"This is your lucky day. You have just won a $20,000 prize. You are setting aside $8,000 for taxes and partying expenses, but you have decided to invest the other $12,000. Upon hearing this news, two different friends have offered you an opportunity to become a partner in two different entrepreneurial ventures, one planned by each friend. In both cases, this investment would involve expending some of your time next summer as well as putting up cash. Becoming a full partner in the first friend's venture would require an investment of $10,000 and 400 hours, and your estimated profit (ignoring the value of your time) would be $9,000. The corresponding figures for the second friend's venture are $8,000 and 500 hours, with an estimated profit to you of $9,000. However, both friends are flexible and would allow you to come in at any fraction of a full partnership you would like. If you choose a fraction of a full partnership, all the above figures given for a full partnership (money investment, time investment, and your profit) would be multiplied by this same fraction.

Because you were looking for an interesting summer job anyway (maximum of 600 hours), you have decided to participate in one or both friends' ventures in whichever combination would maximize your total estimated profit. You now need to solve the problem of finding the best combination.

a. Describe the analogy between this problem and the Wyndor Glass Co. problem discussed in Section 2.1. Then construct and fill in a table like Table 2.1 for this problem, identifying both the activities and the resources.

TABLE 2.1: Data for the Wyndor Glass Co. Product-Mix Problem:

b. Identify verbally the decisions to be made, the constraints on these decisions, and the overall measure of performance for the decisions.
c. Convert these verbal descriptions of the constraints and the measure of performance into quantitative expressions in terms of the data and decisions.
d. Formulate a spreadsheet model for this problem. Identify the data cells, the changing cells, and the objective cell. Also show the Excel equation for each output cell expressed as a SUMPRODUCT function. Then use Solver to solve this model.
e. Indicate why this spreadsheet model is a linear programming model.
f. Formulate this same model algebraically.
g. Identify the decision variables, objective function, non-negativity constraints, functional constraints, and parameters in both the algebraic version and spreadsheet version of the model.
h. Use the graphical method by hand to solve this model. What is your total estimated profit?

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Production Time Used for Each Unit Produced Doors 1 hour Available per Week 4 hours 12 hours 18 hours Windows Plant 2 hours 2 hours $500 3 hours $300 Unit profit