Question

Problem 8-04 (Algorithmic)

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 = 1050 – 1.5p1 + 0.4p2

q2 = 3500 + 0.6p1 – 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 $15,000 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 $5000 per mower. 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?

q1 =

q2 =

c1 = $

c2 = $

Gross Profit = $

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 +

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 = $

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

Answer 1

solution - gelling price of in dollars The Residential mower P.)- $ 2,000 selling price of In dollar's Industrial mower P.)- $ 6.000 no of residential mower sold 9 ) = 1. q i= 1050 - 1,5P + 0.482 qi=1050 -1.5[2000] +0.4 Ľ 6,000] 19, = $450 of Resid Industrial mower sold (92) r2 = 3500 + 0.6 P,-0.5 P2 -> 3500 +0.6 [2000] -0.5[6,000] no 7,-$1,7008 coz 1500 + 1500 x B3 from Given Data, C₂ = 3000 + 5000 x By Grapp profit = B, XB3 + B₂ X Bu - Bg-B6.

expression for Goong profit is followin va G= Revenue from Residential mower of Revenue from Industrial mowry - total production cost of Presidential mowerp - total production cost of Industrial Pix quo + Pe X 92-15000 -1500 91-3000-500092 In above expression substituting Their g, hq with REP Respective expressions in terms of P, AP. we get,

G= p [1050-1.5 P1+0.4Pe] + Pe [3500 +0.6 P,-0.5f2] - 1500 - 1500[1050-1.5P, +0.4 Pa] – 3000-5000[ 3500 +0.37, -0.5P2. to 1050p,-1.5P² +0.4P, P2 + 3500 P2 +0.6P, P2-0.5P ² -1500 -1,575,000 +2250 P, -600 P2 - 3000 -1,75,00,000 - 1500p, + 2500 P2

=> 1-1.5 P2 - 0.5P2? + Pi. Pq +1800 Pi +5900 Pzt - 19,075,000 © using the part (a) enter Shown below. spreadsheet moded develop an stover slo solver parameters as

S Solver Parameters $B$8| value of: O Set Objective: To: Max Min By Changing Variable Cells: $B$1:$B$2 Subject to the Constraints: Add Change Delete Reset All Make Unconstrained variables Non-Negative Select a Solving Method: IPOPT Nonlinear Options Solving Method Select the IPOPT Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems. | Solve Close

Note click that solve this is non-linear model. to generate the solubion. P,= Grogo profit = & The same formulation as asked in this part is used In The above sphreadsheet model objective function is cell Bo, which is max Piq, + P2 92-C, - Cq , The excel formula is: =B* Bag + Bat By - Bg-B6

Cia $ + $qexcel formula is: 1500 +15007B3 C2 = $ + $9 excel formula 19: 3000 +5000* By q, = P +P2 excel formula ip = 1050-1.5 B, 70.4B2 Y = P+P excel Ermula is = 3500+ 0.6B, +0.5B2 o The end