Jim Matthews, vice president for marketing of the J. R. Nickel Company, is planning advertising campaigns for two unrelated products. These two campaigns need to use some of the same resources. Therefore, Jim knows that his decisions on the levels of the two campaigns need to be made jointly after considering these resource constraints. In particular, letting x 1 and x 2 denote the levels of campaigns 1 and 2, respectively, these constraints are 4x1 + x2 ≤ 20 and x1 + 4x2 ≤ 20.

In facing these decisions, Jim is well aware that there is a point of diminishing returns when raising the level of an advertising campaign too far. At that point, the cost of additional advertising becomes larger than the increase in net revenue (excluding advertising costs) generated by the advertising. After careful analysis, he and his staff estimate that the net profit from the first product (including advertising costs) when conducting the first campaign at level x 1 would be 3x1 - (x1 - 1)2 in millions of dollars. The corresponding estimate for the second product is 3x2 - (x2 - 2)2.

Letting P be total net profit, this analysis led to the following nonlinear programming model for determining the levels of the two advertising campaigns:

Maximize P = 3x1 - (x1 - 1)2 + 3x2 - (x2 - 2)2

subject to

4x1 + x2 ≤ 20

x1 + 4x2 ≤ 20

and

x1 ≥ 0 x2 ≥ 0

a. Construct tables to show the profit data for each product when the level of its advertising campaign is x1 = 0, 1, 2, 2.5, 3, 4, 5 (for the first product) or x2 = 0, 1, 2, 3, 3.5, 4, 5 (for the second product).

b. Use these profit data to draw rough-hand a smooth profit graph for each product. (Note that these profit graphs start at negative values when x1 = 0 or x2 = 0 because the products would lose money if there is no advertising to support them.)

c. On the profit graph for the first product, draw an approximation of this profit graph by inserting a dashed-line segment between the profit at x1 = 0 and x1 = 2, between the profit at x1 = 2 and x1 = 4, and between the profit at x1 = 4 and x1 = 5. Then do the same on the profit graph for the second product with x2 = 0, 2, 4, 5.

d. Use separable programming with the approximation of the profit graphs obtained in part c to formulate an approximate linear programming model on a spreadsheet for Jim Matthews's problem. Then solve this model. What does this solution say the levels of the advertising campaigns should be? What would the total net profit from the two products be?

e. Repeat parts c and d except using x1 = 0, 2, 2.5, 3, 5 and x2 = 0, 3, 3.5, 4, 5 for the approximations of the profit graphs in part c. (These particular approximations actually lead to the exact optimal solution for Jim Matthews's problem.)

f. Use Excel and Solver to formulate and solve the original nonlinear programming model directly. Compare with the answers obtained after completing part e.

g. Use calculus to find the value of x1 that maximizes 3x1 - (x1 - 1)2, the net profit from the first product. Also use calculus to find the value of x2 that maximizes 3x2 - (x2 - 2)2, the net profit from the second product. Show that these values satisfy the constraints for the nonlinear programming model. Then compare these values with the answers obtained in parts e and f.