An Introduction to Management Science 13th Edition by David Anderson,Dennis Sweeney ,Thomas Williams ,Jeffrey Camm, Kipp Martin

Exercise 11

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.
Where
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
where c 1 and c 2 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 function. Solve this new constrained optimization problem to sec whether you get the same answer. What arc the advantages of this formulation, if any?