Hi$,$ can you help me with this problem, The Computer Services Department must decide how best to allocate all $23$ new laptops it has purchased. Three different departments have been designated to receive the laptops: the Production Department, Marketing Department, and Finance Department. Each department has requested as many laptops as they can get.

The Production Department can increase its productivity by $5\%$ for each new laptop it gets. The Marketing Department can increase its productivity by $4\%$ for each new laptop it gets. The Finance Department can increase its productivity by $2\%$ for each new laptop it gets.

All of the new laptops must be allocated, subject to the following constraints. The Marketing and Production Departments each must receive at least four of the new laptops. The Finance Department must get at least five of the new laptops. Furthermore, because of company politics, the Finance Department must get at least half as many of the new laptops as are allocated to the Production Department. For the same reason, the Finance Department must also get at least half as many of the new laptops as are given to the Marketing Department. The Production Department is adamant about getting at least four more of the new laptops than the Marketing Department gets. The Marketing Department must not get more than half of combined number of new laptops that are given to both the Finance and Production Departments.

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

$1.$ In new lines entered below this part, define all the decision variables. Be explicit and specific, including the units of measure.

$2.$ In new lines entered below this part, in terms of the decision variables defined above, state mathematically the objective function.

$3.$ In new lines entered below this part, in terms of the decision variables you defined above, state mathematically all the constraints of the situation described.

$4.$ In new lines entered below this part, state whether or not the problem you formulated mathematically in parts $1$ through $3$ is a linear program $($LP$)$ or not? If it is$,$ explain why.

$5.$ In new lines entered below this part, if this is a linear program, then what special kind of LP problem is this and explain why.

$6.$ Solve this problem with a computer using either Excel Solver or LINGO.

In a space created below this part, you must present HERE copies of both the computer input and output screens.

$7.$ Based on your computer solution presented in part $6,$ in new lines entered below this part, state in plain language the best allocation of the new laptops and the total percent productivity increase that would result.

$8.$ Based on your computer solution presented in part $6,$ in new lines entered below this part, state in plain language the precise meaning in the context of the business situation of any and all amounts of slack$/$surplus involved in the optimal solution.

$9.$ Based on your computer solution presented in part $6,$ in new lines entered below this part, state the ranges of optimality on the productivity gain per new laptop for each department $($that is$,$ for each objective function coefficient$).$

$10.$ In a new line entered below this part, state the name of the specific methodology used to solve the given problem as it is stated.