Question

Q4.         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.    Describe why this problem is a weighted goal-programming problem by giving quantitative expressions for the goals and the overall objective. (4 points)

        b.    Formulate and solve this problem as a linear programming model on a spreadsheet (or Python) & get the optimal solution. (6 points)

        c.     Interpret this solution to management in its simple English language. What is your recommendation (3 Points)

Answer 1

(a)

Goal programming model is following:

Standard variables: x1, x2, x3 be the amount of money allocated (in millions of dollars) to the three campaigns.

Deviation variables: pj and nj be the positive and negative deviation variables for j-th goal. There are two goals, one each for market share of the two products.

Minimize n1+n2

s.t.

x1+x2+x3 <= 55

x3 >= 10

.5x1+.2x3-p1+n1=15

.3x2+.2x3-p2+n2=10

x1, x2, x3, pj, nj >= 0

(b)

Both goals cannot be achieved with the $ 55 million budget. So, budget constraint has to be removed for formulation as linear program and the objective of the model is minimization of cost. Resulting LP model is following:

Min x1+x2+x3

s.t.

x3 >= 10

0.5x1+ 0.2x3 >= 15

0.3x2+ 0.2x3 >= 10

x1, x2, x3 >= 0

Excel Solution:

Formulae:

Solver Parameters:

(c)

1. Management should invest $13.33 million in campaign 1, $0 million in campaign 2, and $41.67 million in campaign 3.

2. Two goals cannot be met at the same time with $55 million.

3. The market share of Product 1 and Product 2 is 15% and 8.33%. For product 2, it cannot reach 10%.

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