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 = 3500 + 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 $20,000 and the variable cost is $2500 per mower. The fixed cost of production for the industrial mower is $40,000 and the variable cost is $4000 per mower.

1. Lawn King traditionally priced the lawn mowers at $3000 and $7000 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?
   a.

   	q1 =
   	q2 =
   	c1 = $
   	c2 = $
   	Gross Profit = $

2. 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. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. If constant is “1”, it must be entered in the box.
   
   G = p₁² + p₂² + p₁p₂ + p₁ + p₂ +
3. 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? If required, round your answers to the nearest whole number.
   

   	p1 = $
   	p2 = $
   	q1 =
   	q2 =
   	G = $

4. 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. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank.
   

   	s.t.
   	c1 = $ + $q1
   	c2 = $ + $q2
   	q1 = + p1 + p2
   	q2 = + p1 + p2

   What are the advantages of this formulation, if any?