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 a reasonable price range for the hot tubs is between $1,000 and $1,500, and that within this range, 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 - 0.175 × price of Aqua-Spas

Demand for Hydro-Luxes = 325 - 0.15 × price of Hydro-Luxes

Howie determined that the costs of manufacturing Aqua-Spas and Hydro-Luxes are $850 and $700 per unit, respectively. Ideally, he wants to produce enough hot tubs to meet demand exactly and carry no inventory. Each Aqua-Spa requires 1 pump, 9 hours of labor, and 12 feet of tubing; each Hydro-Lux 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.

a. Formulate an NLP model for this problem.

b. Implement your model in a spreadsheet and solve it.

c. What is the optimal solution?

d. Which of the resource constraints are binding at the optimal solution?

e. What values would you expect the Lagrange multipliers to take on for these constraints? (Create a Sensitivity Report for this problem to verify your answer.)