Lawn King manufactures two types of riding lawn mowers. One is a low-cost mower sold primarily to residential home owners; the other is an industrial model sold to landscaping and lawn service companies. The company is interested in establishing a pricing policy for the two mowers that will maximize the gross profit for the product line. A study of the relationships between sales prices and quantities sold of the two mowers has validated the following price-quantity relationships.

q1 = 950 – 1.5p1 + 0.7p2

q2 = 2500 + 0.3p1 - 0.5p2

Where

q1 = number of residential mowers sold

q2 = number of industrial mowers sold

p1 = selling price of the residential mower in dollars

p2 = selling price of the industrial mower in dollars

The accounting department developed cost information on the fixed and variable cost of producing the two mowers. The fixed cost of production for the residential mower is $10,000 and the variable cost is $1500 per mower. The fixed cost of production for the industrial mower is $30,000 and the variable cost is $4000 per mower.

a. Lawn King traditionally priced the lawn mowers at $2000 and $6000 for the residential and industrial mowers, respectively. Gross profit is computed as the sales revenue minus production cost. How many mowers will be sold, and what is the gross profit with this pricing policy?

b. Following the approach of Section 8.1, develop an expression for gross profit as a function of the selling prices for the two mowers.

c. What are the optimal prices for Lawn King to charge? How many units of each mower will be sold at these prices and what will the gross profit be?

d. Try a different formulation for this problem. Write the objective function as

Max p1q1 + p2q2 – c1 – c2

where c1 and c2 represent the production costs for the two mowers. Then add four constraints to the problem, two based on the price-quantity relationships and two based on the cost functions. Solve this new constrained optimization problem to see whether you get the same answer. What are the advantages of this formulation, if any?