Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows: Number of Fans Number of Cooling Coils Manufacturing Time (hours)
Economy 1 1 8
Standard 1 2 12
Deluxe 1 4 14
For the coming production period, the company has 200 fan motors, 320 cooling coils, and 2400 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 63E + 95S + 135D
s. t.
1E + 1S + 1D ≤ 200 Fan motors
1E + 2S + 4D ≤ 320 Cooling coils
8E + 12S + 14D ≤ 2400 Manufacturing time
E, S, D ≥ 0
The computer solution is shown in the figure below.
Optimal Objective Value = 16440.00000
Variable Value Reduced Cost
E 80.00000 0.00000
S 120.00000 0.00000
D 0.00000 -24.00000
Constraint Slack/Surplus Dual Value
Fan motors 0.00000 31.00000
Cooling coils 0.00000 32.00000
Manufacturing time 320.00000 0.00000
Variable Objective Coefficient Allowable Increase Allowable Decrease
E 63.00000 12.00000 15.50000
S 95.00000 31.00000 8.00000
D 135.00000 24.00000 Infinite
Constraint RHS Value Allowable Increase Allowable Decrease
Fan motors 200.00000 80.00000 40.00000
Cooling coils 320.0000 80.00000 120.00000
Manufacturing time 2400.00000 Infinite 320.00000
What is the optimal solution, and what is the value of the objective function? If required, round your answers to the nearest whole number.
Optimal Solution
Economy models (E)
Standard models (S)
Deluxe models (D)
Value of the objective function $
Which constraints are binding?
Fan motors: Binding
Cooling coils: Binding
Manufacturing time: Non binding
Which constraint shows extra capacity? How much? If constraint shows no extra capacity, enter 0 as number of units. If required, round your answers to the nearest whole number.
Constraints Extra capacity Number of units
Fan motors No ?
Cooling coils No ?
Manufacturing time Yes ?
If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change?
The optimal solution would not change because the profit of the deluxe model can vary from $0 to $159 . $150 is in this range without the optimal solution changing.