This preview has intentionally blurred parts. Sign up to view the full document

View Full Document

Unformatted Document Excerpt

Inventory Probabilistic Models All the inventory models discussed in Chapter 15 require that demand during any period of time be known with certainty. In this chapter, we consider inventory models in which demand over a given time period is uncertain, or random; single-period inventory models, where a problem is ended once a single ordering decision has been made; single-period bidding models; versions of the EOQ model for uncertain demand that incorporate the important concepts of safety stock and service level; the periodic review (R, S) model; the ABC inventory classication system; and exchange curves. 16.1 Single-Period Decision Models In many situations, a decision maker is faced with the problem of determining the value q for a variable (q may be the quantity ordered of an inventoried good, for example, or the bid on a contract). After q has been determined, the value d assumed by a random variable D is observed. Depending on the values of d and q, the decision maker incurs a cost c(d, q). We assume that the person is risk-neutral and wants to choose q to minimize his or her expected cost. Since the decision is made only once, we call a model of this type a single-period decision model. 16.2 The Concept of Marginal Analysis For the single-period model described in Section 16.1, we now assume that D is an integer-valued discrete random variable with P(D d) p(d). Let E(q) be the decision makers expected cost if q is chosen. Then E(q) d p(d)c(d, q) In most practical applications, E(q) is a convex function of q. Let q* be the value of q that minimizes E(q). If E(q) is a convex function, the graph of E(q) must look something like Figure 1. From the gure, we see that q* is the smallest value of q for which E(q* 1) E(q*) 0 (1) Thus, if E(q) is a convex function of q, we can nd the value of q minimizing expected cost by nding the smallest value of q that satises Inequality (1). Note that E(q 1) E(q) is the change in expected cost that occurs if we increase the decision variable q to q 1. E (q) FIGURE 1 q 0 1 q* 1 q* q* + 1 Determination of q* by Marginal Analysis To determine q*, we begin with q 0. If E(1) E(0) 0, we can benet by increasing q from 0 to 1. Now we check to see whether E(2) E(1) 0. If this is true, then increasing q from 1 to 2 will reduce expected cost. Continuing in this fashion, we see that increasing q by 1 will reduce expected costs up to the point where we try to increase q from q* to q* 1. In this case, increasing q by 1 will increase expected cost. From Figure 1 (which is the appropriate picture if E(q) is a convex function), we see that if E(q* 1) E(q*) 0, then for q q*, E(q 1) E(q) 0. Thus, q* must be the value of q that minimizes E(q). If E(q) is not convex, this argument may not work. (See Problem 1 at the end of this section.) Our approach determines q* by repeatedly computing the effect of adding a marginal unit to the value of q. For this reason, it is often called marginal analysis. Marginal analysis is very useful if it is easy to determine a simple expression for E(q 1) E(q). In the next section, we use marginal analysis to solve the classical news vendor problem. PROBLEM Group A 1 Suppose E(q) is E(0) 8, E(1) 6, E(2) 5, E(3) 7, E(4) 6, E(5) 5.5, E(6) 4.5, and E(7) 5. a What value of q minimizes E(q)? b of c of If marginal analysis is used to determine the value q that minimizes E(q), what is the answer? Explain why marginal analysis fails to nd the value q that minimizes E(q). 16.3 The News Vendor Problem: Discrete Demand Organizations often face inventory problems where the following sequence of events occurs: The organization decides how many units to order. We let q be the number of units ordered. 1 2 With probability p(d), a demand of d units occurs. In this section, we assume that d must be a nonnegative integer. We let D be the random variable representing demand. 1 6 . 3 The News Vendor Problem: Discrete Demand 881 3 Depending on d and q, a cost c(d, q) is incurred. Problems that follow this sequence are often called news vendor problems. To see why, consider a vendor who must decide how many newspapers should be ordered each day from the newspaper plant. If the vendor orders too many papers, he or she will be left with many worthless newspapers at the end of the day. On the other hand, a vendor who orders too few newspapers will lose prot that could have been earned if enough newspapers to meet customer demand had been ordered, and customers will be disappointed. The news vendor must order the number of papers that properly balances these two costs. We have already encountered a news vendor problem in the discussion of decision theory in Section 13.1. In this section, we show how marginal analysis can be used to solve news vendor problems when demand is a discrete random variable and c(d, q) has the following form: c(d, q) c(d, q) coq (terms not involving q) cuq (terms not involving q) (d (d q) q (2) 1) (2.1) In (2), co is the per-unit cost of being overstocked. If d q, we have ordered more than was demandedthat is, overstocked. If the size of the order is increased from q to q 1, then (2) shows that the cost increases by co. Hence, co is the cost due to being overstocked by one extra unit. We refer to co as the overstocking cost. Similarly, if d q 1, we have understocked (ordered an amount less than demand). If d q 1 and we increase the size of the order by one unit, we are understocked by one less unit. Then (2.1) implies that the cost is reduced by cu, so cu is the per-unit cost of being understocked. We call cu the understocking cost. To derive the optimal order quantity via marginal analysis, let E(q) be the expected cost if an order is placed for q units. We assume that the decision makers goal is to nd the value q* that minimizes E(q). If c(d, q) can be described by (2) and (2.1), and E(q) is a convex function of q, then marginal analysis can be used to determine q*. Following (1), we must determine the smallest value of q for which E(q 1) E(q) 0. To calculate E(q 1) E(q), we must consider two possibilities: d q. In this case, ordering q 1 units instead of q units causes us to be overstocked by one more unit. This increases cost by co. The probability that Case 1 will occur is simply P(D q), where D is the random variable representing demand. Case 1 Case 2 d q 1. In this case, ordering q 1 units instead of q units enables us to be short one less unit. This will decrease cost by cu. The probability that Case 2 will occur is P(D q 1) 1 P(D q). In summary, a fraction P(D q) of the time, ordering q 1 units will cost co more than ordering q units; and a fraction 1 P(D q) of the time, ordering q 1 units will cost cu less than ordering q units. Thus, on the average, ordering q 1 units will cost co P(D q) cu[1 P(D q)] more than ordering q units. More formally, we have shown that E(q Then E(q (co 1) 1) E(q) cu) P(D E(q) co P(D q) cu[1 (co cu) P(D q) P(D cu q)] 0 will hold if q) cu 0 or P(D q) cu co cu cu Since P(D q) increases as q increases, E(q 1) E(q) will increase as q increases. Hence, if co 0, E(q) is a convex function of q, and our use of marginal analysis is justied. 882 CHAPTER 1 6 Probabilistic Inventory Models Let F(q) P(D q) be the demand distribution function. Since marginal analysis is applicable, we have just shown that E(q) will be minimized by the smallest value of q (call it q*) satisfying F(q*) cu co cu (3) The following example illustrates the use of (3). EXAMPLE 1 Walton Bookstore Calendar Sales In August, Walton Bookstore must decide how many of next years nature calendars should be ordered. Each calendar costs the bookstore $2 and is sold for $4.50. After January 1, any unsold calendars are returned to the publisher for a refund of 75 per calendar. Walton believes that the number of calendars sold by January 1 follows the probability distribution shown in Table 1. Walton wants to maximize the expected net prot from calendar sales. How many calendars should the bookstore order in August? Solution Let q d number of calendars ordered in August number of calendars demanded by January 1 If d q, the costs shown in Table 2 are incurred (revenue is negative cost). From (2), co 1.25. If d q 1, the costs shown in Table 3 are incurred. From (2), cu 2.5, or cu 2.50. Then cu co TA B L E cu 2.50 3.75 2 3 1 Probability Mass Function for Calendar Sales No. of Calendars Sold Probability 100 150 200 250 300 .30 .20 .30 .15 .05 TA B L E 2 Computation of Total Cost If d q Cost Buy q calendars at $2/calendar Sell d calendars at $4.50/calendar Return q d calendars at 75/calendar Total cost 2q 4.50d 0.75(q d) 1.25q 3.75d Based on Barron (1985). 1 6 . 3 The News Vendor Problem: Discrete Demand 883 TA B L E 3 Computation of Total Cost If d q 1 Cost Buy q calendars at $2/calendar Sell d calendars at $4.50/calendar Total cost 2q 4.50q 2.50q From (3), Walton should order q* calendars, where q* is the smallest number for which 2 . As a function of q, P(D q) increases only when q 100, 150, 200, P(D q*) 3 250, or 300. Also note that P(D 100) .30, P(D 150) .50, and P(D 200) 2 .80. Since P(D 200) is greater than or equal to 3 , q* 200 calendars should be ordered. REMARKS 1 In terms of marginal analysis, the probability of selling the 200th calendar that is ordered is P(D 200) .50. This implies that the 200th calendar has a 1 .50 .50 chance of being unsold. Thus, the 200th calendar will increase Waltons expected costs by .50( 2.50) .50(1.25) $0.625. Hence, the 200th calendar should be ordered. On the other hand, the probability that the 201st calendar will be sold is P (D 201) .20, and the probability that the 201st calendar will not be sold is 1 .20 .80. Therefore, the 201st calendar will increase expected costs by .20( 2.50) .80(1.25) $0.50. Thus, the 201st calendar will increase expected costs and should not be ordered. 2 In Example 1, co and cu could easily have been determined without recourse to (2) and (2.1). For example, being one more unit over actual demand increases Waltons costs by 2 0.75 $1.25. Thus, co $1.25. Similarly, being one more unit under actual demand will cost Walton 4.50 2.00 $2.50 in prot. Hence, cu $2.50. If we are able to determine co and cu without using Equations (2) and (2.1), we should do so. In more difcult problems, however, they can be very useful (see Examples 2 and 3). PROBLEMS Group A 1 In August 2003, a car dealer is trying to determine how many 2004 models should be ordered. Each car costs the dealer $10,000. The demand for the dealers 2004 models has the probability distribution shown in Table 4. Each car is sold for $15,000. If the demand for 2004 cars exceeds the number of cars ordered in August, the dealer must reorder at a cost of $12,000 per car. If the demand for 2004 cars falls short, the dealer may dispose of excess cars in an endof-model-year sale for $9,000 per car. How many 2004 models should be ordered in August? 2 Each day, a news vendor must determine how many New York Herald Wonderfuls to order. She pays 15 for each paper and sells each for 30. Any leftover papers are a total loss. From past experience, she believes that the number of papers she can sell each day is governed by the probability distribution shown in Table 5. How many papers should she order each day? 3 If cu is xed, will an increase in co increase or decrease the optimal order quantity? TA B L E 4 Probability TA B L E 5 Probability No. of Cars Demanded No. of Papers Demanded 20 25 30 35 40 .30 .15 .15 .20 .20 50 70 90 110 130 .30 .15 .25 .10 .20 884 CHAPTER 1 6 Probabilistic Inventory Models TA B L E 6 Probability TA B L E 8 Probability No. of Cells Copies Demanded 50 60 70 80 90 100 .20 .15 .30 .10 .15 .10 5,000 6,000 7,000 8,000 .30 .20 .40 .10 TA B L E TA B L E 9 Probability 7 Probability Week of Birth Number Needed 200 275 350 400 450 500 550 600 650 .03 .03 .03 .05 .40 .30 .06 .07 .03 36 37 39 40 41 42 43 .05 .15 .20 .30 .15 .10 .05 4 If co is xed, will an increase in cu increase or decrease the optimal order quantity? 5 The power at Ice Station Lion is supplied via solar cells. Once a year, a plane ies in and sells solar cells to the ice station at a price of $20 per cell. Because of uncertainty about future power needs, the ice station can only guess the number of cells that will be required during the coming year (see probability distribution in Table 6). If the ice station runs out of solar cells, a special order must be placed at a cost of $30 per cell. a Assuming that the news vendor problem is relevant, how many cells should be ordered from the plane? b In part (a), what type of cost is being ignored? 6 The daily demand for substitute teachers in the Los Angeles teaching system follows the distribution given in Table 7. Los Angeles wants to know how many teachers to keep in the substitute teacher pool. Whether or not the substitute teacher is needed, it costs $30 per day to keep a substitute teacher in the pool. If not enough substitute teachers are available on a given day, regular teachers are used to cover classes at a cost of $54 per regular teacher. How many teachers should Los Angeles have in the substitute teacher pool? copies of the book are in stock, and Blockbuster must determine how many copies of the book should be printed for the next year. The sales department believes that sales during the next year are governed by the distribution in Table 8. Each copy of Joy sold during the next year brings the publisher $35 in revenues. Any copies left at the end of the next year cannot be sold at full price but can be sold for $5 to Bonds Ennoble and Gitanos bookstores. The cost of a printing of the book is $50,000 plus $15 per book printed. How many copies of Joy should be printed? Would the answer change if 4,000 copies were currently in stock? 8 Vivian and Wayne are planning on going to Lamaze natural childbirth classes. Lamaze classes meet once a week for ve weeks. Each class gives 20% of the knowledge needed for natural childbirth. If Vivian and Wayne nish their classes before the birth of their child, they will forget during each week 5% of what they have learned in class. To maximize their expected knowledge at the time of childbirth, during which week of pregnancy should they begin classes? Assume that the number of weeks from conception to childbirth follows the probability distribution given in Table 9. 9 Some universities allow an employee to put an amount q into an account at the beginning of each year, to be used for child-care expenses. The amount q is not subject to federal income tax. Assume that all other income is taxed by the federal government at a 40% rate. If child-care expenses for the year (call them d ) are less than q, the employee in effect loses q d dollars in before-tax income. If child-care expenses exceed q, the employee must pay the excess out of his or her own pocket but may credit 25% of that as a savings on his or her state income tax. Suppose Professor Muffy Rabbit believes that there is an equal chance that her child-care expenses for the coming year will be $3,000, $4,000, $5,000, $6,000, or $7,000. At the beginning of the year, how much money should she place in the child-care account? Group B 7 Every four years, Blockbuster Publishers revises its textbooks. It has been three years since the best-selling book, The Joy of OR, has been revised. At present, 2,000 Based on Bruno (1970). Based on Rosenfeld (1986). 1 6 . 3 The News Vendor Problem: Discrete Demand 885 16.4 The News Vendor Problem: Continuous Demand We now consider the news vendor scenario of Section 16.3 when demand D is a continuous random variable having density function f (d ). By modifying our marginal analysis argument of Section 16.3 (or by using Leibnizs rule for differentiating an integralsee Problem 7 at the end of this section), it can be shown that the decision makers expected cost is minimized by ordering q* units, where q* is the smallest number satisfying P(D q*) cu co cu (4) Since demand is a continuous random variable, we can nd a number q* for which (4) holds with equality. Hence, in this case, the optimal order quantity can be determined by nding the value of q* satisfying P(D q*) cu co cu or P(D q*) co co cu (5) From (5), we see that it is optimal to order units up to the point where the last unit ordered has a chance co co cu of being sold. Examples 2 and 3 illustrate the use of (5). EXAMPLE 2 ABA Room Reservations The American Bar Association (ABA) is holding its annual convention in Las Vegas. Six months before the convention begins, the ABA must decide how many rooms should be reserved in the convention hotel. At this time, the ABA can reserve rooms at a cost of $50 per room, but six months before the convention, the ABA does not know with certainty how many people will attend the convention. The ABA believes, however, that the number of rooms required is normally distributed, with a mean of 5,000 rooms and a standard deviation of 2,000 rooms. If the number of rooms required exceeds the number of rooms reserved at the convention hotel, extra rooms will have to be found at neighboring hotels at a cost of $80 per room. It is inconvenient for convention participants to stay at neighboring hotels. We measure this inconvenience by assessing an additional cost of $10 for each room obtained at a neighboring hotel. If the goal is to minimize the expected cost to the ABA and its members, how many rooms should the ABA reserve at the convention hotel? Solution Dene q d number of rooms reserved number of rooms actually required If d q, then the only cost incurred is the cost of the rooms reserved in advance, so if d q, the total cost is 50q. Thus, co 50. If d q 1, the following costs are incurred: Cost of reserving q rooms 50q Cost of renting d q rooms in neighboring hotels Inconvenience cost to overow participants 10(d Total cost 90d 40q and cu 40 80(d q) q) 886 CHAPTER 1 6 Probabilistic Inventory Models Since 40 4 , we see from (5) that the optimal number of rooms to reserve is 90 9 cu co the number q* satisfying cu P(D q*) 4 9 (6) The Excel function NORMINV can be used to calculate q*. Since NORMINV(4/9,5000,2000) yields 4,720.58, the ABA should reserve 4,720 or 4,721 rooms. EXAMPLE 3 Airline Overbooking The ticket price for a New YorkIndianapolis ight is $200. Each plane can hold up to 100 passengers. Usually, some of the passengers who have purchased tickets for a ight fail to show up (no-shows). To protect against no-shows, the airline will try to sell more than 100 tickets for each ight. Federal law states that any ticketed customer who is unable to board the plane is entitled to compensation (say, $100). Past data indicate that the number of no-shows for each New YorkIndianapolis ight is normally distributed, with a mean of 20 and a standard deviation of 5. To maximize expected revenues less compensation costs, how many tickets should the airline sell for each ight? Assume that anybody who doesnt use a ticket receives a $200 refund. Solution Let q d number of tickets sold by airline number of no-shows Observe that q d will be the number of customers actually showing up for the ight. If q d 100, then all customers who show up will board the ight, and the cost to the airline is 200(q d) 200d 200q. If q d 100, then 100 passengers will board the plane (paying the airline 200(100) $20,000), and q d 100 customers will be turned away. These q d 100 customers will receive compensation of 100(q d 100). Hence, if q d 100, the total cost to the airline is given by 100(q d 100) 200(100) 100(q 100) 100d 20,000. In summary, the net cost to the airline may be expressed as shown in Table 10. If q 100 is considered as a decision variable, we have a news vendor problem with cu 200 (or cu 200) and co 100. From (5), we should choose q 100 to satisfy P(D q 100) cu co cu 2 3 (7) The problem can be solved with the help of Excel. Since NORMINV(2/3,120,5) TA B L E 10 Total Cost Computation of Total Cost q q d d 100 (or d 100 (or d q q 100) 100) 100 (q 100) 200d 200 (q 100d 100) 20,000 200 (100) 1 6 . 4 The News Vendor Problem: Continuous Demand 887 yields 122.15, we may conclude that the airline should attempt to sell 122 or 123 tickets. This means that once ticket sales have reached 122 (or 123), no more tickets should be sold for the ight. Of course, if fewer than 122 people want to purchase tickets for the ight, the airline should not refuse to sell anybody a ticket for the ight. PROBLEMS Group A 1 a In Example 3, why is it unrealistic to assume that the distribution of the number of no-shows is independent of q? b If the number of no-shows were normally distributed with a mean of .05q and a standard deviation of .05q, would we still have a news vendor problem? must determine how much production capacity it should have. The cost of building enough production capacity to make 1,000 sets per year is $1,000,000 (equivalent in present value terms to a cost of $100,000 per year forever). Exclusive of the cost of building capacity, each set sold contributes $250 to prots. How much production capacity should Motorama have? 6 I. L. Pea is a well-known mail-order company. During the Christmas rush (from November 1 to December 15), the number of orders that I. L. Pea must ll each day (ve days per week) is normally distributed, with a mean of 2,000 and a standard deviation of 500. I. L. Pea must determine how many employees should be working during the Christmas rush. Each employee works ve days a week, eight hours a day, can process 50 orders per day, and is paid $10 per hour. If the full-time work force cannot handle the days orders during regular hours, some employees will have to work overtime. Each employee is paid $15 per hour for overtime work. For example, if 300 orders are received in a day and there are four employees, then 300 4(50) 100 orders must be processed by employees who are working overtime. Since each employee can ll 580 6.25 orders per hour, 1 I. L. Pea would need to pay workers 6.00 16 hours of 25 overtime for that day. To minimize its expected labor costs, how many full-time employees should I. L. Pea employ during the Christmas rush? 7 Suppose demand is a continuous random variable having a probability density function f (d), and c(d, q) is given by Equation (2). Show that if q units are ordered, the expected cost E(q) may be written as q 2 Condo Construction Company is going to First National Bank for a loan. At the present time, the bank is willing to lend Condo up to $1 million, with interest costs of 10%. Condo believes that the amount of borrowed funds needed during the current year is normally distributed, with a mean of $700,000 and a standard deviation of $300,000. If Condo needs to borrow more money during the year, the company will have to go to Louie the Loan Shark. The cost per dollar borrowed from Louie is 25. To minimize expected interest costs for the year, how much money should Condo borrow from the bank? 3 Joe is selling Christmas trees to pay his college tuition. He purchases trees for $10 each and sells them for $25 each. The number of trees he can sell is normally distributed with a mean of 100 and standard deviation of 30. How many trees should Joe purchase? 4 A hot dog vendor at Wrigley Field sells hot dogs for $1.50 each. He buys them for $1.20 each. All the hot dogs he fails to sell at Wrigley Field during the afternoon can be sold that evening at Comiskey Park for $1 each. The daily demand for hot dogs at Wrigley Field is normally distributed with a mean of 40 and a standard deviation of 10. a If the vendor buys hot dogs once a day, how many should he buy? b If he buys 52 hot dogs, what is the probability that he will meet all of the days demand for hot dogs at Wrigley? E(q) Group B 5 Motorama TV estimates the annual demand for its TVs is (and will be in the future) normally distributed, with a mean of 6,000 and standard deviation of 2,000. Motorama 0 coqf (t)dt q ( cu)qf (t)dt E(q) (terms not involving q in integrand) Now use Leibnizs rule to derive Equation (5). Based on Virts and Garrett (1970). 16.5 Other One-Period Models Many interesting single-period models in operations research cannot be easily handled by marginal analysis. In such situations, we express the decision makers objective function (usually expected prot or expected cost) as a function f (q) of the decision variable q. 888 CHAPTER 1 6 Probabilistic Inventory Models Then we nd a maximum or minimum of f (q) by setting f (q) lustrate this idea by a brief discussion of a bidding model. EXAMPLE 0. In this section, we il- 4 Condo Construction Company Condo Construction Company is bidding on an important construction job. The job will cost $2 million to complete. One other company is bidding for the job. Condo believes that the opponents bid is equally likely to be any amount between $2 million and $4 million. If Condo wants to maximize expected prot, what should its bid be? Solution Let B b random variable representing bid of Condos opponent actual bid of Condos opponent Then f (b), the density function for B, is given by f (b) 1 2,000,000 0 (2,000,000 otherwise b 4,000,000) Let q Condos bid. If b q, Condo outbids the opponent and earns a prot of q 2,000,000. On the other hand, if b q, Condo is outbid by the opponent and earns nothing. The event b q has a zero probability of occurring and may be ignored. Let E(q) be Condos expected prot if it bids q. Then q 4,000,000 E(q) 2,000,000 (0) f (b)db for 2,000,000 E(q) (q b q (q 2,000,000) f (b)db Since f (b) 1 2,000,000 4,000,000, we obtain q) 2,000,000 (4,000,000 q) 6,000,000 2,000,000 2q 2,000,000)(4,000,000 To nd the value of q maximizing E(q), we nd E (q) (q 2,000,000) 2,000,000 Hence, E (q) 0 for q 3,000,000. Since E (q) 2,0002 0, we know that E(q) is a ,000 concave function of q, and q 3,000,000 does indeed maximize E(q). Hence, Condo should bid $3 million. Condos expected prot will be E(3,000,000) $500,000. PROBLEMS Group A 1 The City of Rulertown consists of the unit interval [0, 1] (see Figure 2). Rulertown needs to determine where to build the citys only re station. It knows that for small x, the probability that a given re occurs at a location between x and x x is 2x( x). Rulertown wants to minimize the average distance between the re station and a re. Where should the re station be located? FIGURE 2 x x x 1 0 Group B 2 Assume that the Federal Reserve Board can control the growth rate of the U.S. money supply. Also assume that 1 6 . 5 Other One-Period Models 889 during a year in which the money supply grows by x%, the Gross Domestic Product (GDP) grows by Zx%, where Z is a known random variable. The government has decided it wants the GDP to grow by k% each year. (Too high a growth rate causes excessive ination, and too low a growth rate causes high unemployment.) To model the governments view, the government assesses a cost of (d k)2 during a year in which the GDP grows by d%. a Determine the growth rate of the money supply that should be set by the Federal Reserve Board if the goal is to minimize the expected cost to the government. b Show that for a given value of E(Z), an increase in var Z will decrease the optimal growth rate of the money supply found in part (a). (Hint: Use the fact that var Z E(Z2) E(Z)2.) 16.6 The EOQ with Uncertain Demand: The (r, q) and (s, S ) Models In this section, we discuss a modication of the EOQ that is used when lead time is nonzero and the demand during each lead time is random. We begin by assuming that all demand can be backlogged. As in Chapter 15, we assume a continuous review model, so that orders may be placed at any time, and we dene K h L q ordering cost holding cost/unit/year lead time for each order (assumed to be known with certainty) quantity ordered each time an order takes place We also require the following denitions: D cB OHI(t) random variable (assumed continuous) representing annual demand, with mean E(D), variance var D, and standard deviation sD cost incurred for each unit short, which does not depend on how long it takes to make up stockout on-hand inventory (amount of stock on hand) at time t 100, OHI(0) 200, and OHI(6) OHI(7) 0. From Figure 3, we can see that OHI(1) B(t) I(t) r number of outstanding back orders at time t net inventory level at time t OHI(t) B(t) inventory level at which order is placed (reorder point) I(t) 300 200 L=2 r = 100 q = 240 100 q = 240 0 t (months) 1 2 3 4 5 6 7 FIGURE 3 100 (O1) (O1 + L) (O2) (O2 + L) Evolution of Inventory over Time in Reorder Point Model 890 Cycle 1 Cycle 2 CHAPTER 1 6 Probabilistic Inventory Models In Figure 3, B(t) 0 for 0 t 6 and B(7) 100. I(t) agrees with the inventory concept used in Chapter 15; I(0) 200 0 200, I(3) 260 0 260, and I(7) 0 100 100. The reorder point r 100; whenever the inventory level drops to r, an order is placed for q units. X random variable representing demand during lead time We assume that X is a continuous random variable having density function f (x) and mean, variance, and standard deviation of E(X), var X, and sX, respectively. If we assume that the demands at different points in time are independent, then it can be shown that the random lead time demand X satises E(X) LE(D), var X L(var D), sX sD L (8) We assume that if D is normally distributed, then X will also be normally distributed. Suppose we allow the lead time L to be a random variable (denoted by L), with mean E(L), variance var L, and standard deviation sL. If the length of the lead time is independent of the demand per unit time during the lead time, then E(X) E(L)E(D) and var X E(L)(var D) E(D)2(var L) (8 ) We want to choose q and r to minimize the annual expected total cost (exclusive of purchasing cost). Before showing how optimal values of r and q can be found, we look at an illustration of how inventory evolves over time. Assume that an order of q 240 units has just arrived at time 0. We also assume that L 2. In Figure 3, orders of size q are placed at times O1 1 and O2 5. These orders are received at times O1 L 3 and O2 L 7, respectively. A cycle is dened to be the time interval between any two instants at which an order is received. Figure 3 contains two complete cycles: cycle 1, from arrival of order at time 0 to the instant before order arrives at time O1 L 3; and cycle 2, from arrival of order at time O1 L 3 to the instant before order arrives at time O2 L 7. During cycle 1, demand during lead time is less than r, so no shortage occurs. During cycle 2, however, demand during lead time exceeds r, so stockouts do occur between time 6 and time O2 L 7. It should be clear that by increasing r, we can reduce the number of stockouts. Unfortunately, increasing r will force us to carry more inventory, thereby resulting in higher holding costs. Thus, an optimal value of r must represent some sort of trade-off between holding and stockout costs. We now show how the optimal values of q and r may be determined. Determination of Reorder Point: The Back-Ordered Case The situation in which all demand must eventually be met and no sales are lost is called the back-ordered case, for which we show how to determine the reorder point and order quantity that minimize annual expected cost. We assume each unit is purchased for the same price, so purchasing costs are xed. Dene TC(q, r) expected annual cost (excluding purchasing cost) incurred if each order is for q units and is placed when the reorder point is r. Then TC(q, r) (expected annual holding cost) (expected annual ordering cost) (expected annual cost due to shortages). To determine the optimal reorder point and order quantity, we assume that the average number of back orders is small relative to the average on-hand inventory level. In most cases, this assumption is reasonable, because shortages (if they occur at all) usually occur during only a small portion of a cycle. (See Problem 5 at the end of this section.) Then I(t) OHI(t) B(t) yields Expected value of I(t) expected value of OHI(t) (9) 1 6 . 6 The EOQ with Uncertain Demand: The (r, q) and (s, S ) Models 891 We can now approximate the expected annual holding cost. We know that expected annual holding cost h(expected value of on-hand inventory level). Then from (9), we can approximate expected annual holding cost by h(expected value of I(t)). As in Chapter 3, the expected value of I(t) will equal the expected value of I(t) during a cycle. Since the mean rate at which demand occurs is constant, we may write Expected value of I(t) during a cycle 1 [(expected value of I(t) at beginning of cycle) 2 (expected value of I(t) at end of a cycle)] (10) At the end of a cycle (the instant before an order arrives), the inventory level will equal the inventory level at the reorder point (r) less the demand X during lead time. Thus, expected value of I(t) at end of cycle r E(X). At the beginning of a cycle, the inventory level at the end of the cycle is augmented by the arrival of an order of size q. Thus, expected value of I(t) at beginning of cycle r E(X) q. Now (10) yields Expected value of I(t) during cycle q 1 (r 2 q 2 r E(X) r E(X) E(X) q) Thus, expected annual holding cost h( r E(X)). 2 To determine the expected annual cost due to stockouts or back orders, we must dene Br Now Expected annual shortage cost By the denition of Br, Expected shortage cost cBE(Br) E(D) q random variable representing the number of stockouts or back orders during a cycle if the reorder point is r expected shortage cost expected cycles Since all demand will eventually be met, an average of year. Then Expected shortage cost Finally, Expected annual order cost K orders will be placed each cBE(Br)E(D) q expected orders year cBE(Br)E(D) q KE(D) q KE(D) q Putting together the expected annual holding, shortage, and ordering costs, we obtain TC(q, r) h q 2 r E(X) (11) Using the method described in Section 11.5, we could nd the values of q and r that minimize (11) by determining values q* and r* of q and r satisfying TC(q*, r*) q TC(q*, r*) r 0 (12) In Review Problem 7 we show how LINGO can be used to determine values of q and r that exactly satisfy (12). In most cases, however, the value of q* satisfying (12) is very close 892 CHAPTER 1 6 Probabilistic Inventory Models to the EOQ of ( 2KE(D) )1/2. For this reason, we assume that the optimal order quantity q* may h be adequately approximated by the EOQ. Given a value q for the order quantity, we now show how marginal analysis can be used to determine a reorder point r* that minimizes TC(q, r). If we assume a given value of q, the expected annual ordering cost is independent of r. Thus, in determining a value of r that minimizes TC(q, r), we may concentrate on minimizing the sum of the expected annual holding and shortage costs. Following the marginal analysis approach of Sections 16.216.3, suppose we increase the reorder point (for small) from r to r (with q xed). Will this result in an increase or a decrease in TC(q, r)? If we increase r to r , the expected annual holding cost will increase by h q 2 r E(X) h q 2 r E(X) h If we increase the reorder point from r to r , expected annual stockout costs will be reduced, because of the fact that during any cycle in which lead time demand is at least r, the number of stockouts during the cycle will be reduced by units. In other words, increasing the reorder point from r to r will reduce stockout costs by cB during a D fraction P(X r) of all cycles. Since there are an average of E(q ) cycles per year, increasing the reorder point from r to r will reduce expected annual stockout cost by E(D)cBP(X q r) Observe that as r increases, P(X r) decreases, so as r increases, the expected reduction in expected annual shortage cost resulting from increasing the reorder point by will decrease. This observation allows us to draw Figure 4. Let r* be the value of r for which marginal benet equals marginal cost, or E(D)cBP(X q P(X r*) r*) h hq cBE(D) Suppose that r r*. Then Figure 4 shows that if we increase the reorder point from r to r*, we can save more in shortage cost than we lose in holding cost. Now suppose that r r*. Figure 4 shows that by reducing the reorder point from r to r*, we can save more in holding cost than we lose in increased shortage cost. Thus, r* does attain the optimal trade-off between shortage and holding costs. In summary, if we assume that the order quantity can be approximated by EOQ 2KE(D) h 1/2 then we have the reorder point r* and the order quantity q* for the back-ordered case: q* P(X If hq* cBE(D) r*) 2KE(D) h hq* cBE(D) 1 1/2 (13) Brown (1967) has shown that for approximating the optimal value of q, the EOQ is usually acceptable unless EOQ sX. 1 6 . 6 The EOQ with Uncertain Demand: The (r, q) and (s, S ) Models 893 Cost Decrease in expected annual shortage cost if r is increased to r + FIGURE 4 Increase in expected annual holding cost if r is increased to r + h Trade-off between Holding Cost and Shortage Cost r r* then (13) will have no solution, and holding cost is prohibitively high relative to the stockout cost. Management should set the reorder point at the smallest acceptable level. If (13) yields a negative value of r*, management should also set the reorder point at the smallest acceptable level. REMARKS 1 P(X r) is just the probability that a stockout will occur during a lead time. Also note that for h near zero, (13) yields a stockout probability near zero. For large cB also, (13) yields a stockout probability near zero. Both of these results should be consistent with intuition. 2 After substituting the EOQ for q in (13), we may easily determine an approximately optimal value of r, the reorder point. Note that r E(X) is the amount in excess of expected lead time demand that is ordered to protect against the occurrence of stockouts during the lead time. For this reason, r E(X) is often referred to as safety stock. 3 From (11), we nd that the expected annual cost of holding safety stock is h(r E(X)) h(safety stock level). The following example illustrates the determination of the reorder point and safety stock level in the back-ordered demand case. EXAMPLE 5 Disk Stock Each year, a computer store sells an average of 1,000 boxes of disks. Annual demand for boxes of disks is normally distributed with a standard deviation of 40.8 boxes. The store orders disks from a regional distributor. Each order is lled in two weeks. The cost of placing each order is $50, and the annual cost of holding one box of disks in inventory is $10. The per-unit stockout cost (because of loss of goodwill and the cost of placing a special order) is assumed to be $20. The store is willing to assume that all demand is backlogged. Determine the proper order quantity, reorder point, and safety stock level for the computer store. Assume that annual demand is normally distributed. What is the probability that a stockout occurs during the lead time? Solution We begin by determining the EOQ. Since h we nd that EOQ $10/box/year, K 1/2 $50, and E(D) 1,000, 2(50)(1,000) 10 100 894 CHAPTER 1 6 Probabilistic Inventory Models We now substitute q* 100 in (13) and use (13) to determine the reorder point. To do this, we need to determine the probability distribution of X, the lead time demand. Since L 2 weeks, X will be normally distributed with E(X) Since cB E(D) 26 1,000 26 38.46 and sX sD 26 40.8 26 8 $20, (13) now yields P(X r) 10(100) 20(1,000) .05 (14) We use the Excel function NORMINV Since . NORMINV(0.95,38.46,8) yields 51.62, we nd that the safety stock level is r E(X) 51.62 38.46 13.16. To see how the reorder point and safety stock level would be affected by a variable lead time, suppose that the lead time has a mean of two weeks but also has a standard deviation of one week ( 512 year). Then (8 ) yields s2 X sX ( 216 )(40.8)2 433.84 (1,000)2 ( 512 )2 20.83 64.02 369.82 433.84 Assuming that the lead time demand is normally distributed, we would nd that r 38.46 1.65(20.83) 72.83, and the safety stock held is 1.65(20.83) 34.37. Thus, the variability of the lead time has more than doubled the required safety stock level! Determination of Reorder Point: The Lost Sales Case We now assume that all stockouts result in lost sales and that a cost of cLS dollars is incurred for each lost sale. (In addition to penalties for loss of future goodwill, cLS should include prot lost because of a lost sale.) As in the back-ordered case, we assume that the optimal order quantity can be adequately approximated by the EOQ and attempt to use marginal analysis to determine the optimal reorder point r* (see Problem 6 at the end of this section). The optimal order quantity q* and the reorder point r* for the lost sales case are q* P(X r*) 2KE(D) h hq* hq* cLSE(D) 1/2 (15) The key to the derivation of (15) is to realize that expected inventory in lost sales case (expected inventory in back-ordered case) (expected number of shortages per cycle). This equation follows because in the lost sales case, we nd that during each cycle, an average of (expected shortages per cycle) fewer orders will be lled from inventory, thereby raising the average inventory level by an amount equal to expected shortages per cycle. Observe that the right-hand side of (15) is smaller than the right-hand side of (13). Thus, the lost sales assumption will yield a lower stockout probability (and a larger reorder point and safety stock level) than the back-ordered assumption. To illustrate the use of (15), we continue our discussion of Example 5. Suppose that each box of disks sells for $50 and costs the store $30. Assuming that the stockout cost 1 6 . 6 The EOQ with Uncertain Demand: The (r, q) and (s, S ) Models 895 of $20 given in Example 5 represents lost goodwill, we obtain cLS by adding the lost prot ($50 $30) to the lost goodwill of $20. Thus, cLS 20 20 40. Recall from Example 5 that E(D) 1,000 boxes per year, h $10/box/year, EOQ 100 boxes, and K $50. Now (15) yields P(X r*) 10(100) .024 Excel is used to compute r. Since NORMINV(.976,38.46,8) yields 54.28, we nd that r 54.28 38.46 15.82. 54.28. Thus, in the lost sales case, the safety stock level is Continuous Review (r, q) Policies A continuous review inventory policy, in which we order a quantity q whenever our inventory level reaches a reorder level r, is often called an (r, q) policy. An (r, q) policy is also called a two-bin policy, because it can easily be implemented by using two bins to store an item. For example, to implement a (30, 500) policy, we ll orders from bin 1 as long as bin 1 contains any items. As soon as bin 1 becomes empty, we know that the reorder point r 30 has been reached, and we place an order for q 500 units. When the order arrives, we bring the number of units in bin 2 up to 30, and place the remainder of the 500 units ordered in bin 1. Thus, whenever bin 1 has been emptied, we know that the reorder point has been reached. Continuous Review (s, S) Policies In our derivation of the best (r, q) policy, we assumed that an order could be placed exactly at the point when the inventory level reached the reorder point r. We used this assumption to compute the expected inventory level at the beginning and end of a cycle. Suppose that a demand for more than one unit can arrive at a particular time. Then an order may be triggered when the inventory level is less than r, and our computation of expected inventory level at the end and beginning of a cycle is then incorrect. For example, suppose r 30 and our current inventory level is 35. If an order for 10 units arrives, an order will be placed when the inventory level is 25 (not r 30), and this invalidates the computations that led to (11). From this discussion, we see that it is possible for the inventory level to undershoot the reorder point. Note that this problem could not occur if all demands were for one unit, for then the inventory level would drop from (say) 32 to 31 and then to 30, and each order would be placed when the inventory level equaled the reorder point r. From this example, we see that if demands of size greater than one unit can occur at a point in time, then the (r, q) model may not yield a policy that minimizes expected annual cost. In such situations, it has been shown that an (s, S) policy is optimal. To implement an (s, S) policy, we place an order whenever the inventory level is less than or equal to s. The size of the order is sufcient to raise the inventory level to S (assuming zero lead time). For example, if we were implementing a (5, 40) policy and the inventory level suddenly dropped from 7 to 3, we would immediately place an order for 40 3 37 units. Exact computation of the optimal (s, S) policy is difcult. If we neglect the problem of the undershoots, however, we may approximate the optimal (s, S) policy as follows. Set S s equal to the economic order quantity q. Then set s equal to the reorder point r obtained 896 CHAPTER 1 6 Probabilistic Inventory Models from (13) or (15). Finally, we obtain S r q. Thus, for Example 5 (with back orders allowed), we would set s 51.66 and S 51.66 100 151.66 and use (assuming that fractional demand is possible) a (51.66, 151.66) policy. PROBLEMS Group A 1 A hospital orders its blood from a regional blood bank. Each year, the hospital uses an average of 1,040 pints of Type O blood. Each order placed with the regional blood bank incurs a cost of $20. The lead time for each order is one week. It costs the hospital $20 to hold 1 pint of blood in inventory for a year. The per-pint stockout cost is estimated to be $50. Annual demand for Type O blood is normally distributed, with standard deviation of 43.26 pints. Determine the optimal order quantity, reorder point, and safety stock level. Assume that 52 weeks 1 year and that all demand is backlogged. To use the techniques of this section, what unrealistic assumptions must be made? What (s, S) policy would be used in this situation? 2 Furnco sells secretarial chairs. Annual demand is normally distributed, with mean of 1,040 chairs and standard deviation of 50.99 chairs. Furnco orders its chairs from its agship store. It costs $100 to place an order, and the lead time is two weeks. Furnco estimates that each stockout causes a loss of $50 in future goodwill. Furnco pays $60 for each chair and sells it for $100. The annual cost of holding a chair in inventory is 30% of its purchase cost. a Assuming that all demand is backlogged, what are the reorder point and the safety stock level? b Assuming that all stockouts result in lost sales, determine the optimal reorder point and the safety stock level. 3 We are given the following information for a product: Order cost $50 Annual demand N(960, 3,072.49) Annual holding cost $6/item/year Shortage cost $80 per unit Lead time one month Sales price $40 per unit Product cost $30 per unit a Determine the order quantity and the reorder point under the assumption that all demands are backordered. b Determine the order quantity and reorder point under the lost sales assumption. TA B L E 11 Probability Lead Time Demand 180 190 200 210 220 .30 .30 .15 .10 .15 5 In Figure 3, assume that demand occurs at a constant rate during each cycle. Approximate the average level of onhand inventory between t 0 and t 7. Also approximate the average number of shortages. Does the assumption that the average shortage level is small relative to the average level of on-hand inventory seem valid here? Group B 6 In this problem, use marginal analysis to determine the optimal reorder point for the lost sales case. a Show that the average inventory level for the lost sales case may be written as 1 E(X) E(Br)) (r E(X) E(Br) q)] 2 [(r q r E(X) E(Br) 2 b Although expected orders per year will no longer equal E(qD) (why?), we assume that the expected number of lost sales per year is relatively small. Thus, we may E(D) still assume that expected orders per year q . Now use marginal analysis to derive (15). 7 Suppose that a cost of S dollars (independent of the size of the stockout) is incurred whenever a stockout occurs during a cycle. Under the assumption of backlogged demand, use marginal analysis to determine the reorder point. 8 Explain the following statement: Faster-moving items require larger safety stocks than slower-moving items. (Hint: Does E(qD) large imply that an item is fast-moving or slowmoving?) 9 Suppose annual demand for a product is normally distributed, with a mean of 600 and a variance of 300. Suppose that the lead time for an order is always one month. Show (without using Equation (8)) that the lead time demand has mean 50, variance 25, and standard deviation 5. Assume that the demands during different one-month periods are independent, identically distributed random variables. 4 The lead time demand for bathing suits is governed by the discrete random variable shown in Table 11. The company sells an average of 10,400 suits per year. The cost of placing an order for bathing suits is $30, and the cost of holding one bathing suit in inventory for a year is $3. The stockout cost is $3 per bathing suit. Use marginal analysis to determine the optimal order quantity and the reorder point. 1 6 . 6 The EOQ with Uncertain Demand: The (r, q) and (s, S ) Models 897 16.7 The EOQ with Uncertain Demand: The Service Level Approach to Determining Safety Stock Level As we have previously stated, it is usually very difcult to determine accurately the cost of being one unit short. For this reason, managers often decide to control shortages by meeting a specied service level. In this section, we discuss two measures of service level: Service Level Measure 1 SLM1, the expected fraction (usually expressed as a percentage) of all demand that is met on time. Service Level Measure 2 SLM2, the expected number of cycles per year during which a shortage occurs. Throughout this section, we assume that all shortages are backlogged. The following example illustrates the meaning of the two service level measures. EXAMPLE 6 SLM1 and SLM2 Suppose that for a given inventory situation, average annual demand is 1,000 and the EOQ is 100. Demand during a lead time is random and is described by the probability distribution in Table 12. For a reorder point of 30 units, determine SLM1 and SLM2. Solution 1 1 1 1 The expected demand during a lead time is 1 (20) (30) (40) (50) (60) 5 5 5 5 5 40 units. With a reorder point of 30 units, we will reorder during each cycle at the instant when the inventory level hits 30 units. If the lead time demand during a cycle is 20 or 30 units, we will experience no shortage. During a cycle in which lead time demand is 40, a shortage of 10 units will occur; if lead time demand is 50, a shortage of 20 units will occur; if lead time demand is 60, a shortage of 30 units will occur. Hence, the expected 1 1 1 1 number of units short per cycle is given by 1 (0) (0) (10) (20) (30) 12. 5 5 5 5 5 Since the EOQ 100 and all demand must eventually be met, the average number of E(D) 1,000 orders placed each year will be q 10. Then the average number of shortages 100 that occur during a year will equal 10(12) 120 units. Thus, each year, on the average, 0 the demand for 1,000 120 880 units is met on time. In this case, the SLM1 18800 ,0 0.88 or 88%. This shows that even if the reorder point is less than the mean lead time demand, a relatively high SLM1 may result, because stockouts can only occur during the lead time, which is often a small portion of each cycle. We now determine SLM2 for a reorder point of 30. With a reorder point of 30, a stockout will occur during any cycle in which lead time demand exceeds 30 units. Thus, the 3 probability of a stockout during a cycle P(X 40) P(X 50) P(X 60) . 5 TA B L E 12 Mass Function for Lead Time Demand Lead Time Demand 5 Probability 1 1 5 1 5 1 5 1 5 20 30 40 50 60 898 CHAPTER 1 6 Probabilistic Inventory Models Since there are an average of 10 cycles per year, the expected number of cycles per year 6. Thus, a reorder point of 30 yields SLM2 6 that will result in shortages is 10 ( 3 ) 5 stockouts per year. Determination of Reorder Point and Safety Stock Level for SLM1 Given a desired value of SLM1, how do we determine a reorder point that provides the desired service level? Suppose we order the EOQ (q) and use a reorder point r. From Section 16.6, Expected shortages Expected shortages E(Br) E(Br)E(D) q Here, E(D) is the average annual demand. Let SLM1 be the percentage of all demand that is met on time. Then for given values of q (for the order quantity) and r (for the reorder point), we have 1 SLM1 expected shortages per year E(Br)E(D)/q E(D) E(Br) q (16) Equation (16) can be used to determine the reorder point that yields a desired service level. We now assume that the lead time demand is normally distributed, with mean E(X) and standard deviation sX. To use (16), we need to determine E(Br). If X is normally distributed, the determination of E(Br) requires a knowledge of the normal loss function. DEFINITION s The normal loss function, NL( y), is dened by the fact that sXNL( y) is the expected number of shortages that will occur during a lead time if (1) lead time demand is normally distributed with mean E(X) and standard deviation sX and (2) the reorder point is E(X) ysX. s In short, if we hold y standard deviations (in terms of lead time demand) of safety stock, then NL( y)sX is the expected number of shortages occurring during a lead time. Since a larger reorder point leads to fewer shortages, we would expect NL( y) to be a nonincreasing function of y. This is indeed the case. The function NL( y) is tabulated in Table 13. For example, NL(0) 0.3989 means that if the reorder point equals the expected lead time demand, and the standard deviation of lead time demand is sX, then an average of 0.3989sX shortages will occur during a lead time. Similarly, NL(2) 0.0085 means that if the reorder point exceeds the mean lead time demand by 2sX, then an average of 0.0085sX shortages will occur during a given lead time. NL( y) is not tabulated for negative values of y. This is because it can be shown that for y 0, NL( y) NL( y) y. For example, NL( 2) NL(2) 2 2.0085. This means that if the reorder point is 2sX less than the mean lead time demand, an average of 2.0085sX shortages will occur during each cycle. LINGO with the @PSL function may be used to compute values of the normal loss function. In LINGO, the program MODEL: x = @PSL(2); END will yield x .0085. 1 6 . 7 The EOQ with Uncertain Demand: The Service Level Approach 899 TA B L E 13 NL (x) x NL (x) x NL (x) The Normal Loss Function x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.3989 0.3940 0.3890 0.3841 0.3793 0.3744 0.3697 0.3649 0.3602 0.3556 0.3509 0.3464 0.3418 0.3373 0.3328 0.3284 0.3240 0.3197 0.3154 0.3111 0.3069 0.3027 0.2986 0.2944 0.2904 0.2863 0.2824 0.2784 0.2745 0.2706 0.2668 0.2630 0.2592 0.2555 0.2518 0.2481 0.2445 0.2409 0.2374 0.2339 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.2304 0.2270 0.2236 0.2203 0.2169 0.2137 0.2104 0.2072 0.2040 0.2009 0.1978 0.1947 0.1917 0.1887 0.1857 0.1828 0.1799 0.1771 0.1742 0.1714 0.1687 0.1659 0.1633 0.1606 0.1580 0.1554 0.1528 0.1503 0.1478 0.1453 0.1429 0.1405 0.1381 0.1358 0.1334 0.1312 0.1289 0.1267 0.1245 0.1223 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 0.1202 0.1181 0.1160 0.1140 0.1120 0.1100 0.1080 0.1061 0.1042 0.1023 0.1004 0.09860 0.09680 0.09503 0.09328 0.09156 0.08986 0.08819 0.08654 0.08491 0.08332 0.08174 0.08019 0.07866 0.07716 0.07568 0.07422 0.07279 0.07138 0.06999 0.06862 0.06727 0.06595 0.06465 0.02034 0.06210 0.06086 0.05964 0.05844 0.05726 (Continued) 900 CHAPTER 1 6 Probabilistic Inventory Models TA B L E 13 NL (x) x NL (x) x NL (x) (Continued) x 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 0.05610 0.05496 0.05384 0.05274 0.05165 0.05059 0.04954 0.04851 0.04750 0.04650 0.04553 0.04457 0.04363 0.04270 0.04179 0.04090 0.04002 0.03916 0.03831 0.03748 0.03667 0.03587 0.03508 0.03431 0.03356 0.03281 0.03208 0.03137 0.03067 0.02998 0.02931 0.02865 0.02800 0.02736 0.02674 0.02612 0.02552 0.02494 0.02436 0.02380 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 0.02324 0.02270 0.02217 0.02165 0.02114 0.02064 0.02015 0.01967 0.01920 0.01874 0.01829 0.01785 0.01742 0.01699 0.01658 0.01617 0.01578 0.01539 0.01501 0.01464 0.01428 0.01392 0.01357 0.01323 0.01290 0.01257 0.01226 0.01195 0.01164 0.01134 0.01105 0.01077 0.01049 0.01022 0.009957 0.009698 0.009445 0.009198 0.008957 0.008721 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 0.008491 0.008266 0.008046 0.007832 0.007623 0.007418 0.007219 0.007024 0.006835 0.006649 0.006468 0.006292 0.006120 0.005952 0.005788 0.005628 0.005472 0.005320 0.005172 0.005028 0.004887 0.004750 0.004616 0.004486 0.004358 0.004235 0.004114 0.003996 0.003882 0.003770 0.003662 0.003556 0.003453 0.003352 0.003255 0.003159 0.003067 0.002977 0.002889 0.002804 (Continued) 1 6 . 7 The EOQ with Uncertain Demand: The Service Level Approach 901 TA B L E 13 NL (x) x NL (x) x NL (x) (Continued) x 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 0.002720 0.002640 0.002561 0.002484 0.002410 0.002337 0.002267 0.002199 0.002132 0.002067 0.002004 0.001943 0.001883 0.001826 0.001769 0.001715 0.001662 0.001610 0.001560 0.001511 0.001464 0.001418 0.001373 0.001330 0.001288 0.001247 0.001207 0.001169 0.001132 0.001095 0.001060 0.001026 0.0009928 0.0009607 0.0009295 0.0008992 0.0008699 0.0008414 0.0008138 0.0007870 2.80 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 0.0007611 0.0007359 0.0007115 0.0006879 0.0006650 0.0006428 0.0006213 0.0006004 0.0005802 0.0005606 0.0005417 0.0005233 0.0005055 0.0004883 0.0004716 0.0004555 0.0004398 0.0004247 0.0004101 0.0003959 0.0003822 0.0003689 0.0003560 0.0003436 0.0003316 0.0003199 0.0003087 0.0002978 0.0002873 0.0002771 0.0002672 0.0002577 0.0002485 0.0002396 0.0002311 0.0002227 0.0002147 0.0002070 0.0001995 0.0001922 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 0.0001852 0.0001785 0.0001720 0.0001657 0.0001596 0.0001537 0.0001480 0.0001426 0.0001373 0.0001322 0.0001273 0.0001225 0.0001179 0.0001135 0.0001093 0.0001051 0.0001012 0.00009734 0.00009365 0.00009009 0.00008666 0.00008335 0.00008016 0.00007709 0.00007413 0.00007127 0.00006852 0.00006587 0.00006331 0.00006085 0.00005848 0.00005620 0.00005400 0.00005188 0.00004984 0.00004788 0.00004599 0.00004417 0.00004242 0.00004073 (Continued) 902 CHAPTER 1 6 Probabilistic Inventory Models TA B L E 13 NL (x) x NL (x) x NL (x) (Continued) x 3.60 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.70 3.71 3.72 3.73 3.74 0.00003911 0.00003755 0.00003605 0.00003460 0.00003321 0.00003188 0.00003059 0.00002935 0.00002816 0.00002702 0.00002592 0.00002486 0.00002385 0.00002287 0.00002193 3.75 3.76 3.77 3.78 3.79 3.80 3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89 0.00002103 0.00002016 0.00001933 0.00001853 0.00001776 0.00001702 0.00001632 0.00001563 0.00001498 0.00001435 0.00001375 0.00001317 0.00001262 0.00001208 0.00001157 3.90 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 4.00 0.00001108 0.00001061 0.00001016 0.00000972 0.000009307 0.000008908 0.000008525 0.000008158 0.000007806 0.000007469 0.000007145 Source: From R. Peterson and E. Silver, Decision Systems for Inventory and Production Planning, 1998 John Wiley & Sons, New York. Reprinted with permission. Assuming normal lead time demand, we now determine the reorder point r that will yield a desired level of SLM1 (expressed as a fraction). A reorder point of r corresponds to holding y r E(X) sX standard deviations of safety stock. Now the denition of the normal loss function implies that during a lead time, a reorder point of r will yield an expected number of shortages E(Br) given by E(Br) sXNL r E(X) sX (17) Substituting (17) into (16), we obtain the reorder point for SLM1 with normal lead time demand: sXNL 1 NL r SLM1 E(X) sX q(1 r q SLM1) sX (18) E(X) sX With the exception of r, all quantities in (18) are known. Thus, (18) and Table 13 can be used to determine the reorder point corresponding to a given level of SLM1. EXAMPLE 7 Bads, Inc. Bads, Inc., sells an average of 1,000 food processors each year. Each order for food processors placed by Bads costs $50. The lead time is one month. It costs $10 to hold a food processor in inventory for one year. Annual demand for food processors is normally 1 6 . 7 The EOQ with Uncertain Demand: The Service Level Approach 903 distributed, with a standard deviation of 69.28. For each of the following values of SLM1, determine the reorder point: 80%, 90%, 95%, 99%, 99.9%. Solution Note that E(D) 1,000, K q $50, and h $10, so 1/2 2(50)(1,000) 10 100 Also, E(X) ( 112 )(1,000) 83.33 and sX 69.28 12 20 From (18), the reorder point for an 80% value of SLM1 must satisfy NL r 83.33 20 100(1 20 0.80) 1 From Table 13, we nd that 1 exceeds any of the tabulated values of the normal loss function. Thus, the value of r must make r 83.33 a negative number. A little trial and er20 ror reveals that NL( 0.9) NL(0.9) 0.9 1.004. Hence, r 83.33 20 r 0.9 83.33 20(0.9) 65.33 For SLM1 0.90, Equation (18) shows that the reorder point must satisfy NL r 83.33 20 (1 0.90)100 20 0.5 r 83.33 20 Again, 0.5 exceeds all tabulated values of the normal loss function. Hence, be a negative number. A little trial and error reveals that N( 0.19) N(0.19) 0.5011. Thus, the reorder point for a 90% service level must satisfy r 83.33 20 r 0.19 83.33 20(0.19) 79.53 must 0.19 A 90% service level can be attained by a reorder point that is less than the expected lead time demand. To attain a 95% service level, r must satisfy NL Since NL(0.34) 0.2518, r 83.33 20 r 0.34 83.33 20(0.34) 90.13 r 83.33 20 (1 0.95)100 20 0.25 For a 99% service level, NL Since NL(1.25) r 83.33 20 (1 0.99)100 20 0.05 0.0506, we see that r 83.33 20 r 1.25 83.33 20(1.25) 108.33 904 CHAPTER 1 6 Probabilistic Inventory Models TA B L E 14 Reorder Points for Various Service Levels SLM1 Reorder Point 80% 90% 95% 99% 99.9% 65.33 79.53 90.13 108.33 127.13 Finally, for a 99.9% service level, r must satisfy r Since NL(2.19) 0.005, r 83.33 2.19 20 r 83.33 83.33 20 (1 0.999)100 20 0.005 20(2.19) 127.13 In summary, the reorder points corresponding to the various values of SLM1 are given in Table 14. Notice that to go from an 80% to a 90% service level, we must increase the reorder point by 14.20, but to go from a 90% to a 99.9% service level, the reorder point must be increased by 47.60. For higher service levels, a much greater increase in the reorder point is required to cause a commensurate increase in the service level. Using LINGO to Compute the Reorder Point Level for SLM1 Using the @PSL function in LINGO, it is a simple matter to compute the reorder point level for SLM1. For example, to compute the reorder point for Example 7 corresponding to SLM1 .90 in LINGO, we would use the program MODEL: 1) @PSL((R - 83.33)/20) = 100*(1 - SLM1)/20; 2) SLM1 = .9; This program yields r 79.57. Note that by altering the right-hand side of line 2 we can quickly compute the reorder points for various values of SLM1. Using Excel to Compute the Normal Loss Function It can be shown that NL( y) Normalloss.xls (height of normal density at y) is greater than or equal to y) y*(probability standard normal In the le Normalloss.xls, we therefore compute NL( y) with the Excel formula NORMDIST(D3,0,1,0)-D3*(1-NORMSDIST(D3)) 1 6 . 7 The EOQ with Uncertain Demand: The Service Level Approach 905 B 1 2 3 4 5 6 C D Computing Normal Loss Function y 2 NL(y) 0.008491 E FIGURE 5 FIGURE 6 B 1 2 3 4 5 6 C D Computing Normal Loss Function y 0.344868 NL(y) 0.25 E FIGURE 7 Recall that NORMDIST with last argument 0 computes the density function for a normal random variable, and NORMSDIST( ) computes the standardized normal cumulative probability. For example, we see from Figure 5 that (consistent with Table 14) NL(2) .008491. To illustrate the use of this spreadsheet, recall that in Example 7 we needed to nd a value of y such that NL( y) .25. To do this, we use Excel Goal Seek and ll in the Goal Seek dialog box as shown in Figure 6. This tells Excel to change cell D3 until cell D4 (the normal loss value) reaches .25. The result in Figure 7 shows us that NL(.345) .25. Before doing Goal Seek, you should go to Tools Options Calculation Iteration and change the Maximum Change box to a very small number, such as .0000001. This makes Excel force the Set cell within .000001 of its desired value. Determination of Reorder Point and Safety Stock Level for SLM2 Suppose that a manager wants to hold sufcient safety stock to ensure that an average of s0 cycles per year will result in a stockout. Given a reorder point of r, a fraction P(X E(D) r) of all cycles will lead to a stockout. Since an average of q cycles per year will ocP(X r)E(D) cur (remember we are assuming backlogging), an average of cycles per year q will result in a stockout. Thus, given s0, the reorder point is the smallest value of r satisfying P(X r) E(D) q s0 or P(X r) s0q E(D) 906 CHAPTER 1 6 Probabilistic Inventory Models If X is a continuous random variable, then P(X r) P(X order point r for SLM2 for continuous lead time demand, P(X r) s0q E(D) r). Thus, we obtain the re- (19) and the reorder point for SLM2 for discrete lead time demand, by choosing the smallest value of r satisfying P(X r) s0q E(D) (19 ) To illustrate the determination of the reorder point for SLM2, we suppose that Bads, Inc., wants to ensure that stockouts occur during an average of two lead times per year. Recall from Example 7 that EOQ 100, E(D) 1,000 units per year, and X is N(83.33, 400). 2(100) Now (19) yields P(X r) .2. The reorder point r is calculated using Excel. 1,000 Since NORMINV(.8,83.33,20) yields 100.16, we nd that r 100.16. The safety stock level yielding an average of two stockouts per year would be 100.16 E(X) 16.83. PROBLEMS Group A 1 For Problem 1 of Section 16.6, determine the reorder point that yields 80%, 90%, 95%, and 99% values of SLM1. What reorder point would yield an average of 0.5 stockout per year? 2 For Problem 2 of Section 16.6, determine the reorder point that yields 80%, 90%, 95%, and 99% values of SLM1. What reorder point would yield an average of two stockouts per year? 3 Suppose that the EOQ is 100, average annual demand is 1,000 units, and the lead time demand is a random variable having the distribution shown in Table 15. a What value of SLM1 corresponds to a reorder point of 25? b If we wanted to attain a 95% value of SLM1, what reorder point should we choose? c If we wanted an average of at most two stockouts per year, what reorder point should we choose? TA B L E 15 Probability 1 6 1 4 1 4 1 12 1 4 Lead Time Demand 10 15 20 25 30 4 A rm experiences demand with a mean of 100 units per day. Lead time demand is normally distributed, with a mean of 1,000 units and a standard deviation of 200 units. It costs $6 to hold one unit for one year. If the rm wants to meet 90% of all demand on time, what will be the annual cost of holding safety stock? (Assume that each order costs $50.) 16.8 (R, S ) Periodic Review Policy In this section, we describe a widely used periodic review policy: the (R, S) policy. Before describing the operation of this policy, we need to dene the concept of on-order inventory level. The on-order inventory level is simply the sum of on-hand inventory and inventory on order. Thus, if 30 units of a product are on hand, and we order 70 units (with a lead time of, say, one month), our on-order inventory level is 100. This section covers topics that may be omitted with no loss of continuity. 1 6 . 8 (R, S ) Periodic Review Policy 907 We can now describe the operation of the (R, S) inventory policy. Every R units of time (say, years), we review the on-hand inventory level and place an order to bring the onorder inventory level up to S. For example, if we were using a (.25, 100) policy, we would review the inventory level at the end of each quarter. If i 100 units were on hand, an order for 100 i units would be placed. In general, an (R, S) policy will incur higher holding costs than a cost-minimizing (r, q) policy, but an (R, S) policy is usually easier to administer than a continuous review policy. With an (R, S) policy (unlike a continuous review policy), we can predict with certainty the times when an order will be placed. An (R, S) policy also allows a company to coordinate replenishments. For example, a company could use R 1 month for all products ordered from the same supplier and then order all products from that supplier on the rst day of each month. We now assume that the review interval R has been determined and focus on the determination of a value for S that will minimize expected annual costs. Later in this section, we will discuss how to determine an appropriate value for R. We now assume that all shortages are backlogged and demand is a continuous random variable whose distribution remains unchanged over time. Finally, we assume that the per-unit purchase price is constant. This implies that annual purchasing costs do not depend on our choice of R and S. We dene R D E(D) K J h cB L DL R E(DL R) sDL R time (in years) between reviews demand (random) during a one-year period mean demand during a one-year period cost of placing an order cost of reviewing inventory level cost of holding one item in inventory for one year cost per-unit short in the backlogged case (assumed to be independent of the length of time until the order is lled) lead time for each order (assumed constant) demand (random) during a time interval of length L R mean of DL R standard deviation of DL R Given a value of R, we can now determine a value of S that minimizes expected annual costs. Our derivation mimics the derivation of (13). For a given choice of R and S, our expected costs are given by (Annual expected purchase costs) (annual review costs) (annual ordering costs) (annual expected holding costs) (annual expected shortage costs) 1 J Since R reviews per year are placed, annual review costs are given by R . Also note that whenever an order is placed, the on-order inventory level will equal S. The only way that an order will not be placed at the next review point is if DL R 0. Since DL R is a continuous random variable, DL R 0 will occur with zero probability. Thus, an order is sure to be placed at the next review point (or any review point). This implies that annual ordering cost 1 K is given by K( R ) . Observe that both the annual ordering cost and the review cost are R independent of S. Thus, the value of S that minimizes annual expected costs will be the value of S that minimizes (annual expected holding costs) (annual expected shortage costs). To determine the annual expected holding cost for a given (R, S) policy, we rst dene a cycle to be the time interval between the arrival of orders. If we can determine the expected value of the average inventory level over a cycle, then expected annual holding cost is just h(expected value of on-hand inventory level over a cycle). As in our 908 CHAPTER 1 6 Probabilistic Inventory Models derivation of (11), we now assume that the average number of back orders is small relative to the average on-hand inventory level. Then, as in Section 16.6, Expected value of I(t) expected value of OHI(t) Then expected value of I(t) over a cycle may be approximated by 0.5(expected value of I(t) right before an order arrives) 0.5(expected value of I(t) right after an order arrives). Right before an order arrives, our maximum on-order inventory level (S) has been reduced by an average of E(DL R). Thus, expected value of I(t) right before an order arrives S E(DL R). 1 Since R orders are placed each year and an average of E(D) units must be ordered each year, the average size of an order is E(D)R. Thus, Expected value of I(t) right after an order arrives Then Expected value of I(t) during a cycle Thus, Expected annual holding cost hS E(DL R) S E( DL R) E(D)R S E(DL R) E(D)R 2 E(D)R 2 From this expression, it follows that increasing S to S will increase expected annual holding costs by h . We now focus on how an increase in S to S affects expected annual shortage costs. Then we can use marginal analysis to nd the value of S that minimizes the sum of annual expected holding and shortage costs. Lets dene the shortages associated with each order to be the shortages occurring in the time interval between the arrival of the order and the arrival of the next order. For example, an order placed at time 0 arrives at time L, and the next order will not arrive until time R L. Thus, all shortages occurring between L and R L are associated with the time 0 order. Clearly, the sum of all shortages will equal the sum of the shortages associated with all orders. Lets again focus on the shortages associated with the time 0 order. Since the next order arrives at time R L, and our time 0 order brought the on-order inventory level up to S, a shortage will be associated with the time 0 order if and only if the demand between time 0 and R L exceeds S. If a shortage occurs, the magnitude of the shortage will equal DL R S. We can now use marginal analysis to determine (for a given R) the value of S that minimizes the sum of annual expected holding and shortage costs. If we increase S to S , annual expected holding costs increase by h . Increasing S to S will decrease shortages associated with an order if DL R S. Thus, for a fraction P(DL R S) of all or1 ders, increasing S to S will save cB in shortage costs. Since R orders are placed each year, increasing S to S will reduce expected annual shortage costs by 1 ( R )cB P(DL R S). Marginal analysis then implies that the value of S minimizing the sum of annual expected holding and shortage costs will occur for the value of S satisfying h or P(DL R 1 ( R )cB P(DL R S) S) Rh cB (20) Suppose that all shortages result in lost sales, and a cost of cLS (including shortage cost plus lost prot) is incurred for each lost sale. Then the value of S minimizing the sum of annual expected holding and shortage costs is given by 1 6 . 8 (R, S ) Periodic Review Policy 909 P(DL R S) Rh Rh cLS (21) The following example illustrates the use of (20). EXAMPLE 8 Lowland Appliance Lowland Appliance replenishes its stock of color TVs three times a year. Each order takes 1 year to arrive. Annual demand for color TVs is N(990, 1,600). The cost of holding one 9 color TV in inventory for one year is $100. Assume that all shortages are backlogged, with a shortage cost of $150 per TV. When Lowland places an order, what should the onorder inventory be? Solution We are given that R 1 year, L 1 year, R 3 9 4 mally distributed, with E(DL R) (990) 9 From (20), S should be chosen to satisfy P(DL R L 4 year, and cB 9 440 and sDL R ( 1 ) 100 3 .22 4 9 $150. DL 1,600 R is nor26.67. S) We use the Excel function NORMINV to compute s. Since NORMINV(0.78,440,26.67) yields 460.59, when Lowland places an order for TVs, it should order enough to bring the on-order inventory level up to 460.59 (or 461) TVs. For example, if 160 TVs are in stock when a review takes place, 461 160 301 TVs should be ordered. Determination of R Often, the review interval R is set equal to EOQ . This makes the number of orders placed E(D) per year equal the number recommended if a simple EOQ model were used to determine the size of orders. Since each order is accompanied by a review, however, we must set the cost per order to K J. This yields EOQ 2(K J)E(D) h To illustrate the idea, suppose that it costs $500 to review the inventory level and $5,000 to place an order for TVs. Then EOQ This implies a review interval R 330 990 2(5,500)(990) 100 1 3 330 year. Implementation of an (R, S) System Retail stores (such as J. C. Penneys) often nd an (R, S) policy easy to implement, because the quantity ordered equals the number of sales occurring during the period between reviews. For example, suppose a (1 month, 1,000) policy is being used, and orders are placed on the rst day of each month. If 800 items were sold during January, then an order of 800 items must be placed at the beginning of February to bring the on-order inventory level back up to 1,000. By programming a computer to set monthly orders equal to monthly sales, such a policy can easily be implemented. 910 1 6 Probabilistic Inventory Models CHAPTER PROBLEMS Group A 1 A hospital must order the drug Porapill from Daisy Drug Company. It costs $500 to place an order and $30 to review the hospitals inventory of the drug. Annual demand for the drug is N(10,000, 640,000), and it costs $5 to hold one unit in inventory for one year. Orders arrive one month after being placed. Assume that all shortages are backlogged. a Estimate R and the number of orders per year that should be placed. b Using the answer in part (a), determine the optimal (R, S) inventory policy. Assume that the shortage cost per unit of the drug is $100. 2 Chicagos Treadway Tires Dealer must order tires from its national warehouse. It costs $10,000 to place an order and $400 to review the inventory level. Annual tire sales are N(20,000, 4,000,000). It costs $10 per year to hold a tire in inventory, and each order arrives two weeks after being placed (52 weeks 1 year). Assume that all shortages are backlogged. a Estimate R and the number of orders per year that should be placed. b Using the answer in part (a), determine the optimal (R, S) inventory policy. Assume that the shortage cost is $100 per tire. 3 Suppose we have found the optimal (R, S) policy for the back-ordered case and that S 50. Is the following true or false? The optimal S for the lost sales case has S 50. 16.9 The ABC Inventory Classication System Many companies must develop inventory policies for thousands of items. In such a situation, a company cannot devote a great deal of attention to determining an optimal inventory policy for each item. The ABC classication, devised at General Electric during the 1950s, helps a company identify a small percentage of its items that account for a large percentage of the dollar value of annual sales. These items are called Type A items. Since most of the rms inventory investment is in Type A items, concentrating effort on developing effective inventory control policies for these items should produce substantial savings. Repeated studies have shown that in most companies, 5%20% of all items stocked account for 55%65% of sales; these are the Type A items. It has also been found that 20%30% of all items account for 20%40% of sales; these are called Type B items. Finally, it is often found that 50%75% of all items account for only 5%25% of sales; these are called Type C items. To illustrate how we determine which items are Type A, Type B, and Type C, consider a rm that stocks 100 items. We reorder the items as item 1, item 2, . . . , item 100, where item 1 generates the largest annual sales volume, item 2 generates the second largest annual sales volume, and so on. Then we plot the points (k, percentage of annual sales due to top k% of all items). For example, the point (20, 60) indicates that the top 20 items (from the standpoint of dollar sales) generate 60% of all sales. We then obtain a graph like Figure 8, where items 120 are Type A items, items 2140 are Type B items, and items 41100 are Type C items. Since most of our inventory investment is in Type A items, high service levels will result in huge investments in safety stocks. Therefore, Hax and Candea (1984) recommend that SLM1 be set at only 80%85% for Type A items. Tight management control of ordering procedures is essential for Type A items; individual demand forecasts should be made for each Type A item. Also, every effort should be made to lower the lead time needed to receive orders or produce the item. If an (R, S) policy is used, R should be smallperhaps one week. This enables us to keep a close watch on inventory levels. Parameters such as estimates of annual mean demand, length of lead time, standard deviation of annual demand, and shortage costs should be reviewed fairly often. For Type B items, Hax and Candea (1984) recommend that SLM1 be set at 95%. Inventory policies for Type B items can generally be controlled by computer. Parameters for 1 6 . 9 The ABC Inventory Classication System 911 Type A 100 Type B Type C Cumulative percentage of total dollar value 80 60 40 20 FIGURE 8 Example of ABC Classication of Inventory 0 20 40 60 80 100 Percentage of inventory items Type B items should be reviewed less often than for Type A items. For Type C items, a simple two-bin system is usually adequate. Parameters may be reviewed once or twice a year. Demand for Type C items may be forecast by simple extrapolation methods. A high value of SLM1 (usually 98%99%) is recommended. Little extra investment in safety stock will be required to maintain these high service levels. DEVRO Incorporated, a producer of edible sausage casings, implemented an ABC analysis of its spare parts inventory and found that 2.5% of all items (the Type A items) accounted for 49% of all dollar usage, and 24.7% of all items (the Type B items) accounted for 38% of all dollar usage. By preparing requisition forms in advance for Type A and Type B items, DEVRO was able to substantially reduce the lead time needed to obtain those items. This helped DEVRO effect substantial savings in annual inventory costs. See Flowers and ONeill (1978) for details. PROBLEMS Group A 1 Develop an ABC graph for the data in Table 16. Which items should be classied A, B, and C? TA B L E 16 Annual Usage Unit Cost (in dollars) Item 1 2 3 4 5 6 7 8 9 10 20,000 23,000 20,000 30,000 5,000 10,000 1,000 2,000 3,000 5,000 20 10 3 2 10 7 30 15 10 6 912 CHAPTER 1 6 Probabilistic Inventory Models 16.10 Exchange Curves In many situations, it is difcult to estimate holding and shortage costs accurately. Exchange curves can be used in such situations to identify reasonable inventory policies. Consider a company that stocks two items (1 and 2). Many different ordering policies are possible. For example, the company may order item 1 ve times a year and item 2 ten times a year (policy 1), or it may order each item once per year (policy 2). Clearly, policy 1 will result in higher ordering costs than policy 2, but policy 2 will result in higher holding costs and a higher average inventory level than policy 1. An exchange curve enables us to display graphically the trade-off between annual ordering costs and average inventory investment. To illustrate the construction of an exchange curve, suppose a company stocks two items (item 1 and 2), and suppose that ci h Ki qi Di Then qi 2KiDi hci cost of purchasing each unit of product i cost of holding $1 worth of either product in inventory for one year order cost for product i EOQ for product i annual demand for product i Suppose the company wants to minimize the sum of annual ordering and holding costs. D Then it should follow an EOQ policy for each product and order qi of product i i times qi per year. Two measures of effectiveness for this (or any other) ordering policy are AII AOC average dollar value of inventory cost annual ordering cost If we follow the EOQ policy for each product, then AII q1 2 1 2 2 c1 c1 c2 2 2K1D1 c1h q2 c2 2K2D2 c2h AOC 2 { K1D1c1 K2D2c2} h D1 D K1 K2 2 q1 q2 c1h c2h K1D1 K2D2 2K1D1 2K2D2 2h { K1D1c1 K2D2c2} 2 The expression for AII follows from the fact that the average inventory level of an item D equals half the order quantity. The expression for AOC follows from the fact that i orqi ders per year are placed for item i. 1 6 . 1 0 Exchange Curves 913 Since h is often hard to estimate, lets suppose h is unknown and look at how a change in h affects AII and AOC. A plot of the points (AOC, AII) associated with each value of h is known as an exchange curve. For any point on the exchange curve, we see that AII(AOC) ( 1 ){ K1D1c1 2 K2D2c2}2 (21) This shows that the exchange curve is a hyperbola. Also, any point on the exchange curve AII satises AOC 1 or AOC h. Thus, for any point on the exchange curve, the annual holdh AII ing cost per dollar of inventory is the ratio of the x-coordinate to the y-coordinate. This shows how each point on the exchange curve can be identied with a value of h. We now illustrate the computation of an exchange curve and show how the exchange curve can be used as an aid in decision making. EXAMPLE 9 Exchange Curve A company stocks two products. Relevant information is given in Table 17. 1 2 Draw an exchange curve. Currently, the company is ordering each product ten times per year. Use the exchange curve to demonstrate to management that this is an unsatisfactory ordering policy. 3 Suppose that management limits the companys average inventory investment to $10,000. Use the exchange curve to determine an appropriate ordering policy. Solution 1 From (21), we nd the equation of the exchange curve to be (AII)(AOC) ( 1 ){ 50(10,000)(200) 2 72,000,000 80(20,000)(2.5)}2 Some representative points on the exchange curve, along with the associated value of h, are given in Table 18. The exchange curve is graphed in Figure 9. 2 If the company orders each product ten times per year, AOC AII 10($50) 10($80) $1,300 1 1 (1,000)($200) (2,000)($2.50) 2 2 $102,500 TA B L E 17 Ki Di ci Relevant Information for Example 9 Product 1 Product 2 $50 $80 10,000 20,000 $200 $2.50 TA B L E 18 All h Points on Exchange Curve AOC $2,000 $3,000 $4,000 $5,000 $6,000 $8,000 $36,000 $24,000 $18,000 $14,400 $12,000 $9,000 .06 .13 .22 .35 .50 .89 914 CHAPTER 1 6 Probabilistic Inventory Models 110 C 100 90 80 All (thousands) 70 60 50 40 30 20 10 FIGURE D B A 9 Example of an Exchange Curve 0 702.44 1,300 2,000 3,000 AOC 4,000 5,000 6,000 8,000 This is point A in Figure 9. Observe that point B (1,300, 55,385), corresponding to h .02, yields the same AOC as the current policy, but a much lower AII. Also, point C (702.44, 102,500), corresponding to h .01, yields the same AII as the current policy, but a much lower AOC. Thus, we can use the exchange curve to show the manager how to improve on the current ordering policy. From the exchange curve, we nd that D (7,200, 10,000) is on the exchange curve. Thus, for a $10,000 AII, the best we can do is to hold ordering costs to $7,200. Of course, the manager could opt for AII $9,000 and AOC $8,000 or one of many other possibilities. The point is that the exchange curve claries many of the options available to management. 3 Exchange Curves for Stockouts Exchange curves can also be used to assess the trade-offs between average inventory investment (AII) and the expected number of lead times per year resulting in stockouts. To illustrate, consider a company stocking a single item for which c K h cB E(D) q X E(X) sX r purchase cost per unit setup cost annual cost of holding one unit in inventory cost of a stockout (we assume all items are back-ordered) mean annual demand economic order quantity lead time demand mean lead time demand standard deviation of lead time demand reorder point (determined from Equation (13)) From (13), a fraction qh cBE(D) 1 6 . 1 0 Exchange Curves 915 of all lead times will have a stockout. Since there are an average of year, an average of E(D) q qh cBE(D) h cB E(D) q orders placed per lead times per year will result in stockouts. We let SY expected number of lead times per year resulting in stockouts. From (11), we know that the average inventory level is ( q 2 q r E(X)). Thus, we have AII c( 2 r E(X)). An exchange curve for this situation is a graph of the points (AII, SY) corresponding to different values of cB. To illustrate the construction of an exchange curve, let E(X) 200, sX 50, E(D) 100,000, K $12.50, h $10, and c $100. We will nd four points on the exchange curve by setting cB $1, $5, $10, and $20. First we nd that q 2(12.5)(100,000) 10 500 The stockout probabilities and SY are given in Table 19. Using Table 2 in Chapter 12 or the Excel NORMSDIST( ) function, we can calculate the reorder point r for each value of cB. Then we determine the average inventory level and AII average inventory investment. These calculations are given in Table 20. The exchange curve (based on the four points we have computed) is graphed in Figure 10. For example, the exchange curve shows us that if current AII is $33,250, then for a $3,400 increase in AII, we can reduce SY from 10 to 2, but an additional increase in AII of $3,400 would decrease SY by less than 2. Exchange Surfaces Using more sophisticated techniques (see Gardner and Dannenbring (1979)), an exchange surface involving three or more quantities can be derived. The exchange surface in Figure 11 was derived from a sample of 500 items in a military distribution system. The TA B L E 19 qh cBE (D) h cB Computation of SY cB Stockout Probability 500(10) 1(100,000) 500(10) 5(100,000) 500(10) 10(100,000) 500(10) 20(100,000) SY 10 1 10 5 10 10 10 20 $1 $5 $10 $20 .05 .01 .005 .0025 10 2 1 0.50 TA B L E 20 Reorder Point Average Inventory Level AII Calculation of AII cB $1 $5 $10 $20 200 200 200 200 50(1.65) 50(2.33) 50(2.58) 50(2.81) 282.5 316.5 329.5 340.5 250 250 250 250 282.5 316.5 329.5 340.5 200 200 200 200 332.5 366.5 379.5 390.5 $33,250 $36,650 $37,900 $39,050 916 CHAPTER 1 6 Probabilistic Inventory Models 10 9 8 Stockouts per year FIGURE 7 6 5 4 3 2 1 10 Exchange Curve for AII and SY 0 33 34 35 36 37 38 39 Average inventory investment (thousands) 6.31% Percentage of requisitions short 4.6 6 3.9 5 3.7 1 3.5 5 4.09 2.59 3.15 1.85 1.73 1.28 1.17 1.0 1 2.55 2.39 3.4 2 900 1,000 0.75 1.42 0.69 1,200 0.60 1,5 0 2,0 0 00 2,5 00 3,0 00 3,5 86 1,367 9,0 42 FIGURE 11 Example of an Exchange Surface Workload in annual orders x-coordinate is the annual number of orders placed, the y-coordinate is the average inventory investment (in thousands of dollars), and the z-coordinate is the percentage of requests that yield shortages. For example, suppose the military has xed a $900,000 average inventory investment. By varying the number of orders per year between 1,500 and 9,042, the military can vary the percentage of requests that yield shortages between 6.31% and 3.42%. Also, if annual orders are xed at 3,000, then the percentage of requests yielding shortages can vary between 0.75% and 3.71%. An exchange surface makes it easy to identify the trade-offs involved between improving service, increased inventory investment, and increased work load (orders per year). Reprinted by permission of E. Gardner and D. Dannenbring, Using Optimal Policy Surfaces to Analyze Aggregate Inventory Tradeoffs, Management Science, Vol. 25, No. 8, August 1979. Copyright 1979, the Institute of Management Sciences. 1 6 . 1 0 Exchange Curves In (in ves tm t of hou ent do san lla ds rs) 1,100 0.82 917 PROBLEMS Group A 1 Consider a two-item inventory system with the attributes in Table 21. a Draw an exchange curve for these products (use AOC and AII as the x- and y-coordinates). b Currently, management is ordering each product twice a year. How can it improve on this strategy? c The order costs correspond to machine setup times. Machine time is valued at $50 per hour. If management wants to limit machine setup time to 500 hours per year, what strategies are available? 2 Explain how to draw an exchange curve where the xcoordinate is AII and the y-coordinate is percentage of all requests for stock that result in shortages. 3 Consider the exchange surface in Figure 11. The current inventory policy has yielded 3,586 orders per year, an AII of $1,367,000, and 0.89% shortages. a Without changing orders per year and AII, by how much can shortages be improved? b If AII and shortages are maintained at current levels, by how much can orders per year be reduced? c If shortages and orders per year are maintained at current levels, by how much can AII be reduced? TA B L E 21 Ki Di ci Product 1 Product 2 $500 $800 10,000 20,000 $2,000 $250 SUMMARY Single-Period Decision Models A decision maker begins by choosing a value q of a decision variable. Then a random variable D assumes a value d. Finally, a cost c(d, q) is incurred. The decision makers goal is to choose q to minimize expected cost. News Vendor Problem If c(d, q) has the structure c(d, q) c(d, q) coq (terms not involving q) cuq (terms not involving q) (d (d q) q (2) 1) (2.1) the single-period decision model is a news vendor problem. Here co cu per-unit overstocking cost per-unit understocking cost If D is a discrete random variable, the optimal decision is given by the smallest value of q (q*) satisfying F(q*) cu co cu (3) If D is a continuous random variable, the optimal decision is the value of q (q*) satisfying P(D q*) cu co cu (5) 918 CHAPTER 1 6 Probabilistic Inventory Models Determination of Reorder Point and Order Quantity with Uncertain Demand: Minimizing Annual Expected Cost Let K h L q D cB cLS X Then E(X) LE(D), var X L (var D), sX LsD ordering cost holding cost/unit/year lead time for each order (assumed to be known with certainty) order quantity random variable representing annual demand, with mean E(D), variance var D, and standard deviation sD cost incurred for each unit short if shortages are backlogged cost (including lost prots, lost goodwill) incurred for each lost sale if each shortage results in a lost sale random variable representing lead time demand and r is the reorder point, or inventory level at which an order should be placed. Safety stock, r E(X), is the amount of inventory held in excess of lead time demand to meet shortages that may occur before an order arrives. Assume that the optimal order quantity can be reasonably approximated by the EOQ, D is a continuous random variable, and all shortages are backlogged. Then annual expected cost is minimized by q* and r* given by q* P(X r*) 2KE(D) h hq* cBE(D) 1/2 (13) Assume that the optimal order quantity can be reasonably approximated by the EOQ, D is a continuous random variable, and all shortages result in lost sales. Then annual expected cost is minimized by q* and r* satisfying q* P(X r*) 2KE(D) 1/2 h hq* hq* cLSE(D) (15) Determination of Reorder Point: The Service Level Approach Since it may be difcult to determine the exact cost of a shortage or lost sale, it is often desirable to choose a reorder point that meets a desired service level. Two common measures of service level are Service Level Measure 1 SLM1, the expected fraction (usually expressed as a percentage) of all demand that is met on time. Service Level Measure 2 SLM2, the expected number of cycles per year during which a shortage occurs. Summary 919 If lead time is normally distributed, then for a desired value SLM1, the reorder point r is found from NL r E(X) sX q(1 SLM1) sX (18) where NL( y) is the normal loss function, tabulated in Table 13, and q is the EOQ. If lead time demand is a continuous random variable, and we desire SLM2 s0 shortages per year, the reorder point r is given by P(X r) s0q E(D) (19) Again, q is the EOQ. If lead time demand is a discrete random variable, and we desire SLM2 per year, the reorder point is the smallest value of r satisfying P(X Again, q is the EOQ. r) s0q E(D) s0 shortages (19 ) (R, S) Periodic Review Policy Every R units of time, we review the inventory level and place an order to bring our onhand inventory level up to S. Given a value of R, we determine the value of S from P(DL R S) Rh cB ABC Classication The 5%20% of all items accounting for 55%65% of sales are Type A items; the 20%30% of all items accounting for 20%40% of sales are Type B items; and the 50%75% of all items that account for 5%25% of all sales are Type C items. By concentrating effort on Type A (and possibly Type B) items, we can achieve substantial cost reductions. Exchange Curves Exchange curves (and exchange surfaces) are used to display trade-offs between various objectives. For example, an exchange curve may display the trade-off between annual ordering costs and average dollar level of inventory. An exchange curve can be used to compare how various ordering policies compare with respect to several objectives. REVIEW PROBLEMS Group A 1 The Chocochip Cookie Store bakes its cookies every morning before opening. It costs the store 15 to bake each cookie, and each cookie is sold for 35. At the end of the day, leftover cookies may be sold to a thrift bakery for 5 per cookie. The number of cookies sold each day is described by the discrete random variable in Table 22. 920 CHAPTER 1 6 Probabilistic Inventory Models TA B L E 22 Probability TA B L E 23 Probability Demand (dozens) Cash Needs 20 30 40 50 60 .30 .20 .20 .15 .15 $4,000 $5,000 $6,000 $7,000 $8,000 .30 .20 .10 .30 .10 Group B a How many dozen cookies should be baked before the store opens? b If the daily demand (in dozens) for cookies is N(50, 400), how many dozen cookies should be baked? A description of the N(m, s 2) notation can be found in Section 1.7. c If the daily demand (in dozens) for cookies has a density function e d/50 f (d) (d 0) 50 how many dozen cookies should be baked? 2 An optometrist orders eyeglass frames at a cost of $40 per frame and sells each frame for $70. Annual holding cost is 20% of the optometrists cost of purchasing a frame. Each time frames are ordered, a cost of $200 is incurred. Because of lost goodwill, a cost of $50 is incurred each time a customer wants a frame that is not in stock. Frames are delivered one week after an order is placed. Annual demand for frames is N(1,040, 15.73). a Assuming all shortages are backlogged, determine the order quantity and reorder point. b Assuming all shortages result in lost sales, determine the order quantity and reorder point. c To meet 95% of all orders from stock, what should be the reorder point? d To have shortages occur during an average of two lead times per year, what should be the reorder point? 3 We are given the following information about a product: Cost of placing an order $100 Cost per item $5 Sale price per item $8 Annual holding cost 40% of cost of item Annual demand 5,000 units Lead time demand N(20, 900) a If the reorder point that minimizes expected cost is 80, what is the shortage cost? (Assume backlogging.) b If the reorder point that minimizes expected cost is 80, what is the shortage cost? (Assume lost sales.) c What reorder point would meet 90% of all demand on time? d What reorder point would result in a stockout occurring during an average of 0.5 lead time per year? 4 A business believes that its needs for cash during the next month are described by the random variable shown in Table 23. At the beginning of the month, the business has $10,000 available, and the business manager must determine how much of the money should be placed in an account bearing 24% annual interest. If any money must be withdrawn before the end of the month, all interest on the withdrawn money is forfeited, and a penalty equal to 2% of the withdrawn money must be paid. How much money should be placed in the 24% annual interest account? 5 A fur dealer buys fur coats for $100 each and sells them for $200 each. He believes that the demand for coats is N(100, 100). Any coat not sold can be sold to a discount house for $100, but the fur dealer believes he must charge himself a cost of 10 per dollar invested in a fur coat that is sold at discount. How many coats should the dealer order? If the price at which the dealer sold his coats increased (assuming demand is unchanged), would he buy more or fewer coats? 6 A company currently has two warehouses. Each warehouse services half the companys demand, and the annual demand serviced by each warehouse is N(10,000, 1,000,000). The lead time for meeting demand is 110 year. The company wants to meet 95% of all demand on time. Assume that the EOQ at each warehouse is 2,000. a How much safety stock must be held? b Show that, if the company had only one warehouse, it would hold less safety stock than it does when it has two warehouses. c A young MBA argues, By having one central warehouse, I can reduce the total amount of safety stock needed to meet 95% of all customer demands on time. Therefore, we can save money by having only one central warehouse instead of several branch warehouses. How might this argument be rebutted? 7 Use LINGO to determine the values of q and r that minimize expected annual cost for Example 5. How close are your answers to those given in the text? Review Problems 921 REFERENCES The following references emphasize applications over theory: Brown, R. Decision Rules for Inventory Management. New York: Holt, Rinehart and Winston, 1967. Peterson, R., and E. Silver. Decision Systems for Inventory Management and Production Planning. New York: Wiley, 1998. Tersine, R. Principles of Inventory and Materials Management. New York: North-Holland, 1982. Vollman, T., W. Berry, and C. Whybark. Manufacturing Planning and Control Systems. Homewood, Ill.: Irwin, 1997. The following references contain extensive theoretical discussions as well as applications: Hadley, G., and T. Whitin. Analysis of Inventory Systems. Englewood Cliffs, N.J.: Prentice Hall, 1963. Hax, A., and D. Candea. Production and Inventory Management. Englewood Cliffs, N.J.: Prentice Hall, 1984. Johnson, L., and D. Montgomery. Operations Research in Production, Scheduling, and Inventory Control. New York: Wiley, 1974. Barron, H. Payoff Matrices Pay Off at Hallmark, Interfaces 15(no. 4, 1985):2025. Bruno, J. The Use of Monte-Carlo Techniques for Determining the Size of Substitute Teacher Pools, SocioEconomic Planning Science 4(1970):415428. Flowers, D., and J. ONeill. An Application of Classical Inventory Analysis to a Spare Parts Inventory, Interfaces 8(no. 2, 1978):7679. Rosenfeld, D. Optimal Management of Tax-Sheltered Employment Reimbursement Programs, Interfaces 16(no. 3, 1986):6872. Virts, J., and R. Garrett. Weighting Risk in Capacity Expansion, Harvard Business Review 48(1970). For a discussion of exchange curves and surfaces, see: Gardner, E., and D. Dannenbring. Using Optimal Policy Surfaces to Analyze Aggregate Inventory Tradeoffs, Management Science 25(1979):709720. 922 CHAPTER 1 6 Probabilistic Inventory Models ... View Full Document

End of Preview

Sign up now to access the rest of the document