# Corporate

CD 19-1

CHAPTER 19 INVENTORY MANAGEMENT WITH UNCERTAIN DEMAND

Learning Objectives:

After completing this chapter, you should be able to

1. 2. 3. 4. 5. 6. 7. 8. 9.

Identify some situations where the demand for withdrawing a product from inventory is uncertain. Describe the trade-off that must be considered when developing an inventory policy for a product with uncertain demand. Explain why perishable products with uncertain demand and stable products with uncertain demand require different kinds of inventory models. Describe an inventory model for perishable products with uncertain demand. Apply this model to find the optimal order quantity. Describe a continuous-review inventory model for stable products with uncertain demand. Apply this model to choose an order quantity. Apply this model to determine the inventory level at which an order should be placed with this order quantity. Describe some large inventory systems that arise in practice.

We now continue the focus of the preceding chapter on inventory management, but with one key difference. We have been assuming that the product under consideration in inventory has a known demand, i.e., that we can predict with reasonable certainty when units will need to be withdrawn from inventory. We drop this assumption in this chapter, so we now will consider products with an uncertain demand. The predictability of demand depends greatly on the situation. We certainly have known demand when the product is being withdrawn from inventory at a fixed rate because it actually is one of several components being assembled into a larger product on an assembly line. Similarly, a manufacturer has a known demand for a custom product in inventory when it is producing the product (replenishing inventory) only to satisfy a schedule of orders already received from a particular customer. A wholesaler also has roughly a known demand for a product after its retail customers have developed a well established pattern for purchasing the product month after month. These are the types of situations considered in the preceding chapter.

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By contrast, a retail store manager does not have the luxury of knowing when customers will come in to purchase a given product. If the product is a new one, predicting how well it will catch on may be particularly difficult. Similarly, a wholesaler supplying a number of retailers with a new product may have considerable uncertainty about what the demand will be. Sales can fluctuate widely from one month to the next. Consequently, a manufacturer selling the product to a number of wholesalers (perhaps in competition with other manufacturers) also can have significant uncertainty about the demand. We are considering these kinds of situations in this chapter. Even with uncertainty, it is necessary to make some kind of forecast of the expected demand and what the variability might be. For example, you might use something like the PERT three-estimate approach described in Section 16.4 (making a most likely estimate, an optimistic estimate, and a pessimistic estimate, and then converting these estimates into a probability distribution). In some way, the forecast should be expressed in probabilistic terms. The probabilities might be quite subjective in nature, as with the typical prior probabilities of decision analysis discussed in Chapter 9, or they might be based on considerable historical experience and data. At any rate, the models in this chapter assume that an estimate has been made of the probability distribution of what the demand will be over a given period. A very important consequence of uncertain demand is the great risk of incurring shortages unless the inventory is managed carefully. An order to replenish the inventory needs to be placed while some inventory still remains, because of the lag until the order can be filled. Even the amount of lead time needed to fill the order may be uncertain. However, if too much inventory is replenished too soon, a heavy price is paid because of the high cost of holding a large inventory. A constant theme throughout the chapter is the need to find the best trade-off between the consequences of having too much inventory and of having too little. We will separately discuss inventory management for two types of products. One type is a perishable product, which can be carried in inventory for only a very limited period of time before it can no longer be sold. The second type is a stable product, which will remain sellable indefinitely. These two types need to be handled quite differently. The first two sections present a case study and then a general model for perishable products. Sections 19.3 and 19.4 discuss a case study that involves a stable product. The inventory model that underlies this case study is summarized in Section 19.5. Section 19.6 describes the large inventory systems that commonly arise in practice, including massive systems that have been installed at IBM and Hewlett-Packard.

19.1 A Case Study for Perishable Products — Freddie the Newsboy’s Problem

The problem being addressed in this case study actually is a simplified version of the one for the case study introduced in Section 13.1. For ease of exposition, we have simplified the probability distribution of the demand here by having only three possible values for the demand instead of the 31 considered previously. The technique used in Chapter 13 to analyze this problem was computer simulation with Crystal Ball. In this section and the next, we will develop an inventory model to address the same problem in a relatively straightforward way. For completeness, we repeat the description of the problem below.

Freddie’s Problem

This problem concerns a newsstand in a prominent downtown location of a major city. The newsstand has been there longer than most people can remember. It has always been run by a well known character named Freddie. (Nobody seems to know his last name.) His many customers refer to him affectionately as Freddie the newsboy, even though he is considerably older than most of them.

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Freddie sells a wide variety of newspapers and magazines. The most expensive of the newspapers is a large national daily called the Financial Journal. Our case study involves this newspaper. The day’s copies of the Financial Journal are brought to the newsstand early each morning by a distributor. Any copies unsold at the end of the day are returned to the distributor the next morning. (This is indeed a perishable product). However, to encourage ordering a large number of copies, the distributor does give a small refund for unsold copies. Here are Freddie’s cost figures. Freddie pays $1.50 per copy delivered. Freddie sells it at $2.50 per copy. Freddie’s refund is $0.50 per unsold copy. Partially because of the refund, Freddie always has taken a plentiful supply. However, he has become concerned about paying so much for copies that then have to return unsold, particularly since this has been occurring nearly every day. He now thinks he might be better off by ordering only a minimal number of copies and saving this extra cost. To investigate this further, Freddie has been keeping a record of his daily sales. In contrast to the numbers presented in Chapter 13, we assume now that this is what he has found. Freddie sells 9 copies on 30% of the days. Freddie sells 10 copies on 40% of the days. Freddie sells 11 copies on 30% of the days. So how many copies should Freddie order from the distributor per day? (Think about it before reading on.)

Applying Bayes’ Decision Rule to Freddie’s Problem

One approach to this problem is to apply decision analysis as described in Chapter 9. Specifically, Bayes’ decision rule introduced in Section 9.2 is used as outlined below. The procedure involves filling out the payoff table shown in Figure 19.1. Column B lists the decision alternatives that deserve consideration, namely, to order 9, 10, or 11 copies per day from the distributor. For each of these alternatives, Freddie’s profit on a given day is determined by how many requests to purchase a copy of the Financial Journal occur that day, so these possible numbers of purchase requests (the possible states of nature) are listed in cells D4:F9. The relative likelihood of these numbers of purchase requests are entered in PriorProbability (D9:F9) as the prior probabilities of these states of nature.

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A 1 2 3 4 5 6 7 8 9 B C D E F G H

Bayes' Decision Rule (with Profits) for Freddie's Problem

Payoff Table Alternative (Copies Ordered) 9 10 11 State of Nature (Purchase Requests) 9 10 11 $9 $9 $9 $8 $10 $10 $7 $9 $11 0.3 0.4 0.3

H 3 4 5 6 7

Expected Payoff $9.00 $9.40 $9.00

Prior Probability

Range Name ExpectedPayoff Payoff10 Payoff11 Payoff9 PayoffTable PriorProbability

Cells H5:H7 D6:F6 D7:F7 D5:F5 D5:F7 D9:F9

Expected Payoff =SUMPRODUCT(PriorProbability,Payoff9) =SUMPRODUCT(PriorProbability,Payoff10) =SUMPRODUCT(PriorProbability,Payoff11)

Figure 19.1

This Excel template in your MS Courseware shows that Freddie the newsboy maximizes his expected profit by ordering 10 copies each day.

The payoff from Freddie’s decision, given the state of nature, is the profit for that day. This profit is Profit = sales income - purchase cost + refund. For example, if Freddie orders 11 copies from the distributor and the state of nature turns out to be 9 for that day (9 copies are sold), his profit is Profit = 9 ($2.50) - 11 ($1.50) + 2 ($0.50) = $7.00. Calculating the profit in this way for each of the combinations of a decision alternative and a possible state of nature yields the payoff table shown in cells B3:F9 of Figure 19.1. Applying Bayes’ decision rule with this Excel template involves calculating the expected payoff (EP) for each alternative by using the indicated equations entered into ExpectedPayoff (H5:H7). (If you haven’t studied this topic in Chapter 9, note that each of these equations is simply calculating the statistical average of the payoffs.) The rule then selects the alternative with the largest expected payoff ($9.40). Conclusion: Freddie’s most profitable alternative in the long run is to order 10 copies, since this will provide an average daily profit of $9.40, versus $9.00 for either of the other alternatives. In the next section, you will see a shortcut for drawing this same conclusion.

REVIEW QUESTIONS

1. 2. 3. What is the trade-off that Freddie the newsboy should consider in making his decision? Why are Freddie’s decision alternatives limited to ordering 9, 10, or 11 copies? What is the state of nature when using decision analysis to formulate Freddie’s problem? Why?

CD 19-5 19.2 An Inventory Model for Perishable Products

Freddie the newsboy’s problem illustrates an application of a widely used inventory model for perishable products. Newspapers are just one of the many types of such products to which it can be applied. After summarizing its assumptions and applying Bayes’ decision rule, we will show you a shortcut for solving the model and then describe the various types of perishable products.

The Assumptions of the Model

1. 2. 3. 4. 5. Each application involves a single perishable product. Each application involves a single time period because the product cannot be sold later. However, it will be possible to dispose of any units of the product remaining at the end of the period, perhaps even receiving a salvage value for the units. The only decision to be made is how many units to order (the order quantity) so they can be placed into inventory at the beginning of the period. The demand for withdrawing units from inventory to sell them (or for any other purpose) during the period is uncertain. However, the probability distribution of demand is known (or at least estimated). If the demand exceeds the order quantity, a cost of underordering is incurred. In particular, the cost for each unit short is Cunder = unit cost of underordering = decrease in profit that results from failing to order a unit that could have been sold during the period. 7. If the order quantity exceeds the demand, a cost of overordering is incurred. In particular, the cost for each extra unit is Cover = unit cost of overordering = decrease in profit that results from ordering a unit that could not be sold during the period. These assumptions certainly fit Freddie the newsboy’s problem. The day’s newspaper of concern (the Financial Journal) is a single perishable product that cannot be sold after the day (the single time period), although it can be returned to the distributor for a small refund (the salvage value). Freddie’s only decision is how many copies to order from the distributor for each day, given the probability distribution of how many can be sold (shown in row 9 of Figure 19.1). The last two assumptions also fit, since the definitions of the two unit costs imply that Cunder = unit sale price - unit purchase cost = $2.50 - $1.50 = $1.00, Cover = unit purchase cost - unit salvage value

6.

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= $1.50 - $0.50 = $1.00 for this problem. These expressions for Cunder and Cover will fit any analogous problem where the only cost factors are the unit sale price, unit purchase cost, and unit salvage value. However, the definitions of Cunder and Cover have been expressed more generally as a decrease in profit in order to fit other situations as well. For example, if there is a concern about losing future business due to ill will caused by underordering, then a unit cost of ill will could be added to the expression for Cunder. Similarly, Cover might include an additional term for something like the extra holding cost associated with storing a unit all the way to the end of the period. Another possibility is that it might be necessary to add a unit disposal cost rather than having a refund to subtract. As in Freddie’s problem, it often is not feasible to place and receive an additional order before the period ends if a shortage occurs. However, if it is very important to fill all the demand, arrangements sometimes can be made to do this at an extra cost. In this case, Cunder would equal the extra cost per unit of placing this emergency order plus any reduction in the unit selling price to pacify the customers who had to wait. A key part of assumptions 6 and 7 is that Cunder must be the same for each unit short and Cover must be the same for each extra unit. One way of solving this model is to apply Bayes’ decision rule, as described in a separate box. Another simple method is presented next.

CD 19-7 Applying Bayes’ Decision Rule

Any application of this model can be solved in much the same way as we solved Freddie the newsboy’s problem in the preceding section, namely, by applying Bayes’ decision rule (first introduced in Section 9.2). One alternative is to express the payoffs in terms of profits and then proceed as in Figure 19.1. However, in light of assumptions 6 and 7, a completely equivalent but more direct approach is to focus on just the costs of underordering and overordering. This is what is done in Figure 19.2 for Freddie’s problem.

A 1 2 3 4 5 6 7 8 9

B

C

D

E

F

G

H

Bayes' Decision Rule (with Costs) for Freddie's Problem

Payoff Table Alternative (Copies Ordered) 9 10 11 State of Nature (Purchase Requests) 9 10 11 $0 $1 $2 $1 $0 $1 $2 $1 $0 0.3 0.4 0.3 Expected Payoff $1.00 $0.60 $1.00

Prior Probability

Figure 19.2 This Excel template finds that Freddie the newsboy minimizes his expected cost of underordering or overordering by ordering 10 copies each day. Note that the numbers in cells D5, E6, and F7 are 0, because the order quantity equals the demand in these cases. The costs (in dollars) in cells E5, F5, and F6 are solely the cost of underordering and those in cells D6, D7, and E7 are solely the cost of overordering. The expected cost for each alternative is calculated in column H with the equations given in Figure 19.1. Since we wish to minimize expected cost, the equations in column I now use the MIN function instead of the MAX function, so the conclusion again is that Freddie should order 10 copies. The conclusion here must be the same as in Figure 19.1 since the two approaches are equivalent. All we have done here is eliminate the revenues and cost factors that don’t affect the decision and focus on just those that do, namely, the cost of underordering and the cost of overordering. Both approaches are applying Bayes’ decision rule, but with different payoffs, where one is to be maximized and the other minimized.

A Simple Formula for Solving the Model

The drawback with relying on Bayes’ decision rule as the method for solving the model is that most applications involve many more decision alternatives and states of nature than Freddie’s problem formulated in Figures 19.1 and 19.2. In fact, if Freddie were to apply the same approach to another more popular newspaper where the number of copies sold in a day range from 100 to 200, he then would have 101 decision alternatives and 101 states of nature to consider. Other applications might have thousands. Using Bayes’ decision rule to deal with such large problems would be extremely cumbersome.

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A Quicker Approach: Fortunately, a much quicker way to solve problems of any size has been found. It involves using the service level, defined as follows: Service level = probability that no shortage will occur. A shortage occurs when the demand for the product exceeds the number of units available in inventory, so one or more customers suffer the disappointment of not immediately obtaining the units they wanted. Therefore, the probability of avoiding a shortage is a key measure of the level of service being provided to the customers. Given the prior probabilities in cells D9:F9 of Figure 19.2, the service levels for Freddie’s three alternatives are Service level if order 9 copies = D9 = 0.3,

Service level if order 10 copies = D9 + E9 = 0.3 + 0.4 = 0.7, Service level if order 11 copies = D9 + E9 + F9 = 0.3 + 0.4 + 0.3 = 1. Here is the simple formula for solving this model.

Ordering Rule for the Model for Perishable Products

1. 2.

Optimal service level =

Cunder . Cunder + Cover

Choose the smallest order quantity that provides at least this service level.

Since Cunder = $1.00 and Cover = $1.00 for Freddie’s problem, this formula gives Optimal service level =

$1.00 = 0.5. $1.00 + $1.00

Referring to the service levels for Freddie’s three alternative order quantities, the smallest one that provides at least this optimal service level is to order 10 copies. This of course is the same answer as provided by Bayes’ decision rule. An Excel template is available in your MS Courseware for applying this model for perishable products to any situation where, as for Freddie’s problem, the only cost factors are the unit sale price, unit purchase cost, and unit salvage value. As illustrated in Figure 19.3 for Freddie’s problem, all you need to do is enter these three cost factors. The template then calculates Cover, Cunder, and the optimal service level. What remains is for you to use the probability distribution of demand for your specific problem to apply step 2 of the ordering rule.

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A 1 2 3 4 5 6 B C D E F

Optimal Service Level for Perishable Products

Unit Sales Price Unit Purchase Cost Unit Salvage Value Data $2.50 $1.50 $0.50 Cost of Overordering Cost of Underordering Optimal Service Level Results $1.00 $1.00 0.5

4 5 6

E F Cost of Overordering =UnitPurchaseCost-UnitSalvageValue Cost of Underordering =UnitSalesPrice-UnitPurchaseCost Optimal Service Level =CostOfUnderordering/(CostOfUnderordering+CostOfOverordering)

Cell F4 F5 F6 C5 C4 C6

Range Name CostOfOverordering CostOfUnderordering OptimalServiceLevel UnitPurchaseCost UnitSalesPrice UnitSalvageValue

Figure 19.3

The Excel template for the inventory model for perishable products in your MS Courseware applies step 1 of the ordering rule, as illustrated here for Freddie’s problem.

APPLYING STEP 2 OF THE ORDERING RULE GRAPHICALLY: For any given order quantity, the definition of service level can be restated in another equivalent way as Service level = probability that the demand is less than or equal to the order quantity = P (demand ≤ order quantity). The probability on the right for each of Freddie’s three alternative order quantities is P (demand ≤ 9) P (demand ≤ 10) P (demand ≤ 11) = 0.3, = 0.7, = 1.

Figure 19.4 shows a graph where the horizontal axis is x and the vertical axis is P (demand ≤ x). Since demand is a random variable, this graph is referred to as the cumulative distribution function (or CDF for short) of demand. The point at which the optimal service level of 0.5 (see the horizontal dashed line) hits this CDF gives the optimal order quantity of 10.

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The figure enables visualizing the ordering rule graphically. Furthermore, on larger problems, this graphical approach may find the optimal order quantity more quickly than enumerating the service levels for all the alternatives. This is illustrated by the following example.

A VARIATION OF FREDDIE’S PROBLEM:

Freddie now wishes to find the optimal order quantity for another of his newspapers. This is a more popular newspaper whose daily sales range from 100 copies to 200 copies, with roughly

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equal probabilities over this range. In this case, the relevant unit costs are Cunder = $0.75 and Cover = $0.25. Since the probabilities of the various possible demands from 100 to 200 are roughly equal, a good estimate of the probability distribution of demand is the uniform distribution from 100 to 200. The solid lines in Figure 19.5 show the CDF of this distribution.

With the given unit costs, the ordering rule says that Optimal service level =

Cunder $0.75 = = 0.75. Cunder + Cover $0.75 + $0.25

The corresponding dashed lines in the figure show that the optimal order quantity is 175.

Some Types of Perishable Products

The model presented in this section has traditionally been called the newsboy problem1 because it fits the problems of newsboys like Freddie so well. However, it has always been recognized that the model is just

1

Recently, some writers have been substituting the name, newsvendor problem. Other names include the single-period probabilistic model and single-period stochastic model.

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as applicable to other perishable products as to newspapers. In fact, most of the applications have been to perishable products other than newspapers. As you read through the list below of various types of perishable products, think about how the inventory management of such products is analogous to Freddie’s problem since these products also cannot be sold after a single time period. All that may differ is that the length of this time period may be a week, a month, or even several months rather than just one day. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Periodicals, such as newspapers and magazines. Flowers being sold by a florist. The makings of fresh food to be prepared in a restaurant. Produce, including fresh fruits and vegetables, to be sold in a grocery store. Christmas trees. Seasonal clothing, such as winter coats, where any goods remaining at the end of the season must be sold at highly discounted prices to clear space for the next season. Seasonal greeting cards. Fashion goods that will be out of style soon. New cars at the end of a model year. Any product that will be obsolete soon. Vital spare parts that must be produced during the last production run of a certain model of a product (e.g., an airplane) for use as needed throughout the lengthy field life of that model. Reservations provided by an airline for a particular flight. Reservations provided in excess of the number of seats available (overbooking) can be viewed as the inventory of a perishable product (they cannot be sold after the flight has occurred), where the demand then is the number of no-shows. With this interpretation, the cost of underordering (too little overbooking) would be the lost profit from empty seats and the cost of overordering (too much overbooking) would be the cost of compensating bumped customers. (Section 16.6 presents an example of this type that is addressed by using computer simulation with Crystal Ball.)

This last type is a particularly interesting one because major airlines now are making extensive use of this section’s model to analyze how much overbooking to do. For example, an article in the January-February 1992 issue of Interfaces describes how American Airlines is dealing with overbooking in this way. In addition, the article describes how the company is also using management science to address some related issues (such as the fare structure). These applications of management science are credited with increasing American Airline’s annual revenues by over $500 million. When managing the inventory of these various types of perishable products, it is occasionally necessary to deal with some considerations beyond those discussed in this section. Extensive research has been

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conducted to extend the model to encompass these considerations, and considerable progress has been made. Further information is available in the footnoted references.2

REVIEW QUESTIONS

1. 2. 3. 4. 5. 6. 7. 8. 9. Why does this model for perishable products need only a single time period? What is the only decision to be made with this model? What assumption is made about the demand for the product? How is the unit cost of underordering Cunder defined? The unit cost of overordering Cover? Will Bayes’ decision rule make the same decision when expressing the payoffs in terms of profits to be maximized or in terms of the costs of underordering and overordering to be minimized? What is the definition of service level? What is the formula for the optimal service level? For the graphical application of the ordering rule, what is the point that gives the optimal order quantity? Are there many types of perishable products in addition to newspapers?

19.3 A Case Study for Stable Products — The Niko Camera Corp. Problem

The Niko Camera Corporation is a major Japanese company that specializes in producing high quality cameras with an especially fine lens. It sells many different models to meet the various needs of discriminating photographers (both amateur and professional) around the world.

Background on the Product of Concern

One of Niko’s newer models is an inexpensive disposable panoramic camera. Very light and compact, this camera is designed to be especially convenient for a traveler who wants to take high quality panoramic shots of beautiful scenery without carrying the usual photographic equipment required to do this. The key to this convenience is that the camera is designed to be used for just one series of shots. It comes with special film already loaded at the factory and no provision is made for reloading by the customer. Therefore, the camera is given back to the camera store when the customer wants to have the film developed after completing the allotment of 27 shots. After removing the film, the camera store then returns the camera to the factory so that most of its components can be reused in a recycled camera. The special design for one-time use by the customer (but recycling of the expensive components by the factory) enables selling the camera so cheaply that many customers now think of repeated purchases as a good alternative to repeatedly buying rolls of film to use in an expensive and inconvenient permanent camera.

2 See Lau, H.-S., and A. H.-L. Lau, “The Newsstand Problem: A Capacitated Multiple Product SinglePeriod Inventory Problem,” European Journal of Operational Research, Vol. 94 (Oct. 11, 1996), pp. 2942, and its references. Also see pp. 610–628 in Porteus, E.L., “Stochastic Inventory Theory,” in Heyman, D.P., and M.J. Sobel (eds.), Stochastic Models, North Holland, Amsterdam, 1990.

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Although the cameras are produced initially in Japan, North American camera stores return the cameras for recycling to a factory in the United States run by Niko’s North American Division. Our focus will be on this factory. Niko’s American factory has been selling an average of 8,000 of these recycled cameras per month to a number of wholesale distributors. However, since these distributors only submit purchase orders on a very occasional basis, sales fluctuate widely from month to month (but without any noticeable seasonal pattern). Figure 19.6 shows the pattern of monthly sales over the past year. Note that some months are nearly double the monthly average (e.g., 15,800 in March) while others are almost nil (e.g., 700 in August). This same kind of random fluctuation, with no particular trend or seasonal pattern, also has been observed in the months prior to last year.

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

B

C

D

E

Monthly Sales of Niko's Disposable Cameras

Month January February March April May June July August September October November December Sales 7,000 1,500 15,800 8,600 9,900 4,200 13,600 700 14,100 6,200 5,000 9,400

Figure 19.6

Niko’s sales of disposable panoramic cameras in each month of the past year.

Because of these fluctuations, the camera is only produced on a sporadic basis. Every few months, the needed production facilities are set up to produce this particular model. In one concentrated production run lasting just a few days, a very large number of cameras are produced and placed into final inventory. This run size has been set at 20,000, which covers sales for 2.5 months on the average. (As indicated in the preceding chapter, the number produced or ordered to replenish inventory is called the order quantity.) Although it is only possible to produce these recycled cameras from the cameras returned by camera stores, the factory always has had a plentiful supply of these returned cameras for its production runs. Once the decision has been made to initiate a production run, some time is needed to clear the required production facilities from other uses and set them up for this run. (Recall that this time between ordering a product and receiving it is referred to as the lead time.) The lead time for this camera generally is about one month. Since average sales over a lead time of one month are 8,000, it has become routine to order another production run when the number of cameras in inventory drops to 8,000. (Recall that this inventory level at which an order to replenish is placed is called the reorder point.) To summarize, here are the key data for how the inventory of this camera is being managed. Order quantity = 20,000.

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Lead time = 1 month. Reorder point = 8,000.

Last Year’s Experience

Last year began with 16,500 disposable panoramic cameras in inventory. The January sales of 7,000 reported in Figure 19.6 reduced this inventory level to 9,500 by the end of the month. The February sales of 1,500 then reduced it to 8,000. Since 8,000 is the reorder point, an order was given at the end of February to initiate a production run of 20,000. After the lead time of one month, these 20,000 cameras were received and placed into inventory at the end of March. Since March sales of 15,800 already had depleted the 8,000 in inventory and left 7,800 in backorders, part of the production run immediately was used to fill these backorders. This left 12,200 in inventory to begin April. Figure 19.7 shows the record of what happened throughout the year, where the diamonds in the plot record the beginning inventories in those months. Five orders were placed for production runs of 20,000 (although the last run hadn’t quite been set up by the end of December). Therefore, the beginning-ofmonth inventory levels fluctuated widely, with some values near 15,000 and some others near 0. Only one month (August) had a shortage (called a stockout) at the beginning of the month. This inventory level of 4,100 indicates that backorders for 4,100 cameras had accumulated in late July after the depletion of the inventory and that these backorders would need to be filled from the upcoming production run of 20,000 cameras. (Holding backorders when shortages occur and then filling them when the inventory is replenished is referred to as backlogging.)

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A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 B C D E F G

Monthly Record of Niko's Inventory of Disposable Cameras

Month January February March April May June July August September October November December Sales 7,000 1,500 15,800 8,600 9,900 4,200 13,600 700 14,100 6,200 5,000 9,400 Beginning Inventory 16,500 9,500 8,000 12,200 3,600 13,700 9,500 -4,100 15,200 1,100 14,900 9,900 Ending Inventory 9,500 8,000 12,200 3,600 13,700 9,500 -4,100 15,200 1,100 14,900 9,900 500 Action None Ordered 20,000 at end of month Order received at end of month Ordered 20,000 during month Order received during month None Ordered 20,000 during month Order received during month Ordered 20,000 during month Order received during month None Ordered 20,000 during month

20,000 15,000

Inventory Level

10,000 5,000 0

September

-5,000 -10,000 -15,000

Time

Figure 19.7

Niko’s record of sales, inventories, and orders for each month last year, where the graph shows how the beginning inventory changes from month to month.

However, this figure does not show the full story, because columns D and E only give the inventory levels at the change of a month. Furthermore, the plot at the bottom simply connects the beginning-ofmonth inventories with line segments. By contrast, whereas Figure 19.8 uses dots to show these same inventory levels, the dashed lines then display approximately how the inventory level varied within each month as well. (This still is an approximation since it assumes that each month’s sales occurred evenly throughout the month rather than recording the actual individual sales during the month.) Note that four stockouts occurred during the year. Their sizes ranged from 1,235 to 7,800 cameras backordered. The durations ranged from a few days to a couple weeks. The distributors affected by the longer stockouts were not happy with this shoddy service, and several of them registered complaints with Niko management.

November

December

February

January

October

April

May

June

July

August

March

CD 19-17

CD 19-18 Management’s Concerns

Niko’s management has always taken pride in both the quality of its cameras and the quality of the company’s service to its customers. Therefore, the recent complaints from several distributors about delays in shipping one of the company’s most popular models, the disposable panoramic camera, has caused considerable concern. The North American Division’s Vice President for Marketing, in particular, is urging that something be done about this problem. She is suggesting having more frequent production runs to keep the inventory better stocked. At the same time, complaints have been received from the production floor about the relatively frequent interruptions in the production of other models caused by setting up for a production run for the disposable panoramic camera every two or three months. Although a production run is quick once the setup is completed, the process of setting up is quite complicated. A significant part of the expense in producing this camera is the direct cost of setting up and the additional cost attributable to disrupting other production. Therefore, the Vice President for Production strongly disagrees with the Vice President for Marketing. He recommends instead having much longer production runs much less frequently. He argues that this will solve two problems at once. First, it would provide larger inventories for longer periods of time and thereby greatly reduce the frequency of delayed shipments due to stockouts. Second, it would substantially reduce the annual cost of setting up for production runs, including the cost associated with disrupting other production. However, because it would increase inventory levels, the President of the North American Division is quite skeptical about this recommendation. For some time, he has been pushing the just-in-time philosophy of minimizing inventory by using careful planning and coordinating to provide items just in time to serve their purpose. This philosophy has enabled the company to greatly reduce its work-inprocess inventories while also improving the efficiency of its production processes. Although it has been necessary to maintain some inventories of finished products until they could be sold, the President is proud of the fact that even these inventories have been considerably reduced in recent years. The reductions in inventories throughout the company have provided substantial cost savings, including in the cost of capital tied up in inventory. These economies have been one of the key factors in maintaining Niko’s place as one of the world’s leading producers of cameras. Therefore, the President feels that it should be possible to solve the current problems without increasing the average inventory levels of disposable panoramic cameras. So what should be done? The President has called upon the North American Division’s Management Science Department many times in the past to address similar problems, with excellent results. Therefore, he has instructed this department to form a team to study this problem. The next section describes the management science team’s approach to the problem.

REVIEW QUESTIONS

1. 2. 3. 4. What has been happening in Niko’s North American Division that is causing considerable managerial concern? What is the concern of the Vice President for Marketing of the North American Division about the current situation? What recommendation is she making? What is the main concern of the Division Vice President for Production about the current situation? What recommendation is he making? Why is the Division President skeptical about his Vice President for Production’s recommendation? What company philosophy has he been promoting that relates to the current situation?

CD 19-19 19.4 The Management Science Team’s Analysis of the Case Study

The management science team begins by trying to diagnose why the frequent stockouts were occurring under the current inventory policy (order a production run of 20,000 when the inventory level drops to 8,000). Was this just a string of bad luck? Or did the current policy naturally lead to a high probability of a stockout occurring before the production run takes place? Just what is this probability? If a stockout occurs, what is the probability distribution of the size of the stockout (the number of cameras backordered when the stockout ends)?

Assessing the Stockout Problem

Since the lead time for a production run is approximately one month, the key to answering these questions is to estimate the underlying probability distribution of the number of cameras sold in a month. Examining the pattern of monthly sales over the past year shown in Figure 19.6 (along with similar data for other recent years), the team notes that these sales ranged pretty uniformly from almost nothing up to about 16,000. Therefore, the team’s best estimate is that the number of cameras sold in a month has a uniform distribution over the range from 0 to 16,000. Since this assumes that all the values over this range are equally likely, but that there is no chance of values outside this range, this distribution has the appearance shown in Figure 19.9. The mean of this distribution is 8,000, which corresponds to the observed average monthly sales.

When a production run is ordered with 8,000 cameras left in inventory, and the new cameras arrive a month later, the probability of a stockout occurring is just the probability that a month’s sales exceeds 8,000. With this uniform distribution, these probabilities are

CD 19-20

P (stockout) = P (monthly sales > 8000) = 0.5. With just a 50-50 chance of incurring a stockout after ordering a production run, having this occur four times in a row last year (as shown in Figure 19.8) was indeed a string of bad luck. This should occur only two times out of four on the average. This uniform distribution also indicates that there was further bad luck in terms of the size of the stockouts last year. The inventory level just before the order for the 20,000 new cameras is received is 8,000 minus the month’s sales. Therefore, the probability distribution of this inventory level also is a uniform distribution, but over the range from -8,000 (= 8,000 - 16,000) to 8,000 (= 8,000 - 0), as shown in Figure 19.10. The probability that this inventory level would fall as low as - 7,800 is extremely small (0.0125), but this did indeed occur at the end of March last year.

However, given management’s desire to provide high-quality service to the company’s customers, it seems unacceptable to have a probability of a stockout as high as 0.5 and to have the size of stockouts range as high as 8,000. Even without the bad luck of last year, such high numbers would inevitably lead to occasional significant delays in filling customer orders. Although customers would accept brief delays every once in a while, such frequent and lengthy delays need to be avoided.

CD 19-21

Conclusion: large. Both the probability of a stockout and the maximum size of a stockout are too

Alleviating Stockouts

Why is the probability of a stockout so high? The reason is that the method that was used to set the reorder point is faulty. This point was only set equal to the average sales (8,000) during the lead time (one month) for producing the next batch of cameras. No provision was made for the month’s sales exceeding the average, even though this should be expected to occur half the time. Conclusion: When reordering, a cushion of extra inventory needs to be provided in addition to the amount needed to cover the average sales during the lead time. (This extra inventory is referred to as safety stock.) With safety stock, the formula for setting the reorder point then is Reorder point = average sales during lead time + amount of safety stock = 8,000 + amount of safety stock. For example, Reorder point = 12,000 if amount of safety stock = 4,000. Changing the reorder point from 8,000 to 12,000 would change the probability distribution of the inventory level at the end of the lead time from the one shown in Figure 19.10 to that given in Figure 19.11. Since this new distribution is a uniform distribution over the range from - 4,000 to 12,000, the probability of a stockout now would be 0.25, with a maximum possible size of 4,000. Thus, stockouts now would occur only about once every four times a production run is ordered, on the average, and those shortages that do occur would tend to be somewhat smaller than before.

CD 19-22

Figure 19.12 shows approximately how the inventory level would have evolved throughout the past year if the reorder point had been 12,000. (As with Figure 19.8, the dashed lines in this graph approximate the evolution within each month by treating the month’s sales as having occurred evenly throughout the month.) Under this scenario, an order is fortuitously placed in mid-January and received in mid-February, in time to cover the unusually large sales of 15,800 in March. Consequently, a substantial number of cameras remain in inventory throughout the entire year, except for one very small and brief stockout in October.

CD 19-23

Now compare this figure to Figure 19.8. Note the dramatic improvement in avoiding stockouts and thereby avoiding delays in filling customer orders. It is true that part of the improvement was a matter of luck. Nevertheless, even under the worst circumstances, the higher reorder point of 12,000 would have avoided the more serious stockout problems shown in Figure 19.8. With a maximum possible stockout size of 4,000 instead of 8,000, the stockouts that do occur would tend to be both smaller and briefer, thereby causing much less damage to customer relations.

CD 19-24

Conclusion: Even when the amount of safety stock provided still permits occasional short stockouts, this safety stock can dramatically improve the service to customers by greatly reducing both the number and length of the delays in filling customer orders.

Choosing the Amount of Safety Stock

You have just seen that providing a safety stock of 4,000 cameras is much better than providing none at all. But is 4,000 the right amount? Note that the inventory levels shown in Figure 19.12 are much higher than in Figure 19.8, whereas the President wants to keep inventories down as much as reasonably possible. What is the best trade-off between the costs of holding inventory and the consequences of stockouts? Since choosing such a trade-off is ultimately a management decision, the management science team consults with management about their feelings regarding stockouts. These are the key questions posed to management. 1. 2. How important is it to reduce delays in filling customer orders? Considering that larger inventories would be needed to reduce delays, how would you compare the importance of reducing delays with the importance of holding inventory levels down? Considering that unacceptably large inventories would be needed to completely eliminate any delays, what would you consider tolerable in terms of the frequency, size, and length of stockouts?

3.

To help make these questions more concrete to management, the management science team describes the frequency, size, and length of stockouts that would be expected for each of several alternative amounts of safety stock. For example, here is the description for a safety stock of 4,000 cameras. With a safety stockout of 4,000 disposable panoramic cameras, a new production run would be ordered when the inventory level drops to 12,000 cameras. Since the average sales during the lead time for the production run (one month) are 8,000 cameras, the inventory would be adequate to cover sales in most cases. However, since a month’s sales can range as high as 16,000 cameras, and about a quarter of the months have sales between 12,000 and 16,000 cameras, a stockout would occur about once every four times on the average. With the current production runs of 20,000 cameras occurring about four times every ten months, this means that a stockout would occur about once every ten months. When it does occur, the size would range from very small to about 4,000 cameras backordered, so about 2,000 on the average. Since we sell about 80,000 cameras over ten months, this means that about 2.5% of our customers would incur a delay in having their orders filled. The delays would range from very short (in most cases) up to about a week, with an estimated average of about a third of a week. (The full week would result from a month’s sales of 16,000 cameras after ordering a production run, with the last 4,000 sales occurring during the last week of the month.) Beware, however, that these numbers assume that we can continue holding to a lead time of about one month. If an unexpected delay in a production run should occur, the possible shortages and delays would be extended accordingly. After the management science team elicits management’s views about this scenario and the several alternatives, the following conclusion is drawn. Conclusion: Management feels that providing a safety stock of roughly 4,000 cameras is needed to provide an adequate level of service to customers in minimizing delays in filling their orders.

CD 19-25

Considering the company’s just-in-time philosophy regarding the need to hold down inventories, management does not want the safety stock raised higher than this. Having established that the reorder point should be increased from 8,000 to 12,000, the management science team next wants to investigate what the order quantity (the size of each production run) should be. Realizing that this issue involves a trade-off between several types of costs, the team first turns to estimating these costs.

The Cost Factors

All the relevant cost factors for analyzing inventory problems were described in Section 18.2. The four types are (1) acquisition costs, (2) setup costs, (3) holding costs, and (4) shortage costs. The acquisition cost in this case is the cost of producing the disposable panoramic cameras (when using the recycled components). Excluding setup costs, this production cost is just $7 per camera. However, this cost turns out to be irrelevant for choosing the order quantity. The reason is that the order quantity does not affect sales, and it is sales that determine the number (and so the production cost) of cameras that will be produced eventually. The order quantity only affects the timing of when the fixed production costs will be incurred. However, setup costs are very relevant. The cost of setting up for a production run, plus additional costs associated with disrupting other production in the process, is estimated to be $12,000. The order quantity (size of the production run) determines how frequently this cost will be incurred. The holding costs encompass all the costs associated with holding the cameras in inventory. One important component is the cost of capital tied up in the inventory. The company pays an annual interest rate of about 10% on money borrowed to pay for the production of cameras that are not yet sold. The production cost (including setup cost) per camera is about $7.50, and the company’s cost invested in each set of recycled components of a camera is another $16.50, for a total of $24. Therefore, the cost of capital tied up for each camera in inventory is about $2.40 per year, or 20¢ per month. (This seemingly insignificant cost does add up when there are many thousand cameras in inventory.) Holding costs also include all the costs directly involved with storing the cameras, including the cost of the space, record keeping, protection, insurance, and taxes. When 12,500 cameras are in inventory (a roughly average level), these costs are estimated to add up to just about $1,250 per month, so about 10¢ per camera. Adding in the cost of capital tied up gives a total holding cost per camera of about 30¢ per month. Certain storage costs (e.g., space and protection) may not be directly proportional to the inventory level. However, as an approximation, it is assumed that the total holding cost per month is 30¢ times the current inventory level (except for ignoring negative levels that represent shortages). The shortage costs are more difficult to quantify. The main component is lost future profit from lost future sales caused by customer dissatisfaction with delays in filling current orders. This is difficult to estimate. Other minor components might include the cost of additional record keeping and handling for dealing with backordered cameras. Estimating shortage costs requires a managerial assessment of the seriousness of making customers wait to have their orders filled. To obtain this managerial input, the management science team poses the following question to the Vice President for Marketing. Question: We have discussed the fact that the worst case scenario with a safety stock of 4,000 cameras would be to incur a stockout of about 4,000 cameras with a delay in filling the orders of about one week. If you were to put a dollar figure on the damage that such a stockout would cause the company in terms of lost profit from lost future business, etc., what would that figure be? In other words, if it were possible for the company to pay some money to prevent this one

CD 19-26

stockout completely, how much should we be willing to pay to do so? We realize that it is difficult to pin down an exact dollar figure, but we are only asking you to apply your judgment as best you can in responding to this question. Response: $10,000. As a rough approximation, the management science team assumes that the shortage cost from any stockout should be proportional to both the size of the stockout and the resulting average delay in filling the orders. Thus, with a cost of $10,000 for a shortage of 4,000 cameras for one week, the cost for extending the average delay to one month (four weeks) is assumed to be about $40,000, or about $10 per camera per month of delay. This assumption is definitely questionable. The actual shortage cost for a delay of a month (infuriating the customers) might be considerably more than four times that for a delay of a week (mildly concerning the customers). However, the management science team feels that the assumption of proportionality is a reasonable one over the range of actual delays (up to roughly one week) that would be incurred with a safety stock of 4,000 cameras. Table 19.1 summarizes all these cost factors and their estimated values. The symbols in the table are the same as introduced in Chapter 18 for these types of cost except that the management science team now is measuring costs on a monthly rather than annual basis.

Table 19.1 The Cost Factors for Niko’s Inventory Problem

Type of Cost Setup cost for a production run Symbol K Value $12,000 $0.30 $10

Holding cost per camera per month h Shortage cost per camera per month of p delay in filling the customer order

Choosing the Order Quantity

With these cost factors pinned down, the management science team now is ready to address the problem of determining what order quantity (number of cameras to be produced in a production run) would minimize the sum of all these costs. (Recall from Chapter 18 that the order quantity that minimizes the total average cost is commonly referred to as the economic order quantity or EOQ for short.) Here are the trade-offs involved in making this decision. 1. The average monthly setup costs are decreased by increasing the order quantity, because this decreases the average number of setups required per month. 2. 3. However, the average monthly holding costs are decreased by decreasing the order quantity, since this decreases the average inventory level. However, the average monthly shortage costs are decreased by increasing the order quantity, because this decreases the average number of opportunities for stockouts per month.

Section 18.5 presented the EOQ model with planned shortages to address exactly these same trade-offs. In fact, that model completely fits Niko’s inventory problem with just one important exception. The exception is the model’s assumption that sales occur at a constant rate, with no variation from month to

CD 19-27

month. Thus, for the Niko problem, the model would assume that the average monthly sales of 8,000 disposable panoramic cameras are the actual sales spread evenly through the month for each and every month. The reality, of course, is that Niko’s sales of these cameras vary greatly from month to month. How much effect does this month-to-month variation have on the average total monthly cost that needs to be minimized to determine the economic order quantity? The effect on the average monthly shortage costs is rather considerable, since the month-to-month variation tends to increase the frequency and size of the shortages. However, the effect on the average monthly holding cost is only slight. The fact that no holding costs are incurred during shortages has a slight effect, but otherwise the average inventory level is not significantly affected. Furthermore, there is no effect on the average monthly setup costs, since the variation does not affect the average frequency of setups for production runs. Consequently, the overall effect of the month-to-month variation in sales on the average total monthly cost is modest. Therefore, using the model from Section 18.5 to calculate the economic order quantity (but not the reorder point) for Niko’s problem provides a pretty good approximation. The management science team decides to adopt this approach. To review, the important notation from Section 18.5 (now expressed on a monthly rather than annual basis) is K, h, p as defined in Table 19.1, D = monthly sales rate = 8,000 as the average for Niko’s problem, Q = order quantity, S = shortage just before an order quantity is received.

Using an asterisk to indicate the optimal value of Q and S, their formulas given in Section 18.5 are Q* =

h + p 2KD , p h h $ Q*. " h + p%

! S* = #

An Excel template is available in your MS Courseware for performing these calculations for you. For Niko’s problem, plugging the values of K, h, and p given in Table 19.1 (plus D = 8,000) into this template gives the results shown in Figure 19.13. Since Q* = 25,675 cameras and S* = 748 cameras, the inventory level jumps from a shortage of 748 cameras to a level of Q* - S* = 24,927 cameras when the order quantity (the output of a production run) arrives.

CD 19-28

A 1 2 3 4 5 6 7 8 9 10 11 B C D E F G

EOQ Model with Planned Shortages (Niko Case Study)

D= K= h= p= Data 8000 $12,000 $0.30 $10 (demand/year) (setup cost) (unit holding cost) (unit shortage cost) Max Inventory Level Annual Setup Cost Annual Holding Cost Annual Shortage Cost Total Variable Cost Results 24927.08 $3,739.06 $3,630.16 $108.90 $7,478.12

Q= S=

4 5 6 7 8 9

Decision 25675 (order quantity) 748 (maximum shortage)

F Max Inventory Level =Q-S

G

Annual Setup Cost Annual Holding Cost Annual Shortage Cost Total Variable Cost

Cell G7 G6 G8 C4 C6 C5 G4 C7 C10 C11 G9

=K*D/Q =h*(MaxInventoryLevel^2)/(2*Q) =p*((Q-MaxInventoryLevel)^2)/(2*Q) =AnnualSetupCost+AnnualHoldingCost+AnnualShortageCost

Range Name AnnualHoldingCost AnnualSetupCost AnnualShortageCost D h K MaxInventoryLevel p Q S TotalVariableCost

10 11

B C Q = =SQRT(2*D*K/h)*SQRT((p+h)/p) S = =(h/(h+p))*Q

Figure 19.13

The Excel template for the EOQ model with planned shortages (analytical version) in your MS Courseware is applied here to find the order quantity for the Niko problem.

Using these quantities, Figure 19.14 shows how the EOQ model with planned shortages assumes Niko’s inventory level would evolve over a year, starting with last year’s initial inventory of 16,500 cameras. (Keep in mind that this model assumes that sales occur at a constant rate, which does not fit Niko’s situation.) With a lead time of one month, the model orders a production run each time just one month before a shortage of 748 cameras occurs. Since the assumed sales over this lead time are 8,000 cameras, the model thereby sets the reorder point at 8,000 - 748 = 7,252 cameras.

CD 19-29

CD 19-30

However, it is crucial that only Q* and not this reorder point be used from the model. Because Niko’s actual sales do vary considerably from month to month, a substantial safety stock is needed as a cushion against the sales during the lead time being much higher than average. Therefore, based on the earlier analysis of how much safety stock is needed, the management science team makes the following recommendation. Recommended inventory policy: Whenever the number of disposable panoramic cameras in inventory drops to 12,000, order a production run of 25,675 cameras. Figure 19.15 depicts how this inventory policy would have performed throughout the past year. (Once again, the dashed lines approximate the evolution of the inventory level throughout each month by treating the sales as having occurred evenly throughout the month.) A substantial inventory, ranging from 2,513 to 34,211 cameras, would have been maintained throughout the year. However, in light of the probability distribution shown in Figure 19.11, a bit of good luck was involved. Despite having a probability of 0.25 that a stockout will occur before an order is received, this never happened in the four chances during the year.

CD 19-31

CD 19-32 Management’s Reaction to the Recommended Inventory Policy

This recommended inventory policy provides at least a good approximation of an optimal policy (an optimal combination of a reorder point and an order quantity) that minimizes the total average monthly cost. Or at least it is an “approximately optimal” policy under the conditions given to the management science team (a lead time of one month, the cost factors in Table 19.1, etc.). But is it really a sound policy from a managerial perspective? The three members of management dealing with this problem have expressed their reservations about the recommendation. The President is unhappy about the large increase in inventory levels that would result from the substantial increases in both the reorder point and order quantity. He realizes the need for some safety stock, but feels that there should be a better approach to the problem that is more in line with the company’s just-in-time philosophy of minimizing the use of inventories. The Vice President for Marketing is happy that the new safety stock will considerably alleviate the stockout problem. However, she is somewhat concerned that shortages of various magnitudes still can be expected to occur about once per year. She continues to emphasize that high priority needs to be placed on maintaining and building the company’s reputation for good and prompt service to its customers. The Vice President for Production is quite unhappy. He had emphasized the problems caused in disrupting other production by shifting facilities over so frequently to set up for production runs of the disposable panoramic camera. Although the recommendation would slightly decrease the frequency of production runs, he does not feel that the disruption problems have yet been dealt with adequately. The costs associated with this “approximately optimal” policy are indeed disturbingly high. The profit margin on each camera sold is only about $2, due partially to these high costs. Here is a breakdown of each of the costs of major concern to one of the members of management under the recommended inventory policy.

HOLDING COST CALCULATIONS

Average Inventory Levels: Just before an order is received Just after an order is received = 12,000 - 8,000 = 4,000. = 4,000 + 25,675 = 29,675.

Overall average

! 4, 000 + 29, 675 $ = # & " % 2

= 16,837.

Holding cost per camera per month, h = $0.30. 3 = 16,837 ($0.30) Average monthly holding cost = $5051.

3Actually,

this calculation slightly understates the true average monthly holding cost because it charges a negative holding cost instead of zero when the inventory level is negative because of shortages.

CD 19-33 SHORTAGE COST CALCULATIONS

Average number of orders per month

! $ sales = # " order quantity & %

! 8,000 $ = # = 0.31. " 25,675 & %

Probability of a stockout before order received Expected number of stockouts per month

= 0.25. = 0.25 (0.31) = 0.078.

Average stockout size

! 4,000 $ = # " 2 & %

= 2,000.

Estimate of average delay per camera delayed 4

=

! 1# " 3$ week

= 0.08 month. Shortage cost per camera per month, p Average monthly shortage cost = $10. = 0.078 (2,000) (0.08) ($10) = $125.

SETUP COST CALCULATIONS

Average number of setups per month

! $ sales = # " order quantity & %

! 8,000 $ = # = 0.31. " 25,675 & %

Cost per setup, K Average monthly setup cost

= $12,000. = 0.31 (12,000) = $3,720.

Adding these three average monthly costs gives

4

Although the maximum delay is about a week, most delayed camera orders do not wait nearly this long, eith

CHAPTER 19 INVENTORY MANAGEMENT WITH UNCERTAIN DEMAND

Learning Objectives:

After completing this chapter, you should be able to

1. 2. 3. 4. 5. 6. 7. 8. 9.

Identify some situations where the demand for withdrawing a product from inventory is uncertain. Describe the trade-off that must be considered when developing an inventory policy for a product with uncertain demand. Explain why perishable products with uncertain demand and stable products with uncertain demand require different kinds of inventory models. Describe an inventory model for perishable products with uncertain demand. Apply this model to find the optimal order quantity. Describe a continuous-review inventory model for stable products with uncertain demand. Apply this model to choose an order quantity. Apply this model to determine the inventory level at which an order should be placed with this order quantity. Describe some large inventory systems that arise in practice.

We now continue the focus of the preceding chapter on inventory management, but with one key difference. We have been assuming that the product under consideration in inventory has a known demand, i.e., that we can predict with reasonable certainty when units will need to be withdrawn from inventory. We drop this assumption in this chapter, so we now will consider products with an uncertain demand. The predictability of demand depends greatly on the situation. We certainly have known demand when the product is being withdrawn from inventory at a fixed rate because it actually is one of several components being assembled into a larger product on an assembly line. Similarly, a manufacturer has a known demand for a custom product in inventory when it is producing the product (replenishing inventory) only to satisfy a schedule of orders already received from a particular customer. A wholesaler also has roughly a known demand for a product after its retail customers have developed a well established pattern for purchasing the product month after month. These are the types of situations considered in the preceding chapter.

CD 19-2

By contrast, a retail store manager does not have the luxury of knowing when customers will come in to purchase a given product. If the product is a new one, predicting how well it will catch on may be particularly difficult. Similarly, a wholesaler supplying a number of retailers with a new product may have considerable uncertainty about what the demand will be. Sales can fluctuate widely from one month to the next. Consequently, a manufacturer selling the product to a number of wholesalers (perhaps in competition with other manufacturers) also can have significant uncertainty about the demand. We are considering these kinds of situations in this chapter. Even with uncertainty, it is necessary to make some kind of forecast of the expected demand and what the variability might be. For example, you might use something like the PERT three-estimate approach described in Section 16.4 (making a most likely estimate, an optimistic estimate, and a pessimistic estimate, and then converting these estimates into a probability distribution). In some way, the forecast should be expressed in probabilistic terms. The probabilities might be quite subjective in nature, as with the typical prior probabilities of decision analysis discussed in Chapter 9, or they might be based on considerable historical experience and data. At any rate, the models in this chapter assume that an estimate has been made of the probability distribution of what the demand will be over a given period. A very important consequence of uncertain demand is the great risk of incurring shortages unless the inventory is managed carefully. An order to replenish the inventory needs to be placed while some inventory still remains, because of the lag until the order can be filled. Even the amount of lead time needed to fill the order may be uncertain. However, if too much inventory is replenished too soon, a heavy price is paid because of the high cost of holding a large inventory. A constant theme throughout the chapter is the need to find the best trade-off between the consequences of having too much inventory and of having too little. We will separately discuss inventory management for two types of products. One type is a perishable product, which can be carried in inventory for only a very limited period of time before it can no longer be sold. The second type is a stable product, which will remain sellable indefinitely. These two types need to be handled quite differently. The first two sections present a case study and then a general model for perishable products. Sections 19.3 and 19.4 discuss a case study that involves a stable product. The inventory model that underlies this case study is summarized in Section 19.5. Section 19.6 describes the large inventory systems that commonly arise in practice, including massive systems that have been installed at IBM and Hewlett-Packard.

19.1 A Case Study for Perishable Products — Freddie the Newsboy’s Problem

The problem being addressed in this case study actually is a simplified version of the one for the case study introduced in Section 13.1. For ease of exposition, we have simplified the probability distribution of the demand here by having only three possible values for the demand instead of the 31 considered previously. The technique used in Chapter 13 to analyze this problem was computer simulation with Crystal Ball. In this section and the next, we will develop an inventory model to address the same problem in a relatively straightforward way. For completeness, we repeat the description of the problem below.

Freddie’s Problem

This problem concerns a newsstand in a prominent downtown location of a major city. The newsstand has been there longer than most people can remember. It has always been run by a well known character named Freddie. (Nobody seems to know his last name.) His many customers refer to him affectionately as Freddie the newsboy, even though he is considerably older than most of them.

CD 19-3

Freddie sells a wide variety of newspapers and magazines. The most expensive of the newspapers is a large national daily called the Financial Journal. Our case study involves this newspaper. The day’s copies of the Financial Journal are brought to the newsstand early each morning by a distributor. Any copies unsold at the end of the day are returned to the distributor the next morning. (This is indeed a perishable product). However, to encourage ordering a large number of copies, the distributor does give a small refund for unsold copies. Here are Freddie’s cost figures. Freddie pays $1.50 per copy delivered. Freddie sells it at $2.50 per copy. Freddie’s refund is $0.50 per unsold copy. Partially because of the refund, Freddie always has taken a plentiful supply. However, he has become concerned about paying so much for copies that then have to return unsold, particularly since this has been occurring nearly every day. He now thinks he might be better off by ordering only a minimal number of copies and saving this extra cost. To investigate this further, Freddie has been keeping a record of his daily sales. In contrast to the numbers presented in Chapter 13, we assume now that this is what he has found. Freddie sells 9 copies on 30% of the days. Freddie sells 10 copies on 40% of the days. Freddie sells 11 copies on 30% of the days. So how many copies should Freddie order from the distributor per day? (Think about it before reading on.)

Applying Bayes’ Decision Rule to Freddie’s Problem

One approach to this problem is to apply decision analysis as described in Chapter 9. Specifically, Bayes’ decision rule introduced in Section 9.2 is used as outlined below. The procedure involves filling out the payoff table shown in Figure 19.1. Column B lists the decision alternatives that deserve consideration, namely, to order 9, 10, or 11 copies per day from the distributor. For each of these alternatives, Freddie’s profit on a given day is determined by how many requests to purchase a copy of the Financial Journal occur that day, so these possible numbers of purchase requests (the possible states of nature) are listed in cells D4:F9. The relative likelihood of these numbers of purchase requests are entered in PriorProbability (D9:F9) as the prior probabilities of these states of nature.

CD 19-4

A 1 2 3 4 5 6 7 8 9 B C D E F G H

Bayes' Decision Rule (with Profits) for Freddie's Problem

Payoff Table Alternative (Copies Ordered) 9 10 11 State of Nature (Purchase Requests) 9 10 11 $9 $9 $9 $8 $10 $10 $7 $9 $11 0.3 0.4 0.3

H 3 4 5 6 7

Expected Payoff $9.00 $9.40 $9.00

Prior Probability

Range Name ExpectedPayoff Payoff10 Payoff11 Payoff9 PayoffTable PriorProbability

Cells H5:H7 D6:F6 D7:F7 D5:F5 D5:F7 D9:F9

Expected Payoff =SUMPRODUCT(PriorProbability,Payoff9) =SUMPRODUCT(PriorProbability,Payoff10) =SUMPRODUCT(PriorProbability,Payoff11)

Figure 19.1

This Excel template in your MS Courseware shows that Freddie the newsboy maximizes his expected profit by ordering 10 copies each day.

The payoff from Freddie’s decision, given the state of nature, is the profit for that day. This profit is Profit = sales income - purchase cost + refund. For example, if Freddie orders 11 copies from the distributor and the state of nature turns out to be 9 for that day (9 copies are sold), his profit is Profit = 9 ($2.50) - 11 ($1.50) + 2 ($0.50) = $7.00. Calculating the profit in this way for each of the combinations of a decision alternative and a possible state of nature yields the payoff table shown in cells B3:F9 of Figure 19.1. Applying Bayes’ decision rule with this Excel template involves calculating the expected payoff (EP) for each alternative by using the indicated equations entered into ExpectedPayoff (H5:H7). (If you haven’t studied this topic in Chapter 9, note that each of these equations is simply calculating the statistical average of the payoffs.) The rule then selects the alternative with the largest expected payoff ($9.40). Conclusion: Freddie’s most profitable alternative in the long run is to order 10 copies, since this will provide an average daily profit of $9.40, versus $9.00 for either of the other alternatives. In the next section, you will see a shortcut for drawing this same conclusion.

REVIEW QUESTIONS

1. 2. 3. What is the trade-off that Freddie the newsboy should consider in making his decision? Why are Freddie’s decision alternatives limited to ordering 9, 10, or 11 copies? What is the state of nature when using decision analysis to formulate Freddie’s problem? Why?

CD 19-5 19.2 An Inventory Model for Perishable Products

Freddie the newsboy’s problem illustrates an application of a widely used inventory model for perishable products. Newspapers are just one of the many types of such products to which it can be applied. After summarizing its assumptions and applying Bayes’ decision rule, we will show you a shortcut for solving the model and then describe the various types of perishable products.

The Assumptions of the Model

1. 2. 3. 4. 5. Each application involves a single perishable product. Each application involves a single time period because the product cannot be sold later. However, it will be possible to dispose of any units of the product remaining at the end of the period, perhaps even receiving a salvage value for the units. The only decision to be made is how many units to order (the order quantity) so they can be placed into inventory at the beginning of the period. The demand for withdrawing units from inventory to sell them (or for any other purpose) during the period is uncertain. However, the probability distribution of demand is known (or at least estimated). If the demand exceeds the order quantity, a cost of underordering is incurred. In particular, the cost for each unit short is Cunder = unit cost of underordering = decrease in profit that results from failing to order a unit that could have been sold during the period. 7. If the order quantity exceeds the demand, a cost of overordering is incurred. In particular, the cost for each extra unit is Cover = unit cost of overordering = decrease in profit that results from ordering a unit that could not be sold during the period. These assumptions certainly fit Freddie the newsboy’s problem. The day’s newspaper of concern (the Financial Journal) is a single perishable product that cannot be sold after the day (the single time period), although it can be returned to the distributor for a small refund (the salvage value). Freddie’s only decision is how many copies to order from the distributor for each day, given the probability distribution of how many can be sold (shown in row 9 of Figure 19.1). The last two assumptions also fit, since the definitions of the two unit costs imply that Cunder = unit sale price - unit purchase cost = $2.50 - $1.50 = $1.00, Cover = unit purchase cost - unit salvage value

6.

CD 19-6

= $1.50 - $0.50 = $1.00 for this problem. These expressions for Cunder and Cover will fit any analogous problem where the only cost factors are the unit sale price, unit purchase cost, and unit salvage value. However, the definitions of Cunder and Cover have been expressed more generally as a decrease in profit in order to fit other situations as well. For example, if there is a concern about losing future business due to ill will caused by underordering, then a unit cost of ill will could be added to the expression for Cunder. Similarly, Cover might include an additional term for something like the extra holding cost associated with storing a unit all the way to the end of the period. Another possibility is that it might be necessary to add a unit disposal cost rather than having a refund to subtract. As in Freddie’s problem, it often is not feasible to place and receive an additional order before the period ends if a shortage occurs. However, if it is very important to fill all the demand, arrangements sometimes can be made to do this at an extra cost. In this case, Cunder would equal the extra cost per unit of placing this emergency order plus any reduction in the unit selling price to pacify the customers who had to wait. A key part of assumptions 6 and 7 is that Cunder must be the same for each unit short and Cover must be the same for each extra unit. One way of solving this model is to apply Bayes’ decision rule, as described in a separate box. Another simple method is presented next.

CD 19-7 Applying Bayes’ Decision Rule

Any application of this model can be solved in much the same way as we solved Freddie the newsboy’s problem in the preceding section, namely, by applying Bayes’ decision rule (first introduced in Section 9.2). One alternative is to express the payoffs in terms of profits and then proceed as in Figure 19.1. However, in light of assumptions 6 and 7, a completely equivalent but more direct approach is to focus on just the costs of underordering and overordering. This is what is done in Figure 19.2 for Freddie’s problem.

A 1 2 3 4 5 6 7 8 9

B

C

D

E

F

G

H

Bayes' Decision Rule (with Costs) for Freddie's Problem

Payoff Table Alternative (Copies Ordered) 9 10 11 State of Nature (Purchase Requests) 9 10 11 $0 $1 $2 $1 $0 $1 $2 $1 $0 0.3 0.4 0.3 Expected Payoff $1.00 $0.60 $1.00

Prior Probability

Figure 19.2 This Excel template finds that Freddie the newsboy minimizes his expected cost of underordering or overordering by ordering 10 copies each day. Note that the numbers in cells D5, E6, and F7 are 0, because the order quantity equals the demand in these cases. The costs (in dollars) in cells E5, F5, and F6 are solely the cost of underordering and those in cells D6, D7, and E7 are solely the cost of overordering. The expected cost for each alternative is calculated in column H with the equations given in Figure 19.1. Since we wish to minimize expected cost, the equations in column I now use the MIN function instead of the MAX function, so the conclusion again is that Freddie should order 10 copies. The conclusion here must be the same as in Figure 19.1 since the two approaches are equivalent. All we have done here is eliminate the revenues and cost factors that don’t affect the decision and focus on just those that do, namely, the cost of underordering and the cost of overordering. Both approaches are applying Bayes’ decision rule, but with different payoffs, where one is to be maximized and the other minimized.

A Simple Formula for Solving the Model

The drawback with relying on Bayes’ decision rule as the method for solving the model is that most applications involve many more decision alternatives and states of nature than Freddie’s problem formulated in Figures 19.1 and 19.2. In fact, if Freddie were to apply the same approach to another more popular newspaper where the number of copies sold in a day range from 100 to 200, he then would have 101 decision alternatives and 101 states of nature to consider. Other applications might have thousands. Using Bayes’ decision rule to deal with such large problems would be extremely cumbersome.

CD 19-8

A Quicker Approach: Fortunately, a much quicker way to solve problems of any size has been found. It involves using the service level, defined as follows: Service level = probability that no shortage will occur. A shortage occurs when the demand for the product exceeds the number of units available in inventory, so one or more customers suffer the disappointment of not immediately obtaining the units they wanted. Therefore, the probability of avoiding a shortage is a key measure of the level of service being provided to the customers. Given the prior probabilities in cells D9:F9 of Figure 19.2, the service levels for Freddie’s three alternatives are Service level if order 9 copies = D9 = 0.3,

Service level if order 10 copies = D9 + E9 = 0.3 + 0.4 = 0.7, Service level if order 11 copies = D9 + E9 + F9 = 0.3 + 0.4 + 0.3 = 1. Here is the simple formula for solving this model.

Ordering Rule for the Model for Perishable Products

1. 2.

Optimal service level =

Cunder . Cunder + Cover

Choose the smallest order quantity that provides at least this service level.

Since Cunder = $1.00 and Cover = $1.00 for Freddie’s problem, this formula gives Optimal service level =

$1.00 = 0.5. $1.00 + $1.00

Referring to the service levels for Freddie’s three alternative order quantities, the smallest one that provides at least this optimal service level is to order 10 copies. This of course is the same answer as provided by Bayes’ decision rule. An Excel template is available in your MS Courseware for applying this model for perishable products to any situation where, as for Freddie’s problem, the only cost factors are the unit sale price, unit purchase cost, and unit salvage value. As illustrated in Figure 19.3 for Freddie’s problem, all you need to do is enter these three cost factors. The template then calculates Cover, Cunder, and the optimal service level. What remains is for you to use the probability distribution of demand for your specific problem to apply step 2 of the ordering rule.

CD 19-9

A 1 2 3 4 5 6 B C D E F

Optimal Service Level for Perishable Products

Unit Sales Price Unit Purchase Cost Unit Salvage Value Data $2.50 $1.50 $0.50 Cost of Overordering Cost of Underordering Optimal Service Level Results $1.00 $1.00 0.5

4 5 6

E F Cost of Overordering =UnitPurchaseCost-UnitSalvageValue Cost of Underordering =UnitSalesPrice-UnitPurchaseCost Optimal Service Level =CostOfUnderordering/(CostOfUnderordering+CostOfOverordering)

Cell F4 F5 F6 C5 C4 C6

Range Name CostOfOverordering CostOfUnderordering OptimalServiceLevel UnitPurchaseCost UnitSalesPrice UnitSalvageValue

Figure 19.3

The Excel template for the inventory model for perishable products in your MS Courseware applies step 1 of the ordering rule, as illustrated here for Freddie’s problem.

APPLYING STEP 2 OF THE ORDERING RULE GRAPHICALLY: For any given order quantity, the definition of service level can be restated in another equivalent way as Service level = probability that the demand is less than or equal to the order quantity = P (demand ≤ order quantity). The probability on the right for each of Freddie’s three alternative order quantities is P (demand ≤ 9) P (demand ≤ 10) P (demand ≤ 11) = 0.3, = 0.7, = 1.

Figure 19.4 shows a graph where the horizontal axis is x and the vertical axis is P (demand ≤ x). Since demand is a random variable, this graph is referred to as the cumulative distribution function (or CDF for short) of demand. The point at which the optimal service level of 0.5 (see the horizontal dashed line) hits this CDF gives the optimal order quantity of 10.

CD 19-10

The figure enables visualizing the ordering rule graphically. Furthermore, on larger problems, this graphical approach may find the optimal order quantity more quickly than enumerating the service levels for all the alternatives. This is illustrated by the following example.

A VARIATION OF FREDDIE’S PROBLEM:

Freddie now wishes to find the optimal order quantity for another of his newspapers. This is a more popular newspaper whose daily sales range from 100 copies to 200 copies, with roughly

CD 19-11

equal probabilities over this range. In this case, the relevant unit costs are Cunder = $0.75 and Cover = $0.25. Since the probabilities of the various possible demands from 100 to 200 are roughly equal, a good estimate of the probability distribution of demand is the uniform distribution from 100 to 200. The solid lines in Figure 19.5 show the CDF of this distribution.

With the given unit costs, the ordering rule says that Optimal service level =

Cunder $0.75 = = 0.75. Cunder + Cover $0.75 + $0.25

The corresponding dashed lines in the figure show that the optimal order quantity is 175.

Some Types of Perishable Products

The model presented in this section has traditionally been called the newsboy problem1 because it fits the problems of newsboys like Freddie so well. However, it has always been recognized that the model is just

1

Recently, some writers have been substituting the name, newsvendor problem. Other names include the single-period probabilistic model and single-period stochastic model.

CD 19-12

as applicable to other perishable products as to newspapers. In fact, most of the applications have been to perishable products other than newspapers. As you read through the list below of various types of perishable products, think about how the inventory management of such products is analogous to Freddie’s problem since these products also cannot be sold after a single time period. All that may differ is that the length of this time period may be a week, a month, or even several months rather than just one day. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Periodicals, such as newspapers and magazines. Flowers being sold by a florist. The makings of fresh food to be prepared in a restaurant. Produce, including fresh fruits and vegetables, to be sold in a grocery store. Christmas trees. Seasonal clothing, such as winter coats, where any goods remaining at the end of the season must be sold at highly discounted prices to clear space for the next season. Seasonal greeting cards. Fashion goods that will be out of style soon. New cars at the end of a model year. Any product that will be obsolete soon. Vital spare parts that must be produced during the last production run of a certain model of a product (e.g., an airplane) for use as needed throughout the lengthy field life of that model. Reservations provided by an airline for a particular flight. Reservations provided in excess of the number of seats available (overbooking) can be viewed as the inventory of a perishable product (they cannot be sold after the flight has occurred), where the demand then is the number of no-shows. With this interpretation, the cost of underordering (too little overbooking) would be the lost profit from empty seats and the cost of overordering (too much overbooking) would be the cost of compensating bumped customers. (Section 16.6 presents an example of this type that is addressed by using computer simulation with Crystal Ball.)

This last type is a particularly interesting one because major airlines now are making extensive use of this section’s model to analyze how much overbooking to do. For example, an article in the January-February 1992 issue of Interfaces describes how American Airlines is dealing with overbooking in this way. In addition, the article describes how the company is also using management science to address some related issues (such as the fare structure). These applications of management science are credited with increasing American Airline’s annual revenues by over $500 million. When managing the inventory of these various types of perishable products, it is occasionally necessary to deal with some considerations beyond those discussed in this section. Extensive research has been

CD 19-13

conducted to extend the model to encompass these considerations, and considerable progress has been made. Further information is available in the footnoted references.2

REVIEW QUESTIONS

1. 2. 3. 4. 5. 6. 7. 8. 9. Why does this model for perishable products need only a single time period? What is the only decision to be made with this model? What assumption is made about the demand for the product? How is the unit cost of underordering Cunder defined? The unit cost of overordering Cover? Will Bayes’ decision rule make the same decision when expressing the payoffs in terms of profits to be maximized or in terms of the costs of underordering and overordering to be minimized? What is the definition of service level? What is the formula for the optimal service level? For the graphical application of the ordering rule, what is the point that gives the optimal order quantity? Are there many types of perishable products in addition to newspapers?

19.3 A Case Study for Stable Products — The Niko Camera Corp. Problem

The Niko Camera Corporation is a major Japanese company that specializes in producing high quality cameras with an especially fine lens. It sells many different models to meet the various needs of discriminating photographers (both amateur and professional) around the world.

Background on the Product of Concern

One of Niko’s newer models is an inexpensive disposable panoramic camera. Very light and compact, this camera is designed to be especially convenient for a traveler who wants to take high quality panoramic shots of beautiful scenery without carrying the usual photographic equipment required to do this. The key to this convenience is that the camera is designed to be used for just one series of shots. It comes with special film already loaded at the factory and no provision is made for reloading by the customer. Therefore, the camera is given back to the camera store when the customer wants to have the film developed after completing the allotment of 27 shots. After removing the film, the camera store then returns the camera to the factory so that most of its components can be reused in a recycled camera. The special design for one-time use by the customer (but recycling of the expensive components by the factory) enables selling the camera so cheaply that many customers now think of repeated purchases as a good alternative to repeatedly buying rolls of film to use in an expensive and inconvenient permanent camera.

2 See Lau, H.-S., and A. H.-L. Lau, “The Newsstand Problem: A Capacitated Multiple Product SinglePeriod Inventory Problem,” European Journal of Operational Research, Vol. 94 (Oct. 11, 1996), pp. 2942, and its references. Also see pp. 610–628 in Porteus, E.L., “Stochastic Inventory Theory,” in Heyman, D.P., and M.J. Sobel (eds.), Stochastic Models, North Holland, Amsterdam, 1990.

CD 19-14

Although the cameras are produced initially in Japan, North American camera stores return the cameras for recycling to a factory in the United States run by Niko’s North American Division. Our focus will be on this factory. Niko’s American factory has been selling an average of 8,000 of these recycled cameras per month to a number of wholesale distributors. However, since these distributors only submit purchase orders on a very occasional basis, sales fluctuate widely from month to month (but without any noticeable seasonal pattern). Figure 19.6 shows the pattern of monthly sales over the past year. Note that some months are nearly double the monthly average (e.g., 15,800 in March) while others are almost nil (e.g., 700 in August). This same kind of random fluctuation, with no particular trend or seasonal pattern, also has been observed in the months prior to last year.

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

B

C

D

E

Monthly Sales of Niko's Disposable Cameras

Month January February March April May June July August September October November December Sales 7,000 1,500 15,800 8,600 9,900 4,200 13,600 700 14,100 6,200 5,000 9,400

Figure 19.6

Niko’s sales of disposable panoramic cameras in each month of the past year.

Because of these fluctuations, the camera is only produced on a sporadic basis. Every few months, the needed production facilities are set up to produce this particular model. In one concentrated production run lasting just a few days, a very large number of cameras are produced and placed into final inventory. This run size has been set at 20,000, which covers sales for 2.5 months on the average. (As indicated in the preceding chapter, the number produced or ordered to replenish inventory is called the order quantity.) Although it is only possible to produce these recycled cameras from the cameras returned by camera stores, the factory always has had a plentiful supply of these returned cameras for its production runs. Once the decision has been made to initiate a production run, some time is needed to clear the required production facilities from other uses and set them up for this run. (Recall that this time between ordering a product and receiving it is referred to as the lead time.) The lead time for this camera generally is about one month. Since average sales over a lead time of one month are 8,000, it has become routine to order another production run when the number of cameras in inventory drops to 8,000. (Recall that this inventory level at which an order to replenish is placed is called the reorder point.) To summarize, here are the key data for how the inventory of this camera is being managed. Order quantity = 20,000.

CD 19-15

Lead time = 1 month. Reorder point = 8,000.

Last Year’s Experience

Last year began with 16,500 disposable panoramic cameras in inventory. The January sales of 7,000 reported in Figure 19.6 reduced this inventory level to 9,500 by the end of the month. The February sales of 1,500 then reduced it to 8,000. Since 8,000 is the reorder point, an order was given at the end of February to initiate a production run of 20,000. After the lead time of one month, these 20,000 cameras were received and placed into inventory at the end of March. Since March sales of 15,800 already had depleted the 8,000 in inventory and left 7,800 in backorders, part of the production run immediately was used to fill these backorders. This left 12,200 in inventory to begin April. Figure 19.7 shows the record of what happened throughout the year, where the diamonds in the plot record the beginning inventories in those months. Five orders were placed for production runs of 20,000 (although the last run hadn’t quite been set up by the end of December). Therefore, the beginning-ofmonth inventory levels fluctuated widely, with some values near 15,000 and some others near 0. Only one month (August) had a shortage (called a stockout) at the beginning of the month. This inventory level of 4,100 indicates that backorders for 4,100 cameras had accumulated in late July after the depletion of the inventory and that these backorders would need to be filled from the upcoming production run of 20,000 cameras. (Holding backorders when shortages occur and then filling them when the inventory is replenished is referred to as backlogging.)

CD 19-16

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 B C D E F G

Monthly Record of Niko's Inventory of Disposable Cameras

Month January February March April May June July August September October November December Sales 7,000 1,500 15,800 8,600 9,900 4,200 13,600 700 14,100 6,200 5,000 9,400 Beginning Inventory 16,500 9,500 8,000 12,200 3,600 13,700 9,500 -4,100 15,200 1,100 14,900 9,900 Ending Inventory 9,500 8,000 12,200 3,600 13,700 9,500 -4,100 15,200 1,100 14,900 9,900 500 Action None Ordered 20,000 at end of month Order received at end of month Ordered 20,000 during month Order received during month None Ordered 20,000 during month Order received during month Ordered 20,000 during month Order received during month None Ordered 20,000 during month

20,000 15,000

Inventory Level

10,000 5,000 0

September

-5,000 -10,000 -15,000

Time

Figure 19.7

Niko’s record of sales, inventories, and orders for each month last year, where the graph shows how the beginning inventory changes from month to month.

However, this figure does not show the full story, because columns D and E only give the inventory levels at the change of a month. Furthermore, the plot at the bottom simply connects the beginning-ofmonth inventories with line segments. By contrast, whereas Figure 19.8 uses dots to show these same inventory levels, the dashed lines then display approximately how the inventory level varied within each month as well. (This still is an approximation since it assumes that each month’s sales occurred evenly throughout the month rather than recording the actual individual sales during the month.) Note that four stockouts occurred during the year. Their sizes ranged from 1,235 to 7,800 cameras backordered. The durations ranged from a few days to a couple weeks. The distributors affected by the longer stockouts were not happy with this shoddy service, and several of them registered complaints with Niko management.

November

December

February

January

October

April

May

June

July

August

March

CD 19-17

CD 19-18 Management’s Concerns

Niko’s management has always taken pride in both the quality of its cameras and the quality of the company’s service to its customers. Therefore, the recent complaints from several distributors about delays in shipping one of the company’s most popular models, the disposable panoramic camera, has caused considerable concern. The North American Division’s Vice President for Marketing, in particular, is urging that something be done about this problem. She is suggesting having more frequent production runs to keep the inventory better stocked. At the same time, complaints have been received from the production floor about the relatively frequent interruptions in the production of other models caused by setting up for a production run for the disposable panoramic camera every two or three months. Although a production run is quick once the setup is completed, the process of setting up is quite complicated. A significant part of the expense in producing this camera is the direct cost of setting up and the additional cost attributable to disrupting other production. Therefore, the Vice President for Production strongly disagrees with the Vice President for Marketing. He recommends instead having much longer production runs much less frequently. He argues that this will solve two problems at once. First, it would provide larger inventories for longer periods of time and thereby greatly reduce the frequency of delayed shipments due to stockouts. Second, it would substantially reduce the annual cost of setting up for production runs, including the cost associated with disrupting other production. However, because it would increase inventory levels, the President of the North American Division is quite skeptical about this recommendation. For some time, he has been pushing the just-in-time philosophy of minimizing inventory by using careful planning and coordinating to provide items just in time to serve their purpose. This philosophy has enabled the company to greatly reduce its work-inprocess inventories while also improving the efficiency of its production processes. Although it has been necessary to maintain some inventories of finished products until they could be sold, the President is proud of the fact that even these inventories have been considerably reduced in recent years. The reductions in inventories throughout the company have provided substantial cost savings, including in the cost of capital tied up in inventory. These economies have been one of the key factors in maintaining Niko’s place as one of the world’s leading producers of cameras. Therefore, the President feels that it should be possible to solve the current problems without increasing the average inventory levels of disposable panoramic cameras. So what should be done? The President has called upon the North American Division’s Management Science Department many times in the past to address similar problems, with excellent results. Therefore, he has instructed this department to form a team to study this problem. The next section describes the management science team’s approach to the problem.

REVIEW QUESTIONS

1. 2. 3. 4. What has been happening in Niko’s North American Division that is causing considerable managerial concern? What is the concern of the Vice President for Marketing of the North American Division about the current situation? What recommendation is she making? What is the main concern of the Division Vice President for Production about the current situation? What recommendation is he making? Why is the Division President skeptical about his Vice President for Production’s recommendation? What company philosophy has he been promoting that relates to the current situation?

CD 19-19 19.4 The Management Science Team’s Analysis of the Case Study

The management science team begins by trying to diagnose why the frequent stockouts were occurring under the current inventory policy (order a production run of 20,000 when the inventory level drops to 8,000). Was this just a string of bad luck? Or did the current policy naturally lead to a high probability of a stockout occurring before the production run takes place? Just what is this probability? If a stockout occurs, what is the probability distribution of the size of the stockout (the number of cameras backordered when the stockout ends)?

Assessing the Stockout Problem

Since the lead time for a production run is approximately one month, the key to answering these questions is to estimate the underlying probability distribution of the number of cameras sold in a month. Examining the pattern of monthly sales over the past year shown in Figure 19.6 (along with similar data for other recent years), the team notes that these sales ranged pretty uniformly from almost nothing up to about 16,000. Therefore, the team’s best estimate is that the number of cameras sold in a month has a uniform distribution over the range from 0 to 16,000. Since this assumes that all the values over this range are equally likely, but that there is no chance of values outside this range, this distribution has the appearance shown in Figure 19.9. The mean of this distribution is 8,000, which corresponds to the observed average monthly sales.

When a production run is ordered with 8,000 cameras left in inventory, and the new cameras arrive a month later, the probability of a stockout occurring is just the probability that a month’s sales exceeds 8,000. With this uniform distribution, these probabilities are

CD 19-20

P (stockout) = P (monthly sales > 8000) = 0.5. With just a 50-50 chance of incurring a stockout after ordering a production run, having this occur four times in a row last year (as shown in Figure 19.8) was indeed a string of bad luck. This should occur only two times out of four on the average. This uniform distribution also indicates that there was further bad luck in terms of the size of the stockouts last year. The inventory level just before the order for the 20,000 new cameras is received is 8,000 minus the month’s sales. Therefore, the probability distribution of this inventory level also is a uniform distribution, but over the range from -8,000 (= 8,000 - 16,000) to 8,000 (= 8,000 - 0), as shown in Figure 19.10. The probability that this inventory level would fall as low as - 7,800 is extremely small (0.0125), but this did indeed occur at the end of March last year.

However, given management’s desire to provide high-quality service to the company’s customers, it seems unacceptable to have a probability of a stockout as high as 0.5 and to have the size of stockouts range as high as 8,000. Even without the bad luck of last year, such high numbers would inevitably lead to occasional significant delays in filling customer orders. Although customers would accept brief delays every once in a while, such frequent and lengthy delays need to be avoided.

CD 19-21

Conclusion: large. Both the probability of a stockout and the maximum size of a stockout are too

Alleviating Stockouts

Why is the probability of a stockout so high? The reason is that the method that was used to set the reorder point is faulty. This point was only set equal to the average sales (8,000) during the lead time (one month) for producing the next batch of cameras. No provision was made for the month’s sales exceeding the average, even though this should be expected to occur half the time. Conclusion: When reordering, a cushion of extra inventory needs to be provided in addition to the amount needed to cover the average sales during the lead time. (This extra inventory is referred to as safety stock.) With safety stock, the formula for setting the reorder point then is Reorder point = average sales during lead time + amount of safety stock = 8,000 + amount of safety stock. For example, Reorder point = 12,000 if amount of safety stock = 4,000. Changing the reorder point from 8,000 to 12,000 would change the probability distribution of the inventory level at the end of the lead time from the one shown in Figure 19.10 to that given in Figure 19.11. Since this new distribution is a uniform distribution over the range from - 4,000 to 12,000, the probability of a stockout now would be 0.25, with a maximum possible size of 4,000. Thus, stockouts now would occur only about once every four times a production run is ordered, on the average, and those shortages that do occur would tend to be somewhat smaller than before.

CD 19-22

Figure 19.12 shows approximately how the inventory level would have evolved throughout the past year if the reorder point had been 12,000. (As with Figure 19.8, the dashed lines in this graph approximate the evolution within each month by treating the month’s sales as having occurred evenly throughout the month.) Under this scenario, an order is fortuitously placed in mid-January and received in mid-February, in time to cover the unusually large sales of 15,800 in March. Consequently, a substantial number of cameras remain in inventory throughout the entire year, except for one very small and brief stockout in October.

CD 19-23

Now compare this figure to Figure 19.8. Note the dramatic improvement in avoiding stockouts and thereby avoiding delays in filling customer orders. It is true that part of the improvement was a matter of luck. Nevertheless, even under the worst circumstances, the higher reorder point of 12,000 would have avoided the more serious stockout problems shown in Figure 19.8. With a maximum possible stockout size of 4,000 instead of 8,000, the stockouts that do occur would tend to be both smaller and briefer, thereby causing much less damage to customer relations.

CD 19-24

Conclusion: Even when the amount of safety stock provided still permits occasional short stockouts, this safety stock can dramatically improve the service to customers by greatly reducing both the number and length of the delays in filling customer orders.

Choosing the Amount of Safety Stock

You have just seen that providing a safety stock of 4,000 cameras is much better than providing none at all. But is 4,000 the right amount? Note that the inventory levels shown in Figure 19.12 are much higher than in Figure 19.8, whereas the President wants to keep inventories down as much as reasonably possible. What is the best trade-off between the costs of holding inventory and the consequences of stockouts? Since choosing such a trade-off is ultimately a management decision, the management science team consults with management about their feelings regarding stockouts. These are the key questions posed to management. 1. 2. How important is it to reduce delays in filling customer orders? Considering that larger inventories would be needed to reduce delays, how would you compare the importance of reducing delays with the importance of holding inventory levels down? Considering that unacceptably large inventories would be needed to completely eliminate any delays, what would you consider tolerable in terms of the frequency, size, and length of stockouts?

3.

To help make these questions more concrete to management, the management science team describes the frequency, size, and length of stockouts that would be expected for each of several alternative amounts of safety stock. For example, here is the description for a safety stock of 4,000 cameras. With a safety stockout of 4,000 disposable panoramic cameras, a new production run would be ordered when the inventory level drops to 12,000 cameras. Since the average sales during the lead time for the production run (one month) are 8,000 cameras, the inventory would be adequate to cover sales in most cases. However, since a month’s sales can range as high as 16,000 cameras, and about a quarter of the months have sales between 12,000 and 16,000 cameras, a stockout would occur about once every four times on the average. With the current production runs of 20,000 cameras occurring about four times every ten months, this means that a stockout would occur about once every ten months. When it does occur, the size would range from very small to about 4,000 cameras backordered, so about 2,000 on the average. Since we sell about 80,000 cameras over ten months, this means that about 2.5% of our customers would incur a delay in having their orders filled. The delays would range from very short (in most cases) up to about a week, with an estimated average of about a third of a week. (The full week would result from a month’s sales of 16,000 cameras after ordering a production run, with the last 4,000 sales occurring during the last week of the month.) Beware, however, that these numbers assume that we can continue holding to a lead time of about one month. If an unexpected delay in a production run should occur, the possible shortages and delays would be extended accordingly. After the management science team elicits management’s views about this scenario and the several alternatives, the following conclusion is drawn. Conclusion: Management feels that providing a safety stock of roughly 4,000 cameras is needed to provide an adequate level of service to customers in minimizing delays in filling their orders.

CD 19-25

Considering the company’s just-in-time philosophy regarding the need to hold down inventories, management does not want the safety stock raised higher than this. Having established that the reorder point should be increased from 8,000 to 12,000, the management science team next wants to investigate what the order quantity (the size of each production run) should be. Realizing that this issue involves a trade-off between several types of costs, the team first turns to estimating these costs.

The Cost Factors

All the relevant cost factors for analyzing inventory problems were described in Section 18.2. The four types are (1) acquisition costs, (2) setup costs, (3) holding costs, and (4) shortage costs. The acquisition cost in this case is the cost of producing the disposable panoramic cameras (when using the recycled components). Excluding setup costs, this production cost is just $7 per camera. However, this cost turns out to be irrelevant for choosing the order quantity. The reason is that the order quantity does not affect sales, and it is sales that determine the number (and so the production cost) of cameras that will be produced eventually. The order quantity only affects the timing of when the fixed production costs will be incurred. However, setup costs are very relevant. The cost of setting up for a production run, plus additional costs associated with disrupting other production in the process, is estimated to be $12,000. The order quantity (size of the production run) determines how frequently this cost will be incurred. The holding costs encompass all the costs associated with holding the cameras in inventory. One important component is the cost of capital tied up in the inventory. The company pays an annual interest rate of about 10% on money borrowed to pay for the production of cameras that are not yet sold. The production cost (including setup cost) per camera is about $7.50, and the company’s cost invested in each set of recycled components of a camera is another $16.50, for a total of $24. Therefore, the cost of capital tied up for each camera in inventory is about $2.40 per year, or 20¢ per month. (This seemingly insignificant cost does add up when there are many thousand cameras in inventory.) Holding costs also include all the costs directly involved with storing the cameras, including the cost of the space, record keeping, protection, insurance, and taxes. When 12,500 cameras are in inventory (a roughly average level), these costs are estimated to add up to just about $1,250 per month, so about 10¢ per camera. Adding in the cost of capital tied up gives a total holding cost per camera of about 30¢ per month. Certain storage costs (e.g., space and protection) may not be directly proportional to the inventory level. However, as an approximation, it is assumed that the total holding cost per month is 30¢ times the current inventory level (except for ignoring negative levels that represent shortages). The shortage costs are more difficult to quantify. The main component is lost future profit from lost future sales caused by customer dissatisfaction with delays in filling current orders. This is difficult to estimate. Other minor components might include the cost of additional record keeping and handling for dealing with backordered cameras. Estimating shortage costs requires a managerial assessment of the seriousness of making customers wait to have their orders filled. To obtain this managerial input, the management science team poses the following question to the Vice President for Marketing. Question: We have discussed the fact that the worst case scenario with a safety stock of 4,000 cameras would be to incur a stockout of about 4,000 cameras with a delay in filling the orders of about one week. If you were to put a dollar figure on the damage that such a stockout would cause the company in terms of lost profit from lost future business, etc., what would that figure be? In other words, if it were possible for the company to pay some money to prevent this one

CD 19-26

stockout completely, how much should we be willing to pay to do so? We realize that it is difficult to pin down an exact dollar figure, but we are only asking you to apply your judgment as best you can in responding to this question. Response: $10,000. As a rough approximation, the management science team assumes that the shortage cost from any stockout should be proportional to both the size of the stockout and the resulting average delay in filling the orders. Thus, with a cost of $10,000 for a shortage of 4,000 cameras for one week, the cost for extending the average delay to one month (four weeks) is assumed to be about $40,000, or about $10 per camera per month of delay. This assumption is definitely questionable. The actual shortage cost for a delay of a month (infuriating the customers) might be considerably more than four times that for a delay of a week (mildly concerning the customers). However, the management science team feels that the assumption of proportionality is a reasonable one over the range of actual delays (up to roughly one week) that would be incurred with a safety stock of 4,000 cameras. Table 19.1 summarizes all these cost factors and their estimated values. The symbols in the table are the same as introduced in Chapter 18 for these types of cost except that the management science team now is measuring costs on a monthly rather than annual basis.

Table 19.1 The Cost Factors for Niko’s Inventory Problem

Type of Cost Setup cost for a production run Symbol K Value $12,000 $0.30 $10

Holding cost per camera per month h Shortage cost per camera per month of p delay in filling the customer order

Choosing the Order Quantity

With these cost factors pinned down, the management science team now is ready to address the problem of determining what order quantity (number of cameras to be produced in a production run) would minimize the sum of all these costs. (Recall from Chapter 18 that the order quantity that minimizes the total average cost is commonly referred to as the economic order quantity or EOQ for short.) Here are the trade-offs involved in making this decision. 1. The average monthly setup costs are decreased by increasing the order quantity, because this decreases the average number of setups required per month. 2. 3. However, the average monthly holding costs are decreased by decreasing the order quantity, since this decreases the average inventory level. However, the average monthly shortage costs are decreased by increasing the order quantity, because this decreases the average number of opportunities for stockouts per month.

Section 18.5 presented the EOQ model with planned shortages to address exactly these same trade-offs. In fact, that model completely fits Niko’s inventory problem with just one important exception. The exception is the model’s assumption that sales occur at a constant rate, with no variation from month to

CD 19-27

month. Thus, for the Niko problem, the model would assume that the average monthly sales of 8,000 disposable panoramic cameras are the actual sales spread evenly through the month for each and every month. The reality, of course, is that Niko’s sales of these cameras vary greatly from month to month. How much effect does this month-to-month variation have on the average total monthly cost that needs to be minimized to determine the economic order quantity? The effect on the average monthly shortage costs is rather considerable, since the month-to-month variation tends to increase the frequency and size of the shortages. However, the effect on the average monthly holding cost is only slight. The fact that no holding costs are incurred during shortages has a slight effect, but otherwise the average inventory level is not significantly affected. Furthermore, there is no effect on the average monthly setup costs, since the variation does not affect the average frequency of setups for production runs. Consequently, the overall effect of the month-to-month variation in sales on the average total monthly cost is modest. Therefore, using the model from Section 18.5 to calculate the economic order quantity (but not the reorder point) for Niko’s problem provides a pretty good approximation. The management science team decides to adopt this approach. To review, the important notation from Section 18.5 (now expressed on a monthly rather than annual basis) is K, h, p as defined in Table 19.1, D = monthly sales rate = 8,000 as the average for Niko’s problem, Q = order quantity, S = shortage just before an order quantity is received.

Using an asterisk to indicate the optimal value of Q and S, their formulas given in Section 18.5 are Q* =

h + p 2KD , p h h $ Q*. " h + p%

! S* = #

An Excel template is available in your MS Courseware for performing these calculations for you. For Niko’s problem, plugging the values of K, h, and p given in Table 19.1 (plus D = 8,000) into this template gives the results shown in Figure 19.13. Since Q* = 25,675 cameras and S* = 748 cameras, the inventory level jumps from a shortage of 748 cameras to a level of Q* - S* = 24,927 cameras when the order quantity (the output of a production run) arrives.

CD 19-28

A 1 2 3 4 5 6 7 8 9 10 11 B C D E F G

EOQ Model with Planned Shortages (Niko Case Study)

D= K= h= p= Data 8000 $12,000 $0.30 $10 (demand/year) (setup cost) (unit holding cost) (unit shortage cost) Max Inventory Level Annual Setup Cost Annual Holding Cost Annual Shortage Cost Total Variable Cost Results 24927.08 $3,739.06 $3,630.16 $108.90 $7,478.12

Q= S=

4 5 6 7 8 9

Decision 25675 (order quantity) 748 (maximum shortage)

F Max Inventory Level =Q-S

G

Annual Setup Cost Annual Holding Cost Annual Shortage Cost Total Variable Cost

Cell G7 G6 G8 C4 C6 C5 G4 C7 C10 C11 G9

=K*D/Q =h*(MaxInventoryLevel^2)/(2*Q) =p*((Q-MaxInventoryLevel)^2)/(2*Q) =AnnualSetupCost+AnnualHoldingCost+AnnualShortageCost

Range Name AnnualHoldingCost AnnualSetupCost AnnualShortageCost D h K MaxInventoryLevel p Q S TotalVariableCost

10 11

B C Q = =SQRT(2*D*K/h)*SQRT((p+h)/p) S = =(h/(h+p))*Q

Figure 19.13

The Excel template for the EOQ model with planned shortages (analytical version) in your MS Courseware is applied here to find the order quantity for the Niko problem.

Using these quantities, Figure 19.14 shows how the EOQ model with planned shortages assumes Niko’s inventory level would evolve over a year, starting with last year’s initial inventory of 16,500 cameras. (Keep in mind that this model assumes that sales occur at a constant rate, which does not fit Niko’s situation.) With a lead time of one month, the model orders a production run each time just one month before a shortage of 748 cameras occurs. Since the assumed sales over this lead time are 8,000 cameras, the model thereby sets the reorder point at 8,000 - 748 = 7,252 cameras.

CD 19-29

CD 19-30

However, it is crucial that only Q* and not this reorder point be used from the model. Because Niko’s actual sales do vary considerably from month to month, a substantial safety stock is needed as a cushion against the sales during the lead time being much higher than average. Therefore, based on the earlier analysis of how much safety stock is needed, the management science team makes the following recommendation. Recommended inventory policy: Whenever the number of disposable panoramic cameras in inventory drops to 12,000, order a production run of 25,675 cameras. Figure 19.15 depicts how this inventory policy would have performed throughout the past year. (Once again, the dashed lines approximate the evolution of the inventory level throughout each month by treating the sales as having occurred evenly throughout the month.) A substantial inventory, ranging from 2,513 to 34,211 cameras, would have been maintained throughout the year. However, in light of the probability distribution shown in Figure 19.11, a bit of good luck was involved. Despite having a probability of 0.25 that a stockout will occur before an order is received, this never happened in the four chances during the year.

CD 19-31

CD 19-32 Management’s Reaction to the Recommended Inventory Policy

This recommended inventory policy provides at least a good approximation of an optimal policy (an optimal combination of a reorder point and an order quantity) that minimizes the total average monthly cost. Or at least it is an “approximately optimal” policy under the conditions given to the management science team (a lead time of one month, the cost factors in Table 19.1, etc.). But is it really a sound policy from a managerial perspective? The three members of management dealing with this problem have expressed their reservations about the recommendation. The President is unhappy about the large increase in inventory levels that would result from the substantial increases in both the reorder point and order quantity. He realizes the need for some safety stock, but feels that there should be a better approach to the problem that is more in line with the company’s just-in-time philosophy of minimizing the use of inventories. The Vice President for Marketing is happy that the new safety stock will considerably alleviate the stockout problem. However, she is somewhat concerned that shortages of various magnitudes still can be expected to occur about once per year. She continues to emphasize that high priority needs to be placed on maintaining and building the company’s reputation for good and prompt service to its customers. The Vice President for Production is quite unhappy. He had emphasized the problems caused in disrupting other production by shifting facilities over so frequently to set up for production runs of the disposable panoramic camera. Although the recommendation would slightly decrease the frequency of production runs, he does not feel that the disruption problems have yet been dealt with adequately. The costs associated with this “approximately optimal” policy are indeed disturbingly high. The profit margin on each camera sold is only about $2, due partially to these high costs. Here is a breakdown of each of the costs of major concern to one of the members of management under the recommended inventory policy.

HOLDING COST CALCULATIONS

Average Inventory Levels: Just before an order is received Just after an order is received = 12,000 - 8,000 = 4,000. = 4,000 + 25,675 = 29,675.

Overall average

! 4, 000 + 29, 675 $ = # & " % 2

= 16,837.

Holding cost per camera per month, h = $0.30. 3 = 16,837 ($0.30) Average monthly holding cost = $5051.

3Actually,

this calculation slightly understates the true average monthly holding cost because it charges a negative holding cost instead of zero when the inventory level is negative because of shortages.

CD 19-33 SHORTAGE COST CALCULATIONS

Average number of orders per month

! $ sales = # " order quantity & %

! 8,000 $ = # = 0.31. " 25,675 & %

Probability of a stockout before order received Expected number of stockouts per month

= 0.25. = 0.25 (0.31) = 0.078.

Average stockout size

! 4,000 $ = # " 2 & %

= 2,000.

Estimate of average delay per camera delayed 4

=

! 1# " 3$ week

= 0.08 month. Shortage cost per camera per month, p Average monthly shortage cost = $10. = 0.078 (2,000) (0.08) ($10) = $125.

SETUP COST CALCULATIONS

Average number of setups per month

! $ sales = # " order quantity & %

! 8,000 $ = # = 0.31. " 25,675 & %

Cost per setup, K Average monthly setup cost

= $12,000. = 0.31 (12,000) = $3,720.

Adding these three average monthly costs gives

4

Although the maximum delay is about a week, most delayed camera orders do not wait nearly this long, eith

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