Using discrete distribution: Your business is transporting cars from the factory

Question

Using discrete distribution:

Your business is transporting cars from the factory to dealerships. You do this using car carriers. A car carrier holds 10 cars.

You are paid $10,000 for shipping a carrier-load of cars from the manufacturer to the dealership. During loading, transporting, and unloading, the cars' paint may get chipped or the bodies dented.

You can minimize potential damage by using car shields, and better maintaining the carriers.

Based on past experience, you know that you can purchase top-of-the-line equipment so that no cars get damaged in transit. But top-of-the-line equipment plus labor costs (amortized) $10,000 per shipment. You can reduce your costs by using lesser equipment. From past experience, you estimate that you can save $9 per shipment in exchange for less quality equipment such that 1 more car out of 1,000 (on average) getting damaged. For example, at a cost of $10,000 per shipment, no cars will be damaged. At a cost of $9,991 per shipment, 1 car out of 1,000 will get damaged. At a cost of $9,982, 2 cars out of 1,000 will get damaged.

At no cost to you, the dealership will fix damage, provided only one or two cars are affected. If three or more cars are affected, the dealership will reject the entire shipment.

If the dealership rejects a shipment, you do not get paid for that shipment but still incur the costs of making the shipment.

How many damaged cars (per 1,000) should you tolerate so as to maximize your expected profit?

Explanation / Answer

According to the question, you can afford at most two damaged cars per shipment, i.e. 2 damaged cars per 10 cars, so 2*100(= 200) damaged cars per 10*100 (=1000) cars.

This way we can maximize the profit.

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