Vantage Industries is considering three new products to replace current models that are being discontinued,

so their operations research $($OR$)$ 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 $$125$ million from these products,

$(2)$ maintaining the current employment level of $4,000$ employees, and

$(3)$ holding the capital investment to less than $$55$ million.

However, management realizes that it probably will not be possible to attain all these goals simultaneously,

so it has discussed priorities with the OR 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

below, along with the goals and penalty weights.

Using goal programming:

a$)$ Formulate the mathematical model for this problem.

b$)$ Solve this problem in excel. Attach your excel workbook.