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单纯型算法与目标规划的应用研究

大连海事大学 数学系 刘莹 指导教师 杨晓光

摘 要

线性规划是运筹学的最重要的分支之一。1939年，苏联数学家康托洛维奇出版了“生产组织与计划中的线性规划模型”[1]一书，对列宁格勒胶合板厂的计划任务建立了一个线性规划的数学模型，为用数学方法解决管理并使两者结合进行了开创性的工作。

自1947年由美国的丹兹格提出单纯形法求解一般线性规划问题的方法——单纯形法[2]之后，线性规划在理论上日益成熟，在实际中应用日益广泛与深入，特别是能用计算机来处理成千上万个约束条件和变量的大规模线性规划问题之后，它的适用领域更为广泛。

本文第一章引言部分，简要介绍了线性规划的发展历史，模型建立及求解方法，重点阐述了单纯形算法的原理，步骤，实现等，摘述了近年来单纯形算法的一些新发展；第二章主要阐述的是有关目标规划的一些概念，模型，解法等；第三章通过对具体例题的分析，使用了LINGO,LINDO,MATLAB,EXCEL SOLVER,WinQSB与“运筹学软件2.0”等6种不同的数学软件用于求解线性规划特别是目标规划，并对算法进行了比较。第四章案例分析，通过对一个实际案例的分析，建模，求解，总结，更加直观的反应了本文前三章所要表述的思想。第五章为结束语。

关键词：

线性规划；单纯形法；目标规划；线性规划计算机求解

Abstract

Linear programming is one of the most important branches of Operation Research. In 1937, a mathematician called L.V.Kantorovich in Russia published a book named Mathematical metods of organizing and planning production, in which he proposed a linear programming model for a factory in Lenningrad state. What ‘s more, he created the method of using mathematical ideas to solve problems in the filed of Management and combined them with each other.

Since G..B.Dantzig proposed the Simplex Method for solving general linear programming problems in 1947, the theory of linear programming turns to be perfect and the applications are taken spreadly. Expercially , the opportunitiy of solving problems with thousands of constraints and veriables by computer makes the applications of LP more and more speardly.

In the first chapter of this article, I will show you some introdutions about the history of linear programming and simplex method, the way to build a LP as well as how to solve the problems. In this chapter, I put forcus on the new developments in recent years. In the second chapter, I will give some concepts ,modles and solutions of Goal Programming. In third chapter, through the analysis to a specific example, I use 6 different mathematical software such LINGO, LINDO, MATLAB, EXCEL SOLVER and WinQSB to solve the problem in computer, and compare the adventages and disadventages