GAMS用户指南：建模与求解教程

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"GAMS Manual 是一份详细的用户指南，由Richard E. Rosenthal编写的教程，旨在介绍GAMS（General Algebraic Modeling System）的代码、使用场景和基本特性。GAMS是一种高级建模语言，主要用于解决优化问题，特别是在数学规划领域。这份指南由GAMS Development Corporation出版，适用于华盛顿特区，美国。" 在GAMS用户指南中，内容涵盖了多个关键知识点： 1. **引言**：这部分解释了学习GAMS的动力，包括它在解决复杂优化问题中的应用和优势。 1. **基本特性**： - **一般原则**：GAMS基于代数建模，允许用户以简洁的语法定义模型。 - **文档**：GAMS提供了丰富的文档支持，帮助用户理解和使用系统。 - **可移植性**：GAMS系统是跨平台的，可以在多种操作系统上运行。 - **用户界面**：GAMS提供了一个友好的用户界面，方便用户交互和调试模型。 - **模型库**：GAMS包含一个模型库，用户可以参考和复用已有的模型。 2. **GAMS教程**：Rosenthal的教程逐步引导用户了解GAMS的使用，包括： - **结构化模型**：GAMS模型通常由集合、数据、变量、方程和目标函数等元素组成。 - **集合**：集合是GAMS中的一个重要概念，用于定义模型中的离散元素或索引。 - **数据**：GAMS支持多种数据输入方式，如列表、表格和直接赋值。 - **变量**：变量代表模型中的未知量，可以有上下界限制。 - **方程**：方程定义模型的约束条件，GAMS提供了特殊的求和与乘积表示法。 - **目标函数**：目标函数指定模型要优化的目标。 - **模型和求解语句**：`model`和`solve`语句用于定义模型并求解。 - **显示语句**：`display`语句用于在求解过程中和之后显示变量和方程的状态。 - **'.lo', '.l', '.up', '.m'数据库**：这些数据库记录变量的下界、当前值、上界和边际信息。 - **GAMS输出**：GAMS的输出包括模型的执行日志、错误消息和参考映射，便于用户理解结果和调试。这份手册通过逐步教学和实例演示，帮助用户掌握GAMS的基本用法和高级功能，是学习和使用GAMS进行优化建模的重要参考资料。

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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