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

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## 统计代写|运筹学作业代写operational research代考|Concept of Transshipment Problem

The transshipment problem is an extension of the transportation problem. For the transportation problem, it is assumed that a commodity can only be shipped from an origin to a destination. In many real-life situations, it is also possible to distribute the commodity through the points of origins or through the points of destinations. Sometimes it might be advantageous to distribute a commodity from an origin to an intermediate, or transshipment, point before shipping it to a destination. The transshipment problem allows for these shipments.
The transshipment problem can be described as follows. A manufacturing company has a number of plants, each of which has a limited available capacity, $s_{i}$. After manufacturing, the semifinished products are delivered to the warehouses for final assembly and packaging. Finally, the finished products are shipped to the customers according to their requirements, $d_{j}$. The problem is how to fulfill each customer’s order while not exceeding the capacity of any plant at the minimum cost, $c_{i j \text { – }}$ The problem can be transformed as a conventional transportation model with $(n-b)$ origins and $(n-a)$ destinations, where $n$ is the total number of nodes in the network (i.e., total number of plants, warehouses, and customers), $a$ is the number of node that has supply only (or pure origin), and $b$ is the number of nodes that has demand only (or pure destination). Any node that has both supply and demand is referred to as a transshipment point. The unit transportation costs, $c_{i j}$ are often dependent on the travel distances from node $i$ to node $j$. It is assumed that the cost on a particular route of the network is directly proportional to the amount of products shipped on that route. If there is no route connecting node $i$ and node $j$-or arc $(i, j)$ does not exist-then the cost is considered $\infty$. The cost of delivering 1 unit of product from node $i$ to itself is 0 . By introducing decision variables $x_{i j}$ to represent the amount of product sent from node $i$ to node $j$, the transshipment model can be written as shown in Model 2.3.1.

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

The following example, illustrated in Figure $2.12$, shows a transshipment problem. A manufacturing company has two plants. Plant 1 can produce up to 1000 units of products per day, whereas plant 2 can produce as many as 2000 units of products per day. Instead of shipping the products directly to its customers, each product must be distributed to a warehouse first. After assembling and packaging in the warehouses, the products are shipped to the customers. The demand amounts of customer 1 and customer 2 are 1800 and 1200 , respectively. The problem is how to fulfill each customer’s order

while not exceeding the capacity of any plant at the minimum cost, $c_{i \ddot{ }}$. Unit transportation costs, $c_{i j}$, are shown above the arcs (or arrows).

This transshipment model can be converted into a conventional transportation model with five origins (i.e., P1, P2, W3, W4, and C5) and four destinations (i.e., W3, W4, C5, and C6). The supplies of origins and the demands of destinations are computed as follows. For the five origins, each plant (i.e., $P 1$ and P2) will have a supply equal to its original supply, whereas each warehouse and the first customer (i.e., W3, W4, and C5) will have a supply equal to the total available supply. For the four destinations, each warehouse (i.e., W3 and W4) will have a demand equal to the total available supply. The first customer (i.e, C5) will have a demand equal to the summation of its original demand and total available supply, whereas the second customer (i.e., C6) will have a demand equal to its original demand. The reasons for adding the total available supply to the supply and demand at each transshipment point are to ensure that the total amount of products shipped through each transshipment point will not exceed the total available supply and to also balance the transportation model.
Table $2.5$ details this transshipment problem. Decision variables $x_{i j}$ represent the amount of the products delivered from origin $i$ to destination $j$. The demand of each destination, $d_{j}$, and the supply of each origin, $s_{j}$, are shown. The upperright corner of each cell in the tableau represents the unit transportation cost, $c_{i j}$. By introducing decision variables $x_{i j}$ to represent the shipment from origin $i$ to destination $j$, this transshipment problem can be formulated as shown in Model 2.3.2.

## 统计代写|运筹学作业代写operational research代考| SAS Code for Transshipment Problem

ORTRANS, explained in Section $2.1$ (see also program “sasor_2_3.sas”), is a macro that solves transportation problems, the objective of which is to yield the minimum cost of shipment through a transportation network to destinations. The transshipment problem is an extension of the transportation problem, hence the same macro can be used for this example.

The only difference is that in the transshipment tableau (Table $2.5$ ) some of the costs are set to be $\infty$. Hence in the dataset, we use a large number instead (e.g., $1 \mathrm{E} 10$, which is $10^{10}$ ) because using a large number forces the user to set a variable to 0 . However, such a practice introduces numerical instability, hence we added the code shown next in the program. This constraint explicity fixes the value of the corresponding variables to 0 .
con zero{c in CUSTOMERS, $s$ in SUPPLIERS : cost $[s, c]=1 E 10}$ :
$x[s, c]=0$;
An example data file is seen in Figure 2.13.
Note that the following notations are used in this data file: Cstmer1 (W3), Cstmer2 (W4), Cstmer3 (C5), Cstmer4 (C6), Cstmer5 (P1), and Cstmer6 (P2).
Similar to Section 2.1, two parameters need to be set before calling the \%ortrans macro:
_data: Indicates the name and location of the data file (a text tab delimited file) and contains the cost matrix
_itle: Gives a title in the output of the SAS

## 统计代写|运筹学作业代写operational research代考| SAS Code for Transshipment Problem

ORTRANS，在章节中解释2.1（另见程序“sasor_2_3.sas”），是一个解决运输问题的宏，其目标是通过运输网络产生最小的运输成本到目的地。转运问题是运输问题的扩展，因此本例可以使用相同的宏。

X[s,C]=0;

：指示数据文件的名称和位置（文本制表符分隔的文件）并包含成本矩阵
_itle：在输出中给出标题SAS 的

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## MATLAB代写

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