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. 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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