stochastic processes question inventory theory Suppose the d

stochastic processes question (inventory theory)

Suppose the demand D for a spare airplane part has an exponential distribution with mean 50. This airplane will be obsolete in 1 year, so all production of the spare part is to take place at present. Production costs now are $1000 per item but they become $10000 per item if they must supplied at later dates. The holding costs, charged on the excess after the end of the period, are $300 per item. i) Determine the optimal number of spare parts to produce. ii) Suppose that the manufacturer has 23 parts already in inventory (from a similar, but now obsolete airplane). Determine the optimal inventory policy.

Solution

Mean =50

Production costs = 1000 or 10000

Holding costs =300

Holding cost =300x if x planes are stored

Saving in cost for x planes now = 9000

Hence x =30 both costs becomes equal

IF x >30, holding cost > ordering now instead of later date

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OPtimal quantity = 30

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If already 23 are there extra then saving in cost = (z+23) 300 = 9000

Hence 7 more can be ordered now.