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320 CHAPTER 7 TRANSPORTATION, ASSIGNMENT AND TRANSSHIPMENT PROBLEMS

A General Linear Programming Model of the Transshipment Problem
The general linear programming model of the transshipment problem is:

X
Min c ij x ij
all arcs
s:t:
X X
x ij x ij s i Origin nodes i
arcs out arcs in
X X
x ij x ij ¼ 0 Transshipment nodes
arcs out arcs in
X X
x ij x ij ¼ d j Destination nodes j
arcs in arcs out
x ij 0 for all i and j

where
x ij ¼ number of units shipped from node i to node j
c ij ¼ cost per unit of shipping from node i to node j
s i ¼ supply at origin node i
d j ¼ demand at destination node j

NOTES AND COMMENTS

1 The Management Science in Action, Product constraints for the destination nodes are often
Sourcing Heuristic at Procter & Gamble, written as:
describes how Procter & Gamble used a X X
x ij
transshipment model to redesign its North x ij ¼ d j
arcs out arcs in
American distribution system.
The advantage of writing the constraints this
2 In more advanced treatments of linear
way is that the left-hand side of each constraint
programming and network flow problems, the
then represents the flow out of the node minus
capacitated transshipment problem is called the
the flow in. But such constraints would then
pure network flow problem. Efficient special-
purpose solution procedures are available for have to be multiplied by 1toobtain
nonnegative right-hand sides before the
network flow problems and their special cases.
problem could be solved by many linear
3 In the general linear programming formula-
programming codes.
tion of the transshipment problem, the

7.6 A Production and Inventory Application

The introduction to the transportation and transshipment problems in Sections 7.1
and 7.5 involved applications for the shipment of goods from several supply loca-
tions or origins to several demand sites or destinations. Although the shipment of
goods is the subject of many transportation and transshipment problems, trans-
portation and/or transshipment models can be developed for applications that have
nothing to do with the physical shipment of goods from origins to destinations. In

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