# Question: Consider the following inventory policy for the certain product If

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.

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.

## Relevant Questions

Refer to Selected Reference 5. Read Selected Reference A1 that describes an award-winning OR study done for General Motors. (a) Summarize the background that led to undertaking this study. (b) What was the goal of this study? (c) Describe how software was ...A computer is inspected at the end of every hour. It is found to be either working (up) or failed (down). If the computer is found to be up, the probability of its remaining up for the next hour is 0.95. If it is down, the ...A production process contains a machine that deteriorates rapidly in both quality and output under heavy usage, so that it is inspected at the end of each day. Immediately after inspection, the condition of the machine is ...Suppose that a communications network transmits binary digits, 0 or 1, where each digit is transmitted 10 times in succession. During each transmission, the probability is 0.995 that the digit entered will be transmitted ...Post your question