# Question

Consider the following inventory policy for the certain product. If the demand during a period exceeds the number of items available, this unsatisfied demand is backlogged; i.e., it is filled when the next order is received. Let Zn (n = 0, 1, . . .) denote the amount of inventory on hand minus the number of units backlogged before ordering at the end of period n (Z0 = 0). If Zn is zero or positive, no orders are backlogged. If Zn is negative, then Zn represents the number of backlogged units and no inventory is on hand. At the end of period n, if Zn < 1, an order is placed for 2m units, where m is the smallest integer such that Zn + 2m > 1. Orders are filled immediately.
Let D1, D2, . . . , be the demand for the product in periods 1,
2, . . . , respectively. Assume that the Ds are independent and identically distributed random variables taking on the values, 0, 1, 2, 3, 4, each with probability 1/5. Let Xn denote the amount of stock on hand after ordering at the end of period n (where X0 = 2), so that
When {Xn} (n = 0, 1, . . .) is a Markov chain. It has only two states, 1 and 2, because the only time that ordering will take place is when Zn = 0, 1, 2, or 3, in which case 2, 2, 4, and 4 units are ordered, respectively, leaving Xn = 2, 1, 2, 1, respectively.
(a) Construct the (one-step) transition matrix.
(b) Use the steady-state equations to solve manually for the steady state probabilities.
(c) Now use the result given in Prob. 29.5-2 to find the steady state probabilities.
(d) Suppose that the ordering cost is given by (2 + 2m) if an order is placed and zero otherwise. The holding cost per period is Zn if Zn > 0 and zero otherwise. The shortage cost per period is 4Zn if Zn 0 and zero otherwise. Find the (long-run) expected average cost per unit time.

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