Question

Problem 3-13 (Algorithmic)

Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $59, $89, and $137, 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 240 fan motors, 380 cooling coils, and 2500 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	59E	+	89S	+	137D
s.t.
	1E	+	1S	+	1D	≤	240	Fan motors
	1E	+	2S	+	4D	≤	380	Cooling coils
	8E	+	12S	+	14D	≤	2500	Manufacturing time
E, S, D ≥ 0

The sensitivity report is shown in the figure below.

Optimal Objective Value =      18360.00000
		Variable		Value		Reduced Cost
		E		100.00000		0.00000
		S		140.00000		0.00000
		D		0.00000		12.00000
		Constraint		Slack/Surplus		Dual Value
		Fan motors		0.00000		29.00000
		Cooling coils		0.00000		30.00000
		Manufacturing time		20.00000		0.00000

		Variable		Objective Coefficient		Allowable Increase		Allowable Decrease
		E		59.00000		6.00000		14.50000
		S		89.00000		29.00000		4.00000
		D		137.00000		12.00000		Infinite
		Constraint		RHS Value		Allowable Increase		Allowable Decrease
		Fan motors		240.00000		5.00000		50.00000
		Cooling coils		380.00000		5.00000		140.00000
		Manufacturing time		2500.00000		Infinite		20.00000

1. Identify the range of optimality for each objective function coefficient. If there is no limit, then enter the text "NA" as your answer. If required, round your answers to one decimal place.
   

   	Objective Coefficient Range
   Variable	lower limit	upper limit
   E
   S
   D

2. Suppose the profit for the economy model (E) is increased by $6 per unit, the profit for the standard model (S) is decreased by $2 per unit, and the profit for the deluxe model (D) is increased by $4 per unit. What will the new optimal solution be? If required, round your answers to three decimal places. If your answer is zero, enter "0".
   

   Optimal Solution
   E
   S
   D

   
   If required, round your answer for Total Profit to two decimal places.
   
   Total Profit: $  
   
3. Identify the range of feasibility for the right-hand-side values. If there is no limit, then enter the text "NA" as your answer. If required, round your answers to one decimal place.
   

   	Right-Hand-Side-Range
   Constraints	lower limit	upper limit
   Fan motors
   Cooling coils
   Manufacturing time

4. If the number of cooling coils available for production is increased by 25, will the dual value for that constraint change?
   
   Yes  because the allowable increase for cooling coils is  without changing the optimal solution.

Answer 1

a)

Lower range of optimality - objective coefficient-allowable decrease

Upper range of optimality - objective coefficient+allowable increase

Objective Coefficient Range
Variable	lower limit	upper limit
E	44.5	65
S	85	118
D	NA	149

b)

optimal solution will remain unchanged as all the increase/decrease are within allowable limit

So,

Optimal Solution
E	100
S	140
D	0

Profit will change and new profit = 65*100+87*140+141* 0 = 18680

c)

Upper range of feasibility = RHS+allowable increase

Lower range of feasibility = RHS-allowable decrease

Right-Hand-Side-Range
Constraints	lower limit	upper limit
Fan motors	190	245
Cooling coils	240	385
Manufacturing time	2480	NA

d)

Yes  because the allowable increase for cooling coils is 5 without changing the optimal solution.