Case Study  Auto Assembly
Solve the questions with Reddy Mikks Model.
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Answer 
Introduction
In this study, we are looking into the case study of one of the largest automobile manufacturing company, called Automobile Alliance and the manager of this automobile company is Rachel Rosencrantz. The assembly plant of Automobile Alliance was located outside Detroit, MI, and they are producing two models, namely
Midsized car, intentionally designed as family car, and
Luxury cars that are assembled
Those two cars are called Family Thrillseeker and Classy Cruiser
The car, Family Thrillseeker is designed as a sedan car with four doors and vinyl seats along with plastic interiors and standard interiors which as an excellent gas mileage records. This segment was produced mainly to target the middle class families which was sold at tight budget and this car earns a profit of $ 3600. On the other hand, the two door luxury sedan car which was designed with great leather seats along with wooden interior, custom features and also added with navigation features. . This segment was produced mainly to target the high class families which was sold at luxury budget and this car earns a profit of $ 5400.
Here, Rachel faces a tough decision on deciding the number of cars to be produced on both models which will thus increase the profit to the maximum. The Detroit Plant has a maximum capacity of 48000 labor hours per month and it takes 6 labor hours to assemble one Family Thrill seeker and 10.5 labor hours to assemble one Classy Cruiser. On the other hand, the parts are shipped from Michigan to Detroit Assembly plant. Also, we see that only 10000 right doors and 10000 left doors were only supplied by the door supply and this lesser amount is due to labor strike. Thus, with the available doors, Rachel should decide on producing the number of two model cars. Thus, the demand for Classy Cruiser was fixed to 3500 cars while there is no demand limit for Family Thrill seeker cars.
The table given below shows the information about auto assembly
Profit

Family Thrill seeker

Luxury Cruiser

Quantity

3600

5400

Door

4

2

20000

Labour

6

10.5

48000

Luxury Cruiser

0

1

3500

Maximize Z = 3600 X_{1} + 5400 * X_{2}
Subject to the constraints
Door Constraint = 4 X_{1} + 2 * X_{2} ≤ 20000
Labor Constraint = 6 X_{1} +10.5 * X_{2} ≤ 48000
Luxury Cruiser = X_{2} ≤ 3500
X_{1}, X_{2} ≥ 0 (Non  Negativity restrictions)
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Analysis and Result  What is optimal solution by using both tora & Reddy Mikks Model solve all the case study question.
1. Based on the original information, how many Family Thrillseekers and Classy Cruisers to assemble.
The objective function and the constraints are given below
Objective Function
Maximize Z = 3600 X_{1} + 5400 * X_{2}
Subject to the constraints
Door Constraint = 4 X_{1} + 2 * X_{2} ≤ 20000
Labor Constraint = 6 X_{1} +10.5 * X_{2} ≤ 48000
Luxury Cruiser = X_{2} ≤ 3500
X_{1}, X_{2} ≥ 0 (Non  Negativity restrictions)
The iterations are given below.
Cj

Basic Variables

Quantity

3600 Family Thrillseeker

5400 Luxury

0 slack 1

0 slack 2

0 slack 3

Iteration 1








0

slack 1

20,000

4

2

1

0

0

0

slack 2

48,000

6

10.5

0

1

0

0

slack 3

3,500

0

1

0

0

1


zj

0

0

0

0

0

0


cjzj


3,600

5,400

0

0

0

Iteration 2








0

slack 1

13,000

4

0

1

0

2

0

slack 2

11,250

6

0

0

1

10.5

5400

Luxury

3,500

0

1

0

0

1


zj

18,900,000

0

5400

0

0

5400


cjzj


3,600

0

0

0

5,400

Iteration 3








0

slack 1

5,500

0

0

1

0.6667

5

3600

Family Thrillseeker

1,875

1

0

0

0.1667

1.75

5400

Luxury

3,500

0

1

0

0

1


zj

25,650,000

3600

5400

0

600

900


cjzj


0

0

0

600

900

Iteration 4








0

slack 3

1,100

0

0

0.2

0.1333

1

3600

Family Thrillseeker

3,800

1

0

0.35

0.0667

0

5400

Luxury

2,400

0

1

0.2

0.1333

0


zj

26,640,000

3600

5400

180

480

0


cjzj


0

0

180

480

0

With the initial requirement, we see that the maximum profit of $ 26,640,000 was earned by producing 3800 Family Thrill seekers and 2400 Luxury Cruiser cars.
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2. The marketing department can pursue a targeted $500,000 advertising campaign that will raise the demand for the Classy Cruiser next month by 20 percent. Determine whether or not the advertising campaign should be undertaken.
Another constraint talks about the advertising campaign cost as it amounts to $500,000. Thus, this cost must be detected from the objective function. Thus, the new objective function is
Maximize Z = 3600 X_{1} + 5400 * X_{2}  500000
The main intention for this advertisement is to increase the sale of Luxury Cruiser and it is expected to increase by 20%. Therefore, there will be a change in the third constraint too
Door Constraint = 4 X_{1} + 2 * X_{2} ≤ 20000
Labor Constraint = 6 X_{1} +10.5 * X_{2} ≤ 48000
Luxury Cruiser = X_{2} ≤ 4200
X_{1}, X_{2} ≥ 0 (Non  Negativity restrictions)
Thus, we see that the maximum profit of $ 26,140,000 which seems to be less when compared to part 1 and therefore, this plan of advertising campaign should be dropped
Q3. Rachel can increase next month's plant capacity by using overtime labor; she can increase the plant's laborhour capacity by 25 percent. Determine then how many of each model of car should be assembled.
Another constraint is to increase the labor hours constraint to 25 percent. Thus, the new objective function is
Maximize Z = 3600 X_{1} + 5400 * X_{2}  500000
Now we have 60000 labor hours Therefore, there will be a change in the second constraint too
Door Constraint = 4 X_{1} + 2 * X_{2} ≤ 20000
Labor Constraint = 6 X_{1} +10.5 * X_{2} ≤ 60000
Luxury Cruiser = X_{2} ≤ 4200
X_{1}, X_{2} ≥ 0 (Non  Negativity restrictions)
Thus, we see that the maximum profit of $ 30,600,000. Even though the profit is more, it do not deal with the overtime cost which will bring back the profit to $ 26,640,000.
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4. Determine, based on Part 3, the maximum amount Rachel should be willing to pay for all overtime labor beyond the cost of this labor at regular time rates.
Thus, we see that the maximum profit of $ 30,600,000. Even though the profit is more, it do not deal with the overtime cost which will bring back the profit to $ 26,640,000. This indicates that the manager should be willing to pay $ 3, 96,000 as overtime cost which is of no use.
5. Rachel could use both the targeted advertising campaign and the overtime laborhours. The campaign raises the demand of the Classy Cruiser 20 percent and the overtime laborhours increase the plant's laborhour capacity by 25 percent. Determine the number of each model of car to assemble in this case.
Now, we consider the situation of eliminating the implementation of costs, and therefore, we include both advertising campaign and additional labor hours to the original constraint.
Maximize Z = 3600 X_{1} + 5400 * X_{2}  500000
Subject to the constraints
Door Constraint = 4 X_{1} + 2 * X_{2} ≤ 20000
Labor Constraint = 6 X_{1} +10.5 * X_{2} ≤ 60000
Luxury Cruiser = X_{2} ≤ 4200
Thus, we see that the maximum profit of $ 32,400,000 was earned by producing 3000 Family Thrill seekers and 4000 Luxury Cruiser cars.
6. The advertising campaign costs $500,000 and the overtime laborhours cost $1,600,000 beyond regular time rates. Determine whether or not the actions in Part 5 should be taken.
Here, we see that
The choice of using Advertising campaign results in profit of $26,140,000.
The choice of using labor hours result in profit of $29,000,000.
With the initial requirement, we see that the maximum profit of $ 26,640,000 was earned by producing 3800 Family Thrill seekers and 2400 Luxury Cruiser cars.
Thus, the manager Rachel can select labor hour's constraint ahead of advertisement campaign.
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7. The company has determined that dealerships are heavily discounting the price of the Family Thrillseekers to move them off the lot. Because of a profitsharing agreement with its dealers, the company is now making a profit of only $2800 on the Family Thrillseeker. Given this new discounted price we determine the number of Family Thrillseekers and Classy Cruisers to assemble.
Now, let us limit the profit of selling one Family Thrillseeker to $ 2800 and run the original constraint.
With the initial requirement, we see that the maximum profit of $ 24,150,000 was earned by producing 1875 Family Thrill seekers and 3500 Luxury Cruiser cars.
8. The company discovered quality problems with the Family Thrillseeker through random testing. There are problems with two of the four doors of a Family Thrillseeker sealing properly. Because of the now added quality testing at the end of the line for each Family Thrillseeker, it now takes 7.5 laborhours to assemble a Family Thrillseeker. We determine the number of each model of car to assemble in this case.
Now, let us increase the labor hours of making Family Thrillseeker to 7.5 hours and run the original constraint.
Objective Function
Maximize Z = 3600 X_{1} + 5400 * X_{2}
Subject to the constraints
Door Constraint = 4 X_{1} + 2 * X_{2} ≤ 20000
Labor Constraint = 7.5 X_{1} +10.5 * X_{2} ≤ 48000
Luxury Cruiser = X_{2} ≤ 3500
X_{1}, X_{2} ≥ 0 (Non  Negativity restrictions)
With the initial requirement, we see that the maximum profit of $ 24,300,000 was earned by producing 1500 Family Thrill seekers and 3500 Luxury Cruiser cars.
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9. The board of directors of Automobile Alliance wants to capture a larger share of the luxury sedan market and therefore would like to meet the full demand for Classy Cruisers. They ask Rachel to determine by how much the profit of her assembly plant would decrease as compared to the profit found in Part 1. They then ask her to meet the full demand for Classy Cruisers if the decrease in profit is not more than $2,000,000. We determine whether or not Rachel should meet the full demand for Classy Cruisers.
Now, we fix the restriction that exactly 3500 luxury cruiser should be produced. Thus, we see that
Objective Function
Maximize Z = 3600 X_{1} + 5400 * X_{2}
Subject to the constraints
Door Constraint = 4 X_{1} + 2 * X_{2} ≤ 20000
Labor Constraint = 6 X_{1} +10.5 * X_{2} ≤ 48000
Luxury Cruiser = X_{2} = 3500
X_{1}, X_{2} ≥ 0 (Non  Negativity restrictions)
With the initial requirement, we see that the maximum profit of $ 25,650,000 was earned by producing 1875 Family Thrill seekers and 3500 Luxury Cruiser cars.
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10. Finally, Rachel considers the simultaneous situations described in Parts 6,7, and 8. Determine whether or not she should undertake the advertising campaign and whether or not she should use overtime labor. We then determine the number of each model to be assembled in her final decision.
Now, Rachel has to decide whether or not to use the advertising campaign and (or) labor hours since the dealership is deciding to discontinue assembling of family thrill seekers due to newly detected quality issues in it. Now without using labor hours and advertisement campaign, the new objective function in restricting the profit of Family Thrillseeker to $2800
Objective Function
Maximize Z = 2800 X_{1} + 5400 * X_{2}
Subject to the constraints
Door Constraint = 4 X_{1} + 2 * X_{2} ≤ 20000
Labor Constraint = 7.5 X_{1} +10.5 * X_{2} ≤ 48000
Luxury Cruiser = X_{2} ≤ 3500
X_{1}, X_{2} ≥ 0 (Non  Negativity restrictions)
With the requirement, we see that the maximum profit of $ 23,100,000 was earned by producing 1500 Family Thrill seekers and 3500 Luxury Cruiser cars.
The brief summarize of advertising cost is given below

Advertisement

No Advertisement

Labor Hours, Overtime

26,516,000

25,980,000

Labor Hours, No Overtime

24,136,000

23,100,000

Findings
With the requirement, we see that the maximum profit of $ 26,516,000 was earned by producing 2120 Family Thrill seekers and 4200 Luxury Cruiser cars and it was obtained by using both advertisement campaign and labor hours
Conclusion
In this study, we are looking into the case study of one of the largest automobile manufacturing company, called Automobile Alliance and the manager of this automobile company is Rachel Rosencrantz. The assembly plant of Automobile Alliance was located outside Detroit, MI, and they are producing two models, namely
Midsized car, intentionally designed as family car, and
Luxury cars that are assembled
Those two cars are called Family Thrillseeker and Classy Cruiser
With the requirement, we see that the maximum profit of $ 26,516,000 was earned by producing 2120 Family Thrill seekers and 4200 Luxury Cruiser cars and it was obtained by using both advertisement campaign and labor hours.
This study shows a clear working procedure on linear programming problem along with graphical methods. The graphical procedure shows a clear idea of how the model directs using the objective function and its constraints appropriately to get the feasible solution. On the other hand, the simple method calculations clearly explains the use of slack variables and its purposes. Overall, there is a clear understanding of practical use of this linear programming model.
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