Question

1. Formulate and solve a linear programming model to deter­ mine the number of Family Thrillseekers and the number of Classy Cruisers that should be assembled

Before she makes her final production decisions, Rachel plans to explore the following questions independently, except where otherwise indicated.

2. The marketing department knows that it can pursue a targeted

$500,000 advertising campaign that will raise the demand for the Classy Cruiser next month by 20 percent. Should the campaign be undertaken?

3. Rachel knows that she can increase next month's plant capacity by using overtime labor. She can increase the plant's labor-hour capacity by 25 percent. With the new assembly plant capacity, how many Family Thrillseekers and how many Classy Cruisers should be assembled?

4. Rachel knows that overtime labor does not come without an extra cost. What is the maximum amount she should be willing to pay for all overtime labor beyond the cost of this labor at regular-time rates? Express your answer as a lump sum.

5. Rachel explores the option of using both the targeted advertising campaign and the overtime labor-hours. The advertising campaign raises the demand for the Classy Cruiser by 20 percent, and the overtime labor increases the plant's labor­ hour capacity by 25 percent. How many Family Thrillseekers and how many Classy Cruisers should be assembled using the advertising campaign and overtime labor-hours if the profit from each Classy Cruiser sold continues to be 50 percent more than for each Family Thrillseeker sold ?

CASE Study

Auto Assembly – Case Study Background

Automobile Alliance: large automobile manufacturing company,

• Manufactures three type of vehicles: Trucks, Small cars, Midsized and luxury cars.

• Two models from the family of midsized and luxury cars. Vehicles are made in a plant outside Detroit, Michigan,

1. First model, the Family Thrill seeker, is a four-door sedan with vinyl seats, plastic interior, standard features, and excellent gas mileage. It is marketed as a smart buy for middle-class families with tight budgets.

2. Second model, the Classy Cruiser, is a two-door luxury sedan with leather seats, wooden interior, custom features, and navigational capabilities. It is marketed as a privilege of affluence for upper-middle-class families.

Case Study

Rachel Rosencrantz (Mgr., of the assembly plant) planning prod. Schedule for the next month.

* She must decide how many Family Thrillseekers and how many Classy Cruisers to assemble in the Plant to maximize profit for the company.

Demand

A recent company forecast for monthly demands for the 2 diff. models suggests

• Classy Cruiser - 3,500 cars.

• Family Thrill seeker ∞

Profits

• Family Thrillseeker = $3,600/ea.   

• Classy Cruiser = $5,400/ea.

Labor

48,000 - labor-hours during the month.  

6 hrs. = 1 Family Thrill seekers   

10.5 hrs. = 1 Classy Cruiser   Parts

Rachel knows that she will be able to obtain 20,000 doors from the door supplier.

• Both vehicles use the same door

• 2 doors– Classy Cruisers     

• 4 doors– Thrill seeker        

Answer 1

a) In this case, we have two decision variables: the number of Family Thrillseekers we should assemble and the number of Classy Cruisers we should assemble. We also have the following three constraints:

1. The plant has a maximum of 48,000 labor hours.

2. The plant has a maximum of 20,000 doors available.

3. The number of Cruisers we should assemble must be less than or equal to 3,500.

 

  

Rachel’s plant should assemble 3,800 Thrillseekers and 2,400 Cruisers to obtain a maximum profit of $26,640,000.

b) In part (a) above, we observed that the Cruiser demand constraint was not binding. Therefore, raising the demand for the Cruiser will not change the optimal solution. The marketing campaign should not be undertaken.

c) The new value of the right-hand side of the labor constraint becomes 48,000 * 1.25 = 60,000 labor hours. All formulas and Solver settings used in part (a) remain the same.

Rachel’s plant should now assemble 3,250 Thrillseekers and 3,500 Cruisers to achieve a maximum profit of $30,600,000.

d) Using overtime labor increases the profit by $30,600,000 – $26,640,000 = $3,960,000. Rachel should therefore be willing to pay at most $3,960,000 extra for overtime labor beyond regular time rates.

e) The value of the right-hand side of the Cruiser demand constraint is 3,500 * 1.20 = 4,200 cars. The value of the right-hand side of the labor hour constraint is 48,000 * 1.25 = 60,000 hours. All formulas and Solver settings used in part (a) remain the same. Ignoring the costs of the advertising campaign and overtime labor,

Rachel’s plant should produce 3,000 Thrillseekers and 4,000 Cruisers for a maximum profit of $32,400,000. This profit excludes the costs of advertising and using overtime labor.

f) The advertising campaign costs $500,000. In the solution to part (e) above, we used the maximum overtime labor available, and the maximum use of overtime labor costs $1,600,000. Thus, our solution in part (e) required an extra $500,000 + $1,600,000 = $2,100,000. We perform the following cost/benefit analysis:

Profit in part (e):              $32,400,000

- Advertising and overtime costs: $   2,100,000

              $30,300,000

We compare the $30,300,000 profit with the $26,640,000 profit obtained in part (a) and conclude that the decision to run the advertising campaign and use overtime labor is a very wise, profitable decision.

g) Because we consider this question independently, the values of the right-hand sides for the Cruiser demand constraint and the labor hour constraint are the same as those in part (a). We now change the profit for the Thrillseeker from $3,600 to $2,800 in the problem formulation. All formulas and Solver settings used in part (a) remain the same.

Rachel’s plant should assemble 1,875 Thrillseekers and 3,500 Cruisers to obtain a maximum profit of $24,150,000.

h) Because we consider this question independently, the profit for the Thrillseeker remains the same as the profit specified in part (a). The labor hour constraint changes. Each Thrillseeker now requires 7.5 hours for assembly. All formulas and Solver settings used in part (a) remain the same.

Rachel’s plant should assemble 1,500 Thrillseekers and 3,500 Cruisers for a maximum profit of $24,300,000.

i) Because we consider this question independently, we use the problem formulation used in part (a). In this problem, however, the number of Cruisers assembled has to be strictly equal to the total demand. The formulas used in the problem formulation remain the same as those used in part (a).

The new profit is $25,650,000, which is $26,640,000 – $25,650,000 = $990,000 less than the profit obtained in part (a). This decrease in profit is less than $2,000,000, so Rachel should meet the full demand for the Cruiser.

j) We now combine the new considerations described in parts (f), (g), and (h). In part (f), we decided to use both the advertising campaign and the overtime labor. The advertising campaign raises the demand for the Cruiser to 4,200 sedans, and the overtime labor increases the labor hour capacity of the plant to 60,000 labor hours. In part (g), we decreased the profit generated by a Thrillseeker to $2,800. In part (h), we increased the time to assemble a Thrillseeker to 7.5 hours. The formulas and Solver settings used for this problem are the same as those used in part (a).

Rachel’s plant should assemble 2,120 Thrillseekers and 4,200 Cruisers for a maximum profit of $28,616,000 – $2,100,000 = $26,516,000.