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TRANSSHIPMENT PROBLEM: THE NETWORK MODEL AND A LINEAR PROGRAMMING FORMULATION 319
Figure 7.10 Linear Programming Formulation of The Modified NHS Transshipment Problem
Min 2x 13 +3x 14 +3x 23 +1x 24 +2x 35 +6x 36 +3x 37 +6x 38 +4x 45 +4x 46 +6x 47 +5x 48 +4x 28 +1x 78
s.t.
x 13 + x 14 600 Origin node constraints
400
x 23 + x 24 + x 28
= 0
–x 13 – x 23 + x 35 + x 36 + x 37 + x 38 Transshipment node
– x 14 – x 24 + x 45 + x 46 + x 47 + x 48 = 0 constraints
x 35 + x 45 = 200
= 150
x 36 + x 46 Destination node
x 37 + x 47 – x 78 = 350 constraints
x 38 + x 48 + x 28 + x 78 = 300
x ij 0 for all i and j
Problem Variations
As with transportation and assignment problems, transshipment problems may be
formulated with several variations, including the following:
1 total supply not equal to total demand;
2 maximization objective function;
3 route capacities or route minimums;
4 unacceptable routes.
The linear programming model modifications required to accommodate these var-
iations are identical to the modifications required for the transportation problem
described in Section 7.1. When we add one or more constraints of the form x ij L ij
to show that the route from node i to node j has capacity L ij , we refer to the
transshipment problem as a capacitated transshipment problem.
Figure 7.11 The Excel Solution for the Modified NHS Transshipment Problem
Objective Function Value = 4600.000
Variable Value Reduced Costs
-------------- --------------- -----------------
X13 600.000 0.000
X14 0.000 0.000
X23 0.000 3.000
X24 150.000 0.000
X35 200.000 0.000
X36 0.000 1.000
X37 400.000 0.000
X38 0.000 2.000
X45 0.000 3.000
X46 150.000 0.000
X47 0.000 4.000
X48 0.000 2.000
X28 250.000 0.000
X78 50.000 0.000
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