plot of ?W(??,?r). 
For comparison, we have also shown the surface plot of the difference function 
?W(??,?r), as shown in Figure 3.8(b). Notice in Figure 3.8(a) how the SR is simply the 
projection of points between two parallel planes ??W
0
 cutting through the surface in 
Figure 3.8(b). Notice also that the SR encloses the origin, but is unbounded.  
In Example 3.1 we performed a simple calculation to validate the accuracy of 
Eq. 3.3. We showed that for the pair (??,?r) = (-0.4, 0.05) the pin weight is 501.7 gr (less 
than W+?W
0
), for (??,?r) = (1.0, -0.02) the pin weight is 554.2 gr (greater 
than W+?W
0
), and for (??,?r) = (0.495, -0.04), the pin weight is 512.6 gr (equal 
to W+?W
0
). Observe how these three (??,?r) pairs fall inside, outside, and on the 
boundary of the SR shown in Figure 3.8(a), respectively, and that the changes in the 
weight are as we predicted. ? 
 31
 
3.2.3. Directional Sensitivity 
Figure 3.8(a) shows a drawback in using SR size as a robustness measure, namely 
the asymmetry of the SR. Because of this asymmetry, a design might be very robust if ?p 
moves along certain directions, but very sensitive if it moves along some other directions. 
Besides, if the SR is not fully bounded (like in Figure 3.8(a)), its size is infinite. 
Figure 3.9 shows an example of such a directional sensitivity. In this figure, the SR of 
design A is larger than that of design B and therefore, based on our previous discussions, 
design A is more robust than design B. However, if ?p happens to vary along the 
direction shown, design A will violate the ?f
0
 limit first before design B does. In other 
words, for this ?p direction design B should be considered more robust than design A. 
1
p?
2
p?
SR of design B
?p direction 
SR of design A
 
Figure 3.9: Example of a directional sensitivity. 
From this simple example it is apparent that the size of SR only measures the overall 
sensitivity behavior of a design. If design A has a larger SR than design B, then when ?p 
occurs along all directions, design A will stay within the acceptable ?f
0
 bound more often 
than design B. However, this does not mean that design A will always be more robust 
 32
 
than design B. Therefore, to get a more complete picture of a design?s robustness, we 
need to consider the directional sensitivity of the design as well.  
 
Example 3.3 
Let us return to our piston pin example. Suppose the designer wants to decrease the 
weight of the pin by using an aluminum alloy 6061 (density ? = 2.7 gr.cm
-3
) instead of 
stainless steel. To maintain the strength requirement, however, (s)he finds that the pin 
radius must be increased to r = 3 cm. The pin height is kept at h = 5 cm. The nominal 
weight of the aluminum pin is calculated to be W = 381.7 gr, so it is indeed lighter than 
the stainless steel pin. If the designer still wants the weight variation to be within 
??W
0
 = 10 gr, determine the sensitivity region of this design. Is the aluminum pin more 
or less robust than the stainless steel pin?  
Solution 
The S
f
 equation in Eq. 3.3 was derived independent of the nature of the design, so it 
is also applicable to the aluminum pin design provided we use the appropriate values. 
Substituting ? = 2.7 and r = 3 into Eq. 3.3 and plot the inequalities, we obtain the SR of 
the aluminum pin as shown in Figure 3.10. For comparison, we have also shown the SR 
of the stainless steel pin.  
As can be seen in Figure 3.10, the aluminum pin SR is narrower than the steel pin SR 
and is slightly rotated counter-clockwise. Overall, the steel pin SR is larger than the 
aluminum pin SR, so based on the sensitivity region concept, one will conclude that the 
steel pin is more robust than the aluminum pin. However, this assertion is only partially 
accurate because of the SR rotation. If the changes in the pin radius is between 
 33
 
?r = [?0.017, 0.017], then we can safely assert that the steel pin is more robust than the 
aluminum pin (the steel SR encloses the aluminum SR). However, beyond this range 
there are (??,?r) pairs that the steel pin can absorb but the aluminum pin cannot, and vice 
versa. As such, we cannot make a conclusive comparison between the two designs. 
-0.04 0 0.04
-0.4
-0.2
0.4
r?
??
SR of 
aluminum pin
SR of steel pin
-0.02
0.2
0.02
?
 
Figure 3.10: SR of the aluminum and stainless steel pins. 
Notice in this example how we can easily compare the two SR?s even though the ? 
and r values of the two designs are different. This is because SR is defined in the ?p-
space, and not in the p-space. One advantage of working in the ?p-space is that the SR?s 
are automatically translated to enclose the origin of this space. In addition, the p value 
difference among the designs is also automatically taken care of because we are now 
looking into the changes of those p values, and not their absolute values. ? 
 
An important ?p-direction that must be considered when measuring a design?s 
robustness is the most sensitive (least robust) direction: the ?p-direction along which the 
design can absorb the least amount of ?p before it violates the ?f
0
 bound. In other words, 
this direction represents the worst-case scenario for this design. Geometrically, this 
 34
 
worst-case direction is depicted as a vector from the origin of the ?p-space to the point 
on the SR boundary that is closest (in terms of Euclidean distance) from the origin 
(Figure 3.11). 
1
p?
2
p?
least sensitive
direction
most sensitive
direction
on SR boundary,
closest from origin
on SR boundary,
farthest from origin
 
Figure 3.11: Most and least sensitive directions of a SR. 
Another ?p-direction that might also be of interest is the least sensitive (most robust) 
direction: the ?p-direction along which the design can absorb the most amount of ?p 
before it violates the ?f
0
 bound. This direction represents the best-case scenario for the 
design, and is depicted as a vector from the origin to a point on the SR boundary farthest 
from the origin (Figure 3.11). If a SR is not fully bounded, the least sensitive direction is 
the direction along which the SR is unbounded. For the best-case direction to make sense, 
however, all ?p along this direction must be part of the SR (Figure 3.12). 
?p not part of SR
1
p?
2
p?
SR
farthest 
from origin
 
Figure 3.12: An example where best-case direction not contained in a SR. 
 35
 
3.2.4. Worst Case Sensitivity Region 
It is impossible to consider all possible ?p directions when comparing the robustness 
of two or more designs. This is especially true as the number of parameters under 
consideration increases. For this reason, we choose to measure the robustness of a design 
in the most important direction only: the worst-case direction. Using this worst-case 
approach, the robustness of a design is no longer measured by the size of a SR, but by the 
size of the worst-case approximation to SR. We call this approximation the Worst Case 
Sensitivity Region (WCSR) of the design.  
Geometrically, a WCSR is defined as the smallest hyper-sphere inside a SR that 
touches the SR boundary on at least one point (Figure 3.13). A WCSR does not have to 
be tangent to the SR boundary (e.g., when the point of contact is a cusp), but the point of 
contact between WCSR and SR must necessarily be the point on the SR boundary closest 
from the origin in the ?p-space. This closest-from-origin requirement guarantees that the 
WCSR is indeed the worst-case estimate of the SR. 
1
p?
2
p?
worst-case direction
WCSR 
SR
point of contact
(closest from origin)
 
Figure 3.13: Worst Case Sensitivity Region. 
 36
 
Like SR, the size of a WCSR measures how much ?p a design can absorb before ?f
0
 
is violated, although in the worst-case sense only. So, the size of a WCSR is also a 
measure of a design?s robustness: the larger the size the more robust the design. Unlike 
SR, however, WCSR is always symmetric because it is a hyper-sphere. As such, when 
using a WCSR to compare the robustness of two designs, the directional sensitivity of the 
designs has been taken into account, and the comparison will be conclusive (i.e., one 
design is definitely more or less or as robust as the other). Unless the SR is a hyper-
sphere, using a WCSR to approximate it will always lead to an underestimation of its 
size. In return, however, we are guaranteed that the design is at least as robust as the 
approximation predicts it. 
In addition to its symmetry, WCSR has a computational advantage over SR as well. 
Because of its simple geometric shape, size of a WCSR is much easier to calculate than 
the typically-not-so-simple SR size. In fact, in general the mathematical formulation of a 
SR shape is not known in closed form, so analytical calculation is often not possible 
while simulation-based calculation can be prohibitively expensive.  
To calculate the size of a WCSR, we need to calculate its radius, i.e., Euclidean 
distance from the origin to the point of contact. Using the fact that this point of contact is: 
(i) on the SR boundary and (ii) closest from the origin, the WCSR radius can be obtained 
by solving the following optimization problem: 
[]
),(),()(:where
0
][
)(
1:tosubject
)p()(Rminimize
0000
2
0
2
2
1
G
1
2
pppp
p
p
p
xx fff
f
f
i
if
??+=??
=
?
??
?
?
?
?
?
?
?
? ?=?
=
?
   (3.4) 
 37
 
In this optimization problem, the variables are the G elements of ?p, and the objective is 
to minimize the Euclidian distance from the origin. The constraint in the problem reflects 
the fact that a point on a SR boundary has to satisfy: [f(x
0
,p
0
+?p) ? f(x
0
,p
0
)]
2
 = [?f
0
]
2
. 
The optimum R
f
 value for this problem is the WCSR radius that we seek.  
The optimization problem in Eq. (3.4) is guaranteed to have a solution. The point 
?p = 0 is always a member of S
f
, so the smallest WCSR radius is R
f
 = 0. In the extreme 
case where the design?s SR is unbounded, there will be no feasible solution to Eq. (3.4) 
and R
f
 = ?. However, recall that we require the SR to be bounded along at least one ?p-
direction. So, based on this requirement: R
f
 < ?, and in general 0 ? R
f
 < ?. 
Note that Eq. (3.4) is not the optimization problem for which we are trying to find a 
robust optimum. The optimization problem in Eq. (3.4) is used to measure a design?s 
robustness, which later on will be used to obtain the robust optimum of the original 
optimization problem. 
Mathematically, sometimes it is easier to satisfy an inequality constraint than an 
equality constraint. Since Eq. (3.4) is a minimization problem, its constraint can be 
changed into an inequality, as shown in Eq. (3.5), without changing the optimum R
f
. This 
inequality constraint relaxes the requirement that the point of contact must be on the SR 
boundary. The feasible domain now includes points outside the SR as well. However, 
since we are minimizing a distance function, at the optimum this inequality constraint 
must be active (i.e., satisfied as an equality). For differentiability purposes, it is also 
recommended that we optimize  instead of R
2
R
f
f
. For the rest of our discussions, we will 
use Eq. (3.4) to calculate a WCSR radius because that is a more general formulation. In 
our comparison studies, however, Eq. (3.5) is often used as well.  
 38
 
[]
0
][
)(
1:tosubject
)p(Rminimize
2
0
2
G
1
22
?
?
??
?
? ?=
=
?
f
f
i
if
p
p
        (3.5) 
Once we obtain a WCSR radius, its size can be easily calculated using the following 
formula for the ?hyper-volume? of a G-dimensional hyper-sphere (Apostol, 1957): 
G
2
G
2
G
R
]1[
?
)G(V
f
+?
=      (3.6) 
where  is the gamma function, and R
?
?
??
=?
0
1n
n)( dxex
x
f
 is the WCSR radius. Eq. (3.6) is 
often broken down into more readily usable equations: 
?
?
?
?
?
?
?
=
?
?
odd isG ;R
!G
)!(
?2
even isG ;R
)!(
?
)G(V
G2
1G
G
G
2
G
2
1G
2
G
f
f
   (3.7) 
(As a simple check: if G = 2, V = , area of a circle; if G = 3, V = 
2
R?
f
3
R?
3
4
f
, volume of 
a sphere.) 
It should be pointed out that as a robustness measure, WCSR size is fundamentally 
different from conventional robustness measures discussed in the literature. That is in 
general, our robustness measure cannot determine ?f given ?p (e.g., the gradient measure 
?
p 
f ). The reason is because unlike conventional measures, WCSR size measures the 
design robustness in ?reverse.? Instead of giving an answer to the question: ?If we have 
?p, what is ?f ??, we provide an answer to the question: ?If we can allow ?f
0
, what is 
?p??  
 39
 
Because the size of a WCSR cannot determine ?f given ?p, in this sense WCSR size 
is an ordinal measure. When comparing robustness of two designs, the actual values of 
their WCSR sizes are not important, only the ordering of these values is important. This 
is especially true considering that the actual value of a WCSR size is an abstract indicator 
that does not explicitly represent a physical quantity. On the other hand, the ordering of 
WCSR sizes is not the only important aspect of this measure. The relative ratio among 
WCSR sizes is also important. For instance, if WCSR of say design A is twice as large as 
design B, then we know that design A can absorb 
G
G
B,
A,
2
R
R
=
?
?
?
?
?
?
f
f
?
?
times more ?p than 
design B. In this sense, design A is 2
G
 times more robust than design B. So from this 
angle, the WCSR size measure is cardinal, although not in the traditional sense of 
performance robustness. In other words, WCSR size can be said to be a ?semi-cardinal? 
measure.  
Because the size of a WCSR is a semi-cardinal measure, we do not need to calculate 
its actual value. Rather, we can simply use the WCSR radius R
f
 to measure a design?s 
robustness. Like our previous discussion showed, using only R
f
 is sufficient to calculate 
the relative robustness of two designs. For the rest of our discussions, we will use R
f
 as a 
robustness measure and will not calculate the actual size of the hyper-volume defined by 
this radius. 
 
Example 3.4 
Determine the WCSR?s of our stainless steel and aluminum pin designs using the 
SR?s obtained in Example 3.3. Based on the worst-case notion, which pin design is more 
 40
 
robust? Calculate the radius of each WCSR mathematically using Eq. (3.4). Is it 
consistent with the WCSR graphs obtained?  
Solution 
The SR?s and WCSR?s of the two pin designs are shown in Figure 3.14. We can 
obtain the WCSR radius graphically from Figure 3.14. For the stainless steel pin, the 
point of contact between SR and WCSR is approximately at (??,?r) = (0.11, 0.05) for a 
WCSR radius of R
f
 = 0.12. For the aluminum pin, the point of contact between SR and 
WCSR is approximately at (??,?r) = (0.08, 0.001) for a WCSR radius of R
f
 = 0.08. Since 
the R
f
 of the stainless steel pin is larger than the R
f
 of the aluminum pin, the stainless 
steel pin is more robust than the aluminum pin. 
SR of 
steel pin
-0.4
-0.2
0.4
??
-0.04 0 0.040.02
r?
WCSR of 
steel pin
(a)
-0.02
0.2
SR of 
aluminum pin
-0.04 0.040.02
r?
-0.4
-0.2
0.2
0.4
??
WCSR of 
aluminum pin
(b)
-0.02
??
??
 
Figure 3.14: SR and WCSR of the (a) steel pin, and (b) aluminum pin. 
To obtain the WCSR radius mathematically, we solve the following optimization 
problem (we use  as the objective): 
2
R
f
[]0)r? ,(W
)W(
1
1:tosubject
r?Rminimize
2
2
0
222
r)? ,(
=????
?
?
?+?=
??
f
        (3.8) 
 41
 
Since we know that the SR boundary is defined by ?
)rr(
? h
W
? r
?
2
0
2
?
?+
?
?
=?  and 
?
)rr(
? h
W
? r
?
2
0
2
?
?+
?
+
=?  (recall Figure 3.8(a)), Eq. (3.8) can be broken down into two simpler 
problems, Eq. (3.9) and Eq. (3.10): 
[ ]
?
)rr(
? h
W
? r
r)? (:where
rr)? (Rminimize
2
0
2
222
1,
r
?
?+
?
?
=??
?+??=
?
f
         (3.9) 
[ ]
?
)rr(
? h
W
? r
r)? (:where
rr)? (Rminimize
2
0
2
222
2,
r
?
?+
?
+
=??
?+??=
?
f
             (3.10) 
Where the WCSR radius is ( )
*2
2,
*2
1,
2
)R(,)R(minR
fff
= , and ( and are the 
optima of Eq. (3.9) and Eq. (3.10), respectively.  
*2
1,
)R
f
*2
2,
)R(
f
Eq. (3.9) and Eq. (3.10) are both one-dimensional unconstrained optimization 
problems. Setting the gradient of each function to zero, the optimum ?r is obtained by 
finding the roots of the following sixth order polynomial: 
?
?
?
?
?
?
?
?
+
?
?
=
=??+++
(3.10) Eq.for 
? h
W
? r
(3.9) Eq.for 
? h
W
? r
A:where
0A2)rr(A?2? r)? r(r
0
2
02
225
            (3.11) 
The optimum ?? is obtained by substituting the optimum ?r into the equalities in 
Eq. (3.9) and (3.10). We do not need to check for sufficiency condition because the 
function is unimodal minimum (unimodality of Eq. (3.11) can be verified graphically).  
 42
 
Using r = 2 and ? = 8, the optimum (??,?r) for the stainless steel pin is found to be 
(0.0025, 0.0195) for a WCSR radius of R
f
 = 0.0196. Similarly, using r = 3 and ? = 2.7, 
the optimum (??,?r) value for the stainless steel pin is found to be (0.0166, 0.0297) for a 
WCSR radius of R
f
 = 0.034. Notice that the R
f
 values obtained mathematically are 
different from the R
f
 values obtained graphically. In fact, according to the R
f
?s obtained 
by solving Eq. (3.8), the aluminum pin is more robust than the stainless steel pin, a 
contradiction to our graphical observation before!  
The reason for this inconsistency is because the magnitudes of ?? and ?r are 
different (?? is an order of magnitude larger than ?r). Because of this magnitude 
difference, when minimizing the distance function , ?r numerically dominates ??. 
Graphically this numerical domination translates into an elliptical WCSR, and not a 
circular WCSR (Figure 3.15).   
2
R
f
Mathematical 
WCSR
-0.4
-0.2
0.4
??
(a)
0.2
-0.04 0 0.04
r?
-0.02 0.02
Mathematical
WCSR
-0.4
-0.2
0.2
0.4
??
(b)
-0.04 0.040.02
r?
-0.02
WCSR
?? ??
 
Figure 3.15: Mathematical WCSR of the (a) steel pin, and (b) aluminum pin. 
The reader might be surprised to know that actually the mathematical WCSR 
(Figure 3.15) is a circle and the WCSR in Figure 3.14 is an ellipse. The reason they look 
the way we showed them is because the scales of the ?? and ?r axes are different. The 
 43
 
scale of the ?? axis is much larger than the ?r axis, in effect ?compressing? the WCSR. 
If the WCSR is plotted on an equally scaled coordinate system, the mathematical WCSR 
will be circular (Figure 3.16), and hence is consistent with Eq. (3.8).  
-0.05 0 0.05
0.05
0.1
-0.1
-0.05
WCSR
R
f
= 0.0196
SR
??
r?
(a)
-0.1 -0.05 0 0.05 0.1
-0.1
-0.05
0.05
0.1
??
r?
(b)
WCSR
R
f
= 0.034
SR
??
??
 
Figure 3.16: Equal-scaled SR and WCSR of (a) steel and (b) aluminum pins. 
According to Figure 3.14, the steel pin is more robust than the aluminum pin, but 
according to Figure 3.16, the aluminum pin is more robust. Which one is correct? 
Clearly, when there is a difference in ?p magnitudes, there is an ambiguity between the 
WCSR concept and the mathematics behind it. As it turns out, the optimization problem 
used to calculate the WCSR radius (Eq. (3.4)) needs to be normalized when the scales of 
?p are different. ? 
 
3.2.5. Normalization 
When the magnitudes of ?p are different, the optimum of Eq. (3.4) will be 
numerically driven by those ?p
i
?s whose magnitudes are small. For instance, if ?p are the 
 44
 
normal stress (magnitude in the order of 10
3
) and deflection (magnitude in the order of 
10
-2
) of a beam, then the search for the optimum R
f
 will be numerically driven by the 
deflection factor. However, this is not what we want because this does not reflect the 
actual robustness of the design. If a beam can absorb say 10
3
 MPa of stress variation, an 
R
f
 of 10
-2
 will very much distort the actual robustness of this design. In other words, scale 
importance of the stress and deflection parameters is different.  
Because of this difference in scale importance, an increase (or decrease) of ? along 
one axis is not the same as an increase (or decrease) of the same ? along another axis. As 
a consequence, the most sensitive direction that we are interested in may not necessarily 
be the closest distance from the origin mathematically. Correspondingly, the WCSR 
defined by this direction may not be geometrically spherical. It should be pointed out that 
although the WCSR is not spherical, it is still equally sensitive along all directions 
because now the scale importance is different.  
To obtain the WCSR radius that reflects the scale importance, the optimization 
problem in Eq. (3.4) needs to be normalized. There are many ways to normalize this 
problem. In this dissertation we use a single-valued normalization where we use the 
known variation ranges ?p
0
 as the reference value. The normalized optimization problem 
to obtain the WCSR radius is as follows: 
[]
[]
i
i
i
i
if
f
f
,0
G1
2
0
2
0
2
1
G
1
2
p
p
p;p,...,p:where
0
][
)(
1:tosubject
)p()(Rminimize
?
?
=???=?
=
?
????
?
?
?
?
?
?
?
? ?=?
=
?
p
pp
p
p
  (3.12) 
 45
 
In this formulation, the variables are p?  (the normalized ?p), and the objective and 
constraint are the same except they are modified for p?  instead of ?p. The notation ??? 
refers to a vector operation between p?  and ?p
0
 (recall Section 2.2). As shown before 
we can use 
2
R
f
 for the objective and ??? (recall Eq. (3.5)) for the constraint if it makes 
the problem easier to solve.  
One might point out that the normalization in Eq. (3.12) depends on the ?p
0
 value, 
and that the optimum 
f
R  will change if we use a different ?p
0
. The actual value of 
f
R  
will indeed change along with ?p
0
 changes. However, the ratio between the 
f
R ?s of 
different designs will remain the same as long as the change in ?p
0
 is consistent, i.e., if 
?p
0,1
 is multiplied by 2, ?p
0,2
 is multiplied by 2 also, and so on. This ratio preserving 
property is important because the ratio of 
f
R ?s determines the magnitude of relative 
robustness between different designs. If the ?p
0
 change is not consistent, then obviously 
the 
f
R  ratio will be different because we have implicitly changed the scale importance, 
and correspondingly, the WCSR that we are searching for.  
Aside from the scale importance issue, normalizing Eq. (3.4) into Eq. (3.12) has 
other advantages as well. Computationally, it is easier to obtain a more accurate optimum 
for Eq. (3.12) than for Eq. (3.4). This is because a non-normalized SR might be so 
stretched (i.e., a very long and thin strip) that solving Eq. (3.4) most likely results in 
R
f
 = 0 due to round-off errors. Conceptually, normalizing Eq. (3.4) makes a WCSR easier 
to work with because it is now defined in a unit-less space. A WCSR defined in a space 
with different units for its axes (e.g., MPa vs. cm) promotes human errors when 
comparing it to other WCSR?s.  
 46
 
In almost all cases, ?p magnitudes will be different and therefore we will have to use 
the normalized WCSR. Even for cases when the scales are the same, we still recommend 
using the normalized WCSR for the reasons we just discussed. Strictly speaking, we 
should use the notation 
f
R  to refer to the normalized WCSR radius and R
f
 for the non-
normalized radius. However, since we will be using mostly the normalized radius for the 
rest of our discussions, we will simply use the notation R
f
 to refer to the normalized one.  
 
Example 3.5 
If the ?p
0
 of our piston pin design is known to be (??
0
,?r
0
) = (0.2, 0.02), calculate 
the normalized WCSR radius of the steel and aluminum pins. Which design (steel or 
aluminum) is more robust now? Derive and graph the WCSR?s of the two designs in the 
normalized (??,?r)-space, and compare them to the WCSR?s we obtained in the previous 
examples.  
Solution 
The optimization problem to calculate the normalized WCSR radius is as follows 
(again, we use  instead of R
2
R
f
f
 ): 
[]0)rr,??(W
)W(
1
1:tosubject
r?Rminimize
2
00
2
0
222
)r,?(
=????????
?
?
?+?=
??
f
      (3.13) 
We can use a procedure similar to the one in Example 3.4 to solve Eq. (3.13). The 
optimum r? of this problem is obtained by finding the roots of the following sixth order 
polynomial:  
 47
 
?
?
?
?
?
?
?
?
+
?
?
=
=??+++?
boundary SRupper for 
? h
W
? r
boundary SRlower for 
? h
W
? r
A:where
0A? r2)r? rr(A?? r2)r?? r(rr)??(
02
02
2
0
2
00
5
0
2
0
       (3.14) 
The optimum ??  is obtained by substituting the optimum r?  into Eq. (3.15). 
?
?
?
?
?
?
?
???+?
=? ?
)rrr(
A
?
1
?
2
00
        (3.15) 
Substituting the appropriate values into Eq. (3.14) and solving it, the optimum 
)r?,?(?  for the steel pin is (0.388, 0.4811) for an R
f
 = 0.618. The optimum )r?,?(?  for 
the aluminum pin is (0.0631, 0.342) for an R
f
 = 0.347. So, according to the normalized 
WCSR, the steel pin is more robust than the aluminum pin. 
The graph of the normalized SR and WCSR in the )r?,?(? -space is shown in 
Figure 3.17. Notice that except for the axis values, this graph is identical to the one in 
Figure 3.14, which is what we want. After the normalization, it is verified that our 
mathematical calculation is consistent with the conceptual WCSR. 
For (??
0
,?r
0
) = (0.2, 0.02), the R
f
 ratio between the steel and aluminum pins is 1.77. 
If we change ?p
0
 to (??
0
,?r
0
) = (0.1, 0.01), the R
f
 of the steel and aluminum pin is 1.236 
and 0.695, respectively with a ratio of 1.77. If (??
0
,?r
0
) = (0.5, 0.05), the R
f
 of the steel 
and aluminum pin is 0.247 and 0.139, respectively, again with a ratio of 1.77. Notice how 
the ratios are the same in all three cases. Based on this ratio value, we conclude that the 
steel pin is about (1.77)
2
 = 3.13 times more robust than the aluminum pin. ? 
 48
 
-3 -2 -1 2 30 1
-3
-2
-1
2
3
1 WCSR
R
f
= 0.618
SR
??
r?
(a)
-3 -2 -1 2 30 1
-3
-2
-1
2
3
1
??
r?
WCSR
R
f
= 0.347
SR
(b)
?? ??
 
Figure 3.17: Normalized SR and WCSR of (a) steel pin, and (b) aluminum pin. 
 
3.3. ROBUST OPTIMIZATION 
Now that we have a measure of a design?s robustness, we are ready to tackle the 
problem of finding a robust optimum design for an engineering optimization problem.  
 
3.3.1. Robustness Index 
The quantity R
f
 determines the ranges of ?p that a design can absorb without 
violating the ?f
0
 limit. In other words, it defines the ranges of ?p that must occur if we 
want our design to be as robust as we specified. On the other hand, ?p
0
 defines the ranges 
of ?p that actually do occur in the real world. So, by comparing these two ?p ranges, we 
can see whether or not a design is robust. If the ?p
0
 ranges are larger than the R
f
 ranges, it 
means the design is not robust. There are some ?p variations that will cause the design to 
violate the ?f
0
 limit. Conversely, if the R
f
 ranges are larger than the ?p
0
 ranges, it means 
the design is more robust than we need. It will never violate the ?f
0
 limit, and there are 
 49
 
some ?p variations that the design can still absorb. If the R
f
 ranges are exactly the same 
as the ?p
0
 ranges, it means the design is strictly as robust as we want it to be. It strictly 
satisfies the ?f
0
 limit.  
Because R
f
 is defined in the normalized ?p-space, the ?p
0
 ranges must be 
normalized first before we make any comparison. Since we use ?p
0
 to normalize R
f
, we 
must also use the same reference to normalize ?p
0
 ranges. Normalization of ?p
0
 ranges 
with itself results in a unit hyper-cube in the normalized ?p-space (Figure 3.18). We have 
also shown the interior and exterior hyper-sphere of the unit hyper-cube. The radius of 
the interior and exterior hyper-sphere of this hyper-cube is R
I
 = 1 and R
E
 = (G)
1/2
, 
respectively. For the two-dimensional example shown in Figure 3.18, R
I
 = 1 and 
R
E
 = 1.414. 
1
p?
1+
1?
0
1+1?
E
R
I
R
normalized
?p
0 
ranges
2
p?
 
Figure 3.18: Normalized ?p
0
 ranges. 
Direct comparison between R
f
 and the normalized ?p
0
 is difficult because R
f
 is a 
radius of a hyper-sphere while ?p
0
 is a range. To facilitate this comparison, we use the 
radius of ?p
0
 exterior hyper-sphere (R
E
) instead. Because both R
f
 and R
E
 define a hyper-
sphere, if R
f
 ? R
E
, then R
f
 encloses the ?p
0
 ranges, and our design is robust. If R
f
 < R
E
, on 
 50
 
the other hand, then our design is not robust. If we define a quantity 
E
R
R
?
f
= , then a 
design is robust if ? ? 1, and is not robust if ? < 1. We call ? the ?robustness index? of a 
design. Because R
E
 = (G)
1/2
, the robustness index of a design is 
f
R)G(?
21?
= .  
A design is more robust if R
f
 is larger. Consequently, the larger ?, the more robust 
the design. Figure 3.19 shows the comparison between ? and the normalized ?p
0
 ranges 
in a two-dimensional example. As we can see in this figure, the larger ? the more we 
enclose the ?p
0
 ranges, and thus the more robust the design. 
1
p?
1+
1?
0
1+
1?
normalized
?p
0
ranges
2/1
)2(
1
? < 2/1
)2(
1
? =
1? =
1? >
2
p?
?
?
?
?
 
Figure 3.19: Comparison between ? and ?p
0
. 
 
3.3.2. Constrained Robustness Approach 
In robust optimization, we want to simultaneously optimize performance of a design 
and maximize its robustness, where we use the objective function f(x,p
0
) to measure the 
design performance, and ? to measure its robustness. By doing so, we have essentially 
converted the original single objective optimization problem into a two-objective one. 
 51
 
Because of the trade-off between performance and robustness of a design, in general 
there is a set of optimum solutions to this two-objective problem. It is then up to the 
designer to select the single most preferred design from this set by trading-off the two 
objectives. However, the ? measure that we use here does not have a physical association 
with the design itself and as such it would be difficult for the designer to do such a trade-
off analysis. This difficulty is compounded further by the fact that ? has a semi-cardinal 
scale.  
To avoid these difficulties, in this dissertation we use a constrained approach to 
robustness. Instead of maximizing ?, we constrain it to be greater than or equal to 1 to 
make sure that the WCSR of the optimum design at least encloses the ?p
0
 ranges. The 
formulation for our robust optimization problem is then as follows:  
0?1
K,...,1;0),(h
J,...,1;0),(g:subject to
),(minimize
0
0
0
??
==
=?
k
j
f
k
j
p
p
p
x
x
x
x
   (3.16) 
where the last constraint is the robustness constraint, 
f
R)G(?
21?
= , and R
f
 is calculated 
by solving an inner optimization shown in Eq. (3.12) (re-shown here for convenience).  
[]
0
][
)(
1:tosubject
)p()(Rminimize
2
0
2
0
2
1
G
1
2
=
?
????
?
?
?
?
?
?
?
? ?=?
=
?
f
f
i
if
pp
p
p
   (3.17) 
Notice that although not explicitly shown in Eq. (3.16), ? is a function of x, the design 
variable, because R
f
 is a function of x. In Eq. (3.17), )(
0
pp ????f  is a function of x.  
 52
 
 Using Eq. (3.16), our search for a robust optimum design is performed in the N-
dimensional x-space, in which during this search, we run another search, Eq. (3.17), in 
the G-dimensional p? -space. Although our approach involves two optimization 
problems, this is not a multi-disciplinary optimization approach. Rather, Eq. (3.17) is 
simply a tool to obtain the robustness information needed by Eq. (3.16). As such, 
convergence of this approach depends entirely on the convergence of Eq. (3.16), and has 
nothing to do with the links between the two problems. If Eq. (3.16) has an optimum 
solution before the robustness constraint is added, but does not have a feasible solution 
after it is added, it simply means that there is no design that satisfies our robustness 
requirement (i.e., the addition of the robustness constraint causes the feasible domain of 
Eq. (3.16) to become empty).  
Eq. (3.17) is a simple single objective optimization problem with one constraint. If 
the function f(x,p) is simple enough, Eq. (3.17) can be solved analytically like in our 
piston pin examples. Otherwise, traditional gradient-based optimization methods, such as 
Quasi-Newton methods or the Generalized Reduced Gradient method, can be used to 
solve it. The choice of an optimizer for Eq. (3.16) depends on the complexity of the 
problem. Nevertheless, because of the nature of the robustness constraint, we recommend 
using global optimization methods to solve Eq. (3.16). Unlike conventional constraints, 
the robustness constraint in Eq. (3.16) may not necessarily divide the search space into 
two well-defined feasible and infeasible half-spaces. Rather, it may result in many 
disjointed infeasible domains in the feasible domain, in effect adding ?holes? to the 
original feasible domain (Figure 3.20). Because of this effect, direction-based optimizers 
may have difficulty converging.  
 53
 
x
1
x
2
(a)
= feasible
conventional
constraints
x
1
x
2
(b)
= feasible
robustness
constraint
 
Figure 3.20: Effect of adding robustness constraint. 
Sometimes, the constraint ? ? 1 is too strict, that is no design has that much 
robustness. In this case, the right hand side value of 1 needs to be lowered. To give the 
designer the flexibility to change the desired level of robustness, we use a quantity ?
0
 to 
replace this value of 1, where ?
0
 is determined by the designer. The larger ?
0
, the more 
robust the designer wants the optimum design to be, and vice versa. Using ?
0
 in the 
robustness constraint, our robust optimization formulation becomes:  
0
?
?
1
K,...,1;0),(h
J,...,1;0),(g:subject to
),(minimize
0
0
0
0
??
==
=?
k
j
f
k
j
p
p
p
x
x
x
x
   (3.18) 
where again ? is calculated by solving Eq. (3.17).  
 
3.4. COMPARISON STUDY 
To demonstrate our robust optimization method, we applied it to four examples: one 
numerical example and three engineering examples. The purpose of the numerical 
example is to provide a graphical verification of the results obtained by our method. The 
 54
 
purpose of the three engineering examples is as follows. The welded beam example 
demonstrates an application of our method to a non-differentiable objective function. The 
three-bar truss and compression spring examples compare our method to mean-based and 
worst case analysis-based robust optimization methods, respectively. 
 
3.4.1. Wine-Bottle Function 
This example is originally formulated by Van Veldhuizen and Lamont (1998) as a 
multi-objective optimization problem. We converted it into a single objective problem by 
significantly modifying and optimizing one of the original objectives. The problem has 
two variables x
1
 and x
2
, both continuous, and the objective is to minimize the ?wine-
bottle? function f(x
1
,x
2
). There are variations in the variables (?x
1
 and ?x
2
), and we want 
to minimize the sensitivity of the optimum solution with respect to these variations. 
Because the variability occurs only in the design variables (there are no other noisy 
factors), in this problem p = x and x?=?p . The mathematical formulation of the 
problem is as follows:  
1),(0
36B;36A
)2BAsin(
2
3BA
C:where
32.9
1C
),(minimize
21
21
22
22
21
??
?=?=
+++
++
=
+
=
xx
xx
xxf
   (3.19) 
Figure 3.21 shows a three-dimensional surface mesh of the objective function. As 
shown in this figure, the function is axially symmetric and has a dome-like region in the 
center whose shape resembles the bottom of a wine bottle. The function has an infinite 
number of global minima circularly located around the center point (x
1
,x
2
) = (0.5,0.5), 
 55
 
and they are defined by a circle equation: (x
1
?0.5)
2 
+ (x
2
?0.5)
2
 = (0.246)
2
. The objective 
value at the global minima is f 
*
 = 0.292. Points on the dome-region of the function have 
a slightly higher objective value (f
R
 = 0.365) compared to the global minima. However, 
this region of the function is flat and thus the points in this region are insensitive to the 
variable variations. 
 
Figure 3.21: Surface plot of the wine-bottle function. 
The maximum allowable variation in f is ?f
0
 = 0.01, and the variation ranges are 
known to be ?p = (0.05,0.05). We added the robustness constraint into Eq. (3.19), and 
then solve it using a Genetic Algorithm (GA) for the outer optimization problem, and 
MATLAB?s fmincon function for the inner optimization problem (Eq. (3.17)). 
MATLAB?s fmincon function is an optimizer based on the Sequential Quadratic 
Programming (SQP) method with BFGS formula for the Hessian estimation. We used a 
GA for the outer optimization because the objective function has multiple local minima 
(see Figure 3.21) so that direction-based optimizers may not find the global minima.  
Because GA is a stochastic method, the robust optimum obtained might differ from 
one run to another. To account for this, we solved the problem 50 times (using different 
 56
 
initial population each time). Figures 3.22 and 3.23 show the robust optima obtained from 
solving the problem 50 times each for three different values of ?
0
. Figure 3.22 shows the 
distributions of the optima on the contour plot of the objective function, while 
Figure 3.23 shows those points of the distributions that lie on the cross-section of the 
function when x
2 
= 0.5. 
x
1
x
2
?
0
= (2)
-1/2
= 0.707
?
0
= 0.35
?
0
= 1
x
 
Figure 3.22: Distributions of robust optima on a contour plot. 
x
2
= 0.5
?
0
= 0.35
?
0
= (2)
-1/2
= 0.707
?
0
= 1
x
x
1
f
 
Figure 3.23: Distributions of robust optima on a cross-section plot. 
We observe in Figure 3.22 and 3.23 that for ?
0
 = 1, the optimum obtained is located 
at the dome region of the function for all 50 GA runs. Because this region is insensitive to 
the variations, these results confirm that solving the problem using our robust 
 57
 
optimization method results in a robust optimum. We also observe from Figures 3.22 and 
3.23 that as ?
0
 is reduced (relaxing the robustness constraint), the objective value 
decreases (gets better). The optimum objective value for ?
0
 = 1 is f
1
 = 0.365, while f
0.707
 = 
0.361 and f
0.35
 = 0.292. When ?
0
 = 0.35, the robust optimum obtained is the global 
optimum. We expect to observe these phenomena because in general there are trade-offs 
between performance and robustness. 
 
3.4.2. Design of a Three-Bar Truss 
This example was first formulated by Schmit (1960), and has been thoroughly 
studied in (Sun et al., 1975) and (Haug and Arora, 1979). Here, we modified the problem 
by substituting the original objective with one of the structural constraints and by adding 
some variations into the problem.  
In this problem, we are designing a system of three-bar truss with a constant force 
P = 100 kN acting at an angle ? at the end of the truss as shown in Figure 3.24 
(l = 1.0 m). The truss is symmetric (member 1 and member 3 are identical), and is to be 
designed for minimum stress in members 1 and 3. The variables in this problem are the 
cross section area of members 1 and 3 (A
1
), and the angle ?.  
l
l l
u
v
P
?
132
1 2 3
4
 
Figure 3.24: A three-bar truss. 
 58
 
To prevent failure, the design is subjected to 6 structural constraints. The horizontal and 
vertical deflection at node 4 (u and v) must be less than u
a
 = 0.5 cm and v
a
 = 0.5 cm, 
respectively. The stress on member 2 (?
2
) must be less than ?
a
 = 140 MPa. The buckling 
load on all members must satisfy the buckling constraint. In addition, the total volume of 
the members is constrained to be less than V
a
 = 2000 cm
3
. The mathematical formulation 
of the problem is given in Eq. (3.20).  
2
1
2
a
7
b1
3
6
b2
2
5
b1
1
4
a
2
3
a
2
a
1
211
?,A
cm 01Acm 2;40?30
01
V
V
g
01
F
F
g;01
F
F
g
01
F
F
g;01
?
?
g
01g;01g:subject to
)A2A(
? )sin(
A
)?cos(
2
P
?minimize
1
????
???
??????
??????
??????
?
?
?
?
?
?
+
+=
oo
v
v
u
u
  (3.20) 
The constraints are calculated by the following formulas. In calculating the buckling load, 
the members are considered to be columns with pins at both ends.  
)EA2(A
)?(inP2
;
EA
)?cos(P2
211
+
==
sl
v
l
u     (3.21) 
?
?
?
?
?
?
+
+?=
+
=
)A2(A
)?sin(
A
)?cos(
2
P
F;
)A2(A
)?cos(P2
?
211
1
21
2
 (3.22) 
)A2(A
)?sin(P2
F;
2
AE??
F
21
2
2
1
2
1b
+
?==
l
    (3.23) 
?
?
?
?
?
?
?
+
?==
121
3
2
2
2
2b
A
)?cos(
)A2(A
)?sin(
2
P
F;
AE??
F
l
  (3.24) 
 59
 
)AA22(V
21
+= l      (3.25) 
where: E = Young?s modulus (= 70 GPa) 
 A
2
 = cross section area of member 2 (= 2 cm
2
) 
 ? = non-dimensional constant (= 1.0) 
Using MATLAB?s fmincon to solve Eq. (3.20), we obtained the nominal optimum design 
(A
1
*
,?
*
) = (6.36, 40), and ?
*
 = 134.56 MPa in just 3 iterations. The constraint values of 
this nominal optimum are: g = (-0.5133, -0.7173, -0.2934, -1.6128, -1.716, -0.8375, 0.0), 
where g
7
 is active.  
The two design variables are known to vary by (?A
1
, ??) = (0.1 cm
2
, 5?), and we 
would like to minimize the sensitivity of the optimum design with respect to these 
variations. The allowable variation in the objective is ?? = 2.75 MPa. We added the 
robustness constraint (with ?
0
 = 1) into Eq. (3.20) and then solved it using fmincon for 
both the inner and outer optimization problems. On average (we ran the algorithm using 
many different initial points) the outer optimization converges in 9 iterations. On average, 
the inner optimization converges in 13 iterations. The robust optimum design obtained is 
shown in Table 3.1.  
For comparison, we also solved Eq. (3.20) for the robust optimum using two 
conventional robust optimization approaches: (i) minimizing the mean value of the 
objective function, and (ii) minimizing the sum of mean and standard deviation. The 
mean and standard deviation are calculated by performing 10,000 Monte Carlo 
simulations around the design following a uniform pdf of the ranges 
(?A
1
, ??) = (0.1 cm
2
, 5?). The optimum designs obtained using these two methods are 
also shown in Table 3.1.  
 60
 
Table 3.1: Optimum designs of the three-bar truss. 
Nominal 
Optimum
Robust 
Optimum
Mean 
Optimum
Mean+Std 
Optimum
? (Mpa) 134.56 135.08 134.69 135.06
? 0.826 1.0 0.832 0.885
A
1 
(cm
2
) 6.364 6.364 6.36 6.35
? (degree) 40 36.3 30 38.6
F
call
N/A 40 10000 10000
 
In Table 3.1, the quantity F
call
 is the number of function evaluations needed per 
design to obtain its robustness information. For our method, this quantity is the number of 
function evaluations performed in solving the inner optimization. For the mean-std 
methods, this quantity is the number of samples. In this table we have also shown the ? 
value of each optimum design (remember, each design has a corresponding ? value even 
though it is not obtained by our robust optimization method).  
We see in Table 3.1 that the nominal optimum has the lowest ? value (the objective 
function), but it also has the lowest ? value. In contrast, our robust optimum has the 
highest ? value of the four optima, but it has the highest ?. The ? and ? values of the 
other two optima are somewhere in between. This observation is just what we expected. 
The larger ?, the more robust the design, but at the expense of performance degradation 
(i.e., stress increases). We also see in this table that of the four optima, only the ? value 
of our robust design is equal to 1. So, according to our theoretical development, when the 
variations occur, our robust design should be the only one that satisfies the requirement 
?? ? 2.75 MPa. Let us verify this claim. 
We randomly perturbed (A
1
, ?) 20 times around their nominal values within the 
given ranges, and calculated the new ? value of the design using these perturbed values. 
 61
 
The difference between this new ? value and the previous value is the ??. We did the 
same analysis for all four optima, using the same (?A
1
, ??) for each. Figure 3.25 shows 
the graphs of the ?? of the optima for the 20 random perturbations. In this figure the 
dashed lines are the ??
0
 limit (2.75 MPa).  
-3.25
-2.75
-2.25
-1.75
-1.25
-0.75
-0.25
0.25
0.75
1.25
1.75
2.25
2.75
3.25
16116
Case #
??
(a) Nominal optimum
-3.25
-2.75
-2.25
-1.75
-1.25
-0.75
-0.25
0.25
0.75
1.25
1.75
2.25
2.75
3.25
1 6 11 16
Case #
??
(b) Robust optimum
-3.25
-2.75
-2.25
-1.75
-1.25
-0.75
-0.25
0.25
0.75
1.25
1.75
2.25
2.75
3.25
16116
Case #
??
(c) Mean optimum
-3.25
-2.75
-2.25
-1.75
-1.25
-0.75
-0.25
0.25
0.75
1.25
1.75
2.25
2.75
3.25
1 6 11 16
Case #
??
(d) Mean+std optimum
 
Figure 3.25: Sensitivity analysis of the three-bar truss optima. 
We observe in Figure 3.25 that when the variations occur, the nominal optimum 
design as well as the mean and mean+std optima violates the allowable ?? limit. Only 
our robust optimum design stays within the limit, thus showing that this design satisfies 
our robustness requirement.  
 62
 
To further validate the robustness information provided by the ? value, we calculated 
the SR and WCSR of each of the optima, and compared them to the (?A
1
, ??) range. 
Since the objective function involves trigonometric expressions, analytical derivation of 
SR is impossible, so we derived them numerically instead. We constructed an orthogonal 
grid in the range (?A
1
, ??) = (?0.2 cm
2
, ?10?), and calculated the ?? of the optimum 
designs at each junction. The SR is obtained by determining if (??)
2
 ? (??
0
)
2
 at these 
junctions, and the WCSR is obtained by finding the point on the SR boundary closest 
from origin. The SR and WCSR of the four optima are shown in Figure 3.26 through 
Figure 3.29. In these figures, the dashed rectangles are the known (?A
1,0
, ??
0
) range. 
SR
WCSR
R
f
= 1.16
Region of 
failure
?A
1 
(cm
2
)
??(?)
 
Figure 3.26: SR and WCSR of the nominal optimum. 
SR
WCSR
R
f
= 1.414
No region 
of failure
?A
1 
(cm
2
)
??(?)
 
Figure 3.27: SR and WCSR of the robust optimum. 
 63
 
SR
WCSR
R
f
= 1.17
Region 
of failure
?A
1 
(cm
2
)
??(?)
 
Figure 3.28: SR and WCSR of the mean optimum. 
SR
WCSR
R
f
= 1.25
Region 
of failure
?A
1 
(cm
2
)
??(?)
 
Figure 3.29: SR and WCSR of the mean+std optimum. 
As we can see in the above figures, only the SR of our robust optimum design fully 
encloses the (?A
1,0
, ??
0
) range. The other designs have small regions for which the 
requirement (??)
2
 ? (??
0
)
2
 is not satisfied. The WCSR?s of the designs also reflect this 
observation. Only the WCSR of our robust optimum encloses the (?A
1,0
, ??
0
) range 
completely.  
 
3.4.3. Design of A Welded Beam 
This example is the well-known welded beam problem originally formulated by 
Ragsdell and Phillips (1976) and Reklaitis et al. (1983). We slightly modified this 
 64
 
problem by adding variations to two of the parameters, and by making the objective 
function discontinuous with respect to one of these parameters.  
In this problem, a beam A is to be welded to a rigid support member B. The beam 
has a rectangular cross-section and is to be made out of steel. The beam is designed to 
support a force F = 6000 lbf acting at the tip of the beam, and there are constraints on the 
shear stress, normal stress, deflection, and buckling load on the beam. The problem has 
four (4) continuous design variables, and they are: thickness of the weld (h), length of the 
weld (l), thickness of the beam (t), and width of the beam (b). All variables are in inches. 
The objective of the problem is to minimize the total cost of making such an assembly. 
Figure 3.30 shows a diagram of the welded beam assembly. 
l
L F
h
t
b
B
A
 
Figure 3.30: A welded beam assembly. 
The complete formulation of the problem is shown in Eq. (3.26).  
0.21.0           0.101.0   
0.101.0       0.21.0   
01
125.0
g01g
01
F
g01
0.25
g
01g01g:subject to
)L(c)c1(minimize
65
43
21
4
2
3cost
????
????
??????
??????
??????
+++=
bt
lh
hb
h
P
ltblhf
c
dd
?
?
?
?
?
         (3.26) 
 65
 
where:  
c
3
 = cost of weld material ($0.1047 /inch
3
) 
c
4
 = cost of beam material ($0.0481/inch
3
) 
? 
= maximum shear stress in weld (psi) 
?
d
 
= allowable shear stress of weld (13,600 psi) 
? 
= maximum normal stress in beam (psi) 
?
d
 
= allowable normal stress in beam (30,000 psi) 
? 
= deflection at beam end (inch) 
P
c
 = allowable buckling load (lbf) 
L = length of unwelded beam (14 inch) 
 
The quantities ?, ?, ?, and P
c
 are calculated as shown in Eq. (3.27) through Eq. (3.33). To 
calculate ?, it is assumed that the beam is a cantilever beam with length L, and for steel G 
= 12 ?10
6
 psi and E = 30 ?10
6
 psi.  
() ()
22
"cos"'2' ?????? ++=     (3.27) 
R
l
J
MR
hl 2
cos;";
2
F
' === ???    (3.28) 
2
2
24
;
2
LF
?
?
?
?
?
? +
+=
?
?
?
?
?
?
+=
thl
R
l
M    (3.29) 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? +
+=
2
2
212
707.02
thl
hlJ     (3.0) 
btbt
3
3
2
E
FL4
;
FL6
== ??      (3.1) 
?
?
?
?
?
?
?
?
?=
?
? EI
L2
1
L
EI013.4
2
t
P
c
    (3.2) 
33
G
3
1
;
12
1
I tbtb == ?      (3.3) 
 66
 
The parameters L and c
3
 vary by some amount ?L and ?c
3
, and they affect the total 
cost of the welded beam assembly as shown in Eq. (3.34) (the notation ??? is for rounding 
up to the nearest integer). Notice that Eq. (3.34) reduces to the original f
cost
 in Eq. (3.26) 
when ?L = 0 and ?c
3
 = 0. 
?
?
?
?
?
? ?
=
+++?++=
05.0
L
$0.01cwhere
c)L(c)cc1(
5
54
2
33cost
lltblhf
            (3.34) 
We want to find a design that minimizes f
cost
 and is insensitive to ?L and ?c
3
. We 
added the robustness constraint into Eq. (3.26), and we set ?f
0
 = $0.30 and 
?p
0
 = (0.05, 0.25). In this problem, p = (c
3
,L)
t
 and p
0 
= (0.1047, 14)
t
. We solved the 
problem using a GA for the outer and inner problems (with ?
0
 = 1), and obtained a robust 
optimum value of f
R
 = $2.49. For comparison, we solved Eq. (3.26) using the same GA, 
and obtained a nominal (non-robust) optimum of f
cost
 = $2.39 (which is very close to that 
reported by Ragsdell and Phillips (1976): f
cost
 = $2.38, and by Deb (1991): f
cost
 = $2.43).  
For further comparison, we also solved Eq. (3.26) for robust optimum by minimizing 
the mean value of the design. The mean value was calculated using Monte Carlo 
simulation assuming uniform and normal probability distribution for the parameters. For 
the uniform assumption, ?c
3
 distribution is modeled between [-0.05, 0.05] while ?L 
distribution is modeled between [-0.25, 0.25]. For the normal assumption, ?c
3
 
distribution is modeled to have a mean and standard deviation of [0, 0.05/3] while ?L 
distribution is modeled by [0, 0.25/3]. 
Table 3.2 shows a list of the optimum designs obtained (for the Monte Carlo optima, 
the f
cost
 shown are the mean values). The quantity F
call
 is the number of function 
 67
 
evaluations needed per design to obtain the sensitivity information. As shown in 
Table 3.2, making different assumptions about the pdf of the uncertain parameters can 
lead to significantly different results. We also observe in Table 3.2 that the solution 
obtained by our method is not the same as the one obtained by the Monte Carlo method 
for a uniform pdf. This shows that although our method uses a range of parameter 
variations, it is not the same as assuming a uniform pdf for the parameters. This in fact is 
an advantage of our method in that it does not require a presumed pdf of the uncertain 
parameters. In Chapter 7 we will show how to use the probability distribution 
information, if they are available, with our robust optimization method.  
Table 3.2: Optimum designs of the welded beam. 
Nominal 
Optimum
Robust 
Optimum
Monte Carlo 
(Uniform)
Monte Carlo 
(Normal)
f
cost
2.39 2.49 2.63 2.91
h 0.241 0.246 0.257 0.337
l 6.158 5.461 5.8 5.054
t 8.5 9.138 8.267 7.058
b 0.243 0.248 0.257 0.337
F
call
N/A 250 100000 100000
 
Figure 3.31 shows the SR and WCSR of the nominal optimum and the robust 
optimum obtained using our method. In this figure, the shaded region bounded between 
the straight line on the left and the step function on the right is the SR, the ellipse is the 
WCSR, and the dashed rectangle is the ?p
0
 ranges. We obtain the SR?s analytically by 
substituting the values in Table 3.2 into the f
cost
 function in Eq. (3.34) and obtain the ?f
cost
 
as a function of ?c
3
 and ?L. Then using the given ?f
0
 value, we solve for ?c
3
 at the six 
discrete ?L values. Mathematically, the left boundary of the sensitivity region is also a 
step function similar to that of the right boundary. However, the parameter c
3
 represents a 
 68
 
material cost, so in reality it cannot be negative. For this reason we restrict the ?c
3
 value 
to be ?0.10471 at the lowest (the straight line left boundary). We obtain the WCSR?s by 
solving the inner optimization problem using the values in Table 3.2. The WCSR?s 
shown are ellipses (instead of circles) because Figure 3.31 shows the regions in the non-
normalized ?p-space (for clarity of presentation).  
-0.3
0.3
0
-0.10471 0.09529 0.29529 0.49529 0.69529 0.89529
3
c?
L?
(a) Nominal optimum
-0.2
-0.1
0.1
0.2
SR
WCSR
Region of failure
0
0.89529
3
c?
L?
(b) Robust optimum
0.09529 0.29529 0.49529 0.69529-0.10471
-0.3
-0.2
-0.1
0.1
0.2
0.3
SR
WCSR
No region of failure
 
Figure 3.31: SR and WCSR of the optima: (a) nominal, (b) robust. 
Notice that the SR?s are unbounded. This is because for the range of ?c
3
 values 
where the regions are unbounded, the value c
5
 = $0.05 (|?L|>0.20) still makes the 
inequality (?f)
2
 ? (?f
0
)
2
 satisfied. Notice also that the SR boundary is discontinuous 
because f
cost
 is a step function. However, at the discontinuity, (?f)
2
 = (?f
0
)
2
 is satisfied 
either from the left or from the right, so our SR boundary condition (?f)
2
 = (?f
0
)
2
 is valid 
(recall Figure 3.3).  
We observe from Figure 3.31 that the SR of the nominal optimum does not fully 
enclose the ?p
0
 ranges while the robust optimum?s does. This remains true when using 
 69
 
the WCSR?s as estimates of the SR?s. We conclude that in the worst case sense, the 
robust optimum is less sensitive to p = (c
3
,L)
t
 than the nominal optimum. 
To further assess the sensitivity of the robust optimum design obtained, we 
calculated the f
cost
 value of the design for 15 perturbed values of the two uncertain 
parameters (c
3
 and L), and determined how much the f
cost
 differed from the unperturbed 
f
cost
 value (Table 3.2). For comparison, we performed the same analysis (using the same 
?c
3
 and ?L values) to calculate the ?f
cost
 of the nominal and the Monte Carlo optima. In 
this analysis, the values for ?c
3
 and ?L were randomly sampled from the ?p
0
 range.  
Figure 3.32 and Figure 3.33 show the graphs of the ?f
cost
 of the optimum designs for 
the 15 cases. The dashed-lines in these figures are the ?f
0
. We observe in Figure 3.32 that 
in all 15 cases the ?f
cost
 of the robust optimum is less than that of the nominal optimum, 
i.e., ?f
R
 < ?f
N
. This shows that the robust optimum obtained by our method is less 
sensitive to ?p than the nominal optimum. In addition, we also observe that in all 15 
cases, the ?f
cost
 of the robust optimum stays within the acceptable bound, while the 
nominal optimum violates the bound in cases 6 and 15. In Figure 3.3 we observe that 
both Monte Carlo optima also violate the bound in cases 6 and 15.  
0
0.1
0.2
0.3
0.4
12345678910112131415
Case #
?
f
co
s
t
Nominal optimum Robust optimum
?
f
co
s
t
 
Figure 3.32: Sensitivity analysis of the welded beam optima. 
 70
 
1 2 3 4 5 6 7 8 9 101112131415
0
0.1
0.2
0.3
0.4
Case #
?
f
cost
MC (uniform) MC (normal)
?
f
cost
?
f
cost
 
Figure 3.33: Sensitivity analysis of the welded beam optima. 
In this chapter, our robust optimization method looks only at the objective robustness 
of a design, and we implicitly assume that the robust optimum design remains feasible 
when the variations occur. To verify the validity of this assumption, we calculated the 
constraints of the robust optimum for the 15 random cases. The results are shown in 
Table 3.3. In this table we calculated only the constraints g
1
, g
2
, g
3
, g
4
 because g
5
 and g
6
 
are independent of c
3
 and L. As shown in Table 3.3, in all 15 cases the constraints for the 
robust optimum are still satisfied (g ? 0), so our feasibility assumption holds. 
Table 3.3: Constraints of the robust optimum design. 
Case # g1 g2 g3 g4
1 -0.016 -0.204 -0.956 -0.134
2 -0.004 -0.190 -0.954 -0.109
3 -0.004 -0.190 -0.954 -0.109
4 -0.014 -0.201 -0.955 -0.128
5 -0.010 -0.196 -0.955 -0.120
6 -0.011 -0.198 -0.955 -0.124
7 -0.008 -0.194 -0.954 -0.116
8 -0.014 -0.201 -0.955 -0.128
9 -0.002 -0.188 -0.953 -0.105
10 -0.017 -0.205 -0.956 -0.135
11 -0.009 -0.195 -0.955 -0.119
12 -0.004 -0.190 -0.954 -0.109
13 -0.016 -0.204 -0.956 -0.134
14 -0.014 -0.201 -0.955 -0.129
15 -0.015 -0.202 -0.956 -0.131
 
 71
 
Interestingly, if we look into the constraints of the nominal optimum, we will find 
that constraints g
1
, g
2
, g
3
, g
4
 of this design are active or nearly active, and when c
3
 and L 
change, g
1
 and g
4
 are violated. So, in some sense the robust optimum provides design 
robustness not only in terms of the objective, but in terms of feasibility also; although 
obviously this observation is not general and is valid for this particular example only. We 
will discuss our feasibility robustness approach in Chapter 5. 
 
3.4.4. Design of a Compression Spring 
This example is from Arora (2001) and has been modified to demonstrate the 
application of our robust optimization method. Consider a coil spring loaded in 
compression as shown in Figure 3.34. Before the force P = 10 lbf is applied, the spring is 
at its free length. The spring is to be used as an energy-storing device, and we are 
designing it to have as large a restoring force as possible by maximizing the axial 
deflection ?. There are three variables that affect the design: the wire diameter (d), the 
mean coil diameter (D), and the number of active coils (N). The wire diameter and the 
mean coil diameter are measured in inches, and the number of active coils must be an 
integer between 2 and 15.  
P P
D
d ?
 
Figure 3.34: A compression spring.  
To guarantee a proper design, five (5) constraints are imposed. The wires of a spring 
under compression experience twisting, so to prevent shear failure, the shear stress due to 
 72
 
this twist is constrained to be less than ?
A
 = 80 ksi. In case of a dynamic load P, we would 
like to avoid resonance by requiring that the surge wave frequency of the spring is greater 
than ?
A
 = 100 Hz. The outer diameter of the spring is constrained to be less than 
OD
A
 = 1.5 in, and the total mass of the spring is constrained to be less than 
M
A
 = 2.309 x 10
-5
 lbm. To prevent an unrealistic design, the spring deflection is 
constrained to be less than ?
A
 = 0.75 in (objective constraint). In addition to the design 
constraints, there is also a lower and upper bound constraints on the design variables.  
The mathematical formulation of the problem is shown in Eq. (3.34). In this 
formulation: G = 1.15 x 10
7
 psi is the shear modulus, Q = 2 is the number of inactive 
coils, and ? = 7.383 x 10
-4
 lbm/in
3
 is the mass density. For a more detailed analysis of a 
spring design problem, consult Spott (1953) or Wahl (1963). 
15N2;30.1D25.0;20.0d05.0
01
?
?
g
0
M
M
1g;01
OD
dD
g
0
?
?
1g;01
?
?
g:subject to
Gd
NDP8
?maximize
A
5
A
4
A
3
A
2
A
1
4
3
N)D,(d,
??????
???
?????
+
?
??????
=
          (3.34) 
where:    
?
?
?
?
?
?
+
?
?
=
D
0.615d
4d4D
d4D
? d
8PD
?
3
                (3.35) 
    
2?
G
N2??
d
?
2
=                  (3.36) 
   Q)(N
4
?Dd?
M
22
+=                  (3.37) 
 73
 
We use a GA (Goldberg, 1989) to solve Eq. (3.34) and obtain the nominal optimum 
design (d,D,N)
*
 = (0.0519 in, 0.3616 in, 11) for a maximum deflection of ?
*
 = 0.4985 in. 
At the nominal optimum, constraints g
1
 and g
4
 are active. The lower bound of the 
variable d is also nearly active. In our implementation we use GA to solve this problem 
because one of the variables is an integer. However, other mixed-integer programming 
methods such as Branch and Bound algorithm (Belegundu and Chandrupatla, 1999) may 
also be used. For comparison, if we relax the variable N to be continuous, the nominal 
optimum becomes (d,D,N)
*
 = (0.0517 in, 0.3569 in, 11.293) for a maximum deflection of 
?
*
 = 0.50 in.  
Three of the problem?s parameters have variability in them: the wire diameter d 
varies by ??d
0
 = 0.001 in; the mean coil diameter D varies by ??D
0
 = 0.01 in; and the 
compression force P varies by ??P
0
 = 0.1 lbf. We want to minimize the sensitivity of the 
optimum design with respect to these variations. Using ??
0
 = 0.075 in, we added our 
robustness constraint to Eq. (3.34) and solved it using GA for the inner and outer 
optimization problems (?
0
 = 1). The robust optimum obtained is (d,D,N)
R
*
 = (0.0548 in, 
0.4219 in, 8) for a deflection of ?
*
 = 0.4633 in (a 0.0352 in. difference from the nominal 
optimum). For comparison, we also solved the problem using the constrained worst-case 
gradient approach where we added the gradient robustness constraint: 
?
=
?
?
?
=?
3
1
0,
p
p
?
?
i
i
i
? ??
0
 to Eq. (3.34). The optimum obtained using this worst-case 
gradient approach is (d,D,N)
G
*
 = (0.0560 in, 0.3319 in, 9) for a deflection of 
?
*
 = 0.2321 in. The values of the nominal, robust, and gradient optima are shown in 
Table 3.4. 
 74
 
Table 3.4: Optimum designs of the compression spring. 
Nominal 
Optimum
Robust 
Optimum
Gradient 
Optimum
? (in) 0.4985 0.4633 0.2321
? 0.8248 0.9942 1.9109
d (in) 0.0519 0.0548 0.056
D (in) 0.3616 0.4219 0.3319
N118
F
call
N/A 250 N/A
9
 
We see in Table 3.4 that the nominal optimum has the largest deflection, as expected, 
followed by the robust optimum and then the gradient optimum. If we look at the ? value, 
our robust optimum satisfies the ? ? 1 requirement with an equality (rounded up), the 
nominal optimum does not satisfy the requirement, while the gradient optimum over-
satisfies it. Although the gradient optimum more than satisfies our robustness 
requirement, it comes with a significant reduction in performance (less than half of the 
nominal value).  
This observation brings up a very important fact regarding our robust optimization 
method. Although we label our WCSR measure as a ?worst-case? approach, it is not 
really worst-case per se; at least not in the traditional sense. Traditionally, the term 
?worst-case? refers to a situation where all the worst possible variabilities occur together 
simultaneously. Our WCSR measure is different. Instead of blindly using the worst 
variabilities for each parameters, it has implicitly taken into account the fact that some of 
these variables might cancel out. This is the reason why our robust optimization method 
can obtain a design with such a high performance while still satisfying our robustness 
requirement. This capability comes with a price, however, namely it needs to perform 
 75
 
some function evaluations to gather the robustness information (~250 function calls per 
design in this example).  
To validate the robustness of the optimum designs shown in Table 3.4, we perturbed 
(d,D,P) 20 times within the (?d
0
,?D
0
,?P
0
) range, and each time calculated the deviation 
of the ? value of each design from its original value. The results of our analysis are 
shown in Figures 3.35 through 3.37. In these figures the dashed lines are the ??
0
. We can 
make a couple observations from these figures. We see in Figure 3.35 that the nominal 
optimum violates the ??
0
 requirement in case 11, while the robust and the gradient 
optima never violate ??
0
 (Figure 3.36 and Figure 3.37, respectively). We also observe in 
these figures that on average the ?? of the nominal optimum is large, while the robust 
optimum?s is less, and that of the gradient optimum is least. This observation verifies the 
information provided by their ? values.  
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
1 6 11 16
Case #
????
 
Figure 3.35: Sensitivity analysis of the nominal optimum. 
 76
 
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
1 6 11 16
Case #
????
 
Figure 3.36: Sensitivity analysis of the robust optimum. 
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
1 6 11 16
Case #
????
 
Figure 3.37: Sensitivity analysis of the gradient optimum. 
 
3.5. SUMMARY 
? For each design alternative, there is a unique set of ?p that indicates how much 
variation the design can absorb before the ?f
0
 limit is violated. This set is called the 
sensitivity set (S
f
) of that design.  
? Size of a S
f
 is a measure of how robust a design is: the larger S
f
, the more robust the 
design. In addition, since S
f
 measures how much variation a design can absorb, if we 
 77
 
can control the actual variations such that they are always within S
f
, then we are 
guaranteed that the performance of the design is always within ??f
0
. 
? The plot of S
f
 in the ?p-space is called the Sensitivity Region (SR) of the design. The 
?p points inside, outside, and on the SR boundary satisfy [?f (?p)]
2
 < [?f
0
]
2
, 
[?f (?p)]
2
 > [?f
0
]
2
, and [?f (?p)]
2
 = [?f
0
]
2
, respectively.  
? The size of a SR is a measure of a design?s robustness: the larger SR, the more robust 
the design. However, since a SR is typically asymmetric, we have to account for the 
directional sensitivity of the design as well. 
? We use the most sensitive direction in approximating a SR of a design to account for 
the worst-case situation. This worst-case approximation to a SR is called the Worst 
Case Sensitivity Region (WCSR), and is defined as the smallest hyper-sphere inside 
the SR that touches the SR boundary on at least one point.  
? Size of a WCSR is a worst-case measure of a design?s robustness: the larger the 
WCSR, the more robust the design. The radius of a WCSR (R
f
) can be calculated by 
solving an optimization problem shown in Eq. (3.4). Since WCSR size has a semi-
cardinal scale, it is not necessary to calculate its volume. The radius value is sufficient 
to compare robustness of two or more designs.  
? If the magnitudes of the ?p are significantly different, then we need to normalize the 
optimization problem used to calculate R
f
. We use a single-valued normalization with 
?p
0
 as the reference point.  
? To determine whether or not a design meets our robustness requirement, we use the 
robustness index 
f
R)G(?
21?
= . A design is robust if ? ? 1, and is not robust if ? < 1.  
 78
 
? In robust optimization we want to simultaneously maximize performance and 
robustness. These objectives are often conflicting, so to avoid having to make a trade-
off, we use a constrained robustness approach instead. 
? Using the constrained robustness approach, our robust optimization method searches 
for the robust optimum design by solving two optimization problems: an inner and 
outer optimization. The inner optimization is used to calculate a design?s robustness 
which is then used in the outer optimization.  
 79
 
CHAPTER 4 
MULTI-OBJECTIVE ROBUST OPTIMIZATION 
 
4.1. INTRODUCTION 
When an optimization problem has only one objective, the notion of a robust 
optimum is straightforward because the only trade-off present is between the objective 
and the robustness of the design. When a problem has multiple objectives, however, it is 
not quite so because the same variability might affect some or all of the objectives, and in 
turn, a design will have different robustness behavior with respect to different objectives. 
If we follow the conventional definition of a robust optimum design in the context of a 
multi-objective optimization, we might end up making trade-offs between the 
performance of the design in terms of one objective and its robustness in terms of another 
objective. Clearly, not only this is difficult to do, it is not very useful either.  
The purpose of this chapter is to present the concept of multi-objective robustness, a 
method on how to measure such robustness, and an optimization scheme to obtain a 
design that is optimum and robust multi-objectively by using this measure. The method to 
measure a multi-objective robustness of a design presented in this chapter is a 
generalization of the WCSR measure we presented in Chapter 3, so there will be 
similarities, but with important differences. Like in Chapter 3, here we look into the 
robustness of a design in terms of its objectives only. Feasibility robustness will be 
covered in Chapter 5. Several examples are given at the end of the chapter to demonstrate 
the applications of the method.  
  
 80
 
4.2. BASIC CONCEPTS 
Before we present the method to obtain robust solutions of a multi-objective 
problem, we need to first discuss what it means for a design to be multi-objectively 
robust and optimum. 
 
4.2.1. Multi-Objective Robustness 
When there is only one objective, a design is termed ?robust? if its objective value is 
insensitive to parameter variations. When there are multiple objectives, a design has 
different robustness behaviors depending on the objective in question. So by direct 
extension, a design is termed ?robust? multi-objectively if each of its objective value is 
insensitive to parameter variations. In other words, a design is robust (or insensitive) if 
?f
i
 is small, for all i = 1,?,M. In the f-space, the robustness of a design is depicted 
graphically as the hyper-rectangle constructed from the ?f
i
 ranges (Figure 4.1); the 
smaller this hyper-rectangle (i.e., a rectangle in two-dimension, as shown in Figure 4.1), 
the more robust the design. In Figure 4.1, the solid triangles denote the nominal f values 
of the designs A, B, C and D shown.  
f
1
f
2
A
C
D
B
?f
A,2
?f
A,1
 
Figure 4.1: Graphical illustration of multi-objective robustness. 
 81
 
As shown in Figure 4.1, it is very often the case that a design is less sensitive in one 
objective than the other objective(s) (design B). Also, the negative and positive ranges of 
?f
i
 do not necessarily have to be equal (design A and D).  
It should be noted that the term ?insensitive? is subjective and depends on the 
preferences of the decision maker. If the decision maker is risk averse, then for a design 
to be insensitive, its ?f
i
 must be very small for all objectives i = 1,?,M. On the other 
hand, if the decision maker is risk prone, then even a rather large ?f
i
 is acceptable. In this 
chapter, we deem a design multi-objectively robust if |?f
i
| ? |?f
i,0
| for all i = 1,?,M, 
where ?f
0
 = [?f
1,0
, ?, ?f
M,0
] is the ranges of acceptable f variation determined by the 
decision maker.  
The multi-objective robustness described previously is defined for one design only. 
If we have a set of designs, the overall robustness of the set is determined by looking at 
the robustness of each and every design, one at a time. In other words, our definition of 
multi-objective robustness is for point robustness. Another type of multi-objective 
robustness worth mentioning is that of set robustness. Unlike point robustness, set 
robustness is not defined for one design, but rather for a set of designs, and this set must 
be a trade-off set. A trade-off set is a set of designs in which all designs in the set are 
non-inferior with respect to each other (recall Section 2.2 that this set is different from a 
Pareto set). We define a set of trade-off designs to observe set robustness if the set 
remains a trade-off set when p varies.  
Figure 4.2 shows a comparison between a set of trade-off designs that observes point 
robustness and a set that observes set robustness. We see in Figure 4.2(a) that the ?f of 
each design in the set is small, but for some of the ?f rectangles, there are portions that lie 
 82
 
on either the dominant or inferior region of another ?f rectangle. What this means is that 
when p varies, it is possible that these points are no longer trade-off points (i.e., one 
dominates or is dominated by others). In contrast, the ?f of each design on the set shown 
in Figure 4.2(b) is rather large (i.e., not a point robustness), but all the ?f rectangles lie in 
the non-inferior region of each other. This means that no matter how the points change, 
these points will always remain a trade-off set. It can be readily seen in Figure 4.2 that as 
?f becomes infinitely small, point robustness implies set robustness and vice versa. 
f
1
f
2
a
b
c
?a? is always non-
inferior with ?b?
?c? is always non-
inferior with ?b?
(a) Point robustness (b) Set robustness
f
1
f
2
a
b
c
?a? can be 
inferior to ?b?
?b? can 
dominate ?c?
 
Figure 4.2: Comparison between point and set robustness. 
In multi-objective robust optimization, we are interested in the point robustness of a 
design only, and not its set robustness. This is because practically, the set robustness of a 
design is not important. We will discuss this issue further in the next section.  
 
4.2.2. Multi-Objective Robust Optimality 
The goal of Multi-Objective Robust Optimization (MORO) is to obtain a set of 
design alternatives that are: (i) multi-objectively robust (point robustness), and (ii) Pareto 
optimum for the nominal p. Such a set of design alternatives is termed multi-objectively 
robust and optimum, and we refer to them as robust Pareto solutions. The set of all robust 
Pareto solutions is called the robust Pareto set.  
 83
 
Ideally one wants the robust Pareto set to be the same as the nominal Pareto set. 
However, robust Pareto set is generally inferior to the nominal Pareto set as shown in 
Figure 4.3(a). It is also common for the robust Pareto set and the nominal Pareto set to 
overlap, as in Figure 4.3(b), or for the robust Pareto set to be a subset of the nominal 
Pareto set, as in Figure 4.3(c). Obviously, the robust Pareto set cannot be superior or be a 
superset of the nominal Pareto set. One might argue that in case of Figure 4.3(b), the 
overlap portion of the robust Pareto frontier is better than the non-overlap portion 
because not only this portion is robust, but it also belongs to the nominal Pareto frontier. 
In some sense, it is. However, objective-wise this overlap portion does not dominate the 
non-overlap portion, so multi-objectively the overlap and non-overlap portions are 
equally optimum (i.e., incomparable).  
nominal
Pareto frontier
robust 
Pareto frontier
f
1
f
2
(c)
f
1
f
2
(a)
nominal
Pareto frontier
robust 
Pareto frontier
f
1
f
2
(b)
overlap
nominal
Pareto frontier
robust 
Pareto frontier
 
Figure 4.3: Comparison between nominal and robust Pareto set. 
 84
 
When p varies, f(x,p) deviates from its nominal value. Because the robust Pareto set 
is obtained based on point robustness, it is possible that when p changes the robust Pareto 
set is no longer a trade-off set. However, although the solution to a multi-objective 
optimization problem is a set of trade-off designs, ultimately the designer would choose a 
single design to implement. The purpose of including a design?s robustness as an 
additional optimization criterion is to ensure that the one design selected is robust with 
respect to variations in p. As such, it does not matter if the robust Pareto designs as a set 
are insensitive to p. In other words, the set robustness of the robust Pareto set is of no 
real-world interest beyond our mathematical curiosity.  
 
4.3. TWO-SIDED SENSITIVITY OF MULTIPLE FUNCTIONS 
We are now ready to present our method for measuring multi-objective robustness of 
a design. The method presented here is a generalization of the WCSR method presented 
in Chapter 3 for single objective robustness. As before, the robustness measured here is a 
two-sided robustness, i.e., we account for both the increase and decrease in f.  
 
4.3.1. Generalized Sensitivity Set and Sensitivity Region 
Let x
0
 be the design alternative whose sensitivity we want to measure, and let p
0
 be 
the nominal parameter value with respect to which the objective values of this design is 
defined, i.e., f(x
0
,p
0
). Given the acceptable variation range ?f
0
 = [?f
1,0
, ?f
2,0
, ?, ?f
M,0
], 
there is a set of ?p values such that the ?f due to these ?p values falls within the ranges 
of ?f
i,0
 for all i = 1,?,M. This set is the generalized sensitivity set S
f
, and it is 
mathematically defined as shown in Eq. (4.1). As before, we use the square of each ?f
i
 
 85
 
values to account for negative values, i.e., a two-sided sensitivity. We do not make a 
distinction between single and multiple objective S
f
 because the single objective S
f
 is 
simply the multi-objective version of S
f
 for M = 1.  
{ }
),(),(:where
M,...,1,][][ :R),(
0000
2
0,
2G
00
ppp
ppS
xx
x
iii
if
fff
iff
i
??+=?
=??????=
           (4.1) 
Similar to the single objective case (recall Chapter 3), S
f
 determines how much ?p a 
design can absorb for the given ?f
0
. So, the larger S
f
, the more robust the design, and if 
we can make sure that ?p is always a member of S
f
, then ?f
0
 will always be satisfied. The 
plot of the generalized S
f
 on the ?p-space is the generalized Sensitivity Region (SR) of 
the design.  
If we let the notation S
f,i
 be the set of ?p?s such that [?f
i
]
2
 ? [?f
i,0
]
2
, i.e., the 
sensitivity set with respect to the i-th objective, then it is easily seen that S
f
 is really just 
the intersection of all S
f,i
?s: 
S
f
 = S
f,1
 ? ? ? S
f,M
            (4.2) 
The above observation implies that the generalized SR corresponding to S
f
 is simply the 
intersection of all the SR?s of S
f,i
?s (Figure 4.4). The existence and uniqueness of S
f
 also 
follows directly from the existence and uniqueness of each S
f,i
. 
The fact that the generalized SR is an intersection of each objective?s SR?s has 
another important consequence: it defines the requirements that must be met for a point in 
the ?p-space to be inside, outside, or on the boundary of the generalized SR. A point 
inside the region (shaded region in Figure 4.4) implies that it belongs to all S
f,i
?s, but it is 
not on the boundary of any S
f,i
?s, hence it must satisfy [?f
i
]
2
 < [?f
i,0
]
2
 for all i = 1,?,M. A 
point outside the region implies that it does not belong to at least one S
f,i
, thus for this 
 86
 
point there must be at least one i such that [?f
i
]
2
 > [?f
i,0
]
2
. A point on the boundary of the 
region implies that not only it belongs to all S
f,i
?s, but it is also on the boundary of at least 
one S
f,i
, therefore it must satisfy [?f
i
]
2
 ? [?f
i,0
]
2
 with a strict equality for at least one i. 
1
p?
2
p?
SR
1
SR
2
Intersection,
overall SR
 
Figure 4.4: Graphical definition of the generalized SR. 
Graphically, the points inside, outside, and on the perimeter of the generalized SR 
correspond to points inside, outside, and on the boundary of the hyper-rectangle formed 
by the ?f
0
 ranges respectively (Figure 4.5).  
1
p?
2
p?
),(
00
pxf
M,...,1
)()(:
2
0,
2
=
?>??
i
ffi
ii
2
0,
2
2
0,
2
)()(:
M,...,1;)()(
ii
ii
ffi
iff
?=??
=???
M,...,1
)()(
2
0,
2
=
?<?
i
ff
ii
(a)
A
B
C
(b)
1
f
2
f
1,0
f?
2,0
f?
),(
00
pxf
A
B
C
 
Figure 4.5: (a) Generalized SR, and (b) its corresponding ?f
0
 ranges. 
Theoretically, it is possible that the SR?s intersect at their boundaries only, which 
graphically translates into a degenerate generalized SR (it is now a G-1 dimensional 
 87
 
hyper-surface). When this situation occurs, the points inside the generalized SR satisfy 
the condition for a boundary point as well. Practically however, this implies that one of 
the ?p
j
?s can be expressed as a function of the other ?p
j
?s. In reality this situation almost 
never happens, so for practical purposes we can safely assume that the SR?s intersection 
is a G-dimensional hyper-surface, and that the equations shown in Figure 4.5(a) are valid.  
 
4.3.2. Generalized Worst Case Sensitivity Region 
As seen in Figures 4.4 and 4.5, in general a multi objective SR is asymmetric, and as 
such it is possible that a design is very sensitive along a certain direction of ?p, but is 
much less sensitive along other directions. To account for this directional sensitivity, here 
too we use a worst-case approximation to the generalized SR to measure a design?s 
robustness. We call this approximation the generalized Worst Case Sensitivity Region 
(WCSR), which is a symmetric region that approximates the generalized SR along its 
most sensitive direction. The definition for the generalized WCSR is very much the same 
as the single objective WCSR except that now it is defined for the generalized SR. So, 
mathematically the generalized WCSR is defined as a subset of S
f
 in which for each point 
in the subset, its distance from the origin is less than or equal to the distance of the 
smallest ?p value in S
f
 that causes ?f
0
 to be satisfied. Graphically, the generalized WCSR 
is a hyper-sphere inside the generalized SR that touches the SR boundary at the closest 
point from the origin, as shown in Figure 4.6.  
 88
 
1
p?
2
p?
Generalized SR
Most sensitive
direction
Generalized 
WCSR
0
A
 
Figure 4.6: Generalized WCSR. 
The radius of the generalized WCSR for a design x
0
 can be calculated by solving the 
single objective optimization problem shown in Eq. (4.3). In this problem, the design 
variables are the ?p, the objective function is the WCSR radius, and the constraints 
reflect the fact that the WCSR touches the SR at a boundary point. Notice how the 
constraints of Eq. (4.3) differ significantly from their single objective counterpart 
(Eq. (3.4)). This is because the boundary condition for the generalized SR is different 
than the single objective SR. Despite the difference, however, if M = 1, then Eq. (4.3) 
collapses into Eq. (3.4), thus shows that Eq. (4.3) is simply the generalization of 
Eq. (3.4).  
equalitystrict  oneleast at with 
),(),(
M1,...,;01
][
][
:tosubject
)p()(Rminimize
0000
2
0,
2
2
1
G
1
2
ppp
p
p
xx
iii
i
i
j
jf
fff
i
f
f
??+=?
=??
?
?
?
?
?
?
?
?
? ?=?
=
?
       (4.3) 
Like in single objective WCSR, we can use R  instead of R
2
f
f
 if it makes the problem 
easier to solve. However, this time we cannot choose between either ?=? or ??? for the 
 89
 
constraints. The constraints must all be inequalities, and at the optimum, at least one of 
them must be satisfied as an equality. 
Eq. (4.3) is a single objective optimization problem with M inequality constraints 
(from which at least one is active). Realizing that these constraints simply mean that the 
optimum must lie on the boundary of the rectangle formed by the ?f
0
 ranges (see 
Figure 4.5), we can simplify the constraints by aggregating them into a single constraint 
as follows:  
),(),(:where
01max
0000
0,
M,...,1
ppp xx
iii
i
i
i
fff
f
f
??+=?
=?
?
?
?
?
?
?
?
?
?
?
=
   (4.4) 
The simplified constraint in Eq. (4.4) above restricts the maximum ratio of 
0,i
i
f
f
?
?
 to be 
unity (we use the absolute of ?f
i
 to account for negative values). Graphically, this 
constraint means that the resultant vector of ?f
i
?s must touch the rectangle formed by the 
?f
0
, which is again equivalent to saying that ?p must be on the generalized SR boundary 
(Figure 4.7). 
1
f
2
f
1,0
1
f
f
?
?
),(
00
pxf
2,0
2
f
f
?
?
1,max
1,0
1
2,0
2
1,0
1
=
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
f
f
f
f
f
f
1,0
f?
2,0
f?
 
Figure 4.7: Graphical interpretation of the simplified constraint. 
 90
 
Substituting the constraints in Eq. (4.3) with the simplified constraint in Eq. (4.4), 
the optimization problem to obtain the radius of the generalized WCSR becomes: 
),(),(:where
01max:tosubject
)p()(Rminimize
0000
0,
M,...,1
2
1
G
1
2
ppp
p
p
xx
iii
i
i
i
j
jf
fff
f
f
??+=?
=?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?=?
=
=
?
   (4.5) 
As in the single objective case, Eq. (4.5) is always guaranteed to have a feasible solution. 
At the very least ?p = {0} is guaranteed to be a feasible solution to Eq. (4.5) and R
f
 = 0. 
It is also possible that the generalized WCSR is completely unbounded, and R
f
 = ?. 
However, in general 0 < R
f
 < ?, and solving Eq. (4.5) results in a finite radius of the 
generalized WCSR.  
 
4.3.3. Normalization 
Again, when the magnitudes of ?p are different, Eq. (4.5) needs to be normalized to 
account for the difference in scale importance. Using a single-valued normalization 
where we use the known variation ranges ?p
0
 as the reference value, the normalized 
optimization problem to obtain the generalized WCSR radius is obtained: 
[]
j
j
j
i
i
i
j
jf
f
f
,0
G1
0,
0
M,...,1
2
1
G
1
2
p
p
p;p,...,p:where
01
)(
max:tosubject
)p()(Rminimize
?
?
=???=?
=?
?
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
? ?=?
=
=
?
p
pp
p
p
   (4.6) 
 91
 
In this formulation, the objective and constraint are the same as in Eq. (4.5) except that 
they are modified for p?  instead of ?p. As before we can use 
2
R
f
 for the objective if it 
makes the problem easier to solve.  
To demonstrate the application of Eq. (4.6), let us revisit our stainless steel piston pin 
example from Chapter 3. Previously, we measured the robustness of this pin based on its 
weight only. Here we will add a second objective, the stress acting on the pin. 
 
Example 4.1 
As part of the engine assembly, our stainless steel pin is attached to a piston arm at 
the middle of the pin (Figure 4.8). The length of the pin in contact with the arm is 3 cm, 
and there is a constant force P = 50 kN acting on the piston arm as shown in the figure. 
Determine the normal stress on the pin. If the density and radius of the pin vary by 
(??
0
,?r
0
) = (0.2, 0.02), determine also the normalized radius of the two-objective WCSR 
(weight and stress). The acceptable stress variation is given to be ??
0
 = 0.5 MPa.  
pin
piston
arm
P = 50 kN
3 cm
 
Figure 4.8: A piston-pin-arm assembly. 
 92
 
Solution 
The pin is cylindrical, so the amount of area used to calculate stress is the projection 
of a cylindrical area, i.e., a rectangle. The length of the projection area is the length of 
contact (l = 30 mm), while the width is the diameter of the pin (= 2r). So, the normal 
stress acting on the pin is 
(30)(2r)
P
? = = 41.6 MPa. (Notice that this value is much less 
than the yield stress of stainless steel ~200 MPa, so our pin design is fine.)  
If we use Eq. (4.6) to calculate the normalized WCSR radius of this problem, we 
obtain R
f
 = 0.618. Let?s verify this result. The generalized SR of the pin design is the 
intersection of the weight SR and the stress SR. We have obtained the weight SR from 
the previous examples in Chapter 3, so all we need now is the stress SR. Setting 
(??)
2
 ? (??
0
)
2
, we obtain the following quadratic inequality, where u = 1/(r+?r).  
031.1730
2
P2.83
4
P
2
2
2
?+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
u
l
u
l
          (4.7) 
Substituting the value P = 50 kN and l = 3 cm, and solving the inequality in Eq. (4.7), we 
obtain the stress S
f
 to be as follows (in cm). Notice that the stress S
f
 is independent of ?? 
because ? is independent of ?.  
{ }0.0267r0197.0:)r,?( ??????=
f
S    (4.8) 
The generalized SR and WCSR (based on weight and stress) of this pin design are shown 
in Figure 4.9 (shown in the normalized region). In this graph, the region bounded by the 
solid lines is the weight SR, while the region bounded by the dashed lines is the stress 
SR. The shaded region is the overall SR, while the white circle is the overall WCSR.  
 93
 
-3 -2 -1 2 30 1
-3
-2
-1
2
3
1
Overall WCSR
R
f
= 0.618
Weight SR
??
r?
Stress SR
Overall SR
?
 
Figure 4.9: Overall SR and WCSR of the stainless steel pin. 
From the graph, the point of contact is approximately at )r?,?(?  = (0.4, 0.5), for a 
WCSR radius of 0.64. This value is in good agreement with the one obtained analytically 
(R
f
 = 0.618). Notice also in this graph that the R
f
 value is governed by the weight SR. ? 
 
4.4. ROBUST OPTIMIZATION 
We will now use the generalized R
f
 measure developed previously to obtain robust 
solutions to a multi-objective optimization problem.  
As before, by considering the robustness of a design, we have essentially added an 
additional objective to an optimization problem that already has multiple objectives. 
Unlike other ?real? objectives, however, robustness is a rather abstract concept, and 
therefore it is hard to make trade-offs with robustness as one of the objectives. To avoid 
 94
 
this difficulty, we again use a constrained approach to MORO where we treat the 
robustness of a design as a constraint, and not as an objective.  
With this constrained approach, our MORO method obtains the robust Pareto set by 
first eliminating all non-robust designs and then selecting the best set of trade-off designs 
from the designs that are not eliminated. These optimization steps are illustrated with an 
example in Figure 4.10. In this figure, the triangles indicate the nominal f values of a few 
designs, and the dashed rectangles show the possible ?f. Initially, there are five feasible 
design alternatives (Figure 4.10(a)), but designs A and C do not meet the robustness 
constraint so they are eliminated (recall that in the objective space, the smaller the 
possible ?f range, the more robust the design). From the remaining designs, design E is 
inferior to design B so design E too is eliminated. Designs B and D are our robust Pareto 
designs, as shown in Figure 4.10(b).  
f
1
f
2
A
B
C
D
E
(a) Initially
not robust,
eliminate
f
1
f
2
B
D
E
(b) After elimination
Robust 
Pareto set
 
Figure 4.10: Constrained MORO approach. 
If we define 
f
R)G(?
21?
=  to be the generalized robustness index, the problem to 
obtain robust solutions to a multi objective problem becomes as follows.  
 95
 
[ ]
0
?
?
1
K,...,1;0),(h
J,...,1;0),(g:subject to
),(),...,,(minimize
0
0
0
0M01
??
==
=?
k
j
ff
k
j
p
p
pp
x
x
xx
x
   (4.9) 
Here, ?
0
 is the desired level of robustness and is determined by the designer, and R
f
 is 
calculated using Eq. (4.6). 
Eq. (4.9) can be solved using any traditional multi-objective optimization methods: 
no preference (e.g., global criterion), a priori (e.g., weighting method), or a posteriori 
(e.g., Multi-Objective Genetic Algorithm, Multi-Objective Simulated Annealing). 
However, again due to the ?hole-inducing? nature of the robustness constraint, we do not 
recommend those methods that solve multi-objective problems by converting them into a 
single-objective form. In our implementation, we use Multi-Objective Genetic Algorithm 
(MOGA) as our optimizer (see for example, Deb (2001) or Coello Coello et al. (2002)).  
 
4.5. COMPARISON STUDY 
To demonstrate our MORO method, we applied it to four examples: one numerical 
and three engineering examples. The purpose of the numerical example is to demonstrate 
the behavior of the robust Pareto solutions as we relax our robustness constraint. The 
purpose of the vibrating platform and the speed reducer examples is to demonstrate the 
application of our method to a two-objective optimization problem. The power electronic 
example is presented to show the application of our method to a three-objective 
optimization problem. 
In all examples, the parameter variability of the problems is large (more than 5% of 
the nominal value). In two of the engineering examples, one of the design variables is 
 96
 
discrete. For implementation reasons, we use a GA to solve the inner optimization. Thus, 
in each problem each robustness constraint calculation require ~250 function evaluations.  
 
4.5.1. Numerical Example 
This relatively simple example has three variables, two objectives, and three 
constraints. There are also lower and upper bounds on the variables. All variables are 
continuous, and two of the constraints are objective constraints. Both the objectives and 
the constraints are non-symmetric with respect to the variables, and they are non-linear 
and non-convex. The mathematical formulation of the problem is shown in Eq. (4.10).  
( ) ( ){ }
()
()
0.30.1
25.00001.0
25.00001.0
1000
180
: where,01
100
g
01
100
g;01
100
g:subject to
1000
1620
minimize
1161000minimize
3
2
1
31
5.0
2
3
3
3
3
2
2
1
1
31
5.0
2
3
2
),,(
5.0
2
32
5.0
2
311
),,(
321
321
??
??
??
+
=???
??????
+
=
+++=
x
x
x
xx
x
f
f
ff
xx
x
f
xxxxf
xxx
xxx
  (4.10) 
In this formulation, the objectives and constraints have been numerically scaled to help 
expedite the optimization search. The nominal Pareto solutions to this problem are shown 
in Figure 4.11. These solutions are obtained using the NSGA method (see Srinivas and 
Deb, 1995). 
 97
 
0
20
40
60
80
100
20406080
f
1
f
2
10
Nominal Robust
robust Pareto set
f
2
 
Figure 4.11: Nominal and robust Pareto set of the numerical example. 
All three variables have variations in them, and we want to obtain robust solutions to 
this problem. The variations in each variable are given as (?x
1
,?x
2
,?x
3
)
0
 = (10
-4
,10
-4
,0.1), 
while the acceptable variation of each objective is specified to be (?f
1,0
, ?f
2,0
) = (1.0, 1.0). 
Adding the robustness constraint 0
?
?
0
??1  to Eq. (4.10) and solving it (with ?
0
 = 1), we 
obtain the robust Pareto solutions as shown in Figure 4.11. We see in Figure 4.11 that the 
robust Pareto set is essentially a subset of the nominal Pareto set (recall Figure 4.3(c)).  
If we relax the robustness constraint slightly by using ?
0
 < 1, we will observe an 
interesting behavior. Figure 4.12 shows the robust Pareto set of the problem for ?
0
 = 0.8 
(left) and for ?
0
 = 0.5 (right). As we see in this figure, as the robustness constraint is 
relaxed, the robust Pareto set ?spreads out? along the nominal Pareto frontier, i.e., it 
includes those nominal Pareto solutions that were not robust enough before. This 
behavior is expected because there is a trade-off between the robustness of a design and 
its performance.  
 98
 
0
20
40
60
80
100
2040608010
f
1
f
2
nominal Pareto frontier
robust Pareto set
?
0
= 0.8
0
20
40
60
80
100
20406080
f
1
f
2
nominal Pareto frontier
10
robust Pareto set
?
0
= 0.5
f
2
f
2
f
2
f
2
 
Figure 4.12: Behavior of robust Pareto set as constraint is relaxed. 
Behavior similar to that observed in Figure 4.12 can also be expected for a robust 
Pareto set that is not a subset of the nominal Pareto set. For a robust Pareto set inferior to 
the nominal Pareto set (Figure 4.3(a)), the robust Pareto set will move closer to the 
nominal one as the robustness constraint is relaxed. For a robust Pareto set that overlaps 
with the nominal Pareto set (Figure 4.3(b)), the non-overlap part of the robust set will 
move closer to the nominal set while the overlap part will remain the same.  
For comparison, we also solved Eq. (4.10) using a probabilistic approach by 
minimizing both the mean values of f
1
 and f
2
. The mean values are calculated using 
100000 runs of Monte Carlo simulations assuming two different probability distributions 
of ?x: uniform and normal. For the uniform assumption, the pdf is modeled as an equal 
probability distribution between [?10
-4
, +10
-4
] for ?x
1
, between [?10
-4
, +10
-4
] for ?x
2
, and 
between [?0.1, +0.1] for ?x
3
. For the normal assumption, the pdf is modeled to have a 
mean value of 0, and a standard deviation of 10
-4
/3 for ?x
1
 and ?x
2
, and 0.1/3 for ?x
3
. In 
both models, the variations are assumed to be independent. The solutions obtained using 
this approach are shown in Figure 4.13. 
 99
 
0
20
40
60
80
100
2040608010
f
1
f
2
nominal Pareto frontier
robust Pareto set
?C-Uniform
0
20
40
60
80
100
20406080
f
1
f
2
nominal Pareto frontier
10
robust Pareto set
?C-Normal
f
2
f
2
f
2
f
2
 
Figure 4.13: Robust solutions using Monte Carlo simulations. 
We see in Figure 4.13 that solutions obtained using the Monte Carlo approach are really 
just the nominal Pareto solutions. In other words, minimizing the means of the objectives 
does not result in robust solutions. This is only an intuitive argument, however. Let us 
verify this claim.  
Showing the three-dimensional SR of the designs is difficult, and projecting the SR 
onto a two-dimensional plane does not provide much information. For this reason, we 
will forgo analyzing the SR and WCSR of the solutions obtained, and instead simply 
perform a numerical verification of their robustness. We verify the results by randomly 
perturbing (x
1
,x
2
,x
3
) around their nominal values according to (?x
1
,?x
2
,?x
3
)
0
, and 
calculating the difference between the perturbed f value and the nominal f value. A 
design is robust if ?f is within ?f
0
. Figures 4.14 and 4.15 show the ?f of two of the robust 
designs (denoted by ?robust-1? and ?robust-2?) obtained by our MORO method. The 
figure on the left is for ?f
1
, and the one on the right is for ?f
2
. The dashed lines in the 
figures are the ?f
0
. We see in these figures that the ?f of both designs is within ?f
0
. This 
shows that these designs are indeed robust.  
 100
 
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1 6 11 16
Case #
?
f
1
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1 6 11 16
Case #
?
f
2
?
f
1
?
f
1
?
f
2
?
f
2
 
Figure 4.14: Sensitivity analysis of the robust-1 design. 
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
16116
Case #
?
f
1
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1 6 11 16
Case #
?
f
2
?
f
1
?
f
1
?
f
2
?
f
2
 
Figure 4.15: Sensitivity analysis of the robust-2 design. 
For comparison, we performed the same analysis for the nominal designs. This time 
we picked three designs from the nominal Pareto set: one from the overlap region 
(nominal-1), one from the region where the f
1
 value is high (nominal-2), and one from the 
region where the f
2
 value is high (nominal-3). The ?f of the three designs due to 
perturbations is shown in Figures 4.16, 4.17, and 4.18 for the nominal-1, 2, and 3 designs, 
respectively. As before, the figure on the left is for ?f
1
, while the figure on the right is for 
?f
2
.  
 101
 
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1 6 11 16
Case #
?
f
1
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1 6 11 16
Case #
?
f
2
?
f
1
?
f
1
?
f
2
?
f
2
 
Figure 4.16: Sensitivity analysis of the nominal-1 design. 
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
16116
Case #
?
f
1
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
16116
Case #
?
f
2
?
f
1
?
f
1
?
f
2
?
f
2
 
Figure 4.17: Sensitivity analysis of the nominal-2 design. 
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
16116
Case #
?
f
1
-4
-3
-2
-1
0
1
2
3
4
1 6 11 16
Case #
?
f
2
?
f
1
?
f
1
?
f
2
?
f
2
 
Figure 4.18: Sensitivity analysis of the nominal-3 design. 
 102
 
We see in Figure 4.16 that the nominal-1 design satisfies the ?f
0
 limit, so this further 
shows that the designs in the overlap region are indeed robust. In contrast, Figure 4.17 
shows that the nominal-2 design violates the ?f
1,0
 limit, so this design is not robust with 
respect to f
1
. Similarly, the nominal-3 design violates the ?f
2,0
 limit, so this design is not 
robust with respect to f
2
.  
From Figure 4.13, the solutions obtained using a Monte Carlo approach are 
essentially the same as the nominal Pareto solutions, so robustness of these solutions are 
the same as the robustness of the nominal designs (Figure 4.18), and there is no need to 
redo the sensitivity analysis. 
 
4.5.2. Design of a Vibrating Platform 
For the second example, we applied our MORO approach to a two-objective 
constrained optimization problem as given by Narayanan and Azarm (1999). We 
modified the problem slightly by adding variations to two of the problem?s parameters.  
In this problem, we want to optimize the design of a vibrating platform modeled as a 
pinned-pinned sandwich beam with a vibrating motor on top, as shown in Figure 4.19. 
d
1 d2
d
3
L
   b
Vibrating
   Motor
 
Figure 4.19: A pinned-pinned vibrating platform. 
The platform has three layers of material (the inner layer, two middle layers 
sandwiching the inner layer, and two outer layers sandwiching the inner and middle 
 103
 
layers), and they are to be made out of three different materials: materials A, B, and C. 
The choice of materials for the layers must be mutually exclusive, i.e., no two layers can 
use the same material. However, the layers are allowed to have zero thickness (i.e., there 
is no layer). The properties of each of the materials are shown in Table 4.1. In this table, 
? is the mass density, E is the modulus of elasticity, and c is the cost of the material per 
volume.  
Table 4.1: Material properties. 
Material A Material B Material C
? (kg/m
3
) 100 2770 7780
E (GPa) 1.6 70 200
c ($/m
3
) 500 1500 800
 
We want to minimize the total cost of making such a platform and maximize its 
natural frequency by controlling five sizing variables (continuous) and one combinatorial 
variable (discrete). The sizing variables are the width of the platform (b), the length of the 
beam (L), and the thicknesses of the three layers (d
1
, d
2
, and d
3
). The thicknesses of the 
middle and outer layers are represented as a difference between two sizing variables (e.g., 
thickness of the middle layer is equal to (d
2
-d
1
)).  The combinatorial variable is the choice 
of materials for the layers (M). Since there are three possible material types, there are six 
possibilities for M (starting from the inner layer outward): {A,B,C}, {A,C,B}, {B,A,C}, 
{B,C,A}, {C,A,B}, and {C,B,A}. The platform design is subjected to five constraints: the 
maximum weight of the platform and the lower and upper limits on the thickness of the 
middle and outer layers.  
The optimization formulation for this example is shown in Eq. (4.11).  
 104
 
( ) ( )[ ]
()()[]
()()
()
()
()
() 001.0ddg
0ddg
015.0ddg
0d-dg
02800-Lg:subject to
dddddb2
ddEddEdE
3
b2
EI
EI
L2
?
maximize
ddcddcdcb2costminimize
235
324
123
212
1
23312211
3
2
3
33
3
1
3
22
3
11
5.0
2
n
23312211
????
???
????
??
??
?+?+=
?+?+
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?+?+=
?
????
?
f
   (4.11) 
In Eq. (4.11), the notations (?
1
, ?
2
, ?
3
), (E
1
, E
2
, E
3
), and (c
1
, c
2
, c
3
) refer to the density, 
modulus of elasticity, and material cost for the inner, middle, and outer layer of the 
platform, respectively. The lower and upper bounds for the sizing variables are: 
0.05 ? d
1
 ? 0.5, 0.2 ? d
2
 ? 0.5, 0.2 ? d
3
 ? 0.6, 0.35 ? b ? 0.5, and 3 ? L ? 6. 
There are variations in the density and cost of ?Material A? (?
A
 and c
A
), and we want 
our optimum solutions to be insensitive to these variations. More specifically, we want to 
obtain the robust Pareto solutions of this problem for the nominal parameter values: 
?
A
 = 100 kg/m
3
 and c
A
 = 500 $/m
3
.  
We add the robustness constraint 0
?
?
0
??1  to Eq. (4.11), and solve it (using ?
0
 = 1). 
For the sensitivity requirements, we set the acceptable ?f to be (?f
1,0
, ?f
2,0
) = ($5, 5Hz) 
and the parameter variations are known to be (??
A,0
, ?c
A,0
) = (5 kg/m
3
, 25 $/m
3
); 5% of 
the nominal ?
A
 and c
A
 values. For the optimizer, we use Multi-Objective Genetic 
Algorithm (MOGA) of Fonseca and Fleming (1993) combined with the method of 
Kurapati et al. (2002) to handle constraints.  
 105
 
Figure 4.20 shows the robust Pareto set obtained (shown as a min-min plot by taking 
the negative of the frequency value). The average number of function evaluations 
performed in calculating the sensitivity constraint is 250. For comparison, Figure 4.20 
also shows the nominal Pareto set of the problem (without the sensitivity constraint), and 
the Pareto set obtained using the probabilistic approach. In the probabilistic approach, we 
minimize the sum of mean and standard deviation value for each objective. These values 
are calculated by performing 100,000 Monte Carlo simulations. 
0
50
100
150
200
250
-500 -400 -300 -200 -100 0
- Natural frequency (Hz)
Co
s
t
 (
$
)
Nominal Robust Probabilistic
Point B
Point A
Point C
Co
s
t
 (
$
)
 
Figure 4.20: Pareto sets of the vibrating platform example. 
We observe in Figure 4.20 that for the given sensitivity requirements, the robust 
Pareto set obtained is mostly inferior to the nominal Pareto set. So in designing the 
platform, we must sacrifice some performance to achieve higher robustness. However, 
when the frequency and cost values of the designs are in the range of (150 Hz, 200 Hz) 
and ($75, $100), the nominal and robust Pareto set overlap. Thus, those platform designs 
 106
 
within these ranges are not only insensitive to changes in ?
A
 and c
A
, but are also Pareto 
optimum for the nominal ?
A
 and c
A
 values. Naturally, when making a selection for the 
final design, the designs in the overlap region are good candidates. The Pareto set of the 
probabilistic approach is inferior overall to both the nominal and robust Pareto sets. So, 
based on their multi-objective optimality these solutions are conservative. 
Figure 4.20 shows the optimality of the robust Pareto set obtained, but we have yet to 
make a case for its robustness. To verify the robustness of the obtained robust Pareto set, 
we performed a sensitivity analysis on the design points by arbitrarily perturbing the 
parameters (?
A
,c
A
) around their nominal values. We then computed the changes in the 
objective values of the designs due to these perturbations and compared those changes to 
the acceptable values. Due to space limitations, we present the results of such sensitivity 
analysis only for three randomly selected design points, one point each from the nominal, 
robust, and probabilistic Pareto set (points A, B, and C in Figure 4.20, respectively). 
Figure 4.21 shows the plots of the ?f values of the three design points.  
In Figure 4.21, the inner rectangle is the range of acceptable objective value ?f
0
, and 
the square points are the (?freq, ?cost) value of the design when the parameters (?
A
,c
A
) 
are perturbed. It is clearly shown in Figure 4.21(a) that when (?
A
,c
A
) are perturbed 
around their nominal values, the changes in objective values of the nominal Pareto design 
(point A) are larger than the acceptable range. Thus, according to the sensitivity 
requirements set by the designer, this design is not robust enough. On the other hand, the 
?f of the robust Pareto design, Figure 4.21(b), is within the acceptable range. So this 
design (point B) is indeed robust. The objectives variation of the probabilistic Pareto 
 107
 
design (Figure 4.21(c)) violates the sensitivity requirements as well, so not only this 
design (point C) is inferior, but it is also not robust. 
? frequency
?
co
s
t
(a) Nominal Pareto design
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
? frequency
?
co
s
t
(b) Robust Pareto design
? frequency
?
co
s
t
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
(c) Probabilistic Pareto design
?
co
s
t
?
co
s
t
?
co
s
t
?
co
s
t
?
co
s
t
?
co
s
t
 
Figure 4.21: Sensitivity of: (a) nominal, (b) robust, and (c) probabilistic designs. 
To further assess the sensitivity of the designs, we calculated their sensitivity regions 
and WCSR?s. This is done by substituting their design variable values (b,L,d
1
,d
2
,d
3
,M) 
into the cost and frequency function of Eq. (4.11), and then using the specified ?f
0
 value, 
solve for ??
A
 and ?c
A
. Figure 4.22(a), (b), and (c) show the SR?s and WCSR?s of the 
nominal (point A), robust (point B), and probabilistic (point C) Pareto design, 
respectively.  
 108
 
??
A
?c
A
WSCR
SR
-50
-30
-10
10
30
50
-50-30-1 103050
(a) nominal
 
-50
-30
-10
10
30
50
-50 -30 -10 10 30 50
??
A
?c
A
WSCR
SR
(b) robust
 
-50
-30
-10
10
30
50
-50 -30 -10 10 30 50
??
A
?c
A
WSCR
SR
(c) Probabilistic
 
Figure 4.22: SR and WCSR of: (a) nominal, (b) robust, and (c) probabilistic designs. 
 109
 
The rectangles around the origin in Figure 4.22 are the SR?s of the designs, while the 
circles are their WCSR?s. The SR?s are rectangles because each of the objective functions 
is a function of only one parameter, i.e., cost(?c
A
) and freq(??
A
). Because of this, each 
objective will be sensitive only to a range of one parameter value but not the other, and 
these ranges of values form a rectangle in the ?p-space. The WCSR?s are easily formed 
by finding the point on the SR boundary closest from the origin. 
We can make a few observations from Figure 4.22. First, we observe that both the 
SR and WCSR of the robust Pareto design are larger than that of the nominal and 
probabilistic Pareto designs, verifying the sensitivity analysis shown in Figure 4.21. 
Second, we also observe that the robust design satisfies the (??
A,0
, ?c
A,0
) = (5 kg/m
3
, 25 
$/m
3
) range, while the nominal and probabilistic designs do not. 
 
In our MORO approach, we have implicitly assumed that the constraints are still 
satisfied when the parameters change. To show the validity of this assumption, we 
performed a sensitivity analysis on the constraints of the robust Pareto design under 
study. We conducted the analysis by again arbitrarily perturbing the parameters around 
their nominal values and then calculating the constraints of the design for these perturbed 
parameters. Table 4.2 shows the values of the g
1
 constraint for 20 randomly perturbed ?
A
 
values (c
A
 does not affect the constraints, and g
2
, g
3
, g
4
, and g
5
 are independent of ?
A
). 
We see from the table that g
1
 is still satisfied when ?
A
 changes, so our assumption holds.  
 110
 
Table 4.2: Sensitivity of the constraints. 
Case # g
1
Case # g
1
1 -2196.5 11 -2199.67
2 -2199.8 12 -2195.41
3 -2198.2 13 -2195.33
4 -2198.5 14 -2198.57
5 -2198.6 15 -2197.84
6 -2198.0 16 -2198.77
7 -2198.1 17 -2199.42
8 -2196.9 18 -2195.66
9 -2196.3 19 -2197.14
10 -2195.4 20 -2196.81
 
 
4.5.3. Design of a Speed Reducer 
Our third example is the well-known problem of designing a two-shaft speed reducer 
formulated by Kurapati et al. (2002). Here, we modified it by adding variations to two of 
the design variables.  
Figure 4.23 shows the configuration of the speed reducer to be optimized. The 
objectives of the problem are to minimize the total weight of the speed reducer as well as 
the normal stress on the first gear shaft. Since the speed reducer is to be made of the same 
material throughout, the first objective is the same as minimizing the total volume. The 
problem has seven design variables: the gear face width (x
1
), the teeth module (x
2
), the 
number of teeth pinion (x
3
), the distance between bearings on the first shaft (x
4
) and on 
the second shaft (x
5
), and the diameter of the first shaft (x
6
) and second shaft (x
7
). All 
design variables are continuous except for x
3
 (the number of teeth), which is an integer.  
 111
 
x
7
x
5
x
6
x
4
bearings 1
shaft 1
shaft 2
bearings 2
 
Figure 4.23: A speed reducer. 
A lower and upper bound are imposed on each of the design variables. In addition, 
the design is subject to 11 inequality constraints as required by gear and shaft design 
practices. The constraints are: upper bound on the bending stress of the gear tooth (g
1
), 
upper bound on the contact stress of the gear tooth (g
2
), upper bound on the transverse 
deflection of the first shaft (g
3
) and the second shaft (g
4
), dimensional restrictions based 
on space and/or experience (g
5
, g
6
, and g
7
), design requirements on the shaft based on 
experience (g
8
 and g
9
), and upper bound on the normal stress on the first shaft (g
10
) and 
on the second shaft (g
11
). The constraint g
10
 is an objective constraint. The mathematical 
formulation of the problem is as follows. The units for all the variables are cm (except for 
x
3
 ? the integer variable). The unit for the first and second objective is cm
3
 and kPa, 
respectively.  
() ()
()
3
6
7
2
32
4
2
2
75
2
64
3
7
3
6
2
7
2
61
3
2
32
211
1.0
1069.1
745
minimize
7854.0
477.7 508.1
0934.43933.14
3
10
7854.0minimize
x
x
xx
x
f
xxxx
xxxxx
x
x
xxf
+
?
?
?
?
?
?
?
?
=
++
+++?
?
?
?
?
?
?
?
?
?+=
   (4.12a) 
 112
 
01100g
01300g;01.19.1g
05.19.1g;05g
012g;040g
0
93.1
1
g;0
93.1
1
g
0
5.397
11
g;0
27
11
g:tosubject
311
210759
648
2
1
7
2
1
6325
4
732
3
5
4
4
632
3
4
3
2
3
2
21
2
3
2
21
1
???
????+??
?+?????
??????
??????
??????
f
fxx
xx
x
x
x
x
xx
xxx
x
xxx
x
xxxxxx
  (4.12b) 
3
7
8
2
32
5
3
1.0
10575.1
745
:where
x
x
xx
x
f
+
?
?
?
?
?
?
?
?
=      (4.12c) 
  
5.50.5
9.39.23.83.7
3.83.72817
8.07.06.36.2
7
65
43
21
??
????
????
????
x
xx
xx
xx
      (4.12d) 
Solving Eq. (4.12) using NSGA, we obtain the nominal Pareto solutions as shown in 
Figure 4.24. We see in this figure that the nominal Pareto frontier is composed of two 
regions: a weakly Pareto region (the horizontal portion) and a normally Pareto region (the 
non-horizontal portion). For multi-objective minimization, a design x
a
 is termed weakly 
Pareto if there is no feasible design x
b
 such that f
i
(x
b
) < f
i
(x
a
) for all i=1,?,M. Some 
might argue that a weakly Pareto frontier is not really a Pareto frontier. However, to be 
general we include it in our discussion.  
 113
 
0
200
400
600
800
1000
1200
1400
0 1000 2000 3000 4000 5000 6000
Nominal Robust
nominal Pareto
robust Pareto
f
1
: volume
f
2
:
 str
e
ss
f
2
:
 str
e
ss
 
Figure 4.24: Nominal and robust Pareto solutions of the speed reducer problem. 
Two of the variables, the teeth module (x
2
) and the first shaft diameter (x
6
), vary by 
(?x
2
, ?x
6
) = (0.01, 0.1), and we want to obtain robust optimum solutions to this problem. 
The acceptable f variation is given to be (?f
1,0
, ?f
2,0
) = (150, 75). Using ?
0
 = 1, we added 
the robustness constraint 0
?
?
0
??1  to Eq. (4.12) and solved it using NSGA. The robust 
Pareto solutions obtained are also shown in Figure 4.24. We see in this figure that the 
robust Pareto set is a subset of the nominal Pareto set, and it is located on the weakly 
Pareto portion of the nominal Pareto frontier.  
To verify the robustness of the designs obtained, we randomly perturbed the 
variables x
2
 and x
6
 around their values following (?x
2
, ?x
6
) = (0.01, 0.1), and observed 
the changes in (f
1
, f
2
) due to these perturbations. To ensure the validity of the analysis, we 
performed the analysis on two designs from the robust Pareto set. For reference, we call 
them robust-1 and robust-2 designs. The results of the sensitivity analysis of the two 
 114
 
designs are shown in Figure 4.25. In this figure, the small rectangles are the ?f values of 
the design when x
2
 and x
6
 are perturbed, while the large inner rectangle is the ?f
0
 limit.  
-200 -150 -100 -50 0 50 100 150 200
-100
-80
-60
-40
-20
0
20
40
60
80
100
?f
1
?f
2
-200 -150 -100 -50 0 50 100 150 200
-100
-80
-60
-40
-20
0
20
40
60
80
100
?f
1
?f
2
 
Figure 4.25: Sensitivity analysis of the robust-1 (left) and robust-2 (right) design. 
We observe in Figure 4.25 that for both designs, the variations in f
1
 and f
2
 are always 
within the ?f
0
 limit, thus verifying that these designs are indeed robust.  
For comparison, we performed the same sensitivity analysis for two of the nominal 
Pareto designs. We chose one optimum design from the non-horizontal portion, and 
another design from the horizontal portion to be analyzed. For reference, we call them 
nominal-1 and nominal-2 design, respectively. The results of the sensitivity analysis are 
shown in Figure 4.26. As before, in this figure the small rectangles are the ?f values of 
the designs, while the large inner rectangle is the ?f
0
 limit.  
-200 -150 -100 -50 0 50 100 150 200
-100
-80
-60
-40
-20
0
20
40
60
80
100
?f
1
?f
2
-200 -150 -100 -50 0 50 100 150 200
-100
-80
-60
-40
-20
0
20
40
60
80
100
?f
1
?f
2
 
Figure 4.26: Sensitivity analysis of the nominal-1 (left) and nominal-2 (right) design. 
 115
 
We observe in Figure 4.26 that for the nominal-1 design, the variations in f
2
 violate the 
?f
2,0
 limit, while the ?f of the nominal-2 design is within the ?f
0
 limit. This observation 
shows that the designs in the horizontal portion of the nominal Pareto frontier are robust, 
while those in the non-horizontal portion are not.  
To further confirm the results in Figures 4.25 and 4.26, we computed the SR and 
WCSR of the designs. Since the robustness of the designs has been verified 
experimentally, here we only show the SR and WCSR of the nominal-1 and robust-1 
designs. Analytical derivation of the designs SR is very difficult, if at all possible. So, we 
performed an exhaustive analysis instead. We partition the ranges (?x
2
, ?x
6
) = (0.04, 0.4) 
into an equally spaced orthogonal grid, and calculate the ?f of the designs on each 
intersection point. If (?f)
2
 ? (?f
0
)
2
, then this point is in the SR. The set of such points is 
the SR of the design. To be able to derive the WCSR of the designs graphically, we plot 
the SR in the normalized space. The SR and WCSR of the nominal-1 and robust-1 
designs are shown in Figures 4.27(a) and 4.27(b), respectively (in the normalized space).  
WCSR
SR
6
x?
-3
-2
-1
1
2
3
-4 -2 0 2 4
2
x?
(a) nominal-1 design
 
Figure 4.27(a): SR and WCSR of the nominal-1 design. 
 116
 
-3
-2
-1
1
2
3
-4 -2 0 2 4
WCSR
SR
2
x?
6
x?
(b) robust-1 design
 
Figure 4.27(b): SR and WCSR of the robust-1 design.  
We see in Figure 4.27 that the SR of the robust-1 design encloses the normalized range of 
variable variation ),(
62
xx ?? = (1,1), while that of the nominal-1 design does not. This 
observation again confirms the robustness of each design (or lack thereof).  
If we solve the problem for robust optimum solutions using probabilistic methods 
(by minimizing mean of each objective), we will observe that the probabilistic Pareto 
solutions are essentially the same as the nominal Pareto solutions. So, to avoid repetition 
we do not re-show these results in the dissertation (a summary of this result can be found 
in Table 7.3 in Chapter 7).  
 
4.5.4. Design of a Power Electronic Module 
For our last example, we look into the design optimization of a power electronic 
module as formulated by Palli et al. (1998). The main components of the power module 
to be optimized are shown in Figure 4.28. In this configuration, a chip/die is attached to a 
substrate using solder. The substrate is made of alumina and is coated with copper.  
 117
 
t
Cu t
c
CuAl
2
O
3
solder
diel
c
l
s
= 27.5 mm
t
Al
 
Figure 4.28: A power electronic module. 
The four design variables of the problem are: the length of the chip (l
c
), the thickness of 
the chip (t
c
), the thickness of the alumina substrate (t
Al
), and the thickness of the copper 
coating (t
Cu
). The units for all design variables are mm.  
The power module experiences stresses in the solder layer due to thermal mismatch 
between the die and the substrate. In addition, the solder also undergoes fatigue due to 
cyclic stresses created by thermal changes during power switches. The objectives of the 
optimization problem are to: (1) minimize the maximum shear stress in the solder layer, 
(2) maximize the life expectancy of the module under chip fatigue, and (3) maximize the 
life expectancy of the module under substrate fatigue.  
The shear stress model used in this problem is based on elastic, plastic, and creep 
behavior and is governed by the following differential equation: 
casscc
t
kT
H
nn
ttEtE
dte
GT
AG
G
C
Gdx
d
cp
?????
?
?
?
?
?
?
?
?
+=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
+
?
??
11
0
1
2
2
  (4.13) 
In this equation, ? is the shear stress, G is the shear modulus, x is the distance from the 
center of the chip, ?H is the activation energy, k is the Boltzmann?s constant, and the 
quantities A, ?, n
p
, and n
c
 are constants. The boundary conditions for Eq. (4.13) are: (1) 
the shear stress at the center of the chip is zero, i.e., ?(0) = 0, and (2) the compressive 
 118
 
force caused by the solder displacement is equal to the tensile force caused by the thermal 
mismatch.  
The shear stress is a function of the position from the center of the chip. However, 
because we are interested only in the maximum shear stress, we need only to calculate the 
shear stress at the edge of the chip, i.e., at x = ? l
c
/2. Following the boundary condition, 
the shear stress at this position can be estimated using the following function. Here, we 
have looked into the shear stress at a particular instance of time.  
T
11
A:where
23
1
A4
2
3
2
max
?
?
?
?
?
?
?
?
?
+
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
=
sscc
cs
c
c
c
tEtE
l
l
l
??
?
    (4.14) 
The expected life of the module under chip fatigue is estimated using Eq. (4.15) 
following the physics of failure model of Suhir (1987). 
()() [
)2/1()2/1(
1
2
tensilechip
2
1
pcpc
pc
pcpc
vicvfcpc
pc
AAClife
??
?
??
??
?
??
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?= ]  (4.15a) 
T)(
)cosh(
1
1
4
11
)1(12
)(31
1
1
2
tensile
??
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
+
?
+
?
?
+
+
=
cs
cfactor
sc
cc
c
ss
s
c
cccs
c
lA
F
tt
tEtE
F
tEtt
t
??
??
?
?  (4.15b) 
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
=
)1(12
)(
)1(12
)(
)1(12
)(
2
3
2
3
2
3
1
c
cc
ca
caca
s
ss
tEtEtE
F
???
    (4.15c) 
2/1
1
2
3
2
33
4
)(11
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? +
+
?
?
?
?
?
?
?
? ?
+
?
?
?
?
?
?
?
? ?
=
ca
ca
s
s
c
c
sc
cc
c
ss
s
factor
G
t
G
t
G
t
F
tt
tEtE
A
??
    (4.15d) 
 119
 
Similarly, the expected module life under substrate fatigue is formulated as follows: 
()() [
)2/1()2/1(
1
2
tensilesubstrate
2
1
psps
ps
psps
visvfsps
ps
AAClife
??
?
??
??
?
??
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?= ]  (4.16a) 
T)(
)cosh(
1
1
4
11
)1(12
)(31
2
2
2
tensile
??
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
+
?
+
?
?
+
+
=
scase
sfactor
scase
ss
s
casecase
case
s
ssscase
s
lA
F
tt
tEtE
F
tEtt
t
??
??
?
?  (4.16b) 
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
=
)1(12
)(
)1(12
)(
)1(12
)(
2
3
2
3
2
3
2
s
ss
sa
sasa
case
casecase
tEtEtE
F
???
    (4.16c) 
2/1
2
2
3
2
33
4
)(11
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? +
+
?
?
?
?
?
?
?
? ?
+
?
?
?
?
?
?
?
? ?
=
sa
sa
s
s
case
case
scase
casecase
case
ss
s
factor
G
t
G
t
G
t
F
tt
tEtE
A
??
    (4.16d) 
The constants used in Eq. (4.14), (4.15), and (4.16) are as follows: 
 ?T = temperature difference between chip and substrate (= 10 ?K) 
A
vfc
 = Length of final vertical crack in chip (= 5.08 x 10
-5
 m) 
A
vfs
 = Length of final vertical crack in substrate (= 5.08 x 10
-5
 m) 
A
vic
 = Length of initial vertical crack in chip (= 6.35 x 10
-6
 m) 
A
vis
 = Length of initial vertical crack in substrate (= 6.35 x 10
-6
 m) 
C
pc
 = Paris coefficient for chip fatigue (= 7.53 x 10
-18
) 
C
ps
 = Paris coefficient for substrate fatigue (= 5.15 x 10
-15
) 
E
c
 = elastic modulus of chip (= 120 GPa) 
E
ca
 = elastic modulus of chip attach (= 11.85 GPa) 
E
case
 = elastic modulus of case (= 110 GPa) 
E
s
 = elastic modulus of substrate (= 327 GPa) 
 120
 
E
sa
 = elastic modulus of substrate attach (= 29.8 GPa) 
G
c
 = shear modulus of chip (= 45.1 GPa) 
G
ca
 = shear modulus of chip attach (= 4.2 GPa) 
G
case
 = shear modulus of case (= 42.3 GPa) 
G
s
 = shear modulus of substrate (= 130.8 GPa) 
G
sa
 = shear modulus of substrate attach (= 10.6 GPa) 
l
s
 = length of substrate (= 27.5 mm) 
t
ca
 = thickness of chip attach (= 0.08 mm) 
t
case
 = thickness of case (= 4.0 mm) 
t
s
 = thickness of substrate (= t
Cu
 + t
Al
)  
t
sa
 = thickness of substrate attach (= 0.5 mm) 
?
c
 = coefficient of thermal expansion of chip (= 2.6 x 10
-6
/?K) 
?
case
 = coefficient of thermal expansion of case (= 17 x 10
-6
/?K) 
?
s
 = coefficient of thermal expansion of solder layer (= 25 x 10
-6
/?K) 
?
pc
 = Paris exponent for chip fatigue (= 3) 
?
ps
 = Paris exponent for substrate fatigue (= 3.3) 
?
c
 = Poisson?s ratio of chip (= 0.3) 
?
ca
 = Poisson?s ratio of chip attach (= 0.4) 
?
case
 = Poisson?s ratio of case (= 0.3) 
?
s
 = Poisson?s ratio of substrate (= 0.25) 
?
sa
 = Poisson?s ratio of substrate attach (= 0.4) 
In addition to the three objectives, there are lower and upper bounds on the design 
variables. The complete formulation of the optimization problem is as follows: 
 121
 
substrate3
chip2
max1
maximize
maximize
minimize
lifef
lifef
f
=
=
=?
    (4.17a) 
  
0.21.0                  
5.10.3                  
0.1.10                  
0.100.5  :subject to
Al
Cu
??
??
??
??
c
c
t
t
t
l
    (4.17b) 
The nominal Pareto solutions of the optimization problem, Eq. (4.17), are shown in 
Figure 4.29 (these solution points are obtained using the NSGA method (Srinivas and 
Deb, 1995)). Because we have three objectives, we have shown the points projected onto 
two-dimensional planes. We have also shown the three objectives as all minimization by 
taking the negative of f
2
 and f
3
.  
Three of the design variables and four of the design parameters have variability in 
them. The three design variables are (t
c
, t
Al
, t
Cu
), and their variations are (0.05, 0.1, 0.2), 
respectively (all in mm). The four design parameters are (?
s
, ?
c
, E
c
, E
s
), and their 
variations are (2 x 10
-6
/?K, 0.1 x 10
-6
/?K, 6 GPa, 10 GPa), respectively. It is required that 
the objective values are bounded within (?f
1,0
, ?f
2,0
, ?f
3,0
) = (3 MPa, 1.0 x 10
31
 cycles, 
1.5 x 10
26
 cycles). Adding the robustness constraint 0
?
?
0
??1  to Eq. (4.17) and solving it 
with ?
0
 = 1, we obtain the robust Pareto solutions as shown in Figure 4.29.  
From Figure 4.29, we see that the robust Pareto solutions are a subset of the nominal 
solutions. We also observe that the robust Pareto solutions are concentrated around the 
region close to the origin.  
 
 122
 
-70
-60
-50
-40
-30
-20
-10
0
5 101520253035
Nominal Robust
f
1
(MPa)
f
2
(x
 1
0
30
c
y
cles
)
(a) f
1
-f
2
plane
f
2
(x
 1
0
30
c
y
cles
)
 
-25
-20
-15
-10
-5
0
-70 -60 -50 -40 -30 -20 -10
Nominal Robust
f
3
(x 10
25
c
y
cles
)
f
2
(x 10
30
cycles)
(b) f
2
-f
3
plane
f
3
(x 10
25
c
y
cles
)
f
3
(x 10
25
c
y
cles
)
f
3
(x 10
25
c
y
cles
)
 
-25
-20
-15
-10
-5
0
5 101520253035
Nominal Robust
f
3
(x 10
25
c
y
cles)
f
1
(MPa)
(c) f
1
-f
3
plane
f
3
(x 10
25
c
y
cles)
f
3
(x 10
25
c
y
cles)
 
Figure 4.29: Nominal and robust Pareto solutions of the power electronic problem. 
 123
 
Let us now verify the robustness of the solutions obtained. Since we have a total of 7 
design factors with variability, it is impossible to show their SR and WCSR on a two-
dimensional plane. So, we will perform a numerical sensitivity analysis instead. We 
analyze the sensitivity of one design each from the nominal and robust solution set. For 
the nominal design, we choose a design not located in the region near the origin (the 
robust region). The two designs to be analyzed are shown in Table 4.3. For comparison, 
we have also shown their ? values. Notice in this table that ?
robust
 ? 1.0 while ?
nominal
 is 
not. So according to our WCSR prediction, the robust design satisfies the ?f
0
 limit while 
the nominal design does not.  
Table 4.3: Nominal and robust designs to be analyzed. 
Nominal Robust
l
c
 (mm) 6.071 9.754
t
Cu
 (mm) 0.596 0.153
t
Al
 (mm) 1.005 1.073
t
c
 (mm) 1.962 1.205
f
1
 (MPa) 23.978 9.764
f
2
 (x 10
30
 cycles) 24.478 10.552
f
3
 (x 10
25
 cycles) 9.013 5.789
? 0.418 1.075
 
We randomly perturb the value of the 7 design factors within their variation range, 
and then observe the changes in the objective values of the design. A design is robust if 
the objective changes are within the given limit, for all three objectives. The graphs of the 
?f of the two designs are shown in Figure 4.30. In this figure, the left three graphs are the 
?f
1
, ?f
2
, and ?f
3
 of the nominal design, while the right three graphs are the ?f
1
, ?f
2
, and 
 124
 
?f
3
 of the robust design. The horizontal axis shows the case number, while the dashed 
lines are the ?f
0
 limits.  
-5
-4
-3
-2
-1
0
1
2
3
4
5
16116
?f
1
Case #
-15
-10
-5
0
5
10
15
1 6 11 16
?f
2
Case #
-20
-15
-10
-5
0
5
10
15
20
1 6 11 16
?f
3
Case #
-5
-4
-3
-2
-1
0
1
2
3
4
5
1 6 11 16
?f
1
Case #
-15
-10
-5
0
5
10
15
16116
?f
2
Case #
-20
-15
-10
-5
0
5
10
15
20
1 6 11 16
?f
3
Case #
 
Figure 4.30: Sensitivity analysis of the nominal (left) and robust (right) designs. 
We see in Figure 4.30 that the ?f of the robust design is always within the limits for all 
three objectives. In contrast, the ?f of the nominal design violates the given limits for all 
three objectives. This observation verifies the robustness of the two designs as claimed. 
 125
 
As in the previous example, solving this problem using probabilistic approach 
(Monte Carlo simulation with 100,000 samples) gives us essentially the same results as 
the nominal optimization approach. As such, the optimality and robustness of these 
solutions are the same as the nominal Pareto solutions, and we do not re-perform the 
sensitivity analysis here for clarity.  
 
4.6. SUMMARY 
? A design is termed ?multi-objectively robust? if each of its objective value is 
insensitive to parameter variations.  
? For a set of designs, the overall robustness of the set is determined by the robustness 
of each of the designs in the set. This type of robustness is called point robustness.  
? In Multi Objective Robust Optimization (MORO), we are interested in the point 
robustness of the solutions. Those designs that are multi-objectively optimum (for 
nominal values of parameters) and also multi-objectively robust are called robust 
Pareto solutions.  
? A robust Pareto set is generally inferior to the nominal Pareto set. It is also common 
for the robust Pareto set and the nominal Pareto set to overlap, or for the robust Pareto 
set to be a subset of the nominal Pareto set. 
? The multi objective S
f
 of a design is the intersection of the S
f,i
 of each objective. 
When the number of objectives M = 1, the multi-objective S
f
 reduces to the single 
objective S
f
.  
? The generalized SR is the plot of the generalized S
f
 on the ?p-space. A point inside 
the SR satisfies [?f
i
]
2
 < [?f
i,0
]
2
 for all i = 1,?,M. A point outside satisfies [?f
i
]
2
 
 126
 
> [?f
i,0
]
2 
for at least one i. A point on the boundary satisfies [?f
i
]
2
 ? [?f
i,0
]
2
 with a 
strict equality for at least one i. 
? The generalized WCSR is a hyper-sphere inside the generalized SR that touches the 
SR boundary at the closest point from the origin. The radius of the WCSR can be 
calculated by solving a single-objective optimization problem with an equality 
constraint. If the magnitudes of ?p are different, this optimization problem needs to 
be normalized to account for scale importance.  
? To avoid difficulties in making trade-offs between the objectives, we use a 
constrained approach to MORO where we treat the robustness of a design as a 
constraint instead of an objective.  
? Using the constrained approach, our MORO method obtains the robust Pareto set by 
first eliminating all non-robust designs and then selecting the set of non-inferior 
designs from those designs that are not eliminated.  
 127
 
CHAPTER 5 
FEASIBILITY ROBUST OPTIMIZATION 
 
5.1. INTRODUCTION 
In the previous two chapters we presented a method to measure objective robustness 
of a design, and a scheme to use that measure to obtain a robust optimum design. In those 
chapters, we implicitly assumed that the design will always remain feasible when the 
variations occur. Obviously, this assumption may not be valid in general. Feasibility 
robustness of a design must be explicitly accounted for and enforced if we want the 
design to be always feasible even as the uncontrollable parameters vary.  
The purpose of this chapter is to develop a method to measure the feasibility 
robustness of a design, and develop an optimization scheme to use this measure to 
guarantee the feasibility of a design. Similar to Chapters 3 and 4, the robustness measure 
presented here is also based on the sensitivity region concept, but with a significant 
difference. Unlike objective robustness, which was ?two-sided?, feasibility robustness of 
a design with respect to constraints is ?one-sided?. This one-sided feasibility robustness 
implies that we are interested in limiting the constraint deviation along one direction 
only, either increase or decrease but not both. In our case, we limit the increase in the 
constraints because we use the notation g(x,p) ? 0 for constraints. We will point out this 
important difference in more detail in this chapter.  
For clarity and simplicity, in this chapter we will focus entirely on the feasibility 
robustness of a design, and we will not account for objective robustness. We will discuss 
the combined objective and feasibility robust optimization approach in the next chapter. 
 128
 
In the next few sections we develop the concept of one-side sensitivity region for single 
and multiple constraints, and then present an approach to use it in an optimization 
routine. At the end of the chapter we give a demonstration of the applications of our 
method to numerical and engineering examples.  
 
5.2. ONE-SIDED SENSITIVITY MEASURE 
We begin our discussions by presenting a method to measure the sensitivity 
(robustness) of a design in terms of constraint functions using the sensitivity region 
concept presented in previous chapters. We present our approach for a single constraint 
case first and then extend it to a more general case of multiple constraints. 
 
5.2.1. Single Constraint 
Let x
0
 be the design alternative whose sensitivity we want to measure, and let g(x
0
,p) 
be the constraint whose change in value is of interest. Constraint g(x
0
,p) depends on two 
factors, the design x
0
 itself and parameter p. Our goal is to get a measure of how g 
changes when p varies by some ?p.  
Suppose the value of g(x
0
,p) is allowed to decrease indefinitely, but is allowed to 
increase only by some non-negative amount ?g
0
 (i.e., a one-sided sensitivity). (We 
choose to limit the increase in g, instead of the decrease, to be consistent with our 
constraint notation ???.) For this ?g
0
 increase, there is a set of ?p?s such that: 
{ }
000
G
0g
g),(g),(g:R)( ?+??+??= ppppS xxx        (5.1) 
This set of ?p?s is called the ?feasibility sensitivity set? (S
g
) of design x
0
. Notice that the 
feasibility sensitivity set S
g
 does not have squared terms as in the objective sensitivity set 
 129
 
S
f
 that was presented in Chapters 3 and 4. This is because the feasibility sensitivity set is 
a one-sided sensitivity measure.  
Rearranging the inequality in Eq. (5.1), we obtain:  
g(x
0
,p+?p) ? g(x
0
,p) ? ?g
0
    (5.2) 
?g(?p) ? ?g
0
      (5.3) 
Eq. (5.3) shows an important property of S
g
: for all ?p?s in S
g
, the changes in g due to 
?p?s are always less than or equal to an allowable increase ?g
0
. This implies that S
g
 is an 
indicator of how much ?p?s design x
0
 can ?absorb? for it to remain within the allowable 
limit. As the number of elements in S
g
 increases, the design can allow more changes in p. 
So, like the objective S
f
, the larger S
g
, the more feasibly robust the design, and if we can 
make sure that ?p is always a member of S
g
, then we are guaranteed that ?g
0
 will always 
be satisfied.  
Plotting S
g
 in ?p-space, we obtain the Feasibility Sensitivity Region (FSR) of the 
design. The size of FSR corresponds to the size of S
g
, and therefore FSR size is also a 
measure of a design?s robustness: the larger it is, the more feasibly robust the design. 
Based on Eq. (5.3), a ?p point inside, outside, and on the boundary of FSR must satisfy 
?g(?p) < ?g
0
, ?g(?p) > ?g
0
, and ?g(?p) = ?g
0
, respectively.  
 
Example 5.1 
Let us demonstrate an application of the above concept with a simple constraint 
function. Suppose we have an inequality constraint: , where 
the parameters are p = [0, -0.5] and the allowable increase for the constraint is 
0)p(),g(
21
2
)p(10
1
?+?
x
xpx
 130
 
?g
0
 = 1.125. If the design of interest is x
0
 = [1.1, 3.0], determine the feasibility sensitivity 
set S
g
 for this design, and show its FSR in (?p
1
,?p
2
)-space.   
Solution 
The nominal value of the constraint is g(x
0
,p) = -0.125. Using ?g
0
 = 1.125, we 
obtain: { }0.2)5.0p()1.1(:)p,p()(
3
2
p10
210g
1
???+??=
?
xS . Plotting this inequality in the 
(?p
1
,?p
2
)-space, we obtain the FSR of the design as shown in Figure 5.1. As a quick 
verification, for (?p
1
,?p
2
) = (0,0), inside the FSR, the inequality is 0.875 < 2.0. For 
(?p
1
,?p
2
) = (1.0,1.0), outside the FSR, the inequality is 2.718 > 2.0. For 
(?p
1
,?p
2
) = (0,1.5), on the FSR boundary, the inequality is 2.0 = 2.0. 
 
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0.5
1
1.5
2
?p
1
?p
2
FSR
inside
outside
on boundary
 
Figure 5.1: Feasibility Sensitivity Region for the inequality constraint. 
Notice that the FSR in Figure 5.1 is unbounded and only has one boundary. This is in 
contrast to the objective SR where even when it is unbounded, it still has two boundaries 
(recall Chapters 3 and 4). Again, the reason for this difference is because FSR is a one-
 131
 
sided sensitivity measure. The FSR boundary corresponds to the ?g
0
 allowable increase, 
but since there is no limit for the decrease, there is no other boundary. ? 
 
As seen in Figure 5.1, like objective SR, FSR also has a drawback of being 
asymmetric. Because of this asymmetry, the directional sensitivity of a design becomes 
an important issue. To account for this directional sensitivity, we once again use the 
worst-case representation of FSR, hereafter called ?Feasibility Worst Case Sensitivity 
Region? (FWCSR), as a measure of a design?s feasibility robustness. Graphically, 
FWCSR of a design is a hyper-sphere inside the FSR of that design that touches the 
boundary at the closest point from the origin in ?p-space. The FWCSR is typically, but 
not necessarily, tangent to the FSR at the point of contact (e.g., when there is a cusp at the 
point of contact).  
 
Example 5.2 
Using the FSR obtained in Example 5.1, determine the FWCSR of the design 
x
0
 = [1.1, 3.0] for the constraint .  0)p(),g(
21
2
)p(10
1
?+?
x
xpx
Solution 
The FSR of the design is shown in Figure 5.1. The FWCSR of it is simply the 
smallest hyper-sphere inside it that touches it at the point closest to the origin as shown in 
Figure 5.2. In this example, the FWCSR is derived from the non-normalized (?p
1
,?p
2
)-
space, but it is valid because the scale of ?p
1
 and ?p
2
 is the same. We will discuss 
normalization in Section 5.3. ?
 
 132
 
?p
1
?p
2
FSR
FWCSR
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0.5
1
1.5
2
point of
contact
 
Figure 5.2: Feasibility WCSR for the inequality constraint. 
The point of contact between FWCSR and FSR is a point on FSR boundary that is 
closest from the origin. Since the point of contact satisfies ?g(?p) = ?g
0
, the FWCSR 
radius can be calculated by solving the following optimization problem: 
000
2
1
G
2
g
g),g(),g(:subject to
p)(Rminimize
?=??+
?
?
?
?
?
?
?=?
?
?
ppp
p
p
xx
i
i
    (5.4) 
Using the radius calculated from Eq. (5.4), we can then calculate the size of FWCSR. 
However, like in the objective robustness case, there is really no need to perform this 
calculation. The R
g
 value alone is sufficient for calculating the feasibility robustness of 
the design. 
 
Example 5.3 
Use Eq. (5.4) to calculate the R
g
 in Example 5.2. Compare this value to the FWCSR 
graph shown in Figure 5.2.  
 133
 
Solution 
The problem is simple enough to be solved analytically. For ease of differentiation, 
we use  instead of R
2
g
R
g
 as objective. Taking the first derivative of the Lagrangian with 
respect to ?p
1
 and ?p
2
, we obtain the following equations (? is the Lagrange multiplier):  
0.1)? (0.953)(1p2?
1
p10?
1
=+           (5.5) 
0.5)0p? (?3p2?
2
22
=+ -         (5.6) 
Combined with the equality constraint: ( , we have three 
linearly independent equations with three unknowns (?p
00.20.5)-p()1.1
3
2
)p(10
1
=??+
?
1
,?p
2
,?). Solving these 
equations, we obtain the point of contact to be (?p
1
,?p
2
) = (0.749, 0.147) for a FWCSR 
radius of R
g
 = 0.763 (the Lagrange multiplier is ? = -0.786). The Hessian of the 
Lagrangian at this (?p
1
,?p
2
) point is positive definite, so the sufficiency condition is 
satisfied.  
Graphically (see Figure 5.2), the point of contact is approximately at (?p
1
,?p
2
) 
= (0.75, 0.15) for a FWCSR radius of R
g
 = 0.765. This value is in agreement with the 
analytic value, thus confirming that solving Eq. (5.4) does indeed give us the FWCSR 
radius that we are looking for. ? 
 
5.2.2. Multiple Constraints 
We can easily extend the FWCSR concept described in the last section to measure 
robustness of a design when there are multiple constraints. Let g(x,p) = [g
1
,?,g
J
] be the 
constraint functions of interest, and ?g
0
 = [?g
1,0
,?, ?g
J,0
] ? 0 be the vector of acceptable 
increments. Then there is a set of ?p?s such that:  
 134
 
{ }J,...,1,g),(g),(g:R)(
0,00
G
0g
=??+??+??= j
jjj
ppppS xxx           (5.5) 
{ }J,...,1,g)(g:R)(
0,
G
0g
=???????=? j
jj
ppS x    (5.6) 
This set is the generalized feasibility sensitivity set S
g
. Notice how Eq. (5.6) collapses to 
Eq. (5.1) when J=1.  
Recall from Chapter 4 that the generalized S
f
 is really just the intersection of all S
f,i
?s. 
The same property also exists for the generalized S
g
. If we let the notation S
g,j
 be the set 
of ?p?s such that ?g
j
(?p) ? ?g
j,0
, then it is easily seen that the overall S
g
 is simply the 
intersection of all S
g,j
?s:  
S
g
 = S
g,1
 ? ? ? S
g,J
     (5.7) 
Consequently, the overall FSR of a design is then formed by the intersections of the FSR 
of each constraint, and the overall FWCSR is then defined for this overall FSR.  
Utilizing the fact that the overall FSR is an intersection of all constraints? FSRs, we 
can define the requirements for a ?p point to be inside, outside, or on the boundary of the 
overall FSR. Similar to the generalized SR, a point inside the overall FSR satisfies 
?g
j
(?p) < ?g
j,0
 for all j=1,?,J. For a point to be outside of the FSR, there must exist at 
least one j such that ?g
j
(?p) > ?g
j,0
. A point on the boundary of the FSR satisfies 
?g
j
(?p) ? ?g
j,0
 with a strict equality for at least one j.  
 
Example 5.4 
Going back to our inequality constraint from Examples 5.1-5.3. Suppose there is a 
second constraint 02)p()471.2(),(g
12
2
5.0p
12
2
?++??
?
?
?
?
?
? +
xxpx  with the same parameter 
 135
 
values p = [0,-0.5]. If the increment limit for this new constraint is ?g
2,0
 = 1.0, determine 
the overall FSR and FWCSR of the design x
0
 = [1.1, 3.0].  
Solution 
We already have the FSR of the first constraint, so to get the overall FSR we only 
need to calculate FSR of the second constraint and then find the intersection of the two 
FSRs. From previous examples, the feasibility sensitivity set of the first constraint is 
{ }0.2)5.0p()1.1(:)p,p()(
3
2
p10
210g,1
1
???+??=
?
xS . The nominal value of the second 
constraint is g
2
(x
0
,p) = 0, so the feasibility sensitivity set of the second constraint is 
{ }02)p()718.2(:)p,p()(
1
2/p
210g,2
2
?+?????
?
xS . Plotting these inequalities in the 
(?p
1
,?p
2
)-space, we obtain the overall FSR of the design as shown in Figure 5.3.  
?p
1
?p
2
FSR of g
2
FSR of g
1
Overall FSR
FWCSR
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0.5
1
1.5
2
 
Figure 5.3: Overall FSR and FWCSR for the two constraints.  
As a verification, the point (?p
1
,?p
2
) = (0,0) is inside the overall FSR, and it satisfies 
both S
g,1
 inequality (0.875 < 2.0) and S
g,2
 inequality (-1 < 0). The point (?p
1
,?p
2
) = (1,1) 
is outside the overall FSR. It satisfies S
g,2
 inequality (-1.35 < 0), but not S
g,1
 inequality 
 136
 
(2.718 > 2.0). The point (?p
1
,?p
2
) = (0,2) is also outside the overall FSR, but it does not 
satisfy either S
g,1
 inequality (4.375 > 2.0) or S
g,2
 inequality (0.718 > 0). The point 
(?p
1
,?p
2
) = (-1,0) is on the boundary of the overall FSR. It satisfies S
g,1
 inequality 
(0.26 < 2.0), and satisfies S
g,2
 inequality with a strict equality (0 = 0).  
The overall FWCSR of the design is also shown in Figure 5.3. We see in this figure 
that the overall FWCSR is the same as the FWCSR of the first constraint. This is because 
the worst-case scenario for this particular design is governed by the first constraint. ? 
 
Using the requirement for a point to be on the overall FSR boundary, the FWCSR 
radius of a multiple-constraint design can be calculated by solving the optimization 
problem shown below (notice again how Eq. (5.8) collapses into Eq. (5.4) when J=1): 
,000
,000
2
1
G
2
g
g),(g),(g:
J,...,1
g),(g),(g:subject to
p)(Rminimize
jjj
jjj
i
i
j
j
?=??+?
=
????+
?
?
?
?
?
?
?=?
?
?
ppp
ppp
p
p
xx
xx
        (5.8) 
Notice in Eq. (5.8) that at least one of the J inequality constraints must be active. Based 
on this information, we can simplify these constraints.  
If we rearrange each inequality constraint, we obtain the following: 
J,...,1
01
g
),(g),(g
,0
00
=
??
?
??+
j
j
jj
ppp xx
    (5.9) 
For a feasible ?p point, the maximum of each of the modified constraint is 0, and for a 
constraint to be active, it has to be at its maximum. Therefore, for a feasible ?p point 
with at least one active constraint, Eq. (5.10) below must be satisfied:  
 137
 
01
g
),(g),(g
max
,0
00
J,...,1
=?
?
?
?
?
?
?
?
?
?
??+
=
j
jj
j
ppp xx
   (5.10) 
Substituting Eq. (5.10) into Eq. (5.8), the optimization problem to calculate the FWCSR 
radius of a multiple-constraint design becomes as shown in Eq. (5.11). (Notice again how 
Eq. (5.11) simplifies to Eq. (5.4) when J=1.) 
),(g),(g)(g:where
01
g
)(g
max:subject to
p)(Rminimize
00
,0
J,...,1
2
1
G
1
2
g
pppp
p
p
p
xx
jjj
j
j
j
i
i
??+=??
=?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?=?
=
=
?
?
  (5.11) 
The constraint in Eq. (5.11) is identical to the constraint used in calculating the WCSR 
radius of the objective SR (Eq. (4.5)) except that this time we do not take the absolute 
value of the nominator, the ?g
j
(?p) term. This is again because this constraint is for a 
one-sided sensitivity measure. In the objective WCSR, the absolute value is necessary to 
guarantee that the ?f value is bounded in both increasing and decreasing directions. In the 
feasibility WCSR, we only need to guarantee that the ?g value is bounded in the 
increasing direction, so there is no need for the absolute value.  
Alternatively, we can also justify the simplified constraint in Eq. (5.11) graphically. 
In the g-space, the quantity ?g(?p) is represented as a vector from the point g(x
0
,p) to the 
point g(x
0
,p+?p). Based on the definition of S
g
, if a point ?p is on the FSR boundary, 
then the vector ?g(?p) must touch the boundary of ?g
0
, which implies that at least one of 
its component vectors must touch this boundary. Coupled with the fact that ?g
0
 ? 0, this 
 138
 
implies that 1
g
)(g
max
0,
J,...,1
=
?
?
?
?
?
?
?
?
?
??
=
j
j
j
p
, which is the constraint in Eq. (5.11). Figure 5.4 shows 
an illustration of this derivation.  
0
g
1
g
2
?g
0
boundary
?g
2,0
?g
1,0
g(x
0
,p)
?g
2
?g
1
1
g
g
g
g
,
g
g
max
0,1
1
0,2
2
0,1
1
=
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
 
Figure 5.4: Graphical derivation of the simplified constraint. 
Since the single-constraint FSR and FWCSR is just a simplified version of the 
multiple-constraint case, from hereon our discussion will focus only on the multiple-
constraint case.  
 
5.3. ROBUST OPTIMIZATION  
We will now show how to use the sensitivity measure presented earlier in a 
feasibility robust optimization scheme.  
 
5.3.1. Determination of Increment Limit 
In developing the FWCSR concept, we have assumed there exists a quantity ?g
0
 that 
defines how much g(x
0
,p) is allowed to increase, but we have not yet discussed how this 
 139
 
quantity is determined. Unlike the objective ?f
0
 limit, the ?g
0
 limit is not determined by 
the designer. Rather it is determined by the position of the design relative to the 
constraints boundary.  
Suppose a design x
0
 is feasible, i.e., g(x
0
,p) ? 0. Then the value |g(x
0
,p)| shows how 
much g(x
0
,p) must increase for it to become active. In other words, the quantity |g(x
0
,p)| 
represents the maximum allowable increment in g(x
0
,p) for the design to remain feasible, 
i.e., ?g
0
 = |g(x
0
,p)|. If x
0
 is infeasible, then there is no need to calculate the feasibility 
robustness of this design (feasibility cannot be guaranteed). Here, we must use the 
absolute value because we use the convention g(x,p) ? 0, i.e., a feasible x has a negative 
g value. If we use the convention g(x,p) ? 0 for a feasible design, then the absolute value 
is not necessary (but then we have to limit the decrease in g). 
Substituting ?g
0
 = |g(x
0
,p)| into the constraint in Eq. (5.11), we obtain: 
01
),(g
),(g),(g
max
0
00
J,...,1
=?
?
?
?
?
?
?
?
?
??+
=
p
ppp
x
xx
j
jj
j
          (5.12) 
Since for a feasible design, we have g
j
(x
0
,p) ? 0 and |g
j
(x
0
,p)| ? 0 for all j: 
0
),(g
),(g
max
0
0
J,...,1
=
?
?
?
?
?
?
?
?
?+
=
p
pp
x
x
j
j
j
    (5.13) 
Let us assume for a moment that |g(x
0
,p)| > 0, i.e., when x = x
0
 no constraint is active. 
Then Eq. (5.13) simplifies into: 
0)],(g[max
0
J,...,1
=?+
=
ppx
j
j
    (5.14) 
Recall that the constraint in Eq. (5.11) is the requirement for a point ?p to be on the FSR 
boundary. Eq. (5.14) further simplifies this requirement. Eq. (5.14) states that if a point 
 140
 
?p causes any g
j
(x
0
,p+?p) to be active, then it is on the FSR boundary. This simplified 
requirement is intuitively obvious. Because ?g
j,0
 = |g
j
(x
0
,p)|, if ?g
j
(?p) = ?g
j,0
, then 
g
j
(x
0
,p+?p) = 0. This FSR boundary condition is still valid even when |g(x
0
,p)| = 0. So, 
Eq. (5.14) is valid for |g(x
0
,p)| ? 0. 
Substituting Eq. (5.14) into Eq. (5.11), the optimization problem to calculate the 
FWCSR radius of a design becomes: 
[]0),(gmax:subject to
p)(Rminimize
0
J,...,1
2
1
G
1
2
g
=?+
?
?
?
?
?
?
?=?
=
=
?
?
pp
p
p
x
j
j
i
i
            (5.15) 
 
5.3.2. Normalization and Feasibility Robustness Index 
To account for the scale difference among the parameters, Eq. (5.15) needs to be 
normalized. Even if the scale of the parameters is the same, we still recommend 
normalizing Eq. (5.15) to help with convergence. As in objective robustness, we use the 
known ranges of variations ?p
0
 to normalize Eq. (5.15). Since 
0,
p
p
p
i
i
i
?
?
=?  for all 
i=1,?,G, then 
0,
ppp
iii
???=? ; or in vector notation 
0
ppp ???=? . Substituting this 
equality into Eq. (5.15), the normalized optimization problem to calculate the FWCSR 
radius becomes as follows (for simplicity, we have forgone using the 
g
R  notation).  
[]0),(gmax:subject to
p)(Rminimize
00
J,...,1
2
1
G
1
2
g
=???+
?
?
?
?
?
?
?=?
=
=
?
?
ppp
p
p
x
j
j
i
i
   (5.16) 
 141
 
If we define 
g
21
g
R)G(
?
=?  to be the feasibility robustness index of the design, then 
the value ?
g
 ? 1 implies that the design is always feasible even when the variations occur, 
i.e., it is feasibly robust. In contrast, the value ?
g
 < 1 implies that there will be instances 
during the changes in parameters when the design will become infeasible. Adding the 
robustness constraint ?
g
 ? 1 to an optimization problem guarantees that the optimum 
design is feasibly robust. Eq. (5.17) shows the overall feasibility robust optimization 
problem. 
0
?
)(?
1
J,...,10),(g:subject to
])(,...,)([)(minimize
g,0
g
M1
??
=?
=
x
x
xxxf
x
j
ff
j
p     (5.17) 
Here ?
g
(x) is the feasibility robustness index calculated from Eq. (5.16), and ?
g,0
 is the 
desired level of robustness as determined by the designer.  
There are a few important things needs to be pointed out about Eq. (5.17). First, the 
robustness constraint guarantees feasibility robustness of an optimum design with respect 
to inequality constraints only. Equality constraints are hard constraints in the sense that 
unless ?p variations are such that h(x,p+?p) = 0, there is no way to guarantee these 
constraints will always be satisfied. Second, although we have presented our feasibility 
robust optimization for a multi-objective optimization problem, our approach is also 
applicable to single objective problems. The feasibility robustness constraint in Eq. (5.17) 
does not depend on the number of objectives. Third, implementation-wise, solving 
Eq. (5.17) as it is will give us a feasibly robust optimum design(s). However, we can 
improve the efficiency of the approach, i.e., reduce the number of evaluations, simply by 
not calculating the robustness constraint when the other constraints (both equalities and 
 142
 
inequalities) are not satisfied. A design can be guaranteed to be always feasible (when 
parameter variations occur) only if it is feasible in the first place (i.e., as a nominal 
design). If a nominal design is infeasible, then it does not even meet our design 
requirements, so there is no need to calculate its robustness.  
 
5.4. COMPARISON STUDY 
As a demonstration of our feasibility robust optimization method, we applied it to 
one numerical and three engineering examples. In the numerical example, we show the 
applicability of our method to problems whose parameter variations are large. In the 
explosive actuator and control valve linkage examples, we provide comparison between 
our method and a probabilistic method. In the Belleville spring example, we compare our 
method to the min-max method developed by Hirokawa and Fujita (2002). 
 
5.4.1. Numerical Example 
For our first example, we solve a two-dimensional numerical example presented in 
Hirokawa and Fujita (2002). The problem is a single objective optimization problem with 
two inequality constraints. The mathematical formulation of the problem is as follows: 
03)41.01.0log()(g
075.2sin3)(g:subject to
123252210)4sin()(minimize
2
43
212
2111
2
11
2
2
2211
2
11
3
1
],[
21
21
??+++??
??+?+?
+++++++=
?+?
=
xexx
xxxxx
xxxxxxxxf
xx
xx
x
x
x
x
 (5.18) 
There are no restrictions on the design variables x = [x
1
,x
2
], but they observe some 
variations ?x
0
 = [0.4,0.4]. Our goal is to obtain a feasibly robust optimum design that will 
always remain feasible when the variations in x occur.  
 143
 
Figure 5.5 shows a contour graph for the objective and constraint functions of the 
problem. The two solid lines are the constraint boundaries with the feasible directions 
indicated by the arrows. The dashed ellipses are the contours of the objective function, 
and the arrow on the objective contours indicates the decreasing direction. From this 
graph we can easily see that the nominal optimum of this problem occurs at the point 
where the g
2
 constraint boundary is tangent to the objective contour (indicated in the 
graph).  
-4 -3 -2 -1 0 1 2
-2
-1 .5
-1
-0 .5
0
0.5
1
1.5
2
feasible g
2
feasible g
1
decreasing f
nominal optimum
x
1
x
2
 
Figure 5.5: Objective and constraint contours of the numerical example. 
Solving the problem numerically using MATLAB?s fmincon function, the nominal 
optimum is found to be at x* = [-1.825,0.741] and the optimum objective is f* = -3.287. 
The constraint value of the nominal optimum is g* = [-5.919,0]. This solution is obtained 
in 8 iterations. We see that numerically, at the nominal optimum the constraint g
2
 is 
active just as observed in Figure 5.5.  
 144
 
Since the constraint g
2
 is active, obviously the nominal optimum is not feasibly 
robust. When the variables [x
1
,x
2
] vary, the g
2
 constraint can be violated. Adding the 
feasibility robustness constraint 0
?
)(?
g,0
g
??
x
1  to Eq. (5.18) with ?
g,0
 = 1.0, and then 
solving it using fmincon, we obtain the robust optimum design to be x
R
* = [-1.394,0.272], 
f
R
* = -1.552, and g
R
* = [-6.089,-1.374]. This robust optimum is obtained in 7 iterations. 
The inner optimization problem used to calculate ?
g
 is also solved using fmincon and 
converges on average in 6.28 iterations.  
For comparison, let?s also find the feasibly robust optimum design using the 
conventional robustness methods in the literature. More specifically, let us solve the 
problem for robust optimum using: (1) the worst-case gradient method where the 
constraint  is replaced by 0)(g ?x
j
0
g
)(
N
1
??
?
?
+
?
=i
i
i
j
j
x
x
xg , and (2) the moment 
matching method where the analytic constraint is replaced by a probabilistic constraint 
, where the standard deviation of g0
g
?
j
?)?(g +k
j x
j
 is estimated using a Taylor series 
expansion: 
2
2
?
g
i
x
i
j
x
?
?
?
?
?
N
1
2
g
?
j
i
?
=
?
?
?
= , where ?  is the variance of x
2
i
x
i
. The factor k in the 
probabilistic constraint is specified to be equal to 3.0. According to Parkinson et al. 
(1993), a factor k=3.0 will provide more than 0.99 probability of constraint satisfaction. 
For the probabilistic constraint, the ?x variation is assumed to be normally distributed 
with a [?
x
,?
x
] = [0, 0.4/3].  
 145
 
The robust optimum designs obtained using the two methods are shown in Table 5.1. 
For comparison, we have also re-listed the nominal optimum and the robust optimum 
obtained by our method. 
Table 5.1: Optimum designs of the numerical example. 
Nominal
Robust 
(FWCSR)
Robust 
(Gradient)
Robust 
(Moment)
x
1
-1.825 -1.394 -1.557 -1.599
x
2
0.741 0.272 0.477 0.517
f -3.287 -1.552 -2.266 -2.419
g
1
-5.919 -6.089 -6.077 -6.071
g
2
0 -1.374 -0.98 -0.874
 
If we consider Table 5.1, we see that the nominal optimum has the smallest f value, while 
our robust optimum has the largest. The other two robust optima have roughly the same f 
value, and they are somewhere between the nominal and our robust optimum. However, 
if we look at the value of the g
2
 constraint, we also see that our robust optimum provides 
that largest amount of ?cushion? for the constraint to vary. The other two robust optima 
provide some amount of variation cushion, while the nominal optimum provides no room 
for variation at all (g
2
 = 0).  
At a first glance, it might seem that our robust optimum is too conservative and is an 
overly non-optimal design. However, let?s verify the robustness of each of the optimum 
designs in Table 5.1. The variations occur in the design variables, so we can use the 
feasible region graph in Figure 5.5 to see what happens when x varies. For each optimum 
design, we add a rectangle of ?0.4 around it to indicate the ?x variation. An optimum 
design is truly feasibly robust only if the entire rectangle is inside the feasible region. 
Figure 5.6 shows the plots of the robustness of each optimum design.  
 146
 
-4 -3 -2 -1 0 1 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
?x
2,0
?x
1,0
nominal optimum 
f = -3.287
x
1
x
2
-4 -3 -2 -1 0 1 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
robust optimum 
(FWCSR)
f = -1.552
x
1
x
2
?x
2,0
?x
1,0
 
-4 -3 -2 -1 0 1 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
?x
2,0
?x
1,0
robust optimum 
(Gradient)
f = -2.266
x
1
x
2
-4 -3 -2 -1 0 1 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
?x
2,0
?x
1,0
robust optimum 
(Moment matching)
f = -2.419
x
1
x
2
 
Figure 5.6: Sensitivity analysis of the optimum designs. 
We see in Figure 5.6 that of the four optimum designs, only our robust optimum can 
completely absorb the ?x
0
 variation, i.e., the dashed rectangle is fully inside the feasible 
region. For the other three optima, there are ?x variations that cause the design to fall into 
the infeasible region. This shows that it is not that our robust optimum is too 
conservative; rather it is the other optima that are not robust enough. This also shows that 
the gradient and moment matching methods do not guarantee a feasibly robust optimum.  
One reason the gradient and moment matching method do not result in a feasibly 
robust optimum in this problem is because the ?x
0
 variation is too large for the Taylor 
series expansion to remain valid. This shows the advantage of our robust optimization 
method: it is applicable to problems where the variations are large. Let us verify this 
claim. We re-solve the same problem using four methods: (1) our FWCSR method with 
 147
 
?
g,0
 = 1.0, (2) our FWCSR method with ?
g,0
 = 0.8, (3) gradient method, and (4) moment 
matching method. We solve the problem using these methods for 6 increasing amount of 
?x
0
 variations (the same for both ?x
1
 and ?x
2
): [0.01, 0.1, 0.2, 0.4, 0.8, 1.0]. For each 
optimum obtained, we calculate the probability of constraint satisfaction (P
s
) of the 
design by performing 1000 runs of Monte Carlo simulation assuming a normal pdf of the 
variations. The plot of the ?x
0
 vs. P
s
 for each method is shown in Figure 5.7 and 5.8.  
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
0 0.2 0.4 0.6 0.8 1 1.2
FWCSR ?
0
= 1.0
?x
0
P
s
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
0 0.2 0.4 0.6 0.8 1 1.2
P
s
?x
0
FWCSR ?
0
= 0.8
 
Figure 5.7: Plot of ?x
0
 vs. P
s
 of the FWCSR optima.  
 
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
0 0.2 0.4 0.6 0.8 1 1.2
Gradient method
?x
0
P
s
 
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
0 0.2 0.4 0.6 0.8 1 1.2
Moment matching method
?x
0
P
s
 
Figure 5.8: Plot of ?x
0
 vs. P
s
 of the gradient and moment matching optima. 
 148
 
We see in Figure 5.7 that for the FWCSR method with ?
g,0
 = 1.0, P
s
 is always 1.0 
regardless of how large ?x
0
 is. This shows that our method will always obtain a feasibly 
robust optimum even when the variations are large. Similarly, for the FWCSR method 
with ?
g,0
 = 0.8, P
s
 is relatively the same (between 0.9995 and 1.0) even as ?x
0
 increases. 
This again indicates that our method will still obtain a feasibly robust optimum even as 
the variations grow beyond the linear range (range in which Taylor series expansion is 
valid).  
In contrast, as seen in Figure 5.8, P
s
 of the gradient method is very high for small 
value of ?x
0
, but decreases as ?x
0
 increases. This indicates that this method fails to 
obtain a robust optimum as the variations grow large. Even worse behavior is observed 
for the moment matching method. When is ?x
0
 small, P
s
 is high, but it decreases 
dramatically (to as low as 0.954) as ?x
0
 increases. This also indicates that this method is 
good only for problems with small variations.  
If we plot the optimum f* value obtained by each method vs. the increasing ?x
0
, we 
will obtain a graph as shown in Figure 5.9. We see once again in this figure that for small 
value of ?x
0
, the optimum f* of all methods are relatively the same, but they diverge as 
?x
0 
increases. The FWSR method with ?
g,0
 = 1.0 is most conservative, but as shown in 
Figure 5.7 this method guarantees P
s
 = 1.0. FWSR method with ?
g,0
 = 0.8 guarantees 
P
s
 of at least 0.999, and its f* value is better than when ?
g,0
 = 1.0. The f* value of the 
gradient and moment matching methods are better than the FWCSR method, but this is 
misleading since the P
s
 of these two methods drops significantly as ?x
0
 increases.  
 149
 
-4
-3
-2
-1
0
1
2
0 0.2 0.4 0.6 0.8 1
FWCSR (1.0)
FWCSR (0.8)
Gradient
Moment
Nominal
?x
0
f *
 
Figure 5.9: Plot of ?x
0
 vs. f* of the four robust optimization methods. 
 
5.4.2. Design of an Explosive Actuated Cylinder 
For our second example, we solve the optimization problem of designing an 
explosive actuated cylinder as formulated by Papalambros and Wilde (1980). In this 
problem, we are interested in optimizing the actuated cylinder shown in Figure 5.10 for 
minimum total length by controlling 5 design variables: the unswept cylinder length (x
1
 - 
inch), the working stroke of the piston (x
2
 ? inch), the outside diameter of the cylinder (x
3
 
? inch), the initial pressure of the combustion (x
4
 ? ksi), and the piston diameter (x
5
 ? 
inch). All design variables are restricted to be positive.  
x
1
x
2
x
5
x
3
piston
cylinder body
fixed chamber, v
c
unswept
cylinder 
length
 
Figure 5.10: An explosive actuated cylinder. 
 150
 
In addition, there are constraints on the kinetic energy produced, the maximum piston 
force, and the stress on the cylinder wall, as well as geometric constraints. The 
mathematical formulation of the problem is as follows: 
356
max215
max34
3
max
2
54
3
2
min
?1
1
?1
2
?
14
3
1
21
001.0)(g
L)(g
D)(g
)3/()(g
F
4
?10
)(g
W)(
?1
)10(
)(g:subject to
)(minimize
xx
xx
x
SS
xx
vv
vx
xxf
yyt
?+?
?+?
??
??
?
?
?
?
?
?
?
?
?
?
??
?
?
+=
??
x
x
x
x
x
x
x
x
     (5.19) 
2
511
)4/?(:where xxvv
c
+=  ;       (5.20) 
2
5212
)4/?( xxvv +=
2/12
221
2
1
)( ???? +?=
yt
S        (5.21) 
42
2
5
2
3
2
5
2
34
1
;
)(
)(
x
xx
xxx
?=
?
+
= ??       (5.22) 
The constants used in the problem are as follows: 
D
max
 = maximum allowable cylinder outside diameter (1.0 in) 
F
max
 = maximum piston force (700 lb) 
L
max
 = maximum cylinder total length (2.0 in) 
S
y
 = cylinder material yield stress (125 ksi) 
v
c
 = fixed chamber volume (0.084 in
3
) 
W
min
 = minimum kinetic energy for satisfactory performance (600 lb-in) 
? = specific heat ratio (1.2) 
Solving Eq. (5.19) using MATLAB?s fmincon, we obtain the nominal optimum to be 
x* = [0, 1.042, 1.0, 23.12, 0.196] and f* = 1.042. For this nominal optimum, the 
 151
 
constraints g
1
, g
2
, g
3
, and g
4
 are active. This optimum value is very close to that reported 
in Papalambros and Wilde (1980), f* = 1.036, obtained using monotonicity analysis 
(actually, their optimum point is slightly infeasible. When f* = 1.036, [g
1
, g
2
, g
3
] = 
[0.0026, 0.0017, 0.0049] > 0).  
There are variations in three of the design variables, and they are [?x
3,0
, ?x
4,0
, ?x
5,0
] 
= [0.01, 1.0 , 0.01]. We need to guarantee that the optimum design is feasibly robust with 
respect to these variations. Adding the robustness constraint 0
?
)(?
g,0
g
??
x
1  to Eq. (5.19) 
with ?
g,0
 = 1.0, and then solving it using fmincon, we obtain the robust optimum design to 
be x
R
* = [0.0097, 1.669, 0.823, 17.8, 0.201] and f
R
* = 1.679. For this robust optimum, all 
constraints are inactive. The inactivity of the constraints is one indication of the 
feasibility robustness of this optimum. If the constraints are active, then there is no 
?cushion? for them to vary as [x
3
, x
4
, x
5
] vary. By moving slightly inside the feasible 
region, our robust optimum design provides some safety margin to absorb the ?x. The 
inner optimization problem used to calculate the robustness of a design is also solved 
using fmincon, and on average converges in 15 iterations.  
For comparison, we also solved the problem using a probabilistic method (Du and 
Chen, 2000). We replaced all of the constraints in the problem with a probabilistic one 
and constrained it to be greater than some predetermined value (= 0.99 in this case), 
. The probability is calculated using 100,000 runs of the Monte Carlo 
method assuming two different pdf models for the variations: uniform and normal. For 
the uniform model, the pdf is assumed to be uniformly distributed between [-?x
99.0])([P ?? 0g x
i,0
, ?x
i,0
]. 
For the normal model, the pdf is assumed to be normally distributed with a mean and 
 152
 
standard deviation value of [0, ?x
i,0
/3]. Table 5.2 shows the robust optimum designs 
obtained using the probabilistic method. For ease of comparison, we have again listed the 
values of the nominal optimum and our robust optimum with ?
g,0
 = 1.0, and ?
g,0
 = 0.8. 
Table 5.3 shows the constraint values of these optimum designs. 
Table 5.2: Optimum designs of the explosive actuated cylinder. 
Nominal
Robust 
(Normal)
Robust 
(Uniform)
Robust      
(?
g
 = 0.8)
Robust     
(?
g
 = 1.0)
x
1
0 0.025 0.003 0 0.010
x
2
1.042 1.283 1.501 1.580 1.669
x
3
1.000 0.686 0.534 0.507 0.823
x
4
23.123 20.381 16.882 15.963 17.805
x
5
0.196 0.201 0.214 0.218 0.202
f 1.042 1.307 1.505 1.580 1.679
 
Table 5.3: Constraints of the optimum designs. 
Nominal
Robust 
(Normal)
Robust 
(Uniform)
Robust      
(?
g
 = 0.8)
Robust     
(?
g
 = 1.0)
g
1
0.0 -0.087 -0.114 -0.126 -0.167
g
2
0.0 -0.075 -0.136 -0.150 -0.188
g
3
0.0 -0.057 -0.147 -0.168 -0.199
g
4
0.0 -0.314 -0.466 -0.493 -0.177
g
5
-0.479 -0.346 -0.248 -0.210 -0.160
g
6
-0.803 -0.705 -0.598 -0.569 -0.754
 
We see in Table 5.2 that the nominal optimum has the lowest f value, while our 
robust optimum with ?
g,0
 = 1.0 has the highest. The f value of the probabilistic optima 
and that obtained by our method using ?
g,0
 = 0.8 are in between these two values. 
However, we observe in Table 5.3 that the constraints of the nominal optimum are closest 
to the feasible region boundary (i.e., its constraint values are largest on average) while 
those of our robust ?
g,0
 = 1.0 optimum are the furthest. The constraints of the other 
 153
 
optima are somewhere in between. These constraint values indicate that our robust 
?
g,0
 = 1.0 optimum can absorb the most ?x variation, while the nominal optimum can 
absorb the least. Let us numerically verify the robustness of each of the optimum designs. 
We perturbed the [x
3
, x
4
, x
5
] of each optima 20 times by adding some ?x value 
randomly sampled from the ?x
0
 range, and then calculate the new g(x) value of the 
design. A design is feasibly robust if all the new constraint values are still feasible, i.e., 
g(x) ? 0. We performed the analysis only for g
1
, g
2
, g
3
, and g
4
 constraints because 
constraint g
5
 is independent of [x
3
, x
4
, x
5
], and it is easily observed from Table 5.2 that 
constraint g
6
 will always be satisfied for all optima.  
Figures 5.11 ? 5.13 show the graphs of the g
1
, g , g
2 3
, and g
4
 constraints of each 
optimum under perturbation. For clarity, but without loss of information, the graphs only 
show the max[g
1
, g
2
, g
3
, g
4
] of each perturbation. In these graphs, a design is feasibly 
robust if all the bars are below the horizontal axis, i.e., max[g
j
(x)] ? 0.  
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
16116
 
Figure 5.11: Sensitivity analysis of the nominal optimum. 
 154
 
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
1 6 11 16
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
16116
Robust (normal)
Robust (uniform)
 
Figure 5.12: Sensitivity analysis of the probabilistic optima. 
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
1 6 11 16
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
1 6 11 16
Robust (?
g
= 0.8) Robust (?
g
= 1.0)
 
Figure 5.13: Sensitivity analysis of the robust optima.  
We see in Figures 5.11 and 5.12 that the nominal optimum becomes infeasible in all 20 
cases, while the probabilistic optima become infeasible in five cases for the robust 
(normal) optimum and in two cases for the robust (uniform) optimum. Our robust 
?
g,0
 = 0.8 optimum becomes infeasible in one case, while the robust ?
g,0
 = 1.0 optimum is 
always feasible. These observations verify that our robust ?
g,0
 = 1.0 optimum is the only 
 155
 
design that is feasibly robust. The robust (uniform) and robust ?
g,0
 = 0.8 optima are close, 
but not quite feasibly robust.  
To further verify the robustness of the optima, we calculate their probability of 
constraint satisfaction using 100,000 runs of Monte Carlo simulations. The simulations 
are performed using the uniform and normal ?x pdf models as before. The P
s
 value of 
each design is shown in Table 5.4.  
Table 5.4: Probability of constraint satisfaction of the optima. 
Uniform pdf Normal pdf
Nominal
00
Robust (Normal)
0.815 0.982
Robust (Uniform)
0.983 0.9997
Robust (?
g
 = 0.8)
0.996 0.9999
Robust (?
g
 = 1.0)
1.0 1.0
P
s
 
We see in Table 5.4 that the nominal optimum has a P
s
 = 0, which implies that it has no 
robustness towards variations at all. In contrast, the P
s
 of our robust ?
g,0
 = 1.0 optimum is 
1.0 confirming that this optimum design is indeed feasibly robust. The P
s
?s of the other 
optima confirm the robustness information shown in Figures 5.12 and 5.13 as well. If we 
compare Table 5.4 with Table 5.2, we also see that P
s
 increases as f increases (i.e., the 
performance vs. robustness trade-off).  
One last item of interest is the computational efficiency of each method. For the 
nominal optimum, we did not perform any calculation of a design?s robustness at all. For 
our FWCSR method, we solved an inner optimization using fmincon to calculate a 
design?s robustness. On average the optimization converged in about 15 iterations. 
Because fmincon performed some function calls to estimate gradients, the actual number 
 156
 
of function calls used by our method is approximately close to 50. The probabilistic 
method calculated the design?s robustness by performing 100,000 runs of Monte Carlo 
simulations. So we see that our robust optimization method is more efficient in terms of 
number of function calls.  
 
5.4.3. Design of a Belleville Spring 
Our third example is the problem of optimizing a Belleville spring originally 
formulated by Siddall (1982), and later modified by Hirokawa and Fujita (2002) as a 
robust optimization problem. The problem presented in this chapter is the one formulated 
by Hirokawa and Fujita (2002).  
P
d
e
d
i
h
l
t
 
Figure 5.14: A Belleville spring. 
In this problem, we want to optimize the Belleville spring (made out of steel) shown 
in Figure 5.14 for maximum rated load P. The design variables of the problem are: the 
external diameter (d
e
), the internal diameter (d
i
), the thickness (t), and the free height (h). 
The variables are all continuous, and their units are meter. The optimization is 
constrained by two design constraints, allowable stress and maximum mass, and five 
geometric constraints. The mathematical formulation of the problem is as follows: 
 157
 
0)(3.0)(g
025.1)(g
0)(g
0)()(g
0)(g
0)(g
0??)(g:subject to
P)(maximize
7
6
max5
4
min3
max2
max1
][
????
???
???
??+?
???
???
???
=
=
ie
ei
e
aw
,t,h,dd
ddh
dd
dd
lth
hh
mm
f
ie
x
x
x
x
x
x
x
x
x
   (5.23) 
Here P is the rated load (N), ?
max
 is the maximal stress (Pa), and m is the spring mass 
(kg). The quantities P, ?
max
, and m are calculated using the following equations.  
()
?
?
?
?
?
?
+?
?
?
?
?
?
?
?
?
=
3
max
max
2
2
2
max
)?(
2
?
?)?1(
?
P tthh
E
e
d
  (5.24) 
()
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
= th
E
e
d
?
2
?
?
?)?1(
?
?
max
2
2
2
max
max
   (5.25) 
tddm
ie
)(?
4
?
22
?=       (5.26) 
2
1
ln?
6
?:where
?
?
?
?
?
?
?
=
K
K
K
 ; 
?
?
?
?
?
?
?
?
= 1
ln
1
ln?
6
?
K
K
K
      (5.27) 
?
?
?
?
?
?
?
=
2
1
ln?
6
?
K
K
 ; 
i
e
d
d
K =     (5.28) 
The constants for this problem are: 
d
max
 = maximum allowable diameter (0.3 m) 
E = Young?s modulus (210 GPa) 
h
min
 = minimum height (0.005 m) 
l = maximum allowable total height including t (0.02 m) 
m
max
 = maximum spring mass (2.0 kg) 
 158
 
?
max
 = maximum allowable deflection (= h) 
? = Poisson?s ratio (0.3) 
? = mass density (7850 kg/m
3
) 
?
aw
 = allowable stress (1200 MPa) 
The nominal optimum of the problem obtained using fmincon is [d
e
, d
i
, t, h]* = [0.3, 
0.211, 7.273, 5.0] and the objective value is (in kN) f* = 42.106. Here the values of the 
variables [t, h] are in mm. The constraints of this optimum are g* = [0, 0, 0, -0.386, 0, 
-0.112, -4.214]. This nominal optimum f value is very close to that obtained by Hirokawa 
and Fuijta (2002), f* = 41.9.  
Due to manufacturing errors, all design variables [d
e
, d
i
, t, h] are subject to 
variations. In addition, two of the material properties [?
aw
, E] are also subject to 
variations. So there are six uncontrollable parameters in this problem p = [d
e
, d
i
, t, h, 
?
aw
, E]. The variation ranges of the parameters are ?p
0
 = [8.67 x 10
-5
, 7.67 x 10
-5
, 
3.33 x 10
-5
, 3.33 x 10
-5
, 4.0 x 10
5
, 6.67 x 10
7
] (as given in Hirokawa and Fujita, 2002). 
The ?p
0
 values of the variables [d
e
, d
i
, t, h] are in m. The ?p
0
 values of the material 
properties [?
aw
, E] are in Pa.  
To obtain a robust optimum, we added the robustness constraint to Eq. (5.23). The 
addition of this constraint, however, causes the fmincon algorithm to fail to obtain a 
solution. One reason of this failure is because the feasible region of the problem is 
already very small as it is. Adding the robustness constraint reduces the feasible region 
even further, making it very hard for an SQP algorithm to converge. Because fmincon 
failed to obtain a solution, we used a Genetic Algorithm (GA) (Goldberg, 1989) instead. 
 159
 
Due to implementation reasons, we used the same GA to solve the inner optimization 
problem. Solving the inner optimization using GA takes ~300 function calls.  
The robust optimum obtained (?
g,0
 = 1.0) is shown in Table 5.5. For comparison 
purposes, we have also shown the robust optimum of the problem for ?
g,0
 = 0.8, 0.6 and 
0.3. Table 5.5 also shows the robust optimum reported by Hirokawa and Fujita (2002). 
Their robust optimum is obtained by a min-max strategy where they use the maximum of 
the constraints within a so-called ?variation pattern? of the parameters as the constraints 
of the problem. The constraint values of these optima are shown in Table 5.6.  
Table 5.5: Optimum designs of the Belleville spring. 
Nominal Min-Max
Robust 
(0.3)
Robust 
(0.6)
Robust 
(0.8)
Robust 
(1.0)
d
e 
(m) 0 0.300 0.299 0.285 0.298 0.292
d
i 
(m) 0.213 0.212 0.207 0.187 0.208 0.196
t  (mm) 7.273 7.083 6.994 6.938 6.738 6.780
h (mm) 5.0 5.1 5.219 5.063 5.187 5.152
f (kN) 42.106 37.190 37.662 35.677 34.253 33.488
?
g
0.075 0.032 0.248 0.601 0.824 1.430
 
Table 5.6: Constraint values of the Belleville spring optima.  
Nominal Min-Max
Robust 
(0.3)
Robust 
(0.6)
Robust 
(0.8)
Robust 
(1.0)
g
1
0 0 -0.002 -0.017 -0.018 -0.036
g
2
0 0 -0.008 -0.006 -0.060 -0.018
g
3
0 -0.020 -0.044 -0.013 -0.038 -0.030
g
4
-0.386 -0.391 -0.389 -0.400 -0.404 -0.403
g
5
0 0 -0.003 -0.049 -0.006 -0.025
g
6
-0.112 -0.116 -0.132 -0.181 -0.124 -0.160
g
7
-4.214 -4.176 -4.256 -4.834 -4.163 -4.592
 
 160
 
We see in Table 5.5 that the nominal optimum has a very low ?
g
, but has the highest 
f value (recall that this is a maximization problem). The value ?
g
 progressively increases 
as f decreases (the performance vs. robustness trade-off). This progressive increase in 
robustness can also be observed in the decrease in the constraint values of the optima, 
Table 5.6 (i.e., the optimum is further away from the constraint boundary).  
One important thing to notice in Table 5.5 is that the min-max optimum of Hirokawa 
and Fujita is inferior to the nominal and robust (0.3) optima (i.e., in terms of ?
g
 and f, it is 
dominated). If we plot the value f vs. ?
g
, this inferiority is immediately apparent 
(Figure 5.15). The performance vs. robustness trade-off of the optima can also be 
observed in this figure. 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
30 35 40 45
nominal
robust (0.3)
robust (0.6)
robust (0.8)
robust (1.0)
min-max
f (kN)
?
g
 
Figure 5.15: Plot of f vs. ?
g
 of the Belleville spring.  
To verify the robustness of the optima, we performed a numerical sensitivity analysis 
by perturbing the parameter values following the given ranges. A design is feasibly 
robust if all the constraints remain feasible. Figures 5.16, 5.17, and 5.18 show the graphs 
 161
 
of the max[g(x,p)] of each optimum. In these graphs, a design is feasibly robust if all the 
bars are below the horizontal axis.  
-0.012
-0.008
-0.004
0
0.004
0.008
0.012
161
-0.012
-0.008
-0.004
0
0.004
0.008
0.012
161
nominal min-max
 
Figure 5.16: Sensitivity analysis of the nominal and min-max optima. 
-0.012
-0.008
-0.004
0
0.004
0.008
0.012
161
robust (0.3)
-0.012
-0.008
-0.004
0
0.004
0.008
0.012
161
robust (0.6)
 
Figure 5.17: Sensitivity analysis of the robust (0.3) and robust (0.6) optima. 
-0.025
-0.015
-0.005
0.005
0.015
0.025
161
-0.012
-0.008
-0.004
0
0.004
0.008
0.012
161
robust (0.8) robust (1.0)
 
Figure 5.18: Sensitivity analysis of the robust (0.8) and robust (1.0) optima. 
 162
 
We see in Figures 5.16 ? 5.18 that the nominal, min-max, and robust (0.3) optima are not 
feasibly robust. The robust (0.6) optimum is almost feasibly robust except in one case 
where it becomes infeasible. The robust (0.8) and robust (1.0) optima are always feasible 
regardless of the perturbations, thus they are feasibly robust. These observations confirm 
the ?
g
 values of each design shown in Table 5.5.  
 
5.4.4. Design of a Control Valve Actuator Linkage 
For our last example, we applied our feasibility robust optimization method to the 
engineering design of a control valve actuator linkage. This example is adapted from the 
example in Balling et al. (1986) with some modifications. 
The control valve actuator linkage to be designed is shown in Figure 5.19.  
55?
?=90?
L
c
L
r
s
(a)
x
y
d
c
d
r
d
h
0
(b)
 
Figure 5.19: A control valve actuator linkage. 
The linkage mechanism has two members: a crank and a rod that are connected by a pin 
joint. The other end of the crank is held stationary, while the other end of the rod is 
pinned to a slider. Originally the crank was at a 55? angle from the vertical axis as shown 
in Figure 5.19(a). The dimensions of the linkage are shown in Figure 5.19(b).  
 163
 
There is a constant force F = 1425.5 lbs (6340 N) acting at the end of the rod that 
causes the mechanism to turn (see Figure 5.20). The design objective is to maximize the 
torque (T) at the end of the crank as it turns from ? = 0? to ? = 90?, averaged over 10? 
interval (Figure 5.20). The weights of the crank and rod are assumed negligible.  
F
y
F
x
FF
y
F
x
T
 
Figure 5.20: Forces acting on the linkage. 
The design variables in this problem are the crank length (L
c
), the rod length (L
r
), 
and the center distance (d). The crank length and the rod length are constrained to be 
within 0 and 10 inch (0 and 25.4 cm), while the center distance is constrained to be 
within 5 and 7 inch (12.7 and 17.78 cm). The problem has three design constraints: (1) 
the vertical position of the slider when ? = 0? (h
0
) is constrained to be less than 6.5 inch 
(16.51 cm), (2) the movement of the slider (s) is constrained to be less than 4.5 inch 
(11.43 cm), and (3) the side force (F
x
) averaged over 10? interval is less than 800 lbs 
(3558 N). In addition, the problem has two geometric constraints: (1) the horizontal 
length of the crank (d
c
) must not be greater than the center distance (d), and (2) the 
difference between d and d
c
 must be less than or equal to the rod length (L
r
). The 
mathematical formulation of the problem is shown below (for English units). 
 164
 
7d5;10L,L0
01
L
d
g
01
d
d
g;01
800
)?(F
9
1
g
01
4.5
s
g;01
5.6
h
g:subject to
)?(T
9
1
)d,L,L(maximize
rc
r
r
5
c
4
90
0?
x
3
2
0
1
90
0?
rc
????
???
??????
??????
=
?
?
?
?=
?
?=
f
       (5.29) 
The quantities T(?), F
x
(?), s, h
0
, d
c
, and d
r
 are calculated as follows: 
45)cos(?Ld
cc
??=  ; 
cr
d-dd =    (5.30) 
radians)in (
L
d
cos?
45)-sin(?L?sinL)?(h
)0h(?h
r
r1-
cr
0
?
?
?
?
?
?
?
?
=
??=
?==
-     (5.31) 
)0h(?)09h(?s ?=?== -      (5.32) 
?cosF)?(F
x
?=       (5.33) 
?sinFF
)?(F)?(h-Fy.d)?(T
y
x
?=
?=
     (5.34) 
Because of manufacturing tolerances, L
c
 and L
r
 vary by ?0.1 inch (?0.254 cm), and 
we need to guarantee the feasibility of the optimum design under these variations. We 
add the sensitivity constraint 0
?
)(?
g,0
g
??
x
1  to Eq. (5.29) and then optimize it. In this 
problem ?p
0
 = [0.1, 0.1] and ?
g,0
 = 1.0. For comparison, we also optimize the original 
problem (nominal optimum), and Eq. (5.29) with its constraints replaced by a 
probabilistic constraint P(g
j
 ? 0) ? 0.99, j=1,?,5. The probabilistic constraint is 
 165
 
calculated using Monte Carlo simulation assuming two probability distribution models: 
uniform and normal. For the uniform model, the lower and upper bound of the 
distribution are specified to be ?0.1 and 0.1, respectively. For the normal model, the 
distribution is specified to have a mean of 0 and standard deviation of 0.033 (= 0.1/3).  
The results obtained from this comparison study are shown in Table 5.7. In this table 
the quantity F
call
 is the number of function evaluations needed per design to calculate its 
feasibility robustness. Table 5.7 also shows the ?
g
 value of each optimum design.  
Table 5.7: Optimum designs of the control valve actuator linkage. 
Nominal Robust
Monte Carlo 
(Uniform)
Monte Carlo 
(Normal)
Torque (in.lb) 3592 3363 3429 3453
L
c
 (in) 3.182 3.03 3.078 3.093
L
r
 (in) 5.061 5.009 4.994 4.999
d (in) 5.0 5.0 5.0 5.0
F
call
N/A 250 100000 100000
?
g
0.005 1.02 0.73 0.62
 
We observe from Table 1 that the nominal optimum has the highest torque but the 
lowest ?
g
. In contrast, the robust optimum obtained by our method has the lowest torque 
but the highest ?
g
. The Monte Carlo optima are somewhere in between the two. This 
observation is expected because generally we have to sacrifice some performance to gain 
an increase in robustness. In fact, if we solve the optimization problem as a two-objective 
problem where we maximize both torque and ?
g
, we will obtain a trade-off frontier as 
shown in Figure 5.21. Notice in this figure how torque decreases as ?
g
 increases. Points 
corresponding to the optima in Table 5.7 are also shown.  
 166
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
3200 3300 3400 3500 3600 3700
Torque (in.lb)
?
g
Robust optimum
Monte Carlo
(uniform)
Monte Carlo
(normal)
Nominal
optimum
 
Figure 5.21: Trade-off frontier of the linkage problem. 
Note also in Table 5.7 that although the two Monte Carlo optima were obtained by 
enforcing a 0.99 probability of constraints satisfaction, their ?
g
 value is less than 1.0. In 
fact, the ?
g
 values of the two optima are quite different. This is because the robustness of 
the optimum design is sensitive to the assumed probability distributions, and the 0.99 
constraints satisfaction probability is valid only if the assumed distribution is valid. We 
will further discuss this important issue next. 
Table 5.7 (and Figure 5.21) also shows that the robust optimum is different than the 
Monte Carlo optimum using a uniform distribution. This shows that our method does not 
presume a uniform probability distribution of the parameters (although it may seem so). 
The fact is our method does not presume any distribution at all, and this is reflected in the 
fact that we cannot provide probability information for the obtained optimum. We lack 
the information to do so. Our robustness constraint guarantees that the optimum design 
will remain feasible if the parameter variations are as specified. If the parameter 
 167
 
distribution changes, the probability of constraint satisfaction will change, but this 
guarantee still holds.  
We also observe in Table 5.7 that Monte Carlo method requires 100,000 function 
evaluations to calculate a design?s robustness. In contrast, our method requires only ~250 
evaluations, comparable to those more efficient probabilistic methods (MPP for instance 
? Du and Chen, 2000). It should be noted, however, that this number (i.e., 250) is an 
upper bound value because we used GA to solve the inner optimization problem. GA is 
an optimizer that needs a lot of function evaluation, but it is applicable to a wide range of 
optimization problem and does not require gradient information. If gradient information 
is available, we can use a more efficient optimizer to solve our inner optimization 
problem, and the number of function evaluations our method needs would be much lower 
(in the order of 10
1
).  
To validate the results in Table 5.7, we conducted a sensitivity analysis on each 
optimum design. We performed 100,000 Monte Carlo simulations on each design, and 
based on the result, calculate its probability of constraint satisfaction. The simulations are 
performed using two probability distribution models of the parameters: uniform and 
normal. In the uniform distribution model, ?L
c
 and ?L
r
 are jointly uniformly distributed 
in the interval [-0.1,0.1]. In the normal distribution model, ?L
c
 and ?L
r
 are bi-normally 
distributed with a mean and standard deviation of [0, 0.033]. In both models, ?L
c
 and ?L
r
 
are assumed independent. The results of this sensitivity study are shown in Table 5.8. For 
comparison purposes, we have also re-listed the ?
g
 value of each optimum. 
 168
 
Table 5.8: Sensitivity analysis of the optima. 
Uniform model Normal model
Nominal 0.376 0.381 0.005
Robust 1.0 1.0 1.02
Monte Carlo (Uniform) 0.989 0.999 0.73
Monte Carlo (Normal) 0.925 0.996 0.62
Probability of constraint satisfaction
?
g
 
We observe in Table 5.8 that the nominal optimum has a poor probability of 
constraint satisfaction in both models. This is not surprising since this optimum is 
obtained by strict optimization of the linkage?s torque, neglecting the variations in L
c
 and 
L
r
. The robust optimum obtained by our method, on the other hand, has a 1.0 probability 
of constraint satisfaction in both models, much more robust than the nominal optimum. 
This observation confirms the information provided by the ?
g
 values of the two optima. 
The robust optimum has a much larger ?
g
 value (1.02) than the nominal optimum 
(0.005). 
The Monte Carlo (uniform) optimum has a 0.989 probability of constraint 
satisfaction for the uniform distribution model (same value as imposed by the 
probabilistic constraint). But this probability value increases to 0.999 for the normal 
model. The same pattern is also observed for the Monte Carlo (normal) optimum. It has a 
0.996 probability of constraint satisfaction for the normal model (same as imposed by the 
constraint), but this value reduces to 0.925 for the uniform model. This observation 
shows that the information provided by the probabilistic constraints is dependent on the 
accuracy of the assumed distribution model. As such, unless the distribution of the 
uncertain parameters is known with relative certainty, such probability information must 
be used with utmost caution since it can be misleading.  
 169
 
To further validate our method, we calculated and compared the FSR and FWCSR of 
the optima in Table 5.7. For simplicity, we only show the comparison for the nominal and 
robust optima. We form the FSR (S
g
) of each design by first forming the FSR of each 
constraint (S
g,j
) and then forming the intersection. S
g,j
 is obtained by constructing the 
difference function ?g
j
(?L
c
, ?L
r
) = g
j
(L
c
+?L
c
, L
r
+?L
r
) ? g
j
(L
c
, L
r
) from Eq. (5.29), 
substituting the L
c
, L
r
, and d values of the design into this function, and then setting it to 
?g
j
 ? ?g
j,0
. The ?g
j,0
 is obtained by taking the absolute value of the j-th constraint of the 
design. The constraint values of the nominal and robust optima are shown in Table 5.9. 
Table 5.9: Constraint values of the nominal and robust optima.  
Nominal Robust
g
1
-0.00026 -0.037
g
2
-0.000031 -0.048
g
3
-0.173 -0.169
g
4
-0.366 -0.396
g
5
-0.457 -0.429
 
Observe in Table 5.9 that only g
1
 and g
2
 are active (or nearly active) for the nominal 
optimum while g
3
, g
4
, and g
5
 are relatively the same for the two optima. This implies that 
the ?g
1
 and ?g
2
 functions are critical components of S
f
, while ?g
3
, ?g
4
, and ?g
5
 are not. 
So, in constructing the FWCSR, using only ?g
1
 and ?g
2
 suffices. From Eq. (5.29), we 
obtain the ?g
1
 and ?g
2
 difference functions: 
 
()[ ]
)LL(707.0dd
?
radians)(in 
LL
d
?
cos?
?
L314.0g
L707.0?sinL?
?
sinLL
6.5
1
g
ccr
rr
r1-
c2
crrr1
?+?=
?
?
?
?
?
?
?
?
?+
=
??=?
?+??+=?
  (5.35) 
 170
 
Figure 5.22 shows the FSR and FWCSR of the nominal and robust optima. In this 
figure only the critical components, i.e., ?g
1
 and ?g
2
, are shown. Note that S
g,1
 is not a 
linear function (although it may seem so from the figure). S
g,2
 is a linear function, while 
S
g,1
 is a trigonometric function as shown in Eq. (5.35). Note also that Figure 5.22 shows 
the regions in the non-normalized space (but has same scale). We do so to better relate to 
the actual values of L
c
 and L
r
.  
(a) nominal (b) robust
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
?L
c
?L
r
FWCSR
S
g,1
S
g,2
S
g
= S
g,1
? S
g,2
?L
c
?L
r
FWCSR
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
S
g,1
S
g,2
S
g
= S
g,1
? S
g,2
 
Figure 5.22: FSR and FWCSR of nominal and robust optima. 
We observe in Figure 5.22 that the FWCSR of the nominal optimum is very small, 
close to a zero radius. This is because g
1
 and g
2
 are almost active for this optimum, and as 
such there is very little ?cushion? for L
c
 and L
r
 variation (in the worst case sense). In 
contrast, the FWCSR of the robust optimum is much larger (R
g
 = 1.44), and it allows for 
more variations in L
c
 and L
r
. This is also reflected by the larger g
1
 and g
2
 values of the 
robust optimum in Table 5.9. Observe also that the FWCSR of the robust optimum covers 
the area bounded by [-0.1, 0.1] as specified.  
 
 171
 
5.5. SUMMARY 
? The feasibility robustness of a design is indicated by its feasibility sensitivity set (S
g
). 
Like the objective sensitivity set (S
f
), the feasibility sensitivity set S
g
 shows how 
much ?p a design can absorb before it violates a prescribed limit. Thus, the larger S
g
, 
the more robust the design.  
? However, unlike S
f
, S
g
 is a one-sided sensitivity measure because we only need to 
limit the increase, but not the decrease, in the constraints.  
? The plot of S
g
 in the ?p-space is the Feasibility Sensitivity Region (FSR) of a design. 
To account for directional sensitivity, we use the worst-case estimate of FSR, the 
Feasibility Worst Case Sensitivity Region (FWCSR) as a robustness measure. The 
FWCSR radius (R
g
) can be calculated by solving a single-objective optimization with 
one equality constraint.   
? Unlike objective robustness, the increment limits for constraints are determined by 
how far a design is from the constraint boundary, and not by the designer.  
? In addition, a design has to be feasible nominally for the FWCSR measure to make 
sense. If a design is infeasible, then feasibility robustness cannot be guaranteed. 
? The inner optimization to calculate R
g
 must be normalized if the scale of ?p is 
different. After the normalization, a design is guaranteed to be feasibly robust if its 
feasibility robustness index, ?
g
 = (G)
-1/2
R
g
, is greater than or equal to 1.0.  
 172
 
CHAPTER 6 
DISCUSSIONS 
 
6.1. INTRODUCTION 
In the last three chapters, we presented methods for objective robust optimization 
and feasibility robust optimization separately. The purpose of this chapter is to show how 
to combine these methods for both objective and feasibility robust optimization. In 
addition, this chapter also aims to address those issues in our robust optimization methods 
that we have not yet addressed, or so far only briefly discussed. More specifically, this 
chapter will discuss issues regarding: (i) use of one-sided sensitivity measure for 
objective robustness, (ii) asymmetrical two-sided sensitivity measure, (iii) asymmetrical 
parameter variations, and (iv) a comparison between robustness index and robustness 
probability.  
 
6.2. OBJECTIVE AND FEASIBILITY ROBUST OPTIMIZATION 
In Chapters 3 and 4, we introduced an index ? to measure objective robustness of a 
design, where ? = (G)
-1/2
R
f
, and R
f
 calculated by solving an optimization problem in 
Eq. (4.6), restated here for convenience: 
01
)(
max:tosubject
)p()(Rminimize
0,
0
M,...,1
2
1
G
1
2
=?
?
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
? ?=?
=
=
?
i
i
i
j
jf
f
f pp
p
p
   (6.1) 
Similarly, in Chapter 5 we introduced an index ?
g
 to measure feasibility robustness of a 
design, where ?
g
 = (G)
-1/2
R
g
, and R
g
 calculated by Eq. (5.16), restated below. 
 173
 
[]0),(gmax:subject to
p)(Rminimize
00
J,...,1
2
1
G
1
2
g
=???+
?
?
?
?
?
?
?=?
=
=
?
?
ppp
p
p
x
j
j
i
i
   (6.2) 
Adding the constraint 0
?
?
0
??1  to an optimization problem guarantees objective 
robustness of an optimum design, while adding the constraint 0
?
?
g,0
g
??1  guarantees its 
feasibility robustness. Accordingly, adding both of these robustness constraints to an 
optimization problem will guarantee both objective and feasibility robustness of an 
optimum design. Eq. (6.3) shows the overall formulation of a general robust optimization 
problem, where ? and ?
g
 are calculated by Eq. (6.1) and (6.2), respectively.  
[ ]
)constraint robustnessty (feasibili0
?
?
1
)constraint robustness (objective0
?
?
1
K,...,1;0),(h
J,...,1;0),(g:subject to
),(),...,,(minimize
g,0
g
0
0
0
0M01
??
??
==
=?
k
j
ff
k
j
p
p
pp
x
x
xx
x
 (6.3) 
Notice in Eq. (6.1) and (6.2) that R
f
 and R
g
 are defined in the same space ( p? -space), 
and are of the same scale (normalized by 
0
p? ). Based on this observation, then the two 
robustness constraints in Eq. (6.3) will be satisfied if the larger of the two constraints is 
satisfied, i.e., if 0
?
?
1,
?
?
1max
g,0
g
0
?
?
?
?
?
?
?
?
?
?? . Using this fact, the robustness constraints in 
Eq. (6.3) can be stated more compactly as: 0
?
?
,
?
?
min
g,0
g
0
?
?
?
?
?
?
?
?
?
1 . ?
 174
 
The advantage of keeping the objective and feasibility robustness constraints 
separate like shown in Eq. (6.3) (they are still separate in the compact form as well) is 
that it provides flexibility for a designer to specify his/her preference towards the two 
types of robustness. By setting different values for ?
0
 and ?
g,0
, a designer can specify that 
(s)he considers one type of robustness (i.e., objective robustness or feasibility robustness) 
more important than the other type. For example, if a designer specifies ?
0
 = 0.8 but 
?
g,0
 = 1.0, then it implies that the feasibility robustness of an optimum design is 
considered to be more important than its objective robustness. 
If a designer is indifferent towards either the objective or the feasibility robustness 
(i.e., ?
0
 = ?
g,0
), then the robustness constraint 0
?
?
,
?
?
min
g,0
g
0
?
?
?
?
?
?
?
?
?
?1  simplifies into 
[]0R,Rmin
?
G
1
g
0
2/1
?
?
?
?
?
?
?
?
?
?
?
f
. This last inequality is of particular interest to us. In this 
inequality, we only need to determine the smaller of R
f
 and R
g
, but not both. Let?s discuss 
the meaning of min[R
f
,R
g
] in more detail next.  
Recall from previous chapters that conceptually R
f
 is the normalized WCSR radius 
of a design, and WCSR is the worst-case estimate of the corresponding SR. Likewise, R
g
 
is the normalized FWCSR radius of the design, and FWCSR is the worst-case estimate of 
the FSR. Since SR and FSR are defined in the same p? -space and are of the same scale, 
min[R
f
,R
g
] implies that we are looking for the radius of worst-case estimate of an 
intersection of SR and FSR, as shown in Figure 6.1. In Figure 6.1, the region between the 
two solid lines is the SR, and the region bounded by the dashed line is the FSR. The R
f
 
 175
 
and R
g
 of the SR and FSR are also shown in Figure 6.1. The shaded region is the 
intersection of SR and FSR. 
R
g
FSR
SR and FSR
intersection
1
p?
2
p?
SR
0
R
f
min[R
f
,R
g
] = R
f
 
Figure 6.1: Intersection of SR and FSR. 
Mathematically, a ?p point is inside the intersection region if it satisfies all of the 
following inequalities: . A ?p point is outside the 
intersection region if either [  or 
?
?
?
?
?
=?<?
=?<?
J,...,1gg
M,...,1][][
0,
2
0,
2
j
iff
jj
ii
2
0,
2
][]
ii
ff ?>? g
j 0,
g
j
?>?  is true for at least one 
i = 1,?,M or j = 1,?,J, respectively. For a ?p point to be on the boundary of the 
intersection region, it needs to satisfy [  and 
2
0,
2
][]
ii
ff ???
0,
g
j
g
j
??? , with at least one 
strict equality (one i or j from i = 1,?,M or j = 1,?,J, respectively). Using the simplified 
condition for SR and FSR boundaries developed previously (recall Section 4.3.2 and 
5.2.2), the condition for a ?p point to be on the boundary of the intersection region can be 
 176
 
simplified into: 01
|),(g|
g
max,
||
maxmax
0
J,...,1
0,
M,...,1
=?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
==
px
j
j
j
i
i
i
f
f
, where we have used the 
fact that ?g
j,0
 = |g
j
(x
0
,p)|. Notice also that in this simplified condition we have used 
|),(g|
g
0
px
j
j
?
 instead of g
j
(x
0
,p) to define the FSR boundary (recall Section 5.3). Using the 
ratio 
|),(g|
g
0
px
j
j
?
 is recommended because we are comparing it to a normalized ?f
i
 value. 
Using g
j
(x
0
,p) in the formulation may create difficulty in terms of numerical comparison.  
 Using the above mathematical definitions, the radius (R) of the worst-case estimate 
of the intersection region can be calculated by solving the following optimization 
problem (normalized): 
 
01
|),(g|
)(g
max,
|)(|
maxmax
:subject to
p)R(minimize
0
0
J,...,1
0,
0
M,...,1
2
1
G
1
2
=?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ???+?
?
?
?
?
?
?
?
?
?
???+?
?
?
?
?
?
?
?=?
==
=
?
?
p
ppp
ppp
p
p
x
j
j
j
i
i
i
i
i
f
f
 (6.4) 
Recall from Figure 6.1 that this radius value (R) is equal to min[R
f
,R
g
]. So, Eq. (6.4) 
shows that we can find min[R
f
,R
g
] by solving just one inner optimization problem instead 
of two (i.e., Eq. (6.1) and (6.2)).  
If we define a quantity ?
min
 = (G)
-1/2
 min[R
f
,R
g
], then the overall objective and 
feasibility robust optimization problem becomes as shown in Eq. (6.5) in which 
min[R
f
,R
g
] is calculated by solving Eq. (6.4). Here, we call ?
min
 the overall robustness 
index. We have used ?
0
 for the robustness constraint in Eq. (6.5), but since ?
0
 = ?
g,0
, we 
could have used ?
g,0
 as well.  
 177
 
[ ]
)constraint robustness (overall0
?
?
1
J,...,1;0),(g:subject to
),(),...,,(minimize
0
min
0
0M01
??
=? j
ff
j
p
pp
x
xx
x
  (6.5) 
One advantage of having to solve just one inner optimization problem is that 
practically the optimization algorithm to solve Eq. (6.5) (the outer and inner problems 
and the interface between them) will be much easier to implement. In addition, it helps 
the outer problem to converge faster because it has one fewer constraint to satisfy. 
Solving only one inner problem also helps reduce numerical errors transmitted from the 
inner problem to the outer problem. One potential disadvantage of combining Eq. (6.1) 
and (6.2) into a single inner problem is that solving Eq. (6.4) might be less efficient 
computationally than solving Eq. (6.1) and (6.2) separately. For instance, if Eq. (6.1) and 
Eq. (6.2) can be solved in say T iterations each, it is possible that Eq. (6.4) may need 
more than T iterations to converge because its constraint is more complex.  
Before we continue further, it is important to point out again that Eq. (6.4) and (6.5) 
are valid only if the designer is indifferent to either objective or feasibility robustness 
(i.e., ?
0
 = ?
g,0
). If one type of robustness is preferred to the other type, then we must 
solve Eq. (6.1) and (6.2) separately. 
 
6.2.1. Design of a Payload for an Undersea Autonomous Vehicle (UAV) 
To demonstrate our combined objective and feasibility robust optimization method, 
we apply it to an engineering example: the design of a UAV payload. The description of 
the problem is as follows.  
 178
 
Typically, the payload of a UAV must be effective in several different uses, called 
?scenarios.? Effectiveness in a scenario is measured by the probability of success, P
S
, of 
payload delivery in that scenario. The design goal is to simultaneously maximize the 
individual P
S
?s for all scenarios. The payload design is constrained by upper limits on the 
weight of the payload and on the radiated noise generated by the payload. 
There are six design variables: the payload length (PL), the hull diameter (DH), the 
material of the hull (HM), the payload type (PT), the first inner material type (I1), and the 
second inner material type (I2). Four of the variables are discrete: HM, PT, I1, and I2. 
The choices for HM, PT and I1 are [6061AL, 7075AL], [BULK, MULTI_MISS], and 
[TYPE_1A, TYPE_1B], respectively. For discrete variable I2, the options available are 
[TYPE_2A, TYPE_2B, TYPE_1B], but I2 can be TYPE_1B only if the variable I1 is 
TYPE_1B also. The other two variables are continuous and they are bounded as: 
6.0 ? DH ? 12.75 and 1.0(DH) ? PL ? 5.0(DH). In addition to the six design variables, 
there is a fixed continuous design parameter, the maximum depth (= 3000 ft), at which 
the payload operates. Unlike our other design examples, there are no closed-form 
relationships to map the design variables to the constraints and to the P
S
?s. Rather, we are 
provided with a design analyzer (a computer program) that maps the design variables to 
the payload weight, the radiated noise, and the P
S
?s for the scenarios.  
In this example, we address a two objective payload design optimization with two 
constraints. The two objectives are to maximize P
S1
 and P
S2
 for two different scenarios. 
The two constraints are an 85 lb upper bound on the payload weight and a 0.16 Watt/m
2
 
upper bound on the radiated noise generated. The problem is mathematically formulated 
as follows. 
 179
 
( )
()
016.0I2I1,PT,HM,DH,PL,Noise
085I2I1,PT,HM,DH,PL,Weight:subject to
I2I1,PT,HM,DH,PL,Pmaximize
I2I1,PT,HM,DH,PL,Pmaximize
S2
S1
??
??
         (6.6) 
There are some uncertainties in the formulation of the problem, and it is modeled by 
assuming that a parameter (internal to the design analyzer), C
eq
, which is used in 
calculating the weight of the payload has an uncontrollable variation. Since the value of 
C
eq
 depends on the discrete combination of [PT, I1, I2], its variation is taken to be a 
percentage of the actual value: ?C
eq,0
 = (0.10)C
eq
. In addition, we also assume that two 
internal parameters [A
1
, A
2
] used in calculating P
S
 vary by 0.01 each. The problem we 
are solving, Eq. (6.6), has two P
S
 objectives, so we have a pair of [A
1
, A
2
] variations in 
calculating P
S
. It is also assumed that two of the design variables have uncontrollable 
variations as well: [?PL
0
, ?DH
0
] = [0.01, 0.01]. In total, there are 7 uncertain parameters 
in this problem: [Ceq, A
1,s1
, A
2,s1
, A
1,s2
, A
2,s2
, PL, DH]. The maximum allowable 
variations in the P
S
?s are specified to be [?P
S1,0
, ?P
S2,0
] = [0.025, 0.025].  
For the designer, the objective and feasibility robustness of the payload designs are 
equally important, and the desired robustness is specified to be: ?
0
 = ?
g,0
 = 1.0. Since the 
designer is indifferent towards the two types of robustness, we can use the combined 
inner problem, Eq. (6.4), to search for the robust Pareto optima of this problem. Adding 
our overall robustness constraint 0
?
?
0
min
??1  to Eq. (6.6) and solving it, we obtain the 
robust Pareto optima of the problem as shown in Figure 6.2. For comparison, Figure 6.2 
also shows the nominal Pareto optima of the problem (i.e., Eq. (6.6) without the 
robustness constraint), and the Pareto optima obtained from solving the problem using a 
 180
 
probabilistic approach. In the probabilistic approach, we minimize the worst-case P
S
 of 
each objective in the form of the sum of mean and standard deviation of the PS?s. The 
mean and standard deviation values are calculated by running 10,000 Monte Carlo 
simulations assuming uniform probability distribution.  
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
P
S1
P
S2
Nominal
Robust
Probabilistic
nominal optimum
to be analyzed
robust optimum
to be analyzed
probabilistic optimum
to be analyzed
P
S2
 
Figure 6.2: Pareto sets of the payload problem. 
We see in Figure 6.2 that overall the robust Pareto optima are inferior to the nominal 
Pareto optima (this is a maximization problem), although there seems to be some overlap 
between the two Pareto frontiers. This observation is expected because there is a trade-off 
between the performance of an optimum and its robustness. We also see that the 
probabilistic Pareto solutions are very close to the nominal Pareto frontier, suggesting 
that these points do not meet our robustness requirement.  
To verify the robustness of the designs, we performed a sensitivity analysis on three 
of the Pareto optimum designs obtained, one each from the nominal, robust, and 
 181
 
probabilistic Pareto set (randomly selected). The objective and constraint values of the 
three designs are shown in Table 6.1. Table 6.2 shows their design variable values.   
Table 6.1: Objective and constraint values of the optima. 
Nominal Robust Probabilistic
P
S1
0.067 0.295 0.514
P
S2
0.695 0.295 0.135
Weight (lb) 85.000 84.433 84.95
Noise (W/m
2
) 0.158 0.158 0.157
 
Table 6.2: Design variables of the optima. 
Nominal Robust Probabilistic
PL (inch) 19.679 24.262 24.237
DH (inch) 9.683 9.048 10.216
HM 7075AL 7075AL 7075AL
PT BULK MULTI_MISS MULTI_MISS
I1 TYPE_1B TYPE_1B TYPE_1B
I2 TYPE_2A TYPE_1B TYPE_2B
 
We perform the sensitivity analysis by perturbing the 7 uncertain parameters 
[Ceq, A
1,s1
, A
2,s1
, A
1,s2
, A
2,s2
, PL, DH] around their original values, and then observing 
the changes in the objective and constraint values of the two designs. The perturbation 
values used in this analysis are randomly sampled from the given ranges of the parameter 
variations. A design meets the robustness criterion if these changes are within the 
specified limits. The results of the analysis are shown in Figure 6.3, 6.4, and 6.5 for the 
nominal, robust, and probabilistic optimum, respectively. In these figures, the dashed 
lines are the variation limits, and a design is robust if the bars do not cross these lines.  
 182
 
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
161
Case #
?Ps2
82
83
84
85
86
87
161
Case #
W
e
ight
0.155
0.156
0.157
0.158
0.159
0.16
0.161
0.162
161
Case #
No
i
s
e
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
161
Case #
?Ps1
W
e
ight
W
e
ight
No
i
s
e
No
i
s
e
 
Figure 6.3: Sensitivity analysis of the nominal optimum payload. 
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
161
Case #
?Ps1
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
161
Case #
?Ps2
0.155
0.156
0.157
0.158
0.159
0.16
0.161
0.162
161
Case #
No
i
s
e
82
83
84
85
86
87
161
Case #
We
i
g
h
t
No
i
s
e
No
i
s
e
We
i
g
h
t
We
i
g
h
t
 
Figure 6.4: Sensitivity analysis of the robust optimum payload. 
 183
 
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
161
Case #
?Ps2
0.155
0.156
0.157
0.158
0.159
0.16
0.161
0.162
161
Case #
No
i
s
e
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
161
Case #
?Ps1
82
83
84
85
86
87
161
Case #
We
i
g
h
t
No
i
s
e
No
i
s
e
We
i
g
h
t
We
i
g
h
t
 
Figure 6.5: Sensitivity analysis of the probabilistic optimum payload. 
We observe in Figure 6.3 that the nominal design satisfies both of the objective 
variation limits and the noise constraint. However, it does not satisfy the weight 
constraint. Similarly, the probabilistic design also satisfies the objective and noise 
robustness requirements, but not the weight constraint robustness (Figure 6.5). In 
contrast, we observe in Figure 6.4 that the robust design satisfies the variation limits for 
both objectives and constraints. This shows that the nominal and probabilistic designs do 
not meet the robustness criteria specified while the robust design does. In turn, these 
observations verify that using our robust optimization method to solve Eq. (6.6) indeed 
results in optimum designs that are robust with respect to both objectives and constraints.  
 
 184
 
6.3. ONE-SIDED SENSITIVITY FOR OBJECTIVE ROBUSTNESS 
Up to this point, we have used a two-sided sensitivity measure to calculate objective 
robustness of a design. The comparison studies presented in Chapters 3 and 4 showed 
applications of this two-sided objective robustness measure to several examples. 
However, in some cases it may be more appropriate to use a one-sided sensitivity 
measure to calculate objective robustness of a design. For instance, if the objective of an 
optimization problem is to minimize the total cost of a design, then we are only interested 
in preventing the cost increase, but not the decrease, due to parameter variations. For this 
type of optimization problems, we should use a one-sided sensitivity measure to calculate 
objective robustness of a design alternative, and not a two-sided one.  
A one-sided sensitivity measure for objective robustness is essentially the same as 
the one-sided sensitivity measure for feasibility robustness (recall Chapter 5) except that 
now we are looking at objective functions instead of constraint functions. Suppose ?f
0
 = 
[?f
1,0
, ?f
2,0
, ?, ?f
M,0
] is the maximum allowable increase in the objective values of a 
design x
0
 due to parameter variations. The one-sided S
f
 of x
0
 is then as follows (notice the 
absence of the square terms): 
{ }
),(),()(:where
M,...,1,)( :R)(
0000
0,
G
0
pppp
ppS
xx
x
iii
if
fff
iff
i
??+=??
=???????=
  (6.7) 
In Eq. (6.7) we have constrained the increase in ?f
i
 and not the decrease because we are 
minimizing f
i
. Since the smaller f
i
 the better, it is only logical that an increase in f
i
 is 
undesirable.  
The plot of this S
f
 in the ?p-space is the one-sided SR of x
0
, and the worst-case 
estimate of this SR is the one-sided WCSR of x
0
. The radius of this one-sided WCSR 
 185
 
( ) is obtained by solving the inner optimization problem shown in Eq. (6.8). Notice 
that Eq. (6.8) is the same as the inner problem to calculate feasibility robustness 
(Eq. (5.11)), except that now the equality constraint involves f
+
f
R
i
 instead of g
j
.  
01
)(
max:subject to
p)(Rminimize
,0
M,...,1
2
1
G
1
2
=?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?=?
=
=
+
?
?
i
i
i
i
if
f
f p
p
p
    (6.8) 
Eq. (6.9) shows the normalized version of Eq. (6.8). 
01
)(
max:tosubject
)p()(Rminimize
0,
0
M,...,1
2
1
G
1
2
=?
?
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
? ?=?
=
=
+
?
i
i
i
j
jf
f
f pp
p
p
   (6.9) 
Using the R  value obtained from Eq. (6.9), the one-sided objective robustness of x
+
f
0
 
is then indicated by the one-sided robustness index , where a value 
 shows that x
++
=
f
-
R(G)?
1/2
0.0.1? ?
+
0
 is robust, while a value  shows that x1? <
+
0
 is not robust. 
Formulating the one-sided objective robustness constraint 0
?
?
0
??
+
+
1 , where  is the 
desired level of robustness, the optimization problem to obtain an optimum design that is 
objectively robust one-sidedly is as shown in Eq. (6.10). 
+
0
?
[ ]
)robustness objective sided-one(0
?
?
1
K,...,1;0),(h
J,...,1;0),(g:subject to
),(),...,,(minimize
0
0
0
0M01
??
==
=?
+
+
k
j
ff
k
j
p
p
pp
x
x
xx
x
 (6.10) 
 
 186
 
6.4. ASYMMETRICAL TWO-SIDED SENSITIVITY MEASURE 
In calculating the two-sided sensitivity of a design?s objectives, we have used a 
single positive value ?f
i,0
 to limit both the increase and decrease in f
i
, i.e., a symmetrical 
two-sided sensitivity. Sometimes, however, it is desired to have different limits for the 
increase and decrease in f
i
, i.e., an asymmetrical two-sided sensitivity. The calculation of 
the asymmetrical robustness index of a design is a straightforward extension of the 
symmetrical one.  
Let  and  be the maximum allowable 
decrease and increase in f, respectively. Here we assume that ?  and 
 for all i=1,?,M. If ?  then we are essentially looking at a 
symmetrical two-sided sensitivity (recall Chapter 4). Using these two allowable limits, 
the asymmetrical S
],...,[
0,M0,10
???
??=? fff
+
0,i
f
],...,[
0,M0,10
+++
??=? fff
+?
?=
00
ff
0,
0,0,
>?
+?
ii
ff
?
???
0,i
f
f
 of a design x
0
 is then as follows: 
{ }
),(),()(:where
M,...,1,)(:R)(
0000
0,0,
G
0
pppp
ppS
xx
x
iii
iif
fff
ifff
i
??+=??
=??????????=
+?
  (6.11) 
Notice in Eq. (6.11) that the square terms are now replaced by two inequalities.  
The plot of this S
f
 in the ?p-space is the asymmetrical SR of x
0
, while the worst-case 
estimate to this SR is the asymmetrical WCSR. The inner optimization problem to 
calculate the radius of the asymmetrical WCSR ( ) is shown in Eq. (6.12) (this is the 
normalized version).  
+?/
R
f
01
)(
max,
)(
maxmax:tosubject
)p()(Rminimize
0,
0
M,...,1
0,
0
M,...,1
2
1
G
1
2/
=?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
? ?=?
+
=
?
=
=
+?
?
i
i
i
i
i
i
j
jf
f
f
f
f
pp
pp
p
p
   (6.12) 
 187
 
The equality constraint in Eq. (6.12) shows the asymmetry of this two-sided measure. 
The first and second terms in the equality constraint in Eq. (6.12) correspond to the 
decrement and increment limits in f
i
, respectively.  
The asymmetric robustness index of x
0
 is then calculated as where 
 is obtained by solving Eq. (6.12). As before, the value  shows that design 
x
+?+?
=
/1/2/
R(G)?
f
-
0.1?
+?/
R
f
?
/+?
0
 is robust while ?  shows that it is not robust. To obtain an optimum design that 
is objectively robust asymmetrically, we only need to add the robustness constraint 
0.1
/
<
+?
0
/
/
?
+
+
?
?
1
0
?
?
?
 to the optimization problem of interest (  is the desired level of 
robustness, and is specified by the designer).  
+?/
0
?
 
6.5. ASYMMETRICAL PARAMETER VARIATIONS 
One of the assumptions of our robust optimization method is that the parameter 
variation ranges are symmetric: -?p
0,i
 ? ?p
i
 ? ?p
0,i
, i=1,?,G (?p
0,i
 > 0). This assumption 
is necessary because we are using a single point normalization to account for the scale 
difference of the ?p when measuring the robustness of a design. Sometimes, however, 
the variation ranges are not symmetric, and when this occurs, our robustness measure 
must be modified accordingly to account for this asymmetry.  
Let  and  be the lower and upper 
bounds of the parameter variation ranges, respectively. We assume that ?  
for all i=1,?,G (otherwise it is a symmetric range). The purpose of normalizing ?p
]p,...,p[
G,01,00
???
??=?p ]p,...,p[
G,01,00
+++
??=?p
0pp
,0,0
>??
+?
ii
i
 is to 
map it to the [-1,1] range. When the parameter variation (?p
0
) is symmetric, the 
 188
 
normalization 
i
i
i
,0
p
p
p
?
?
=?  provides such a mapping. When ?p
0
 is not symmetric, 
however, there are two cases to consider: the negative and positive ?p
i
. If ?p
i
 is negative, 
the normalization must be performed with respect to . If ?p
?
?
0
p
+
0
p
i
 is positive, the 
normalization must be performed with respect to . Based on this fact, the 
normalization for the asymmetric ?p is then: 
?
?
?
?
?
?
?
?
>?
?
?
??
?
?
=
+
?
0p,
p
p
0p,
p
p
p
,0
,0
i
i
i
i
i
i
i
?  or 
?
?
?
?
?
>???
????
=?
+
?
0p,)p)(p(
0p,)p)(p(
p
,0
,0
iii
iii
i
. Using the ternary operator ???  defined in Chapter 2, this 
normalization can be written in a compact form as: ??=?
+?
00
, ppp ?? ,p? .  
01
,,
)p
0,
0
2
1
2
=?
?
?
?
?
?
??
?
?
?
?
i
j
f
p )
0
?
+
p
Because now the ?p
0
 ranges have been normalized to be [-1,1], the radius of the 
exterior hyper-sphere of these ranges are still = (G)
-1/2
. So, the calculation of the objective 
and feasibility robustness index remain the same as before, i.e., ? = (G)
-1/2
R
f
 and 
?
g
 = (G)
-1/2
R
g
, respectively. However, the calculation of the radius of the WCSR and 
FWCSR have to be modified as shown in Eq. (6.13) and (6.14), respectively.   
  (6.13) 
(
max:tosubject
()(Rminimize
M,...,1
G
1
?
?
?
?
???
?
?
?
? ?=?
=
=
?
i
i
j
f
f p
p
p
[]0),,,(gmax:subject to
p)(Rminimize
000
J,...,1
2
1
G
1
2
g
=?????+
?
?
?
?
?
?
?=?
+?
=
=
?
?
pppp
p
p
x
j
j
i
i
  (6.14) 
 189
 
6.6. ROBUSTNESS INDEX VS. ROBUSTNESS PROBABILITY 
Our robust optimization method measures the robustness of a design based on the 
values of its robustness index: ? and ?
g
 for objective and feasibility robustness, 
respectively. The magnitude of a robustness index tells us the degree of robustness of a 
design: the larger ? (and/or ?
g
), the more robust the design. However, since we are only 
provided with ranges of ?p
0
, ? (and/or ?
g
) does not tell us the actual robustness 
probability of the design. (For objective robustness, the robustness probability is the 
probability that the objective values of the design stay within the acceptable limits. For 
feasibility robustness, it is the probability of constraint satisfaction of the design.) 
Nevertheless, if we know the probability distribution of ?p
0
, we can use the value of 
? and/or ?
g
 to calculate a lower bound on the robustness probability of the design.  
The procedure to calculate the robustness probability is described next. Here we only 
show the calculation for independent uniform and normal distributions. Probability 
calculations for other distributions and when there are correlations will be somewhat 
more involved, but the basic procedure is the same.  
Uniform Distribution 
Suppose ?p is distributed uniformly within [-1,+1] and there is no correlation among 
its ?p
i
 components. If the distribution is not within [-1,+1], it can be normalized to fall 
into this range (it has to be normalized anyway since we are going to compare it with the 
normalized robustness index). The probability density function of this distribution is 
f
x
(?p) = (2)
-G
, and the probability that a random variable X is between [?p
1
,?p
2
] is:  
)p()...p()(...][P
G1
p
p
p
p
21
1,2
1,1
G,2
G,1
???=????
??
?
?
?
?
ddf
x
ppXp     (6.15) 
 190
 
Substituting f
x
(?p) = (2)
-G
 into Eq. (6.15), P .  )pp()2(][
,1,2
G
1
G
21 ii
i
???=????
?
=
?
pXp
A value of ? = 1.0 indicates that the WCSR of a design encloses the [-1,+1] range in 
?p-space. (In our discussion we use ?, but it is applicable to ?
g
 as well.) More generally, 
a robustness index of ? tells us that the WCSR of the design encloses the [-?,+?] range in 
?p-space. Recall that the ?p range defined by a WCSR tells us the ?p that must occur if 
we want the ?f
0
 limit to be satisfied, while those ?p defined by the probability 
distributions are the ?p that actually does occur. In other words, the probability that the 
design will satisfy the ?f
0
 limit is equal to the probability that a random ?p falls in the 
[-?,+?] range, i.e., ][P ?p? +????
G
1
G
)? )(?( =??
?
=
?
i
. Using the formula obtained previously, the 
probability that a design will satisfy the ?f
0
 limit is then: 
. Table 6.3 shows the probability values for 
several instances of ? and G.  
G
?)2(][P =+??? ?X?
Table 6.3: Probability values for uniform distribution. 
1234
0.01 0.01 0 0 0
0.1 0.10 0.01 0.001 0
0.5 0.5 0.25 0.125 0.063
0.8 0.8 0.64 0.512 0.410
0.9 0.9 0.81 0.729 0.656
1.0 1.0 1.0 1.0 1.0
G
?
 
It is very important to point out that the probability calculation above is valid only if 
each ?p
i
 is uniformly distributed and they are independent. In addition, the formula 
P[.] = ?
G
 is only a lower bound (worst case) of the actual robustness probability value of 
 191
 
the design. This is because we are comparing the uniform distribution (which is a hyper-
cube) with a WCSR (which is a hyper-sphere). So, there are some ?p values that are part 
of WCSR (and hence ?), but are not included in the probability calculation. Another 
reason our probability calculation is only a lower bound is because WCSR is a worst-case 
estimate of the actual SR, so again, there are some ?p values that in reality the design can 
absorb, but are not included in our calculation.  
 
Normal Distribution 
Suppose ?p is multi-variate normally and independently distributed with a mean 
? = 0 and a covariance matrix . We use a standard deviation of 
?
?
?
?
?
?
?
?
?
?
?
=
9/10
09/1
L
MOM
L
?
ii
 = 1/3 so that the range [-1,1] covers 3?
ii
 of the distribution. The probability density 
function of this distribution is 
?
?
?
?
?
?
?????
?
?
?
?
?
)()(
2
1
exp
1T
2/1
?p??p
?
?
?
?
=?
? )2(
1
)(
G
?
p
x
f  
where |?| and ?
-1
 are the determinant and inverse of ?, respectively. The probability that a 
random variable X is between [?p
1
,?p
2
] is obtained by substituting this density function 
into Eq. (6.15) and then performing the integration. There is no closed-form solution to 
this integration, so we must numerically calculate it.  
As with the uniform distribution, given the robustness index ?, the robustness 
probability of a design is equal to the probability that a random ?p falls into the [-?,+?] 
range: . Table 6.4 shows the probability values for several instances of ? 
and G (these values are obtained numerically). 
][P ?p? +????
 192
 
Table 6.4: Probability values for normal distribution. 
1234
0.01 0.024 0 0 0
0.1 0.236 0.056 0.013 0.004
0.5 0.866 0.742 0.639 0.551
0.8 0.984 0.950 0.926 0.903
0.9 0.993 0.968 0.952 0.936
1.0 0.997 0.975 0.963 0.951
G
?
 
Keep in mind again that the values in Table 6.4 are valid only if the ?p distribution is 
normal. Also, these values are the lower bounds of the actual values of the robustness 
probability of the design. Unlike the uniform distribution, however, the lower bound for 
? = 1.0 is not 1.0, rather it decreases as G increases. The reason for this is because we 
only use ?
ii
 = 1/3 for the distribution. As G increases, the ?tail? region of the normal 
distribution is becoming larger, and this region is not included in our probability 
calculation.  
 
Example 6.1 
Let us revisit the control valve actuator linkage example from Chapter 5. Previously, 
we obtained a set of optimum designs for various values of ? (Figure 5.21). Using the 
probability values in Tables 6.3 and 6.4, estimate the probability of constraint satisfaction 
(P
s
) of each design assuming uniform and normal ?p distribution. Calculate the actual P
s
 
of the designs, and compare them to the estimated values. 
Solution 
We can easily estimate the P
s
 of the designs using the strategy explained previously. 
To calculate the actual P
s
 of the designs, we performed 100,000 runs of Monte Carlo 
 193
 
simulations for each design, for each distribution model. Figure 6.6(a) and (b) show the 
plots of the estimate and actual P
s
 of the designs for the uniform and normal distribution 
models, respectively.  
We see in both Figure 6.6(a) and (b) that the estimate P
s
 values are always lower 
than the actual P
s
 values. This is because the estimated values are only a lower bound of 
the actual values. Figure 6.6 also shows that the difference between the estimate and the 
actual P
s
 values decreases as ? increases. Intuitively, this observation is expected. The 
discrepancy between the estimated and the actual values is mainly caused by the fact that 
we have used a worst-case estimate of the FSR to measure a design?s robustness. As the 
actual P
s
 approaches 1.0, ? increases, and the FWCSR of the design will become more 
and more like the actual ?p distribution. Therefore, as ? increases, the lower bound P
s
 
will approach that of the actual P
s
. ? 
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
?
P
s
Actual
Estimate
(a) uniform
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
?
P
s
Actual
Estimate
(b) normal
P
s
P
s
P
s
P
s
 
Figure 6.6: Comparison between the estimate and actual P
s
. 
 
 194
 
6.7. SUMMARY 
? Adding objective and feasibility robustness constraints to an optimization problem 
results in optimum designs that are robust both objectively and feasibly.  
? The advantage of keeping the objective and feasibility robustness separate is that it 
provides the designer the flexibility to state his/her preference towards either of the 
two types of robustness (i.e., by changing ?
0
 and ?
g,0
). 
? When a designer is indifferent towards the two types of robustness, the overall 
robustness of the design can be calculated by solving just one inner optimization 
problem instead of two.  
? Practically, it is easier to solve just one inner optimization problem (to calculate the 
design?s robustness) than to solve two inner problems. Also, it helps the outer 
problem converges faster as well as reduces the error transmitted from the inner to the 
outer problem. However, solving two inner problems might be computationally more 
efficient. 
? Sometimes it is more appropriate to use a one-sided sensitivity measure for objective 
robustness. This one-sided measure is the same as the one-sided measure used in 
feasibility robustness, except that in this case we use objective functions instead of 
constraint functions in the formulation. 
? For a two-sided objective robustness, sometimes it is desired to have different 
allowable limits for the decrease and increase in f
i
, i.e., an asymmetric two-sided 
sensitivity. This asymmetry can be easily incorporated into our robustness 
formulation by changing the square terms in S
f
 by two inequalities.  
 195
 
? In our robust optimization method, we had made a necessary assumption that 
parameter variations are symmetric. However, in the event they are not symmetric, 
our symmetric normalization can be easily modified to account for it. 
? Since our method does not use a presumed probability density function (pdf), the 
value of the robustness index does not provide us with the probability information of 
the degree of robustness of the design. However, if the pdf of ?p is known, we can 
use the ? value to calculate a lower bound of the robustness probability of the design.   
 196
 
CHAPTER 7 
CONCLUSIONS 
 
7.1. CONCLUDING REMARKS 
In this dissertation, we have presented a step-by-step development of a novel method 
for robust design optimization. After presenting our research objective and review of 
previous work in Chapters 1 and 2, we developed our method for objective robust 
optimization for a single objective problem in Chapter 3, and then extended it to multi-
objective problems in Chapter 4. In Chapter 5, we developed a method for feasibility 
robust optimization of a design. Chapter 6 presented our combined objective and 
feasibility robust optimization method.  
The essence of our robust optimization method is the robustness measurement of a 
design alternative using a sensitivity region concept. A sensitivity region is an inherent 
property of a design that shows how much parameter variations the design can absorb 
given a limit on its performance variation. The more parameter variations a design can 
absorb (i.e., the more we can allow the parameters to vary), the more robust the design is. 
In the method, we use the worst-case estimate of the sensitivity region as a measure of a 
design?s robustness. Based on this worst-case estimate, we calculate a robustness index 
for the design, which we then constrain and add to the original optimization problem to 
guarantee the robustness of the optimum design solution obtained.  
In Chapters 3-6, we demonstrated the application of our robust optimization method 
to several numerical and engineering examples. For comparison, we also solved some of 
the problems using several other well-known robust optimization methods. We showed in 
 197
 
these examples that our method indeed obtains a design that is robust and optimum, and 
that our method is computationally efficient.  
In the next few subsections, we provide some additional concluding remarks 
regarding the results of our research.  
 
7.1.1. Verification 
In Chapters 3 through 6 we solved several numerical and engineering examples and 
analyzed the results obtained to verify our robust optimization methods.  
In the wine-bottle function example (Section 3.4.1), we can graphically verify the 
validity of our robust optimization method. The function has a unique property in that 
there is a flat region around the middle at which the function is insensitive to the variable 
variations, i.e., a robust region. A robust optimization method is valid if its solutions are 
within this flat region. We observe that the solutions from our method indeed fall into the 
flat region, thus verifies that our method is a valid robust optimization method for this 
example.  
For the other comparison studies, graphical verification is difficult or impossible to 
do. So instead, we performed a sensitivity analysis on the results obtained to see if they 
are indeed robust. The results of our sensitivity analysis are summarized in Table 7.1. In 
this table, the symbol ??? means that the optimum designs obtained are robust, while the 
symbol ??? means that they are not. We show in Table 7.1 four categories of robust 
optimization methods. ?Nominal? method refers to a regular optimization method 
without robustness consideration. ?Robust? method refers to our robust optimization 
method with ?
0
 = ?
g,0
 = 1.0. ?Sampling? method refers to those methods that perform a 
 198
 
localized sampling around the nominal parameter values (e.g., Monte Carlo). ?Gradient? 
method refers to those methods that use the gradient of the functions in calculating a 
design?s robustness (e.g., worst-case gradient, moment matching).  
Table 7.1: Summary of sensitivity analysis results. 
Nominal Robust Sampling Gradient
Three-bar truss ? ? ?
Welded beam ? ? ?
Compression spring ? ? ?
Numerical (multi-objective) ? ? ?
Vibrating platform ? ? ?
Speed reducer ? ? ?
Power electronic module ? ? ?
Numerical (feasibility) ? ? ?
Explosive actuated cylinder ? ? ?
Belleville spring ? ? ?
Control valve actuator linkage ? ? ?
Payload for an UAV ? ? ?
 
We see in Table 7.1 that optimum designs obtained by our robust optimization 
method always satisfy the robustness requirements. This observation verifies that our 
method is indeed a valid robust optimization method. In contrast, optimum designs 
obtained by a regular optimization (the ?Nominal? method) are not robust for these 
examples. This is expected since this method does not account for a design?s robustness. 
The optima of the ?Sampling? method also do not satisfy the robustness requirement. 
This is because the method uses probability to measure a design?s robustness, so there is 
a chance (albeit small) that the optimum design will violate the requirement. In the 
sensitivity analysis, a design is termed robust only if it never violates the requirement. 
The optimum of the ?Gradient? method is robust in the compression spring example, but 
it has a very poor objective value (recall Section 3.4.4). In the numerical example in 
 199
 
Chapter 5 and Belleville spring design examples, the ?Gradient? method fails to obtain a 
robust optimum design.  
 
7.1.2. Computational Efficiency  
Our robust optimization method calculates the robustness of a design by solving an 
inner optimization problem, which is a single objective problem with an equality 
constraint. When the objective/constraint functions of the outer problem (the original 
optimization problem) are simple enough, analytic solutions to the inner problem may be 
possible, in which case our method does not need to perform any function evaluations. 
When analytic solutions are not possible, the inner problem may be solved by a gradient-
based optimization algorithm such as Sequential Quadratic Programming. The 
computational cost of such algorithms is generally in the order of 10
1
. When gradient-
based algorithms are not applicable (e.g., when the functions are non-differentiable), 
stochastic algorithms such as GA may be used instead. For high solution accuracy, the 
computational cost for stochastic algorithms is generally in the order of more than 10
3
. 
However, throughout our comparison studies, we found that GA can solve the inner 
optimization problem using only ~200-300 function evaluations. This is because the inner 
problem is not too difficult an optimization problem to solve. Its objective function is 
convex and unimodal, and the search space is not large.  
Table 7.2 shows a summary of the number of function evaluations (F
call
) performed 
by the four methods to calculate the robustness of one design alternative.  
 200
 
Table 7.2: Summary of average number of function evaluations. 
Nominal Robust Sampling Gradient
Wine-bottle N/A 30
Three-bar truss N/A 39 10000
Welded beam N/A 250 100000
Compression spring N/A 250 0
Numerical (multi-objective) N/A 250 100000
Vibrating platform N/A 250 100000
Speed reducer N/A 300 100000
Power electronic module N/A 300 100000
Numerical (feasibility) N/A 24 0
Explosive actuated cylinder N/A 45 100000
Belleville spring N/A 300 0
Control valve actuator linkage N/A 250 100000
Payload for an UAV N/A 300 10000
 
In Table 7.2, the F
call
 for the ?Nominal? method is 0 because this method does not 
calculate the design?s robustness. The F
call
 for the ?Gradient? method is also 0, but this is 
because the gradient of the functions in the examples are known in closed form. If the 
gradient has to be numerically estimated, the F
call
 will be non-zero, depending on the 
dimension of the inner problem. The F
call
 of our robust optimization method varies from 
24 to 300. The F
call
 is either 250 or 300 when we used GA to solve the inner problem, and 
it is less than 50 when we used fmincon. Overall, the F
call
 of our method is much lower 
than the ?Sampling? method, whose F
call
 ranges from 10,000 to 100,000.  
Table 7.3 shows a summary of the computational cost of the four methods in terms 
of absolute time (?hr? for hour, ?m? for minute, and ?s? for second), and the 
optimization algorithms used in each problem. These time values are obtained using a 
Pentium III 866 MHz computer with 384 RAM.  
 201
 
Table 7.3: Summary of computational time. 
Nominal Robust Sampling Gradient Algorithm
Wine-bottle 2 s 30 m 43 s GA
Three-bar truss 0.25 s 3 s 3 m 17 s fmincon
Welded beam 18 s 8 m 25 s 1 hr 15 m GA
Compression spring 3 s 8 m 14 s 3 s GA
Numerical (multi-objective) 4 s 18 m 3 s 2 hr 3 m NSGA
Vibrating platform 4 s 19 m 8 s 1 hr 32 m MOGA
Speed reducer 5 s 12 m 58 s 1 hr 33 m NSGA
Power electronic module 5 s 33 m 6 s 7 hr 42 m NSGA
Numerical (feasibility) 0.22 s 2.5 s 4 s fmincon
Explosive actuated cylinder 1 s 18 s 1 m 28 s fmincon
Belleville spring 4 s 12 m 47 s N/A GA
Control valve act linkage 11 s 16 m 17 s 2 hr 28 m GA
Payload for an UAV 5 s 22 m 29 s 1 hr 52 m NSGA
 
The values in Table 7.3 confirm the data shown in Table 7.2. Overall, the ?Nominal? 
method is the fastest, requiring at maximum only 18 sec to solve the problem. This is not 
surprising since this method does not perform any additional function evaluations. The 
?Gradient? method is also very fast since the gradient information is available 
analytically. The ?Sampling? method is the slowest and very computationally extensive. 
For the power electronic module example, it took more than 7 hours to complete the 
optimization process. In contrast, the computation time of our robust optimization 
method is much less. It ranges from a few seconds to several minutes depending on the 
algorithms used to solve the inner optimization problem (fmincon and GA, respectively).  
 
7.1.3. Advantages and Disadvantages 
The main difference between our robust optimization method and the other methods 
is that our method calculates the robustness of a design in a ?reverse? mode. That is, 
 202
 
instead of calculating ?f (or ?g for constraints) for a given ?p, our method calculates ?p 
for a given ?f (or ?g). The advantage of working in this reverse mode is that the 
robustness information provided by our method does not depend on the actual ?p. Should 
the actual uncontrollable ?p change, the sensitivity region of the design will not change, 
so we can still use this information to determine the design?s robustness with respect to 
the new ?p. In contrast, for the conventional ?forward? methods, if ?p changes, then the 
design?s robustness previously calculated is no longer valid, and we will have to re-
evaluate it.  
Because our robustness calculation does not depend on the actual ?p, our method is 
independent of the probability distribution of ?p. As long as ?p falls within the 
sensitivity region of the design, the design is guaranteed to satisfy the ?f
0
 (or ?g
0
) 
requirement. If the probability distribution of ?p changes, the robustness probability of 
the design will change, but this guarantee stays the same.  
Another advantage of our method is that it does not use the gradient information of 
the objective/constraint functions. As a result, our robustness calculation is valid even if 
the ?p variations are large, beyond the linear range in which gradient estimation is valid. 
This is in contrast with those methods that use gradient calculations, such as a Taylor 
series expansion, to calculate a design?s robustness. The numerical example in Chapter 5 
(Section 5.4.1) showed how gradient-based robustness methods fail when ?p becomes 
large, while our method is still valid. Since our method does not use gradient information, 
it is also applicable to optimization problems whose objective/constraint functions are not 
differentiable everywhere with respect to ?p. This is demonstrated in the welded beam 
 203
 
example in Section 3.4.3, where the objective function of the problem is a step-function 
with respect to ?p.  
Our method is also computationally efficient. As experimentally showed, the upper 
bound for the number of function evaluations needed by our method is in the order of 10
2
 
when stochastic algorithms are necessary to solve the inner problem. Our method uses 
more F
call
 than the gradient-based methods. However, the applications of the gradient-
based methods are limited to small range of ?p variations. Our method is more efficient 
than sampling-based methods whose F
call
 is in the order of 10
3
 or more. Even for the 
more efficient sampling-based methods (such as MPP (Du and Chen, 2000)), the F
call
 is 
still in the order of 10
2
 or above. Besides, our method does not need a presumed 
probability distribution to calculate a design?s robustness. 
One shortcoming of our method is that it is conservative because it uses only the 
worst-case estimate of the sensitivity region to determine a design?s robustness. So, there 
are some ?p variations that in reality the design can absorb, but they are not included in 
the calculations. Our method also does not provide probability information regarding a 
design?s robustness. However, this is not because we cannot calculate the probability, but 
rather because we do not assume a probability distribution of the ?p.  If the pdf of ?p is 
known, we can numerically calculate a lower bound of the probability (recall 
Section 6.6). If the actual probability of the design is necessary, it may be interpolated 
experimentally. We first solve the problem for the robust optimum using several values 
of ?
0
, and then calculate the actual probability of these optima. The probability of an 
optimum for other values of ?
0
 can then be interpolated from the results (recall 
Figure 6.5).  
 204
 
7.2. CONTRIBUTIONS 
In this dissertation, we have introduced and developed several new and innovative 
concepts for robust optimization of a design. The contributions of the research presented 
in this dissertation are summarized below.  
? Introduced and developed the notion of ?reverse? robustness measure of a design 
alternative. This robustness measure does not require a presumed probability 
distribution of parameter variations. It also does not use gradient information so 
that it is valid for large variations of parameters, and is applicable to non-
differentiable objective/constraint functions.  
? Introduced and developed the concept of ?one-sided? and ?two-sided? 
sensitivity set and sensitivity region of a design alternative for single and 
multiple objective/constraint functions. The concept of an asymmetrical 
two-sided sensitivity of a design has also been introduced and developed.  
? Introduced the notion of directional sensitivity of a design, and developed 
an approach to account for it using the worst-case estimation of sensitivity 
region. 
? Developed a mathematical formulation to calculate the radius of worst-
case sensitivity region, and an approach to normalize the formulation to 
account for scale importance of parameters.  
? Developed an approach to calculate the lower bound of the robustness 
probability of a design when the probability distribution of the uncertain 
parameters is known. 
 205
 
? Developed an efficient constraint-based robust optimization method using the 
sensitivity region concept. The method is applicable to both single and multi-
objective optimization problems, and can account for both objective and 
feasibility robustness of an optimum design.  
? Introduced and developed the concept of a robustness index for a design 
alternative, which is a measure of robustness calculated based on the 
radius of worst-case sensitivity region. 
? Developed an inner-outer optimization framework to efficiently search for 
design alternatives that are optimum and robust.  
? Introduced and developed the concepts of multi-objective robustness and 
multi-objective robust optimality of a design alternative.  
 
7.3. FUTURE RESEARCH DIRECTIONS 
The robust optimization method presented in this dissertation addresses many of the 
shortcomings of previous works in robust optimization. However, there are still many 
important research issues left unresolved. In this last section we briefly discuss some of 
these issues and provide some general research directions to address them. Some of the 
discussions presented here are based on our experience during the development of this 
dissertation. Some others are based on the inputs and comments from colleagues and 
active researchers from other institutions.  
? One very important issue that has received little attention so far is in determining 
if a robust optimization is needed in the first place. We have assumed in our 
research that there is a trade-off between performance and robustness of a design. 
 206
 
However, it is not uncommon that robustness of a design increases as its 
performance increases. For a situation like this, robust optimization is not needed 
since the optimum design is already guaranteed to be also the most robust. To 
avoid wasting time and resources, we need some sort of indicators that can tell us 
from the beginning if the performance vs. robustness trade-off exists. The 
gradient of a function may be such an indicator. If the gradient of a function is 
monotonically decreasing with the function?s value, then as the function is 
minimized the gradient is minimized as well, so there is no performance-
robustness trade-off. Other inherent properties, such as the concavity or modality 
of the function, may also indicate such a trade-off.  
? One shortcoming of our robust optimization method is that it is conservative. This 
is because we have used the worst-case estimate of the sensitivity region of a 
design as a measure of the overall robustness of the design. If we can incorporate 
those portions of the sensitivity region that are not included in the worst-case 
estimate into our robustness calculations, we can obtain a more accurate 
description of the design?s robustness. An experiment-based regression analysis 
potentially can be used to numerically approximate the sensitivity region of a 
design so that we have the entire region as a robustness measure, and not just the 
worst-case region.  
? In this dissertation, we assume that parameter variations are continuous. In many 
engineering design problems, parameters such as temperature variations or 
dimensional tolerances are continuous. However, in some cases the variations 
might be discrete, and one wants to find a design alternative that is optimum and 
 207
 
robust over a range of discrete scenarios. Examples of discrete variations include 
changes in material type, or number of teeth in a gear for a power tool. 
Theoretically, the sensitivity set concept should still be applicable to the discrete 
variations case; however, the notion of a sensitivity region may no longer apply. 
The possibility that the parameter variations have both continuous and discrete 
elements should also be investigated.  
? An important issue that has not been addressed in this dissertation is the fact that 
the notion of a robust design is a subjective matter. A design that is considered 
robust by one designer may not satisfy the robustness preference or requirements 
of another designer. A design that is considered robust for each designer in a 
group may not be robust enough for the group collectively. The topic of 
preferences and decision-making is a very active area of research by itself. 
Nevertheless, if we are somehow able to incorporate some of the preference 
capturing methods into our robust optimization method, it will make the method 
more practical.  
? Practically speaking, design optimization should be fitted within an iterative and 
collaborative process with various disciplines in which the information regarding 
the design is constantly updated and improved after each iteration. The robust 
optimization method presented in this dissertation does not account for this 
collaborative model. Integration of our method with some sort of systematic 
design techniques might be beneficial to improve the applicability of the method 
to more practical and real-world problems.  
 208
 
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