统计代写|运筹学作业代写operational research代考|Minimum-Cost Capacitated Flow Problem

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• Foundations of Data Science 数据科学基础

统计代写|运筹学作业代写operational research代考|Concept of Minimum-Cost Capacitated Flow Problem

Transportation, assignment, and transshipment problems are all special cases of the minimum-cost capacitated flow problem. The minimum-cost capacitated flow problem aims at finding the minimum cost flow through a network while satisfying the supply and demand requirements of the origins and destinations, respectively, and also satisfying the flow restrictions through the network. However, there is no such flow restriction in transportation, assignment, and transshipment problems.

Consider a directed and connected network with $n$ nodes in which there is at least one origin and one destination. Any node that has both supply and demand is referred to as a transshipment point. The minimum-cost capacitated flow problem determines how to meet the supply requirement, $s_{i}$, and the demand requirement, $d_{j}$, while not violating the capacity restrictions of any arc at the minimum cost, $c_{i j}$. If an arc $(i, j)$ does not exist, the cost is considered infinite $(\infty)$. The unit cost of flow from node $i$ to itself is 0 . The lower bound on flow through $\operatorname{arc}(i, j)$ is $L_{i j}$, where $L_{i j}=0$ if there is no lower bound. The upper bound on flow through arc $(i, j)$ is $U_{i j}$, where $U_{i j}=\infty$ if there is no upper bound. By introducing decision variables $x_{i j}$ to represent the number of units of flow sent from node $i$ to node $j$ through arc $(i, j)$, the minimum-cost capacitated flow model can be written as shown in Model 3.1.1.

统计代写|运筹学作业代写operational research代考|Example of Minimum-Cost Capacitated Flow Problem

Figure $3.1$ shows a minimum-cost capacitated flow problem. There are eight nodes; nodes 1 and node 2 are origins, and nodes 6,7 , and 8 are destinations. The amount of supply, amount of demand, unit cost on each arc, and upper bound on flow through each arc are shown in the figure.
This minimum-cost capacitated flow network (or problem) can be represented by a tableau as shown in Table 3.1. The upper-right corner of each cell in the tableau represents the unit transportation $\operatorname{cost} c_{i j i}$. The upper bound on flow through $\operatorname{arc}(i, j), U_{i j}$, is shown in Table 3.2.

By introducing decision variables $x_{i j}$ to represent the number of units of flow sent from node $i$ to node $j$ through $\operatorname{arc}(i, j)$, the minimum-cost capacitated flow problem can be formulated as shown in Model 3.1.2.

Model 3.1.2 Example of formulation of minimum-cost capacitated flow problem
$$\begin{gathered} \text { Minimize } 5 x_{13}+9999 x_{14}+9999 x_{15}+9999 x_{16}+9999 x_{17}+9999 x_{18} \ +9999 x_{23}+4 x_{24}+9999 x_{25}+9999 x_{26}+9999 x_{27}+9999 x_{28} \ +0 x_{33}+2 x_{34}+6 x_{35}+5 x_{36}+9999 x_{37}+9999 x_{38} \ +9999 x_{43}+0 x_{44}+x_{45}+9999 x_{46}+9999 x_{47}+2 x_{48} \end{gathered}$$

统计代写|运筹学作业代写operational research代考| SAS Code for Minimum

ORMCFLOW is a macro that solves minimum-cost capacitated flow problems, the objective of which is to find the minimum cost flow through a network while satisfying the supply and demand requirements of the origins and destinations, respectively, and also satisfying the flow restrictions through the network (see program “sasor_3_1.sas”). The primary procedure used for minimum-cost capacitated flow problem is PROC NETFLOW. A full syntax of this procedure is available in Appendix $4 .$
Figure $3.2$ illustrates the data flow in the ORMCFLOW. It shows:

• The cost matrix that is required for ORMCFLOW, in which the cost, capacity, minimum demand, and maximum supply of any origin $i$ and destination $j$ are specified
• The macros (\%data, \%model, and \%report)
• The results datasets that are available for print or can be used for further analysis
In the rest of this section, the procedure used for solving the minimum-cost capacitated flow problem (ORMCFLOW) in SAS, together with an example, is explained. The ORMCFLOW runs three macros: data-handling (\%data), model-building (\%model), and report-writing (\%report).

统计代写|运筹学作业代写operational research代考|Example of Minimum-Cost Capacitated Flow Problem

最小化 5X13+9999X14+9999X15+9999X16+9999X17+9999X18 +9999X23+4X24+9999X25+9999X26+9999X27+9999X28 +0X33+2X34+6X35+5X36+9999X37+9999X38 +9999X43+0X44+X45+9999X46+9999X47+2X48

统计代写|运筹学作业代写operational research代考| SAS Code for Minimum

ORMCFLOW 是一个解决最小成本容量的流量问题的宏，其目标是在满足始发地和目的地的供需需求的同时，找到通过网络的最小成本流量，同时满足通过网络的流量限制。网络（参见程序“sasor_3_1.sas”）。用于最小成本容量流问题的主要程序是 PROC NETFLOW。附录中提供了此过程的完整语法4.

• ORMCFLOW 所需的成本矩阵，其中包含任何来源的成本、容量、最小需求和最大供应一世和目的地j被指定
• 宏（\%data、\%model 和 \%report）
• 可用于打印或可用于进一步分析的结果数据集
在本节的其余部分中，将解释用于解决 SAS 中的最小成本容量流问题 (ORMCFLOW) 的过程以及示例。ORMCFLOW 运行三个宏：数据处理 (\%data)、模型构建 (\%model) 和报告编写 (\%report)。

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MATLAB代写

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