Find optimal solution linear programming problem

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

E

Expert

##### Verified

 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,

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