T HE F RIENDLY C AR F ARM
Buddy Friendly and his wife, Vera, own and manage the Friendly Car Farm, a large new and used car dealership . Buddy and Vera primarily stock family-oriented cars like vans and midsize sedans and pickup trucks . The profit margin on these types of cars is not much, but the Friendlys have a very high volume of business, which offsets their low per-unit profit margin. They feel like they know their customer preferences, and they have tended to stock accordingly . However, for several years one of the automobile manufacturers the Friendlys deal with has repeatedly tried to get them to sell its new, high-priced sports car, the Zephyr AK2000. Buddy and Vera have finally agreed to stock the AK2000, and they have developed a tentative order policy as follows . If the number of Zephyrs they have on the lot is one or less at the end of the month, they will order either two or three Zephyrs from the manufacturer so that they will never have more than three of the cars on the lot at the beginning of the month. (The new cars will arrive in the week following the order, at the first of the next month.) The Friendlys paid a statistical consultant from a nearby university to develop the following transition matrix:
(The consultant used a Poisson distribution with l = one sale per month to compute each transition probabilitya fact completely lost on the Friendlys.)
This transition matrix shows the probability of having either zero, one, two, or three cars in stock one month, given either zero, one, two, or three cars in stock the previous month. For example, if the Friendlys have no Zephyrs in stock in one month, the probability is only .08 that they would have none in stock the next month. The reason this probability is so small is because if the stock level is zero, a new order is placed for three cars. Thus, .08 is actually the probability that three cars were ordered and they were sold in the next month.
The Friendlys pay the manufacturer for cars using loans from a local bank. The interest on these loans is part of the inventory holding cost, which also includes maintenance costs while the car is on the lot. The inventory holding cost for the Zephyr is relatively high compared with that for other cars the Friendlys sell. In any month the holding cost for one Zephyr is $75, for two Zephyrs it is $175, and for three Zephyrs it is $310. (Notice that the holding cost per car increases at an increasing rate because of accumulating interest charges.)
Determine the probability of the Friendlys having zero, one, two, or three cars in stock in a month in the long run, the expected number of cars in stock in a month in the long-run future, and the average inventory holding cost per month.