Consider the following inventory policy for a 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 a product in periods 1, 2, . . . , respectively. Assume that the Dn 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 Xn (n 1, 2, . . .), 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 steadystate probabilities.

(c) Now use the result given in Prob. 16.5-2 to find the steadystate 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.