Mathematical modeling

2.4 Second Example: Market AllocationsA marketing company is planning a one week advertising campaign for their new knife.The ads have been designed and produced and now they wish to determine how muchmoney to spend in each advertising outlet. They have hundreds of possible outlets tochoose from. Let’s consider two outlets: Prime-time TV, and newsmagazines. Theproblem of optimally spending advertising money can be formulated in many ways. Forinstance, given a fixed budget, the goal might be to maximize the number of targetcustomers reached (customer with a reasonable chance of purchasing the product). Analternative approach, which we adopt here, is to define targets for reaching each marketsegment and to minimize the money spent to reach those targets. For this product, thetarget segments are teenage boys, women (ages 40-49), and retired men. Each minute ofprimetime TV and page of newsmagazine advertisement reaches the following number ofpeople (in millions):Outlet Boys Women Men CostTV 5 1 3 600Magazine 2 6 3 500Target 24 18 24Again, the marketing company is willing to know how many units of each outlet topurchase to meet the goals.2.4.1 ModelingThe process of modeling using linear programming is fairly consistent: define thedecision variables, objective, and constraints.Decision Variables: In this case, they are the number of units of each outlet to purchase.Let x 1 be the number of TV spots, and x 2 the number of magazine pages.Objective: Our objective is to minimize the amount we spend: 600x 1 + 500x 2 .Systems Analysis 25 Dr. Emad Elbeltagi