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Introduction to Management Science (ISOM 3123) Darrell Hicks Practice Homework Problems Chapter 4 Spring 2013

Chapter 4 – Linear Programming: Modeling Examples (#3)(a) In the investment example in this chapter, how would the solution become affected if the requirement that the entire $70,000 be invested was relaxed such that it is the maximum amount available for investment? Answer: Original solution: x3 = $38,181.82 invested in treasury bonds, x4 = $31,818.18 invested in growth stock fund (max return on investments)

Z = $6,618.18 (as shown in Exhibit 4.7).

(a) Changes to constraints #1 and #3 must be made into algebraic expressions (pgs. 122, 123). Changing constraint #1 to: 0.8x1 - 0.2x2 - 0.2x3 - 0.2x4 ≤ 0, constraint #3 to: 0.3x1+ 0.7x2+ 0.7x3 - 0.3x4≥ 0, and constraint #5 to: x1 + x2 + x3 + x4 ≤ 70,000produces the following results: x1 = $0.00, x2 = $0.00, x3 = $38,181.82 invested in treasury bills, x4 = $31,818.18 invested in growth stock fund Z = $6,618.18 (no change from the original problem) (b) If the entire amount available for investment does not have to be invested and the amount available is increased by $10,000 (to $80,000), how much will the total optimal return increase? Answer: Changing only constraint #5 to: x1 + x2 + x3 + x4 ≤ 80,000produces the following results: x3 = $43,636.36 invested in treasury bills, x4 = $36,363.64 invested in growth stock fund (max return on investments)

Z = $7,563.64; the increase is $7,563.64 -

$6,618.18 = $945.46 increase (c) Will the entire $10,000 increase be invested in one alternative? Answer: (c) No, both x3 and x4 increase in the amounts invested respectively: x3 invests $354.54 more than before and x4 invests $590.91 more (than with $70,000 total). The sum of these two investments (treasury bills + stock funds) is $7,563.63. The optimal investments are now in the same arrangement as in the original problem. Treasury bills

Introduction to Management Science (ISOM 3123) Darrell Hicks Practice Homework Problems Chapter 4 Spring 2013 and growth stock funds should be the only investments:x3 = $43,636.36 invested in treasury bonds, x4 = $36,363.64 invested in growth stock fund (with $80,000 total). The sum is $43,636.36 + $36,363.64 = $80,000 total investments; no municipal bonds & no certificates of deposit.

(#5) For the transportation example in this chapter, suppose that television sets not shipped were to incur storage costs of $9.00 at Cincinnati, $6.00 at Atlanta, and $7.00 at Pittsburgh. How would these storage costs be reflected in the linear programming model for this example problem, and what would the new solution be if any? Original Transportation Example(pg.127)

Warehouse 1 Cincinnati 2 Atlanta 3 Pittsburgh Actual TV Sets Shipped Constraints City Demand Cost = $

New York A 0 0 150 150 = 150 7300

Store Dallas B 0 200 50 250 = 250

Detroit C 200 0 0 200 = 200

TV Sets Shipped 200 200 200

Constraint