Lawn King manufactures two types of riding lawn mowers. One is a low-cost mower sold primarily to residential home owners; the other one 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 relationship 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, and 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 $1,500 per mower. The fixed cost of production for the industrial mower is $30,000 and the variable cost is $4,000 per mower.
a) Lawn King traditionally priced the lawn mowers at $2,000 and $6,000 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 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?