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# 数学代写|运筹学代写Operations Research代考|PROTOTYPE EXAMPLE

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## 数学代写|运筹学代写Operations Research代考|PROTOTYPE EXAMPLE

The WYNDOR GLASS CO. produces high-quality glass products, including windows and glass doors. It has three plants. Aluminum frames and hardware are made in Plant 1, wood frames are made in Plant 2, and Plant 3 produces the glass and assembles the products.
Because of declining earnings, top management has decided to revamp the company’s product line. Unprofitable products are being discontinued, releasing production capacity to launch two new products having large sales potential:
Product 1: An 8-foot glass door with aluminum framing
Product 2: A $4 \times 6$ foot double-hung wood-framed window
Product 1 requires some of the production capacity in Plants 1 and 3 , but none in Plant 2. Product 2 needs only Plants 2 and 3 . The marketing division has concluded that the company could sell as much of either product as could be produced by these plants. However, because both products would be competing for the same production capacity in Plant 3 , it is not clear which mix of the two products would be most profitable. Therefore, an OR team has been formed to study this question.

The OR team began by having discussions with upper management to identify management’s objectives for the study. These discussions led to developing the following definition of the problem:
Determine what the production rates should be for the two products in order to maximize their total profit, subject to the restrictions imposed by the limited production capacities available in the three plants. (Each product will be produced in batches of 20 , so the production rate is defined as the number of batches produced per week.) Any combination of production rates that satisfies these restrictions is permitted, including producing none of one product and as much as possible of the other.
The OR team also identified the data that needed to be gathered:

1. Number of hours of production time available per week in each plant for these new products. (Most of the time in these plants already is committed to current products, so the available capacity for the new products is quite limited.)
2. Number of hours of production time used in each plant for each batch produced of each new product.
3. Profit per batch produced of each new product. (Profit per batch produced was chosen as an appropriate measure after the team concluded that the incremental profit from each additional batch produced would be roughly constant regardless of the total number of batches produced. Because no substantial costs will be incurred to initiate the production and marketing of these new products, the total profit from each one is approximately this profit per batch produced times the number of batches produced.)

## 数学代写|运筹学代写Operations Research代考|THE LINEAR PROGRAMMING MODEL

The Wyndor Glass Co. problem is intended to illustrate a typical linear programming problem (miniature version). However, linear programming is too versatile to be completely characterized by a single example. In this section we discuss the general characteristics of linear programming problems, including the various legitimate forms of the mathematical model for linear programming.

Let us begin with some basic terminology and notation. The first column of Table 3.2 summarizes the components of the Wyndor Glass Co. problem. The second column then introduces more general terms for these same components that will fit many linear programming problems. The key terms are resources and activities, where $m$ denotes the number of different kinds of resources that can be used and $n$ denotes the number of activities being considered. Some typical resources are money and particular kinds of machines, equipment, vehicles, and personnel. Examples of activities include investing in particular projects, advertising in particular media, and shipping goods from a particular source to a particular destination. In any application of linear programming, all the activities may be of one general kind (such as any one of these three examples), and then the individual activities would be particular alternatives within this general category.

As described in the introduction to this chapter, the most common type of application of linear programming involves allocating resources to activities. The amount available of each resource is limited, so a careful allocation of resources to activities must be made. Determining this allocation involves choosing the levels of the activities that achieve the best possible value of the overall measure of performance.

Certain symbols are commonly used to denote the various components of a linear programming model. These symbols are listed below, along with their interpretation for the general problem of allocating resources to activities.
$Z=$ value of overall measure of performance.
$x_j=$ level of activity $j$ (for $j=1,2, \ldots, n$ ).
$c_j=$ increase in $Z$ that would result from each unit increase in level of activity $j$.
$b_i=$ amount of resource $i$ that is available for allocation to activities (for $i=$ $1,2, \ldots, m)$
$a_{i j}=$ amount of resource $i$ consumed by each unit of activity $j$.
The model poses the problem in terms of making decisions about the levels of the activities, so $x_1, x_2, \ldots, x_n$ are called the decision variables. As summarized in Table 3.3 , the values of $c_j, b_i$, and $a_{i j}$ (for $i=1,2, \ldots, m$ and $j=1,2, \ldots, n$ ) are the input constants for the model. The $c_j, b_i$, and $a_{i j}$ are also referred to as the parameters of the model.
Notice the correspondence between Table 3.3 and Table 3.1 .

## 数学代写|运筹学代写Operations Research代考|THE LINEAR PROGRAMMING MODEL

Wyndor Glass公司的问题旨在说明一个典型的线性规划问题(微型版本)。然而，线性规划太通用了，不能完全用一个例子来描述。在本节中，我们讨论线性规划问题的一般特征，包括线性规划数学模型的各种合法形式。

$Z=总体性能度量的$值。
$x_j=$活动水平$j$(对于$j=1,2， \ldots, n$)。
$c_j=$ Z$的$增加，这将导致$j$的单位活动水平的增加。
$b_i=$可用的资源$i$的数量

## 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。