The Computer Services Department must decide how to allocate 21 new PC's. Three different departments have been designated to receive the PC's: the Production Department, Marketing Department, and Finance Department. Each department has requested as many PC's as they can get.

The Production Department can increase its productivity by 5% for each new PC it gets. The Marketing Department can increase its productivity by 3% for each new PC it gets. The Finance Department can increase its productivity by 2% for each new PC it gets.

All of the new PCs must be allocated, subject to the following constraints. The Production Department needs at least five PCs. Because of company politics, the Finance Department must have no less than half as many new PCs as the Marketing Department. It also must have at least one-third as many new PCs as the Production Department.

The problem is to allocate all the new PC's to the Production, Finance, and Marketing Departments so as to maximize the total increase in productivity.

1. Define the decision variables. Be explicit and specific (including units).
a. In terms of the decision variables defined above, state mathematically the objective function.
b.In terms of the decision variables you defined above, state mathematically all the constraints of the situation described.
c.Is the problem you formulated mathematically, inclusive, above a linear program (LP) or not? If it is, explain why.
d. What special kind of LP problem is this?

e. Solve this problem with a computer using Lingo.

f. On the basis of your computer solution in part e, a state in plain language the best allocation of the new laptops and the total percent productivity increase that would result.

g. On the basis of your computer solution in part e, interpret, in plain language any amounts of slack/surplus involved in the optimal solution.

h. On the basis of your computer solution in part e, determine the ranges of optimality on the productivity gain per new Laptop for each department.

i. State the name of the specific methodology used by your computer program to solve this problem.