Problem 3-12 (Algorithmic) Quality Air Conditioning manufactures three home air
ID: 393455 • Letter: P
Question
Problem 3-12 (Algorithmic) Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $61, s99, and $135, respectively. The production requirements per unit are as Number of Cooling Coils Manufacturing Time (hours) Number of Fans Economy Standard Deluxe 12 14 For the coming production period, the company has 250 fan motors, 360 cooling coils, and 2600 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows: Max 61E99S135D S.t. 1E1S1D S 2S0 Fen motors 1E254D S360 Cooling coils BE 12S14D 2600 Manufacturing time E, S, D20 The computer solution is shown in the figure below. Optimal Objective Value19430.00000 Variable Value 140.00000 110.00000 Reduced Cost 0.00000 0.00000 40.00000 0.00000 Constraint Slack/Surplus Dual Value 0.00000 0.00000 160.00000 23.00000 38.00000 0.00000 Objective Coefficient Allowable Decrease Variable 61.00000 99.00000 135.00000 Increase 20.00000 23.00000 40.00000 11.50000 13.33333 InfiniteExplanation / Answer
Ans a)
From the computer solution, the Optimal solution is as follows-
Economy model = 140 unit
Standard model = 110 unit
Deluxe model = 0 unit
Value of objective function = $19430
Ans b)
Fan motors constraint is binding as it has a shadow price or dual value of 23. Changes in the right-hand side of the constraint within allowable limits would change the objective function value. Hence it is binding.
Cooling coil constraint is binding as it has a shadow price or dual value of 38. Changes in the right-hand side of the constraint within allowable limits would change the objective function value. Hence it is binding.
Manufacturing time constraint is not binding as it has a shadow price or dual value of 0. Changes in the right-hand side of the constraint within allowable limits would not change the objective function value. Hence it is not binding.
Ans c)
Fan motor constraint has no extra capacity as it has 0 surplus.
Cooling coils constraint has no extra capacity as it has 0 surplus.
Manufacturing time constraint has an extra capacity as it has 160 hours of surplus.
Ans d)
Current profit per unit of deluxe model = $135
The allowable increase of coefficient = 40
Due to this allowable increase, the optimal solution would not change because the profit of the deluxe model can vary from $0 to $175. $150 is in this range without the optimal solution changing.
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