Engineering Optimization: Theory and Practice, Fourth Edition

244 Linear Programming II: Additional Topics and Extensions

4.25 Assume that productsA, B, C, andDrequire, in addition to the stated amounts of copper

and zinc, 4, 3, 2 and 5 lb of nickel per unit, respectively. If the total quantity of nickel

available is 2000 lb, in what way the original optimum solution is affected?

4.26 If productArequires 5 lb of copper and 3 lb of zinc (instead of 4 lb of copper and 2 lb

of zinc) per unit, find the change in the optimum solution.

4.27 If productCrequires 5 lb of copper and 4 lb of zinc (instead of 7 lb of copper and 3 lb

of zinc) per unit, find the change in the optimum solution.

4.28 If the available quantities of copper and zinc are changed to 8000 lb and 5000 lb, respec-

tively, find the change in the optimum solution.

4.29 Solve the following LP problem:

Minimizef= 8 x 1 − 2 x 2

subject to

− 4 x 1 + 2 x 2 ≤ 1

5 x 1 − 4 x 2 ≤ 3

x 1 ≥ 0 , x 2 ≥ 0

Investigate the change in the optimum solution of Problem 4.29 when the following changes are

made(a)by using sensitivity analysis and(b)by solving the new problem graphically:

4.30 b 1 = 2

4.31 b 2 = 4

4.32 c 1 = 10

4.33 c 2 = − 4

4.34 a 11 = − 5

4.35 a 22 = − 2

4.36 Perform one iteration of Karmarkar’s method for the LP problem:

Minimizef= 2 x 1 − 2 x 2 + 5 x 3

subject to

x 1 −x 2 = 0

x 1 +x 2 +x 3 = 1

xi≥ 0 , i= 1 , 2 , 3

4.37 Perform one iteration of Karmarkar’s method for the following LP problem:

Minimizef= 3 x 1 + 5 x 2 − 3 x 3

subject to

x 1 −x 3 = 0

x 1 +x 2 +x 3 = 1

xi≥ 0 , i= 1 , 2 , 3