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IE426 Problem Set #3—Answers Prof Jeff Linderoth IE 426 – Problem Set #3–Answers 1 Selling Swoosh Shoes Swoosh Shoes, Inc. has established goals for the market share it wants two new products to capture in their respective markets. Specifically, Swoosh management wants Product 1 to capture 25% of its market and Product 2 to capture 20% of its market. Three advertising campaigns are being planned to achieve to try and achieve these market shares. The first is targeted directly on the first product. The second targets the second product, and the third is intended to enhance the general reputation of the company and its products. Letting x 1 , x 2 , and x 3 be the amount of money (in millions of dollars) allocated to each campaign, management estimates that the resulting market shares can be expressed (in percentages) as Market Share for Product 1: 0 . 5 x 1 + 0 . 2 x 3 (1) Market Share for Product 2: 0 . 3 x 2 + 0 . 2 x 3 . (2) A total of \$55M is available for the three advertising campaigns, but management wants at least \$10M devoted to the third campaign. 1.1 Problem Formulate a linear program that will determine the proper allocation of money to advertising cam- paigns. Answer: minimize x 1 + x 2 + x 3 subject to x 1 + x 2 + x 3 55 0 . 5 x 1 + 0 . 2 x 3 25 0 . 3 x 2 + 0 . 2 x 3 20 x 1 0 x 2 0 x 3 10 1.2 Problem Build your model from Problem 1.1 in the Mosel modeling language and solve it. What is the result of solving the instance? Answer: A Mosel model for this is given here: Problem 1 Page 1

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IE426 Problem Set #3—Answers Prof Jeff Linderoth !\$Id: swoosh-1.mos,v 1.1 2006-10-22 12:13:14 jeff Exp \$ ! Model: swoosh-1 ! Author: Jeff Linderoth, Lehigh University model "swoosh-1" uses "mmxprs" forward procedure print_status forward procedure print_solution declarations x1: mpvar ! Amount invested in campaign 1 x2: mpvar ! Amount invested in campaign 2 x3: mpvar ! Amount invested in campaign 3 end-declarations InvestAmount := x1 + x2 + x3 MoneyAvailable := x1 + x2 + x3 <= 55 Product1MS := 0.5 * x1 + 0.2 * x2 >= 25 Product2MS := 0.3 * x2 + 0.2 * x3 >= 20 InvestInC3 := x3 >= 10 minimize(InvestAmount) print_status if(getprobstat = XPRS_OPT) then print_solution end-if procedure print_status declarations status: array({XPRS_OPT,XPRS_UNF,XPRS_INF,XPRS_UNB, XPRS_OTH}) of string end-declarations status:= ["Optimum found", "Unfinished", "Infeasible", "Unbounded", "Failed"] writeln("Problem status: ", status(getprobstat)) end-procedure procedure print_solution writeln("Invest: ", getsol(x1), " in Campaign 1") writeln("Invest: ", getsol(x2), " in Campaign 2") writeln("Invest: ", getsol(x3), " in Campaign 3") end-procedure The output from running this model is the following: Problem status: Infeasible 1.3 Problem Change your Problem 1.2 to a linear program that will come as close as possible to meeting market share for Product 2 while satisfying all others constraints. Solve this instance. What is the result?
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