a powerful tool for economic modeling: GAMS User book

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GAMS was developed to improve on this situation by:
· Providing a high-level language for the compact representation of large and complex
· Allowing changes to be made in model specifications simply and safely
· Allowing unambiguous statements of algebraic relationships
· Permitting model descriptions that are independent of solution algorithms

Table of Contents1

Introduction 1
1.1 MOTIVATION 1
1.2 BASIC FEATURES OF GAMS 1
1.2.1 GENERAL PRINCIPLES 1
1.2.2 DOCUMENTATION 2
1.2.3 PORTABILITY 2
1.2.4 USER INTERFACE 2
1.2.5 MODEL LIBRARY 2
1.3 ORGANIZATION OF THE BOOK 3
A Gams Tutorial 5
2.1 INTRODUCTION 5
2.2 STRUCTURE OF A GAMS MODEL 7
2.3 SETS 9
2.4 DATA 10
2.4.1 DATA ENTRY BY LISTS 10
2.4.2 DATA ENTRY BY TABLES 11
2.4.3 DATA ENTRY BY DIRECT ASSIGNMENT 11
2.5 VARIABLES 12
2.6 EQUATIONS 13
2.6.1 EQUATION DECLARATION 13
2.6.2 GAMS SUMMATION (AND PRODUCT) NOTATION 14
2.6.3 EQUATION DEFINITION 14
2.7 OBJECTIVE FUNCTION 16
2.8 MODEL AND SOLVE STATEMENTS 16
2.9 DISPLAY STATEMENTS 17
2.10 THE '.LO, .L, .UP, .M' DATABASE 17
2.10.1 ASSIGNMENT OF VARIABLE BOUNDS AND/OR INITIAL VALUES 17
2.10.2 TRANSFORMATION AND DISPLAY OF OPTIMAL VALUES 18
2.11 GAMS OUTPUT 19
2.11.1 ECHO PRINTS 19
2.11.2 ERROR MESSAGES 21
2.11.3 REFERENCE MAPS 23
2.11.4 EQUATION LISTINGS 24
2.11.5 MODEL STATISTICS 25
2.11.6 STATUS REPORTS 25
2.11.7 SOLUTION REPORTS 26
2.12 SUMMARY 27

Gams Programms 29
3.1 INTRODUCTION 29
3.2 THE STRUCTURE OF GAMS PROGRAMS 29
3.2.1 FORMAT OF GAMS INPUT 29
3.2.2 CLASSIFICATION OF GAMS STATEMENTS 30
3.2.3 ORGANIZATION OF GAMS PROGRAMS 30
3.3 DATA TYPES AND DEFINITIONS 31
3.4 LANGUAGE ITEMS 32
3.4.1 CHARACTERS 32
3.4.2 RESERVED WORDS 33
3.4.3 IDENTIFIERS 33
3.4.4 LABELS 34
3.4.5 TEXT 34
3.4.6 NUMBERS 35
3.4.7 DELIMITERS 35
3.4.8 COMMENTS 36
3.5 SUMMARY 36
Set Definitions 37
4.1 INTRODUCTION 37
4.2 SIMPLE SETS 37
4.2.1 THE SYNTAX 37
4.2.2 SET NAMES 37
4.2.3 SET ELEMENTS 38
4.2.4 ASSOCIATED TEXT 38
4.2.5 SEQUENCES AS SET ELEMENTS 39
4.2.6 DECLARATIONS FOR MULTIPLE SETS 40
4.3 THE ALIAS STATEMENT: MULTIPLE NAMES FOR A SET 40
4.4 SUBSETS AND DOMAIN CHECKING 41
4.5 MULTI-DIMENSIONAL SETS 41
4.5.1 ONE-TO-ONE MAPPING 41
4.5.2 MANY-TO-MANY MAPPING 42
4.6 SUMMARY 43
Data Entry: Parameters, Scalars & Tables 45
5.1 INTRODUCTION 45
5.2 SCALARS 45
5.2.1 THE SYNTAX 45
5.2.2 AN ILLUSTRATIVE EXAMPLE 45
5.3 PARAMETERS 46
5.3.1 THE SYNTAX 46
5.3.2 AN ILLUSTRATIVE EXAMPLES 46
5.3.3 PARAMETER DATA FOR HIGHER DIMENSIONS 47
5.4 TABLES 48
5.4.1 THE SYNTAX 48
5.4.2 AN ILLUSTRATIVE EXAMPLE 48
5.4.3 CONTINUED TABLES 49

5.4.4 TABLES WITH MORE THAN TWO DIMENSIONS 49
5.4.5 CONDENSING TABLES 50
5.4.6 HANDLING LONG ROW LABELS 50
5.5 ACRONYMS 51
5.5.1 THE SYNTAX 51
5.5.2 ILLUSTRATIVE EXAMPLE 51
5.6 SUMMARY 51
Data Manipulations With Parameters 53
6.1 INTRODUCTION 53
6.2 THE ASSIGNMENT STATEMENT 53
6.2.1 SCALAR ASSIGNMENTS 53
6.2.2 INDEXED ASSIGNMENTS 53
6.2.3 USING LABELS EXPLICITLY IN ASSIGNMENTS 54
6.2.4 ASSIGNMENTS OVER SUBSETS 54
6.2.5 ISSUES WITH CONTROLLING INDICES 54
6.2.6 EXTENDED RANGE IDENTIFIERS IN ASSIGNMENTS 55
6.2.7 ACRONYMS IN ASSIGNMENTS 55
6.3 EXPRESSIONS 55
6.3.1 STANDARD ARITHMETIC OPERATIONS 56
6.3.2 INDEXED OPERATIONS 56
6.3.3 FUNCTIONS 58
6.3.4 EXTENDED RANGE ARITHMETIC AND ERROR HANDLING 58
6.4 SUMMARY 60
Variables 61
7.1 INTRODUCTION 61
7.2 VARIABLE DECLARATIONS 61
7.2.1 THE SYNTAX 61
7.2.2 VARIABLE TYPES 62
7.2.3 STYLES FOR VARIABLE DECLARATION 62
7.3 VARIABLE ATTRIBUTES 63
7.3.1 BOUNDS ON VARIABLES 64
7.3.2 FIXING VARIABLES 64
7.3.3 ACTIVITY LEVELS OF VARIABLES 64
7.4 VARIABLES IN DISPLAY AND ASSIGNMENT STATEMENTS 64
7.4.1 ASSIGNING VALUES TO VARIABLE ATTRIBUTES 64
7.4.2 VARIABLE ATTRIBUTES IN ASSIGNMENTS 65
7.4.3 DISPLAYING VARIABLE ATTRIBUTES 65
7.5 SUMMARY 66
Equations 67
8.1 INTRODUCTION 67
8.2 EQUATION DECLARATIONS 67
8.2.1 THE SYNTAX 67
8.2.2 AN ILLUSTRATIVE EXAMPLE 67
8.3 EQUATION DEFINITIONS 68

8.3.1 THE SYNTAX 68
8.3.2 AN ILLUSTRATIVE EXAMPLE 68
8.3.3 SCALAR EQUATIONS 69
8.3.4 INDEXED EQUATIONS 69
8.3.5 USING LABELS EXPLICITLY IN EQUATIONS 69
8.4 EXPRESSIONS IN EQUATION DEFINITIONS 70
8.4.1 ARITHMETIC OPERATORS IN EQUATION DEFINITIONS 70
8.4.2 FUNCTIONS IN EQUATION DEFINITIONS 70
8.4.3 PREVENTING UNDEFINED OPERATIONS IN EQUATIONS 71
8.5 DATA HANDLING ASPECTS OF EQUATIONS 72
8.6 SUMMARY 72
Model And Solve Statements 73
9.1 INTRODUCTION 73
9.2 THE MODEL STATEMENT 73
9.2.1 THE SYNTAX 73
9.2.2 CLASSIFICATION OF MODELS 74
9.2.3 MODEL ATTRIBUTES 74
9.3 THE SOLVE STATEMENT 76
9.3.1 THE SYNTAX 76
9.3.2 REQUIREMENTS FOR A VALID SOLVE STATEMENT 77
9.3.3 ACTIONS TRIGGERED BY THE SOLVE STATEMENT 77
9.4 PROGRAMS WITH SEVERAL SOLVE STATEMENTS 77
9.4.1 SEVERAL MODELS 77
9.4.2 SENSITIVITY OR SCENARIO ANALYSIS 78
9.4.3 ITERATIVE IMPLEMENTATION OF NON-STANDARD ALGORITHMS 79
9.5 MAKING NEW SOLVERS AVAILABLE WITH GAMS 80
Gams Output 81
10.1 INTRODUCTION 81
10.2 THE ILLUSTRATIVE MODEL 81
10.3. COMPILATION OUTPUT 82
10.3.1 ECHO PRINT OF THE INPUT FILE 82
10.3.2 THE SYMBOL REFERENCE MAP 83
10.3.3 THE SYMBOL LISTING MAP 85
10.3.4 THE UNIQUE ELEMENT LISTING - MAP 85
10.3.5 USEFUL DOLLAR CONTROL DIRECTIVES 86
10.4 EXECUTION OUTPUT 87
10.5 OUTPUT PRODUCED BY A SOLVE STATEMENT 87
10.5.1 THE EQUATION LISTING 87
10.5.2 THE COLUMN LISTING 89
10.5.3 THE MODEL STATISTICS 89
10.5.4 THE SOLVE SUMMARY 90
10.5.5 SOLVER REPORT 93
10.5.6 THE SOLUTION LISTING 94
10.5.7 REPORT SUMMARY 95
10.5.8 FILE SUMMARY 96
10.6 ERROR REPORTING 96

10.6.1 COMPILATION ERRORS 97
10.6.2 COMPILATION TIME ERRORS 98
10.6.3 EXECUTION ERRORS 99
10.6.4 SOLVE ERRORS 99
10.7 SUMMARY 100
Conditional Expressions, Assignments And Equations 101
11.1 INTRODUCTION 101
11.2 LOGICAL CONDITIONS 101
11.2.1 NUMERICAL EXPRESSIONS AS LOGICAL CONDITIONS 101
11.2.2 NUMERICAL RELATIONSHIP OPERATORS 102
11.2.3 LOGICAL OPERATORS 102
11.2.4 SET MEMBERSHIP 102
11.2.5 LOGICAL CONDITIONS INVOLVING ACRONYMS 103
11.2.6 NUMERICAL VALUES OF LOGICAL CONDITIONS 103
11.2.7 MIXED LOGICAL CONDITIONS - OPERATOR PRECEDENCE 103
11.2.8 MIXED LOGICAL CONDITIONS - EXAMPLES 104
11.3 THE DOLLAR CONDITION 104
11.3.1 AN EXAMPLE 105
11.3.2 NESTED DOLLAR CONDITIONS 105
11.4 CONDITIONAL ASSIGNMENTS 105
11.4.1 DOLLAR ON THE LEFT 106
11.4.2 DOLLAR ON THE RIGHT 106
11.4.3 FILTERING CONTROLLING INDICES IN INDEXED OPERATIONS 107
11.4.4 FILTERING SETS IN ASSIGNMENTS 107
11.5 CONDITIONAL INDEXED OPERATIONS 109
11.5.1 FILTERING CONTROLLING INDICES IN INDEXED OPERATIONS 109
11.6 CONDITIONAL EQUATIONS 110
11.6.1 DOLLAR OPERATORS WITHIN THE ALGEBRA 110
11.6.2 DOLLAR CONTROL OVER THE DOMAIN OF DEFINITION 110
11.6.3 FILTERING THE DOMAIN OF DEFINITION 111
Dynamic Sets 113
12.1 INTRODUCTION 113
12.2 ASSIGNING MEMBERSHIP TO DYNAMIC SETS 113
12.2.1 THE SYNTAX 113
12.2.2 ILLUSTRATIVE EXAMPLE 113
12.2.3 DYNAMIC SETS WITH MULTIPLE INDICES 114
12.2.4 ASSIGNMENTS OVER THE DOMAIN OF DYNAMIC SETS 114
12.2.5 EQUATIONS DEFINED OVER THE DOMAIN OF DYNAMIC SETS 115
12.3 USING DOLLAR CONTROLS WITH DYNAMIC SETS 115
12.3.1 ASSIGNMENTS 115
12.3.2 INDEXED OPERATIONS 116
12.3.3 EQUATIONS 116
12.3.4 FILTERING THROUGH DYNAMIC SETS 116
12.4 SET OPERATIONS 117
12.4.1 SET UNION 117
12.4.2 SET INTERSECTION 117

12.4.3 SET COMPLEMENT 117
12.4.4 SET DIFFERENCE 117
12.5 SUMMARY 118
Sets As Sequences: Ordered Sets 119
13.1 INTRODUCTION 119
13.2 ORDERED AND UNORDERED SETS 119
13.3 ORD AND CARD 120
13.3.1 THE ORD OPERATOR 120
13.3.2 THE CARD OPERATOR 121
13.4 LAG AND LEAD OPERATORS 121
13.5 LAGS AND LEADS IN ASSIGNMENTS 122
13.5.1 LINEAR LAG AND LEAD OPERATORS - REFERENCE 122
13.5.2 LNEAR LAG AND LEAD OPERATORS - ASSIGNMENT 123
13.5.3 CIRCULAR LAG AND LEAD OPERATORS 123
13.6 LAGS AND LEADS IN EQUATIONS 124
13.6.1 LINEAR LAG AND LEAD OPERATORS - DOMAIN CONTROL 124
13.6.2 LINEAR LAG AND LEAD OPERATORS - REFERENCE 125
13.6.3 CIRCULAR LAG AND LEAD OPERATORS 125
13.7 SUMMARY 126
The Display Statement 127
14.1 INTRODUCTION 127
14.2 THE SYNTAX 127
14.3 AN EXAMPLE 127
14.4 THE LABEL ORDER IN DISPLAYS 128
14.4.1 EXAMPLE 129
14.5 DISPLAY CONTROLS 129
14.5.1 GLOBAL DISPLAY CONTROLS 130
14.5.2 LOCAL DISPLAY CONTROL 130
14.5.3 DISPLAY STATEMENT TO GENERATE DATA IN LIST FORMAT 131
The Put Writing Facility 133
15.1 INTRODUCTION 133
15.2 THE SYNTAX 133
15.3 AN EXAMPLE 134
15.4 OUTPUT FILES 135
15.4.1 DEFINING FILES 135
15.4.2 ASSIGNING FILES 136
15.4.3 CLOSING A FILE 136
15.4.4 APPENDING TO A FILE 137
15.5 PAGE FORMAT 137
15.6 PAGE SECTIONS 138
15.6.1 ACCESSING VARIOUS PAGE SECTIONS 139
15.6.2 PAGING 139
15.7 POSITIONING THE CURSOR ON A PAGE 140
15.8 SYSTEM SUFFIXES 140

15.9 OUTPUT ITEMS 141
15.9.1 TEXT ITEMS 141
15.9.2 NUMERIC ITEMS 142
15.9.3 SET VALUE ITEMS 142
15.10 GLOBAL ITEM FORMATTING 143
15.10.1 FIELD JUSTIFICATION 143
15.10.2 FIELD WIDTH 143
15.11 LOCAL ITEM FORMATTING 144
15.12 ADDITIONAL NUMERIC DISPLAY CONTROL 144
15.12.1 ILLUSTRATIVE EXAMPLE 145
15.13 CURSOR CONTROL 146
15.13.1 CURRENT CURSOR COLUMN 146
15.13.2 CURRENT CURSOR ROW 147
15.13.3 LAST LINE CONTROL 147
15.14 PAGING CONTROL 148
15.15 EXCEPTION HANDLING 148
15.16 SOURCE OF ERRORS ASSOCIATED WITH THE PUT STATEMENT 149
15.16.1 SYNTAX ERRORS 149
15.16.2 PUT ERRORS 149
15.17 SIMPLE SPREADSHEET/DATABASE APPLICATION 150
15.17.1 AN EXAMPLE 150
Programming Flow Control Features 151
16.1 INTRODUCTION 151
16.2 THE LOOP STATEMENT 151
16.2.1 THE SYNTAX 151
16.2.2 EXAMPLES 152
16.3 THE IF-ELSEIF-ELSE STATEMENT 152
16.3.1 THE SYNTAX 153
16.3.2 EXAMPLES 153
16.4 THE WHILE STATEMENT 154
16.4.1 THE SYNTAX 154
16.4.2 EXAMPLES 154
16.5 THE FOR STATEMENT 155
16.5.1 THE SYNTAX 155
16.5.2 EXAMPLES 155
Special Language Features 157
17.1 INTRODUCTION 157
17.2 SPECIAL MIP FEATURES 157
17.2.1 TYPES OF DISCRETE VARIABLES 157
17.2.2 SPECIAL ORDER SETS OF TYPE 1 (SOS1) 157
17.2.3 SPECIAL ORDER SETS OF TYPE 2 (SOS2) 159
17.2.4 SEMI-CONTINUOUS VARIABLES 159
17.2.5 SEMI-INTEGER VARIABLES 160
17.2.6 SETTING PRIORITIES FOR BRANCHING 160
17.3 MODEL SCALING - THE SCALE OPTION 161
17.3.1 THE SCALE OPTION 161

17.3.2 VARIABLE SCALING 161
17.3.3 EQUATION SCALING 162
17.3.4 SCALING OF DERIVATIVES 163
Glossary 165
Appendix A: The Gams Model Library 171
A.1 INTRODUCTION 171
A.2 USING THE MODEL LIBRARY 171
A.3 THE MODELS 172
Appendix B: The Gams Call 177
B.1 INTRODUCTION 177
2.1.1 THE GENERIC GAMS CALL 177
2.1.2 SPECIFYING OPTIONS THROUGH THE COMMAND LINE 177
B.2 LIST OF COMMAND LINE PARAMETERS 178
B.2.1 PARAMETERS CONTROLLING THE SPECIFIC GAMS RUN 178
B.2.2 PARAMETERS CONTROLLING SYSTEM SETTINGS 178
B.2.3 PARAMETERS AFFECTING INPUT FILE PROCESSING 179
B.2.4 PARAMETERS AFFECTING OUTPUT IN LISTING FILE 179
B.2.5 PARAMETERS AFFECTING OTHER FILES 179
B.3 DETAILED DESCRIPTION OF COMMAND LINE PARAMETERS 180
B.3.1 ACTION 180
B.3.2 APPENDLOG 180
B.3.3 APPENDOUT 181
B.3.4 BOTMARGIN 181
B.3.5 CASE 181
B.3.6 CERR 181
B.3.7 CTRLM 182
B.3.8 CTRLZ 182
B.3.9 CURDIR 182
B.3.10 DFORMAT 182
B.3.11 DUMPOPT 183
B.3.12 DUMPPARMS 185
B.3.13 EOLONLY 185
B.3.14 ERROR 185
B.3.15 ERRMSG 186
B.3.16 EXPAND 187
B.3.17 FERR 187
B.3.18 FORCEWORK 188
B.3.19 G205 188
B.3.20 INPUT 189
B.3.21 INPUTDIR 189
B.3.22 INPUTDIR1 TO INPUTDIR18 190
B.3.23 LEFTMARGIN 190
B.3.24 LIBINCDIR 191
B.3.25 LICENSE 191

B.3.26 LOGFILE 191
B.3.27 LOGLINE 192
B.3.28 LOGOPTION 193
B.3.29 MULTIPASS 193
B.3.30 NOCHECK 194
B.3.31 OPTFILE 195
B.3.32 OUTPUT 195
B.3.33 PAGECONTR 196
B.3.34 PAGESIZE 196
B.3.35 PAGEWIDTH 196
B.3.36 PROFILE 197
B.3.37 PUTDIR 198
B.3.38 REFERENCE 198
B.3.39 RELPATH 198
B.3.40 RESTART 199
B.3.41 SAVE 199
B.3.42 SCRDIR 199
B.3.43 STEPSUM 200
B.3.44 STRINGCHK 200
B.3.45 SUBSYS 201
B.3.46 SUPPRESS 201
B.3.47 SYSDIR 201
B.3.48 SYSINCDIR 201
B.3.49 TABIN 202
B.3.50 TFORMAT 202
B.3.51 TOPMARGIN 202
B.3.52 WORKDIR 203
Appendix C: Dollar Control Options 205
C.1 INTRODUCTION 205
C.1.1 SYNTAX 205
C.1.2 EXAMPLE 205
C.2 LIST OF DOLLAR CONTROL OPTIONS 206
C.2.1 OPTIONS AFFECTING INPUT COMMENT FORMAT 206
C.2.2 OPTIONS AFFECTING INPUT DATA FORMAT 206
C.2.3 OPTIONS AFFECTING OUTPUT FORMAT 207
C.2.4 OPTIONS AFFECTING LISTING OF REFERENCE MAPS 207
C.2.5 OPTIONS AFFECTING PROGRAM CONTROL 208
C.3 DETAILED DESCRIPTION OF DOLLAR CONTROL OPTIONS 208
C.3.1 ABORT ‘TEXT’ 208
C.3.2 BATINCLUDE FILE ARG1 ARG2 … 209
C.3.3 CALL COMMAND 210
C.3.4 CLEAR ID1 ID2 … 211
C.3.5 COMMENT C 212
C.3.6 DOLLAR C 212
C.3.7 DOUBLE 213
C.3.8 ECHO ‘TEXT’ > FILE OR ECHO ‘TEXT’ >> FILE 213
C.3.9 EJECT 213
C.3.10 EOLCOM C 214

D.1.2 EXAMPLE 240
D.2 LIST OF OPTIONS 240
D.2.1 OPTIONS CONTROLLING OUTPUT DETAIL 240
D.2.2 OPTIONS CONTROLLING SOLVER SPECIFIC PARAMETERS 241
D.2.3 OPTIONS CONTROLLING CHOICE OF SOLVER 241
D.2.4 OPTIONS AFFECTING INPUT PROGRAM CONTROL 241
D.3 DETAILED DESCRIPTION OF OPTIONS 241
Appendix E: The Save And Restart Feature 247
E.1 INTRODUCTION 247
E.2 THE SAVE AND RESTART FEATURES 247
E.2.1 SAVING WORK FILES 248
E.2.2 RESTARTING FROM WORK FILES 248
E.3 WAYS IN WHICH WORK FILES ARE USEFUL 249
E.3.1 SEPARATION OF MODEL AND DATA 249
E.3.2 INCREMENTAL PROGRAM DEVELOPMENT 250
E.3.3 TACKING SEQUENCES OF DIFFICULT SOLVES 250
E.3.4 MULTIPLE SCENARIOS 251
Index 253