Use the information below to answer the questions that follow. The car company f

Question

Use the information below to answer the questions that follow.

The car company for which you work sells 2 models, the Super Model and the Ultimate Model. To promote the safety of these cars the company has installed roll bars in both of the models. The roll bar for the Super Model is composed of 20 pounds of high alloy steel. It requires 45 minutes of manufacturing time and 60 minutes of assembly time. The roll bar for the Ultimate Model is composed of 35 pounds of high alloy steel. It requires 90 minutes of manufacturing time and 50 minutes of assembly time. The steel mill from which the car company obtains the high alloy steel has indicated that they can only manufacture at most 40,000 pounds of the high alloy steel for the next quarter. Your company has scheduled 2000 hours for manufacturing time and 1600 hours of assembly time for the next quarter. If the Super Model has a profit of $2,000 and the Ultimate Model has a profit of $2,700, answer the following:

Question 1

What is the maximum profit (optimal solution)? How many Super Models? How many Ultimate Models?

Question 2

Suppose we could change the profitability of the Super Model so that it is now $2300. Will this change the optimal solution? What is the new maximum profit? (Round to the nearest cent)

Question 3

From the original model, it we were able to add 450 hours to assembly would this change the optimal solution? What is the new maximum profit? (Round to the nearest cent)

Explanation / Answer

Let

x be the number of super model cars

y be the number of ultimate model cars

Maximise: z=2000x+2700y

Subject to : 1) 20x + 35y<= 40000

2) 45x + 90y <= 120000

3) 60x + 50y <= 96000

The corner points after solving (1,2) , (2,3) and (1,3) are (-8000/3,8000/3) , (17600/21,6400/7) and (13600/11,4800/11).

So z= 1,866,667, 4,144,762 , 3,650,909

As the second point leads to the maximum profit, it is the optimal solution

Super = 17600/21 = 838 cars

Ultimate = 6400/7 = 914 cars

Profit = 4,143,800

2) Now, z = 2300x + 2700y

Using the solutions,

z = 1,066,667 , 4,396,191, 4,021,818

So this doesn't change the optimal solution

The profit will be same as in 1)

3) The new LPP becomes

Maximise: z=2000x+2700y

Subject to : 1) 20x + 35y<= 40000

2) 45x + 90y <= 120000

3) 60x + 50y <= 123000

The corner points after solving (1,2) , (2,3) and (1,3) are (-8000/3,8000/3) , (33800/21,3700/7) and (23050/11,-600/11).

So z= 1,866,667, 4,646,191 , 4,043,636

As the second point leads to the maximum profit, it is the optimal solution

Super = 1609 cars

Ultimate = 528 cars

Profit = 4,643,600

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