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SENSITIVITY ANALYSIS WITH THE SIMPLEX TABLEAU 263

12 þ 0:32 b 1 0 (6:3)
8 0:32 b 1 0 (6:4)
30 0:20 b 1 0 (6:5)

The left-hand sides of these inequalities represent the new values of the basic
variables after b 1 has been changed by b 1 .
Solving for b 1 in inequalities (6.3), (6.4), and (6.5), we obtain:
b 1 ð3:125Þð 12Þ¼ 37:5
b 1 ð 3:125Þð 8Þ¼ 25
b 1 ð 5:0Þð 30Þ¼ 150
Because all three inequalities must be satisfied, the most restrictive limits on b 1 must
be satisfied for all the current basic variables to remain nonnegative. Therefore, b 1
must satisfy

37:5 b 1 25 (6:6)

The initial amount of assembly time available was 150 hours. Therefore,
b 1 ¼ 150 + b 1 ,where b 1 is the amount of assembly time available. We add 150
to each of the three terms in expression (6.6) to obtain:

150 37:5 150 þ b 1 150 þ 25 (6:7)

Replacing 150 + b 1 with b 1 , we obtain the range of feasibility for b 1 :
112:5 b 1 175

This range of feasibility for b 1 indicates that as long as the available assembly time is
between 112.5 and 175 hours, the current optimal basis will remain feasible, which is
why we call this range the range of feasibility.
Because the dual price for b 1 (assembly time) is 2.80, we know profit can be
increased by E2.80 by obtaining an additional hour of assembly time. Suppose then
that we increase b 1 by 25; that is, we increase b 1 to the upper limit of its range of
feasibility, 175. The profit will increase to E1980 + (E2.80)25 ¼ E2050, and the
values of the optimal basic variables become:
x 2 ¼ 12 þ 25ð0:32Þ¼ 20
s 2 ¼ 8 þ 25ð 0:32Þ¼ 0
x 1 ¼ 30 þ 25ð 0:20Þ¼ 25
What happened to the solution? The increased assembly time caused a revision in
the optimal production plan. HighTech should produce more of the UltraPortable
and less of the Deskpro. Overall, the profit will be increased by (E2.80)(25) ¼ E70.
Note that although the optimal solution changed, the basic variables that were
optimal before are still optimal.
The procedure for determining the range of feasibility has been illustrated with
the assembly time constraint. The procedure for calculating the range of feasibility
for the right-hand side of any less-than-or-equal-to constraint is the same. The first
step for a general constraint i is to calculate the range of values for b i that satisfies
the following inequalities.

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