MODULE 3 – LINEAR PROGRAMMING PROBLEMS

MODULE 3 – LINEAR PROGRAMMING PROBLEMS 3.1

Suppose you have just inherited $50,000 and you want to invest all. Upon hearing this news, two of your friends have offered you opportunities to become a partner in two different business ventures; both require time and cash. Becoming a full partner in the first friend’s venture would require an investment of $45,000 and 400 hours with an expected profit of $40,000. The corresponding figures for the second friend’s venture are $40,000 and 500 hours with an expected profit of $45,000. However, both are flexible and would allow you to come in with any fraction of partnership, and your investment (cash and time) and the profit would be proportional to this fraction. You have restricted yourself to a maximum of 600 hours and decided to participate in both ventures in any combination that would maximize your total expected profit. (a) Formulate the problem as a linear programming model. (b) Construct the corresponding initial simplex tableau. (c) Given below is the final simplex tableau for the problem.

Solution X1 X2 A S1 S2 S3 Cj Quantity Mix (40000) (45000) (0) (0) (0) (0) 45000 X2 0 1 0 0 0 1 1 0 S1 0 0 -0.0089 1 0 -144.4444 11.1111 40000 X1 1 0 0 0 0 -0.8889 0.2222 0 S2 0 0 0 0 1 0.8889 0.7778 Zj 40000 45000 0.8889 0 0 9444.444 53888.89 Cj - Z j 0 0 -0.8889 0 0 -9444.4444 State the optimal solution for this problem.

3.2

A pharmaceutical firm is about to begin the production of three new drugs. An objective function designed to minimize ingredient costs and three production constraints are given below: minimize Cost = 50X + 10Y + 75Z subject to X - Y = 1000 2Y + 2Z = 2000 X  1500 X, Y, Z  0.

Convert this problem to a proper form for the use in the initial simplex tableau.

3.3

Given a linear programming model as follows: maximize Z = 2X1 + 3X2 – X3 subject to 4X1 + 2X2 + X3  22 (Resource I) 3X1 + 6X2  30 (Resource II) X2 + X3 10 (Resource III) X1, X2, X3  0.

The optimal simplex tableau is shown below:

2 3 -1 0 0 0 Basic Qty. variables X1 X2 X3 S1 S2 S3 2 X1 1 0 1/3 1/3 -1/9 0 4 3 X2 0 1 -1/6 -1/6 2/9 0 3 0 S3 0 0 7/6 1/6 -2/9 1 7 Zj Cj-Zj (a) Fill in the values in Zj and Cj-Zj rows. (b) State the optimal values of the decision variables and slack variables and interpret your result.

Cj

3.4

Consider the following optimal tableau for a linear programming problem: Solution X1 X2 X3 S1 S2 S3 S4 Qty Mix Cj 2 4 6 0 0 0 0 S3 0 0 0 2 4 -2 1 0 8 X2 4 0 1 2 2 -1 0 0 6 X1 2 1 0 -1 1 2 0 0 4 S4 0 0 0 1 3 2 0 1 12 Zj 2 4 6 10 0 0 0 32 Cj-Zj 0 0 0 -10 0 0 0 (a) State and interpret the optimal solution for the tableau. (b) Is there any alternative optimal solution exist in the final tableau? Explain.

3.5

Consider the following problem: maximize 2x1 + 5x2 + 3x3 subject to 3x1 + x2 + 2x3 ≤ 30 x1 + x2 + x3 ≥ 5 x1, x2, x3 ≥ 0. (a) Solve the problem using the simplex method. (b) Perform a sensitivity analysis on the right-hand side of the first constraint. (c) Perform a sensitivity analysis on the right-hand side of the second constraint.

3.6

Write down the dual problem corresponding to the following primal problem: maximize 4x1 + 7x2 subject to 3x1 + 2x2 ≥ 64 x1 + 3x2 ≤ 40 x2 ≤ 12 x1, x2 ≥ 0.

3.7

Solve the following linear programming problem by considering the dual problem: minimize 10x1 + 8x2 + 6x3 subject to x1 + x2 + x3 ≥ 25 3x1 + 2x2 + 3x3 ≥ 60 x1, x2, x3 ≥0.