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TRANSSHIPMENT PROBLEM: THE NETWORK MODEL AND A LINEAR PROGRAMMING FORMULATION 315

Table 7.31 Transportation Costs Per Unit for the Ryan Transshipment
Problem, £s per unit
Warehouse

Plant London Glasgow
Geneva 2 3
Stockholm 3 1
Hospital
Warehouse Edinburgh Manchester Southampton Newcastle
London 2 6 3 6
Glasgow 4 4 6 5

medical supplies that Ryan provides to the NHS are first transported to two regional
warehouses, one in London and one in Glasgow. From there, supplies are then
shipped as needed to specialist hospitals and clinics in Edinburgh, Manchester,
Try part (a) of Problem Southampton and Newcastle. The transportation cost per unit for each distribution
11 for practise in route is shown in Table 7.31. The key features of the problem are shown in the
developing a network
representation of a network model in Figure 7.6. Note that the supply at each origin and demand at each
transshipment problem. destination are shown in the left and right margins, respectively. Nodes 1 and 2 are
the origin nodes; nodes 3 and 4 are the transshipment nodes; and nodes 5, 6, 7 and 8
are the destination nodes.
As with the transportation and assignment problems, we can formulate a linear
programming model of the transshipment problem from a network representation.
Again, we need a constraint for each node and a variable for each arc. Let x ij denote
the number of units shipped from node i to node j. For example, x 13 denotes the
number of units shipped from the Geneva plant to the London warehouse, x 14
denotes the number of units shipped from the Geneva plant to the Glasgow ware-
house and so on. If the supply at the Geneva plant is 600 units, the amount shipped
from the Geneva plant must be less than or equal to 600. Mathematically, we write
this supply constraint as:
x 13 þ x 14 600
Similarly, for the Stockholm plant we have:
x 23 þ x 24 400
We now consider how to write the constraints corresponding to the two trans-
shipment nodes. For node 3 (the London warehouse), we must guarantee that the
number of units shipped out must equal the number of units shipped into the
warehouse. Because:
Number of units
shipped out of node 3 ¼ x 35 þ x 36 þ x 37 þ x 38
and
Number of units

shipped into node 3 ¼ x 13 þ x 23

we obtain
x 35 þ x 36 þ x 37 þ x 38 ¼ x 13 þ x 23

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