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MSC IN OPERATIONAL RESEARCH

LECTURE 8

TRANSSHIPMENT MODEL

In generalized transshipment model, items are supplied from different sources to different destination. It is sometimes economical if the shipment passes through some transient nodes in between sources and destinations. Unlike in transportation problem, where shipments are sent directly to a particular source to a particular destination, in transshipment problem, the objective is to minimize the total cost of

shipments, and thus the shipment passes through one or more intermediate nodes before it reaches its desired destination. There are mainly two types of the transshipment problem discussed in the following section Transshipment Problem with Sources and Destinations Acting Transient Nodes A schematic diagram of a simple form of transshipment problem in which the sources and destinations act as transient nodes is shown in the Figure 1

Figure 1Schematic diagram of simple transshipment model.

Lecture Notes

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In the figure, consider the shipment of items from source 1 to destination 2. The shipment from the source I can pass through the source 2 and the destination I before it reaches the specified destination 2. Since, in this case the shipment passes through some transient nodes, the arrangement is termed as transshipment model. The objective of the transshipment problem is to find the optimal shipping pattern such that the total cost of transportation is minimized. A different view of the Figure 1 is shown in Figure 2 in which the number of starting nodes as well as the number of ending nodes is the sum of the number of sources and the number of destinations of the original problem. Let B be the buffer which must be maintained at each of the transient sources and transient destinations. At the minimum the buffer, B can be equal to the sum of the supplies or the sum of the demands, assuming that it is a balanced problem. So, a constant B is added to all the starting nodes and all the ending nodes as shown in Figure 2. Thus we have

Sources Destinations Figure 2 Modified view of simple transshipment problem.

The destinations D1, D2, D3,..., Di,..., Dn are included as additional starting nodes in Figure 2 mainly to act as transient nodes. So, they are not having any original supply. The supply of each of these transient nodes should be at least equal to B. Hence, each of these transient nodes is assigned with B units as the supply value. Similarly, the sources S,, S2, 53,..., Sj..., Sm are included as additional ending nodes in Figure 3.3, mainly to act as transient nodes. These nodes are not having any original demand. But, each of these transient nodes is assigned with B units as the demand value. So, just to have a balance, B is added to each a, of the starting nodes and to each b, of the ending nodes in Figure 2. Then the problem in Figure 2 is

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similar to any conventional transportation problem for which one can use U-V method to get the optimum shipping plan.

Example 1 Consider the following transshipment problem involving 4 sources and 2 destinations The supply values of the sources S,, S,, S3 and S4 are 100 units, 200 units, 150 units and 350 units respectively. The demand values of destinations D, and D, are 350 units and 450 units, respectively The transportation cost per unit between different sources and destinations are summarized as in Table 1. Solve the transshipment problem. Table 1 Cij Values for Example 1 Destination S1 S1 S2 Source S3 S4 D1 D2 0 10 15 20 20 10 S2 4 0 20 25 18 25 S3 20 6 0 10 60 30 S4 5 10 8 0 15 23 D1 25 5 45 30 0 4 D2 12 20 7 6 10 0

Solution Here, the number of sources is 4, and the number of destinations is 2. Therefore, ...