使用LINGO_12解决线性与非线性优化问题

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"Lingo_12_Users_Manual使用手册" Lingo是一款强大的数学建模软件，主要用于解决线性、非线性优化问题。这款工具以其高效求解器著称，能够快速处理复杂的数学模型，并提供详尽的分析结果。Lingo支持多种类型的优化问题，包括线性规划（LP）、整数规划（IP）、动态规划（DP）、随机规划、非线性规划（NLP）以及多目标优化问题。在版权方面，Lingo软件及其相关文档受到法律保护。用户只能按照文档授权的方式或得到Lindo Systems Inc.的书面许可后进行复制。Lingo是Lindo Systems Inc.的商标，而Lindo是其注册商标。其他产品和公司名称可能提及的，属于各自所有者的财产。在免责声明中，Lindo Systems Inc.保证在收到付款时，提供的光盘内包含准确的Lingo软件副本，且相关文档也准确无误。然而，由于计算机程序和模型的复杂性，Lingo软件可能存在无法完全避免的错误。因此，用户在基于软件结果做出决策前，应先验证答案的准确性。Lindo Systems Inc.及其与创建、生产或分发Lingo软件相关的任何个人或实体，都不提供任何形式的额外明示保修，也不对磁盘或文档做任何暗示性保修，无论是否涉及适用性、商业性或特定用途的保修。 Lingo的用户手册通常会涵盖以下几个关键部分： 1. **软件安装与启动**：指导用户如何下载、安装和启动Lingo软件，包括系统需求和兼容性信息。 2. **建模语言介绍**：详细介绍Lingo的建模语法，包括变量定义、约束设置、目标函数等，帮助用户构建优化模型。 3. **求解器功能**：解释Lingo的求解引擎如何处理不同类型的优化问题，如线性、非线性和混合整数问题。 4. **求解过程与结果分析**：阐述如何使用Lingo求解模型、监控求解过程以及解读输出结果。 5. **高级特性**：可能包括灵敏度分析、灵敏度报告、多区域优化、遗传算法等高级功能的使用方法。 6. **用户界面指南**：描述图形用户界面（GUI）的操作流程，包括输入模型、设置参数和查看结果的步骤。 7. **实例与案例研究**：提供实际应用的示例，帮助用户理解和应用Lingo解决实际问题。 8. **技术支持与资源**：列出获取帮助的途径，如用户手册、在线文档、论坛或直接联系客服。 9. **许可证和使用条款**：详细说明软件的许可证限制，包括可安装的计算机数量、商业使用的规定等。通过学习和掌握Lingo，用户可以有效地将数学模型应用于实际业务问题，如生产计划、物流调度、投资决策等领域，从而实现更优的决策。

4 CHAPTER 1

Developing a LINGO Model in Windows

The Problem

For our sample model, we will create a small product-mix example. Let’s imagine that the

CompuQuick Corporation produces two models of computers—Standard and Turbo. CompuQuick can

sell every Standard unit it produces for a profit contribution of $100, and each Turbo unit for a

contribution of $150. At the CompuQuick factory, the Standard computer production line can produce,

at most, 100 computers per day. At the same time, the Turbo computer production line can turn out

120 computers per day. Furthermore, CompuQuick has a limited supply of daily labor. In particular,

there is a total of 160 hours of labor available each day. Standard computers require 1 hour of labor,

while Turbo computers are relatively more labor intense requiring 2 hours of labor. The problem for

CompuQuick is to determine the mix of Standard and Turbo computers to produce each day to

maximize total profit without exceeding line and labor capacity limits.

In general, an optimization model will consist of the following three items:

• Objective Function - The objective function is a formula that expresses exactly

what it is you want to optimize. In business oriented models, this will usually be

a profit function you wish to maximize, or a cost function you want to minimize.

Models may have, at most, one objective function. In the case of our

CompuQuick example, the objective function will compute the company’s profit

as a function of the output of Standards and Turbos.

• Variables - Variables are the quantities you have under your control. You must

decide what the best values of the variables are. For this reason, variables are

sometimes also called decision variables. The goal of optimization is to find the

values of a model’s variables that generate the best value for the objective

function, subject to any limiting conditions placed on the variables. We will

have two variables in our example−one corresponding to the number of

Standards to produce and the other corresponding to the number of Turbos to

produce.

• Constraints - Almost without exception, there will be some limit on the values

the variables in a model can assume—at least one resource will be limited (e.g.,

time, raw materials, your department’s budget, etc.). These limits are expressed

in terms of formulas that are a function of the model’s variables. These formulas

are referred to as constraints because they constrain the values the variables can

take. In our CompuQuick example, we will have one constraint for each

production line and one constraint on the total labor used.

GETTING STARTED 5

Entering the Model

We will now construct the objective function for our example. We will let the variables STANDARD

and TURBO denote the number of Standard and Turbo computers to produce, respectively.

CompuQuick’s objective is to maximize total profit. Total profit is calculated as the sum of the profit

contribution of the Standard computer ($100) multiplied by the total Standard computers produced

(STANDARD) and the profit contribution of the Turbo computer ($150) multiplied by the total Turbo

computers produced (TURBO). Finally, we tell LINGO we want to maximize an objective function by

preceding it with “MAX =”. Therefore, our objective function is written on the first line of our model

window as:

MAX = 100 * STANDARD + 150 * TURBO;

Note: Each mathematical expression in LINGO is terminated with a semicolon. These semicolons

are required. Your model will not solve without them. For more information on the syntax of

LINGO, see below.

Next, we must input our constraints on line capacity and labor supply. The number of Standard and

Turbo computers produced must be constrained to the production line limits of 100 and 120,

respectively. Do this by entering the following two constraints just below the objective function:

STANDARD <= 100;

TURBO <= 120;

In words, the first constraint says the number of Standard computers produced daily (STANDARD)

must be less-than-or-equal-to (<=) the production line capacity of 100. Likewise, the second constraint

says the number of Turbo computers produced daily (TURBO) must be less-than-or-equal-to (<=) its

line capacity of 120.

Note: Since most computers do not have less-than-or-equal-to keys (≤), LINGO has adopted the

convention of using the two character symbol <= to denote ≤. As an alternative, you may

simply enter < to signify less-than-or-equal-to. In a similar manner, >= or > are used to

signify greater-than-or-equal-to (≥).

The final constraint on the amount of labor used can be expressed as:

STANDARD + 2 * TURBO <= 160;

Specifically, the total number of labor hours used (STANDARD + 2 * TURBO) must be

less-than-or-equal-to (<=) the amount of labor hours available of 160.

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