The Dorwyn Company has two new products (special kinds of doors and windows) that will compete with the two new products for the Wyndor Glass Co. (described in Section 2.1). Using units of hundreds of dollars for the objective function, the linear programming model in algebraic form shown below has been formulated to determine the most profitable product mix. Maximize Profit = 4D + 6W
subject to
D + 3W ≤ 8
5D + 2W ≤ 14
and
D ≥ 0 W ≥ 0
However, because of the strong competition from Wyndor, Dorwyn management now realizes that the company will need to make a strong marketing effort to generate substantial sales of these products. In particular, it is estimated that achieving a production and sales rate of D doors per week will require weekly marketing costs of D3 hundred dollars (so $100 for D = 1, $800 for D = 2, $2,700 for D = 3, etc.). The corresponding marketing costs for windows are estimated to be 2W2 hundred dollars. Thus, the objective function in the model should be
Profit = 4D + 6W − D3 − 2W2
Dorwyn management now would like to use the revised model to determine the most profitable product mix.
a. Formulate and solve this nonlinear programming model on a spreadsheet.
b. Construct tables to show the profit data for each product when the production rate is 0, 1, 2, 3.
c. Draw a figure that plots the weekly profit points for each product when the production rate is 0, 1, 2, 3. Connect the pairs of consecutive points with (dashed) line segments.
d. Use separable programming based on this figure to formulate an approximate linear programming model on a spreadsheet for this problem. Then solve the model. What does this say to Dorwyn management about which product mix to use?
e. Compare the solution based on a separable programming approximation in part d with the solution obtained in part a for the exact nonlinear programming model.