### 统计代写|运筹学作业代写operational research代考|Shortest Path Problem

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• 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 Shortest Path Problem

The shortest path problem aims at finding a shortest path between a starting node and a terminal node through a network. The problem can be regarded as a special case of the transshipment problem, which was discussed in Section 2.3. Consider a directed and connected network with $n$ nodes in which there is exactly one origin and one destination. All the remaining nodes are transshipment points. The shortest path problem is to minimize the cost of shipping 1 unit of product from node $i$ to node $j, c_{i j}$. If arc $(i, j)$ exists, the unit transportation cost, $c_{i j,}$, is the same as the length of such an arc. Otherwise, $c_{i j}=\infty$. The cost of delivering 1 unit of product from node $i$ to itself is 0 . As mentioned in Section 2.3, the number of nodes that has supply only, or pure origin, is denoted as $a$, whereas the number of node, that has demand only, or pure destination, is denoted as $b$. Because there is exactly one origin and one destination in the shortest path problem, both $a$ and $b$ equal 1 . By introducing decision variables $x_{i j}$ to represent the flow from node $i$ to node $j$, the shortest path model can be written as shown in Model 3.3.1.
Model 3.3.1 Standard shortest path model
$$\text { Minimize } z=\sum_{i=j}^{n-b} \sum_{j=a+1}^{n} c_{i j} x_{i j}$$
subject to
\begin{aligned} &\sum_{i=a+1}^{n} x_{i j}=1 \quad i=1,2, \ldots, n-b \ &\sum_{i=1}^{n-b} x_{i j}=1 \quad j=a+1, a+2, \ldots, n \end{aligned}
All $x_{i j} \geq 0$.
Model 3.3.1 is referred to as the shortest path model. Objective function 3.3.1 finds a path that connects the origin and the destination and requires the minimum total transportation cost. Constraint set $3.3 .2$ is an availability constraint, which guarantees that the total maximum amount of products shipped from node $i$ equals 1. Constraint set $3.3 .3$ is a requirement constraint, which ensures that the total maximum amount of products received by node jequals $1 .$

## 统计代写|运筹学作业代写operational research代考|Example of Shortest Path Problem

Figure $3.13$ shows a shortest path network with seven nodes. Node 1 is an origin, whereas node 7 is a destination. All the remaining nodes are transshipment points. The unit transportation cost is shown above each arc.

The shortest path network is a special case of transshipment problem and can be transformed into a tableau as shown in Table 3.9. Nodes 1 to 6 can be regarded as origins, whereas nodes 2 to 7 can be treated as

destinations. Decision variables $x_{i j}$ represent the quantity of the products delivered from origin $i$ to destination $j$. The demand of each destination is denoted as $d_{j}$, whereas the supply of each origin is denoted as $s_{i}$. Because we want to ship 1 unit of product from node 1 to node 7 , all $d_{j}$ and $s_{i}$ equal 1. The upper-right corner of each cell in the tableau represents the unit transportation cost, $c_{i j}$. If arc $(i, j)$ does not exist, the cost $c_{i j}$ is $\infty$. For any $\operatorname{arc}(i, i)$, the cost $c_{i i}$ is 0 .

By introducing decision variables $x_{i j}$ to represent the shipment from origin $i$ to destination $j$, this shortest path problem can be formulated as shown in Model 3.3.2.Constraint sets $3.3 .5$ to $3.3 .10$ are the availability constraints. For example, constraint set $3.3 .5$ ensures that the total maximum amount of products shipped from node 1 equals 1. Constraint sets $3.3 .11$ to $3.3 .16$ are the requirement constraints. For example, constraint set $3.3 .11$ ensures that the total maximum amount of products received by node 2 equals 1 . Because of the integrality property that the transshipment problem has, we can be sure that this shortest path flow through each arc will be 0 or 1 , even when we solve the problem as the LP problem.

## 统计代写|运筹学作业代写operational research代考| ORSHORTPATH: SAS Code for Shortest Path Problem

ORMCFLOW (see Section 3.1) is a macro that can also be used to solve the shortest path problem, which is a special case of the minimum-cost capacitated flow problem, and it aims at finding a shortest path between a starting node and a terminal node. Hence, we use a similar macro with some minor changes to solve the shortest path problem. The new macro is called ORSHORTPATH (see program “sasor_3_3.sas”).

The only difference is that the dataset only contains the name of origins and destinations and the cost of each arc. An example of such a dataset is shown in Figure 3.14.This code determines the results based on the specified parameters and the cost of each arc saved in the text file; it also produces a macro variable (_ORNETFL) at termination. The SAS code for the shortest path problem contains 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|Concept of Shortest Path Problem

最小化 和=∑一世=jn−b∑j=一种+1nC一世jX一世j

∑一世=一种+1nX一世j=1一世=1,2,…,n−b ∑一世=1n−bX一世j=1j=一种+1,一种+2,…,n

## 统计代写|运筹学作业代写operational research代考| ORSHORTPATH: SAS Code for Shortest Path Problem

ORMCFLOW（见第 3.1 节）是一个宏，也可以用来解决最短路径问题，它是最小成本容量流问题的一个特例，它旨在找到起始节点和终端之间的最短路径节点。因此，我们使用一个类似的宏并进行一些小的改动来解决最短路径问题。新宏称为 ORSHORTPATH（参见程序“sasor_3_3.sas”）。

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## MATLAB代写

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