For Problem $1$ and $2,$ formulate a linear program and solve the problem using Excel Solver.

Please include an algebraic $($mathematical$)$ formulation of the linear program. Specify your decision variables, followed by the objective function and the complete set of constraints.

After solving the problem with Excel, please generate an answer report and copy your solution & objective value to the end of your formulation.

Problem $1$: PROTRAC produces the largest member of its earthmoving equipment line $($the E$-9)$ and the largest member of its forestry equipment line $($the F$-9)$ with essentially the same equipment. Demand for these items is very high at this time, and management believes it is possible to sell as many of both as the firm can produce. The profit is $$5000$ for an E$-9$ and $$4000$ for an F$-9.$ In planning production for the next $2$ weeks, management knows it is necessary to produce a total of at least five E$-9$s and F$-9$s in order to meet previous commitments.

Each of these products requires processing in department A and department B$.$ Over the next $2$ weeks, these departments have $150$ and $160$ hours of available time, respectively. Each E$-9$ requires $10$ hours in department A and $20$ hours in department B$,$ whereas each F$-9$ requires $15$ hours in department A and $10$ hours in department B$.$ In order to honor the labor contract, the total labor hours used in testing finished products cannot fall more than $10\%$ below the agreed goal of $150$ hours. $($This is independent of the processing requirements in departments A and B$.)$ Each E$-9$ requires $30$ hours of testing and each F$-9$ requires $10.$ Finally, in order to maintain the current market position, management feels that it is necessary to build at least one F$-9$ for every three E$-9$s produced. How much of each product should PROTRAC produce in the next $2$ weeks to maximize profit?