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DUALITY 275

3 Refer to the problem formulation and optimal simplex tableau given in Problem 1.
a. Find the dual price for the first constraint.
b. Find the dual price for the second constraint.
c. Find the dual price for the third constraint.
d. Suppose the right-hand side of the first constraint is increased from 120 to 125. Find the
new optimal solution and its value.
e. Suppose the right-hand side of the first constraint is decreased from 120 to 110. Find
the new optimal solution and its value.
4 Refer again to the problem formulation and optimal simplex tableau given in Problem 1.
a. Find the range of feasibility for b 1 .
b. Find the range of feasibility for b 2 .
c. Find the range of feasibility for b 3 .
5 For the HighTech problem, we found the range of feasibility for b 1 , the assembly time
available (see Section 6.1).
a. Find the range of feasibility for b 2 .
b. Find the range of feasibility for b 3 .
c. How much will HighTech’s profit increase if there is a 20-square-metre increase in the
amount of warehouse space available (b 3 )?
6 Recall the GulfGolf problem introduced in Chapter 2. The linear programme for this
problem is:
Max 10x 1 þ 9x 2
s:t:
0:7x 1 þ 1x 2 630 Cutting and dyeing time
0:5x 1 þ 0:8333x 2 600 Sewing time
1x 1 þ 0:6667x 2 708 Finishing time
0:1x 1 þ 0:25x 2 135 Inspection and packaging time
x 1 ; x 2 0

where:
x 1 ¼ number of standard bags produced
x 2 ¼ number of deluxe bags produced
The final simplex tableau is:

x 1 x 2 s 1 s 2 s 3 s 4
Basis c B 10 9 0 0 0 0

x 2 9 0 1 1.875 0 1.3125 0 252
s 2 0 0 0 0.9375 1 0.15625 0 120
x 1 10 1 0 1.875 0 1.875 0 540
0 0 0 0.34375 0 0.140625 1 18
s 4
10 9 4.375 0 6.9315 0 7 668
z j
0 0 4.375 0 6.9375 0
c j – z j

a. Calculate the range of optimality for the profit contribution of the standard bag.
b. Calculate the range of optimality for the profit contribution of the deluxe bag.

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