Problem $2$: Pricing for Blue Ridge Hot Tubs

Howie Jones, owner of Blue Ridge Hot Tubs, is facing a new problem. Although sale of the two hot tubs manufactured by his company $($Aqua$-$Spas and Hydro$-$Luxes$)$ have been brisk, the company is not earning the level of profits that Howie wants to achieve. Having established a reputation for high quality and reliability, Howie believes he can increase profits by increasing the prices of the hot tubs.

However, he is concerned that a price increase might have a detrimental effect on demand, so Howie has engaged a marketing research firm to estimate the level of demand for Aqua$-$Spas and Hydro$-$Luxes at various prices. The marketing research firm used the technique of regression analysis to develop a model of the relationship between the prices and demand for the hot tubs. After analyzing the situation, the marketing research firm concluded that Howie can expect the demand for hot tubs in the next quarter to vary with price in the following way:

Demand for Aqua$-$Spas $=300-0\mathrm{.}175$ price of Aqua$-$Spas

Demand for Hydro$-$Luxes $=325-0\mathrm{.}15$ price of Hydro$-$Luxes

Howie Jones also needs to determine the number of each product to produce. Apparently, he cannot sell more than the anticipated demand for a product. For example, suppose Aqua$-$Spas are priced at $$900$ each, the demand will be $300-(0\mathrm{.}175)(900)=142\mathrm{.}5.$ That means Howie Jones can sell at most $142\mathrm{.}5$ products. Let's assume he can produce and sell fractional number of products.

Howie determined that the costs of manufacturing Aqua$-$Spas and Hydro$-$Luxes are $$850$ and $$700$ per unit, respectively.. Each Aqua$-$Spa requires $1$ pump, $9$ hours of labor, and $12$ feet of tubing; each HydroLux requires $1$ pump, $6$ hours of labor, and $16$ feet of tubing. Howie's suppliers have committed to supplying him with $200$ pumps and $2,800$ feet of tubing. Also, $1,566$ hours of labor are available for production. Howie wants to determine how much to charge for each type of hot tub and how many of each type to produce to maximize Howie's profit. Remember the company can only produce integer number of products.

Please provide a mathematical formulation for this problem $($defining decision variables, objective function and constraints$).$ You do NOT need to solve it$.$

Suppose there is a setup cost of $$1,000$ to produce any number of Aqua$-$Spas. That is$,$ the cost will be incurred as long as any Aqua$-$Spas is produced, and the cost is $$1,000$ regardless of the number of products produced. How would you revise your formulation to incorporate this change?

$($Not a continuum of part $2)$ If Howie Jones chooses to produce Hydro$-$Luxes, he needs to produce at least $300$ of it$,$ how would you revise your formulation to incorporate this change?

$($Not a continuum of parts $2$ and $3)$ The production manager of the factory told Howie that their facility is not designed for production of large volume. As a result, they can produce more than $300$ units for at most one of the two products. How would you revise your formulation to incorporate this change?