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TRANSPORTATION SIMPLEX METHOD: A SPECIAL-PURPOSE SOLUTION PROCEDURE 293

Table 7.9 Initial Feasible Solution to the Foster Electronics Problem

Boston Dubai Singapore London Supply
3 2 7 6
Czech 1 000 4 000 5 000
Republic

7 5 2 3

Brazil 2 500 2 000 1 500 6 000

2 5 4 5
China 2 500 2 500

Demand 6 000 4 000 2 000 1 500

Solving these equations leads to the following values for u 1 , u 2 , u 3 , v 1 , v 2 , v 3 and v 4 :

u 1 ¼ 0 v 1 ¼ 3
u 2 ¼ 4 v 2 ¼ 2
u 3 ¼ 1 v 3 ¼ 2
v 4 ¼ 1

Management scientists have shown that for each unoccupied cell, e ij ¼ c ij u i v j
shows the change in total cost per unit that will be obtained by allocating one unit of
flow to the corresponding arc. Thus, we will call e ij the net evaluation index. Because of
the way u i and v j are calculated the net evaluation index for each occupied cell equals
zero.
Rewriting the tableau containing the initial feasible solution for the Foster
Electronics problem and replacing the previous marginal information with the
values of u i and v j , we obtain Table 7.10. We calculated the net evaluation index
(e ij ) for each unoccupied cell, which is the circled number in the cell. So, shipping
one unit over the route from origin 1 to destination 3 (Czech Republic–Singapore)
will increase total cost by E9; shipping one unit from origin 1 to destination 4 (Czech
Republic–London) will increase total cost by E7; shipping one unit from origin 2 to
destination 2 (Brazil–Dubai) will decrease total cost by E1; and so on.
On the basis of the net evaluation indexes, the best arc in terms of cost reduction
(a net evaluation index of 1) is associated with the Brazil–Dubai route (origin 2–
destination 2); thus, the cell in row 2 and column 2 is chosen as the incoming cell.
Total cost decreases by E1 for every unit of flow assigned to this arc. The question
now is: How much flow should we assign to this arc? Because the total cost
decreases by E1 per unit assigned, we want to allocate the maximum possible flow.
A. Charnes and W. W. To find that maximum, we must recognize that, to maintain feasibility, each unit of
Cooper published their flow assigned to this arc will require adjustments in the flow over the other currently
article ‘The Stepping
Stone Method of used arcs. The stepping-stone method can be used to determine the adjustments
Explaining Linear necessary and to identify an outgoing arc.
Programming
Calculations in The Stepping-Stone Method Suppose that we allocate one unit of flow to the
Transportation Problems’ incoming arc (the Brazil–Dubai route). To maintain feasibility – that is, not exceed
in the first publication of
the Management Science the number of units to be shipped to Dubai – we would have to reduce the flow
journal in 1954. assigned to the Czech Republic–Dubai arc to 3999. But then we would have to

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