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数学代写|运筹学代写Operations Research代考|The Transportation Problem Model

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数学代写|运筹学代写Operations Research代考|The Transportation Problem Model

To describe the general model for the transportation problem, we need to use terms that are considerably less specific than those for the components of the prototype example. In particular, the general transportation problem is concerned (literally or figuratively) with distributing any commodity from any group of supply centers, called sources, to any group of receiving centers, called destinations, in such a way as to minimize the total distribution cost. The correspondence in terminology between the prototype example and the general problem is summarized in Table 8.4.

As indicated by the fourth and fifth rows of the table, each source has a certain supply of units to distribute to the destinations, and each destination has a certain demand for units to be received from the sources. The model for a transportation problem makes the following assumption about these supplies and demands.
The requirements assumption: Each source has a fixed supply of units, where this entire supply must be distributed to the destinations. (We let $s_i$ denote the number of units being supplied by source $i$, for $i=1,2, \ldots, m$.) Similarly, each destination has a fixed demand for units, where this entire demand must be received from the sources. (We let $d_j$ denote the number of units being received by destination $j$, for $j=1,2, \ldots, n$.)
This assumption that there is no leeway in the amounts to be sent or received means that there needs to be a balance between the total supply from all sources and the total demand at all destinations.
The feasible solutions property: A transportation problem will have feasible solutions if and only if
$$\sum_{i=1}^m s_i=\sum_{j=1}^n d_j$$

数学代写|运筹学代写Operations Research代考|Using Excel to Formulate and Solve Transportation Problems

To formulate and solve a transportation problem using Excel, two separate tables need to be entered on a spreadsheet. The first one is the parameter table. The second is a solution table, containing the quantities to distribute from each source to each destination. Figure 8.4 shows these two tables in rows $3-9$ and 12-18 for the P\&T Co. problem.

The two types of functional constraints need to be included in the spreadsheet. For the supply constraints, the total amount shipped from each source is calculated in column $\mathrm{H}$ of the solution table in Fig. 8.4. It is the sum of all the decision variable cells in the corresponding row. For example, the equation in cell H15 is ” $=\mathrm{D} 15+\mathrm{E} 15+\mathrm{F} 15+\mathrm{G} 15$ ” or “=SUM(D15:G15).” The supply at each source is included in column J. Hence, the cells in column H must equal the corresponding cells in column J.

For the demand constraints, the total amount shipped to each destination is calculated in row 18 of the spreadsheet. For example, the equation in cell D18 is “=SUM(D15:D17).” The demand at each destination is then included in row 20 .

The total cost is calculated in cell H18. This cost is the sum of the products of the corresponding cells in the main bodies of the parameter table and the solution table. Hence, the equation contained in cell H18 is “=SUMPRODUCT(D6:G8,D15:G17).”

Now let us look at the entries in the Solver dialogue box shown at the bottom of Fig. 8.4. These entries indicate that we are minimizing the total cost (calculated in cell H18) by changing the shipment quantities (in cells D15 through G17), subject to the constraints that the total amount shipped to each destination equals its demand (D18:G18=D20:G20) and that the total amount shipped from each source equals its supply (H15:H17=J15:J17). One of the selected Solver options (Assume Non-Negative) specifies that all shipment quantities must be nonnegative. The other one (Assume Linear Model) indicates that this transportation problem is a linear programming problem.

The values of the $x_{i j}$ decision variables (the shipment quantities) are contained in the changing cells (D15:G17). To begin, any value (such as 0) can be entered in each of these cells. After clicking on the Solve button, the Solver will use the simplex method to solve the problem. The optimal solution obtained in this way is shown in the changing cells in Fig. 8.4, along with the resulting total cost in cell H18.

Note that the Solver simply uses the general simplex method to solve a transportation problem rather than a streamlined version that is specially designed for solving transportation problems very efficiently, such as the transportation simplex method presented in the next section. Therefore, a software package that includes such a streamlined version should solve a large transportation problem much faster than the Excel Solver.
We mentioned earlier that some problems do not quite fit the model for a transportation problem because they violate the requirements assumption, but that it is possible to reformulate such a problem to fit this model by introducing a dummy destination or a dummy source. When using the Excel Solver, it is not necessary to do this reformulation since the simplex method can solve the original model where the supply constraints are in $\leq$ form or the demand constraints are in $\geq$ form. However, the larger the problem, the more worthwhile it becomes to do the reformulation and use the transportation simplex method (or equivalent) instead with another software package.
The next two examples illustrate how to do this kind of reformulation.

数学代写|运筹学代写Operations Research代考|The Transportation Problem Model

$$\sum_{i=1}^m s_i=\sum_{j=1}^n d_j$$

数学代写|运筹学代写Operations Research代考|Using Excel to Formulate and Solve Transportation Problems

$x_{i j}$决策变量(装运数量)的值包含在变化单元格中(D15:G17)。首先，可以在每个单元格中输入任何值(例如0)。单击Solve按钮后，求解器将使用单纯形法求解问题。通过这种方法得到的最优解如图8.4变化的单元所示，以及得到的总成本在H18单元中。

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。