Introduction to operations research

COVER......Page 1
ABOUT THE AUTHORS......Page 7
ABOUT THE CASE WRITERS......Page 9
DEDICATION......Page 10
A WEALTH OF SOFTWARE OPTIONS......Page 11
NEW EMPHASES......Page 13
OTHER FEATURES......Page 14
ACKNOWLEDGMENTS......Page 16
TABLE OF CONTENTS......Page 18
1.1 THE ORIGINS OF OPERATIONS RESEARCH......Page 26
1.2 THE NATURE OF OPERATIONS RESEARCH......Page 27
1.3 THE IMPACT OF OPERATIONS RESEARCH......Page 28
1.4 ALGORITHMS AND OR COURSEWARE......Page 30
2.1 DEFINING THE PROBLEM AND GATHERING DATA......Page 32
2.2 FORMULATING A MATHEMATICAL MODEL......Page 35
2.3 DERIVING SOLUTIONS FROM THE MODEL......Page 39
2.4 TESTING THE MODEL......Page 41
2.5 PREPARING TO APPLY THE MODEL......Page 43
2.6 IMPLEMENTATION......Page 45
2.7 CONCLUSIONS......Page 46
3 Introduction to Linear Programming......Page 49
3.1 PROTOTYPE EXAMPLE......Page 50
Formulation as a Linear Programming Problem......Page 51
Graphical Solution......Page 52
Continuing the Learning Process with Your OR Courseware......Page 55
3.2 THE LINEAR PROGRAMMING MODEL......Page 56
A Standard Form of the Model......Page 57
Terminology for Solutions of the Model......Page 58
Proportionality......Page 61
Additivity......Page 65
Divisibility......Page 67
The Assumptions in Perspective......Page 68
Design of Radiation Therapy......Page 69
Regional Planning......Page 71
Controlling Air Pollution......Page 75
Reclaiming Solid Wastes......Page 78
Personnel Scheduling......Page 82
Distributing Goods through a Distribution Network......Page 84
3.5 SOME CASE STUDIES......Page 86
Choosing the Product Mix at Ponderosa Industrial......Page 87
Personnel Scheduling at United Airlines......Page 88
Planning Supply, Distribution, and Marketing at Citgo Petroleum Corporation......Page 90
3.6 DISPLAYING AND SOLVING LINEAR PROGRAMMING MODELS ON A SPREADSHEET......Page 92
Displaying the Model on a Spreadsheet......Page 93
Using the Excel Solver to Solve the Model......Page 94
Modeling Languages......Page 98
The Structure of the Resulting Model......Page 99
Formulation of the Model in MPL......Page 101
The LINGO Modeling Language......Page 103
3.8 CONCLUSIONS......Page 104
Formulation of the Model in LINGO......Page 107
Importing and Exporting Spreadsheet Data with LINGO......Page 111
Importing and Exporting from a Database with LINGO......Page 112
More about LINGO......Page 114
Supplement to Appendix 3.1:......Page 115
4.1 THE ESSENCE OF THE SIMPLEX METHOD......Page 134
Solving the Example......Page 136
The Key Solution Concepts......Page 137
4.2 SETTING UP THE SIMPLEX METHOD......Page 139
Optimality Test......Page 143
Determining Where to Stop (Step 2 of an Iteration)......Page 145
Solving for the New BF Solution (Step 3 of an Iteration)......Page 146
Iteration 2 and the Resulting Optimal Solution......Page 147
4.4 THE SIMPLEX METHOD IN TABULAR FORM......Page 148
Summary of the Simplex Method (and Iteration 1 for the Example)......Page 150
Iteration 2 for the Example and the Resulting Optimal Solution......Page 152
Tie for the Entering Basic Variable......Page 153
No Leaving Basic Variable—Unbounded......Page 154
Multiple Optimal Solutions......Page 155
Equality Constraints......Page 157
Negative Right-Hand Sides......Page 161
Form......Page 162
Solving the Radiation Therapy Example......Page 165
The Two-Phase Method......Page 167
No Feasible Solutions......Page 173
Variables Allowed to Be Negative......Page 175
Reoptimization......Page 177
Shadow Prices......Page 178
Sensitivity Analysis......Page 181
Using Excel to Generate Sensitivity Analysis Information......Page 182
Parametric Linear Programming......Page 184
Implementation of the Simplex Method......Page 185
Linear Programming Software Featured in This Book......Page 186
4.9 THE INTERIOR-POINT APPROACH TO SOLVING LINEAR PROGRAMMING PROBLEMS......Page 188
The Key Solution Concept......Page 189
Comparison with the Simplex Method......Page 190
The Complementary Roles of the Simplex Method and the Interior-Point Approach......Page 192
4.10 CONCLUSIONS......Page 193
Files (Chapter 3) for Solving the Wyndor and Radiation Therapy Examples:......Page 197
Terminology......Page 215
Adjacent CPF Solutions......Page 218
Properties of CPF Solutions......Page 220
Extensions to the Augmented Form of the Problem......Page 223
5.2 THE REVISED SIMPLEX METHOD......Page 227
Solving for a Basic Feasible Solution......Page 229
Matrix Form of the Current Set of Equations......Page 231
The Overall Procedure......Page 233
General Observations......Page 236
5.3 A FUNDAMENTAL INSIGHT......Page 237
Mathematical Summary......Page 241
Adapting to Other Model Forms......Page 243
Applications......Page 244
5.4 CONCLUSIONS......Page 245
Files (Chapter 3) for Solving the Wyndor Example:......Page 246
6 Duality Theory and Sensitivity Analysis......Page 255
6.1 THE ESSENCE OF DUALITY THEORY......Page 256
Origin of the Dual Problem......Page 257
Summary of Primal-Dual Relationships......Page 261
Applications......Page 263
Interpretation of the Dual Problem......Page 264
Interpretation of the Simplex Method......Page 266
Complementary Basic Solutions......Page 267
Relationships between Complementary Basic Solutions......Page 270
6.4 ADAPTING TO OTHER PRIMAL FORMS......Page 272
Changes in the Coefficients of a Nonbasic Variable......Page 277
Introduction of a New Variable......Page 278
6.6 THE ESSENCE OF SENSITIVITY ANALYSIS......Page 279
Case 1—Changes in......Page 287
—Changes in the Coefficients of a Nonbasic Variable......Page 294
—Introduction of a New Variable......Page 298
Case 3—Changes in the Coefficients of a Basic Variable......Page 299
Case 4—Introduction of a New Constraint......Page 303
Systematic Sensitivity Analysis—Parametric Programming......Page 305
6.8 CONCLUSIONS......Page 309
Files (Chapter 3) for Solving the Wyndor Example:......Page 310
7.1 THE DUAL SIMPLEX METHOD......Page 334
7.2 PARAMETRIC LINEAR PROGRAMMING......Page 337
Parameters......Page 338
Parameters......Page 340
7.3 THE UPPER BOUND TECHNIQUE......Page 342
7.4 AN INTERIOR-POINT ALGORITHM......Page 345
The Relevance of the Gradient for Concepts 1 and 2......Page 346
Using the Projected Gradient to Implement Concepts 1 and 2......Page 348
A Centering Scheme for Implementing Concept 3......Page 350
Summary and Illustration of the Algorithm......Page 352
7.5 LINEAR GOAL PROGRAMMING AND ITS SOLUTION PROCEDURES......Page 357
Prototype Example for Nonpreemptive Goal Programming......Page 358
Preemptive Goal Programming......Page 360
The Sequential Procedure for Preemptive Goal Programming......Page 361
The Streamlined Procedure for Preemptive Goal Programming......Page 363
7.6 CONCLUSIONS......Page 364
An Automatic Routine:......Page 365
“Ch. 7—Other Algorithms for LP” Files for Solving the Examples:......Page 366
8 The Transportation and Assignment Problems......Page 375
Prototype Example......Page 376
The Transportation Problem Model......Page 379
Using Excel to Formulate and Solve Transportation Problems......Page 383
An Example with a Dummy Destination......Page 384
An Example with a Dummy Source......Page 387
Setting Up the Transportation Simplex Method......Page 390
Initialization......Page 393
Optimality Test......Page 400
An Iteration......Page 401
8.3 THE ASSIGNMENT PROBLEM......Page 406
Prototype Example......Page 407
The Assignment Problem Model and Solution Procedures......Page 408
Example—Assigning Products to Plants......Page 411
8.4 CONCLUSIONS......Page 416
Supplement to this Chapter:......Page 417
9 Network Optimization Models......Page 430
9.1 PROTOTYPE EXAMPLE......Page 431
9.2 THE TERMINOLOGY OF NETWORKS......Page 432
9.3 THE SHORTEST-PATH PROBLEM......Page 436
Applying This Algorithm to the Seervada Park Shortest-Path Problem......Page 437
Using Excel to Formulate and Solve Shortest-Path Problems......Page 438
9.4 THE MINIMUM SPANNING TREE PROBLEM......Page 440
Some Applications......Page 441
An Algorithm......Page 442
Applying This Algorithm to the Seervada Park Minimum Spanning Tree Problem......Page 443
9.5 THE MAXIMUM FLOW PROBLEM......Page 445
Some Applications......Page 446
An Algorithm......Page 447
Applying This Algorithm to the Seervada Park Maximum Flow Problem......Page 449
Finding an Augmenting Path......Page 451
Using Excel to Formulate and Solve Maximum Flow Problems......Page 453
Some Applications......Page 454
Formulation of the Model......Page 456
An Example......Page 458
Using Excel to Formulate and Solve Minimum Cost Flow Problems......Page 459
Special Cases......Page 460
Incorporating the Upper Bound Technique......Page 463
Correspondence between BF Solutions and Feasible Spanning Trees......Page 464
Selecting the Entering Basic Variable......Page 466
Finding the Leaving Basic Variable and the Next BF Solution......Page 469
9.8 CONCLUSIONS......Page 473
An Interactive Routine:......Page 474
“Ch. 9—Network Opt Models” Files for Solving the Examples:......Page 475
10 Project Management with PERT/CPM......Page 493
10.1 A PROTOTYPE EXAMPLE—THE RELIABLE CONSTRUCTION CO. PROJECT......Page 494
10.2 USING A NETWORK TO VISUALLY DISPLAY A PROJECT......Page 495
Project Networks......Page 496
Using Microsoft Project......Page 497
The Critical Path......Page 500
Scheduling Individual Activities......Page 502
Identifying Slack in the Schedule......Page 507
10.4 DEALING WITH UNCERTAIN ACTIVITY DURATIONS......Page 510
The PERT Three-Estimate Approach......Page 511
Three Simplifying Approximations......Page 512
Approximating the Probability of Meeting the Deadline......Page 516
10.5 CONSIDERING TIME-COST TRADE-OFFS......Page 517
Time-Cost Trade-Offs for Individual Activities......Page 518
Which Activities Should Be Crashed?......Page 519
Using Linear Programming to Make Crashing Decisions......Page 521
Mr. Perty’s Conclusions......Page 526
10.6 SCHEDULING AND CONTROLLING PROJECT COSTS......Page 527
Scheduling Project Costs......Page 528
Controlling Project Costs......Page 531
The Value of PERT/CPM......Page 533
Approximating the Means and Variances of Activity Durations......Page 534
Approximating the Probability of Meeting the Deadline......Page 535
Incorporating the Allocation of Resources to Activities......Page 536
10.8 CONCLUSIONS......Page 537
MS Project Folder:......Page 539
11.1 A PROTOTYPE EXAMPLE FOR DYNAMIC PROGRAMMING......Page 558
Solving the Problem......Page 559
11.2 CHARACTERISTICS OF DYNAMIC PROGRAMMING PROBLEMS......Page 563
11.3 DETERMINISTIC DYNAMIC PROGRAMMING......Page 566
A Prevalent Problem Type—The Distribution of Effort Problem......Page 572
11.4 PROBABILISTIC DYNAMIC PROGRAMMING......Page 587
“Ch. 11—Dynamic Programming” LINGO File......Page 593
12 Integer Programming......Page 601
12.1 PROTOTYPE EXAMPLE......Page 602
The BIP Model......Page 603
Software Options for Solving Such Models......Page 604
Capital Budgeting with Fixed Investment Proposals......Page 605
Designing a Production and Distribution Network......Page 606
Dispatching Shipments......Page 607
Scheduling Interrelated Activities......Page 608
Airline Applications......Page 609
12.3 INNOVATIVE USES OF BINARY VARIABLES IN MODEL FORMULATION......Page 610
Either-Or Constraints......Page 611
Constraints Must Hold......Page 612
Possible Values......Page 613
The Fixed-Charge Problem......Page 614
Binary Representation of General Integer Variables......Page 615
12.4 SOME FORMULATION EXAMPLES......Page 616
12.5 SOME PERSPECTIVES ON SOLVING INTEGER PROGRAMMING PROBLEMS......Page 625
12.6 THE BRANCH-AND-BOUND TECHNIQUE AND ITS APPLICATION TO BINARY INTEGER PROGRAMMING......Page 629
Branching......Page 630
Bounding......Page 631
Fathoming......Page 632
Completing the Example......Page 634
Other Options with the Branch-and-Bound Technique......Page 638
12.7 A BRANCH-AND-BOUND ALGORITHM FOR MIXED INTEGER PROGRAMMING......Page 641
Background......Page 647
Automatic Problem Preprocessing for Pure BIP......Page 649
Generating Cutting Planes for Pure BIP......Page 653
12.9 CONCLUSIONS......Page 655
“Ch. 12—Integer Programming” Files for Solving the Examples:......Page 656
13 Nonlinear Programming......Page 679
The Product-Mix Problem with Price Elasticity......Page 680
The Transportation Problem with Volume Discounts on Shipping Costs......Page 681
Portfolio Selection with Risky Securities......Page 683
13.2 GRAPHICAL ILLUSTRATION OF NONLINEAR PROGRAMMING PROBLEMS......Page 684
13.3 TYPES OF NONLINEAR PROGRAMMING PROBLEMS......Page 689
Quadratic Programming......Page 690
Separable Programming......Page 692
Fractional Programming......Page 693
The Complementarity Problem......Page 694
The One-Dimensional Search Procedure......Page 695
13.5 MULTIVARIABLE UNCONSTRAINED OPTIMIZATION......Page 698
The Gradient Search Procedure......Page 699
13.6 THE KARUSH-KUHN-TUCKER (KKT) CONDITIONS FOR CONSTRAINED OPTIMIZATION......Page 704
13.7 QUADRATIC PROGRAMMING......Page 708
The KKT Conditions for Quadratic Programming......Page 710
The Modified Simplex Method......Page 711
Some Software Options......Page 714
13.8 SEPARABLE PROGRAMMING......Page 715
Reformulation as a Linear Programming Problem......Page 717
Extensions......Page 721
13.9 CONVEX PROGRAMMING......Page 722
A Sequential Linear Approximation Algorithm (Frank-Wolfe)......Page 723
13.10 NONCONVEX PROGRAMMING......Page 727
Sequential Unconstrained Minimization Technique (SUMT)......Page 728
13.11 CONCLUSIONS......Page 731
“Ch. 13—Nonlinear Programming” Files for Solving the Examples:......Page 732
14.1 THE FORMULATION OF TWO-PERSON, ZERO-SUM GAMES......Page 751
Formulation as a Two-Person, Zero-Sum Game......Page 753
Variation 1 of the Example......Page 754
Variation 2 of the Example......Page 756
Variation 3 of the Example......Page 757
14.3 GAMES WITH MIXED STRATEGIES......Page 758
14.4 GRAPHICAL SOLUTION PROCEDURE......Page 760
14.5 SOLVING BY LINEAR PROGRAMMING......Page 763
14.6 EXTENSIONS......Page 766
14.7 CONCLUSIONS......Page 767
“Ch. 14—Game Theory” Files for Solving the Examples:......Page 768
15 Decision Analysis......Page 774
15.1 A PROTOTYPE EXAMPLE......Page 775
15.2 DECISION MAKING WITHOUT EXPERIMENTATION......Page 776
The Maximin Payoff Criterion......Page 777
The Maximum Likelihood Criterion......Page 778
Bayes’ Decision Rule......Page 779
Sensitivity Analysis with Bayes’ Decision Rule......Page 780
Continuing the Prototype Example......Page 783
Posterior Probabilities......Page 784
The Value of Experimentation......Page 787
15.4 DECISION TREES......Page 789
Constructing the Decision Tree......Page 790
Performing the Analysis......Page 792
Helpful Software......Page 794
15.5 UTILITY THEORY......Page 795
Utility Functions for Money......Page 796
Applying Utility Theory to the Goferbroke Co. Problem......Page 798
)......Page 800
Using a Decision Tree to Analyze the Goferbroke Co. Problem with Utilities......Page 801
15.6 THE PRACTICAL APPLICATION OF DECISION ANALYSIS......Page 803
15.7 CONCLUSIONS......Page 806
Excel Add-Ins:......Page 807
16.1 STOCHASTIC PROCESSES......Page 827
16.2 MARKOV CHAINS......Page 828
Formulating the Inventory Example as a Markov Chain......Page 830
Additional Examples of Markov Chains......Page 832
16.3 CHAPMAN-KOLMOGOROV EQUATIONS......Page 833
-Step Transition Matrices for the Inventory Example......Page 834
16.4 CLASSIFICATION OF STATES OF A MARKOV CHAIN......Page 835
Recurrent States and Transient States......Page 836
Steady-State Probabilities......Page 837
Expected Average Cost per Unit Time......Page 839
Expected Average Cost per Unit Time for Complex Cost Functions......Page 841
16.6 FIRST PASSAGE TIMES......Page 843
16.7 ABSORBING STATES......Page 845
Formulation......Page 847
Some Key Random Variables......Page 848
Steady-State Probabilities......Page 850
Automatic Routines in OR Courseware:......Page 853
17 Queueing Theory......Page 859
Input Source (Calling Population)......Page 860
Queue Discipline......Page 861
An Elementary Queueing Process......Page 862
Terminology and Notation......Page 864
17.3 EXAMPLES OF REAL QUEUEING SYSTEMS......Page 865
17.4 THE ROLE OF THE EXPONENTIAL DISTRIBUTION......Page 866
17.5 THE BIRTH-AND-DEATH PROCESS......Page 873
17.6 QUEUEING MODELS BASED ON THE BIRTH-AND-DEATH PROCESS......Page 877
Model......Page 878
Model)......Page 886
Model......Page 889
A Model with State-Dependent Service Rate and/or Arrival Rate......Page 891
/1 Model......Page 896
Model......Page 897
Model......Page 898
Models without a Poisson Input......Page 901
Other Models......Page 903
17.8 PRIORITY-DISCIPLINE QUEUEING MODELS......Page 904
The Models......Page 905
Results for the Nonpreemptive Priorities Model......Page 906
Results for the Preemptive Priorities Model......Page 907
The County Hospital Example with Priorities......Page 908
17.9 QUEUEING NETWORKS......Page 910
Infinite Queues in Series......Page 911
Jackson Networks......Page 912
17.10 CONCLUSIONS......Page 914
“Ch. 17—Queueing Theory” Excel File:......Page 915
“Ch. 17—Queueing Theory” LINGO File for Selected Examples......Page 916
Example 1—How Many Repairers?......Page 932
Example 3—How Many Tool Cribs?......Page 933
18.2 DECISION MAKING......Page 934
) Form......Page 937
) Form......Page 939
and......Page 942
and......Page 945
18.5 SOME AWARD-WINNING APPLICATIONS OF QUEUEING THEORY......Page 948
Supplement to This Chapter:......Page 951
19 Inventory Theory......Page 960
19.1 EXAMPLES......Page 961
19.2 COMPONENTS OF INVENTORY MODELS......Page 963
19.3 DETERMINISTIC CONTINUOUS-REVIEW MODELS......Page 966
The Basic EOQ Model......Page 967
The EOQ Model with Planned Shortages......Page 968
The EOQ Model with Quantity Discounts......Page 971
Some Useful Excel Templates......Page 972
Observations about EOQ Models......Page 973
A Broader Perspective of the Speaker Example......Page 974
19.4 A DETERMINISTIC PERIODIC-REVIEW MODEL......Page 976
An Algorithm......Page 978
Application of the Algorithm to the Example......Page 980
19.5 A STOCHASTIC CONTINUOUS-REVIEW MODEL......Page 981
The Assumptions of the Model......Page 982
Choosing the Reorder Point......Page 983
19.6 A STOCHASTIC SINGLE-PERIOD MODEL FOR PERISHABLE PRODUCTS......Page 986
Some Types of Perishable Products......Page 987
An Example......Page 988
Analysis of the Model......Page 990
Application to the Example......Page 992
The Model with Initial Stock Level......Page 993
A Single-Period Model with a Setup Cost......Page 997
19.7 STOCHASTIC PERIODIC-REVIEW MODELS......Page 1000
A Stochastic Two-Period Model with No Setup Cost......Page 1001
Stochastic Multiperiod Models—An Overview......Page 1005
Multiproduct Inventory Systems......Page 1008
Multiechelon Inventory Management at IBM......Page 1009
Supply Chain Management at Hewlett-Packard......Page 1010
“Ch. 19—Inventory Theory” Excel File:......Page 1012
“Ch. 19—Inventory Theory” LINGO File for Selected Examples......Page 1013
20 Forecasting......Page 1034
Forecasting the Need for Spare Parts......Page 1035
Forecasting Economic Trends......Page 1036
Forecasting Staffing Needs......Page 1037
20.2 JUDGMENTAL FORECASTING METHODS......Page 1038
20.3 TIME SERIES......Page 1039
Last-Value Forecasting Method......Page 1041
Exponential Smoothing Forecasting Method......Page 1042
20.5 INCORPORATING SEASONAL EFFECTS INTO FORECASTING METHODS......Page 1043
The General Procedure......Page 1045
20.6 AN EXPONENTIAL SMOOTHING METHOD FOR A LINEAR TREND MODEL......Page 1046
Adapting Exponential Smoothing to This Model......Page 1047
Application of the Method to the CCW Example......Page 1048
20.7 FORECASTING ERRORS......Page 1050
20.8 BOX-JENKINS METHOD......Page 1051
Causal Forecasting......Page 1053
Linear Regression......Page 1054
Method of Least Squares......Page 1055
*)......Page 1058
Predictions......Page 1059
20.10 FORECASTING IN PRACTICE......Page 1061
“Ch. 20—Forecasting” Excel File:......Page 1063
“Ch. 20—Forecasting” LINGO File for Selected Examples......Page 1064
21.1 A PROTOTYPE EXAMPLE......Page 1078
21.2 A MODEL FOR MARKOV DECISION PROCESSES......Page 1081
Solving the Prototype Example by Exhaustive Enumeration......Page 1082
21.3 LINEAR PROGRAMMING AND OPTIMAL POLICIES......Page 1084
Randomized Policies......Page 1085
A Linear Programming Formulation......Page 1086
Solving the Prototype Example by Linear Programming......Page 1088
Preliminaries......Page 1089
Summary of the Policy Improvement Algorithm......Page 1090
Solving the Prototype Example by the Policy Improvement Algorithm......Page 1091
21.5 DISCOUNTED COST CRITERION......Page 1094
A Policy Improvement Algorithm......Page 1095
Linear Programming Formulation......Page 1098
Finite-Period Markov Decision Processes and the Method of Successive Approximations......Page 1100
21.6 CONCLUSIONS......Page 1102
“Ch. 21—Markov Decision Proc” Files for Solving the Linear Programming Formulations:......Page 1103
22.1 ESSENCE OF SIMULATION......Page 1109
The Role of Simulation in Operations Research Studies......Page 1110
Discrete-Event versus Continuous Simulation......Page 1111
Design and Operation of Queueing Systems......Page 1122
Estimating the Probability of Completing a Project by the Deadline......Page 1123
Financial Risk Analysis......Page 1124
Applications to Other Service Industries......Page 1125
22.3 GENERATION OF RANDOM NUMBERS......Page 1126
Characteristics of Random Numbers......Page 1127
Congruential Methods for Random Number Generation......Page 1128
Simple Discrete Distributions......Page 1130
The Inverse Transformation Method......Page 1131
Exponential and Erlang Distributions......Page 1132
Normal and Chi-Square Distributions......Page 1133
The Acceptance-Rejection Method......Page 1134
22.5 OUTLINE OF A MAJOR SIMULATION STUDY......Page 1135
Step 2: Collect the Data and Formulate the Simulation Model......Page 1136
Step 4: Select the Software and Construct a Computer Program......Page 1137
Step 5: Test the Validity of the Simulation Model......Page 1138
Step 7: Conduct the Simulation Runs and Analyze the Results......Page 1139
22.6 PERFORMING SIMULATIONS ON SPREADSHEETS......Page 1140
Inventory Management—Freddie the Newsboy’s Problem......Page 1141
Improving PERT—Revisiting the Reliable Construction Co. Problem......Page 1143
Financial Risk Analysis—The Think-Big Development Co. Problem......Page 1147
22.7 VARIANCE-REDUCING TECHNIQUES......Page 1151
Stratified Sampling......Page 1152
Method of Complementary Random Numbers......Page 1154
Conclusions......Page 1155
The Regenerative Method Approach......Page 1156
Statistical Formulas......Page 1160
Application of the Statistical Formulas to the Example......Page 1161
22.9 CONCLUSIONS......Page 1163
Demonstration Examples in OR Tutor:......Page 1165
Excel Add-Ins:......Page 1166
OR TUTOR......Page 1181
EXCEL ADD-INS......Page 1182
UPDATES......Page 1183
OF A SINGLE VARIABLE......Page 1184
OF SEVERAL VARIABLES......Page 1186
CONVEX SETS......Page 1188
FUNCTION OF A SINGLE VARIABLE......Page 1190
FUNCTION OF SEVERAL VARIABLES......Page 1191
THE DERIVATIVE OF A DEFINITE INTEGRAL......Page 1192
MATRIX OPERATIONS......Page 1194
VECTORS......Page 1196
SOME PROPERTIES OF MATRICES......Page 1197
APPENDIX 5 TABLES......Page 1199
CHAPTER 3......Page 1201
CHAPTER 4......Page 1202
CHAPTER 6......Page 1203
CHAPTER 7......Page 1204
CHAPTER 8......Page 1205
CHAPTER 10......Page 1206
CHAPTER 11......Page 1207
CHAPTER 12......Page 1208
CHAPTER 13......Page 1209
CHAPTER 14......Page 1210
CHAPTER 15......Page 1211
CHAPTER 17......Page 1213
CHAPTER 19......Page 1214
CHAPTER 21......Page 1216
CHAPTER 22......Page 1217
AUTHOR INDEX......Page 1219
SUBJECT INDEX......Page 1222