Applied integer programming: modeling and solution
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Hoboken, N.J.
Wiley
2010
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 468 S. graph. Darst. |
| ISBN: | 9780470373064 |
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| 245 | 1 | 0 | |a Applied integer programming |b modeling and solution |c Der-San Chen ; Robert G. Batson ; Yu Dang |
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| _version_ | 1804140993423867904 |
|---|---|
| adam_text | Titel: Applied integer programming
Autor: Chen, Der-San
Jahr: 2010
CONTENTS
PREFACE xvii
PART I MODELING I
1 Introduction 3
1.1 Integer Programming, 3
1.2 Standard Versus Nonstandard Forms, 5
1.3 Combinatorial Optimization Problems, 7
1.4 Successful Integer Programming Applications, 8
1.5 Text Organization and Chapter Preview, 8
1.6 Notes, 17
1.7 Exercises, 18
2 Modeling and Models 21
2.1 Assumptions on Mixed Integer Programs, 22
2.2 Modeling Process, 28
2.3 Project Selection Problems, 30
2.3.1 Knapsack Problem, 30
2.3.2 Capital Budgeting Problem, 31
2.4 Production Planning Problems, 32
2.4.1 Uncapacitated Lot Sizing, 33
2.4.2 Capacitated Lot Sizing, 34
2.4.3 Just-in-Time Production Planning, 34
vii
viii CONTENTS
2.5 Workforce/Staff Scheduling Problems, 36
2.5.1 Scheduling Full-Time Workers, 36
2.5.2 Scheduling Full-Time and Part-Time Workers, 37
2.6 Fixed-Charge Transportation and Distribution Problems, 38
2.6.1 Fixed-Charge Transportation, 38
2.6.2 Uncapacitated Facility Location, 40
2.6.3 Capacitated Facility Location, 41
2.7 Multicommodity Network Flow Problem, 41
2.8 Network Optimization Problems with Side Constraints, 43
2.9 Supply Chain Planning Problems, 44
2.10 Notes, 47
2.11 Exercises, 48
3 Transformation Using 0-1 Variables 54
3.1 Transform Logical (Boolean) Expressions, 55
3.1.1 Truth Table of Boolean Operations, 55
3.1.2 Basic Logical (Boolean) Operations on Variables, 56
3.1.3 Multiple Boolean Operations on Variables, 58
3.2 Transform Nonbinary to 0-1 Variable, 58
3.2.1 Transform Integer Variable, 58
3.2.2 Transform Discrete Variable, 60
3.3 Transform Piecewise Linear Functions, 60
3.3.1 Arbitrary Piecewise Linear Functions, 60
3.3.2 Concave Piecewise Linear Cost Functions:
Economy of Scale, 63
3.4 Transform 0-1 Polynomial Functions, 64
3.5 Transform Functions with Products of Binary and Continuous
Variables: Bundle Pricing Problem, 66
3.6 Transform Nonsimultaneous Constraints, 69
3.6.1 Either/Or Constraints, 69
3.6.2 ñ Out of m Constraints Must Hold, 70
3.6.3 Disjunctive Constraint Sets, 71
3.6.4 Negation of a Constraint, 71
3.6.5 If/Then Constraints, 71
3.7 Notes, 72
3.8 Exercises, 73
4 Better Formulation by Preprocessing 79
4.1 Better Formulation, 79
4.2 Automatic Problem Preprocessing, 86
4.3 Tightening Bounds on Variables, 87
4.3.1 Bounds on Continuous Variables, 87
4.3.2 Bounds on General Integer Variables, 88
4.3.3 Bounds on 0-1 Variables, 90
CONTENTS ix
4.3.4 Variable Fixing, Redundant Constraints,
and Infeasibility, 91
4.4 Preprocessing Pure 0-1 Integer Programs, 93
4.4.1 Fixing 0-1 Variables, 93
4.4.2 Detecting Redundant Constraints And Infeasibility, 95
4.4.3 Tightening Constraints (or Coefficients Reduction), 96
4.4.4 Generating Cutting Planes from Minimum Cover, 97
4.4.5 Rounding by Division with GCD, 98
4.5 Decomposing a Problem into Independent Subproblems, 99
4.6 Scaling the Coefficient Matrix, 100
4.7 Notes, 101
4.8 Exercises, 101
5 Modeling Combinatorial Optimization Problems I 105
5.1 Introduction, 105
5.2 Set Covering and Set Partitioning, 106
5.2.1 Set Covering Problem, 107
5.2.2 Set Partitioning and Set Packing, 111
5.2.3 Set Covering in Networks, 111
5.2.4 Applications of Set Covering Problem, 113
5.3 Matching Problem, 115
5.3.1 Matching Problems in Network, 115
5.3.2 Integer Programming Formulation, 116
5.4 Cutting Stock Problem, 117
5.4.1 One-Dimensional Case, 117
5.4.2 Two-Dimensional Case, 120
5.5 Comparisons for Above Problems, 121
5.6 Computational Complexity of COP, 121
5.6.1 Problem Versus Problem Instance, 123
5.6.2 Computational Complexity of an Algorithm, 123
5.6.3 Polynomial Versus Nonpolynomial Function, 124
5.7 Notes, 125
5.8 Exercises, 126
6 Modeling Combinatorial Optimization Problems II 130
6.1 Importance of Traveling Salesman Problem, 130
6.2 Transformations to Traveling Salesman Problem, 133
6.2.1 Shortest Hamiltonian Paths, 133
6.2.2 TSP with Repeated City Visits, 134
6.2.3 Multiple Traveling Salesmen Problem, 135
6.2.4 Clustered TSP, 137
6.2.5 Generalized TSP, 137
6.2.6 Maximum TSP, 139
÷ CONTENTS
6.3 Applications of TSP, 139
6.3.1 Machine Sequencing Problems in Various
Manufacturing Systems, 140
6.3.2 Sequencing Problems in Electronic Industry, 140
6.3.3 Vehicle Routing for Delivery/Dispatching, 141
6.3.4 Genome Sequencing for Genetic Study, 142
6.4 Formulating Asymmetric TSP, 142
6.4.1 Subtour Elimination by Dantzig-Fulkerson-
Johnson Constraints, 143
6.4.2 Subtour Elimination by Miller-Tucker-Zemlin
(MTZ) Constraints, 144
6.5 Formulating Symmetric TSP, 146
6.6 Notes, 148
6.7 Exercises, 149
PART II REVIEW OF LINEAR PROGRAMMING
AND NETWORK FLOWS 153
7 Linear Programming—Fundamentals 155
7.1 Review of Basic Linear Algebra, 155
7.1.1 Euclidean Space, 155
7.1.2 Linear and Convex Combinations, 156
7.1.3 Linear Independence, 156
7.1.4 Rank of a Matrix, 156
7.1.5 Basis, 157
7.1.6 Matrix Inversion, 157
7.1.7 Determinant of a Matrix, 157
7.1.8 Upper and Lower Triangular Matrices, 158
7.2 Uses of Elementary Row Operations, 159
7.2.1 Finding the Rank of a Matrix, 159
7.2.2 Calculating the Inverse of a Matrix, 160
7.2.3 Converting to a Triangular Matrix, 161
7.2.4 Calculating the Determinant of a Matrix, 162
7.2.5 Solving a System of Linear Equations, 162
7.3 The Dual Linear Program, 165
7.3.1 The Linear Program in Standard Form, 166
7.3.2 Formulating the Dual Problem, 167
7.3.3 Economic Interpretation of the Dual, 170
7.3.4 Importance of the Dual, 171
7.4 Relationships Between Primal and Dual Solutions, 171
7.4.1 Relationships Between All Primal and All
Dual Feasible Solutions, 171
7.4.2 Relationship Between Primal and Dual
Optimum Solutions, 172
CONTENTS XI
7.4.3 Relationships Between Each Complementary Pair
of Variables at Optimum, 173
7.5 Notes, 175
7.6 Exercises, 176
8 Linear Programming: Geometric Concepts 180
8.1 Geometric Solution, 180
8.1.1 Objective Function, 181
8.1.2 Solution Space, 181
8.1.3 Requirement Space, 183
8.2 Convex Sets, 188
8.2.1 Convex Sets and Polyhedra, 188
8.2.2 Directions of Unbounded Convex Sets, 191
8.2.3 Convex and Polyhedral Cones, 191
8.2.4 Convex and Concave Functions, 192
8.3 Describing a Bounded Polyhedron, 194
8.3.1 Representation by Extreme Points, 194
8.3.2 Example Application of Representation Theorem, 194
8.4 Describing Unbounded Polyhedron, 195
8.4.1 Finding Extreme Direction Algebraically, 195
8.4.2 Representing by Extreme Points and Extreme Directions, 199
8.4.3 Example of Representation Theorem, 199
8.5 Faces, Facets, and Dimension of a Polyhedron, 199
8.6 Describing a Polyhedron by Facets, 201
8.7 Correspondence Between Algebraic and Geometric Terms, 202
8.8 Notes, 203
8.9 Exercises, 203
9 Linear Programming: Solution Methods 207
9.1 Linear Programs in Canonical Form, 207
9.2 Basic Feasible Solutions and Reduced Costs, 209
9.2.1 Basic Feasible Solution, 209
9.2.2 Adjacent Basic Feasible Solution, 211
9.2.3 Reduced Costs, 212
9.3 The Simplex Method, 213
9.3.1 Better and Feasible Solution, 213
9.3.2 Updating Simplex Tableau by Pivoting, 215
9.3.3 Optimality Test, 216
9.3.4 Initial Basic Feasible Solution, 216
9.4 Interpreting the Simplex Tableau, 218
9.4.1 Entire Simplex Tableau, 218
9.4.2 Rows of Simplex Tableau, 218
9.4.3 Columns of Simplex Tableau, 219
9.4.4 Pivot Column and Pivot Row, 219
xii CONTENTS
9.4.5 Predicting the New Objective Value Before Updating, 219
9.5 Geometric Interpretation of the Simplex Method, 220
9.5.1 Basic Feasible Solution Versus Extreme Point, 220
9.5.2 Explanation of Simplex Method Nomenclature, 222
9.5.3 Identifying an Extreme Ray in a Simplex Tableau, 223
9.6 The Simplex Method for Upper Bounded Variables, 227
9.7 The Dual Simplex Method, 231
9.8 The Revised Simplex Method, 233
9.9 Notes, 239
9.10 Exercises, 240
10 Network Optimization Problems and Solutions 246
10.1 Network Fundamentals, 247
10.2 A Class of Easy Network Problems, 248
10.2.1 The Minimum Cost Network Flow Problem, 249
10.2.2 Formulating the Transportation-Assignment Problem
as an MCNF Problem, 249
10.2.3 Formulating the Transshipment Problem
as an MCNF Problem, 251
10.2.4 Formulating the Maximum Flow Problem
as an MCNF Problem, 251
10.2.5 Formulating the Shortest Path Problem
as an MCNF Problem, 251
10.3 Totally Unimodular Matrices, 252
10.3.1 Definition, 252
10.3.2 Sufficient Condition for a Totally Unimodular Matrix, 252
10.3.3 Some Properties of Totally Unimodular Matrices, 254
10.3.4 Matrix Structure of the MCNF Problem, 254
10.3.5 Lower Triangular Matrix and Forward Substitution, 255
10.3.6 Naturally Integer Solution for the MCNF Problem, 255
10.4 The Network Simplex Method, 256
10.4.1 Feasible Spanning Trees Versus Basic Feasible Solutions, 256
10.4.2 The Network Algorithm, 257
10.4.3 Numerical Example, 258
10.5 Solution via LINGO, 264
10.6 Notes, 264
10.7 Exercises, 265
PART III SOLUTIONS 269
11 Classical Solution Approaches 271
11.1 Branch-and-Bound Approach, 272
11.1.1 Basic Concepts, 272
11.1.2 Branch-and-Bound Algorithm, 278
CONTENTS xiii
11.2 Cutting Plane Approach, 280
11.2.1 Dual Cutting Plane Approach, 280
11.2.2 Fractional Cutting Plane Method, 281
11.2.3 Mixed Integer Cutting Plane Method, 285
11.3 Group Theoretic Approach, 286
11.3.1 Group Theory Terminology, 287
11.3.2 Deriving the Group (Minimization) Problem, 288
11.3.3 Formulating a Group Problem, 290
11.3.4 Solving Group Problem as a Shortest
Route Problem, 291
11.3.5 Solving the Original Integer Program, 293
11.4 Geometric Concepts, 294
11.4.1 Various Polyhedrons in Original Space, 295
11.4.2 Corner Polyhedron in Solution Space
of Nonbasic Variables, 297
11.5 Notes, 299
11.6 Exercises, 300
12 Branch-and-Cut Approach 305
12.1 Introduction, 306
12.1.1 Basic Concept, 306
12.1.2 Branch-and-Cut Algorithm, 306
12.1.3 Generating Valid Cuts and Preprocessing, 307
12.2 Valid Inequalities, 308
12.2.1 Valid Inequalities for Linear Programs, 308
12.2.2 Valid Inequalities for Integer Programs, 308
12.2.3 Types of Valid Inequalities, 308
12.3 Cut Generating Techniques, 309
12.3.1 Rounding Technique, 310
12.3.2 Disjunction Technique, 310
12.3.3 Lifting Technique, 312
12.4 Cuts Generated from Sets Involving Pure Integer Variables, 313
12.4.1 Gomory Fractional Cut, 313
12.4.2 Chvátal-Gomory Cut, 313
12.4.3 Pure Integer Rounding Cut, 314
12.4.4 Objective Integrality Cut, 315
12.5 Cuts Generated from Sets Involving Mixed Integer Variables, 315
12.5.1 Gomory Mixed Integer Cut, 315
12.5.2 Mixed Integer Rounding Cut, 319
12.6 Cuts Generated from 0-1 Knapsack Sets, 320
12.6.1 Knapsack Cover, 320
12.6.2 Lifted Knapsack Cover, 321
12.6.3 GUB Cover, 323
12.7 Cuts Generated from Sets Containing 0-1 Coefficients
and 0-1 Variables, 324
xiv CONTENTS
12.8 Cuts Generated from Sets with Special Structures, 326
12.8.1 Flow Cover from Fixed-Charge Flow Network, 326
12.8.2 Plant/Facility Location (Fixed-Charge Transportation), 327
12.9 Notes, 329
12.10 Exercises, 330
13 Branch-and-Price Approach 334
13.1 Concepts of Branch-and-Price, 334
13.2 Dantzig-Wolfe Decomposition, 335
13.3 Generalized Assignment Problem, 344
13.3.1 Conventional Formulation, 345
13.3.2 Column Generation Formulation, 345
13.3.3 Initial Solution, 348
13.4 GAP Example, 348
13.4.1 GAP Branching Scheme, 353
13.4.2 Tailing-Off Effect of Column Generation, 353
13.4.3 Treatment of Identical Machines, 354
13.4.4 Branch-and-Price Algorithm, 356
13.5 Other Application Areas, 356
13.6 Notes, 357
13.7 Exercises, 358
14 Solution via Heuristics, Relaxations, and Partitioning 359
14.1 Introduction, 359
14.2 Overall Solution Strategy, 359
14.2.1 Better Formulation by Preprocessing, 360
14.2.2 LP-Based Branch-and-Bound Framework, 361
14.2.3 Heuristics for Tightening Lower Bounds, 361
14.2.4 Relaxations for Tightening Upper Bounds, 362
14.2.5 Strong Cuts for Tightening Solution Polyhedron, 362
14.3 Primal Solution via Heuristics, 363
14.3.1 Local Search Approaches, 364
14.3.2 Artificial Intelligence Approaches, 366
14.4 Dual Solution via Relaxation, 373
14.4.1 Linear Programming Relaxation, 373
14.4.2 Combinatorial Relaxation, 374
14.4.3 Lagrangian Relaxation, 376
14.5 Lagrangian Dual, 377
14.5.1 Lagrangian Dual in LP, 378
14.5.2 Lagrangian Dual in IP, 378
14.5.3 Properties of the Lagrangian Dual, 379
14.6 Primal-Dual Solution via Benders Partitioning, 380
14.7 Notes, 383
14.8 Exercises, 383
CONTENTS XV
15 Solutions with Commercial Software 386
15.1 Introduction, 387
15.2 Typical IP Software Components, 388
15.2.1 Solvers, 388
15.2.2 Presolvers, 389
15.2.3 Modeling Languages, 389
15.2.4 User s Options/Intervention, 390
15.2.5 Data and Application Interfaces, 391
15.3 The AMPL Modeling Language, 392
15.3.1 Components of the AMPL Modeling Language, 392
15.3.2 An AMPL Example: the Diet Problem, 393
15.3.3 Enhanced AMPL Modeling Techniques, 397
15.3.4 AMPL Compatible MIP Solvers, 400
15.4 LINGO Modeling Language, 400
15.4.1 Prescription of Tolerances, 401
15.4.2 Presolver—Automatic Problem Reduction, 402
15.4.3 Solvers for Linear/Integer Programming, 402
15.4.4 Interfacing with the User, 403
15.4.5 LINGO Modeling Conventions, 403
15.4.6 LINGO Model for the Diet Problem, 404
15.5 MPL Modeling Language, 405
15.5.1 MPL Modeling Conventions, 406
15.5.2 MPL Model for the Diet Problem, 408
15.5.3 MPL Compatible MIP Solvers, 409
REFERENCES 411
APPENDIX: ANSWERS TO SELECTED EXERCISES 423
INDEX 459
|
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| author | Chen, Der-San 1940- Batson, Robert G. 1950- Dang, Yu 1977- |
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| spelling | Chen, Der-San 1940- Verfasser (DE-588)141509481 aut Applied integer programming modeling and solution Der-San Chen ; Robert G. Batson ; Yu Dang Hoboken, N.J. Wiley 2010 XIX, 468 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Integer programming Ganzzahlige Optimierung (DE-588)4155950-2 gnd rswk-swf Ganzzahlige Optimierung (DE-588)4155950-2 s DE-604 Batson, Robert G. 1950- Verfasser (DE-588)141509414 aut Dang, Yu 1977- Verfasser (DE-588)141509252 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018864798&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chen, Der-San 1940- Batson, Robert G. 1950- Dang, Yu 1977- Applied integer programming modeling and solution Integer programming Ganzzahlige Optimierung (DE-588)4155950-2 gnd |
| subject_GND | (DE-588)4155950-2 |
| title | Applied integer programming modeling and solution |
| title_auth | Applied integer programming modeling and solution |
| title_exact_search | Applied integer programming modeling and solution |
| title_full | Applied integer programming modeling and solution Der-San Chen ; Robert G. Batson ; Yu Dang |
| title_fullStr | Applied integer programming modeling and solution Der-San Chen ; Robert G. Batson ; Yu Dang |
| title_full_unstemmed | Applied integer programming modeling and solution Der-San Chen ; Robert G. Batson ; Yu Dang |
| title_short | Applied integer programming |
| title_sort | applied integer programming modeling and solution |
| title_sub | modeling and solution |
| topic | Integer programming Ganzzahlige Optimierung (DE-588)4155950-2 gnd |
| topic_facet | Integer programming Ganzzahlige Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018864798&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT chendersan appliedintegerprogrammingmodelingandsolution AT batsonrobertg appliedintegerprogrammingmodelingandsolution AT dangyu appliedintegerprogrammingmodelingandsolution |