Operations Research: A Practical Approach

471 Pages·2019·4.93 MB·English

by  Michael W. Carter

Operations Research
Advances in Applied Mathematics
S e : Daniel Zwillinger
eries ditor
CRC Standard Curves and Surfaces with Mathematica®, Second Edition
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Stochastic Partial Differential Equations, Second Edition
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CRC Standard Mathematical Tables and Formulas, 32nd Edition
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Advanced Engineering Mathematics with MATLAB, Third Edition
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CRC Standard Curves and Surfaces with Mathematica®, Third Edition
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Handbook of Peridynamic Modeling
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Advanced Engineering Mathematics with MATLAB, Fourth Edition
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Linear and Complex Analysis for Applications
John P. D’Angelo
Quadratic Programming with Computer Programs
Michael J. Best
Green’s Functions with Applications, Second Edition
Dean G. Duffy
Introduction to Radar Analysis, Second Edition
Bassem R. Mahafza
CRC Standard Mathematical Tables and Formulas, 33rd Edition
Dan Zwillinger
The Second-Order Adjoint Sensitivity Analysis Methodology
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Operations Research: A Practical Introduction, Second Edition
Michael W. Carter, Camille C. Price, and Ghaith Rabadi
http://operationsresearch.us
Operations Research
A Practical Introduction
Second Edition
Michael W. Carter
Camille C. Price
Ghaith Rabadi
CRC Press
Taylor & Francis Group
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Contents
Preface ...........................................................................................................................................xiii
About the Authors .......................................................................................................................xix
1. Introduction to Operations Research ..................................................................................1
1.1 The Origins and Applications of Operations Research ...........................................1
1.2 System Modeling Principles ........................................................................................3
1.3 Algorithm Efficiency and Problem Complexity .......................................................5
1.4 Optimality and Practicality .........................................................................................9
1.5 Software for Operations Research ............................................................................10
1.6 Illustrative Applications ............................................................................................14
1.6.1 Analytical Innovation in the Food and Agribusiness Industries ............14
1.6.2 Humanitarian Relief in Natural Disasters .................................................15
1.6.3 Mining and Social Conflicts .........................................................................17
1.7 Summary ......................................................................................................................18
Key Terms ...............................................................................................................................19
References and Suggested Readings ...................................................................................20
2. Linear Programming ............................................................................................................23
2.1 The Linear Programming Model ..............................................................................23
2.2 The Art and Skill of Problem Formulation ..............................................................24
2.2.1 Integer and Nonlinear Models ....................................................................30
2.3 Graphical Solution of Linear Programming Problems ..........................................30
2.3.1 General Definitions........................................................................................30
2.3.2 Graphical Solutions .......................................................................................31
2.3.3 Multiple Optimal Solutions..........................................................................33
2.3.4 No Optimal Solution .....................................................................................34
2.3.5 No Feasible Solution .....................................................................................35
2.3.6 General Solution Method .............................................................................36
2.4 Preparation for the Simplex Method .......................................................................36
2.4.1 Standard Form of a Linear Programming Problem ..................................36
2.4.2 Solutions of Linear Systems .........................................................................38
2.5 The Simplex Method ..................................................................................................39
2.6 Initial Solutions for General Constraints .................................................................46
2.6.1 Artificial Variables .........................................................................................46
2.6.2 The Two Phase Method ................................................................................48
2.7 Information in the Tableau ........................................................................................50
2.7.1 Multiple Optimal Solutions..........................................................................51
2.7.2 Unbounded Solution (No Optimal Solution) ............................................51
2.7.3 Degenerate Solutions ....................................................................................53
2.7.4 Analyzing the Optimal Tableau: Shadow Prices.......................................55
2.8 Duality and Sensitivity Analysis ..............................................................................56
2.8.1 The Dual Problem ..........................................................................................56
2.8.2 Postoptimality and Sensitivity Analysis ....................................................60
v
vi Contents
2.9 Revised Simplex and Computational Efficiency ..................................................63
2.10 Software for Linear Programming .........................................................................64
2.10.1 Extensions to General Simplex Methods ...............................................65
2.10.2 Interior Methods ........................................................................................67
2.10.3 Software for Solving Linear Programming ...........................................69
2.11 Illustrative Applications ..........................................................................................71
2.11.1 Forest Pest Control Program ....................................................................71
2.11.2 Aircraft and Munitions Procurement .....................................................72
2.11.3 Grape Processing: Materials Planning and Production .......................73
2.12 Summary ....................................................................................................................74
Key Terms ...............................................................................................................................75
Exercises ..................................................................................................................................76
References and Suggested Readings ...................................................................................85
3. Network Analysis .................................................................................................................89
3.1 Graphs and Networks: Preliminary Definitions ..................................................90
3.2 Maximum Flow in Networks ..................................................................................92
3.2.1 Maximum Flow Algorithm ......................................................................93
3.2.2 Extensions to the Maximum Flow Problem ..........................................96
3.3 Minimum Cost Network Flow Problems ..............................................................97
3.3.1 Transportation Problem ............................................................................97
3.3.1.1 Northwest Corner Rule ............................................................99
3.3.1.2 Minimum Cost Method ..........................................................101
3.3.1.3 Minimum “Row” Cost Method .............................................102
3.3.1.4 Transportation Simplex Method ...........................................103
3.3.1.5 Transportation Simplex ..........................................................107
3.3.2 Assignment Problem and Stable Matching .........................................109
3.3.2.1 Stable Matching .......................................................................113
3.3.3 Capacitated Transshipment Problem ...................................................114
3.4 Network Connectivity ............................................................................................116
3.4.1 Minimum Spanning Trees ......................................................................116
3.4.2 Shortest Network Problem: A Variation on Minimum
Spanning Trees .........................................................................................118
3.5 Shortest Path Problems ..........................................................................................119
3.5.1 Shortest Path through an Acyclic Network .........................................120
3.5.2 Shortest Paths from Source to All Other Nodes ..................................121
3.5.3 Problems Solvable with Shortest Path Methods .................................123
3.6 Dynamic Programming .........................................................................................125
3.6.1 Labeling Method for Multi-Stage Decision Making...........................126
3.6.2 Tabular Method .......................................................................................127
3.6.3 General Recursive Method ....................................................................130
3.7 Project Management ...............................................................................................132
3.7.1 Project Networks and Critical Paths .....................................................133
3.7.2 Cost versus Time Trade-Offs ..................................................................137
3.7.3 Probabilistic Project Scheduling ............................................................139
3.8 Software for Network Analysis ............................................................................141
Contents vii
3.9 Illustrative Applications ........................................................................................142
3.9.1 DNA Sequence Comparison Using a Shortest Path Algorithm .......142
3.9.2 Multiprocessor Network Traffic Scheduling .......................................142
3.9.3 Shipping Cotton from Farms to Gins ...................................................143
3.10 Summary ..................................................................................................................144
Key Terms .............................................................................................................................145
Exercises ................................................................................................................................146
References and Suggested Readings .................................................................................154
4. Integer Programming .........................................................................................................157
4.1 Fundamental Concepts ..........................................................................................157
4.2 Typical Integer Programming Problems..............................................................159
4.2.1 General Integer Problems .......................................................................159
4.2.2 Zero–One (0–1) Problems .......................................................................159
4.2.3 Mixed Integer Problems .........................................................................160
4.3 Zero–One (0–1) Model Formulations ...................................................................161
4.3.1 Traveling Salesman Model .....................................................................161
4.3.2 Knapsack Model ......................................................................................162
4.3.3 Bin Packing Model ..................................................................................162
4.3.4 Set Partitioning/Covering/Packing Models .......................................163
4.3.5 Generalized Assignment Model ............................................................164
4.4 Branch-and-Bound ..................................................................................................165
4.4.1 A Simple Example ...................................................................................165
4.4.2 A Basic Branch-and-Bound Algorithm .................................................169
4.4.3 Knapsack Example ..................................................................................169
4.4.4 From Basic Method to Commercial Code ............................................171
4.4.4.1 Branching Strategies ...............................................................172
4.4.4.2 Bounding Strategies ................................................................174
4.4.4.3 Separation Rules ......................................................................175
4.4.4.4 The Impact of Model Formulation ........................................175
4.4.4.5 Representation of Real Numbers ..........................................177
4.5 Cutting Planes and Facets .....................................................................................177
4.6 Cover Inequalities ...................................................................................................180
4.7 Lagrangian Relaxation ...........................................................................................187
4.7.1 Relaxing Integer Programming Constraints........................................187
4.7.2 A Simple Example ...................................................................................188
4.7.3 The Integrality Gap .................................................................................191
4.7.4 The Generalized Assignment Problem.................................................192
4.7.5 A Basic Lagrangian Relaxation Algorithm ..........................................194
4.7.6 A Customer Allocation Problem ...........................................................194
4.8 Column Generation ................................................................................................197
4.9 Software for Integer Programming ......................................................................201
4.10 Illustrative Applications ........................................................................................202
4.10.1 Solid Waste Management .......................................................................202
4.10.2 Timber Harvest Planning .......................................................................204
4.10.3 Propane Bottling Plants ..........................................................................205
viii Contents
4.11 Summary ..................................................................................................................206
Key Terms .............................................................................................................................207
Exercises ................................................................................................................................208
References and Suggested Readings .................................................................................213
5. Nonlinear Optimization ....................................................................................................217
5.1 Preliminary Notation and Concepts ....................................................................218
5.2 Unconstrained Optimization ................................................................................223
5.2.1 One-Dimensional Search ..........................................................................223
5.2.1.1 One-Dimensional Search Algorithm .......................................223
5.2.2 Multivariable Search: Gradient Method ................................................225
5.2.2.1 Multivariable Gradient Search ................................................226
5.2.3 Newton’s Method ......................................................................................228
5.2.4 Quasi-Newton Methods ...........................................................................229
5.3 Constrained Optimization .....................................................................................229
5.3.1 Lagrange Multipliers (Equality Constraints) ........................................229
5.3.2 Karush–Kuhn–Tucker Conditions (Inequality Constraints) ...............230
5.3.3 Quadratic Programming ..........................................................................231
5.3.4 More Advanced Methods .........................................................................236
5.4 Software for Nonlinear Optimization ..................................................................236
5.5 Illustrative Applications ........................................................................................239
5.5.1 Gasoline Blending Systems ......................................................................239
5.5.2 Portfolio Construction ..............................................................................240
5.5.3 Balancing Rotor Systems ..........................................................................241
5.6 Summary ..................................................................................................................242
Key Terms .............................................................................................................................242
Exercises ................................................................................................................................243
References and Suggested Readings .................................................................................245
6. Markov Processes ................................................................................................................249
6.1 State Transitions ......................................................................................................250
6.2 State Probabilities....................................................................................................256
6.3 First Passage Probabilities .....................................................................................259
6.4 Properties of the States in a Markov Process ......................................................261
6.5 Steady-State Analysis .............................................................................................263
6.6 Expected First Passage Times................................................................................265
6.7 Absorbing Chains ...................................................................................................267
6.8 Software for Markov Processes .............................................................................271
6.9 Illustrative Applications ........................................................................................272
6.9.1 Water Reservoir Operations .....................................................................272
6.9.2 Markov Analysis of Dynamic Memory Allocation ..............................273
6.9.3 Markov Models for Manufacturing Production Capability ................274
6.9.4 Markov Decision Processes in Dairy Farming ......................................275
6.10 Summary ..................................................................................................................276
Key Terms .............................................................................................................................276
Exercises ................................................................................................................................277
References and Suggested Readings .................................................................................281
Contents ix
7. Queueing Models ...............................................................................................................285
7.1 Basic Elements of Queueing Systems ..................................................................285
7.2 Arrival and Service Patterns .................................................................................288
7.2.1 The Exponential Distribution ..................................................................288
7.2.2 Birth-and-Death Processes .......................................................................290
7.3 Analysis of Simple Queueing Systems ................................................................291
7.3.1 Notation and Definitions ..........................................................................291
7.3.2 Steady State Performance Measures .......................................................292
7.3.3 Practical Limits of Queueing Models .....................................................298
7.4 Software for Queueing Models .............................................................................299
7.5 Illustrative Applications ........................................................................................300
7.5.1 Cost Efficiency and Service Quality in Hospitals .................................300
7.5.2 Queueing Models in Manufacturing ......................................................302
7.5.3 Nurse Staffing Based on Queueing Models ..........................................304
7.6 Summary ..................................................................................................................305
Key Terms .............................................................................................................................306
Exercises ................................................................................................................................306
References and Suggested Readings .................................................................................309
8. Simulation ............................................................................................................................311
8.1 Simulation: Purposes and Applications ..............................................................311
8.2 Discrete Simulation Models ..................................................................................314
8.2.1 Event-Driven Models ................................................................................314
8.2.2 Generating Random Events .....................................................................317
8.3 Observations of Simulations .................................................................................321
8.3.1 Gathering Statistics ...................................................................................321
8.3.1.1 Average Time in System ...........................................................321
8.3.1.2 Average Waiting Time ...............................................................322
8.3.1.3 Average Number in Queue ......................................................322
8.3.1.4 Server Utilization .......................................................................323
8.3.2 Design of Simulation Experiments .........................................................324
8.4 Software for Simulation .........................................................................................325
8.5 Illustrative Applications ........................................................................................328
8.5.1 Finnish Air Force Fleet Maintenance ......................................................328
8.5.2 Simulation of a Semiconductor Manufacturing Line .............................329
8.5.3 Simulation of Eurotunnel Terminals ......................................................331
8.5.4 Simulation for NASA’s Space Launch Vehicles Operations................332
8.6 Summary ..................................................................................................................334
Key Terms .............................................................................................................................334
Exercises ................................................................................................................................335
References and Suggested Readings .................................................................................337