An Introduction to Optimization Techniques

432 Pages·2021·12.325 MB·English

by  Vikrant Sharma, Vinod Kumar Jain, Atul Kumar

An Introduction to Optimization
Techniques

An Introduction to
Optimization Techniques
Vikrant Sharma, Vinod Kumar Jain and
Atul Kumar
First edition published 2021
by CRC Press
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Library of Congress Cataloging‑in‑Publication Data
ISBN: 978-0-367-49324-0 (hbk)
ISBN: 978-0-367-49326-4 (pbk)
ISBN: 978-1-003-04576-2 (ebk)
Typeset in Palatino
by SPi Global, India
Contents
Preface ......................................................................................................................................xi
Acknowledgements .............................................................................................................xiii
Authors ....................................................................................................................................xv
1. Introduction to Optimization Techniques .........................................................................1
1.1 Introduction ...................................................................................................................1
1.2 Need for Optimization .................................................................................................2
1.3 Historical Perspective ...................................................................................................2
1.4 Optimization Terms/Parameters ................................................................................3
1.5 Types of Optimization ..................................................................................................4
1.5.1 Based on Constraints .......................................................................................4
1.5.2 Based on the Physical Configuration ............................................................4
1.5.3 Based on the Nature of the Variables ............................................................4
1.5.4 Based on the Number of Objective Functions ..............................................5
1.5.5 Geometric Programming Problem .................................................................5
1.5.6 Integer Programming Problem.......................................................................5
1.5.7 Real-Value Programming Problem ................................................................5
1.5.8 Non-linear Programming (NLP) ....................................................................5
1.6 Advanced Optimization Techniques ..........................................................................5
1.7 Optimization by Design of Experiments (DOE) (Experimental Designs) ............6
1.7.1 Types of Experimental Design ........................................................................6
1.8 Applications of Optimization Techniques .................................................................8
1.9 Limitations of Optimization Techniques ...................................................................9
1.10 Optimization Method in Engineering and Management Applications ..............10
Summary .................................................................................................................................10
Questions ................................................................................................................................10
2. Linear Programming ............................................................................................................13
2.1 Introduction .................................................................................................................13
2.2 Examples of LPP ..........................................................................................................14
2.3 Formulation of LPP .....................................................................................................15
2.4 General Form of LPP ..................................................................................................15
2.4.1 Important Concepts of LP .............................................................................16
2.4.2 Variables Used in LP ......................................................................................16
2.5 Basic Assumptions of the LPP Model ......................................................................16
Examples based on LP formulation ..........................................................................17
2.6 Solutions to LPP ..........................................................................................................22
2.6.1 Graphical Method ..........................................................................................22
2.6.2 Procedure of Graphical Method ...................................................................22
2.6.3 Types of Graphical Methods to Solve LPP .................................................23
2.7 Types of Solutions to LPP ..........................................................................................23
Examples based on the LP graphical solution.........................................................23
v
vi Contents
2.8 Simplex Method ..........................................................................................................26
2.8.1 Some Important Terms of the Simplex Technique .....................................27
2.8.2 Simplex Method Steps ...................................................................................27
Examples based on the LP simplex method ............................................................27
2.9 Big M Method/Method of Penalty ...........................................................................32
Example based on the big M method .......................................................................32
2.10 Two-Phase Method .....................................................................................................35
Example based on the LP two-phase method .........................................................35
2.11 Duality in LPP .............................................................................................................39
Examples based on duality in LLP............................................................................40
2.12 Sensitivity Analysis .....................................................................................................42
2.13 Advantages of the LP Model .....................................................................................43
2.14 Limitations of the LP Model ......................................................................................43
Chapter Summary ..................................................................................................................44
Questions ................................................................................................................................44
3. Transportation Problems and Assignment Problem ......................................................45
3.1 Introduction .................................................................................................................45
3.2 Mathematical Form of the Transportation Problem ...............................................46
3.3 Solution of a Transportation Problem ......................................................................47
3.3.1 Methods to Find an Initial Basic Feasible Solution (BFS) .........................47
3.3.2 Unit Cost Penalty Method (Vogel’s Approximation Method) .................48
Examples based on the initial BFS ............................................................................48
3.3.3 Methods to Find an Optimal Solution .........................................................54
Examples based on an optimal solution ..................................................................55
3.4 Degeneracy in Transport Problems ..........................................................................62
Examples based on degeneracy .................................................................................63
3.5 Unbalanced Transportation Problem .......................................................................66
Examples based on an unbalanced transportation problem .................................66
3.6 Introduction to an Assignment Problem .................................................................68
3.7 The Nature of an Assignment Problem ...................................................................68
3.8 Mathematical Formulation of an Assignment Problem ........................................69
3.8.1 Fundamental Theorems .................................................................................69
3.9 Assignment Algorithm (Hungarian Assignment Method) ...................................70
Examples based on an assignment problem............................................................72
3.10 The Maximal Assignment Problem ..........................................................................83
Examples based on maximization.............................................................................84
3.11 Unbalanced Assignment Problem ............................................................................88
Examples based on an unbalanced assignment problem ......................................88
Chapter Summary ..................................................................................................................98
Questions ................................................................................................................................99
4. Network Models .................................................................................................................101
4.1 Introduction ...............................................................................................................101
4.1.1 Network .........................................................................................................102
4.2 Dummy Activity ........................................................................................................102
Examples based on drawing a network diagram .................................................104
Contents vii
4.3 Critical Path Method (CPM)/Program Evaluation and
Review Technique (PERT) ........................................................................................106
4.3.1 CPM Computations .....................................................................................107
4.3.2 Forward Pass and a Backward Pass ..........................................................109
Examples based on CPM ..........................................................................................110
4.3.3 PERT Computations .....................................................................................115
4.3.4 Probability Computations ...........................................................................115
Examples based on PERT .........................................................................................116
4.3.5 Project Crashing ............................................................................................121
Example based on the crashing of the project .......................................................122
4.4 Introduction to the Shortest Path Method .............................................................127
4.4.1 Dijkstra’s Shortest Path Method.................................................................128
4.4.2 Minimum Spanning Tree Algorithm .........................................................129
Examples based on the shortest path method .......................................................130
4.5 The Maximum Flow Problem..................................................................................135
Example based on the maximum flow problem ...................................................136
4.6 A Travelling Salesman Problem (TSP) ...................................................................137
4.6.1 TSP Solution by Branch and Bound Method ............................................138
4.7 Chinese Postman Problem (CPP) ............................................................................139
Examples based on TSP and CPP ............................................................................140
Chapter Summary ................................................................................................................148
Questions ..............................................................................................................................148
Practice Problem ..................................................................................................................149
5. Sequencing ...........................................................................................................................153
5.1 Introduction ...............................................................................................................153
5.1.1 Assumptions Made while Solving Sequencing Problems ......................155
5.1.2 Terms Used in Sequencing ..........................................................................156
5.2 Gantt Chart.................................................................................................................157
5.3 Sequencing of n Jobs through One Machine .........................................................159
Examples based on sequencing of n jobs through one machine ........................160
5.4 Sequencing of n Jobs through Two Machines .......................................................172
Examples based on sequencing of n jobs through two machines ......................173
5.5 Sequencing of n Jobs through Three Machines .....................................................180
Examples based on sequencing of n jobs through three machines ....................181
5.6 Sequencing of n Jobs through m Machines ............................................................184
Examples based on the sequencing of n jobs through m machines ...................184
Typical Examples .................................................................................................................187
Chapter Summary ................................................................................................................190
Questions ..............................................................................................................................191
Practice Problem ..................................................................................................................191
6. Replacement Models ..........................................................................................................193
6.1 Introduction ...............................................................................................................193
6.2 Replacement of Items That Deteriorate with Time ..............................................194
6.2.1 Replacement Policy When the Value of Money Does Not Change
with Time .......................................................................................................194
Examples based on the replacement policy when the value of
money does not change with time .............................................................197
viii Contents
6.2.2 Replacement Policy When the Value of Money
Changes with Time ......................................................................................202
Examples based on the replacement policy when value of
money changes with time ...........................................................................204
6.3 Replacement When the Equipment/Assets Fail Completely
All of a Sudden ..........................................................................................................207
Examples based on the replacement policy when the equipment
fails completely all of a sudden ..................................................................208
Chapter Summary ................................................................................................................214
Questions ..............................................................................................................................215
Practice Problem ..................................................................................................................215
7. Game Theory .......................................................................................................................219
7.1 Introduction ...............................................................................................................219
7.1.1 Terminology in Game Theory ....................................................................220
7.2 Two-Person, Zero-Sum Games (with Saddle Point) ............................................221
7.3 The Maximin-Minimax Principle............................................................................222
Examples based on the two-person, zero-sum game with saddle point ...........223
7.4 Two-Person, Zero-Sum Games (without Saddle Point) ......................................226
Examples based on the two-person, zero-sum game without saddle point .....228
7.5 The Principle of Dominance ....................................................................................231
Examples based on the principle of dominance ...................................................232
7.6 Graphical Method for Solving Games ...................................................................236
Examples of the graphical method for solving games .........................................238
Chapter Summary ................................................................................................................243
Questions ..............................................................................................................................244
Practice Problem ..................................................................................................................244
8. Queuing Theory ..................................................................................................................247
8.1 Introduction ...............................................................................................................247
8.1.1 Elements of the Queuing System ...............................................................248
8.1.2 Kendall Notation ..........................................................................................250
8.2 Single-Server Queuing Model with Poisson Arrivals and
Exponential Service Times (M/M/1) .....................................................................251
Examples based on the single-server queuing model ..........................................253
8.3 Multiple-Server Queuing Model with Poisson Arrivals and
Exponential Service Times (M/M/m) ....................................................................257
Examples based on the multiple-server queuing model .....................................258
8.4 Little’s Relationships for Queuing Models............................................................264
Chapter Summary ................................................................................................................265
Questions ..............................................................................................................................265
Practice Problem ..................................................................................................................266
9. Dynamic and Integer Programming ................................................................................269
9.1 Introduction to Dynamic Programming ................................................................269
9.2 Terms Used in Dynamic Programming .................................................................270
9.3 Characteristics of Dynamic Programming ............................................................271
Examples based on dynamic programming ..........................................................272
Contents ix
9.4 Introduction to Integer Programming ...................................................................281
9.5 Formulating Integer Programming Problems ......................................................283
Examples based on formulating the integer programming problem ................283
9.6 Solution of Integer Programming Using the Branch-and-Bound Method .......288
Examples based on the branch-and-bound method ............................................289
9.7 Solution of Integer Programming Using the Cutting Plane Method ................295
Examples based on the cutting plane method ......................................................295
Chapter Summary ................................................................................................................303
Questions ..............................................................................................................................303
Practice Problem ..................................................................................................................304
10. Goal Programming and Decision Making .....................................................................307
10.1 Goal Programming Introduction ............................................................................307
10.2 Goal Programming Formulation ............................................................................308
Examples based on goal programming formulation............................................311
10.3 Graphical Solution Method of Goal Programming ..............................................315
Examples based on the graphical solution method of goal programming .......315
10.4 The Analytic Hierarchy Process (AHP) .................................................................321
10.4.1 Steps in AHP .................................................................................................321
Example based on AHP ............................................................................................323
10.5 Introduction to Decision Making ............................................................................328
10.5.1 Decision-Making Components ...................................................................328
10.6 Decision-Making Environments .............................................................................329
10.6.1 Decision Making under Uncertainty .........................................................329
Examples based on decision making under uncertainty .....................................331
10.6.2 Decision Making under Risk ......................................................................337
Examples based on decision making under risk ..................................................338
10.7 Decision Trees ............................................................................................................342
Examples based on decision tree .............................................................................342
Chapter Summary ................................................................................................................345
Questions ..............................................................................................................................346
Practice Problem ..................................................................................................................346
11. Optimization Modelling with an Open Source Tool (Excel) ......................................351
11.1 Introduction ...............................................................................................................351
Examples based on the linear programming graphical method using Excel ...352
Examples based on the linear programming simplex method using Excel ......359
Examples based on transportation and assignment problems using Excel ......365
Examples based on network models using Excel .................................................374
Example based on sequencing using Excel ...........................................................385
Example based on integer programming using Excel .........................................388
Example based on decision making using Excel ..................................................391
Example based on dynamic programming using Excel ......................................399
MCQ ......................................................................................................................................401
Bibliography ........................................................................................................................411
Index ......................................................................................................................................413