Management of the Albert Franko Co. has established goals for the market share it wants each of the company’s two new products to capture in their respective markets. Specifically, management wants Product 1 to capture at least 15 percent of its market and Product 2 to capture at least 10 percent of its market. Three advertising campaigns are being planned to try to achieve these market shares. One is targeted directly on the first product. The second targets the second product. The third is intended to enhance the general reputation of the company and its products. Letting x₁, x₂, and x₃ be the amount of money allocated (in millions of dollars) to these ­respective campaigns, the resulting market share ­(expressed as a ­percentage) for the two products are estimated to be

Market share for Product 1 = 0.5x1 + 0.2x3,

Market share for Product 2 = 0.3x2 + 0.2x3.

A total of $55 million is available for the three advertising campaigns, but management wants at least $10 million devoted to the third campaign. If both market share goals cannot be achieved, management considers each 1 percent decrease in the market share from the goal to be equally serious for the two products. In this light, management wants to know how to most effectively allocate the available money to the three campaigns.

(a) Formulate a goal programming model for this problem.

(b) Reformulate this model as a linear programming model.

(c) Use the simplex method to solve this model.