REPAIRABLE SERVICE PARTS INVENTORY MANAGEMENT lecture note

REPAIRABLE SERVICE PARTS INVENTORY MANAGEMENT lecture note Morris A. Cohen The Wharton School University of Pennsylvania Philadelphia, PA 19104 Vinayak Deshpande Krannert School of Management Purdue University W Lafayette IN 47907 Nils Rudi The Simon School University of Rochester Rochester, NY 14627 November 24, 2004 Do not quote, cite, or reproduce without permission. c 2004 1

Prerequisites The reader of this note should have some basic knowledge of inventory models and probability. Introductory courses at college level that touch on these areas or an introductory course on operations management are sufficient. A large portion of the material is rather quantitative in nature, but the presentation attempts to limit the mathematical aspects as much as possible and emphasizes the applications. 1 Introduction For many firms, after sales support is a critical component of their competitive strategy. This is especially true in industries where availability of the product is considered to be mission critical by the customer. Examples of industries where effective after sales support is a necessity include: Transportation (cars, rail, ships, air planes, space shuttle) Computer systems (banks, air traffic control) Consumer electronics (TV, freezer) Process industry (oil refining, mining, food processing, nuclear power, offshore oil platforms) Telecommunication (satellite systems, radio lines) Military applications (weapon systems) Construction equipment (bulldozers, cranes) Seminconductor fabrication (process machine, testers) Medical (X-ray, CAT Scan, MRI) In order to provide such support, firms must make significant investments in assets that include infrastructure (repair depots, warehouses, communication and logistics), spare parts inventory and highly trained personnel (customer engineers and repair technicians). For the most part these assets are not highly utilized since the demand for service support can be highly erratic and is often very difficult to predict. Service parts management is an important requirement for meeting customer expectations for product support in a cost effective manner. Most customers are dependent on the uptime of their products, and if a break down occurs, they expect rapid restoration of service through repair and/or replacement. To 2

perform such rapid repair, it is necessary to have the required service parts available. This note is an introduction to the basic management procedures required to achieve rapid response, in a cost effective manner which limits investment in spare parts inventory. 1.1 Characteristics of service parts management The challenges faced by managers of service parts inventory are very different from those encountered in finished product and direct material supply chains. There are a number of distinct characteristics that make it especially difficult to provide a high level of customer service at a low cost. These characteristics include: 1.1.1 Criticality Criticality refers to the requirement for a part to be functioning in order for the product to be available for use. Typically there are different degrees of such criticality. A car, for example, is not driveable without a working fuel pump. The car works well enough, however, under most conditions, with a defective right-hand-side wiper blade. The model introduced in this note focuses on highly critical parts, that are often characterized by high cost, long repair/procurement lead times and the fact that they are usually repaired rather than being scrapped if they fail. 1.1.2 Random break-down/planned maintenance There are two main triggers that initiate the replacement of a service part; i) it breaks down unexpectedly or ii) it reaches a point of planned replacement regardless of its condition. An example of the first case is a starter mechanism in a car that suddenly stops working. An example of the second case is an oil filter that is planned for replacement after a new car has been driven for 5,000 miles. Product owners have little control in the first case, and since the cost of delay can be quite high, it is necessary to keep high levels of service parts inventory in the event that a break down occurs. The second case, however, is something we often know about in advance, or can be predicted with some level of certainty (e.g based on product age or usage), and hence the appropriate service part can be made available specifically for this purpose at the appropriate time and place. When managing service parts, we often separate these two kinds of demand and when deciding on safety stock levels, we base this decision only on the demand due to random break-downs. 1.1.3 Slow moving Many repairable spare parts have extremely low demand (i.e. slow moving), often averaging less than one unit per year. Consider Figure 1, which is a 3

frequency plot of global demand for parts over a 12 month period at KLA- Tencor, a leading manufacturer of equipment that is used in semiconductor fabrication plants. Note that in their environment 75% of the parts planned had one demand or less and that 60% of the parts with demand had 3 or less pieces requested. Such low demand requires very different control methods than those used for high-volume parts, which often turn tens of times a year. Figure 1: Demand for parts at KLA-Tencor over a 12 month period Low demand also makes forecasting challenging, since one has very few observations of historical usage. This is especially true in the initial stage of a part s life cycle, i.e. upon its introduction as a new product, where one needs to rely on engineering (mean time between failure) estimates. 1.1.4 Repairable parts Is is often worthwhile to repair more expensive service parts after a breakdown. For example, a motherboard for a main-frame computer can cost tens of thousands of dollars. If it breaks down, it is replaced by a good motherboard. But the cause of the break-down might be just a small chip that costs $10 and is easy to change. By repairing the defective board, one generates a good motherboard that can be used as a service part for potential future break-downs. Figure 2 illustrates the cycle of repairable service parts. 4

Figure 2: Closed part cycle for repairables We see here that a broken part goes to the repair queue if it is worth repairing, and otherwise it is scrapped. When the broken part s turn comes, it is repaired and then placed in the inventory of good parts. The inventory of good parts can also be replenished by purchases from the supplier of the part. These good parts are then used to replace parts that break down in a product, which we will henceforth refer to as a system (consisting of a collection of parts). 1.1.5 Long lead times Lead times for service parts, especially for repairable parts, are often very long (e.g. up to 18 months in aerospace and defense) and highly variable. The repair of a high-tech part often needs to take place at a central depot which may be very distant from the installed base system location. Such repair may require specialized resources, which include extensive testing both before and after the repair itself. The return time delays for defectives, procurement lead times for components used to repair defective parts and the actual repair/testing time all contribute to the overall part lead time. 1.1.6 Parts relationships (system perspective) In finished goods supply chains, each part is a product and relevant service measures such as fill rate, can be evaluated at an part level. The purpose of service parts, however, is to support maintenance and repair of systems, meaning 5

that in the case of a system breakdown the necessary service part should be available so that it can be replaced immediately. This ensures that the system will be up and running again with minimal interruptions (downtime). Hence, the relevant service measure is defined at a system level, since this is what the customer cares about, e.g. the owner of a car is concerned with the driveability of the vehicle and not the service level of parts at the repair shop. This leads to use of system availability service metrics and that are driven by the level of service for the parts. If, for example, one wants a 95% availability for a particular system (i.e. the product is down no more than 5% of the time awaiting delivery of a part), then one may choose to set service level (fill rate) targets for parts to be on average 95%. This average part fill rate could be realized by setting a lower fill rate for very expensive parts and a much higher fill rate for very inexpensive parts. Figure 3: Target service level is on system rather than on part basis 1.1.7 Cost differences Service parts supporting the same system are often very diverse in terms of costs. For example, an F16 fighter jet is supported by parts ranging from fuses that cost virtually nothing to field replacable units, such as the radar system that might cost more than $2,000,000. One will, obviously, make sure not to have an F16 grounded on an aircraft carrier due to the lack of a fuse, while the 6

radar system is so expensive that one may be able to tolerate a lower service level for it at the carrier and stock a limited number at a central depot only. 1.1.8 Large number of parts When optimizing a system of service parts, one needs to evaluate the parts in relation to each other. As we shall see there are significant interactions among all of the parts in terms of the total budget for inventory and the overall system (product) service level. This makes the problem of managing service parts inventory challenging in terms of the underlying mathematics as well as the solution times required to generate an optimal solution. Indeed, service parts management for inventory systems where each part can be optimized independently of the other parts are considerably easier to deal with. 1.2 Advanced characteristics We will now turn to some advanced characteristics that are found in some service parts environments. 1.2.1 Multiple locations In the case that customer systems are located at different locations, one often also locates service parts inventories at multiple locations to avoid shipping delays and to pool risk 1. The forward locations are typically supported by a central location that also keeps inventory in addition to performing repairs. Also, all locations might support each other through transshipments, meaning that if one location has excess stock while another has a stock-out, the former can transship to the latter. Figure 4 illustrates this type of network. 1.2.2 Cannibalization If two systems of the same type are waiting for different types of parts, then one of them can cannibalize the other for parts. For example, if one F16 is waiting for a radar system to become available, while another F16 is waiting for a new catapult mechanism to become available, then the former can get the working radar system from the latter. This reduces the number of systems that are down from two to one. In the military, aircraft used as a part source through cannibalization are referred to as hanger queens. 1.2.3 Express shipments Some suppliers of service parts utilize express shipments in addition to regular shipments, especially in those cases where a customer system is down and is 1 Placement of stock at a central location to fill demand originating from several local sites can reduce the overall variability in the system. 7

Figure 4: Schematic example of multi-location service parts network awaiting delivery of a part. Teradyne, for example, offers two classes of repair services to its customers, one short and the other long at high and low cost respectively. Customers will opt for the long delivery time option in cases where they have sufficient on-hand inventory and/or spare system capacity. Options of this type further complicate the analysis since multiple delivery modes have different lead times and could affect stocking levels. 1.2.4 Multiple Customer Segments In many service support environments different customers have different priorities and needs for support. For example a squadron of military aircraft engaged in combat require much higher levels of availability that a squadron engaged in training exercises. Similarly a company trading in foreign exchange futures cannot tolerate downtime for its internet routers while a company with a collection of redundant servers used to support non-mission critical processes will not be willing to pay a premium for rapid repair. Such differentiated service needs can affect both target inventory stocking levels and real time fulfillment priorities. 1.2.5 Multiple Indentures All assembled systems can be described by a Bill of Materials (BOM), which specifies how each part is used in putting the final system together. Typically 8

BOMs are described by multiple levels or indentures which indicate the hierarchy of part usage. Thus at the highest level we have the final system. At the next level we have the major assemblies used to manufacture the system. Each assembly can be broken down into sub-assemblies. This process repeats until we get to the bottom level of components. A BOM can also be specified for service support. At the level below the system we have Field Replacable Units (FRUs), which are the major assemblies used to restore a system to running condition (e.g. the radar system in an F-16 plane). The various levels are referred to as indentures. An investment in service parts could be made at all indenture levels. Having FRUs available leads to faster system restoration. These are the most expensive parts however. Thus there is a tradeoff in determining stocking levels across indentures between cost and speed. Overall repair lead times will be determined by the service level of parts at all indenture levels. 2 Repairable service parts Repairable service parts are often those parts with low demand rates, long new buy procurement lead times and high unit costs. Hence it is especially important to manage repairables well. This chapter presents a powerful procedure for optimizing repairable service parts inventory decisions. This procedure is also the foundation for the methods used in more advanced scenarios (e.g. multilocation). 2.1 Fundamentals of service parts modelling This chapter provides the foundations for modeling and optimizing service parts systems. We start by considering aspecifictype of service part. 2.1.1 Failures of parts By failure rate, we mean the rate of failure occurrence. We will characterize here the failure rate for each service part. This rate can be measured either by the number of failures per unit of time or by the time between failures. In this note, we will denote the average number of failures per year by λ. If, in one s house, an average of 4 light bulbs fail in a year, then λ =4. Thetime between failures canthenbeexpressedas 1 λ. The average time between failures for light bulbs is then 0.25 years, or 3 months. Memoryless failure rate One of the fundamentals in modeling service parts is the assumption of a memoryless failure rate. This means that what will happen in the future does not depend on what happened in the past, or in other words what has happened does 9

not affect what will happen. To illustrate the memoryless property, let s look at some examples that are not memoryless: a. Wear and tear A part is more likely to fail as it gets older and becomes more worn. Consequently, changing a part makes it less likely that this part will fail in the near future. This is often the case for mechanical systems (i.e. nonelectronic). The population of installed systems and the service parts included in these systems are often of different ages. Hence, the fact that one part is replaced could affect the total demand, (failure rate), for the system. b. Failures due to external events Some systems are exposed to external forces. One example would be telecommunication centers that are exposed to lightning. During periods with lightning there is a higher probability of parts in a telecommunication center failing than during periods without lightning. To link it to the definition of memoryless failures, let s look at it from a slightly different angle: if a part in a telecommunication system has just failed, it is likely that there is currently lightning in the area, and hence it is more likely that another copy of the same part will fail at a different telecommunication center in the same area. It follows then that what will happen in the future depends on what has (recently) happened. Other examples of systems that are exposed to external forces include electricity networks that are exposed to wind and rain. c. Detection due to an external event Similar to failures due to external events, we also might have detection due to external events. An example of this might be windshield wipers. When the sky is blue, no one thinks about changing windshield wipers. In a heavy rain storm, however, many people that have poor windshield wipers will drive by a gas station and have them replaced. The individual demands then come together at the station, and are not independent of each other or the past. d. Demand correlation Demand for parts are triggered by system failures. Typically multiple parts will fail together (e.g. a power supply and control boards). In such cases demand for parts are not independent. Both replenishment and repair can also require kits which contain a collection of parts used together. This also leads to correlated demand pattern. e. Small stuff Consumables are often used in batches. When one is replaced, one also changes several others since they don t cost much releative to the fixed cost of ordering a batch. 10

We see then that the conditions where the failure rate is not memoryless could often appear in real life settings. It follows that, for repairables, the assumption of a memoryless failure rate could represent a strong assumption. Empirical research has shown, however, that the results derived from models that make this assumption lead to very reasonable (near optimal) decisions in most real settings. As you will see, this assumption also simplifies the analysis for repairables. For the remainder of this note we will therefore make the memoryless failure rate assumption. Number of failures in a time interval: Poisson distribution We consider here the number of failures in a given time period, (measured in years). It turns out that the only probability distribution for the number of events (failures) in a period of time where the events are memoryless is the Poisson distribution. Let S(t) be the number of failures in t years, which are assumed to follow a Poisson distribution. The probability that the number of failures in t years is equal to s is then given by 2 : λt (λt)s Pr (S(t) =s) =e (1) s! The probability that 3 60W light bulbs will fail during the next half year (with λ = 4) is then: 4 0.5 (4 0.5)3 Pr (S(0.5) = 3) = e =0.18 3! or 18%. This means that the probability that we will need exactly 360W light bulbs is 18%. In the management of service parts, we are more interested, however, in how many spares are needed to avoid a shortage. The probability that no more than s failures will occur during t years gives the foundation for this, and is given by the formula: Pr (S(t) s) = s λt (λt)j e j! j=0 = s e λt (λt) j 2 Note that e represents the number 2.71828 and that s! = 1 2... (s 1) s, e.g. 3! = 1 2 3 = 6, and by definition we have that 0! = 1. j=1 j! (2) 11

The probability that no more than 3 60W light bulbs will fail during the next half year is then: [ ] (4 0.5) Pr (S(0.5) 3) = e 4 0.5 0 (4 0.5)1 (4 0.5)2 (4 0.5)3 + + + 0! 1! 2! 3! = 0.1353 [1 + 2 + 2 + 1.3333] = 0.86 or 86%. This means that the probability that 3 light bulbs will be sufficient for one half year s use is 86%. 2.1.2 The repairables cycle As discussed in the introduction, repairable service parts follow a closed cycle as illustrated in Figure 2. In a closed cycle no parts are added to or taken out of the system. Following Figure 2, we see that if a part fails, it is replaced by a good part from the service parts inventory, if available (otherwise the system waits until a good part becomes available). The failed part is then sent to the repair queue. When its turn comes, it is repaired and then placed in the good parts inventory. Sometimes there are restrictions on whether a part can be repaired. Among these restrictions are physical restrictions (e.g. the part may be too severely damaged to be repaired) and policy restrictions (e.g. maximum number of three repairs is allowed for a specific part). In such cases, the cycle will be open, meaning that material leaks out through disposal at a testing stage and gets resupplied through additional acquisitions. Figure 5 illustrates such a cycle. Modelling the repair process If part failures are memoryless with an annual rate of λ per year, then the arrivals into the repair queue are also memoryless. The repair lead-time is defined as the time spent by the failed part to reach the repair-depot, plus the time spent in queue at the repair depot, plus the time spent in actual repair process, plus the time to return the fixed part to the warehouse. It is typically assumed that the repair-lead times for parts are independent and identically ditributed with a mean of L years. Thus L years is the average time from the part failure until the time the part is placed in the inventory of good parts. Under this assumption, the number of parts in the repair pipeline (i.e. parts in transit to the repair shop, parts waiting in the repair queue, and parts in actual repair process) also follows a Poisson distribution. Let R denote the number of units that are in the repair pipeline. Thus R is distributed as a Poisson random variable with mean equal to λl. The probability of j units being due from repair at the same time is then: 12

Figure 5: Open cycle for repairables λl (λl)j Pr (R = j) =e (3) j! The above result is remarkable because the actual shape of the repair leadtime distribution does not matter. Only the mean of the repair lead-time is important to measure the repair pipeline. Although in practice repair leadtimes are often not independent (because of interactions in the repair-shop), this approximation has been validated by a number of practioneers (Sherbrooke, 2004). This assumption is usually well justified when there exists ample capacity to meet demand for repair. Note that if not all parts of a specific type are repairable (i.,e. a percentage of failed parts are scrapped), then the part repair leadtime is a weighted average of the repair and new buy procurement lead times. 2.2 Measuring Customer Service Many alternative measures of customer service can be defined. We will consider here two main types of service measures that are considered especially relevant for service parts management. These two types define part and system service performance. 13

Part service Service metrics at an part level measure (and specify) the service level of parts independently of each other. This is the traditional service approach for tracking customer service in inventory systems, and it applies primarily when the part is a finished product. Part service targets are typically specified for groups of parts, where the groups are made up of similar products. Examples include retailers and mail order companies where a group could be a product family, e.g. men s dress shirts. The specific service measures that are most often used at an part level are: Probability of stockout This measures the probability that a stockout will occur during a specified time interval at a location when either a customer demands an part or the system generates a replenishment order to be filled from that location. It does not reflect the number of replenishments per year or the size of the stockout. While stockout probability is very easy to compute and to understand, in terms of the underlying mathematics, it does not relate well to the final customer s perception of service quality, especially when it is measured at an internal location (e.g. the central warehouse). Fill rate This measures the expected value of the percentage of demand that can be fulfilled directly from available inventory, i.e. off-the-shelf. This metric is most often used by supply chain managers when they refer to service level. It is defined by the (average) ratio of the quantity shipped to the quantity demanded. Backorders Fill rate only considers if an part is immediately available from inventory, i.e. at the point in time when the demand for the part is realized - it does not reflect how long the customer has to wait in case the part is not available. For spare parts this is an important issue, since delay in receiving a part needed for a system repair contributes to the unavailability of the system for customer use (downtime). The backorder rate specifically measures the average number of parts that systems/customers are waiting to receive. If, for example, one has no backorders half of the year, a backorder of 1 unit a quarter of the year, and a backorder of 2 units for the remaining quarter of the year, the average number of backorders for the year is 0.75. The problem with the backorder rate is that it is not very intuitive, and hence it is not easy for customers to relate it to their overall service level needs. Another issue is that in some environments excess demands are not backordered at the point where they enter the inventory system. Rather shortages may be transferred to emergency backup locations (either laterally to other forward locations or vertically to a central warehouse). 14

Response time/backorder delay This is the average amount of time that a customer has to wait for their order to be fulfilled. If the part is available off the shelf, at the point where demand is realized then there is no (or minimal) delay. If the demand order cannot be filled at that location and the demand is either backordered or transferred, then the customer will have to wait until the part arrives. It should be clear that for a given part, all of the part service metrics introduced here are inter-related and are determined by the interaction between the level of inventory of the part, the process that generates demands and any delays or lags in the system. System service From a customer/system user perspective, individual part service levels are not that relevant. Consider for example a hot dog stand - having a really high service level on buns might not help all that much if one is out of sausages half of the time. Service measured at a system level is typically more relevant for service parts since the purpose of the after sales service supply chain is to keep a system up and running. As we shall see system service metrics are based on some combination of part service performance that is weighted by the importance of each part to the system. Different service measures at the system level that are relevant for service parts management include: Average fill rate When measuring fill rate at a system level, one considers the percentage of parts that are required for repair that are available from on the shelf inventory. The system fill rate is then the weighted average of the fill rates for the parts that supports the system, with the weights being the relative demand rates for each part per year. Total (System) Backorders Thesystemwidebackordersissimplythesum of expected backorders of all parts that are used to support the system. System availability As we already have noted, the purpose of service parts is to keep systems up and running - in other words as available as possible. It can be proved that the average availability of a system is determined by the total number of backorders. The resulting availability metric is a measure that is both intuitive and directly reflects the customer goal of generating value through the use of the system. When choosing service measures, the following considerations are useful: The service measure should be understood by both the customer and the service supply chain organization. It also should be closely related to the intuitive understanding of service as perceived by the end user customer. 15

It should reflect the consequences that a stock out has on customer service and not just on internal process measures of efficiency and performance. When specifying a target for any service level, the value should be based on: The cost of a system being down (one hour down for an oil platform can be of enormous cost). The expectations and requirements for availability (many people cannot tolerate being without their TV for very long). Inventory position We define inventory position for a specific part as the total number of parts in the system. This includes the number of parts on the shelf and immediately available for customer use as well as the number currently in repair or in transit minus the number of units backordered. Let s be the target number of units for the inventory position for a specific part type. If there currently is a system that is waiting for this part, it implies that there are no such parts on the shelf, (otherwise they would have been used to repair the system). Hence, in the case of one or more systems waiting for a specific part, (i.e. backorder) it means that more than s units are currently being repaired, or waiting in the repair queue or are in transit. We shall refer to the total number of such units to be the number of units in repair and denote it by R. If one system is waiting for a specific part, then the number of units in repair of this part is R = s +1, if two systems are waiting for the type of part then R = s +2,andsoon. Figures 6 and 7 illustrates how part failures, repair completions and the target inventory position drive the number of units in repair. Probability of No Stockout Following the previous discussion, the probability of filling a demand for a part is equal to 1 - Probability of a stockout and can be expressed as: PNS(s) = Pr(R =0)+Pr(R =1)+...+Pr(R = s 1) = Pr(R s 1) (4) In order to deal with a system of spare parts, we need some more notation. Let n be the number of spare parts that supports the system. We index these spare parts by i, so when a variable has subscript i, it means that we are talking about a specific service part out of the total of n service parts. System fill rate probability is defined as: PNS(s 1,s 2,...,s n )= 1 n i=1 λ i n λ i Pr (R i s i 1) (5) i=1 16

Figure 6: Inventory Dynamics a. S=2 Figure 7: Inventory Dynamics b. S=3 17

where the probability of not stocking out for each part is weighted by its relative contribution to the total demand for all parts 3. Backorders As has been noted, the expected number of backorders for an part can be expressed as: BO(s) =Pr(R = s +1)+2Pr(R = s +2)+3Pr(R = s +3)... = (j s)pr(r = j) (6) j=s To obtain the expected or average number of backorders, the different realizations of backorder values are weighted by their respective probability. So the expected number of backorders is 1 times the probability of having 1 number of backorders at any time, plus 2 times the probability of having 2 number of backorders at any time, and so on. One special case of this expression that is useful, is the expected number of backorders when the inventory stocking level is equal to zero units. This is given by the simple expression: BO(0) = λl (7) The expected number of backorders on a system level is simply the sum of the expected number of backorders of all parts that are supporting the system. Let BO i (s i ) be the expected backorders for part i when the inventory position is equal to s i. The system-wide expected backorders is then: System availability SBO(s 1,s 2,...,s n )= n BO i (s i ) (8) Let K be the number of systems being supported, i.e. installed and being used by customers. A simple estimate of the availability due to service parts is then: A = K SBO (9) K If for example, there are 100 systems with an average number of system backorders of 2.5 units, then the system availability is 0.975 or 97.5%. This 3 Note that Fill rate =1 P {Stockout} for the special case of a Poisson distribution. i=1 18

measure is not completely accurate, but it is a reasonably good approximation for most practical cases. To summarize, both part and system service metrics are based on management s selection of a vector of target inventory positions s 1,s 2,..., s n.thevalues for each metric are also affected by the probability distribution for each part at each location (Poisson with mean λ i and demand lead time L i ). Variation in s i leads to variation in service metric levels for part i alone. System service metrics on the other hand, are a function of all of the target stocking levels, s 1,s 2,..., s n. 2.3 System optimization We are now ready to put things together. When optimizing a system of service parts, we seek to allocate the inventory investment so that it gives maximum system availability. In other words, we want to spend the next dollar on inventory on the part that provides the highest impact on reducing backorders (which, as noted, can be shown to be equivalent to improving availability). The overall optimization problem can be stated as the following: Maximize Availability S. T. Total Inventory Investment Inventory Budget or, Minimize Inventory Budget S.T. Availability Target Availability In practice, these optimization problems can be complicated by the existence of additional service constraints (by location, system type, customer etc.) and additional resource constraints (local budget, cash flow, etc.). The following summarizes the procedure of system optimization: 19

1. Start with target inventory position of zero for all parts. 2. Calculate expected backorders for each part. 3. Calculate the reduction in backorders per dollar invested by increasing the inventory position by one unit for each part. 4. Choose the part with the largest reduction in backorders per dollar invested, and increase the inventory position of this part by one unit. For this part, recalculate the two measures of expected backorders and reduction in backorders per dollar invested caused by increasing the inventory level of this part by one unit. 5. Continue to increase the inventory position of the most promising candidate by one unit until the desired service level (e.g. target availability) is reached or until all of the inventory budget is used up. Let δ i (s i ) be defined as the reduction in expected backorders for part i when increasing the inventory position from s i to s i +1 units. Wethenhave: δ i (s i )=BO i (s i +1) BO i (s i ) (10) By writing this out long-hand, we get: δ i (s i )=Pr(R i = s i +2)+2Pr(R i = s i +3)+... Pr (R i = s i +1) 2Pr(R i = s i +2) 3Pr(R i = s i +3)... (11) By collecting similar terms, we get: δ i (s i )= Pr (R i = s i +1) Pr (R i = s i +2) Pr (R i = s i +2)... = Pr (R i s +1) =Pr(R i s) 1 (12) This is a very simple and compact expression for the reduction in expected back orders resulting from increasing the inventory position by one unit for part i. The measure we are interested in, however, is reductioninexpectedback orders per dollar invested in increasing inventory by one unit. This expression for part i is given by: δ i (s i ) (13) c i We have now defined all the components required to optimize target inventory levels for a system of service parts at a single stocking location. In a 20

subsequent chapter we will introduce an extension of the model and solution methodology to cover multiple locations and echelons 4. To clarify the approach, let s consider an example. 2.3.1 An example The Medini School of Business is an American business school who recently aquired new telephone switching boards for all five departments from an European producer. It is important to have high availability for the telephone system, and Medini needs to secure this. The appropriate goal of availability is set to 99.8%, which translates into one out of 500 days when the system is down on average. There are only three different parts that are critical in terms of failure to the telephone system. The motherboard, denoted MOM, is the most expensive of these at a unit cost $9000. The parallell port handling unit, denoted PAP, is a lot cheaper at $50. And finally, the main fuse denoted by FUS costs only $1 per unit. In the case that a MOM or PAP part fails, it needs to be returned to Europe for repair. The lead time for this is 3 months. If a FUS fails, a new one can be ordered with a lead time of 1 1/2 month (the FUS is not worth repairing). From the experience of the producer, on average 0.115 MOM s and 0.265 PAP s are needed to be replaced per switching board per year. FUS s fail at the rate of 0.5 per year per switching board. We summarize the critical information for service parts management for this example in the following table (note that the demand rates are multiplied by 5 which is the number of switching boards to be supported): part# λ L c MOM 0.575 0.25 9000 PAP 1.325 0.25 50 FUS 2.5 0.125 1 Analysis of the historical failure data indicates that a memoryless failure rate assumption is reasonable and so we can model the number of units in repair as being distributed according to a Poisson distribution. 4 Note that this optimization process is bared on marginal analysis. It is sometimes referred to as the greedy method. It results in an optimal solution for the problem of interest with n parts at one location. In more complicated problems (e.g. multiple locations and/or indentures) the greedy method is not guaranteed to generate an optimal solution and is therefore a huristic (approximate) method. 21

Part approach We start by finding the inventory positions necessary to achieve the desired (target) level of 99.8% availability using an approach where each part is treated independently. To do this, we first need to figure out the maximum number of backorders that are associated with a system availability level of 99.8%. Let x denote the (unknown) expected system backorders. Based on (9), x must satisfy the following equation: By solving for x, weget: 5 x 5 =0.998 x =5 5 0.998 = 0.01 i.e. 0.01 units of expected backorders are allowed for the system per year. Since we have three parts, the simplest approach is to require each of these parts to have an expected backorder rate equal to one third of x, i.e. BO = 0.0034, for each part, to achieve the goal of 99.8% system availability. The key question is to determine how a maximum BO level of 0.0034 determines a (target) stocking level for each part. To illustrate how this is done, consider the details for MOM. First, based on (7), we find the expected number of backorders for an inventory position of 0 units: BO MOM (0) = 0.575 0.25 = 0.14375 When finding the expected backorders for inventory position of 1, we adjust the expected number of backorders for inventory position of 0 by δ MOM (0). The δ s are computed here directly, but they also can be easily evaluated using built in Excel functions. We have: BO MOM (1) = BO MOM (0) + δ MOM (0) =0.14375 + (Pr (R MOM 0) 1) 0.14375 0.143750 =0.14375 + e 1 0! =0.00985425 In this calculation, note that 0.14375 0 = 1 and that 0! = 1. We continue in the same way, to compute the expected backorder for MOM when the target inventory position is equal to 2: 22

BO MOM (2) = BO MOM (1) + δ MOM (1) =0.00985425 + (Pr (R MOM 1) 1) ( 0.14375 =0.00985425 + e 0.14375 0 =0.00046098 0! + 0.143751 1! ) 1 We see that this value is less than the target backorder rate of 0.0034 and so we can stop the procedure for MOM. We conclude that for a target inventory level of 2 units, the expected number of backorders for MOM is less than the maximum target based on equal allocation of system backorders. This means that we should choose an inventory position of 2 for MOM in order for this part to support the system goal of availability. The same procedure is carried out for PAP and FUS. The results are given in the following table with the expected backorders for the different appropriate inventory positions indicated. A target inventory position of 3 is required for both PAP and FUS in order to reduce the expected backorders to a value below 0.0034. part BO(0) BO(1) BO(2) BO(3) MOM 0.14375 0.00985425 0.00046098 0.000016328 PAP 0.33125 0.04927564 0.00514727 0.00041215 FUS 0.3125 0.04411563 0.00436114 0.00033007 SYSTEM 0.7875 0.10324552 0.00996939 0.00075854 Note that this results in expected system backorders of: SBO(2, 3, 3) = BO MOM (2) + BO PAP (3) + BO FUS (3) =0.00046098 + 0.00041215 + 0.00033007 =0.00120322 < 0.01 The (2,3,3) solution results in an estimated availability level of: A = 5 0.00120322 =0.999759356 5 Thus with an inventory investment of $18153 (2 $9000 + 3 $50 + 3 $1), we get an estimated system availability of 99.98% when using stock levels based on the part approach 5. 5 Since we care here about whether the total expected backorders is lower than the target 23

System approach With the system approach we look at backorders for all parts simultaneously while also taking cost into account. The calculations of δ, BO and A follow the same procedure as for the part approach, and hence we will not describe the details here. We start with inventory positions of 0 for all parts, and calculate δ(s)/c for each of the parts. part s δ(s)/c A MOM 0-0.0000148773 PAP 0-0.0056394872 FUS 0-0.2683843711 SYS 0.8425 From the table, we clearly see that FUS is the part that will improve expected backorders per dollar invested, i.e. it provides the greatest bang per buck, a backorder reduction of 0.268 per dollar of additional inventory. part s δ(s)/c A MOM 0-0.0000148773 PAP 0-0.0056394872 FUS 1-0.0397544870 SYS 0.89617687 In the next iteration, FUS is still the most promise candidate for increasing the inventory position. part s δ(s)/c A MOM 0-0.0000148773 PAP 0-0.0056394872 FUS 2-0.0040310676 SYS 0.90412777 At the next step, we see that PAP has become the most promising candidate for getting an additional unit in inventory. part s δ(s)/c A MOM 0-0.0000148773 PAP 1-0.0008825673 FUS 2-0.0040310676 SYS 0.96052264 level, we note that a stocking level of 2 units for each part yields a system backorder level of 0.00996939 which is less than 0.01 and has less inventory than the (2,3,3) solution. 24

FUS has again become the most promising candidate. part s δ(s)/c A MOM 0-0.0000148773 PAP 1-0.0008825673 FUS 3-0.0003098781 SYS 0.96132886 Now PAP is the candidate for a unit increase in inventory position. part s δ(s)/c A MOM 0-0.0000148773 PAP 2-0.0000947025 FUS 3-0.0003098781 SYS 0.97015453 Now FUS is the candidate for a unit increase in inventory position. part s δ(s)/c A MOM 0-0.0000148773 PAP 2-0.0000947025 FUS 4-0.0000191602 SYS 0.97021651 Now PAP is the candidate for a unit increase in inventory position. part s δ(s)/c A MOM 0-0.0000148773 PAP 3-0.0000077091 FUS 4-0.0000191602 SYS 0.97116353 We go back to FUS as the candidate for a unit increase in inventory position. part s δ(s)/c A MOM 0-0.0000148773 PAP 3-0.0000077091 FUS 5-0.0000009903 SYS 0.97116736 Finally, MOM is the candidate for a unit increase in inventory position, and gets its first unit. 25

part s δ(s)/c A MOM 1-0.0000010437 PAP 3-0.0000077091 FUS 5-0.0000009903 SYS 0.99794651 Here, PAP is the candidate. part s δ(s)/c A MOM 1-0.0000010437 PAP 4-0.0000005049 FUS 5-0.0000009903 SYS 0.99802360 With this increase in PAP s inventory position, we have arrived at the desired target of 99.8% availability. Using the systems procedure we have generated the optimal stocking policy (lowest inventory investment) that achieves an availabiltiy of 99.8% with a total inventory cost of 1 $9, 000 + 4 $50 + 5 $1 = $9, 205. Note that the same availability is achieved with almost half of the inventory investment required by the part approach ($18,53). This is of course a constructed example, so the savings in real life may not be that high. Empirical analysis indicates, however, that the system approach often leads to significant savings. Also note that the improvement to availability was rather large at the beginning of the procedure and smaller at the end. This is because the first units added have a greater impact since they are more likely to be needed. As more units are added the marginal benefit (reduction in backorder rate per dollar invested in inventory) decreases as we add stock to protect against shortages with lower probability, (i.e. based on seeing many failures during one lead time). This indicates that investment in parts inventory exhibits decreasing returns to scale. 26

2.4 Another example We will consider here another simple example of the system approach. Consider a situation where we have 10 systems, that are supported by two different service parts indexed by 1 and 2 with a target availability of at least 99.7%. The necessary data are given in the following table: part# λ L c 1 1.5 0.15 200 2 2 0.1 50 When analyzing this system from a system perspective, we start by evaluating the performance associated with having no service parts in inventory. The expected number of backorders for part 1 is then: BO 1 (0) = 1.5 0.15 = 0.225 And similarly for part 2: BO 2 (0) = 2 0.1 =0.2 The total expected system backorders is then: SBO(0, 0) = BO 1 (0) + BO 2 (0) = 0.425 The resulting availability of having no service parts inventory is then: A(0, 0) = 10 0.425 10 = 0.9575 We see that having no inventory does not achieve the target availability. We then need to find the part that provides the highest reduction in backorders per dollar invested in an additional unit. We first find the reduction in expected backorders by increasing the inventory by one unit for both parts. For part 1 we have: And similarly for part 2: δ 1 (0) = Pr(R 1 0) 1 1.5 0.15 (1.5 0.15)0 = e 1 0! = 0.2014838 δ 2 (0) = Pr(R 2 0) 1 2 0.1 (2 0.1)0 = e 1 0! = 0.1812692 27

But we are interested in the reduction per dollar invested in inventory. Thisis obtained by dividing by the unit cost: δ 1 (0) = 0.2014838 = 0.001007419 c 1 200 and δ 2 (0) = 0.1812692 = 0.003625384 c 2 50 This can be summarized in the following table: part# s δ(s)/c 1 0-0.001007419 2 0-0.003625384 We see that part 2 has the highest reduction in backorders per dollar invested, and hence we choose to increase its inventory from 0 to 1 units. This will result in an inventory investment of $50. The system backorders would decrease to: SBO(0, 1) = SBO(0, 0) + δ 2 (0) = 0.425 + ( 0.1812692) = 0.2437308 This results in the availability of: A(0, 1) = 10 0.2437308 10 = 0.97563 So, we have not yet reached the desired availability of 99.7%. This means that we need to make another round of adding a unit of inventory of the part that has the highest marginal reduction in backorders per dollar invested. Since we have added a unit of part 2, we need to update δ 2.Weget: δ 2 (1) = Pr(R 2 1) 1 ( (2 0.1) = e 2 0.1 0 + 0! = 0.0175231 Dividing this by the unit cost of $50 gives: ) (2 0.1)1 1 1! δ 2 (1) = 0.0175231 = 0.000350462 c 2 50 28

We then get a new table for choosing the candidate for the additional inventory increase: part# s δ(s)/c 1 0-0.001007419 2 1-0.000350462 Now we see that part 1 is the best candidate for an additional unit in inventory. By adding this, our inventory investment amounts to $250 (one of each part). The system backorders are updated to: SBO(1, 1) = SBO(0, 1) + δ 1 (0) = 0.2437308 + ( 0.2014838) = 0.042247 This results in the following availability: A(1, 1) = 10 0.042247 10 = 0.99578 This is still not sufficient to meet our availability criterion. We must therefore conduct another round of adding a unit of inventory, where we need to update the necessary values for part 1 (which was our most recent candidate). We then have: δ 1 (1) = Pr(R 1 1) 1 ( (1.5 0.15) = e 1.5 0.15 0 + 0! = 0.02181763 ) (1.5 0.15)1 1 1! and δ 1 (1) = 0.02181763 = 0.000109088 c 1 200 The updated table then becomes: part# s δ(s)/c 1 1-0.000109088 2 1-0.000350462 part 2 is now the candidate for adding an additional unit (meaning an increase in inventory position from 1 unit to 2 units). The updated system backorders then becomes: SBO(1, 2) = SBO(1, 1) + δ 2 (1) 29

= 0.042247 + ( 0.0175231) = 0.024724 With availability: A(1, 2) = 10 0.024724 10 = 0.99753 We have thus reached the availability target by having one unit of part 1 and two units of part 2 with a total inventory investment of $300. These calculations can be easily done in Excel. For example δ 1 (1) is calculated by entering: in a cell. POISSON(1, 1.5 0.15,TRUE) 1 30

References If you are interested in learning more about this area, the following readings are suggested. J. J. Chamberlain, J. Nunes, Service Parts Management: A Real Life Success Story, Supply Chain Management Review, Sep 2004, pp38-44. M. A. Cohen, S. Whang Competing in Product and Service: A Product Life- Cycle Model. Special Issue of Management Science on Frontier Research in Manufacturing and Logistics, Vol. 43, No. 4, April 1997, pp. 535-545. M. A. Cohen, H. Lee, C. Cull and D. Willen, Supply Chain Innovation: Delivering Values in After Sales Service Sloan Management Review, Vol. 41, No. 4, Summer 2000, pp. 93-101. M. A. Cohen, A.Tekerian, P. Kamesam, H. Lee and P. Kleindorfer, OPTI- MIZER: IBMs Multi-Echelon Inventory System for Managing Service Logistics. Interfaces, Vol. 20, No. 1, January-February 1990, pp. 65-82. M. A. Cohen, Y. Wang and Y-S. Zheng Identifying Opportunities for Improving Teradynes Service Parts Logistics System. Interfaces, Vol. 29, No. 4, July-August 1999, pp.1-18. M. J. Dennis and A. Kambil, Service Management: Building Profits After the Sale, Supply Chain Management Review, Jan.-Feb. 2003, 42-49. V. Deshpande, M.A. Cohen and K. Donohue A Threshold Inventory Rationing Policy for Service Differentiated Demand Classes, Management Science, Volume 49, No. 6, 2003. V. Deshpande, M. A. Cohen, and K. Donohue, An Empirical Study of Service Differentiation for Weapon System Service Parts. Operations Research, Volume 51, No. 4, 2003. C. C. Sherbrooke. Optimal Inventory Modeling of Systems: Multi-Echelon Techniques. Kluwer Academic Publishers; 2nd edition, 2004. R. Wise and P. Baumgartner, Go Downstream: The New Profit Imperative in Manufacturing, Harvard Business Review, Sept.-Oct. 1999, 133-141. 31