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 $69, 91, and $127, respectively. The production requirements per unit are as follows: Number ofNumber of Manufacturing Fans Cooling Coils Time (hours) Economy 8 Standard 12 Deluxe 14 4 For the coming production period, the company has 200 fan motors, 340 cooling coils, and 2800 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 69E91S 127D s.t 1E 1SD 200 Fan motors 1E 2s +4D 340 Cooling coils 8E 12S + 14D 2800 Manufacturing time E,S,D20 The computer solution is shown in the figure below Optimal Objective Value16880.00000 16880.00000 Optimal Objective Value - Value Variable Reduced Cost 60.00000 0.00000 140.00000 0.00000 0.00000 8.00000 Constraint Slack/Surplus Dual Value 0.00000 47.00000 0.00000 22.00000 2 0.00000 3 640.00000 Objective Allowable Allowable Variable Coefficient Increasee Decrease 69.00000 23.50000 4.00000 91.00000 47.00000 2.66667 127.00000 8.00000 Infinite RHS Allowable Allowable Constraint Value Increasee Decrease 200.00000 140.00000 30.00000 340.00000 60.00000 2 140.00000 Infinite 3 2800.00000 640.00000 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 60 Economy models (E) Standard models (s) 140 Deluxe models (D) 0 Value of the objective function16,880 b. Which constraints are binding? Binding Fan motors Binding Cooling coils: Manufacturing time Non binding 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, Constraints Extra capacity Number of units Fan motors No 0 Cooling coils No 640 Manufacturing time Yes d. If the profit for the deluxe model were increased to $142 per unit, would the optimal solution change? The optimal solution would change because the profit of the deluxe model can vary from $127 this range without the to $135 $142 is not in optimal solution changing

Answer 1

a) Optimal solution

Optimal Solution Economy models 60 Standard models 140 Deluxe models C) Value of the objective function 16880

b) Constraints having slack/surplus equal to 0 are binding, otherwise non-binding

Binding Fan motors: Binding Cooling coils: Manufacturing Non binding

c) Slack/surplus is the number of units

Constaints Fan motors No C) Cooling coils No C) Manufacturing time Yes 640

d) Allowable increase in coefficient of variable D is 8, which means it be increased up to 135(=127+8) without changing the optimal solution. 142 is within the limit, so changing the coefficient to 142 will not change the optimal solution.