$2.$ Campbell Motors $(20$ points$)$

Campbell Motors is an auto dealership that specializes in the sale of station wagons and light trucks. Because of its reputation for quality and service, Campbell has a strong position in the regional market, but demand remains sensitive to price. While evaluating the new models, Campbell's marketing consultant has come up with the following demand$/$sales response curves:

Truck Demand $=550-17($Truck Price in thousands$)$ Wagon Demand $=\backslash (450-12\backslash )($Wagon Price in thousands$)$

The dealership's unit costs are $\backslash$$$21,000$ for each truck and $\backslash (\backslash$$ $24,000\backslash )$ for each wagon. Each truck requires $2\mathrm{.}5$ hours of prep labor, and each wagon requires $2$ hours of prep labor. The current staff can supply at most a total of $300$ hours of prep labor for the wagons and trucks purchased.

In order to feel good about the environment, Campbell has decided that the vehicles sold by Campbell Motors will meet the EPA emissions limit for NOx. Thus, Campbell wants the average emissions of NOx over the vehicles sold by Campbell Motors to be at most $\mathrm{.}07$ grams per mile. From company data Campbell knows that a truck emits $.13$ grams per mile and that a wagon emits $\mathrm{.}05$ grams per mile. Campbell must determine how many cars and trucks to purchase to maximize his profit.

Hint: See Unit $6$ Practice Problem $\backslash$#$2$

$($a$)$ State the decisions to be made in Campbell's problem, the objective and the constraints in words using or $=$ in the constraints. $($Submit copy of Handwritten answer, $5$ pts$)$

$($b$)$ Define your inputs algebraically as decision variables. Then, using your objective and constraints in part $($a$),$ formulate Campbell's problem as an algebraic linear optimization model. $($Submit copy of Handwritten answer, $5$ pts$)$

$($c$)$ Construct an EXCEL model corresponding to your algebraic model in part $($b$).($Submit answer in an Excel file, $5$ pts$)$

$($d$)$ Optimize your EXCEL model using Solver and give the Answer Report. What is Campbell's maximum profit? What prices should Campbell set for the trucks and wagons to maximize profit? How many trucks and wagons should he sell? $($Submit answer in an Excel file, $5$ pts$)$