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TRANSPORTATION SIMPLEX METHOD: A SPECIAL-PURPOSE SOLUTION PROCEDURE 295

Table 7.12 Stepping-Stone Path With The Brazil–Dubai Route as The Incoming
Arc

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

Brazil 2 000 1 500 6 000
2 500

2 5 4 5
China 2 500 2 500

Demand 6 000 4 000 2 000 1 500

An occupied cell An unoccupied cell
not on the stepping-stone path
An occupied cell
on the stepping-stone path

the stones at the corners of the path; the objective is to step from stone to stone and
return to the incoming cell where we started. To focus attention on which occupied
cells are part of the stepping-stone path, we draw each occupied cell in the stepping-
stone path as a cylinder, which should reinforce the image of these cells as stones
sticking up in the pond. Table 7.12 depicts the stepping-stone path associated with
the incoming arc of the Brazil–Dubai route.
In Table 7.12 we place a plus sign (+) or a minus sign ( ) in each occupied cell
on the stepping-stone path. A plus sign indicates that the allocation to that cell will
increase by the same amount we allocate to the incoming cell. A minus sign indicates
that the allocation to that cell will decrease by the amount allocated to the incoming
cell. So, to determine the maximum amount that may be allocated to the incoming
cell, we simply look to the cells on the stepping-stone path identified with a minus
sign. Because no arc can have a negative flow, the minus-sign cell with the smallest
amount allocated to it will determine the maximum amount that can be allocated to
the incoming cell. After allocating this maximum amount to the incoming cell, we
then make all the adjustments necessary on the stepping-stone path to maintain
feasibility. The incoming cell becomes an occupied cell, and the outgoing cell is
dropped from the current solution.
In the Foster Electronics problem, the Brazil–Boston and Czech Republic–Dubai
cells are the ones where the allocation will decrease (the ones with a minus sign) as
flow is allocated to the incoming arc (Brazil–Dubai). The 2500 units currently
assigned to Brazil–Boston is less than the 4000 units assigned to Czech Republic–
Dubai, so we identify Brazil–Boston as the outgoing arc. We then obtain the new
solution by allocating 2500 units to the Brazil–Dubai arc, making the appropriate
adjustments on the stepping-stone path and dropping Brazil–Boston from the
solution (its allocation has been driven to zero). Table 7.13 shows the tableau
associated with the new solution. Note that the only changes from the previous
tableau are located on the stepping-stone path originating in the Brazil–Dubai cell.

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