Find optimal solution linear programming problem

Question

a)      Let X is the no of Thrill seeker car and Y is the Classy Cruiser, profit per car is 3600 and 5400 respectively. One needs to maximize the function i.e. 

Max (3600*X + 5400*Y)

The constraints are that the labor hours are restricted to 48000 and the no of doors are restricted to 20000.  Also the maximum demand of Classy Cruiser is 3500. Keeping this constraints one solves the problem. Following is the output of the problem.

	Variables to be changed
	Family Thrill Seekers	Classy Cruiser
No of Cars to be produced	3800	2400

 

Constraints			Cars constraint total	Company constraint
Labour Hours	6	10.5	48000	48000
No of Doors	4	2	20000	20000
Limit on the cars		2400	2400	3500

 

Objective function

Total profit

26640000

b)      Optimal solution gives 2400 Classy Cruiser to be produced even when the demand is 3500. So this is more than 20% of the cars that should be produced. So spending on advertisement is waste and will not affect the output as the constraint is a non binding.

c)      Following is the solution if the no of labor hours are increased

	Variables to be changed
	Family Thrill Seekers	Classy Cruiser
No of Cars to be produced	3250	3500

 

Constraints			Cars constraint total	Company constraint
Labor Hours	6	10.5	56250	60000
No of Doors	4	2	20000	20000
Limit on the cars		3500	3500	3500

 

Objective function

Total profit

30600000

 

d)     Increase in profit by using extra 25%labor hours is 30600000 - 26640000 which equals to 3960000. So ideally this increase in profit should be the maximum cost which Rachel should be willing to shell out.

e)      If the labor hour increases by 25% then the value becomes 60000 and similarly for the demand the value becomes 4200. Solving for these values one gets the following the answers:

	Variables to be changed
	Family Thrill Seekers	Classy Cruiser
No of Cars to be produced	3000	4000

 

Constraints			Cars constraint total	Company constraint
Labor Hours	6	10.5	60000	60000
No of Doors	4	2	20000	20000
Limit on the cars		4000	4000	4200

 

Objective function

Total profit

32400000

 

f)       The increase in profit is 32400000- 26640000 = 5760000, so this is greater than the cost of advertising which is 500,000 and extra labor which is 1600000, the total is 2100000.

g)      Following is the output of the solution

	Variables to be changed
	Family Thrill Seekers	Classy Cruiser
No of Cars to be produced	1875	3500

 

Constraints			Cars constraint total	Company constraint
Labor Hours	6	10.5	48000	48000
No of Doors	4	2	14500	20000
Limit on the cars		3500	3500	3500

 

Objective function

Total profit

24150000

h)      Following is the solution

	Variables to be changed
	Family Thrill Seekers	Classy Cruiser
No of Cars to be produced	1500	3500

 

Constraints			Cars constraint total	Company constraint
Labor Hours	7.5	10.5	48000	48000
No of Doors	4	2	13000	20000
Limit on the cars		3500	3500	3500

 

Objective function

Total profit

24300000

 

i)        In this one needs to modify one constraint. Instead of making the constraint for the cars as less than equal to one need to change the constraint to equal to 3500 cars. Following are the solutions

	Variables to be changed
	Family Thrill Seekers	Classy Cruiser
No of Cars to be produced	1875	3500

 

Constraints			Cars constraint total	Company constraint
Labor Hours	6	10.5	48000	48000
No of Doors	4	2	14500	20000
Limit on the cars		3500	3500	3500

 

Objective function

Total profit

25650000

 

The decrease in profit if all the demands are met is equal to 25650000- 26640000 = 990000 which is less than the specified limit of 2000000. So one can produce 3500 Cruiser cars and meet all the required demand.

j)        Following is the output

	Variables to be changed
	Family Thrill Seekers	Classy Cruiser
No of Cars to be produced	2120	4200

 

Constraints			Cars constraint total	Company constraint
Labor Hours	7.5	10.5	60000	60000
No of Doors	4	2	16880	20000
Limit on the cars		4200	4200	4200

 

Objective function

Total profit

28616000

The profit finally obtained will be 28616000 - advertising cost and overtime labor cost. So the profit is 28616000 -500000-1600000 which is equal to 26516000

Answer 1

Adjustable Cells
			Final	Reduced	Objective	Allowable	Allowable
	Cell	Name	Value	Cost	Coefficient	Increase	Decrease
	$B$3	No of Cars to be produced Family Thrill Seekers	3800	0	3600	7200	514.2857143
	$C$3	No of Cars to be produced Classy Cruiser	2400	0	5400	900	3600
Constraints
			Final	Shadow	Constraint	Allowable	Allowable
	Cell	Name	Value	Price	R.H. Side	Increase	Decrease
	$D$7	Labour Hours Cars constraint total	48000	480	48000	1E+30	1E+30
	$D$8	No of Doors Cars constraint total	20000	180	20000	1E+30	1E+30
	$D$9	Limit on the cars Cars constraint total	2400	0	3500	1E+30	1100

Complete solution available in Excel,