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 $63, $97, and $133, 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 350 fan motors, 340 cooling coils, and 2000 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	+	97S	+	133D
s.t.
	1E	+	1S	+	1D	≤	350	Fan motors
	1E	+	2S	+	4D	≤	340	Cooling coils
	8E	+	12S	+	14D	≤	2000	Manufacturing time
E, S, D ≥ 0

The sensitivity report is shown in the figure below.

Optimal Objective Value =      16660.00000
		Variable		Value		Reduced Cost
		E		180.00000		0.00000
		S		0.00000		2.55556
		D		40.00000		0.00000
		Constraint		Slack/Surplus		Dual Value
		Fan motors		130.00000		0.00000
		Cooling coils		0.00000		10.11111
		Manufacturing time		0.00000		6.61111

		Variable		Objective Coefficient		Allowable Increase		Allowable Decrease
		E		63.00000		13.00000		2.30000
		S		97.00000		2.55556		Infinite
		D		133.00000		119.00000		11.50000
		Constraint		RHS Value		Allowable Increase		Allowable Decrease
		Fan motors		350.00000		Infinite		130.00000
		Cooling coils		340.00000		231.42860		90.00000
		Manufacturing time		2000.00000		720.00000		810.00000

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

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: $  

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

If the number of manufacturing time available for production is increased by 740, will the dual value for that constraint change?

Yes because the allowable increase for manufacturing time is  without changing the optimal solution.

Answer 1

Q - What is the optimal solution, and what is the value of the objective function? If required, round your answers to the nearest whole number.

Answer - Refer the first table of computer solution (Variable, Value, Reduced Cost)

	Optimal Solution
Economy models (E)	80
Standard models (S)	120
Deluxe models (D)	0
Value of the objective function	$ 16440

Q - Which constraints are binding?

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

Fan motors:	Binding
Cooling coils:	Binding
Manufacturing time:	Non binding

Q - 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.

Answer – Refer table (Constraint, Slack/Surplus, Dual Value)

Slack/Surplus is the number of units

Constraints	Extra capacity	Number of units
Fan motors	No	0
Cooling coils	No	0
Manufacturing time	Yes	320

Q - If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change?
Answer - Refer table (variable, objective coefficient, allowable increase, allowable decrease)

Allowable increase in coefficient of Variable D is 24, which means it be increased up to 159 (=135+24) without changing the optimal solution. 150 is within the limit, so changing the coefficient to 150 will not change the optimal solution.