Question

2. In a Air Conditioning manufacturing problem, the firm is maximizing the profit by selling three different products. Linear programming model for this problem will be as follows:

Decision Variables: E := Number of economy models to be produced

S := Number of standard models to be produced

D := Number of deluxe models to be produced

Linear Program: max 68E + 86S + 118D

s.t.

E + S + D ≤ 230 (Fan Motors)

E + 2S + 4D ≤ 320 (Cooling Coils)

8E + 12S + 15D ≤ 2250 (Manufacturing time)

E, S, D ≥ 0

Variable Cells

Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
Economy	140	0	68	2	25
Standard	90	0	86	50	1.33
Deluxe	0	-4	118	4	1E+30

Constraints

Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable decrease
Fan Motors	230	50	230	12.5	70
Cooling Coils	320	18	320	12.5	90
Manufacturing Time	2200	0	2250	1E+30	50

Use the Sensitivity Report given above to answer the following questions. Show your work!

a) What are the optimal solution and optimal value?

b) What should be the profit of Deluxe at least so that they can sell Deluxe?

c) What would be the new optimal solution and optimal value when the profit of Economy becomes $62?

d) What would be the new optimal solution and optimal value when the profit of Standard becomes $140?

e) What would be the new optimal value when cooling coils amount becomes 260?

f) Suppose the profit for the economy model is decreased by $5, the profit for the standard model is increased by $4, and the profit for the deluxe model is decreased by $3. What will be the new optimal value?

g) The firm is considering using overtime to increase available Manufacturing time. What would you advise the firm to do regarding this option? Explain.

h) The firm has an option to purchase additional 10 units of cooling coils at a total cost of $250. Should the firm purchase the additional units of cooling coils? Explain.

Answer 1

a ) From Sensitivity Analysis Optimal solution can be find out in final values in Variable cells

E := Number of economy models to be produced =140

S := Number of standard models to be produced = 90

D := Number of deluxe models to be produced = 0

Profit = 68E + 86S + 118D = 68*140 + 118*90 = 20140

b) As we can see from variable cell value of D=0 in optimal solution and allowable increase is 4 that means coefficient of D in profit function can be increase to 4 without changing the optimal solution so till 122 Delux will be 0 in optimal solution. Profit =>122

c) Allowable decrease for Economy coefficent is 25 so any profit of economy less than 68-25 = 43 will have no change in optimal solution. at Profit = 65, optimal solution will unchage

E := Number of economy models to be produced =140

S := Number of standard models to be produced = 90

D := Number of deluxe models to be produced = 0

d) What would be the new optimal solution and optimal value when the profit of Standard becomes $140?

So allowable increase for standard is 50 .optimal solution will remain unchange till profit of standard 86+50 = 136.

So the optimal solution will change Run the LP agin with new coefficent of profit

new Optimal solution will be

E	S	D
0	160	0

Optimal Value = 22400

e) What would be the new optimal value when cooling coils amount becomes 260?

	E	S	D
	200	30	0
	E	S	D
Fan Motors	1	1	1	230	<=	230
Cooling Coils	1	2	4	260	<=	260
Manufacturing times	8	12	15	1960	<=	2250
	E	S	D	Profit
Profit	68	86	118	16180

f)Suppose the profit for the economy model is decreased by $5, the profit for the standard model is increased by $4, and the profit for the deluxe model is decreased by $3. What will be the new optimal value?

No change in optimal solution as the changes for profit coefficienta are with in allowable increase or decrease of profit coeffficient

Optimal Solution	E	S	D
	140	90	0
	E	S	D
Fan Motors	1	1	1	230	<=	230
Cooling Coils	1	2	4	320	<=	320
Manufacturing times	8	12	15	2200	<=	2250
	E	S	D	Profit
Profit	63	90	115	16920

g ) manufacturing time constraint is nonbinding as all time is not consumed in optimal solution and buffer left is 50. So no need to increase further manufacturing time

h ) Increasing the Cooling constraint by 10 units

As we can see Shadow price for colling constraint is 18 and it is binding constraint so within allowable increase every increase in value with increase the total profit by 18

so with 10 unit increase profit will increase by 10*18 = 180