Question

Problem 3-12 (Algorithmic) uality Air Conditioning manufactures three home air conditioners: an economy mode a standard model and a deluxe mo e The profits per nit are S,, 4 , and respectively. The production requirements per unit are as follows: Number of Number of Manufacturing Cooling Coils Time (hours) Fans Economy 8 Standard 12 Deluxe 4 14 For the coming production period, the company has 320 fan motors, 380 cooling coils, and 3200 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 57E93S141D s.t. 1E 1S 1D 320 Fan motors 1E 2SAD s 380 Cooling coils 8E12514D 3200 Manufacturing time E, S, D2 0 The computer solution is shown in the figure below Optimal Objective Value20400.00000 Variable Value Reduced Cost 260.00000 0.00000 60.00000 0.00000 0.00000 24.00000 Slack/Surplus Dual Value Constraint 0.00000 21.00000 0.00000 36.00000 400.00000 0.00000 Objective Allowablee Allowable Variable Coefficient Increase Decrease 57.00000 12.00000 10.50000 93.00000 21.00000 8.00000 Infinite 141.00000 24.00000 RHS Allowable Allowable Constraint Value Increase Decrease 320.00000 60.00000 130.00000 380.00000 60.00000 2 100.00000 3200.00000 Infinite 400.00000 a. What is the optimal solution, and what is the value of the objective function? If required, round your answers to the nearest whole number. If your answer is zero, enter "O Optimal Solution Economy models (E) Standard models (S) Deluxe models (D) Value of the objective function $ b. Which constraints are binding? Fan motors: Cooling coils Manufacturing time: c. 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. Extra capacity Number of units Constraints Fan motors Cooling coils Manufacturing time d. If the profit for the deluxe model were increased to $156 per unit, would the optimal solution change? change because the profit of the deluxe model can vary from $156 is The optimal solution this range without the optimal solution changing to

Answer 1

a.

Optimal solution

Economy models(E) = 260
Standard models (S) = 60
Deluxe models(D) = 0

Value of the objective function = 20400

b.

Fan Motors => Binding
Cooling Coils => Binding
Manufacturing time => Not binding

Binding constraints do not have slack

c.

Extra Capacity Number of units Constraints Fan Motors No C) Cooling Coils No C) Manufacturing Time Yes

d.

The optimal solution will not change because the profit of the deluxe model can vary from $141 to $165.$156 is within this range without the optimal solution changing