# 数学代写|运筹学作业代写operational research代考|LINEAR GOAL PROGRAMMING AND ITS SOLUTION PROCEDURES

#### Doug I. Jones

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## 数学代写|运筹学作业代写operational research代考|LINEAR GOAL PROGRAMMING AND ITS SOLUTION PROCEDURES

We have assumed throughout the preceding chapters that the objectives of the organization conducting the linear programming study can be encompassed within a single overriding objective, such as maximizing total profit or minimizing total cost. However, this assumption is not always realistic. In fact, as we discussed in Sec. 2.1, studies have found that the management of U.S. corporations frequently focuses on a variety of other objectives, e.g., to maintain stable profits, increase (or maintain) market share, diversify products, maintain stable prices, improve worker morale, maintain family control of the business, and increase company prestige. Goal programming provides a way of striving toward several such objectives simultaneously.

The basic approach of goal programming is to establish a specific numeric goal for each of the objectives, formulate an objective function for each objective, and then seek a solution that minimizes the (weighted) sum of deviations of these objective functions from their respective goals. There are three possible types of goals:

1. A lower, one-sided goal sets a lower limit that we do not want to fall under (but exceeding the limit is fine).
1. An upper, one-sided goal sets an upper limit that we do not want to exceed (but falling under the limit is fine).
2. A two-sided goal sets a specific target that we do not want to miss on either side.
Goal programming problems can be categorized according to the type of mathematical programming model (linear programming, integer programming, nonlinear programming, etc.) that it fits except for having multiple goals instead of a single objective. In this book, we only consider linear goal programming – those goal programming problems that fit linear programming otherwise (each objective function is linear, etc.) and so we will drop the adjective linear from now on.

Another categorization is according to how the goals compare in importance. In one case, called nonpreemptive goal programming, all the goals are of roughly comparable importance. In another case, called preemptive goal programming, there is a hierarchy of priority levels for the goals, so that the goals of primary importance receive firstpriority attention, those of secondary importance receive second-priority attention, and so forth (if there are more than two priority levels).

We begin with an example that illustrates the basic features of nonpreemptive goal programming and then discuss the preemptive case.

## 数学代写|运筹学作业代写operational research代考|Prototype Example for Nonpreemptive Goal Programming

The DEWRIGHT COMPANY is considering three new products to replace current models that are being discontinued, so their OR department has been assigned the task of determining which mix of these products should be produced. Management wants primary consideration given to three factors: long-run profit, stability in the workforce, and the level of capital investment that would be required now for new equipment. In particular, management has established the goals of (1) achieving a long-run profit (net present value) of at least \$125 million from these products, (2) maintaining the current employment level of 4,000 employees, and (3) holding the capital investment to less than$\$55$ million. However, management realizes that it probably will not be possible to attain all these goals simultaneously, so it has discussed priorities with the OR department. This discussion has led to setting penalty weights of 5 for missing the profit goal (per \$1 million under), 2 for going over the employment goal (per 100 employees), 4 for going under this same goal, and 3 for exceeding the capital investment goal (per$\$1$ million over). Each new product’s contribution to profit, employment level, and capital investment level is proportional to the rate of production. These contributions per unit rate of production are shown in Table 7.5 , along with the goals and penalty weights.

Formulation. The Dewright Company problem includes all three possible types of goals: a lower, one-sided goal (long-run profit); a two-sided goal (employment level); and an upper, one-sided goal (capital investment). Letting the decision variables $x_1, x_2, x_3$ be the production rates of products 1,2 , and 3 , respectively, we see that these goals can be stated as
\begin{aligned} 12 x_1+9 x_2+15 x_3 & \geq 125 & & \text { profit goal } \ 5 x_1+3 x_2+4 x_3 & =40 & & \text { employment goal } \ 5 x_1+7 x_2+8 x_3 & \leq 55 & & \text { investment goal. } \end{aligned}

More precisely, given the penalty weights in the rightmost column of Table 7.5 , let $Z$ be the number of penalty points incurred by missing these goals. The overall objective then is to choose the values of $x_1, x_2$, and $x_3$ so as to
\text { Minimize } \quad \begin{aligned} Z & =5 \text { (amount under the long-run profit goal) } \ & +2(\text { amount over the employment level goal) } \ & +4(\text { amount under the employment level goal) } \ & +3(\text { amount over the capital investment goal), } \end{aligned}
where no penalty points are incurred for being over the long-run profit goal or for being under the capital investment goal. To express this overall objective mathematically, we introduce some auxiliary variables (extra variables that are helpful for formulating the model) $y_1, y_2$, and $y_3$, defined as follows:
\begin{aligned} & y_1=12 x_1+9 x_2+15 x_3-125 \ & y_2=5 x_1+3 x_2+4 x_3-40 \ & y_3=5 x_1+7 x_2+8 x_3-55 \end{aligned}
(long-run profit minus the target).
(employment level minus the target).
(capital investment minus the target).

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|Prototype Example for Nonpreemptive Goal Programming

DEWRIGHT公司正在考虑三种新产品来取代目前停产的型号，因此他们的OR部门被分配了一项任务，即确定应该生产这些产品的哪种组合。管理层希望主要考虑三个因素:长期利润，劳动力的稳定性，以及现在新设备所需的资本投资水平。特别是，管理层制定了以下目标:(1)从这些产品中获得至少1.25亿美元的长期利润(净现值)，(2)保持目前4,000名员工的就业水平，(3)将资本投资控制在$\$ 55$百万以下。然而，管理层意识到同时实现所有这些目标可能是不可能的，因此它已经与手术室部门讨论了优先级。这种讨论导致了对未能达到利润目标(每100万美元)设置5个惩罚权重，对超过就业目标(每100名员工)设置2个惩罚权重，对低于同一目标设置4个惩罚权重，对超过资本投资目标设置3个惩罚权重(每$\$1$百万美元)。每个新产品对利润的贡献、就业水平和资本投资水平与生产率成正比。表7.5显示了单位生产率的贡献，以及目标和惩罚权重。

\begin{aligned} 12 x_1+9 x_2+15 x_3 & \geq 125 & & \text { profit goal } \ 5 x_1+3 x_2+4 x_3 & =40 & & \text { employment goal } \ 5 x_1+7 x_2+8 x_3 & \leq 55 & & \text { investment goal. } \end{aligned}

\text { Minimize } \quad \begin{aligned} Z & =5 \text { (amount under the long-run profit goal) } \ & +2(\text { amount over the employment level goal) } \ & +4(\text { amount under the employment level goal) } \ & +3(\text { amount over the capital investment goal), } \end{aligned}

\begin{aligned} & y_1=12 x_1+9 x_2+15 x_3-125 \ & y_2=5 x_1+3 x_2+4 x_3-40 \ & y_3=5 x_1+7 x_2+8 x_3-55 \end{aligned}
(长期利润减去目标)。
(就业水平减去目标)。
(资本投资减去目标)。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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