Question

A car dealership gets an allocation for 100 cars from the manufacturer. Model A can be sold at $4250 above factory invoice, Model B at $3500, Model C at $5500, Model D at $3250. In addition, since the dealership sold plenty of cars last year, the manufacturer also includes the halo model (specialized high end model), the Type GT RS-Rt Limited, in that allocation. This model can be marked up at $15000 above invoice. However, the manufacturer imposes these conditions:

• At least 30% of the total order must be the fuel-efficient Model D to fulfill the federal average fuel-economy regulation a.k.a. CAFÉ (Car Average Fuel Economy).
• The dealership will get one Type GT RS-Rt Limited for every 22 Model D ordered (hint: constrain this as a ratio e.g. 1:22).
• To attract foot traffics to the dealership, the dealer has to order at least one Type GT RS-Rt Limited
• Since Model D and Model B share many common parts, the manufacturer produces plenty of them. So, they must account for 2/3 of the order.
• Model C is a gigantic SUV. So, due to the aforementioned CAFÉ regulation, it can’t count no more than 10% of the total order.

a. Model D has the least profit, yet it is the most ordered model by the dealer because?

b. Which model is not ordered?

c. Suppose that the economy turns to be really bad and nobody buys sports cars, including Type GT RS-Rt, such that the profit for this type becomes zero. Which model will the dealer order the most?

Answer 1

Let the model-wise allocation of cars ordered by the dealership be:

Model A - x1

Model B - x2

Model C - x3

Model D - x4

Model GT RS-Rt - x5

So, Objective Function becomes:

Max Profit = 4250*x1 + 3500*x2 + 5500*x3 + 3250*x4 + 15000*x5

subject to constraints:

x1 + x2 + x3 + x4 + x5 = 100

x4 >= 30 [30%*100]

x5 = (1/22)*x4

x5 >= 1

x2 + x4 >= 67 [2/3rd of 100 = 66.67, considering interger = 67]

x3 <= 10 [10% of 100]

Above LP is formulated and solved as:

Thus, dealership will make a total optimal profit of $ 403,000 by ordering cars of

Model A - 20 cars

Model B - 1 cars

Model C - 10 cars

Model D - 66 cars

Model GT RS-Rt - 3 cars

Answer a:

Even though Model D has least profit but it is still the most ordered model by the dealer because of a couple of reasons. Firstly, CAFE regulation requires atleast 30% of the order should be made up of Model D. Secondly, both Model B and Model D generate lowest profits but they together should be more than 66.67% of the order but on top of that, every 22 Model D cars give 1 Model GT RS-Rt car which generates approx 5 times profit from Model D and approx. 3 times profit from highest profit yielding Model C.

Answer b:

None. From the above LP, we can see that every Model is ordered for at least once.

Answer c:

When profit for Model GT RS-Rt becomes 0, modified LP returns the following Optimal Solution:

Thus, we can see that the profit for dealer will decrease and share of other cars in the order increases, but the dealer will still order Model D the most with 44 cars.

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