Optimization For Engineers By Kalyanmoy Deb

7/21/2019 Optimization For Engineers By Kalyanmoy Deb

1/45


2/45


3/45


4/45

1 \ ' \I C L\ e

Rs 195.00

OPTIMIZATION FOR ENGINEERING DESIGN: Algorithms and Examplesby Kalyanmoy Deb

1995 by Prentice-Hall of India Private Limited New Delhi. All rights reserved.No part of this book may be reproduced in any form by mimeograph or anyother means without permission in writing from the pUblisher.ISBN81203 o943XThe export rights of this book are vested solely with the publisher.Eighth Printing July 2005

Published by Asoke K Ghosh Prentice-Hall of India Private Limited M-97Connaught Circus New Delhi110001 and Printed by Rajkamal Electric PressB-35/9 G.T. Kamal Road Industrial Area Delhi-110033.


5/45

oMy P RENTS


6/45

Contents

Preface IXAcknowledgements xiii1 Introduction 1

1.1 Optimal Problem Formulation 21.1.1 Design variables 41.1.2 Constraints 41.1.3 Objective function 61.1.4 Variable bounds 7

1.2 Engineering Optimization Problems 101.2.1 Optimal design of a truss structure 101.2.2 Optimal design of an ammonia reactor 141.2.3 Optimal design of a transit schedule 171.2.4 Optimal design of a car suspension 21

1.3 Optimization lgorithms 261.4 Summary 29References . . . . . . . . . . . 30

2 Single-variable Optimization Algorithms 312.1 Optimality Criteria . 322.2 Bracketing Methods . . . . . . . 34

2.2.1 Exhaustive search method 352.2.2 Bounding phase method 38

2.3 Region-Elimination Methods . 402.3.1 Interval halving method 41

v


7/45

VI ptimization for Engineering Design Algorithms and Examples

2.3.2 Fibonacci search method. . . 442.3.3 Golden section search method 472.4 Point-Estimation Method . . . . . . 502.4.1 Successive quadratic estimation method 50

2.5 Gradient-based Methods 542.5.1 Newton-Raphson method 542.5.2 Bisection method . . 562.5.3 Secant method 582.5.4 Cubic search method . . . 602.6 Root-finding Using Optimization Techniques 63

2.7 Summary 65References. . . . . . 66Problems 66Computer Programs 69

Bounding phase method . 69Simulation run . . . . . . 72Golden section search method 72Simulation run . . . . . . . . . 75

3 Multivariable Optimization Algorithms 773.1 Optimality Criteria 773.2 Unidirectional Search . . . . . . . . . . . 793.3 Direct Search Methods 823.3.1 Evolutionary optimization method 83

3.3.2 Simplex search method. . . . . . . 883.3.3 Hooke-Jeeves pattern search method 923.3.4 Powell's conjugate direction method 98

3.4 Gradient-based Methods 1043.4.1 Cauchy's (steepest descent) method. 1093.4.2 Newton's method . . . . . . 1113.4.3 Marquardt s method . . . . . . . . . 1163.4.4 Conjugate gradient method 1193.4.5 Variable-metric method (DFP method) . 123

3.5 Summary 128References. . ... . . . . . . . . . . . . . . . . . . . 129


8/45

ONTENTS

4.1 Kuhn-Tucker Conditions .4.2 Transformation Methods .

4.2.1 Penalty function method4.2.2 Method of multipliers. .

4.3 Sensitivity Analysis . . . . . .4.4 Direct Search for Constrained Minimization.

4.4.1 Variable elimination method.4.4.2 Complex search method4.4.3 Random search methods4.5 Linearized Search Techniques4.5.1 Frank-Wolfe method4.5.2 Cutting plane method

4.6 Feasible Direction Method4.7 Generalized Reduced Gradient Method4.8 Gradient Projection Method4.9 Summary

Simulation Run . . . . . . . . . . . . . . . . . .

5.1 Integer Programming .5.1.1 Penalty function method ..5.1.2 BranGh-and-bound method.

5.2 Geometric Programming

References .

vii

Problems .Computer Program-Steepest Descent MethodSimulation Run . . . . . . . . . . . . . . . .Constrained Optimization Algorithms

References . . . . . . . . . . . . .Problems .Computer Program-Penalty Function Method

Specialized Algorithms

5.3 Summary

130133. 141143

. 144152153162167174174178184188

189 196 208 219 228 236 239.240.246.254

257 258 259 265 273 285 286


9/45

Vlll Optimization or Engineering Design Algorithms and Examples

Nontraditional Optimization Algorithms 2 906.1 Genetic Algorithms 2906.1.1 Working principles 291

6.1.2 Differences between GAs andtraditional methods 2966.1.3 Similarities between GAs and traditional methods 3016.1.4 GAs for constrained optimization . 3126.1.5 Other GA operators 3146.1.6 Real-coded GAs. . 3166.1.7 Advanced GAs . 317

6.2 Simulated Annealing. . . 3206.3 Global Optimization . . . 325

6.3.1 Using the steepest descent method . 3266.3.2 Using genetic algorithms . . 3286.3.3 Using simulated annealing . 331

6.4 Summary . 333References . . . . . . . . . . . . . . . . 333Problems . 336Computer Program Genetic Algorithms . 339Simulation Run . . . . . . . . . . . . . . . . 356

AppendixLinear Programming Algorithms 360A.l Linear Programming Problem . 360A.2 Simplex Method . . . . . . . . . . . . . . . . . 363A.3 Artificial Variables and Dual Phase Method. . 369A.4 Summary . 373References. 374Problems . 375

Index 377


10/45

Preface

Many engineers and researchers in industries and academics facedifficulty in understanding the role of optimization in engineeringdesign. To many of them, optimization is an esoteric techniqueused only in m.sJbematks and operations research related activities.With the advent of computers, optimization has become a part ofcomputer-aided desigrractIvItles. t is primarily being used in thosedesJgn activities in whictcthe goal is not only to achieve just afeasible design, but also a design objective. n most engineeringdesign activities; the design objective could be simply to minimize thecostoLp:roduction or to maximize the efficiency of production. Ano p t i m i ~ a t i o n algorithm is a procedure which is executed iterativelyby comparing various solutions till the optimum or a satisfactorysolution is found. n many industrial design activities, optimizationis achieved indirectly by comparing a few chosen design solutionsand accepting the best solution. This simplistic approach neverguarantees an optimal solution. On the contrary, optimizationalgorithms begin with one or more design solutions supplied bythe user and then iteratively check new design solutions in orderto achieve the true optimum solution. n this book, have puttogether and discussed a few popular optimization algorithms anddemonstrated their working principles by hand-simulating on asimple example problem. Some working computer codes are alsoappended for limited use.

There are two distinct types of o g L Z 2 : t Q I l . , ~ ~ o ~ t h m s whichare in use today. First, there are algorithms vvhich are d e t e r m ~ n i s t i c , with specific rules for moving froml1fle solution to the otne:r.-Thesealgorithms have been in use for quite some time and have q ~ e n successfully applied tomany engineering design E I9l>lems ..Secondly,there are algorithms which are smtiaSfiCill nature, with probabilistictransition rules. These algorithms are comparatively n e w a : ~ r e g a i n i n i - p o p ~ l a r i t y due to certain properties which the deterministic


11/45

reface

algorithms do not have. In this book probably for the first time anattempt has been made to present both these types of algg.rit1Hns ina single volume. Because of the growing complexity in engineeringdesign p r ~ l e m s the designer can no longer afford to rely on aparticular method. The designer must know the advantages andlimitations of various methods and choose the one that is moreefficient to the problem at hand.

An important aspect of the optimal design process is theformulation of the design problem in a mathematical format whichis acceptable to an optimization algorithm. However there is nounique way of formulating every engineering design problem. Toillustrate the variations encountered in the formulation processI have presented four different engineering design problems inChapter 1. Optimization problems usually contain multiple designvariables but I have begun by first presenting a number of single-variable function optimization algorithms in Chapter 2. The workingprinciples of these algorithms are simpler and therefore easier tounderstand. Besides these algorithms are used in multivariableoptimization algorithms as unidirectional search methods. Chapter 3presents a number of algorithms for optimizing unconstrainedobjective functions having multiple variables. Chapter 4 is animportant one in that it discusses a number of algorithms forsolving constrained optimization problems-most engineering designoptimization problems are constrained. Chapter 5 deals with twospecialized algorithms for solving integer programming problems andgeometric programming problems. Two nontraditional optimizationalgorithms which are very different in principle than the abovealgorithms are covered in Chapter 6. Genetic algorithmssearch and optimization algorithms that mimic natural evolutionand genetics-are potential optimization algorithms and have beenapplied to many engineering design problems in the recent past.Due to their population approach and parallel processing thesealgorithms have been able to obtain global optimal solutions incomplex optimization problems. Simulated annealing methodmimics the cooling phenomenon ofmolten metals. Due to its inherentstochastic approach and availability of a convergence proof thistechnique has also been used in many engineering design problems.Chapter 6 also discusses the issue of finding the global optimalsolution in a multioptimal problem where the problem containsa number of local and global optimal solutions and the objectiveis to find the global optimal solution. To compare the power ofvarious algorithms one of the traditional constrained optimizationtechniques is compared with both the nontraditional optimizationtechniques in a multioptimal problem.


12/45

reface Xl

Some algorithms in Chapter 4 use linear programming methodswhich are usually taught in operations research and transportationengineering related courses. Sometimes linear programmingmethods are also taught in first or second-year undergraduateengineering courses. Thus a detailed discussion of linearprogramming methods is avoided in this book. Instead a briefanalysis of the simplex search technique of the linear programmingmethod is given in Appendix A.

The algorithms are presented in a step-by-step format so thatthey can be easily understood and coded in a computer language.The working principle of each algorithm is also illustrated by showinghand calculations up to a f w iterations of the algorithms on anumerical optimization problem. The hand calculations providea better insight into the working of the optimization algorithms.Moreover in order to compare the efficiency of different algorithmsas far as possible the same numerical example is chosen for eachalgorithm.

Most of the chapters contain at least one working computer codeimplementing optimization algorithms presented in the chapter. Thecomputer codes are written in FORTRAN programming languageand sample simulation runs obtained under the Microsoft FORTRANcompiler on a PC platform are presented. These codes are alsotested with a Unix FORTRAN compiler on a SUN machine. Theydemonstrate the ease and simplicity with which other optimizationalgorithms can also be coded. The computer codes presented inthe text can be available by sending an e-mail to the author [email protected].

The primary objective of this book is to introduce differentoptimization algorithms to students and design engineers andprovide them with a few computer codes for easy understanding.The mathematical treatment of the algorithms is. kept at a lessrigorous level so that the text can be used as an introductorybook on optimization by design engineers as well as practitionersin industries and by the undergraduate and postgraduate studentsof engineering institutions. An elementary knowledge of matrixalgebra and calculus would be sufficient for understanding most ofthe materials presented in the book.

Instructors may find this text useful in explaining optimizationalgorithms and solving numerical examples in the class althoughoccasional reference to a more theoretical treatment on optimizationmay be helpful. The best way to utilize this book is to beginwith Chapter 1. This chapter helps the reader to correlate thedesign problems to the optimization problems already discussed.
mailto:[email protected]:[email protected]


13/45

Xll reface

Thereafter subsequent chapters may be read one at a time. To havea better understanding of the algorithms the reader must follow thesteps of the solved exercise problems as they are outlined in the givenalgorithm. Then the progress of each algorithm may be understoodby referring to the accompanying figure. For better comprehensionthe reader may use the FORTRAN code given at the end of thechapters to solve the example problems.

Any comments and suggestions for improving the text would bealways welcome.

Kalyanmoy eb


14/45

Acknowledgements

The person who introduced me to the field of optimizationand who has had a significant role in moulding my career sProfessor David E. Goldberg of the University of illinois at UrbanaChampaign. On a lunch table, he once made me understandthat probably the most effective way of communicating one's ideass through books. That discussion certainly motivated me intaking up this project. The main inspiration for writing thisbook came from Professor Amitabha Ghosh, Mechanical EngineeringDepartment, IIT Kanpur, when in one tutorial class I showed himthe fifty-page handout I prepared for my postgraduate course entitled

Optimization Methods in Engineering Design . Professor Ghoshlooked at the handout and encouraged me to revise it in the form ofa textbook. Although it took me about an year-and-half to executethat revision, I have enjoyed every bit of my experience.

Most of the algorithms presented in this text are collected fromvarious books and research papers related to engineering designoptimization. My sincere thanks and appreciation are due toall authors of those books and papers. I have been particularlyinfluenced by the concise and algorithmic approach adopted in thebook entitled 'Engineering Optimization-Methods and Applications'by G. V. Reklaitis, A. Ravindran, and K. M. Ragsdell. Manyalgorithms presented here are modified abstractions from that book.I am also grateful to Professor Brahma Deo and Dr. Partha

Chakroborty for their valuable comments which significantlyimproved the contents of this book. The computer facility of theComputer Aided Design (CAD) Project,generously provided byProfessor Sanjay Dhande, s highly appreciable. My special thanksare due to two of my students N. Srinivas and Ram Bhusan Agrawalfor helping me in drawing some of the diagrams and checking someexercise problems. The computer expertise provided by P. V. M.Rao, Samir Kulkarni, and Sailesh Srivastava in preparing one of

Xlll


15/45

XIV cknowledgements

the computer codes is also appreciated. Discussions with ProfessorsDavid Blank and M. P. Kapoor on different issues of optimizationwere also helpful. I am thankful to my colleagues and staff of theCAD Project for their constant support. t would. have taken atleast twice the time to complete this book, if I did not have theprivilege to meet Dr.Subhransu Roy who generously provided mewith his text-writing and graph plotting softwares. My visits toTELCO, TISCO, Hindustan Motors and Engineers.India Ltd, andthe discussions had with many design engineers were valuable inwriting some of the chapters. The financial assistance providedby the Continuing Education Centre at the Indian Institute ofTechnology Kanpur to partially compensate for the preparation ofthe manuscript is gratefully acknowledged. I also wish to thank thePublishers Prentice-Hall of India for the meticulous care they tookin processing the book.

This book could not have been complete without the lovingsupport and encouragement of my wife Debjani. Her help in typinga significant portion of the manuscript, in proof-reading and inpreparing the diagrams has always kept me on schedule. FinallyI take this opportunity to express my gratitude to my parents andmy loving affection to my brothers.

Indian Institute of Technology Kalyanmoy ebKanpur


16/45

Introduction

Optimization algorithms are becoming increasingly .popular inengineering design activities, primarily because of the availabilityand affordability of high speed computers. They are extensivelyused in those engineering design problems where the emphasisis on maximizing or minimizing a certain goal. For example,optimization algorithms are routinely used in aerospace designactivities to minimize the overall weight, simply because everyelement or component adds to the overall weight of the aircraft.Thus, the minimization of the weight of aircraft components is ofmajor concern to aerospace designers. Chemical engineers, on theother. hand, are interested in designing and operating a processplant for an optimum rate of production. Mechanical engineersdesign mechanical components for the purpose of achieving eithera minimum manufacturing cost or a maximum component life.Production engineers are interested in designing optimum schedulesof various machining operations to minimize the idle time of machinesand the overall job completion time. Civil engineers are involvedin designing buildings, bridges, dams, and other structures inorder to achieve a minimum overall cost or maximum safety orboth. Electrical engineers are interested in designing communicationnetworks so as to achieve minimum time for communication from onenode to another.

All the above-mentioned tasks involve either minimization ormaximization (collectively known as optimization of an objective. tis clear from the spectrum of the above problems that it s difficult todiscuss the formu lation of various engineering optimization problemsin a single book. Fortunately, a designer spr.cializE'd in a particular

1


17/45

2 Optimization for Engineering Design lgorithms and Examples

design is usually more informed about different factors governingthat design than anyone else. Thus as far as the formulationof the optimal problem is concerned, the designer can acquire itwith some practice. However, every designer should know a fewaspects of the formulation procedure which would help him or her tochoose a proper optimization algorithm for the chosen optimal designproblem. This requires a knowledge about the working principles ofdifferent optimization methods. n subsequent chapters, w discussvarious optimization methods which would hopefully provide some ofthat knowledge. In this chapter, we demonstrate the optimal problemformulation procedures of four different engineering optimal designproblems.1 1 Optimal Problem ormulationIn many industrial design activities, a naive optimal design isachieved by comparing a few limited up to ten or so) alternativedesign solutions created by using a priori problem knowledge. In suchan activity, the feasibility of each design solution is first investigated.Thereafter an estimate ofthe underlying objective cost, profit, etc.)of each design solution is computed and the best design solution isadopted. This naive method is often followed because of the timeand resource limitations. But in many cases this method is followedsimply because of the lack of knowledge of the existing optimizationprocedures. But whatever may be the reason, the purpose of eitherachieving a quality product or of achieving a competing productis never guaranteed to be fulfilled with the above naive method.Optimization algorithms described in this book provide systematicand efficient ways of creating and comparing new design solutionsin order to achieve an optimal design. Since an optimizationalgorithm requires comparison of a number of design solutions, it isusually time-consuming and computationally expensive. Thus theoptimization procedure must only be used in those problems wherethere is a definite need of achieving a quality product or a competitiveproduct. t is expected that the design solution obtained through anoptimization procedure is better than other solutions in terms of thechosen objective-cost efficiency, safety, or others.

We begin our discussion with the formulation procedure bymentioning that it is almost impossible to apply a single formulationprocedure for all engineering design problems. Since the objectivein a design problem and the associated design parameters vary fromproduct to product different techniques need to be used in differentproblems. The purpose of the formulation procedure is to create a


18/45

3ntroduction

mathematical model of the optimal design problem, which then canbe solved using an optimization algorithm. Since an optimizationalgorithm accepts an optimization problem in a particular format,every optimal design problem must be formulated in that format. Inthis section, w discuss different components of that format.Figure 1.1 shows an outline of the steps usually involved in an

optimal design formulation process. As mentioned earlier, the first

Need for optimization1

Choose design variables1,. Formulate constraints14

L .. . Formulate objective function1

, . - - - Set up variable bounds1

Choose an optimization algorithm-1

Obtain solution(s)Figure 1 1 A flowchart of the optimal design procedure.

step is to realize the need for using optimization in a specific designproblem. Thereafter, the designer needs to choose the importantdesign variables associated with the design problem. The formulationof optimal design problems involves other considerations, such asconstraints, objective function, and variable bounds. As shown inthe figure, there is usually a hierarchy in the optimal design process;although one consideration may get influenced by the other Wediscuss all these aspects in the following subsections.


19/45

Optimization for Engineering Design: lgorithms and Examples

1 1 1 Design variablesThe formulation of an optimization problem begins with identifyingthe underlying design variables which are primarily varied duringthe optimization process. A design problem -usually involves manydesign parameters, of which some are highly sensitive to the properworking of the design. These parameters are called design variables inthe parlance of optimization procedures. Other not so important)design parameters usually remain fixed or vary in relation to thedesign variables. There is no rigid guideline to choose a priorithe parameters which may be important in a problem because oneparameter may be more important with respect to minimizing theoverall cost of the design while it may be insignificant with respect tomaximizing the life of the product. Thus, the choice of the importantparameters in an optimization problem largely depends on the user.However it is important to understand that the efficiency and speedof optimization algorithms depend to a large extent, on the numberof chosen design variables. n subsequent chapters, w shall discusscertain algorithms which work very efficiently when the number ofdesign variables is small but do not work that well for a largenumber of design variables. Thus, by selectively choosing the designvariables the efficacy of the optimization process can be increased.The first thumb rule of the formulation of an optimization problemis to choose as few design variables as possible. The outcome ofthat optimization procedure may indicate whether to include moredesign variables in a revised formulation or to replace some previouslyconsidered design variables with new design variables.1 1 2 ConstraintsHaving chosen the design variables the next task is to identifythe constraints associated with the optimization problem. Theconstraints represent some functional relationships among the designvariables and other design parameters satisfying certain physicalphenomenon and certain resource limitations. Some of theseconsiderations require that the design remain in static or dynamicequilibrium. n many mechanical and civil engineering problems theconstraints are formulated to satisfy stress and deflection limitations.Often a component needs to be designed in such a way that itcan be placed inside a fixed housing thereby restricting the sizeof the component. There is however no unique way to formulatea constraint in all problems. The nature and number of constraintsto be included in the formulation depend on the user. n manyalgorithms discussed in this book, it is not necessary to have an


20/45

ntroduction

explicit mathematical expression of a constraint; but an algorithm ora mechanism to calculate the constraint is mandatory. For example,a mechanical engineering component design problem may involve aconstraint to restrain the maximum stress developed anywhere in thecomponent to the strength of the material. In an irregular-shapedcomponent, there may not exist an exact mathematical expression forthe maximum stress developed in the component. A finite elementsimulation software may be necessary to compute the maximumstress. But the simulation procedure and the necessary input to thesimulator and the output from the simulator must be understood atthis step.

There are usually two types of constraints that emerge from mostconsiderations. Either the constraints are of an inequality type or ofan equality type. Inequality constraints state that the functionalrelationships among design variables are either greater than, smallerthan, or equal to, a resource value. For example, the stress a x))developed anywhere in a component must be smaller than or equal tothe allowable strength (Sallowable) of the material. Mathematically,

a x) ::; SallowableMost of the constraints encountered in engineering design problemsare of this type. Some constraints may be of greater-than-equalto type: for example, the natural frequency v x)) of a systemmay required to be be greater than 2 Hz, or mathematically,v x) 2 Fortunately, one type of inequality constraints can betransformed into the other type by multiplying both sides by 1or by interchanging the left and right sides. For example, the formerconstraint can be transformed into a greater-than-equal-to type byeither -a x) -Sallowable or Sallowable a x).

Equality constraints state that the functional relationships shouldexactly match a resource value. For example, a constraint mayrequire that the deflection 8 x)) of a point in the component mustbe exactly equal to 5 mm, or mathematically,8 x)=5.

Equality constraints are usually more difficult to handle and,therefore, need to be avoided wherever possible. f the functionalrelationships of equality constraints are simpler, it may be possibleto reduce the number of design variables by using the equalityconstraints. In such a case, the equality constraints reducethe complexity of the problem, thereby making it easier for theoptimization algorithms to solve the problem. In Chapter 4, we


21/45

Optimization for Engineering Design: lgorithms and Examples

discuss a number of algorithms which are specially designed tohandle equality constraints. Fortunately, in many engineering designoptimization problems, it may be possible to relax an equalityconstraint by including two inequality constraints. The abovedeflection equality constraint can be replaced by two constraints:

< 5 x ~ 4 ,


22/45

7ntroduction

design problem. The designer may be interested in minimizing theoverall weight of the structure and simultaneously be concerned inminimizing the deflection of a specific point in the truss. n theoptimal problem formulation, the designer may like to use the weightof the truss as a function of the cross-sections of the members) asthe objective function and have a constraint with the deflection ofthe concerned point to be less than a specified limit. n general, theobjective function is not required to be expressed in a mathematicalform. A simulation package may be required to evaluate the objectivefunction. But whatever may be the way to evaluate the objectivefunction, it must be clearly understood.The objective function can be of two types. Either theobjective function is to be maximized or it has to be minimized.Unfortunately, the optimization algorithms are usually written eitherfor minimization problems or for maximization problems. Althoughin some algorithms, some minor structural changes would enable toperform either minimization or maximization, this requires extensiveknowledge of the algorithm. Moreover, if an optimization software isused for the simulation, the modified software needs to be compiledbefore it can be used for the simulation. Fortunately, the dualityprinciple helps by allowing the same algorithm to be used forminimization or maximization with a minor change in the objectivefunction instead of a change in the entire algorithm. the algorithmis developed for solving a minimization problem, it can also beused to solve a maximization problem by simply multiplying theobjective function by 1 and vice versa. For example, considerthe maximization of the single-variable function f x) = x 2 1 x)shown by a solid line in Figure 1.2. The maximum point happensto be at x* = 0.667. The duality principle suggests that the aboveproblem is equivalent to minimizing the function F x) = 2 1 x),which is shown by a dashed line in Figure 1.2. The figure showsthat the minimum point of the function F x) is also at x = 0.667.Thus the optimum solution remains the same. But once w obtainthe optimum solution by minimizing the function F x , we need tocalculate the optimal function value of the original function f x bymultiplying F x by -1.1 1 4 Variable bounds

The final task of the formulation procedure is to set the minimum andthe maximum bounds on each design variable. Certain optimizationalgorithms do not require this information. n these problems, theconstraints completely surround the feasible region. Other problems


23/45

8 Optimization for Engineering Design: Algorithms and Examples

f(x)0.3

0.2

0.1: x*0.0 1 I o.........--I.o......l.. ............ L...-..I.o.-I..-.l--Io.......,I-'-......l..............L...-........ X

0.25 0.50 0.75 1.00-0.1 , ,;-0.2 , ;-- _ - --'-0 .3 F x)

Figure 1.2 Illustration of the duality principle. The maximumpoint of f x) is the same s the minimum point of F(x).require this information in order to confine the search algorithm

within these bounds. In general, all N design variables are restrictedto lie within the minimum and the maximum bounds as follows:

i x ~ U for i = 1,2, . . . ,N.In any given problem, the determination of the variables bounds x ~and x ~ U may be difficult. One way to remedy this situation is tomake a guess about the optimal solution and set the minimum andmaximum bounds so that the optimal solution lies within these twobounds. After simulating the optimization algorithm once, i theoptimal solution is found to lie within the chosen variable bounds,there is no problem. On the other hand, if any design variablecorresponding to the optimal solution is found to lie on or near theminimum or the maximum bound, the chosen bound may not becorrect. The chosen bound may be readjusted and the optimizationalgorithm may be simulated again. Although this strategy may seemto work only with linear problems, it has been found useful in manyreal-world engineering optimization problems.

After the above four tasks are completed, the optimizationproblem can be mathematically written in a special format, known asnon inear program" ting (NLP) format. Denoting the design variables


24/45

9ntroduction

as a column vector x = Xl, X2, , xN )T, the objective function asa scalar quantity f x), J inequality constraints as 9j x) ? 0, and Kequality constraints as hk x = 0, we write the NLP problem:

Minimize f x)subject to

9j X) ? 0, j = 1,2, ... ,J; (1.1 )hk X) =0 k=1,2, ... ,K;< x < xW)t t t i =1 2 . . . ,N.

Note that the above formulation can represent a formulationfor maximization problems by using the duality principle and canrepresent a formulation for problems with the lesser-than-equalto type inequality constraints by using the techniques describedearlier. However, the optimization algorithm used to solve the aboveNLP problem depends on the type of the objective function andconstraints. t is important to note that the constraints must bewritten in a way so that the right-side of the inequality or equalitysign is zero.

t is worth mentioning here that all the above four tasks are notindependent of each other. While formulating the objective function,the designer may decide to include or delete some constraints. Inmany problems, while the constraints are being formulated, it isnecessary to add some artificial design variables, which make theoverall formulation easier. The update of the design variables, theconstraints, the objective function, and the variable bounds maycontinue for a few iterations until the designer is finally satisfiedwith a reasonable formulation. Certain possible iterations areshown in Figure 1.1. We may mention here that this update alsodepends on the knowledge of the optimization algorithms to beused to solve the problem. But this requires some practice of usingoptimization algorithms before such input may be incorporated intothe formulat ion procedure. Nevertheless, after the optimizationproblem is formulated, an optimization algorithm is chosen and anoptimal solution of the NLP problem is obtained. We now illustratethe above four steps of the formulation procedure in four engineeringoptimal design problems.

lThe representation of the design variables in the above column vector helpsto achieve some matrix operations in certain multivariable optimization methodsdescribed in Chapters 3 and 4


25/45


1 2 Engineering Optimization ProblemsBecause of the variety of engineering design problems it is notpossible to discuss the formulation of various optimization problemsthat are usually encountered in engineering design. To illustratethe formulation procedure w formulate four different optimizationproblems. The first is a truss structure design problem the secondis a chemical reactor design problem the third problem is aninteresting transport scheduling problem and the fourth probleminvolves the optimal design of a car suspension system. Theformulation procedures show different considerations often adoptedin formulating engineering optimization problems.

1 2 1 Optimal design of a truss structureA truss structure is used in many civil engineering applicationsincluding bridges buildings and roofs. There are two different typesof optimization problems in a truss structure design. Firstly thetopology of the truss structure the connectivity of the elements ina truss) could be an optimization problem. In this problem theobjective is to find the optimal connectivity of truss elements so asto achieve the minimal cost of materials and construction. Secondlyonce the optimal layout of the truss is known the determinationof every element cross-section is another optimization problem. Inthis problem the objective is to find the optimal cross-sections ofall elements in order to achieve a minimum cost of materials andconstruction. Although both these problems attempt to achieve thesame objective the search space and the optimization algorithmrequired to solve each problem are different. Here w discuss thelatter problem only. However there exist certain algorithms whichcan be used to solve both the above problems simultaneously. Wediscuss more about these algorithms in Chapter 6.

Consider the seven-bar truss structure shown in Figure 1.3. Theloading is also shown in the figure. The length of the membersAC CE i 1 m. Once the connectivity of the truss is giventhe cross-sectional area and the material properties of the membersare the design parameters. Let us choose the cross-sectionalarea of members as the design variables for this problem. Thereare seven design variables each specifying the cross-section of amember AI to 7 . Using the symmetry of the truss structure andloading w observe that for the optimal solution 7 AI, 6 2 and As A3 Thus, there ~ practically four design variablesAI to 4 . This completes the first task of the optimization


26/45

ntroduction

procedure.

P= k NFigure 1 3 A typical seven-bar truss structure.

The next task is to formulate the constraints. In order for thetruss to carry the given load P = 2 kN, the tensile and compressivestress generated in each member must not be more than thecorresponding allowable strength Byt and Byc of the material. Letus assume that the material strength for all elements is Byt = Byc= 500 MPa and the modulus of elasticity E = 200 GPa For thegiven load, we can compute the axial force generated in each elementTable 1.1). The positive force signifies tensile load and the negativeforce signifies compressive load acting on the member. Thereafter,Table 1 1 Axial Force in Each Member of the TrussMember Force Member ForceAB ~ csc BC ~ csc aAC ~ cot BD ~ c o t + cot a

the axial stress can be calculated by dividing the axial load by thecross-sectional area of that member. Thus, the first set of constraintscan be written as

PcscO < BA - ycPcot < BA - y2


27/45


Pcsco: < S2A - yt3P2A (cot 0 +cot 0: ::; Sye4

n the above truss structure, tan 0 = 1.0 and tan 0: = 2/3. Theother set of constraints arises from the stability consideration of thecompression members AB BD and DE. Realizing that each of thesemembers is connected by pin joints, we can write the Euler bucklingconditions for the axial load in members AB and BD (Shigley, 1986):

P r E ~ < - - -= 2 sin 0 - 1.2812P rEA22(cot 0+ cot 0: ::; 7 6 i ~

n most structures, deflection is a major consideration. n the abovetruss structure, let us assume that the maximum vertical deflectionat C is omax = 2 mm. By using Castigliano s theorem (Timoshenko,1986), we obtain the deflection constraint:

Pi 0.566 0.500 2.236 2.700) 0E ~ ~ ~ ; maxAll the above constraints are of less-than-equal-to type. Once theconstraints are formulated, the next task is to formulate the objectivefunction. n this problem, we are interested in minimizing the weightof the truss structure. Since we have assumed the same materialfor all members, the minimization of the total volume of materialwill yield the same optimal solution as the minimization of the totalweight. Thus, we write the objective function as

The fourth task is to set some lower and upper bounds for thefour cross-sectional areas. We may choose to make all four areas lie2between 10 and 500 mm . Thus, the variable bounds are as


28/45

13ntroduction

In the following, we present the above truss structure problemin NLP form, which is suitable for solving by using an optimizationalgorithm described in Chapter 4:

subject toPyc - AIsm J :: 0,P >yt - A cot J PSyt - 2 A :: 0,3 sm o

Pyc - A (cot J + cot 0: :: 0,41 l E A ~ P= : : - >.2812 2 sin J -

1l EA2 P5 7 6 i ~ - 2 (cot(J+ cot 0: :: 0,6 _ i (0.566 0.500 2.236 2.700) >

max E Al + A2 + A3 + A 410 X 10-6 ::; All A 2 , A3' A4 ::; 500 X 10-6

This shows the formulation of the truss structure problem. Theseven-bar truss shown in Figure 1.3 is statically determinate and theaxial force, stress, and deflection were possible to compute exactly.n cases where the truss is statically indeterminate and large (forhand calculations), the exact computations of stress and deflectionmay not be possible. A finite element software may be necessary tocompute the stress and deflection in any member and at any point in

the truss. Although similar constraints can then be formulated withthe simulated stresses and deflections, the optimization algorithmwhich may be used to solve the above seven-bar truss problem maynot be efficient to solve the resulting NLP problem for staticallyindeterminate or large truss problems. The difficulty arises due tothe inability to compute the gradients of the constraints. We shalldiscuss more about this aspect in Chapter 4.


29/45

14 Optimization for Engineering Design Algorithms and Examples

1 2 2 Optimal design of an ammonia reactorn the ammonia reactor design problem, feed gas containing nitrogen,hydrogen, methane argon, and a small percentage of ammonia enters

the bottom of the reactor Figure 1.4). Thereafter, the feed gas rises

Catalyst

Bottom

Product Gas, NH3Figure 1 4 A typical ammonia synthesis reactor.

till it reaches the top of the reactor. Then, while moving downward,the nitrogen and hydrogen present in the feed gas undergo reactionto form ammonia in the presence of a catalyst placed in the reactor.

The production of ammonia depends on the temperature ofthe feed gas, the temperature at the top of the reactor, thepartial pressures of the reactants nitrogen and hydrogen), and thereactor length. The optimal design problem requires to achieveof the optimal reactor length yielding maximum economic returnsprofits) from the reactor operation corresponding to various toptemperatures.

In this problem, w identify four design variables-the reactorlength x molar flow rate of nitrogen per unit catalyst area NN2 thefeed gas temperature Tj and the reacting gas temperature T Inorder to maintain the energy balance of reactions in the reactor, threecoupled differential equations must be satisfied Murase, Roberts,and Converse, 1970; Upreti and Deb, 1994). First, the decrease inthe feed gas temperature must be according to the heat loss to thereaction gas:


30/45

15ntroduction

dTf USIdx = - WCpf Tg - Tf) (1.2)In Equation 1.2), U is the overall heat transfer coefficient, SI is

the surface area of the catalyst tubes per unit reactor length, W isthe total mass flow rate and Cpf is the specific heat capacity of thefeed gas. Secondly, the change in the reaction gas temperature mustbe according to the heat gain from the feed gas and heat generatedin the reaction:

dTg = _ USI Tg _ Tf) -bJI)S2 fadx WCpg WCpgK 1.5PN2PH2 K PNHa )X 1 - , 1.3)PNHa 1.5pH2

where the parameters K l and K are as follows:K = 1.78954 X 1 4 exp [-20800j RTg)],K = 2.5714 X 1 16 exp [-47400j RTg)].

The parameter S denotes the cross-sectional area of the catalyst.zone, AH is the heat of reaction, Cpg is the specific heat capacity ofthe reacting gas, fa is the catalyst activity, and R is the ideal gasconstant. The parameters PN2 PH2 and PNHa are the partial pressuresof nitrogen, hydrogen, and ammonia, respectively. Thirdly, the massbalance of nitrogen yields

dNN2 = _ fa K 1.5PN2PH2 _ K PNHa . 1.4)ldx PNHa 1.5PH2The partial pressures in the above differential equations are

computed as286NN2PN2 2.598NN2 2NN2

PH2 = 3PN2286(2.23NN2 - 2NN2 )PNHa =


31/45


Cpf = 0.707 kcalj(kg K), 81 = 10 m,Cp = 0.719kcalj kgK), 82 = 0.78 m.fj,H = -26,600 kcaljkmol nitrogen, R = 1.987 kcalj(kmol K),

U = 500 kcalj(m2 h K), W = 26,400 kgjh,fa = 1.Note that all the above three differential equations (1.2)-(1.4) arecoupled to each other. In order to solve these equations, we use thefollowing boundary conditions:

Tf(X = 0) = To,Tg x = 0) = 694 K,

NN2(X = 0) = 701.2 kmolj m2 h).The three constraints (Equations (1.2) to (1.4)) can be usedto eliminate three of the four design variables and express theoptimization problem in terms of only one design variable. We choose

to keep the reactor length X) as the design variable.The objective of the reactor design problem is to achieve as muchprofit as possible in the production of ammonia. Considering the netvalue of ammonia production as the difference between the valueof the product gas (heating value and the ammonia value) and thevalue of feed gas (as a source of heat only) less the amortisation ofreactor capital costs, we formulate the profit function as the objectivefunction (in jyear):f(x,NN2,Tj,Tg) = 1.33563 x 107 -1.70843 X 104NN

+704.09 Tg - To) - 699.27 Tf - To)-[3.45663 X 107 1.98365 X 109xP/2 1.5)

Although the above function depends on four variables, three?ariables (NN2 Tf and Tg) are computed from equality constraints.As in the previous example, this problem also requires numericalsolution of coupled differential equations. Thus, the NLP problem isas follows:


32/45

17ntroduction

subject to

1 2 3 Optimal design of a transit scheduleFigure 1.5 shows a typical transit system network. The solid lines

Station B

igure 1 5 A typical transit system network.represent different routes, the points on the lines represent the stopsand the circled intersections of the routes represent the transferstations. The problem is to determine schedules for the routes suchthat the transit system provides the best level of service LOS)to its passengers, within the resources available. One of the goodmeasures of the LOS is the amount of time -passengers wait duringtheir )ou:rne-y-tn.e lesser tb.e waiting time, the better is the LOSChakroborty, et al., 1994). On any transit network, passengerswait either to board the vehicle at the station of origin or theywait at a transfer station at which they transfer from one vehicle

to another. For example, a passenger wishing to travel from station


33/45


A to station B in the network shown in Figure 1.5) will have to waitat station A to board a vehicle on Route 1. Further, the passengerwill have to wait at transfer station C to board a vehicle on Route3 which will take him/her to the destination). We will refer to thewait at station A as the initial wait time IWT) and the wait atstation C as the transfer time TT). A good schedule is one whichminimizes the sum of IWT and TT for ll passengers. Thus, theoptimization problem involves finding a schedule of vehicles on allroutes arrival and departure times) such that the total waiting timefor the passengers is minimum.

The design variables in this problem are the arrival time af anddeparture time df for the k-th vehicle at i-th route. Thus, if in aproblem, there are a total of M routes and each route has K vehicles,the total number of design variables is 2MK In addition, there area few more artificial variables which we shall discuss later.

The constraints in this problem appear from different servicerelated limitations. Some of these constraints are formulated in thefollowing:Minimum stopping time A vehicle cannot start as soon as itstops; it has to wait at the stop for a certain period of time, or

d7 - a7) : Smin for all i and kMaximum stopping time A vehicle cannot stop for more thana certain period of time even if it means increasing the total transfertime on the network, or

d7 - a7) ; Smax for all i and kMaximum allowable transfer time No passenger on the transitnetwork should have to wait more than a certain period of time T atany transfer station. This can be enforced by checking all possibledifferences between departure and arrival times and limiting thosevalues to T Mathematically, this is difficult to achieve. We simplify the formulation of this constraint by introducing a new set of

variables tl between the k-th vehicle of the i-th route and the l-thvehicle of the j-th route. These variables can take either a zero or aone. A value of zero means that the transfer of passengers betweenthose two vehicles is not feasible. A value of one means otherwise.Consider the arrival and departure times of vehicles in two differentroutes at a particular station, as shown in Figure 1.6. The parame


34/45

19

~ ~ ime

IntroductionIea.

i th route/

//

J th route1 1 d ~ la.:J :J

Figure 1 6 Transfers from k-th vehicle on the i-route to threeconsecutive vehicles in the j-th route.

ters af and df are the arrival and departure times of the k-th vehiclein the i-th route. A passenger from the k-th vehicle in the i-th routecan only transfer to a vehicle in the j-th route, which is arriving atthe station after af According to the figure, the transfer of a passenger from the k-th vehicle in the i-th route is not possible to thel-1)-th vehicle in the j-th route, because the departure time of thelatter vehicle d ~ - l is earlier than af Thus, the parameter ~ t l

k Itakes a value zero, whereas the parameter i ,j takes a value one. norder to simplify the model, we assume that transfers to vehiclesdeparting after l th vehicle in the j-th route are also not possible.All parameters 6U for q = 1 + 1 , 1 + 2 , . . . are also zero. Thus,between any two vehicles, the following condition must be satisfied:

- a7 6tf o T for all i, j k and 1t is clear that the left side expression of the above condition is zerofor those transfers that are not feasible. Since transfers only to the

next available vehicle are assumed, only one 6tf for l = 1,2, . . .is one and the rest all are zeros for fixed values of i, j and kMathematically,

I: = 1 for all i, j and kI ,JThe introduction of the artificial variables 6: f makes the formulationeasier, but causes a difficulty. Many optimization algorithms cannothandle discrete design variables efficiently. Since the artificial designvariables 6: 1 can only take a value zero or one, another set of


35/45


constraints is added to enforce the binary values:k 8ki - a ) M l - , l > 0 for a i, J , k and 1J t t,J -

where M is a large positive number. The above constraint ensuresthat the variable 8: ;1 always takes a value one whenever a transferis possible and the value zero whenever transfer is not possible.This constraint is derived purely from the knowledge of the availableoptimization algorithms. There may be other ways to formulate theconcept of feasible transfers, but inclusion of such artjfidal designvariables often make the understanding of the problem easier.

Maximum headway: The headway between two consecutivevehicles should be less than or equal to the policy headway, hi, ora ~ - a k ) < h for a i and k. t _ 1

The objective function consists of two terms: the first termrepresents the total transfer time TT) over a the passengers andthe second term represents the initial waiting time IWT) for a thepassengers. The objective is to minimize the foUowing function:

The parameter Wfj is the number of passengers transferring fromthe k-th vehicle of the i-th route to the j-th route. The first term isobtained by summing the individual transfer time - af) over apassengers for a the vehicles for every pair of routes. The parameterVi,k t) is the number of passengers arriving at the stop for the k-thvehicle in the i-th route at a given time t. Since the arrival time forpassengers can be anywhere between t = 0 to t = af - a7- 1 ) theheadway , the initial waiting time also differs from one passenger toanother. For example, a passenger arriving at the stop just afterthe previous vehicle has left has to wait for the fu headway timeaf - a7- 1 ) before the next vehicle arrives. On the other hand, apassenger arriving at the stop later has to wait for a shorter time.The calculation of the second term assumes that passengers arrive atthe stop during the time interval a7- 1 to af according to the known


36/45

21ntroduction

time-varying function Vi,k t), where t is measured from af l. Thenthe quantity

gives the sum of the initial waiting times for all passengers who boardthe k-th vehicle of the i-th route. We then sum it over all the routesand vehicles to estimate the network total of the W ~ Thus thecomplete NLP problem can be written as follows:

Minimize

subject to

max - df - af) 0 for all i and k,(d F - a F) - S lDJD > 0 for all i and k,_T - d l - a F ) b ~ , 1 > 0 for all i, J., k and l,J J -

- af) M l - btj) 0 for all i, j k and l,h - a k+1 - a F) > 0 for all i and k,. . .L b ~ , l = 1 for all i, j and k.I ,J

In the above NLP problem, the variables btj are binary variablestaking only a value zero or a one and other variables f and df arereal-valued. Thus a mixed integer programming technique describedin Chapter 5 or genetic algorithms described in Chapter 6 can be usedto solve the above NLP problem Chakroborty, et al., 1994).1 2 4 Optimal designof car suspensionThe comfort in riding a car largely depends on the suspensioncharacteristics. The car body is usually supported by a suspensioncoil spring and a damper at each wheel Figure 1.7). n some cars,the, axle assembly is directly supported on the wheel. The tyre of


37/45

22 Optimization or Engineering Design Algorithms and Examples

Figure 1.7 A two-dimensional model of a car suspension system.the wheel can also be assumed to have some stiffness in the verticaldirection. A two-dimensional dynamic model of a car suspensionsystem is shown in Figure 1.8. n this model, only two wheels oneeach at rear and front) are considered. The sprung mass of the car isconsidered to be supported on two axles front and rear) by means ofa suspension coil spring and a shock absorber damper). Each axlecontains some unsprung mass which is supported by the tyre.

In order to formulate the optimal design problem, the first taskz

Rear Front

Figure 1.8 The dynamic model of the car suspension system. Theabove model has four degrees-of-freedom ql to Q4 .


38/45

23ntroduction

is to identify the important design variables. Let us first identify allthe design parameters that could govern the dynamic behaviour ofthe car vibration. In the following, we list all these parameters:

Sprung; mass m s , Front coil stiffness k j s,Front unsprung mass mju, Rear coil stiffness k rsRear unsprung mass m ru Front tyre stiffness k j tRear damper coefficient Or, Rear tyre stiffness k r tFront damper coefficient OJ Axle-to-axle distance ,Polar moment of inertia of the car J.

We may consider all the above parameters as design variables, butthe time taken for convergence of the optimization algorithm may betoo much. In order to simplify the formulation, we consider only fourof the above parameters-front coil stiffness k j s, rear coil stiffnessk rs front damper coefficient oj and rear damper coefficient or a sdesign variables. We keep the other design parameters as constant:

1000 kg, mju = 70 kg, m ru = 150 kg,20 kg/mm k r t = 20 kg/mm J = 550 kg-m2 ,

= 3.2 m.The parameters and 2 are the horizontal distance of the front andrear axle from the centre of gravity of the sprung mass. Using theseparameters differential equations governing the vertical motion ofthe unsprung mass at the front axle (ql), the sprung mass Q2), andthe unsprung mass at the rear axle Q4), and the angular motion ofthe sprung mass Q3) are written Figure 1.8):

ql = {F2 F3 - F1 )/mju, 1.6)q2 = - F2 F3 F4 Fs)/m s , 1.7)q3 = [(F4 FS2 - F2 F3)I] / J, 1.8)q4 = F4 Fs - F6 /mru 1.9)


39/45


where the forces F1 to F6 are calculated as follows:F1 = k t d1 , F2 = k s d2 , F3 = f d2 ,F4 = krs d4 , Fs = r d4 , F6 = kr td3

The parameters d1 , d2 , d3 , and d4 are the relative deformations inthe front tyre, the front spring, the rear tyre, and the rear spring,respectively. Figure 1.8 shows all the four degrees-of-freedom of theabove system q1 to Q4). The relative deformations in springs andtyres can be written as follows:d1 = q1 - h t),d2 = q q3 - ql,d3 = q4 - 2(t),d4 = q - q3 - q4

The time-varying functions h t) and h t) are road irregularitiesas functions of time. Any function can be used for h t). Forexample, a bump can be modelled s h t) = Asin(1rt/T), whereA is the amplitude of the bump and T the time required to cross thebump. When a car is moving forward, the front wheel experiences.the bump first, while the rear wheel experiences the same bumpa little later, depending upon the speed of the car. Thus, thefunction h t) can be written as f2(t) = f1(t - /v), where isthe axle-to-axle distance and v is the speed of the car. For theabove bump, h t) = Asin(1r(t - /v)/T). The coupled differentialequations specified in Equations 1.6) to 1.9) can be solved using anumerical integration technique for example, a fourth-order RungeKutta method can be used) to obtain the pitching and bouncingdynamics of the sprung mass m s Equations can be integrated for atime range from zero to tmax .

After the design variables are chosen, the next task is to formulatethe constraints associated with the above car suspension problem.n order to simplify the problem, we consider only one constraint.The jerk the rate of change of the vertical acceleration of the

sprung mass) is a major factor concerning the comfort of the ridingpassengers. The guideline used in car industries suggests that themaximum jerk experienced by the passengers should not be morethan about 18 mls3 . Mathematically,

max qz (t) ::; 18.


40/45

5ntroduction

When the four coupled differential equations (1.6) to (1.9) are solved,the above constraint can be computed by numerically differentiatingthe vertical movement of the sprung mass q2) thrice with respect totime.

The next task is to formulate the objective function. n thisproblem, the primary objective is to minimize the transimissibilityfactor which is calculated as the ratio of the bouncing amplitudeq2 t) of the sprung mass to the road excitation amplitude A. Thus,we write the objective function as

max abs q2 t)Minimize AThe above objective function can be calculated from the solution ofthe four differential equations mentioned earlier. A minimum valueof the transmissibility factor suggests the minimum transmission ofroad vibration to the passengers. This factor is also directly relatedto the ri e characteristics as specified by the SO standard. Thus, theoptimized design of the above car suspension system would providethe minimum transmissibility of the road vibration to the passengers with a limited level of jerk.

Finally, a minimum and maximum limit for each design variablecan be set. This may require some previous experience with a carsuspension design, but the following limits for the above car mayinclude the optimal solution:

o ; a j a ; 300 kgj mjs).Thus, the above optimal car suspension design problem can be

written in NLP form as follows:max abs q2 t)Minimize A

subject to18 - max II t)

o ; a j a ; 300.


41/45

26 Optimization for Engineering Design: lgorithms and Examples

1 3 Optimization AlgorithmsThe above optimization problems reveal the fact that the formulationof engineering design problems could differ from problem to problem.Certain problems involve linear terms for constraints and objectivefunction but certain other problems involve nonlinear terms for them.In some problems, the terms are not explicit functions of the designvariables. Unfortunately, there does not exist a single optimizationalgorithm which will work in all optimization problems equallyefficiently. Some algorithms perform better on one problem, but mayperform poorly on other problems. That is why the optimizationliterature contains a large number of algorithms, each suitable tosolve a particular type of problem. The choice of a suitable algorithmfor an optimization problem is, to a large extent dependent on theuser s experience in solving similar problems. This book providesa number of optimization algorithms used in engineering designactivities.

Since the optimization algorithms involve repetitive applicationof certain procedures, they need to be used with the help of acomputer. That s why the algorithms are presented in a step-bystep format so that they can be easily coded. To demonstrate theease of conversion of the given algorithms into computer codes, mostchapters contain a representative working computer code. Further inorder to give a clear understanding of the working of the algorithms,they are hand-simulated on numerical exercise problems. Simulationsare performed for two to three iterations following the steps outlinedin the algorithm sequentially. Thus for example, when the algorithmsuggests to move from Step 5 to Step 2 in order to carry outa conditional statement the exercise problem demonstrates thisby performing Step 2 after Step 5. For the sake of clarity, theoptimization algorithms are classified into a number of groups, whichare now briefly discussed.

Single variable optimization algorithms. Because of their simplicity, single-variable optimization techniques are discussed first.These algorithms provide a good understanding of the propertiesof the minimum and maximum points in a function and how optimization algorithms work iteratively to find the optimum pointin a problem. The algorithms are classified into two categoriesdirect methods and gradient-based methods. Direct methods do notuse any derivative information of the objective function; only objective function values are used to guide the search process. However, gradient-based methods use derivative information (first and/orsecond-order) to guide the search process. Although engineering op


42/45

27ntroduction

timization problems usually contain more than one design variablesingle-variable optimization algorithms are mainly used as unidirectional search methods in multivariable optimization algorithms.

Multi variable optimization algorithms. A number of algorithmsfor unconstrained multivariable optimization problems are discussednext. These algorithms demonstrate how the search for the optimumpoint progresses in multiple dimensions. Depending on whether thegradient information is used or not used these algorithms are alsoclassified into direct and gradient-based techniques.

Constrained optimization algorithms. Constrained optimizationalgorithms are described next. These algorithms use the singlevariable and multivariable optimization algorithms repeatedly andsimultaneously maintain the search effort inside the feasible searchregion. Since these algorithms are mostly used in engineering optimization problems the discussion of these algorithms covers most ofthe material of this book.

Specialized optimization algorithms. There exist a number ofstructured algorithms which are ideal for only a certain classof optimization problems. Two of these algorithms-integerprogramming and geometric programming-are often used inengineering design problems and are discussed. Integer programmingmethods can solve optimization problems with integer designvariables. Geometric programming methods solve optimizationproblems with objective functions and constraints written in a specialform.There exist quite a few variations of each of the above algorithms.These algorithms are being used in engineering design problems sincesixties. Because of their existence and use for quite some years wecall these algorithms as traditional optimization algorithms.

Nontraditional optimization algorithms. There exist a numberof other search and optimization algorithms which are comparativelynew and are becoming popular in engineering design optimizationproblems in the recent past. Two such algorithms-geneticalgorithms and simulated annealing-are discussed in this book.

We have put together about 34 different optimization algorithms.Over the years researchers and practitioners have modifiedthese algorithms to suit their problems and to increase theefficiency of the algorithms. However there exist a few otheroptimization algorithms-stochastic programming methods anddynamic programming method-which are very different than theabove algorithms. Because of the space limitation and occasional


43/45


use of these algorithms in engineering design problems, w have not.included them in this book. A detailed discussion of these algorithmscan be found elsewhere Rao, 1984).

Many engineering optimization problems contain multipleoptimum solutions, among which one or more may be the absoluteminimum or maximum solutions. These absolute optimum solutionsare known as global optimal solutions and other optimum solutionsare known as local optimum solutions. Ideally, w are interestedin the global optimal solutions because they correspond to theabsolute optimum objective function value. Unfortunately, none ofthe traditional algorithms are guaranteed to find the global optimalsolution, but genetic algorithms and simulated annealing algorithmare found to have a better global perspective than the traditionalmethods. The global optimality issues are discussed in Chapter 6.

Moreover, when an optimal design problem contains multipleglobal solutions, designers are not only interested in finding justone global optimum solution, but as many as possible for variousreasons. Firstly, a design suitable in one situation may not be validin another situation. Secondly, it is also not possible to includeall aspects of the design in the optimization problem formulation.Thus, there always remains some uncertainty about the obtainedoptimal solution. Thirdly, designers may not be interested in findingthe absolute global solution, instead may be interested in a solutionwhich corresponds to a marginally inferior objective function valuebut is more amenable to fabrication. Thus, it is always prudent toknow about other equally good solutions for later use. However, ithe traditional methods are used to find multiple optimal solutions,they need to be applied a number of times, each time starting froma different initial solution and hoping to achieve a different optimalsolution each time. Genetic algorithms described in Chapter 6 allowan easier way to find multiple optimal solutions simultaneously in asingle simulation.

Another class of optimization problems deals with simultaneousoptimization of multiple objective functions. In formulating anoptimal design problem, designers are often faced with a numberof objective functions. For example, the truss structure problemdescribed earlier should really be reformulated as the minimizationof both the weight of the truss and the deflection at the point C.Multiobjective optimization problems give rise to a set of optimalsolutions known as Pareto optimal solutions Chankong and Haimes,1983), ll of which are equally important as far as all objectivesare concerned. Thus, the aim in these problems is to find as manyPareto-optimal solutions as possible. Because of the complexity


44/45

9ntroduction

involved in the multiobjective optimization algorithms designersusually choose to consider only one objective and formulate otherobjectives as constraints. Genetic algorithms described in Chapter 6demonstrate one way to handle multiple objectives and help findmultiple Pareto-optimal solutions simultaneously.

At the end of the optimization process one obvious questionmay arise: Is the obtained solution a true optimum solution?Unfortunately there s no easy answer to this question for alloptimization problems. In problems where the objective functionsand constraints can be written in simple explicit mathematicalforms the Kuhn-Tucker conditions described in Chapter 4 may beused to check the optimality of the obtained solution. Howeverthose conditions are valid only for a few classes of optimizationproblems. In a generic problem this question s answered in a morepractical way. In many engineering design problems a good solutionis usually known either from the previous studies or from experience.After formulating the optimal problem and applying the optimizationalgorithm if a better solution s obtained the new solution becomesthe current best solution. The optimality of the obtained solution susually confirmed by applying the optimization algorithms a numberof times from different initial solutions.

1 4 SummaryIn order to use optimization algorithms in engineering designactivities the first task s to formulate the optimization problem.The formulation process begins with identifying the importantdesign variables that can be changed in a design. The otherdesign parameters are usually kept fixed. Thereafter constraintsassociated with the design are formulated. The constraints may arisedue to resource limitations such as deflection limitations strengthlimitations frequency limitations and others. Constraints may alsoarise due to codal restrictions that govern the design. The next tasks to formulate the objective function which the designer s interestedin minimizing or maximizing. The final task of the formulation phases to identify some bounding limits for the design variables.The formulation of an optimization problem can be moredifficult than solving the optimization problem. Unfortunatelyevery optimization problem requires different considerations forformulating objectives constraints and variable bounds. Thus it isnot possible to describe all considerations in a single book. Howevermany of these considerations require some knowledge about the


45/45

30 Optimiiation for Engineering Design: lgorithms and Examples

problem, which is usually available with the experienced designersdue to their involvement with similar other design problems.The rest of the book assumes that the formulation of anoptimization problem is available. Chapters 2 to 6 describea number of different optimization algorithms-traditional andnontraditional-in step-by-step format. To demonstrate the working

of each algorithm, hand-simulations on a numerical example problemare illustrated. Sample computer codes for a number of optimizationalgorithms are also appended to demonstrate the ease of conversionof other algorithms into similar computer codes.

REFEREN ES

Chakroborty, P., Deb, K., and Subrahmanyam, P. 1995): Optimalscheduling of urban transit systems using genetic algorithms. SCEJournal of Transportation Engineering 121 6), 544-553.

Chankong, V. and Haimes, Y Y. 1983): Multiobjective Decision MakingTheory and Methodology. New York: North-Holland.

Murase, A., Roberts, H. 1. and Converse, A. O. 1970): Optimal thermaldesign of an autothermal ammonia synthesis reactor. Ind. Eng. Chem.Process Des. Develop. 9, 503-513.

Rao, S S. 1984): Optimization Theory and Applications. New Delhi:Wiley Eastern.Shigley, J. E. 1986): Mechanical Engineering Design. New York: McGrawHill.Timoshenko, S 1986): Strength of Materials Part 1: Elementary Theory

and Problems. Delhi: CBS Publishers.Upreti, S. and Deb, K in press): Optimal design of an ammonia synthesisreactor using genetic algorithms. Computers f j Chemical EngineeringAlso avaliable as Technical Report No. IlTK/ME/SMD-940015).Kanpur: Department of Mechanical Engineering, Indian Institute ofTechnology, Kanpur.

Documents

Optimization For Engineers By Kalyanmoy Deb