The Dewright Company is considering three new products to replace current models that are being discontinued, so their chief systems engineer has been assigned the task of determining which mix of these products should be produced. Management wants primary consideration given to three factors: long-run profit, stability in the work force, and the level of capital investment that would be required now for new equipment. In particular, they have established the goals of (1) achieving a long-run profit (net present value) of at least $125,000,000 from these products, (2) maintaining the current employment level of 4,000 employees, and (3) holding the capital investment to less than $55,000,000. However, management realizes that it probably won't be possible to attain all of these goals simultaneously, so they have discussed their priorities with the chief systems engineer. This discussion has led to setting penalty weights of 5 for missing the profit goal (per million dollars under), 2 for going over the employment goal (per hundred employees) and 4 for going under this same goal, and 3 for exceeding the capital investment goal (per million dollars over). Each new product's contribution to profit, employment level, and capital investment level is proportional to the numbers produced. These contributions per unit rate of production are shown in the table below, along with the goals and penalty weights.
Data for Dewright Company Goal Programming Problem
|
Factor
|
Unit Contribution
Product
1 2 3
|
Goal (Units)
|
Penalty
Weight
|
|
Long-run profit
Employment level
Capital investment
|
12 9 15
5 3 4
5 7 8
|
≥ 125 (millions of dollars)
= 40 (hundreds of employees)
≤ 55 (millions of dollars)
|
5
2(+), 4(-)
3
|
Formulate and solve the Dewright Company's problem.