# Transshipment Model

In a transportation problem, consignments are always transported from an origin to a destination.

But, there could be several situations where it might be economical to transport items via one or more intermediate centres (or stages). In a transshipment problem, the available commodity is not sent directly from sources to destinations, i.e., it passes through one or more intermediate points before reaching the actual destination.

For instance, a company may have regional warehouses that distribute the products to smaller district warehouses, which in turn ship to the retail stores. Succinctly, the transshipment model is an extension of the classical transportation model where an item available at point i is shipped to demand point j through one or more intermediate points.

The transshipment model helps the management of a company in deciding the optimal number and location of its warehouses.

## Example of Transshipment Model

A company has nine large stores located in several states. The sales department is interested in reducing the price of a certain product in order to dispose all the stock now in hand. But, before that the management wants to reposition its stock among the nine stores according to its sales expectations at each location.

The above figure shows the numbered nodes (9 stores). A positive value next to a store represents the amount of inventory to be redistributed to the rest of the system. A negative value represents the shortage of stock. Thus, stores 1 and 4 have excess stock of 10 & 2 items respectively. Stores 3, 6 & 8 need 3, 1, and 8 more items respectively. The inventory positions of stores 2, 5 & 7 are to remain unchanged.

An item may be shipped through stores 2, 4, 5, 6, 7 & 8. These locations are known as transshipment points. Each remaining store is a source if it has excess stock, and a sink if it needs stock. In the above figure, store 1 is a source and store 3 is a sink.

The value cij is the cost of transporting items. To transport an item from store 1 to store 3, the total shipping cost is
c12 + c23

In the following example, you will learn how to convert a transshipment problem to a standard transportation problem.

#### Example

Consider a transportation problem where the origins are plants and destinations are depots. The unit transportation costs, capacity at the plants, and the requirements at the depots are indicated below:

Table 1

Plant Depot
X Y Z
A 1 3 15 150
B 3 5 25 300
150 150 150 450

When each plant is also considered a destination and each depot is also considered an origin, there are altogether five origins and five destinations. Some additional cost data are also necessary. These are presented in the following Tables.

Table 2

Unit Transportation Cost from Plant to Plant
From Plant To
Plant A Plant B
A 0 65
B 1 0

Table 3

Unit Transportation Cost from Depot to Depot
From Depot To
Depot X Depot Y Depot Z
X 0 23 1
Y 1 0 3
Z 65 3 0

Table 4

Unit Transportation Cost from Depot to Plant
Depot Plant
A B
X 3 15
Y 25 3
Z 45 55

Solution.

From Table 1, Table 2, Table 3 and Table 4 we obtain the transportation formulation of the transshipment problem.

Table 5

Use Horizontal Scrollbar to View Full Table Calculation.

Transshipment Table
A B X Y Z Capacity
A 0 65 1 3 15 150 + 450 = 600
B 1 0 3 5 25 300 + 450 = 750
X 3 15 0 23 1 450
Y 25 3 1 0 3 450
Z 45 55 65 3 0 450
Requirement 450 450 150 + 450 =600 150 + 450 =600 150 + 450 =600 2700

The transportation model is extended and now it includes five supply points & demand points. To have a supply and demand from all the points, a fictitious supply and demand quantity (buffer stock) of 450 is added to both supply and demand of all the points. An initial basic feasible solution is obtained by the Vogel's Approximation method and is shown in the final table.

Final Table

Transshipment Table
A B X Y Z Capacity
A 65 15 600
B 3 5 25 750
X 3 15 23 450
Y 25 3 1 3 450
Z 45 55 65 3 450
Requirement 450 450 600 600 600 2700

The total transhipment cost is:
0 X 150 + 1 X 300 + 3 X 150 + 1 X 300 + 0 X 450 + 0 X 300 + 1 X 150 + 0 X 450 + 0 X 450 = 1200

The transportation problem is a special type of linear programming problem in which the objective is to transport a homogeneous product manufactured at several plants (origins) to a number of different destinations at a minimum total cost. In this chapter, you learned several different techniques for computing an initial basic feasible solution to a transportation problem such as north west corner rule, matrix minimum method and vogel approximation method. Further, you learned how to compute an optimal solution with the help of MODI & stepping stone method. In the end of the chapter, we discussed the transshipment model, which is an extension of the classical transportation model where an item available at point i is shipped to demand point j through one or more intermediate points.