Kane Manufacturing has a division that produces two models of fireplace grates, model A and model B. To produce each model A grate requires 3 lb of cast iron and 6 min of labor. To produce each model B grate requires 4 lb of cast iron and 3 min of labor. The profit for each model A grate is $2, and the profit for each model B grate is $1.50. 1,000 lb of cast iron and 22 labor-hours are available for the production of grates each day. Because of an excess inventory of model A grates, management has decided to limit the production of model A grates to no more than 200 grates per day. How many grates of each model should the division produce daily to maximize Kane's profits? The range of values that the coefficient of x can assume without changing the optimal solution is If the contribution to the profit of a model A grate is changed to $1.90/grate, will the original optimal solution still hold? What will be the new optimal solution?
A) No; 136 grates of model A; 136 grates of model B; optimal profit is $486.70
B) Yes; 152 grates of model A; 152 grates of model B; optimal profit is $486.70
C) Yes; 152 grates of model A; 136 grates of model B; optimal profit is $492.80
D) No; 136 grates of model A; 152 grates of model B; optimal profit is $492.80

Question

Kane Manufacturing has a division that produces two models of fireplace grates, model A and model B. To produce each model A grate requires 3 lb of cast iron and 6 min of labor. To produce each model B grate requires 4 lb of cast iron and 3 min of labor. The profit for each model A grate is $2, and the profit for each model B grate is $1.50. 1,000 lb of cast iron and 22 labor-hours are available for the production of grates each day. Because of an excess inventory of model A grates, management has decided to limit the production of model A grates to no more than 200 grates per day. How many grates of each model should the division produce daily to maximize Kane's profits? The range of values that the coefficient of x can assume without changing the optimal solution is  If the contribution to the profit of a model A grate is changed to $1.90/grate, will the original optimal solution still hold? What will be the new optimal solution?
A) No; 136 grates of model A; 136 grates of model B; optimal profit is $486.70 
B) Yes; 152 grates of model A; 152 grates of model B; optimal profit is $486.70 
C) Yes; 152 grates of model A; 136 grates of model B; optimal profit is $492.80 
D) No; 136 grates of model A; 152 grates of model B; optimal profit is $492.80

Quizplus · Accepted Answer

To maximize profits, we can set up the following linear programming model:

Maximize 2A + 1.5B
subject to
3A + 4B <= 1000 (cast iron constraint)
6A + 3B <= 1320 (labor constraint)
A <= 200 (limit on model A production)

where A is the number of model A grates produced, and B is the number of model B grates produced.

Using a linear programming calculator or software, we can find that the optimal solution is A = 152 and B = 136, with a profit of $492.80.

To answer the second part of the question, we can simply change the coefficient of A in the objective function from 2 to 1.9, and solve the same linear programming problem. Using a calculator or software, we find that the new optimal solution is still A = 152 and B = 136, with a profit of $492.80. Therefore, the answer is C.

Kane Manufacturing Has a Division That Produces Two Models of Fireplace