Question

Consider the following blood inventory problem facing a hospital. There is a need for a rare blood type, namely, type AB, Rh negative blood. The demand D in pints over any 3-day period is given by P{D = 0} = 0.4, P{D = 1} = 0.3, P{D = 2} = 0.2, P{D = 3} = 0.1 Note that the expected demand is 1 pint, since E(D) = 0.4 times 0 + 0.3 times 1 + 0.2 times 2 + 0.1 times 3 = 1. Suppose that there are 3 days between deliveries. The hospital proposes a policy of receiving 1 pint at each delivery and using the oldest blood first. If more blood is required than is on hand, an expensive emergency delivery is made. Blood is discarded if it is still on the shelf after 21 days. Denote the state of the system as the number of pints on hand just after a delivery.
a) Find the steady-state probabilities of the state of the Markov chain.
b) The emergency order costs $100 per pint, and the inventory holding cost is $10 per pint and per day. Calculate the expected long-run cost per day.

Answer 1

Answer:

Step-by-step explanation:

A. The steady state probability vectorπ= (π1, π2, π3, π4, π5, π6, π7) satisfies the system ofequations:πP=π,7Xi=1πi= 1which has solutionπ= (π1, π2, π3, π4, π5, π6, π7) = (0.1389,0.1389,0.1390,0.1383,0.1411,0.1302,0.1736)

(B) At any time period, an emergency delivery will be needed ifXn= 1 and the demands either 2 or 3, or ifXn= 2 and the demand is 3. The long-run proportion of time the chain will be in state 1 isπ1 and in-state 2π2.