Question
Problem 1: A furniture manufacturer produces two types of tables (country and contemporary) using three types of machines. The time required to produce the tables on each machine is given in the following table:
Machine Country Contemporary Available Hours
Router 1.5 2.0 1000
Sander 3.0 4.5 2000
Polisher 2.5 1.5 1500
Country tables sell for $350 and contemporary tables sell for $450. Management has determined that at least 20% of the tables made should be country and at least 30% should be contemporary. Formulate a linear programming model to determine the number of each table to make in order to maximize revenue.
Explanation / Answer
Formulate an LP model for this problem Create a spreadsheet model for this problem what is the optimal solution Use solver to create a sensitivity report and answer the following questions: If the company could get 50 more units of routing capacity, should they do it? If so how much should they be willing to pay for it? If the company could get 50 more units of sanding capacity, should they do it? If so how much should they be willing to pay for it? Suppose the polishing time on country tables could be reduced from 2.5 to 2 units per table. How much should the company be willing to pay to achieve this improvement in efficiency? Contemporary tables sell for $450. By how much would the selling price have to decrease before we would no longer be willing to produce contemporary tables? Does this make sense? Explain Let x be the number of country tables and y be the number of contemporary tables manufactured. The total revenue z obtained from selling x country tables and y contemporary tables is z = 350 x + 450 y. The constraints from machine availability are 1.5 x +2.0 y = 0.30(x + y) or -0.30 x + 0.70 y >= 0 Since negative quantities cannot be manufactured, we have x >= 0 and y >= 0. Since tables cannot be manufactured in fractions, x and y are integers. So the LPP is Maximize z = 350 x + 450 y subject to 1.5 x +2.0 y = 0, y >= 0 x and y are integers. The optimal solution is x = 405, y = 174, and z = 220,050 If the company could get 50 more units of routing capacity, should they do it? If so how much should they be willing to pay for it? They should not do it. At present, there is a slack of 43.5. the Lagrangian multiplier is 0. So they should not pay anything for the additional capacity because 50 *0 = 0. If the company could get 50 more units of sanding capacity, should they do it? If so how much should they be willing to pay for it? They should do it. They will be willing to pay a maximum of 50 * 110.1449273 = $5507.25 Suppose the polishing time on country tables could be reduced from 2.5 to 2 units per table. How much should the company be willing to pay to achieve this improvement in efficiency? At present, there is a slack of 224.64 slack in polishing time. The Lagrangian multiplier is 0. So there is no improvement in revenue. The company should not pay anything for this. Contemporary tables sell for $450. By how much would the selling price have to decrease before we would no longer be willing to produce contemporary tables? Does this make sense? Explain Since the management has decided that at least 30% of the tables should be contemporary, we have to produce contemporary tables as long we are producing tables.