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304 CHAPTER 7 TRANSPORTATION, ASSIGNMENT AND TRANSSHIPMENT PROBLEMS

Figure 7.4 A Network Model of the Fowle Marketing Research Assignment Problem

Project Leaders Completion Clients
(origin nodes) Time in Days (destination nodes)

1 10 Client
1 Terry 1 1
15
9

9
2 18 Client
1 Karl 5 2 1

6
14
3 3 Client
1 1
Mustafa 3

Supplies Possible Assignments Demands
(arcs)

the similarity between the network models of the assignment problem (Figure 7.4)
and the transportation problem (Figure 7.1). The assignment problem is a special
case of the transportation problem in which all supply and demand values equal 1
Try part (a) of Problem 9 and the amount shipped over each arc is either 0 or 1.
for practise in developing Because the assignment problem is a special case of the transportation problem, a
a network model for an
assignment problem. linear programming formulation can be developed. Again, we need a constraint for
each node and a variable for each arc. As in the transportation problem, we use double-
subscripted decision variables, with x 11 denoting the assignment of project leader 1
(Terry) to client 1, x 12 denoting the assignment of project leader 1 (Terry) to client 2
and so on. So, we define the decision variables for Fowle’s assignment problem as:

1 if project leader i is assigned to client j
x ij ¼
0 if otherwise
where i ¼ 1; 2; 3; j ¼ 1; 2; 3

Using this notation and the completion time data in Table 7.22, we develop com-
pletion time expressions:
Days required for Terry’s assignment ¼ 10x 11 þ 15x 12 þ 9x 13
Days required for Karl’s assignment ¼ 9x 21 þ 18x 22 þ 5x 23
Days required for Mustafa’s assignment ¼ 6x 31 þ 14x 32 þ 3x 33
The sum of the completion times for the three project leaders will provide the total
days required to complete the three assignments. Thus, the objective function is:
Min 10x 11 þ 15x 12 þ 9x 13 þ 9x 21 þ 18x 22 þ 5x 23 þ 6x 31 þ 14x 32 þ 3x 33

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