GAMS用户手册详解：从入门到高级特性

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GAMS（General Algebraic Modeling System）是一个强大的数学建模语言和求解器集成环境，专为解决复杂的经济优化问题而设计。《GAMS用户手册》是一本全面的教程，由Richard E. Rosenthal撰写，旨在帮助用户深入了解GAMS的功能、组织结构以及高效使用方法。以下部分知识点概述： 1. **入门介绍**： - **动机**：手册提供了GAMS的基本概念，解释了为何选择GAMS作为建模工具，特别是在处理大型系统优化问题时的优势。 - **核心特性**：包括通用原则，如模型清晰表达，文档支持的重要性，以及GAMS的跨平台兼容性。 - **用户界面**：详细介绍了GAMS的图形用户界面，如何进行模型构建和数据管理。 2. **教程内容**： - **章节结构**：以逐步学习的方式展开，从基础到深入，帮助读者熟悉GAMS的工作流程。 - **模型结构**：讲解了GAMS模型的基本组成部分，如集合（sets）、数据（data）、变量（variables）、方程（equations）等。 - **数据输入**： - 列表（lists）：介绍如何通过列表形式输入数据。 - 表格（tables）：展示表格形式的数据录入方法。 - 直接赋值（direct assignment）：演示如何直接在代码中设置变量和常量值。 - **方程和目标函数**：方程的声明、GAMS特有的求和与积符号以及定义；目标函数的设定和优化目的。 - **模型和求解**：如何编写模型指令和调用求解器，以及模型的执行流程。 - **显示和数据库**：'.lo', '.l', '.up', '.m'数据库的用途，涉及变量边界和初始值的设定，以及最优值的处理和输出。 - **GAMS输出**：包括echo打印、错误消息的解读以及参考映射的使用，这些都是调试和理解模型结果的关键。通过阅读这份用户手册，用户不仅能够掌握GAMS的基础操作，还能了解其高级特性和最佳实践，从而更有效地利用这个工具进行复杂问题的建模和求解。无论是初次接触GAMS还是希望提升技能的用户，这本手册都是不可或缺的参考资料。

4 Introduction

A GAMS Tutorial by Richard E. Rosenthal

2.1 Introduction

The introductory part of this book ends with a detailed example of the use of GAMS for formulating, solving,

and analyzing a small and simple optimization problem. Richard E. Rosenthal of the Naval Postgraduate School

in Monterey, California wrote it. The example is a quick but complete overview of GAMS and its features. Many

references are made to other parts of the book, but they are only to tell you where to look for more details; the

material here can be read proﬁtably without reference to the rest of the book.

The example is an instance of the transportation problem of linear programming, which has historically served

as a ’laboratory animal’ in the development of optimization technology. [See, for example, Dantzig (1963)

.] It is

a good choice for illustrating the power of algebraic modeling languages like GAMS because the transportation

problem, no matter how large the instance at hand, possesses a simple, exploitable algebraic structure. You will

see that almost all of the statements in the GAMS input ﬁle we are about to present would remain unchanged if

a much larger transportation problem were considered.

In the familiar transportation problem, we are given the supplies at several plants and the demands at several

markets for a single commodity, and we are given the unit costs of shipping the commodity from plants to

markets. The economic question is: how much shipment should there be between each plant and each market so

as to minimize total transport cost?

The algebraic representation of this problem is usually presented in a format similar to the following.

Indices:

i = plants

j = markets

Given Data:

= supply of commodity of plant i (in cases)

= demand for commodity at market j

= cost per unit shipment between plant i and market j ($/case)

Decision Variables:

= amount of commodity to ship from plant i to market j (cases),

where x

≥ 0, for all i, j

Constraints:

Observe supply limit at plant i:

≤ a

for all i (cases)

Satisfy demand at market j:

≥ b

for all j (cases)

Objective Function: Minimize

($K)

Note that this simple example reveals some modeling practices that we regard as good habits in general and

that are consistent with the design of GAMS. First, all the entities of the model are identiﬁed (and grouped) by

type. Second, the ordering of entities is chosen so that no symbol is referred to before it is deﬁned. Third, the

Dantzig, George B. (1963). Linear Programming and Extensions, Princeton University Press, Princeton N.J.

6 A GAMS Tutorial by Richard E. Rosenthal

units of all entities are speciﬁed, and, fourth, the units are chosen to a scale such that the numerical values to be

encountered by the optimizer have relatively small absolute orders of magnitude. (The symbol $K here means

thousands of dollars.)

The names of the types of entities may diﬀer among modelers. For example, economists use the terms exogenous

variable and endogenous variable for given data and decision variable, respectively. In GAMS, the terminology

adopted is as follows: indices are called sets, given data are called parameters, decision variables are called

variables, and constraints and the objective function are called equations.

The GAMS representation of the transportation problem closely resembles the algebraic representation above.

The most important diﬀerence, however, is that the GAMS version can be read and processed by the computer.

Plants Shipping Distances to Markets (1000 miles) Supplies

New York Chicago Topeka

Seattle 2.5 1.7 1.8 350

San Diego 2.5 1.8 1.4 600

Demands 325 300 275

Table 2.1: Data for the transportation problem (adapted from Dantzig, 1963)

As an instance of the transportation problem, suppose there are two canning plants and three markets, with the

data given in table 2.1. Shipping distances are in thousands of miles, and shipping costs are assumed to be $90.00

per case per thousand miles. The GAMS representation of this problem is as follows:

Sets

i canning plants / seattle, san-diego /

j markets / new-york, chicago, topeka / ;

Parameters

a(i) capacity of plant i in cases

/ seattle 350

san-diego 600 /

b(j) demand at market j in cases

/ new-york 325

chicago 300

topeka 275 / ;

Table d(i,j) distance in thousands of miles

new-york chicago topeka

seattle 2.5 1.7 1.8

san-diego 2.5 1.8 1.4 ;

Scalar f freight in dollars per case per thousand miles /90/ ;

Parameter c(i,j) transport cost in thousands of dollars per case ;

c(i,j) = f * d(i,j) / 1000 ;

Variables

x(i,j) shipment quantities in cases

z total transportation costs in thousands of dollars ;

Positive Variable x ;

Equations

cost define objective function

supply(i) observe supply limit at plant i

demand(j) satisfy demand at market j ;

cost .. z =e= sum((i,j), c(i,j)*x(i,j)) ;

supply(i) .. sum(j, x(i,j)) =l= a(i) ;

2.2 Structure of a GAMS Model 7

demand(j) .. sum(i, x(i,j)) =g= b(j) ;

Model transport /all/ ;

Solve transport using lp minimizing z ;

Display x.l, x.m ;

If you submit a ﬁle containing the statements above as input to the GAMS program, the transportation model

will be formulated and solved. Details vary on how to invoke GAMS on diﬀerent of computers, but the simplest

(’no frills’) way to call GAMS is to enter the word GAMS followed by the input ﬁle’s name. You will see a number

of terse lines describing the progress GAMS is making, including the name of the ﬁle onto which the output is

being written. When GAMS has ﬁnished, examine this ﬁle, and if all has gone well the optimal shipments will

be displayed at the bottom as follows.

new-york chicago topeka

seattle 50.000 300.000

san-diego 275.000 275.000

You will also receive the marginal costs (simplex multipliers) below.

chicago topeka

seattle 0.036

san-diego 0.009

These results indicate, for example, that it is optimal to send nothing from Seattle to Topeka, but if you insist

on sending one case it will add .036 $K (or $36.00) to the optimal cost. (Can you prove that this ﬁgure is correct

from the optimal shipments and the given data?)

2.2 Structure of a GAMS Model

For the remainder of the tutorial, we will discuss the basic components of a GAMS model, with reference to the

example above. The basic components are listed in table 2.2.

There are optional input components, such as edit checks for bad data and requests for customized reports of

results. Other optional advanced features include saving and restoring old models, and creating multiple models

in a single run, but this tutorial will discuss only the basic components.

Before treating the individual components, we give a few general remarks.

1. A GAMS model is a collection of statements in the GAMS Language. The only rule governing the

ordering of statements is that an entity of the model cannot be referenced before it is declared to exist.

2. GAMS statements may be laid out typographically in almost any style that is appealing to the user.

Multiple lines per statement, embedded blank lines, and multiple statements per line are allowed. You

will get a good idea of what is allowed from the examples in this tutorial, but precise rules of the road

are given in the next Chapter.

3. When you are a beginning GAMS user, you should terminate every statement with a semicolon, as in

our examples. The GAMS compiler does not distinguish between upper-and lowercase letters, so you

are free to use either.

4. Documentation is crucial to the usefulness of mathematical models. It is more useful (and most likely

to be accurate) if it is embedded within the model itself rather than written up separately. There are

at least two ways to insert documentation within a GAMS model. First, any line that starts with an

asterisk in column 1 is disregarded as a comment line by the GAMS compiler. Second, perhaps more

important, documentary text can be inserted within speciﬁc GAMS statements. All the lowercase

words in the transportation model are examples of the second form of documentation.

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