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DUALITY 277

a. Solve this problem using the Simplex method.
b. The value of the optimal solution is a measure of the riskiness of the portfolio. What effect
will increasing the annual income requirement have on the riskiness of the portfolio?
c. Find the range of feasibility for b 2 .
d. How will the optimal solution and its value change if the annual income requirement is
increased from E60 000 to E65 000?
e. How will the optimal solution and its value change if the risk measure for the stock fund
is increased from 8 to 9?

12 Suppose that in a product-mix problem x 1 , x 2 , x 3 and x 4 indicate the units of products 1, 2, 3
and 4, respectively, and we have:

Max 4x 1 þ 6x 2 þ 3x 3 þ 1x 4
s:t:
1:5x 1 þ 2x 2 þ 4x 3 þ 3x 4 550 Machine A hours
4x 1 þ 1x 2 þ 2x 3 þ 1x 4 700 Machine B hours
2x 1 þ 3x 2 þ 1x 3 þ 2x 4 200 Machine C hours
x 1 ; x 2 ; x 3 ; x 4 0
a. Formulate the dual to this problem.
b. Solve the dual. Use the dual solution to show that the profit-maximizing product mix is
x 1 ¼ 0, x 2 ¼ 25, x 3 ¼ 125 and x 4 ¼ 0.
c. Use the dual variables to identify the machine or machines that are producing at
maximum capacity. If the manager can select one machine for additional production
capacity, which machine should have priority? Why?
13 Find the dual for the following linear programme:

Min 2800x 1 þ 6000x 2 þ 1200x 3
s:t:
15x 1 þ 15x 2 þ 1x 3 5
4x 1 þ 8x 2 5
þ 8x 3 24
12x 1
x 1 ; x 2 ; x 3 0

14 Write the following primal problem in canonical form, and find its dual.
Max 3x 1 þ 1x 2 þ 5x 3 þ 3x 4
s:t:
3x 1 þ 1x 2 þ 2x 3 ¼ 30
2x 1 þ 1x 2 þ 3x 3 þ 1x 4 15
2x 2 þ 3x 4 25
x 1 ; x 2 ; x 3 ; x 4 0

15 Photo Chemicals produces two types of photograph-developing fluids at a cost of E1.00
per litre. Let:

x 1 ¼ litre of product 1
x 2 ¼ litre of product 2

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