. Problem 2.10* (Goal programing with penalties): A manufacturing company in Houston is considering three new products to replace current models that are being discontinued, so their Analytics department 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 workforce, and the level of capital investment that would be required now for new equipment. In particular, management has established the goals of (1) achieving a long-run profit (net present value) of at least $120 million from these products, (2) maintaining the current employment level of 4,000 employees, and (3) holding the capital investment to less than $60 million. However, management realizes that it probably will not be possible to attain all these goals simultaneously, so it has discussed priorities with the Analytics department. This discussion has led to setting penalty weights of 5 for missing the profit goal (per $1 million under), 2 for going over the employment goal (per 100 employees), 4 for going under this same goal, and 3 for exceeding the capital investment goal (per $1 million over). Each new product's contribution to profit, employment level, and capital investment level is proportional to the rate of production. These contributions per unit rate of production are shown in Table, along with the goals and penalty weights. Unit contribution Goal Penalty Factor Product 1 Product 2 Product 3 (units) weight Long-run profit 10 20 5 120 (millions of dollars) 5 Employment level 6 10 4 - 40 (hundreds of employees) 2(+), 4(-) Capital investment 4 12 2