Question

An Air Conditioning manufactures three varieties of home air conditioners: an economy model, a standard model, and a deluxe model. The profit per unit is SAR 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

Based on the computer solution is provided on the next page answer the following questions. a. What is the optimal solution, and what is the value of the objective function?

b. Which constraints are binding?

c. How much of extra hours are available when the company produce at the optimal solution?

d. If the profit for the deluxe model were increased to 180 SAR per unit, would the optimal solution change?

e. What would be the effect on the optimal solution if the company could give at least 200 hours in manufacturing time?

f. What is the range of optimality for the objective function coefficients? Interpret.

g. Suppose the profit for the economy model is increased by SAR 10 per unit, the profit for the standard model is decreased by SAR 3 per unit, and the profit for the deluxe model is increased by SAR 10 per unit. What will the new optimal solution be?

h. Identify the range of feasibility for the right-hand-side values.

i. If the number of fan motors available for production is increased by 50, will the dual value for that constraint change? Explain.

j. If the number of coiling coil is decreased by 150, will the dual value for that constraint change? Explain.

k. Identify and explain the shadow prices for each of the requirements mentioned for making the products.
Optimal Objective Value = 16440.00000

Variable Value Reduced Cost E 80.00000 0.00000 S 120.00000 0.00000 D 0.00000 -24.00000 Constraints Slack/Surplus Dual Value 1 0.00000 31.00000 2 0.00000 32.00000 3 320.00000 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 Constraints RHS Value Allowable Increase Allowable Decrease 1 200.00000 80.00000 40.00000 2 320.00000 80.00000 120.00000 3 2400.00000 Infinite 320.0000

Can you answer this questions please?

Thank you

Answer 1

a) Value of objective function is 16440

b) Fan motors & cooling coils are the binding constraints

c) 320 extra hours are available when company produces at optimal level

d) If the profitability of deluxe model incresed to 180, the optimal siolution will change as the range starts from 159 onwards.

e) There will be no change in optimal solution, even if company were to give 200 hours in manufacturing time.

f) The range of optimality for profit coefficients in which the optimal solutions do not change are:-

	Lower Bound	Z coefficient	Upper Bound
E	48	63	75
S	87	95	126
D	0	135	159

g)

Profit for economy model incresed by 10 = 63+10 = 73

Profit for economy model decresed by 3 = 95-3 = 92

Profit for economy model incresed by 10 = 135+10 = 145

Since the changed profit coefficients are within the lower bound and upper boound, there is no change in optimal solution.

h) Range for feasibility for RHS values

	Lower Bound	RHS coefficient	Upper Bound
Fan motors	160	200	230
Cooling coil	200	320	350
Manufacturing time	2280	2400	infinity

i) Since increasing fan motors by 50 , will take it out of the fan motor range, hence the dual value will change.

j) Since decreasing cooling coil by 150 , will take it out of the colling coil range, hence the dual value will change.

k)

fan motor = 31

Cooling coil = 32

Manufacturing time = 0