10. In Problem 9, suppose that the daily demand can exceed supply, which gives rise to shortage

(negative inventory). The end-of-day inventory level for the past 30 days is given as:

1,2,0, -2,2,2, -1, -1,3,0,0,1, -1, -2,3,3, -2, -1,0,2,0, -1, 3,0,0,3, -1, 1,2, -2.

(a) Express the situation as a Markov chain.

(b) Determine the long-term probability of a surplus inventory in anyone day.

(c) Determine the long-term probability of a shortage inventory in anyone day.

(d) Determine the long-term probability of the daily supply meeting the daily demand exactly.

(e) If the holding cost per (end-of-day) surplus book is $.15 per day and the penalty

·cost per shortage book is $4.00 per day, determine the expected inventory cost per day.