# ELECTRICAL ENGINEERING

Homework #2 Due Thursday, September 17, 2015 ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

Part I. (Modification of Problem #3.2-1 in page 79 from Hillier & Lieberman’s OR text, 9th edition) The Primo Insurance Company is introducing two new product lines: special risk insurance and mortgages. The expected profit is \$5 per unit on special risk insurance and \$2 per unit on mortgages. Management wishes to establish sales quotas for the new product lines to maximize total expected profit. The work requirements are as follows:

Department Work-Hours per Unit Work-Hours Available Special Risk Mortgage Underwriting 3 2 2400

Administration 0 1 800 Claims 2 0 1200

The linear programming model for this problem is:

Maximize Z = 5X1 + 2X2 Subject to 3X1 + 2X2 ≤ 2400 X2 ≤ 800 2X1 ≤ 1200 X1 ≥ 0, X2 ≥ 0

a. Draw the feasible region of the above LP (1 point). b. Use the simplex algorithm to find the optimal solution of the above LP (Show the initial and

each succeeding tableau) (1 point). c. On the sketch of the feasible region in (a), indicate the initial basic solution and the basic

solution at each succeeding iteration (1 point). d. Describe the optimal solution briefly in “plain English” (1 point). e. Use the graphical method to find the optimal solution of this problem (1 point). Part II. (Modification of Problem #3.2-3 in page 83 from Hillier & Lieberman’s OR text, 10th edition) This is your lucky day. You have just won a \$20,000 prize. You are setting aside \$8,000 for taxes and partying expenses, but you have decided to invest the other \$12,000. Upon hearing this news, two different friends have offered you an opportunity to become a partner in two different entrepreneurial ventures, one planned by each friend. In both cases, this investment would involve expending some of your time next summer as well as putting up cash. Becoming a full partner in the first friend’s venture would require an investment of \$10,000 and 400 hours, and your estimated profit (ignoring the value of your time) would be \$9,000. The corresponding figures for the second friend’s venture are \$8,000 and 500 hours, with an estimated profit to you

of \$8,700. However, both friends are flexible and would allow you to come in at any fraction of a full partnership you would like. If you choose a fraction of a full partnership, all the above figures given for a full partnership (money investment, time investment, and your profit) would be multiplied by this same fraction. Because you were looking for an interesting summer job anyway (maximum of 600 hours), you have decided to participate in one or both friends’ ventures in whichever combination would maximize your total estimated profit. You now need to solve the problem of finding the best combination. The linear programming model for this problem is:

Maximize Z = 9000 X1 + 9000 X2 Subject to X1 ≤ 1 (Fraction of partnership #1) X2 ≤ 1 (Fraction of partnership #2) 10000 X1 + 8000 X2 ≤ 12000 (Money investment) 400 X1 + 500 X2 ≤ 600 (Time investment)

X1 ≥ 0 , X2 ≥ 0 Work through the simplex method step by step in tabular form (Show the initial and each succeeding tableau) (2 point). Part III. (Modification of Problem #4.3-7 in page 154 from Hillier & Lieberman’s OR text, 10th edition) Consider the following LP problem:

Maximize Z = 2X1 + 4X2 + 3X3 Subject to X1 + 3X2 + 2X3 <= 30

X1 + X2 + X3 <= 24 3X1 + 5X2 + 3X3 <= 60

X1 ≥ 0 , X2 ≥ 0 , X3 ≥ 0 You are given the information that X1 > 0, X2 = 0, and X3 > 0 in the optimal solution. a. Describe how you can use this information to adapt the simplex method to solve this problem

in the minimum possible number of iterations (when you start from the usual initial basic feasible solution). Do not actually perform any iterations (2 points).

b. Use the procedure developed in part (a) to solve this problem in tabular form (Show the

initial and each succeeding tableau) (1 point).

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