## 数学代写|运筹学作业代写operational research代考|PROTOTYPE EXAMPLE

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|PROTOTYPE EXAMPLE

SEERVADA PARK has recently been set aside for a limited amount of sightseeing and backpack hiking. Cars are not allowed into the park, but there is a narrow, winding road system for trams and for jeeps driven by the park rangers. This road system is shown (without the curves) in Fig. 9.1, where location $O$ is the entrance into the park; other letters designate the locations of ranger stations (and other limited facilities). The numbers give the distances of these winding roads in miles.

The park contains a scenic wonder at station $T$. A small number of trams are used to transport sightseers from the park entrance to station $T$ and back.

The park management currently faces three problems. One is to determine which route from the park entrance to station $T$ has the smallest total distance for the operation of the trams. (This is an example of the shortest-path problem to be discussed in Sec. 9.3.)
A second problem is that telephone lines must be installed under the roads to establish telephone communication among all the stations (including the park entrance). Because the installation is both expensive and disruptive to the natural environment, lines will be installed under just enough roads to provide some connection between every pair of stations. The question is where the lines should be laid to accomplish this with a minimum total number of miles of line installed. (This is an example of the minimum spanning tree problem to be discussed in Sec. 9.4.)

The third problem is that more people want to take the tram ride from the park entrance to station $T$ than can be accommodated during the peak season. To avoid unduly disturbing the ecology and wildlife of the region, a strict ration has been placed on the number of tram trips that can be made on each of the roads per day. (These limits differ for the different roads, as we shall describe in detail in Sec. 9.5.) Therefore, during the peak season, various routes might be followed regardless of distance to increase the number of tram trips that can be made each day. The question pertains to how to route the various trips to maximize the number of trips that can be made per day without violating the limits on any individual road. (This is an example of the maximum flow problem to be discussed in Sec. 9.5.)

## 数学代写|运筹学作业代写operational research代考|THE TERMINOLOGY OF NETWORKS

A relatively extensive terminology has been developed to describe the various kinds of networks and their components. Although we have avoided as much of this special vocabulary as we could, we still need to introduce a considerable number of terms for use throughout the chapter. We suggest that you read through this section once at the outset to understand the definitions and then plan to return to refresh your memory as the terms are used in subsequent sections. To assist you, each term is highlighted in boldface at the point where it is defined.

A network consists of a set of points and a set of lines connecting certain pairs of the points. The points are called nodes (or vertices); e.g., the network in Fig. 9.1 has seven nodes designated by the seven circles. The lines are called arcs (or links or edges or branches); e.g., the network in Fig. 9.1 has 12 arcs corresponding to the 12 roads in the road system. Arcs are labeled by naming the nodes at either end; for example, $A B$ is the $\operatorname{arc}$ between nodes $A$ and $B$ in Fig. 9.1.

The arcs of a network may have a flow of some type through them, e.g., the flow of trams on the roads of Seervada Park in Sec. 9.1. Table 9.1 gives several examples of flow in typical networks. If flow through an arc is allowed in only one direction (e.g., a oneway street), the arc is said to be a directed arc. The direction is indicated by adding an arrowhead at the end of the line representing the arc. When a directed arc is labeled by listing two nodes it connects, the from node always is given before the to node; e.g., an arc that is directed from node $A$ to node $B$ must be labeled as $A B$ rather than $B A$. Alternatively, this arc may be labeled as $A \rightarrow B$.

If flow through an arc is allowed in either direction (e.g., a pipeline that can be used to pump fluid in either direction), the arc is said to be an undirected arc. To help you distinguish between the two kinds of arcs, we shall frequently refer to undirected arcs by the suggestive name of links.

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|PROTOTYPE EXAMPLE

SEERVADA公园最近被划为有限数量的观光和背包徒步旅行。汽车不允许进入公园，但有一条狭窄蜿蜒的道路供有轨电车和公园护林员驾驶的吉普车通行。该道路系统如图9.1所示(没有曲线)，其中位置$O$为进入公园的入口;其他字母标明了护林站(和其他有限设施)的位置。这些数字以英里为单位给出了这些蜿蜒道路的距离。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|运筹学作业代写operational research代考|THE TRANSPORTATION PROBLEM

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|THE TRANSPORTATION PROBLEM

One of the main products of the P \& T COMPANY is canned peas. The peas are prepared at three canneries (near Bellingham, Washington; Eugene, Oregon; and Albert Lea, Minnesota) and then shipped by truck to four distributing warehouses in the western United States (Sacramento, California; Salt Lake City, Utah; Rapid City, South Dakota; and Albuquerque, New Mexico), as shown in Fig. 8.1. Because the shipping costs are a major expense, management is initiating a study to reduce them as much as possible. For the upcoming season, an estimate has been made of the output from each cannery, and each warehouse has been allocated a certain amount from the total supply of peas. This information (in units of truckloads), along with the shipping cost per truckload for each cannery-warehouse combination, is given in Table 8.2. Thus, there are a total of 300 truckloads to be shipped. The problem now is to determine which plan for assigning these shipments to the various cannery-warehouse combinations would minimize the total shipping cost.

By ignoring the geographical layout of the canneries and warehouses, we can provide a network representation of this problem in a simple way by lining up all the canneries in one column on the left and all the warehouses in one column on the right. This representation is shown in Fig. 8.2. The arrows show the possible routes for the truckloads, where the number next to each arrow is the shipping cost per truckload for that route. A square bracket next to each location gives the number of truckloads to be shipped out of that location (so that the allocation into each warehouse is given as a negative number).

The problem depicted in Fig. 8.2 is actually a linear programming problem of the transportation problem type. To formulate the model, let $Z$ denote total shipping cost, and let $x_{i j}(i=1,2,3 ; j=1,2,3,4)$ be the number of truckloads to be shipped from cannery $i$ to warehouse $j$. Thus, the objective is to choose the values of these 12 decision variables (the $x_{i j}$ ) so as to
\begin{aligned} & \text { Minimize } Z=464 x_{11}+513 x_{12}+654 x_{13}+867 x_{14}+352 x_{21}+416 x_{22} \ & +690 x_{23}+791 x_{24}+995 x_{31}+682 x_{32}+388 x_{33}+685 x_{34}, \ & \end{aligned}

## 数学代写|运筹学作业代写operational research代考|An Award Winning Application of a Transportation Problem

Except for its small size, the P \& T Co. problem is typical of the problems faced by many corporations which must ship goods from their manufacturing plants to their customers.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|运筹学作业代写operational research代考|The Relevance of the Gradient for Concepts 1 and 2

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|The Relevance of the Gradient for Concepts 1 and 2

The algorithm begins with an initial trial solution that (like all subsequent trial solutions) lies in the interior of the feasible region, i.e., inside the boundary of the feasible region. Thus, for the example, the solution must not lie on any of the three lines $\left(x_1=0, x_2=0\right.$, $x_1+x_2=8$ ) that form the boundary of this region in Fig. 7.3. (A trial solution that lies on the boundary cannot be used because this would lead to the undefined mathematical operation of division by zero at one point in the algorithm.) We have arbitrarily chosen $\left(x_1, x_2\right)=(2,2)$ to be the initial trial solution.

To begin implementing concepts 1 and 2, note in Fig. 7.3 that the direction of movement from $(2,2)$ that increases $Z$ at the fastest possible rate is perpendicular to (and toward) the objective function line $Z=16=x_1+2 x_2$. We have shown this direction by the arrow from $(2,2)$ to $(3,4)$. Using vector addition, we have
$$(3,4)=(2,2)+(1,2),$$
where the vector $(1,2)$ is the gradient of the objective function. (We will discuss gradients further in Sec. 13.5 in the broader context of nonlinear programming, where algorithms similar to Karmarkar’s have long been used.) The components of $(1,2)$ are just the coefficients in the objective function. Thus, with one subsequent modification, the gradient $(1,2)$ defines the ideal direction to which to move, where the question of the distance to move will be considered later.

The algorithm actually operates on linear programming problems after they have been rewritten in augmented form. Letting $x_3$ be the slack variable for the functional constraint of the example, we see that this form is
$$\text { Maximize } Z=x_1+2 x_2 \text {, }$$
subject to
$$x_1+x_2+x_3=8$$
and
$$x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0 .$$
In matrix notation (slightly different from Chap. 5 because the slack variable now is incorporated into the notation), the augmented form can be written in general as
\begin{aligned} & \text { Maximize } Z=\mathbf{c}^T \mathbf{x}, \ & \text { subject to } \ & \mathbf{A x}=\mathbf{b} \end{aligned}
and
$$\mathbf{x} \geq \mathbf{0},$$
where
$$\mathbf{c}=\left[\begin{array}{l} 1 \ 2 \ 0 \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right], \quad \mathbf{A}=\left[\begin{array}{lll} 1, & 1, & 1 \end{array}\right], \quad \mathbf{b}=[8], \quad \mathbf{0}=\left[\begin{array}{l} 0 \ 0 \ 0 \end{array}\right]$$
for the example. Note that $\mathbf{c}^T=[1,2,0]$ now is the gradient of the objective function.

## 数学代写|运筹学作业代写operational research代考|Using the Projected Gradient to Implement Concepts 1 and 2

In augmented form, the initial trial solution for the example is $\left(x_1, x_2, x_3\right)=(2,2,4)$. Adding the gradient $(1,2,0)$ leads to
$$(3,4,4)=(2,2,4)+(1,2,0)$$
However, now there is a complication. The algorithm cannot move from $(2,2,4)$ toward $(3,4,4)$, because $(3,4,4)$ is infeasible! When $x_1=3$ and $x_2=4$, then $x_3=8-x_1-$ $x_2=1$ instead of 4 . The point $(3,4,4)$ lies on the near side as you look down on the feasible triangle in Fig. 7.4. Therefore, to remain feasible, the algorithm (indirectly) projects the point $(3,4,4)$ down onto the feasible triangle by dropping a line that is perpendicular to this triangle. A vector from $(0,0,0)$ to $(1,1,1)$ is perpendicular to this triangle, so the perpendicular line through $(3,4,4)$ is given by the equation
$$\left(x_1, x_2, x_3\right)=(3,4,4)-\theta(1,1,1) \text {, }$$
where $\theta$ is a scalar. Since the triangle satisfies the equation $x_1+x_2+x_3=8$, this perpendicular line intersects the triangle at $(2,3,3)$. Because
$$(2,3,3)=(2,2,4)+(0,1,-1),$$
the projected gradient of the objective function (the gradient projected onto the feasible region) is $(0,1,-1)$. It is this projected gradient that defines the direction of movement for the algorithm, as shown by the arrow in Fig. 7.4.

A formula is available for computing the projected gradient directly. By defining the projection matrix $\mathbf{P}$ as
$$\mathbf{P}=\mathbf{I}-\mathbf{A}^T\left(\mathbf{A} \mathbf{A}^T\right)^{-1} \mathbf{A},$$
the projected gradient (in column form) is
$$\mathbf{c}_p=\mathbf{P c} .$$

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|The Relevance of the Gradient for Concepts 1 and 2

$$(3,4)=(2,2)+(1,2),$$

$$\text { Maximize } Z=x_1+2 x_2 \text {, }$$

$$x_1+x_2+x_3=8$$

$$x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0 .$$

\begin{aligned} & \text { Maximize } Z=\mathbf{c}^T \mathbf{x}, \ & \text { subject to } \ & \mathbf{A x}=\mathbf{b} \end{aligned}

$$\mathbf{x} \geq \mathbf{0},$$

$$\mathbf{c}=\left[\begin{array}{l} 1 \ 2 \ 0 \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right], \quad \mathbf{A}=\left[\begin{array}{lll} 1, & 1, & 1 \end{array}\right], \quad \mathbf{b}=[8], \quad \mathbf{0}=\left[\begin{array}{l} 0 \ 0 \ 0 \end{array}\right]$$

## 数学代写|运筹学作业代写operational research代考|Using the Projected Gradient to Implement Concepts 1 and 2

$$(3,4,4)=(2,2,4)+(1,2,0)$$

## 数学代写|运筹学作业代写operational research代考|Interpretation of the Simplex Method

$$\sum_{i=1}^m a_{i j} y_i$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|运筹学作业代写operational research代考|A FUNDAMENTAL INSIGHT

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|A FUNDAMENTAL INSIGHT

We shall now focus on a property of the simplex method (in any form) that has been revealed by the revised simplex method in the preceding section. ${ }^1$ This fundamental insight provides the key to both duality theory and sensitivity analysis (Chap. 6), two very important parts of linear programming.

The insight involves the coefficients of the slack variables and the information they give. It is a direct result of the initialization, where the $i$ th slack variable $x_{n+i}$ is given a coefficient of +1 in Eq. (i) and a coefficient of 0 in every other equation [including Eq. (0)] for $i=1,2, \ldots, m$, as shown by the null vector $\mathbf{0}$ and the identity matrix $\mathbf{I}$ in the slack variables column for iteration 0 in Table 5.8. (For most of this section, we are assuming that the problem is in our standard form, with $b_i \geq 0$ for all $i=1,2, \ldots, m$, so that no additional adjustments are needed in the initialization.) The other key factor is that subsequent iterations change the initial equations only by

1. Multiplying (or dividing) an entire equation by a nonzero constant
2. Adding (or subtracting) a multiple of one entire equation to another entire equation
As already described in the preceding section, a sequence of these kinds of elementary algebraic operations is equivalent to premultiplying the initial simplex tableau by some matrix. (See Appendix 4 for a review of matrices.) The consequence can be summarized as follows.

## 数学代写|运筹学作业代写operational research代考|Mathematical Summary

Because its primary applications involve the final tableau, we shall now give a general mathematical expression for the fundamental insight just in terms of this tableau, using matrix notation. If you have not read Sec. 5.2, you now need to know that the parameters of the model are given by the matrix $\mathbf{A}=\left|a_{i j}\right|$ and the vectors $\mathbf{b}=\left|b_i\right|$ and $\mathbf{c}=\left|c_j\right|$, as displayed at the beginning of that section.

The only other notation needed is summarized and illustrated in Table 5.10. Notice how vector $\mathbf{t}$ (representing row 0) and matrix $\mathbf{T}$ (representing the other rows) together correspond to the rows of the initial tableau in Table 5.9 , whereas vector $\mathbf{t}^$ and matrix $\mathbf{T}^$ together correspond to the rows of the final tableau in Table 5.9. This table also shows these vectors and matrices partitioned into three parts: the coefficients of the original variables, the coefficients of the slack variables (our focus), and the right-hand side. Once again, the notation distinguishes between parts of the initial tableau and the final tableau by using an asterisk only in the latter case.

For the coefficients of the slack variables (the middle part) in the initial tableau of Table 5.10 , notice the null vector $\mathbf{0}$ in row 0 and the identity matrix $\mathbf{I}$ below, which provide the keys for the fundamental insight. The vector and matrix in the same location of the final tableau, $\mathbf{y}^$ and $\mathbf{S}^$, then play a prominent role in the equations for the fundamental insight. $\mathbf{A}$ and $\mathbf{b}$ in the initial tableau turn into $\mathbf{A}^$ and $\mathbf{b}^$ in the final tableau. For row 0 of the final tableau, the coefficients of the decision variables are $\mathbf{z}^-\mathbf{c}$ (so the vector $\mathbf{z}^$ is what has been added to the vector of initial coefficients, $-\mathbf{c}$ ), and the right-hand side $Z^*$ denotes the optimal value of $Z$.

# 运筹学代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|运筹学作业代写operational research代考|Functional Constraints in $\geq$ Form

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|Functional Constraints in $\geq$ Form

To illustrate how the artificial-variable technique deals with functional constraints in $\geq$ form, we will use the model for designing Mary’s radiation therapy, as presented in Sec. 3.4. For your convenience, this model is repeated below, where we have placed a box around the constraint of special interest here.
Radiation Therapy Example The graphical solution for this example (originally presented in Fig. 3.12) is repeated here in a slightly different form in Fig. 4.5. The three lines in the figure, along with the two axes, constitute the five constraint boundaries of the problem. The dots lying at the intersection of a pair of constraint boundaries are the corner-point solutions. The only two corner-point feasible solutions are $(6,6)$ and $(7.5,4.5)$, and the feasible region is the line segment connecting these two points. The optimal solution is $\left(x_1, x_2\right)=(7.5,4.5)$, with $Z=5.25$

We soon will show how the simplex method solves this problem by directly solving the corresponding artificial problem. However, first we must describe how to deal with the third constraint.

Our approach involves introducing both a surplus variable $x_5$ (defined as $x_5=$ $\left.0.6 x_1+0.4 x_2-6\right)$ and an artificial variable $\bar{x}_6$, as shown next.
\begin{aligned} & 0.6 x_1+0.4 x_2 \quad \geq 6 \ & \rightarrow \quad 0.6 x_1+0.4 x_2-x_5 \quad=6 \quad\left(x_5 \geq 0\right) \ & \rightarrow \quad 0.6 x_1+0.4 x_2-x_5+\bar{x}_6=6 \quad\left(x_5 \geq 0, \bar{x}_6 \geq 0\right) \text {. } \ & \end{aligned}
Here $x_5$ is called a surplus variable because it subtracts the surplus of the left-hand side over the right-hand side to convert the inequality constraint to an equivalent equality constraint. Once this conversion is accomplished, the artificial variable is introduced just as for any equality constraint.

## 数学代写|运筹学作业代写operational research代考|Minimization

One straightforward way of minimizing $Z$ with the simplex method is to exchange the roles of the positive and negative coefficients in row 0 for both the optimality test and step 1 of an iteration. However, rather than changing our instructions for the simplex method for this case, we present the following simple way of converting any minimization problem to an equivalent maximization problem:
Minimizing $\quad Z=\sum_{j=1}^n c_j x_j$
is equivalent to
$$\text { maximizing } \quad-Z=\sum_{j=1}^n\left(-c_j\right) x_j \text {, }$$
i.e., the two formulations yield the same optimal solution(s).
The two formulations are equivalent because the smaller $Z$ is, the larger $-Z$ is, so the solution that gives the smallest value of $Z$ in the entire feasible region must also give the largest value of $-Z$ in this region.

Therefore, in the radiation therapy example, we make the following change in the formulation:
\begin{aligned} & \text { Minimize } & Z & =0.4 x_1+0.5 x_2 \ \rightarrow & \text { Maximize } & -Z & =-0.4 x_1-0.5 x_2 . \end{aligned}
After artificial variables $\bar{x}_4$ and $\bar{x}_6$ are introduced and then the Big $M$ method is applied, the corresponding conversion is
$$\begin{array}{lrr} & \text { Minimize } & Z=0.4 x_1+0.5 x_2+M \bar{x}_4+M \bar{x}_6 \ \rightarrow & \text { Maximize } & -Z=-0.4 x_1-0.5 x_2-M \bar{x}_4-M \bar{x}_6 . \end{array}$$

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|Functional Constraints in $\geq$ Form

\begin{aligned} & 0.6 x_1+0.4 x_2 \quad \geq 6 \ & \rightarrow \quad 0.6 x_1+0.4 x_2-x_5 \quad=6 \quad\left(x_5 \geq 0\right) \ & \rightarrow \quad 0.6 x_1+0.4 x_2-x_5+\bar{x}_6=6 \quad\left(x_5 \geq 0, \bar{x}_6 \geq 0\right) \text {. } \ & \end{aligned}

## 数学代写|运筹学作业代写operational research代考|Minimization

$$\text { maximizing } \quad-Z=\sum_{j=1}^n\left(-c_j\right) x_j \text {, }$$

\begin{aligned} & \text { Minimize } & Z & =0.4 x_1+0.5 x_2 \ \rightarrow & \text { Maximize } & -Z & =-0.4 x_1-0.5 x_2 . \end{aligned}

$$\begin{array}{lrr} & \text { Minimize } & Z=0.4 x_1+0.5 x_2+M \bar{x}_4+M \bar{x}_6 \ \rightarrow & \text { Maximize } & -Z=-0.4 x_1-0.5 x_2-M \bar{x}_4-M \bar{x}_6 . \end{array}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|运筹学作业代写operational research代考|Optimality Test for the New BF Solution

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|Optimality Test for the New BF Solution

The current Eq. (0) gives the value of the objective function in terms of just the current nonbasic variables
$$Z=30+3 x_1-\frac{5}{2} x_4$$
Increasing either of these nonbasic variables from zero (while adjusting the values of the basic variables to continue satisfying the system of equations) would result in moving toward one of the two adjacent BF solutions. Because $x_1$ has a positive coefficient, increasing $x_1$ would lead to an adjacent BF solution that is better than the current BF solution, so the current solution is not optimal.
Iteration 2 and the Resulting Optimal Solution
Since $Z=30+3 x_1-\frac{5}{2} x_4, Z$ can be increased by increasing $x_1$, but not $x_4$. Therefore, step 1 chooses $x_1$ to be the entering basic variable.

For step 2, the current system of equations yields the following conclusions about how far $x_1$ can be increased (with $x_4=0$ ):
$$\begin{array}{ll} \boldsymbol{x}_3=4-x_1 \geq 0 & \Rightarrow x_1 \leq \frac{4}{1}=4 . \ \boldsymbol{x}_2=6 \geq 0 & \Rightarrow \text { no upper bound on } x_1 . \ \boldsymbol{x}_5=6-3 x_1 \geq 0 & \Rightarrow x_1 \leq \frac{6}{3}=2 \quad \leftarrow \text { minimum. } \end{array}$$
Therefore, the minimum ratio test indicates that $x_5$ is the leaving basic variable.

## 数学代写|运筹学作业代写operational research代考|THE SIMPLEX METHOD IN TABULAR FORM

The algebraic form of the simplex method presented in Sec. 4.3 may be the best one for learning the underlying logic of the algorithm. However, it is not the most convenient form for performing the required calculations. When you need to solve a problem by hand (or interactively with your OR Courseware), we recommend the tabular form described in this section. ${ }^1$

The tabular form of the simplex method records only the essential information, namely, (1) the coefficients of the variables, (2) the constants on the right-hand sides of the equations, and (3) the basic variable appearing in each equation. This saves writing the symbols for the variables in each of the equations, but what is even more important is the fact that it permits highlighting the numbers involved in arithmetic calculations and recording the computations compactly.

Table 4.3 compares the initial system of equations for the Wyndor Glass Co. problem in algebraic form (on the left) and in tabular form (on the right), where the table on the right is called a simplex tableau. The basic variable for each equation is shown in bold type on the left and in the first column of the simplex tableau on the right. [Although only the $x_j$ variables are basic or nonbasic, $Z$ plays the role of the basic variable for Eq. (0).] All variables not listed in this basic variable column $\left(x_1, x_2\right)$ automatically are nonbasic variables. After we set $x_1=0, x_2=0$, the right side column gives the resulting solution for the basic variables, so that the initial BF solution is $\left(x_1, x_2, x_3, x_4, x_5\right)=(0,0,4,12$, 18) which yields $Z=0$.
The tabular form of the simplex method uses a simplex tableau to compactly display the system of equations yielding the current $\mathrm{BF}$ solution. For this solution, each variable in the leftmost column equals the corresponding number in the rightmost column (and variables not listed equal zero). When the optimality test or an iteration is performed, the only relevant numbers are those to the right of the $\mathrm{Z}$ column. The term row refers to just a row of numbers to the right of the $Z$ column (including the right side number), where row $i$ corresponds to Eq. ( $i$ ).
We summarize the tabular form of the simplex method below and, at the same time, briefly describe its application to the Wyndor Glass Co. problem. Keep in mind that the logic is identical to that for the algebraic form presented in the preceding section. Only the form for displaying both the current system of equations and the subsequent iteration has changed (plus we shall no longer bother to bring variables to the right-hand side of an equation before drawing our conclusions in the optimality test or in steps 1 and 2 of an iteration).

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|Optimality Test for the New BF Solution

$$Z=30+3 x_1-\frac{5}{2} x_4$$

$$\begin{array}{ll} \boldsymbol{x}_3=4-x_1 \geq 0 & \Rightarrow x_1 \leq \frac{4}{1}=4 . \ \boldsymbol{x}_2=6 \geq 0 & \Rightarrow \text { no upper bound on } x_1 . \ \boldsymbol{x}_5=6-3 x_1 \geq 0 & \Rightarrow x_1 \leq \frac{6}{3}=2 \quad \leftarrow \text { minimum. } \end{array}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|运筹学作业代写operational research代考|AUTO ASSEMBLY

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|AUTO ASSEMBLY

Automobile Alliance, a large automobile manufacturing company, organizes the vehicles it manufactures into three families: a family of trucks, a family of small cars, and a family of midsized and luxury cars. One plant outside Detroit, MI, assembles two models from the family of midsized and luxury cars. The first model, the Family Thrillseeker, is a four-door sedan with vinyl seats, plastic interior, standard features, and excellent gas mileage. It is marketed as a smart buy for middle-class families with tight budgets, and each Family Thrillseeker sold generates a modest profit of $\$ 3,600$for the company. The second model, the Classy Cruiser, is a two-door luxury sedan with leather seats, wooden interior, custom features, and navigational capabilities. It is marketed as a privilege of affluence for upper-middle-class families, and each Classy Cruiser sold generates a healthy profit of$\$5,400$ for the company.

Rachel Rosencrantz, the manager of the assembly plant, is currently deciding the production schedule for the next month. Specifically, she must decide how many Family Thrillseekers and how many Classy Cruisers to assemble in the plant to maximize profit for the company. She knows that the plant possesses a capacity of 48,000 laborhours during the month. She also knows that it takes 6 labor-hours to assemble one Family Thrillseeker and 10.5 labor-hours to assemble one Classy Cruiser.

Because the plant is simply an assembly plant, the parts required to assemble the two models are not produced at the plant. They are instead shipped from other plants around the Michigan area to the assembly plant. For example, tires, steering wheels, windows, seats, and doors all arrive from various supplier plants. For the next month, Rachel knows that she will be able to obtain only 20,000 doors (10,000 left-hand doors and 10,000 right-hand doors) from the door supplier. A recent labor strike forced the shutdown of that particular supplier plant for several days, and that plant will not be able to meet its production schedule for the next month. Both the Family Thrillseeker and the Classy Cruiser use the same door part.

In addition, a recent company forecast of the monthly demands for different automobile models suggests that the demand for the Classy Cruiser is limited to 3,500 cars. There is no limit on the demand for the Family Thrillseeker within the capacity limits of the assembly plant.

## 数学代写|运筹学作业代写operational research代考|CUTTING CAFETERIA COSTS

A cafeteria at All-State University has one special dish it serves like clockwork every Thursday at noon. This supposedly tasty dish is a casserole that contains sautéed onions, boiled sliced potatoes, green beans, and cream of mushroom soup. Unfortunately, students fail to see the special quality of this dish, and they loathingly refer to it as the Killer Casserole. The students reluctantly eat the casserole, however, because the cafeteria provides only a limited selection of dishes for Thursday’s lunch (namely, the casserole).
Maria Gonzalez, the cafeteria manager, is looking to cut costs for the coming year, and she believes that one sure way to cut costs is to buy less expensive and perhaps lower-quality ingredients. Because the casserole is a weekly staple of the cafeteria menu, she concludes that if she can cut costs on the ingredients purchased for the casserole, she can significantly reduce overall cafeteria operating costs. She therefore de

cides to invest time in determining how to minimize the costs of the casserole while maintaining nutritional and taste requirements.

Maria focuses on reducing the costs of the two main ingredients in the casserole, the potatoes and green beans. These two ingredients are responsible for the greatest costs, nutritional content, and taste of the dish.

Maria buys the potatoes and green beans from a wholesaler each week. Potatoes cost $\$ 0.40$per pound, and green beans cost$\$1.00$ per pound.

All-State University has established nutritional requirements that each main dish of the cafeteria must meet. Specifically, the total amount of the dish prepared for all the students for one meal must contain 180 grams (g) of protein, 80 milligrams (mg) of iron, and $1,050 \mathrm{mg}$ of vitamin C. (There are $453.6 \mathrm{~g}$ in $1 \mathrm{lb}$ and $1,000 \mathrm{mg}$ in $1 \mathrm{~g}$.) For simplicity when planning, Maria assumes that only the potatoes and green beans contribute to the nutritional content of the casserole.

Because Maria works at a cutting-edge technological university, she has been exposed to the numerous resources on the World Wide Web. She decides to surf the Web to find the nutritional content of potatoes and green beans. Her research yields the following nutritional information about the two ingredients:
\begin{tabular}{l|c|c}
\hline & Potatoes & Green Beans \
\hline Protein & $1.5 \mathrm{~g}$ per $100 \mathrm{~g}$ & $5.67 \mathrm{~g}$ per 10 ounces \
Iron & $0.3 \mathrm{mg}$ per $100 \mathrm{~g}$ & $3.402 \mathrm{mg}$ per 10 ounces \
Vitamin C & $12 \mathrm{mg}$ per $100 \mathrm{~g}$ & $28.35 \mathrm{mg}$ per 10 ounces \
\hline
\end{tabular}
(There are $28.35 \mathrm{~g}$ in 1 ounce.)

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|CUTTING CAFETERIA COSTS

\begin{tabular}{l|c|c}
\hline & Potatoes & Green Beans \hline Protein & $1.5 \mathrm{~g}$ per $100 \mathrm{~g}$ & $5.67 \mathrm{~g}$ per 10 ounces \Iron &$0.3 \mathrm{mg}$ per $100 \mathrm{~g}$ & $3.402 \mathrm{mg}$ per 10 ounces \Vitamin C &$12 \mathrm{mg}$ per $100 \mathrm{~g}$ & $28.35 \mathrm{mg}$ per 10 ounces \hline
\end{tabular}
(每盎司含有$28.35 \mathrm{~g}$。)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|运筹学作业代写operational research代考|DISPLAYING AND SOLVING LINEAR PROGRAMMING MODELS ON A SPREADSHEET

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|DISPLAYING AND SOLVING LINEAR PROGRAMMING MODELS ON A SPREADSHEET

Spreadsheet software, such as Excel, is a popular tool for analyzing and solving small linear programming problems. The main features of a linear programming model, including all its parameters, can be easily entered onto a spreadsheet. However, spreadsheet software can do much more than just display data. If we include some additional information, the spreadsheet can be used to quickly analyze potential solutions. For example, a potential solution can be checked to see if it is feasible and what $Z$ value (profit or cost) it achieves. Much of the power of the spreadsheet lies in its ability to immediately see the results of any changes made in the solution.

In addition, the Excel Solver can quickly apply the simplex method to find an optimal solution for the model.

To illustrate this process, we now return to the Wyndor example introduced in Sec. 3.1.
Displaying the Model on a Spreadsheet
After expressing profits in units of thousands of dollars, Table 3.1 in Sec. 3.1 gives all the parameters of the model for the Wyndor problem. Figure 3.14 shows the necessary additions to this table for an Excel spreadsheet. In particular, a row is added (row 9, labeled “Solution”) to store the values of the decision variables. Next, a column is added (column E, labeled “Totals”). For each functional constraint, the number in column E is the numerical value of the left-hand side of that constraint. Recall that the left-hand side represents the actual amount of the resource used, given the values of the decision variables in row 9. For example, for the Plant 3 constraint in row 7 , the amount of this resource used (in hours of production time per week) is
Production time used in Plant $3=3 x_1+2 x_2$.
In the language of Excel, the equivalent equation for the number in cell E7 is
$$\mathrm{E} 7=\mathrm{C} 7 * \mathrm{C} 9+\mathrm{D} 7 * \mathrm{D} 9 .$$
Notice that this equation involves the sum of two products. There is a function in Excel, called SUMPRODUCT, that will sum up the product of each of the individual terms in two different ranges of cells. For instance, SUMPRODUCT(C7:D7,C9:D9) takes each of the individual terms in the range C7:D7, multiplies them by the corresponding term in the range C9:D9, and then sums up these individual products, just as shown in the above equation. Although optional with such short equations, this function is especially handy as a shortcut for entering longer linear programming equations.

## 数学代写|运筹学作业代写operational research代考|Using the Excel Solver to Solve the Model

Excel includes a tool called Solver that uses the simplex method to find an optimal solution. (A more powerful version of Solver, called Premium Solver, also is available in your OR Courseware.) Before using Solver, all the following components of the model need to be included on the spreadsheet:

1. Each decision variable
2. The objective function and its value
3. Each functional constraint
The spreadsheet layout shown in Fig. 3.14 includes all these components. The parameters for the functional constraints are in rows 5,6 , and 7 , and the coefficients for the objective function are in row 8 . The values of the decision variables are in cells $\mathrm{C} 9$ and $\mathrm{D} 9$, and the value of the objective function is in cell E8. Since we don’t know what the values of the decision variables should be, they are just entered as zeros. The Solver will then change these to the optimal values after solving the problem.

The Solver can be started by choosing “Solver” in the Tools menu. The Solver dialogue box is shown in Fig. 3.15. The “Target Cell” is the cell containing the value of the objective function, while the “Changing Cells” are the cells containing the values of the decision variables.

Before the Solver can apply the simplex method, it needs to know exactly where each component of the model is located on the spreadsheet. You can either type in the cell addresses or click on them. Since the target cell is cell E8 and the changing cells are in the range C9:D9, these addresses are entered into the Solver dialogue box as shown in Fig. 3.15. (Excel then automatically enters the dollar signs shown in the figure to fix these addresses.) Since the goal is to maximize the objective function, “Max” also has been selected.

# 运筹学代考

## DISPLAYING AND SOLVING LINEAR PROGRAMMING MODELS ON A SPREADSHEET

$$\mathrm{E} 7=\mathrm{C} 7 * \mathrm{C} 9+\mathrm{D} 7 * \mathrm{D} 9 .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。