Introduction to Optimum Design phần 1 pdf

Introduction to
Optimum Design
Introduction to
Optimum Design
Second Edition
Jasbir S. Arora
The University of Iowa
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Library of Congress Cataloging-in-Publication Data
Arora, Jasbir S.
Introduction to optimum design / Jasbir S. Arora.—2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-12-064155-0 (acid-free paper)
1. Engineering design—Mathematical models. I. Title.

TA174.A76 2004
620¢.0042¢015118—dc22 2004046995
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Printed in the United States of America
040506070809 987654321
Jasbir S. Arora
F. Wendell Miller Distinguished Professor of Engineering
Department of Civil and Environmental Engineering
Department of Mechanical and Industrial Engineering
Center for Computer Aided Design
College of Engineering
The University of Iowa
Iowa City, Iowa 52242-1527
To
Ruhee
Rita
Balwant Kaur
Wazir Singh
Preface
I have based the material of the Second Edition on the comments that I had received from
the students over the years and on input from colleagues around the world. The text has been
rewritten, reorganized, and expanded for the second edition. Particular attention has been
paid to the pedagogical aspect of the material. Each chapter starts with a list of learning
objectives that the students can keep in mind while studying the material of the chapter. The

basic philosophy of the text remains the same as before: to describe an organized approach
to engineering design optimization in a rigorous and yet simplified manner, illustrate various
concepts and procedures with simple examples, and demonstrate their applicability to engi-
neering design problems. Formulation of a design problem as an optimization problem is
emphasized and illustrated throughout the text. Some computational algorithms are presented
in a step-by-step format to give the students a flavor of the calculations needed for solving
optimum design problems. The new material covered in the second edition includes: use of
Excel and MATLAB as learning and teaching aids, discrete variable optimization, genetic
algorithms, multiobjective optimization, and global optimization.
The text can be used in several ways to design different types of courses for undergradu-
ate and graduate studies. For undergraduate students, the key question is, “What should be
taught on the subject of optimization?” I feel that the material thoroughly covered should be:
optimum design problem formulation, basic concepts that characterize an optimum design,
basic concepts of numerical methods for optimization, and simple but illustrative examples
of optimum design. In addition, some exposure to the use of optimization software would be
quite beneficial. With this background, the students would be able to formulate and use soft-
ware properly to optimize problems once they go into industry. The basic knowledge gained
with this material can serve as a life-long learning tool on the subject of optimum design.
Such a course for junior and senior students in most branches of engineering can include the
following material, augmented with 2- to 3-week-long team projects (project type exercises
and sections with advanced material are marked with an “*” in the text):
Appendix A. Economic Analysis
Chapter 1. Introduction to Design
Chapter 2. Optimum Design Problem Formulation
Chapter 3. Graphical Optimization Method
Chapter 4. Optimum Design Concepts
Chapter 6. Linear Programming Methods for Optimum Design
Chapter 8. Numerical Methods for Unconstrained Optimum Design
Chapter 10. Numerical Methods for Constrained Optimum Design
ix

Another intermediate level course for seniors and first year graduate students can be
designed to augment the above material with Chapter 12 on MATLAB along with more
advanced design projects and introduction to discrete variable optimization using the mate-
rial contained in Chapters 15 and 16. The pace of material coverage can be a little faster than
the course designed for undergraduates only. A two-course sequence for graduate students
may be designed using the material from Chapters 1 to 10 and 12 in the first course and the
material from Chapters 11 and 13 to 18 for the second course.
I have been fortunate to have received advice, encouragement, and help from numerous
people around the globe to undertake and complete this project. Without that, a project of
this magnitude would not have been possible. I would like sincerely to thank all of them for
their input, in particular, Professor Tae Hee Lee of Hanyang University, and my graduate stu-
dents Tim Marler and Qian Wang for their special contributions to the text material. Pro-
fessor Tae Hee Lee provided me with a first draft of the material for Chapter 12 on
Introduction to Optimization with MATLAB. He developed all the examples and the corre-
sponding m-files. Tim Marler provided me with first draft of the material for Chapter 17 on
Multiobjective Optimum Design Concepts and Methods, and Qian Wang provided me with
material related to the use of Excel. Without their contributions this material would not be in
the good shape it is now. In addition, Tim Marler, Qian Wang, and Ashok Govil proofread several
chapters and provided me with suggestions for improving the presentation of the material.
Along with all the individuals mentioned in the first edition, I would like to sincerely thank
the following colleagues and friends who provided me with specific suggestions on the material
for the second edition of the text: Rick Balling, Ashok Belegundu, Scott Burns, Alex Diaz, Dan
Frangopol, Ramana Grandhi, Don Grierson, Rafi Haftka, Gene Hou, Tae Hee Lee, T.C. Lin, Kuni
Matsui, Duc Nguyen, Makoto Ohsaki, G.J. Park, Subby Rajan, David Thompson, Mats Tinnsten,
and Ren-Jye Yang. In addition, the useful exchange of ideas on the subject of optimum design
over the years with many colleagues are acknowledged: Santiago Hernández, Hans Eschenauer,
Ed Haug, Niels Olhoff, H. Furuta, U. Kirsch, J. Sobieski, Panos Papalambros, Colby Swan, V.K.
Goel, F.Y. Cheng, S. Pezeshk, D.H. Choi, Dan Tortorelli, H. Yamakawa, C.M. Chan, Lucien
Schmit, V. Kumar, Kwan Rim, Hasan Kamil, Mike Poldneff, Bob Benedict, John Taylor, Marek
Rysz, Farrokh Mistree, M.H. Abolbashari, Achille Messac, J. Herskovits, M. Kamat, V. Venkayya,

N. Khot, Gary Vanderplaats, B.M. Kwak, George Rozvany, N. Kikuchi, Prabhat Hajela, Z.
Gürdal, Nielen Stander, Omar Ghattas, Peter Eriksson, Olof Friberg, Jan Snyman, U. Kirsch, P.
Pedersen, K. Truman, C. Mota Soares, Igbal Rai, Rajbir Samra, Jagir Sooch, and many more.
I appreciate my colleagues at The University of Iowa who used the first edition of the
book to teach an undergraduate course on optimum design: Karim Abdel-Malek, Asghar
Bhatti, Kyung Choi, Ray Han, Harry Kane, George Lance, and Emad Tanbour. Their dis-
cussions and suggestions have greatly helped in improving the presentation of the material
of first 11 chapters of the second edition.
I would like to acknowledge all my former graduate students whose thesis work on various
topics of optimization contributed to the broadening of my horizon on the subject. The recent
work of Mike Huang, C.C. Hsieh, Fatos Kocer, and Ossama Elwakeil has formed the basis
for the material of Chapters 15, 16, and 18.
I would also like to thank Carla Kinney, Christine Kloiber, Joel Stein, Shoshanna Gross-
man and Brandy Palacios of Elsevier Science, and Dan Fitzgerald of Graphic World Pub-
lishing Services for their support and superb handling of the manuscript for the book.
I am grateful to the Department of Civil and Environmental Engineering, College of Engi-
neering, and The University of Iowa for providing me with time, resources, and support for
this very satisfying endeavor.
Finally, I would like to thank all my family and friends for their love and support.
Jasbir Singh Arora
Iowa City
x Preface
Table of Contents
Preface
ix
Chapter 1 Introduction to Design 1
1.1 The Design Process 2
1.2 Engineering Design versus Engineering Analysis 4
1.3 Conventional versus Optimum Design Process 4
1.4 Optimum Design versus Optimal Control 6

1.5 Basic Terminology and Notation 7
1.5.1 Sets and Points 7
1.5.2 Notation for Constraints 9
1.5.3 Superscripts/Subscripts and Summation Notation 9
1.5.4 Norm/Length of a Vector 11
1.5.5 Functions 11
1.5.6 U.S British versus SI Units 12
Chapter 2 Optimum Design Problem Formulation 15
2.1 The Problem Formulation Process 16
2.1.1 Step 1: Project/Problem Statement 16
2.1.2 Step 2: Data and Information Collection 16
2.1.3 Step 3: Identification/Definition of Design
Variables 16
2.1.4 Step 4: Identification of a Criterion to
Be Optimized 17
2.1.5 Step 5: Identification of Constraints 17
2.2 Design of a Can 18
2.3 Insulated Spherical Tank Design 20
2.4 Saw Mill Operation 22
2.5 Design of a Two-Bar Bracket 24
2.6 Design of a Cabinet 30
2.6.1 Formulation 1 for Cabinet Design 30
2.6.2 Formulation 2 for Cabinet Design 31
2.6.3 Formulation 3 for Cabinet Design 31
xi
2.7 Minimum Weight Tubular Column Design 32
2.7.1 Formulation 1 for Column Design 33
2.7.2 Formulation 2 for Column Design 34
2.8 Minimum Cost Cylindrical Tank Design 35
2.9 Design of Coil Springs 36

2.10 Minimum Weight Design of a Symmetric
Three-Bar Truss 38
2.11 A General Mathematical Model for Optimum Design 41
2.11.1 Standard Design Optimization Model 42
2.11.2 Maximization Problem Treatment 43
2.11.3 Treatment of “Greater Than Type” Constraints 43
2.11.4 Discrete and Integer Design Variables 44
2.11.5 Feasible Set 45
2.11.6 Active/Inactive/Violated Constraints 45
Exercises for Chapter 2 46
Chapter 3 Graphical Optimization 55
3.1 Graphical Solution Process 55
3.1.1 Profit Maximization Problem 55
3.1.2 Step-by-Step Graphical Solution Procedure 56
3.2 Use of Mathematica for Graphical Optimization 60
3.2.1 Plotting Functions 61
3.2.2 Identification and Hatching of Infeasible
Region for an Inequality 62
3.2.3 Identification of Feasible Region 62
3.2.4 Plotting of Objective Function Contours 63
3.2.5 Identification of Optimum Solution 63
3.3 Use of MATLAB for Graphical Optimization 64
3.3.1 Plotting of Function Contours 64
3.3.2 Editing of Graph 64
3.4 Design Problem with Multiple Solutions 66
3.5 Problem with Unbounded Solution 66
3.6 Infeasible Problem 67
3.7 Graphical Solution for Minimum Weight
Tubular Column 69
3.8 Graphical Solution for a Beam Design Problem 69

Exercises for Chapter 3 72
Chapter 4 Optimum Design Concepts 83
4.1 Definitions of Global and Local Minima 84
4.1.1 Minimum 84
4.1.2 Existence of Minimum 89
4.2 Review of Some Basic Calculus Concepts 89
4.2.1 Gradient Vector 90
4.2.2 Hessian Matrix 92
4.2.3 Taylor’s Expansion 93
4.2.4 Quadratic Forms and Definite Matrices 96
4.2.5 Concept of Necessary and
Sufficient Conditions 102
xii Contents
4.3 Unconstrained Optimum Design Problems 103
4.3.1 Concepts Related to Optimality Conditions 103
4.3.2 Optimality Conditions for Functions of
Single Variable 104
4.3.3 Optimality Conditions for Functions of
Several Variables 109
4.3.4 Roots of Nonlinear Equations Using Excel 116
4.4 Constrained Optimum Design Problems 119
4.4.1 Role of Constraints 119
4.4.2 Necessary Conditions: Equality Constraints 121
4.4.3 Necessary Conditions: Inequality Constraints—
Karush-Kuhn-Tucker (KKT) Conditions 128
4.4.4 Solution of KKT Conditions Using Excel 140
4.4.5 Solution of KKT Conditions Using MATLAB 141
4.5 Postoptimality Analysis: Physical Meaning of
Lagrange Multipliers 143
4.5.1 Effect of Changing Constraint Limits 143

4.5.2 Effect of Cost Function Scaling on
Lagrange Multipliers 146
4.5.3 Effect of Scaling a Constraint on Its
Lagrange Multiplier 147
4.5.4 Generalization of Constraint Variation
Sensitivity Result 148
4.6 Global Optimality 149
4.6.1 Convex Sets 149
4.6.2 Convex Functions 151
4.6.3 Convex Programming Problem 153
4.6.4 Transformation of a Constraint 156
4.6.5 Sufficient Conditions for Convex
Programming Problems 157
4.7 Engineering Design Examples 158
4.7.1 Design of a Wall Bracket 158
4.7.2 Design of a Rectangular Beam 162
Exercises for Chapter 4 166
Chapter 5 More on Optimum Design Concepts 175
5.1 Alternate Form of KKT Necessary Conditions 175
5.2 Irregular Points 178
5.3 Second-Order Conditions for
Constrained Optimization 179
5.4 Sufficiency Check for Rectangular Beam
Design Problem 184
Exercises for Chapter 5 185
Chapter 6 Linear Programming Methods for Optimum Design 191
6.1 Definition of a Standard Linear Programming
Problem 192
6.1.1 Linear Constraints 192
6.1.2 Unrestricted Variables 193

6.1.3 Standard LP Definition 193
Contents xiii
6.2 Basic Concepts Related to Linear Programming
Problems 195
6.2.1 Basic Concepts 195
6.2.2 LP Terminology 198
6.2.3 Optimum Solution for LP Problems 201
6.3 Basic Ideas and Steps of the Simplex Method 201
6.3.1 The Simplex 202
6.3.2 Canonical Form/General Solution of Ax = b 202
6.3.3 Tableau 203
6.3.4 The Pivot Step 205
6.3.5 Basic Steps of the Simplex Method 206
6.3.6 Simplex Algorithm 211
6.4 Two-Phase Simplex Method—Artificial Variables 218
6.4.1 Artificial Variables 219
6.4.2 Artificial Cost Function 219
6.4.3 Definition of Phase I Problem 220
6.4.4 Phase I Algorithm 220
6.4.5 Phase II Algorithm 221
6.4.6 Degenerate Basic Feasible Solution 226
6.5 Postoptimality Analysis 228
6.5.1 Changes in Resource Limits 229
6.5.2 Ranging Right Side Parameters 235
6.5.3 Ranging Cost Coefficients 239
6.5.4 Changes in the Coefficient Matrix 241
6.6 Solution of LP Problems Using Excel Solver 243
Exercises for Chapter 6 246
Chapter 7 More on Linear Programming Methods for
Optimum Design 259

7.1 Derivation of the Simplex Method 259
7.1.1 Selection of a Basic Variable That Should
Become Nonbasic 259
7.1.2 Selection of a Nonbasic Variable That Should
Become Basic 260
7.2 Alternate Simplex Method 262
7.3 Duality in Linear Programming 263
7.3.1 Standard Primal LP 263
7.3.2 Dual LP Problem 264
7.3.3 Treatment of Equality Constraints 265
7.3.4 Alternate Treatment of Equality Constraints 266
7.3.5 Determination of Primal Solution from
Dual Solution 267
7.3.6 Use of Dual Tableau to Recover Primal Solution 271
7.3.7 Dual Variables as Lagrange Multipliers 273
Exercises for Chapter 7 275
Chapter 8 Numerical Methods for Unconstrained Optimum Design 277
8.1 General Concepts Related to Numerical Algorithms 278
8.1.1 A General Algorithm 279
8.1.2 Descent Direction and Descent Step 280
xiv Contents
8.1.3 Convergence of Algorithms 282
8.1.4 Rate of Convergence 282
8.2 Basic Ideas and Algorithms for Step Size Determination 282
8.2.1 Definition of One-Dimensional
Minimization Subproblem 282
8.2.2 Analytical Method to Compute Step Size 283
8.2.3 Concepts Related to Numerical Methods to
Compute Step Size 285
8.2.4 Equal Interval Search 286

8.2.5 Alternate Equal Interval Search 288
8.2.6 Golden Section Search 289
8.3 Search Direction Determination: Steepest
Descent Method 293
8.4 Search Direction Determination: Conjugate
Gradient Method 296
Exercises for Chapter 8 300
Chapter 9 More on Numerical Methods for Unconstrained
Optimum Design 305
9.1 More on Step Size Determination 305
9.1.1 Polynomial Interpolation 306
9.1.2 Inaccurate Line Search 309
9.2 More on Steepest Descent Method 310
9.2.1 Properties of the Gradient Vector 310
9.2.2 Orthogonality of Steepest Descent Directions 314
9.3 Scaling of Design Variables 315
9.4 Search Direction Determination: Newton’s Method 318
9.4.1 Classical Newton’s Method 318
9.4.2 Modified Newton’s Method 319
9.4.3 Marquardt Modification 323
9.5 Search Direction Determination:
Quasi-Newton Methods 324
9.5.1 Inverse Hessian Updating: DFP Method 324
9.5.2 Direct Hessian Updating: BFGS Method 327
9.6 Engineering Applications of Unconstrained Methods 329
9.6.1 Minimization of Total Potential Energy 329
9.6.2 Solution of Nonlinear Equations 331
9.7 Solution of Constrained Problems Using Unconstrained
Optimization Methods 332
9.7.1 Sequential Unconstrained Minimization

Techniques 333
9.7.2 Multiplier (Augmented Lagrangian) Methods 334
Exercises for Chapter 9 335
Chapter 10 Numerical Methods for Constrained Optimum Design 339
10.1 Basic Concepts and Ideas 340
10.1.1 Basic Concepts Related to Algorithms for
Constrained Problems 340
10.1.2 Constraint Status at a Design Point 342
10.1.3 Constraint Normalization 343
Contents xv
10.1.4 Descent Function 345
10.1.5 Convergence of an Algorithm 345
10.2 Linearization of Constrained Problem 346
10.3 Sequential Linear Programming Algorithm 352
10.3.1 The Basic Idea—Move Limits 352
10.3.2 An SLP Algorithm 353
10.3.3 SLP Algorithm: Some Observations 357
10.4 Quadratic Programming Subproblem 358
10.4.1 Definition of QP Subproblem 358
10.4.2 Solution of QP Subproblem 361
10.5 Constrained Steepest Descent Method 363
10.5.1 Descent Function 364
10.5.2 Step Size Determination 366
10.5.3 CSD Algorithm 368
10.5.4 CSD Algorithm: Some Observations 368
10.6 Engineering Design Optimization Using Excel Solver 369
Exercises for Chapter 10 373
Chapter 11 More on Numerical Methods for Constrained
Optimum Design 379
11.1 Potential Constraint Strategy 379

11.2 Quadratic Programming Problem 383
11.2.1 Definition of QP Problem 383
11.2.2 KKT Necessary Conditions for
the QP Problem 384
11.2.3 Transformation of KKT Conditions 384
11.2.4 Simplex Method for Solving QP Problem 385
11.3 Approximate Step Size Determination 388
11.3.1 The Basic Idea 388
11.3.2 Descent Condition 389
11.3.3 CSD Algorithm with Approximate Step Size 393
11.4 Constrained Quasi-Newton Methods 400
11.4.1 Derivation of Quadratic Programming
Subproblem 400
11.4.2 Quasi-Newton Hessian Approximation 403
11.4.3 Modified Constrained Steepest
Descent Algorithm 404
11.4.4 Observations on the Constrained
Quasi-Newton Methods 406
11.4.5 Descent Functions 406
11.5 Other Numerical Optimization Methods 407
11.5.1 Method of Feasible Directions 407
11.5.2 Gradient Projection Method 409
11.5.3 Generalized Reduced Gradient Method 410
Exercises for Chapter 11 411
Chapter 12 Introduction to Optimum Design with MATLAB 413
12.1 Introduction to Optimization Toolbox 413
12.1.1 Variables and Expressions 413
xvi Contents
12.1.2 Scalar, Array, and Matrix Operations 414
12.1.3 Optimization Toolbox 414

12.2 Unconstrained Optimum Design Problems 415
12.3 Constrained Optimum Design Problems 418
12.4 Optimum Design Examples with MATLAB 420
12.4.1 Location of Maximum Shear Stress for Two
Spherical Bodies in Contact 420
12.4.2 Column Design for Minimum Mass 421
12.4.3 Flywheel Design for Minimum Mass 425
Exercises for Chapter 12 429
Chapter 13 Interactive Design Optimization 433
13.1 Role of Interaction in Design Optimization 434
13.1.1 What Is Interactive Design Optimization? 434
13.1.2 Role of Computers in Interactive
Design Optimization 434
13.1.3 Why Interactive Design Optimization? 435
13.2 Interactive Design Optimization Algorithms 436
13.2.1 Cost Reduction Algorithm 436
13.2.2 Constraint Correction Algorithm 440
13.2.3 Algorithm for Constraint Correction
at Constant Cost 442
13.2.4 Algorithm for Constraint Correction
at Specified Increase in Cost 445
13.2.5 Constraint Correction with Minimum Increase
in Cost 446
13.2.6 Observations on Interactive Algorithms 447
13.3 Desired Interactive Capabilities 448
13.3.1 Interactive Data Preparation 448
13.3.2 Interactive Capabilities 448
13.3.3 Interactive Decision Making 449
13.3.4 Interactive Graphics 450
13.4 Interactive Design Optimization Software 450

13.4.1 User Interface for IDESIGN 451
13.4.2 Capabilities of IDESIGN 453
13.5 Examples of Interactive Design Optimization 454
13.5.1 Formulation of Spring Design Problem 454
13.5.2 Optimum Solution for the Spring
Design Problem 455
13.5.3 Interactive Solution for Spring Design Problem 455
13.5.4 Use of Interactive Graphics 457
Exercises for Chapter 13 462
Chapter 14 Design Optimization Applications with
Implicit Functions 465
14.1 Formulation of Practical Design
Optimization Problems 466
14.1.1 General Guidelines 466
14.1.2 Example of a Practical Design
Optimization Problem 467
Contents xvii
14.2 Gradient Evaluation for Implicit Functions 473
14.3 Issues in Practical Design Optimization 478
14.3.1 Selection of an Algorithm 478
14.3.2 Attributes of a Good Optimization Algorithm 478
14.4 Use of General-Purpose Software 479
14.4.1 Software Selection 480
14.4.2 Integration of an Application into General-
Purpose Software 480
14.5 Optimum Design of Two-Member Frame with
Out-of-Plane Loads 481
14.6 Optimum Design of a Three-Bar Structure for Multiple
Performance Requirements 483
14.6.1 Symmetric Three-Bar Structure 483

14.6.2 Asymmetric Three-Bar Structure 484
14.6.3 Comparison of Solutions 490
14.7 Discrete Variable Optimum Design 491
14.7.1 Continuous Variable Optimization 492
14.7.2 Discrete Variable Optimization 492
14.8 Optimal Control of Systems by Nonlinear Programming 493
14.8.1 A Prototype Optimal Control Problem 493
14.8.2 Minimization of Error in State Variable 497
14.8.3 Minimum Control Effort Problem 503
14.8.4 Minimum Time Control Problem 505
14.8.5 Comparison of Three Formulations for
Optimal Control of System Motion 508
Exercises for Chapter 14 508
Chapter 15 Discrete Variable Optimum Design Concepts
and Methods 513
15.1 Basic Concepts and Definitions 514
15.1.1 Definition of Mixed Variable Optimum
Design Problem: MV-OPT 514
15.1.2 Classification of Mixed Variable Optimum
Design Problems 514
15.1.3 Overview of Solution Concepts 515
15.2 Branch and Bound Methods (BBM) 516
15.2.1 Basic BBM 517
15.2.2 BBM with Local Minimization 519
15.2.3 BBM for General MV-OPT 520
15.3 Integer Programming 521
15.4 Sequential Linearization Methods 522
15.5 Simulated Annealing 522
15.6 Dynamic Rounding-off Method 524
15.7 Neighborhood Search Method 525

15.8 Methods for Linked Discrete Variables 525
15.9 Selection of a Method 526
Exercises for Chapter 15 527
Chapter 16 Genetic Algorithms for Optimum Design 531
16.1 Basic Concepts and Definitions 532
16.2 Fundamentals of Genetic Algorithms 534
xviii Contents
16.3 Genetic Algorithm for Sequencing-Type Problems 538
16.4 Applications 539
Exercises for Chapter 16 540
Chapter 17 Multiobjective Optimum Design Concepts and Methods 543
17.1 Problem Definition 543
17.2 Terminology and Basic Concepts 546
17.2.1 Criterion Space and Design Space 546
17.2.2 Solution Concepts 548
17.2.3 Preferences and Utility Functions 551
17.2.4 Vector Methods and Scalarization Methods 551
17.2.5 Generation of Pareto Optimal Set 551
17.2.6 Normalization of Objective Functions 552
17.2.7 Optimization Engine 552
17.3 Multiobjective Genetic Algorithms 552
17.4 Weighted Sum Method 555
17.5 Weighted Min-Max Method 556
17.6 Weighted Global Criterion Method 556
17.7 Lexicographic Method 558
17.8 Bounded Objective Function Method 558
17.9 Goal Programming 559
17.10 Selection of Methods 559
Exercises for Chapter 17 560
Chapter 18 Global Optimization Concepts and Methods for

Optimum Design 565
18.1 Basic Concepts of Solution Methods 565
18.1.1 Basic Concepts 565
18.1.2 Overview of Methods 567
18.2 Overview of Deterministic Methods 567
18.2.1 Covering Methods 568
18.2.2 Zooming Method 568
18.2.3 Methods of Generalized Descent 569
18.2.4 Tunneling Method 571
18.3 Overview of Stochastic Methods 572
18.3.1 Pure Random Search 573
18.3.2 Multistart Method 573
18.3.3 Clustering Methods 573
18.3.4 Controlled Random Search 575
18.3.5 Acceptance-Rejection Methods 578
18.3.6 Stochastic Integration 579
18.4 Two Local-Global Stochastic Methods 579
18.4.1 A Conceptual Local-Global Algorithm 579
18.4.2 Domain Elimination Method 580
18.4.3 Stochastic Zooming Method 582
18.4.4 Operations Analysis of the Methods 583
18.5 Numerical Performance of Methods 585
18.5.1 Summary of Features of Methods 585
18.5.2 Performance of Some Methods Using
Unconstrained Problems 586
Contents xix
18.5.3 Performance of Stochastic Zooming and
Domain Elimination Methods 586
18.5.4 Global Optimization of Structural
Design Problems 587

Exercises for Chapter 18 588
Appendix A
Economic Analysis 593
A.1 Time Value of Money 593
A.1.1 Cash Flow Diagrams 594
A.1.2 Basic Economic Formulas 594
A.2 Economic Bases for Comparison 598
A.2.1 Annual Base Comparisons 599
A.2.2 Present Worth Comparisons 601
Exercises for Appendix A 604
Appendix B Vector and Matrix Algebra 611
B.1 Definition of Matrices 611
B.2 Type of Matrices and Their Operations 613
B.2.1 Null Matrix 613
B.2.2 Vector 613
B.2.3 Addition of Matrices 613
B.2.4 Multiplication of Matrices 613
B.2.5 Transpose of a Matrix 615
B.2.6 Elementary Row–Column Operations 616
B.2.7 Equivalence of Matrices 616
B.2.8 Scalar Product–Dot Product of Vectors 616
B.2.9 Square Matrices 616
B.2.10 Partitioning of Matrices 617
B.3 Solution of n Linear Equations in n Unknowns 618
B.3.1 Linear Systems 618
B.3.2 Determinants 619
B.3.3 Gaussian Elimination Procedure 621
B.3.4 Inverse of a Matrix: Gauss-Jordan Elimination 625
B.4 Solution of m Linear Equations in n Unknowns 628
B.4.1 Rank of a Matrix 628

B.4.2 General Solution of m ¥ n Linear Equations 629
B.5 Concepts Related to a Set of Vectors 635
B.5.1 Linear Independence of a Set of Vectors 635
B.5.2 Vector Spaces 639
B.6 Eigenvalues and Eigenvectors 642
B.7 Norm and Condition Number of a Matrix 643
B.7.1 Norm of Vectors and Matrices 643
B.7.2 Condition Number of a Matrix 644
Exercises for Appendix B 645
Appendix C A Numerical Method for Solution of
Nonlinear Equations 647
C.1 Single Nonlinear Equation 647
C.2 Multiple Nonlinear Equations 650
Exercises for Appendix C 655
xx Contents
Appendix D Sample Computer Programs 657
D.1 Equal Interval Search 657
D.2 Golden Section Search 660
D.3 Steepest Descent Method 660
D.4 Modified Newton’s Method 669
References
675
Bibliography
683
Answers to Selected Problems
687
Index
695
Contents xxi

1 Introduction to Design
1
Upon completion of this chapter, you will be able to:
•
Describe the overall process of designing systems
•
Distinguish between engineering design and engineering analysis activity
•
Distinguish between the conventional design process and optimum design process
•
Distinguish between the optimum design and optimal control problems
•
Understand the notations used for operations with vectors, matrices, and functions
Engineering consists of a number of well established activities, including analysis, design,
fabrication, sales, research, and the development of systems. The subject of this text—the
design of systems—is a major field in the engineering profession. The process of designing
and fabricating systems has been developed over centuries. The existence of many complex
systems, such as buildings, bridges, highways, automobiles, airplanes, space vehicles, and
others, is an excellent testimonial for this process. However, the evolution of these systems
has been slow. The entire process has been both time-consuming and costly, requiring
substantial human and material resources. Therefore, the procedure has been to design,
fabricate, and use the system regardless of whether it was the best one. Improved systems
were designed only after a substantial investment had been recovered. These new systems
performed the same or even more tasks, cost less, and were more efficient.
The preceding discussion indicates that several systems can usually accomplish the same
task, and that some are better than others. For example, the purpose of a bridge is to provide
continuity in traffic from one side to the other. Several types of bridges can serve this purpose.
However, to analyze and design all possibilities can be a time-consuming and costly affair.
Usually one type has been selected based on some preliminary analyses and has been
designed in detail.

The design of complex systems requires data processing and a large number of calcula-
tions. In the recent past, a revolution in computer technology and numerical computations
has taken place. Today’s computers can perform complex calculations and process large
amounts of data rapidly. The engineering design and optimization processes benefit greatly
from this revolution because they require a large number of calculations. Better systems can
now be designed by analyzing and optimizing various options in a short time. This is highly
desirable because better designed systems cost less, have more capability, and are easy to
maintain and operate.
The design of systems can be formulated as problems of optimization in which a measure
of performance is to be optimized while satisfying all constraints. Many numerical methods
of optimization have been developed and used to design better systems. This text describes
the basic concepts of optimization methods and their applications to the design of engineer-
ing systems. Design process is emphasized rather than optimization theory. Various theorems
are stated as results without rigorous proofs. However, their implications from an engineer-
ing point of view are studied and discussed in detail. Optimization theory, numerical methods,
and modern computer hardware and software can be used as tools to design better engineer-
ing systems. The text emphasizes this theme throughout.
Any problem in which certain parameters need to be determined to satisfy constraints can
be formulated as an optimization problem. Once this has been done, the concepts and the
methods described in this text can be used to solve the problem. Therefore, the optimiza-
tion techniques are quite general, having a wide range of applicability in diverse fields. The
range of applications is limited only by the imagination or ingenuity of the designers. It is
impossible to discuss every application of optimization concepts and techniques in this
introductory text. However, using simple applications, we shall discuss concepts, funda-
mental principles, and basic techniques that can be used in numerous applications. The
student should understand them without getting bogged down with the notation, terminol-
ogy, and details of the particular area of application.
1.1 The Design Process
The design of many engineering systems can be a fairly complex process. Many assumptions
must be made to develop models that can be subjected to analysis by the available methods

and the models must be verified by experiments. Many possibilities and factors must be
considered during the problem formulation phase. Economic considerations play an impor-
tant role in designing cost-effective systems. Introductory methods of economic analysis
described in Appendix A are useful in this regard. To complete the design of an engineering
system, designers from different fields of engineering must usually cooperate. For example,
the design of a high-rise building involves designers from architectural, structural, mechan-
ical, electrical, and environmental engineering as well as construction management experts.
Design of a passenger car requires cooperation among structural, mechanical, automotive,
electrical, human factors, chemical, and hydraulics design engineers. Thus, in an interdisci-
plinary environment considerable interaction is needed among various design teams to com-
plete the project. For most applications the entire design project must be broken down into
several subproblems which are then treated independently. Each of the subproblems can be
posed as a problem of optimum design.
The design of a system begins by analyzing various options. Subsystems and their com-
ponents are identified, designed, and tested. This process results in a set of drawings, calcu-
lations, and reports by which the system can be fabricated. We shall use a systems engineering
model to describe the design process. Although a complete discussion of this subject is
beyond the scope of the text, some basic concepts will be discussed using a simple block
diagram.
Design is an iterative process. The designer’s experience, intuition, and ingenuity are
required in the design of systems in most fields of engineering (aerospace, automotive, civil,
chemical, industrial, electrical, mechanical, hydraulic, and transportation). Iterative implies
analyzing several trial designs one after another until an acceptable design is obtained. The
concept of trial designs is important to understand. In the design process, the designer
2 INTRODUCTION TO OPTIMUM DESIGN
How do I
begin to
design a
system?