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TRANSPORTATION PROBLEM: A NETWORK MODEL AND A LINEAR PROGRAMMING FORMULATION 281

Figure 7.1 The Network Representation of the Foster Electronics Transportation
Problem

Distribution Centres
(destination nodes)
Transportation
Cost per Unit
Plants 1 6000
(origin nodes) Boston
3
1
5000 Czech 2
Republic 7

6
2 4000
7 Dubai
5
2
6000 Brazil 2
3
3
2 2000
Singapore
5
4
3
2500 5
China
4
London 1500

Supplies Distribution Routes Demands
(arcs)

represented by an arc. The amount of the supply is written next to each origin node,
and the amount of the demand is written next to each destination node. The goods
shipped from the origins to the destinations represent the flow in the network. Note
that the direction of flow (from origin to destination) is indicated by the arrows.
Try Problem 1 for For Foster’s transportation problem, the objective is to determine the routes to
practise in developing a be used and the quantity to be shipped via each route that will provide the minimum
network model of a
transportation problem. total transportation cost. The cost for each unit shipped on each route is given in
Table 7.1 and is shown on each arc in Figure 7.1.
Clearly, it looks as if we can develop an LP model for this problem – we have an
objective function and we have a set of constraints. We will use double-subscripted
decision variables, with x 11 denoting the number of units shipped from origin 1
(Czech Republic) to destination 1 (Boston), x 12 denoting the number of units
The first subscript shipped from origin 1 (Czech Republic) to destination 2 (Dubai) and so on. In
identifies the ‘from’ node general, the decision variables for a transportation problem having m origins and n
of the corresponding arc destinations are written as follows:
and the second subscript
identifies the ‘to’ node of x ij ¼ number of units shipped from origin i to destination j
the arc.
where i ¼ 1; 2; 3; .. . ; m; and j ¼ 1; 2; ... ; n

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