Suppose that the demand D for a spare airplane part has an exponential distribution with mean 50, that is,

5 1

0

e /50 for   0

D()

0 otherwise.

This airplane will be obsolete in 1 year, so all production of the spare part is to take place at present. The production costs now are

$1,000 per item—that is, c 1,000—but they become $10,000 per item if they must be supplied at later dates—that is, p 10,000.

The holding costs, charged on the excess after the end of the period, are $300 per item.

T

(a) Determine the optimal number of spare parts to produce.

(b) Suppose that the manufacturer has 23 parts already in inventory (from a similar, but now obsolete airplane). Determine the optimal inventory policy.

(c) Suppose that p cannot be determined now, but the manufacturer wishes to order a quantity so that the probability of a shortage equals 0.1. How many units should be ordered?

(d) If the manufacturer were following an optimal policy that resulted in ordering the quantity found in part (c), what is the implied value of p?