GAMS入门教程：数学规划与优化关键

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GAMS（General Algebraic Modeling System）使用指南是一份详尽的教程，由Richard E. Rosenthal撰写，于2014年发布。该指南旨在介绍GAMS这款高级数学规划和优化建模系统的概念、功能和操作流程，帮助用户充分利用其进行高效建模。 **1. 引言** GAMS的动机在于提供一个强大而直观的工具，使非专业数学背景的用户也能编写复杂的数学模型，解决实际问题。它具备通用原则，支持文档编写，注重跨平台兼容性，并具有用户友好的界面。此外，GAMS还拥有丰富的模型库，为学习者和实践者提供了现成模型的参考。 **2. GAMS的基本特征** - **通用原则**：GAMS采用统一的语法，允许用户用自然语言描述数学模型，简化了复杂模型的构建。 - **文档**：系统提供了全面的文档支持，包括用户手册、在线帮助和教程，便于理解和使用。 - **便携性**：GAMS可在多个操作系统上运行，确保模型可以在不同环境之间无缝转移。 - **用户界面**：设计简洁明了，通过图形化界面和命令行交互，方便模型输入和结果查看。 - **模型库**：GAMS包含大量的预定义模型，有助于快速学习和复用解决方案。 **3. 教程结构** 教程分为两部分：概述和具体操作步骤。首先，介绍GAMS的整体结构和书籍组织方式，随后逐步引导读者进入GAMS的世界。 **4. GAMS教程内容概览** - **模型结构**：章节讲解了GAMS模型的基本组成部分，如集合（Sets）、数据输入（列表、表格和直接赋值）、变量（包括边界和初始值设定）、方程（定义和使用GAMS求和和积符号）、目标函数、模型定义和求解声明、显示语句等。 - **数据库管理**：详细介绍了'.lo','.l','.up','.m'数据库的使用，涉及变量值的绑定、优化结果的转换和显示，以及错误处理和参考映射的功能。 - **输出分析**：包括echo打印（程序输出）、错误消息解读和参考地图，帮助用户理解模型执行过程中的关键信息。通过这份使用指南，读者将掌握如何使用GAMS进行模型设计、数据处理、求解优化问题，以及如何有效解析和利用GAMS的输出结果。无论是初次接触GAMS的初学者，还是希望深入学习和提升技能的专业人员，都能从中获益匪浅。

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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