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316 CHAPTER 7 TRANSPORTATION, ASSIGNMENT AND TRANSSHIPMENT PROBLEMS

Figure 7.6 Network Representation of The NHS Transshipment Problem

Hospital
(destination nodes)

5
Edinburgh 200
Plants Warehouses
(origin nodes) (transshipment nodes)
2
1 2 3
600 Geneva London 6
6
3 150
3 Manchester
6

4
3
4
7 350
2 1 4 6 Southampton
400 Stockholm Glasgow
5

8
Newcastle 300

Supplies Distribution Routes Demands
(arcs)

Placing all the variables on the left-hand side provides the constraint corresponding
to node 3 as:
x 13 x 23 þ x 35 þ x 36 þ x 37 þ x 38 ¼ 0

Similarly, the constraint corresponding to node 4 is:
x 14 x 24 þ x 45 þ x 46 þ x 47 þ x 48 ¼ 0
To develop the constraints associated with the destination nodes, we recognize
that for each node the amount shipped to the destination must equal the demand.
For example, to satisfy the demand for 200 units at node 5 (the Edinburgh hospital),
we write:
x 35 þ x 45 ¼ 200

Similarly, for nodes 6, 7 and 8, we have:
x 36 þ x 46 ¼ 150
x 37 þ x 47 ¼ 350
x 38 þ x 48 ¼ 300

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