AMPL : a modeling language for mathematical programming

AMPL is a language for large-scale optimization and mathematical programming problems in production, distribution, blending, scheduling, and many other applications. Combining familiar algebraic notation and a powerful interactive command environment, AMPL makes it easy to create models, use a wide variety of solvers, and examine solutions. Though flexible and convenient for rapid prototyping and development of models, AMPL also offers the speed and generality needed for repeated large-scale production runs. This book, written by the creators of AMPL, is a complete guide for modelers at all levels of experience. It begins with a tutorial on widely used linear programming models, and presents all of AMPL's features for linear programming with extensive examples. Additional chapters cover network, nonlinear, piecewise-linear, and integer programming; database and spreadsheet interactions; and command scripts. Most chapters include exercises. Download free versions of AMPL and several solvers from www.ampl.com for experimentation, evaluation, and education. The Web site also lists vendors of the commercial version of AMPL and numerous solvers.

Introduction	p. xv
Chapter 1. Production Models: Maximizing Profits	p. 1
1.1 A two-variable linear program	p. 2
1.2 The two-variable linear program in AMPL	p. 5
1.3 A linear programming model	p. 6
1.4 The linear programming model in AMPL	p. 7
The basic model	p. 8
An improved model	p. 10
Catching errors	p. 12
1.5 Adding lower bounds to the model	p. 13
1.6 Adding resource constraints to the model	p. 15
1.7 AMPL interfaces	p. 18
Chapter 2. Diet and Other Input Models: Minimizing Costs	p. 27
2.1 A linear program for the diet problem	p. 27
2.2 An AMPL model for the diet problem	p. 30
2.3 Using the AMPL diet model	p. 32
2.4 Generalizations to blending, economics and scheduling	p. 37
Chapter 3. Transportation and Assignment Models	p. 43
3.1 A linear program for the transportation problem	p. 44
3.2 An AMPL model for the transportation problem	p. 45
3.3 Other interpretations of the transportation model	p. 49
Chapter 4. Building Larger Models	p. 55
4.1 A multicommodity transportation model	p. 56
4.2 A multiperiod production model	p. 59
4.3 A model of production and transportation	p. 63
Chapter 5. Simple Sets and Indexing	p. 73
5.1 Unordered sets	p. 73
5.2 Sets of numbers	p. 75
5.3 Set operations	p. 76
5.4 Set membership operations and functions	p. 78
5.5 Indexing expressions	p. 79
5.6 Ordered sets	p. 82
Predefined sets and interval expressions	p. 86
Chapter 6. Compound Sets and Indexing	p. 91
6.1 Sets of ordered pairs	p. 91
6.2 Subsets and slices of ordered pairs	p. 93
6.3 Sets of longer tuples	p. 96
6.4 Operations on sets of tuples	p. 98
6.5 Indexed collections of sets	p. 100
Chapter 7. Parameters and Expressions	p. 109
7.1 Parameter declarations	p. 110
7.2 Arithmetic expressions	p. 111
7.3 Logical and conditional expressions	p. 114
7.4 Restrictions on parameters	p. 116
7.5 Computed parameters	p. 118
7.6 Randomly generated parameters	p. 121
7.7 Logical parameters	p. 122
7.8 Symbolic parameters	p. 123
Chapter 8. Linear Programs: Variables, Objectives and Constraints	p. 129
8.1 Variables	p. 129
8.2 Linear expressions	p. 132
8.3 Objectives	p. 134
8.4 Constraints	p. 137
Chapter 9. Specifying Data	p. 143
9.1 Formatted data: the data command	p. 143
9.2 Data in lists	p. 145
Lists of one-dimensional sets and parameters	p. 145
Lists of two-dimensional sets and parameters	p. 146
Lists of higher-dimensional sets and parameters	p. 148
Combined lists of sets and parameters	p. 151
9.3 Data in tables	p. 154
Two-dimensional tables	p. 154
Two-dimensional slices of higher-dimensional data	p. 156
Higher-dimensional tables	p. 157
Choice of format	p. 159
9.4 Other features of data statements	p. 160
Default values	p. 160
Indexed collections of sets	p. 161
Initial values for variables	p. 162
9.5 Reading unformatted data: the read command	p. 163
Chapter 10. Database Access	p. 169
10.1 General principles of data correspondence	p. 169
10.2 Examples of table-handling statements	p. 174
10.3 Reading data from relational tables	p. 180
Reading parameters only	p. 180
Reading a set and parameters	p. 182
Establishing correspondences	p. 184
Reading other values	p. 185
10.4 Writing data to relational tables	p. 186
Writing rows inferred from the data specifications	p. 186
Writing rows inferred from a key specification	p. 189
10.5 Reading and writing the same table	p. 191
Reading and writing using two table declarations	p. 192
Reading and writing using the same table declaration	p. 193
10.6 Indexed collections of tables and columns	p. 193
Indexed collections of tables	p. 193
Indexed collections of data columns	p. 196
10.7 Standard and built-in table handlers	p. 197
Using the standard ODBC table handler	p. 198
Using the standard ODBC table handler with Access and Excel	p. 200
Built-in table handlers for text and binary files	p. 201
Chapter 11. Modeling Commands	p. 203
11.1 General principles of commands and options	p. 203
Commands	p. 204
Options	p. 204
11.2 Setting up and solving models and data	p. 206
Entering models and data	p. 206
Solving a model	p. 207
11.3 Modifying data	p. 209
Resetting	p. 209
Resampling	p. 209
The let command	p. 210
11.4 Modifying models	p. 212
Removing or redefining model components	p. 213
Changing the model: fix, unfix; drop, restore	p. 214
Relaxing integrality	p. 215
Chapter 12. Display Commands	p. 219
12.1 Browsing through results: the display command	p. 219
Displaying sets	p. 220
Displaying parameters and variables	p. 220
Displaying indexed expressions	p. 224
12.2 Formatting options for display	p. 227
Arrangement of lists and tables	p. 227
Control of line width	p. 229
Suppression of zeros	p. 231
12.3 Numeric options for display	p. 232
Appearance of numeric values	p. 233
Rounding of solution values	p. 236
12.4 Other output commands: print and printf	p. 238
The print command	p. 238
The printf command	p. 239
12.5 Related solution values	p. 240
Objective functions	p. 240
Bounds and slacks	p. 241
Dual values and reduced costs	p. 243
12.6 Other display features for models and instances	p. 245
Displaying model components: the show command	p. 246
Displaying model dependencies: the xref command	p. 247
Displaying model instances: the expand command	p. 247
Generic synonyms for variables, constraints and objectives	p. 249
Resource listings	p. 250
12.7 General facilities for manipulating output	p. 251
Redirection of output	p. 251
Output logs	p. 251
Limits on messages	p. 252
Chapter 13. Command Scripts	p. 255
13.1 Running scripts: include and commands	p. 255
13.2 Iterating over a set: the for statement	p. 258
13.3 Iterating subject to a condition: the repeat statement	p. 262
13.4 Testing a condition: the if-then-else statement	p. 264
13.5 Terminating a loop: break and continue	p. 266
13.6 Stepping through a script	p. 268
13.7 Manipulating character strings	p. 270
String functions and operators	p. 270
String expressions in AMPL commands	p. 273
Chapter 14. Interactions with Solvers	p. 275
14.1 Presolve	p. 275
Activities of the presolve phase	p. 276
Controlling the effects of presolve	p. 278
Detecting infeasibility in presolve	p. 279
14.2 Retrieving results from solvers	p. 282
Solve results	p. 282
Solver statuses of objectives and problems	p. 286
Solver statuses of variables	p. 287
Solver statuses of constraints	p. 291
AMPL statuses	p. 293
14.3 Exchanging information with solvers via suffixes	p. 295
User-defined suffixes: integer programming directives	p. 296
Solver-defined suffixes: sensitivity analysis	p. 298
Solver-defined suffixes: infeasibility diagnosis	p. 299
Solver-defined suffixes: direction of unboundedness	p. 300
Defining and using suffixes	p. 302
14.4 Alternating between models	p. 304
14.5 Named problems	p. 309
Defining named problems	p. 311
Using named problems	p. 314
Displaying named problems	p. 315
Defining and using named environments	p. 316
Chapter 15. Network Linear Programs	p. 319
15.1 Minimum-cost transshipment models	p. 319
A general transshipment model	p. 320
Specialized transshipment models	p. 323
Variations on transshipment models	p. 326
15.2 Other network models	p. 328
Maximum flow models	p. 328
Shortest path models	p. 329
Transportation and assignment models	p. 330
15.3 Declaring network models by node and arc	p. 333
A general transshipment model	p. 334
A specialized transshipment model	p. 335
Variations on transshipment models	p. 336
Maximum flow models	p. 337
15.4 Rules for node and arc declarations	p. 340
node declarations	p. 340
arc declarations	p. 340
Interaction with objective declarations	p. 341
Interaction with constraint declarations	p. 342
Interaction with variable declarations	p. 342
15.5 Solving network linear programs	p. 343
Chapter 16. Columnwise Formulations	p. 353
16.1 An input-output model	p. 354
Formulation by constraints	p. 354
A columnwise formulation	p. 355
Refinements of the columnwise formulation	p. 357
16.2 A scheduling model	p. 359
16.3 Rules for columnwise formulations	p. 362
Chapter 17. Piecewise-Linear Programs	p. 365
17.1 Cost terms	p. 366
Fixed numbers of pieces	p. 366
Varying numbers of pieces	p. 368
17.2 Common two-piece and three-piece terms	p. 369
Penalty terms for "soft" constraints	p. 369
Dealing with infeasibility	p. 373
Reversible activities	p. 377
17.3 Other piecewise-linear functions	p. 379
17.4 Guidelines for piecewise-linear optimization	p. 382
Forms for piecewise-linear expressions	p. 382
Suggestions for piecewise-linear models	p. 383
Chapter 18. Nonlinear Programs	p. 391
18.1 Sources of nonlinearity	p. 392
Dropping a linearity assumption	p. 393
Achieving a nonlinear effect	p. 396
Modeling an inherently nonlinear process	p. 397
18.2 Nonlinear variables	p. 397
Initial values of variables	p. 398
Automatic substitution of variables	p. 399
18.3 Nonlinear expressions	p. 400
18.4 Pitfalls of nonlinear programming	p. 403
Function range violations	p. 403
Multiple local optima	p. 407
Other pitfalls	p. 410
Chapter 19. Complementarity Problems	p. 419
19.1 Sources of complementarity	p. 419
A complementarity model of production economics	p. 420
Complementarity for bounded variables	p. 423
Complementarity for price-dependent demands	p. 425
Other complementarity models and applications	p. 426
19.2 Forms of complementarity constraints	p. 427
19.3 Working with complementarity constraints	p. 428
Related solution values	p. 428
Presolve	p. 429
Generic synonyms	p. 431
Chapter 20. Integer Linear Programs	p. 437
20.1 Integer variables	p. 438
20.2 Zero-one variables and logical conditions	p. 439
Fixed costs	p. 440
Zero-or-minimum restrictions	p. 444
Cardinality restrictions	p. 445
20.3 Practical considerations in integer programming	p. 448
Appendix A. AMPL Reference Manual	p. 453
A.1 Lexical rules	p. 453
A.2 Set members	p. 454
A.3 Indexing expressions and subscripts	p. 455
A.4 Expressions	p. 455
A.4.1 Built-in functions	p. 458
A.4.2 Strings and regular expressions	p. 459
A.4.3 Piecewise-linear terms	p. 460
A.5 Declarations of model entities	p. 461
A.6 Set declarations	p. 461
A.6.1 Cardinality and arity functions	p. 462
A.6.2 Ordered sets	p. 463
A.6.3 Intervals and other infinite sets	p. 463
A.7 Parameter declarations	p. 465
A.7.1 Check statements	p. 465
A.7.2 Infinity	p. 466
A.8 Variable declarations	p. 466
A.8.1 Defined variables	p. 467
A.9 Constraint declarations	p. 468
A.9.1 Complementarity constraints	p. 469
A.10 Objective declarations	p. 470
A.11 Suffix notation for auxiliary values	p. 470
A.11.1 Suffix declarations	p. 471
A.11.2 Statuses	p. 473
A.12 Standard data format	p. 473
A.12.1 Set data	p. 473
A.12.2 Parameter data	p. 475
A.13 Database access and tables	p. 477
A.14 Command language overview	p. 479
A.14.1 Options and environment variables	p. 481
A.15 Redirection of input and output	p. 481
A.16 Printing and display commands	p. 482
A.17 Reading data	p. 484
A.18 Modeling commands	p. 485
A.18.1 The solve command	p. 485
A.18.2 The solution command	p. 487
A.18.3 The write command	p. 487
A.18.4 Auxiliary files	p. 487
A.18.5 Changing a model: delete, purge, redeclare	p. 488
A.18.6 The drop, restore and objective commands	p. 489
A.18.7 The fix and unfix commands	p. 489
A.18.8 Named problems and environments	p. 489
A.18.9 Modifying data: reset, update, let	p. 490
A.19 Examining models	p. 491
A.19.1 The show command	p. 491
A.19.2 The xref command	p. 492
A.19.3 The expand command	p. 492
A.19.4 Generic names	p. 492
A.19.5 The check command	p. 492
A.20 Scripts and control flow statements	p. 492
A.20.1 The for, repeat and if-then-else statements	p. 493
A.20.2 Stepping through commands	p. 495
A.21 Computational environment	p. 495
A.21.1 The shell command	p. 495
A.21.2 The cd command	p. 495
A.21.3 The quit, exit and end commands	p. 496
A.21.4 Built-in timing parameters	p. 496
A.21.5 Logging	p. 496
A.22 Imported functions	p. 497
A.23 AMPL invocation	p. 499
Index	p. 501

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