Engineering Optimization: Theory and Practice, Fourth Edition

Problems 243

4.14 Express the dual of the following LP problem:

Maximizef= 2 x 1 +x 2

subject to

x 1 − 2 x 2 ≥ 2

x 1 + 2 x 2 = 8

x 1 −x 2 ≤ 11

x 1 ≥ 0 , x 2 is unrestricted in sign

4.15 Find the effect of changingb=

{ 1200

800

}

to

{ 1180

120

}

in Example 4.5 using sensitivity analysis.

4.16 Find the effect of changing the cost coefficientsc 1 andc 4 from−45 and−50 to− 40

and−60, respectively, in Example 4.5 using sensitivity analysis.

4.17 Find the effect of changingc 1 from−45 to−40 andc 2 from−100 to−90 in Example

4.5 using sensitivity analysis.

4.18 If a new product,E, which requires 10 min of work on lathe and 10 min of work on

milling machine per unit, with a profit of $120 per unit is available in Example 4.5,

determine whether it is worth manufacturingE.

4.19 A metallurgical company produces four products,A, B, C, andD, by using copper and

zinc as basic materials. The material requirements and the profit per unit of each of the

four products, and the maximum quantities of copper and zinc available are given below:

Product

A B C D

Maximum quantity

available

Copper (lb) 4 9 7 10 6000

Zinc (lb) 2 1 3 20 4000

Profit per unit ($) 15 25 20 60

Find the number of units of the various products to be produced for maximizing the

profit.

Solve Problems 4.20–4.28 using the data of Problem 4.19.

4.20 Find the effect of changing the profit per unit of productDto $30.

4.21 Find the effect of changing the profit per unit of productAto $10, and of productBto

$20.

4.22 Find the effect of changing the profit per unit of productBto $30 and of productCto

$25.

4.23 Find the effect of changing the available quantities of copper and zinc to 4000 and

6000 lb, respectively.

4.24 What is the effect of introducing a new product,E, which requires 6 lb of copper and

3 lb of zinc per unit if it brings a profit of $30 per unit?