repairable service parts inventory

8/13/2019 Repairable Service Parts Inventory

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REPAIRABLE SERVICE PARTS INVENTORY

MANAGEMENT

lecture note

Morris A. CohenThe Wharton School

University of PennsylvaniaPhiladelphia, PA 19104

Vinayak DeshpandeKrannert School of Management

Purdue UniversityW Lafayette IN 47907

Nils RudiThe Simon School

University of RochesterRochester, NY 14627

November 24, 2004

Do not quote, cite, or reproduce without permission. c 2004

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Prerequisites

The reader of this note should have some basic knowledge of inventory modelsand probability. Introductory courses at college level that touch on these areasor an introductory course on operations management are sufficient. A largeportion of the material is rather quantitative in nature, but the presentationattempts to limit the mathematical aspects as much as possible and emphasizesthe applications.

1 Introduction

For many rms, after sales support is a critical component of their competitivestrategy. This is especially true in industries where availability of the product is

considered to be mission critical by the customer. Examples of industries whereeffective 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 rening, mining, food processing, nuclear power, off-shore 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, rms must make signicant investmentsin assets that include infrastructure (repair depots, warehouses, communicationand logistics), spare parts inventory and highly trained personnel (customerengineers and repair technicians). For the most part these assets are not highlyutilized since the demand for service support can be highly erratic and is oftenvery difficult to predict.

Service parts management is an important requirement for meeting customerexpectations for product support in a cost effective manner. Most customersare 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

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perform such rapid repair, it is necessary to have the required service parts avail-able. This note is an introduction to the basic management procedures requiredto achieve rapid response, in a cost effective manner which limits investment inspare parts inventory.

1.1 Characteristics of service parts management

The challenges faced by managers of service parts inventory are very differentfrom those encountered in nished product and direct material supply chains.There are a number of distinct characteristics that make it especially difficultto provide a high level of customer service at a low cost. These characteristicsinclude:

1.1.1 Criticality

Criticality refers to the requirement for a part to be functioning in order forthe product to be available for use. Typically there are different degrees of suchcriticality. A car, for example, is not driveable without a working fuel pump.The car works well enough, however, under most conditions, with a defectiveright-hand-side wiper blade. The model introduced in this note focuses on highlycritical parts, that are often characterized by high cost, long repair/procurementlead times and the fact that they are usually repaired rather than being scrappedif 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 replacementregardless of its condition. An example of the rst case is a starter mechanism ina car that suddenly stops working. An example of the second case is an oil lterthat is planned for replacement after a new car has been driven for 5,000 miles.Product owners have little control in the rst case, and since the cost of delaycan be quite high, it is necessary to keep high levels of service parts inventory inthe event that a break down occurs. The second case, however, is something weoften 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 canbe made available specically for this purpose at the appropriate time and place.When managing service parts, we often separate these two kinds of demand andwhen deciding on safety stock levels, we base this decision only on the demanddue 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

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frequency plot of global demand for parts over a 12 month period at KLA-Tencor, a leading manufacturer of equipment that is used in semiconductorfabrication plants. Note that in their environment 75% of the parts plannedhad one demand or less and that 60% of the parts with demand had 3 or lesspieces requested. Such low demand requires very different control methods thanthose 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 ob-servations of historical usage. This is especially true in the initial stage of aparts life cycle, i.e. upon its introduction as a new product, where one needsto 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 break-down. 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 mother-board that can be used as a service part for potential future break-downs. Figure2 illustrates the cycle of repairable service parts.

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Figure 2: Closed part cycle for repairables

We see here that a broken part goes to the repair queue if it is worth re-pairing, and otherwise it is scrapped. When the broken parts turn comes, it isrepaired and then placed in the inventory of good parts. The inventory of goodparts can also be replenished by purchases from the supplier of the part. Thesegood 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 repairof a high-tech part often needs to take place at a central depot which may bevery distant from the installed base system location. Such repair may requirespecialized resources, which include extensive testing both before and after therepair itself. The return time delays for defectives, procurement lead times forcomponents used to repair defective parts and the actual repair/testing time allcontribute to the overall part lead time.

1.1.6 Parts relationships (system perspective)In nished goods supply chains, each part is a product and relevant servicemeasures such as ll rate, can be evaluated at an part level. The purpose of service parts, however, is to support maintenance and repair of systems , meaning

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that in the case of a system breakdown the necessary service part should beavailable so that it can be replaced immediately. This ensures that the systemwill be up and running again with minimal interruptions (downtime). Hence,the relevant service measure is dened at a system level, since this is what thecustomer cares about, e.g. the owner of a car is concerned with the driveabilityof the vehicle and not the service level of parts at the repair shop. This leadsto 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 particularsystem (i.e. the product is down no more than 5% of the time awaiting deliveryof a part), then one may choose to set service level (ll rate) targets for partsto be on average 95%. This average part ll rate could be realized by settinga lower ll rate for very expensive parts and a much higher ll rate for veryinexpensive 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 ghter jet is supported by parts ranging from fusesthat cost virtually nothing to eld replacable units, such as the radar systemthat might cost more than $2,000,000. One will, obviously, make sure not tohave an F16 grounded on an aircraft carrier due to the lack of a fuse, while the

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radar system is so expensive that one may be able to tolerate a lower servicelevel 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 partsin relation to each other . As we shall see there are signicant interactionsamong all of the parts in terms of the total budget for inventory and the overallsystem (product) service level. This makes the problem of managing serviceparts inventory challenging in terms of the underlying mathematics as well asthe solution times required to generate an optimal solution. Indeed, serviceparts management for inventory systems where each part can be optimizedindependently 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 serviceparts environments.

1.2.1 Multiple locations

In the case that customer systems are located at different locations, one oftenalso locates service parts inventories at multiple locations to avoid shippingdelays and to pool risk 1 . The forward locations are typically supported by acentral location that also keeps inventory in addition to performing repairs.Also, all locations might support each other through transshipments , meaningthat 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 oneof them can cannibalize the other for parts. For example, if one F16 is waitingfor a radar system to become available, while another F16 is waiting for a newcatapult mechanism to become available, then the former can get the workingradar system from the latter. This reduces the number of systems that aredown from two to one. In the military, aircraft used as a part source throughcannibalization are referred to as hanger queens.

1.2.3 Express shipments

Some suppliers of service parts utilize express shipments in addition to regularshipments, especially in those cases where a customer system is down and is

1 Placement of stock at a central location to ll demand originating from several local sitescan reduce the overall variability in the system.

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Figure 4: Schematic example of multi-location service parts network

awaiting delivery of a part. Teradyne, for example, offers two classes of repairservices to its customers, one short and the other long at high and low costrespectively. Customers will opt for the long delivery time option in cases wherethey have sufficient on-hand inventory and/or spare system capacity. Optionsof this type further complicate the analysis since multiple delivery modes havedifferent lead times and could affect stocking levels.

1.2.4 Multiple Customer Segments

In many service support environments different customers have different priori-ties and needs for support. For example a squadron of military aircraft engagedin combat require much higher levels of availability that a squadron engaged intraining exercises. Similarly a company trading in foreign exchange futures can-not tolerate downtime for its internet routers while a company with a collectionof redundant servers used to support non-mission critical processes will not bewilling to pay a premium for rapid repair. Such differentiated service needs canaffect both target inventory stocking levels and real time fulllment priorities.

1.2.5 Multiple Indentures

All assembled systems can be described by a Bill of Materials (BOM), whichspecies how each part is used in putting the nal system together. Typically

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BOMs are described by multiple levels or indentures which indicate the hierarchyof part usage. Thus at the highest level we have the nal system. At thenext level we have the major assemblies used to manufacture the system. Eachassembly can be broken down into sub-assemblies. This process repeats until weget to the bottom level of components. A BOM can also be specied for servicesupport. 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 toas indentures. An investment in service parts could be made at all indenturelevels. Having FRUs available leads to faster system restoration. These are themost expensive parts however. Thus there is a tradeoff in determining stockinglevels across indentures between cost and speed. Overall repair lead times willbe 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 newbuy procurement lead times and high unit costs. Hence it is especially impor-tant to manage repairables well. This chapter presents a powerful procedure foroptimizing repairable service parts inventory decisions. This procedure is alsothe foundation for the methods used in more advanced scenarios (e.g. multi-location).

2.1 Fundamentals of service parts modelling

This chapter provides the foundations for modeling and optimizing service partssystems. We start by considering a specic type of service part.

2.1.1 Failures of parts

By failure rate, we mean the rate of failure occurrence. We will characterizehere the failure rate for each service part. This rate can be measured either bythe number of failures per unit of time or by the time between failures . In thisnote, we will denote the average number of failures per year by . If, in oneshouse, an average of 4 light bulbs fail in a year, then = 4. The time between failures can then be expressed as 1 . The average time between failures for lightbulbs is then 0.25 years, or 3 months.

Memoryless failure rate

One of the fundamentals in modeling service parts is the assumption of a mem-oryless 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

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not affect what will happen. To illustrate the memoryless property, lets lookat some examples that are not memoryless:

a. Wear and tear A part is more likely to fail as it gets older and becomesmore worn. Consequently, changing a part makes it less likely that this part willfail in the near future. This is often the case for mechanical systems (i.e. non-electronic). The population of installed systems and the service parts includedin these systems are often of different ages. Hence, the fact that one part isreplaced could affect the total demand, (failure rate), for the system .

b. Failures due to external events Some systems are exposed to externalforces. One example would be telecommunication centers that are exposed tolightning. During periods with lightning there is a higher probability of parts

in a telecommunication center failing than during periods without lightning. Tolink it to the denition of memoryless failures, lets look at it from a slightlydifferent angle: if a part in a telecommunication system has just failed, it islikely that there is currently lightning in the area, and hence it is more likelythat another copy of the same part will fail at a different telecommunicationcenter in the same area. It follows then that what will happen in the futuredepends on what has (recently) happened.

Other examples of systems that are exposed to external forces include elec-tricity networks that are exposed to wind and rain.

c. Detection due to an external event Similar to failures due to externalevents, 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 aboutchanging windshield wipers. In a heavy rain storm, however, many peoplethat have poor windshield wipers will drive by a gas station and have themreplaced. The individual demands then come together at the station, and arenot 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 controlboards). In such cases demand for parts are not independent. Both replenish-ment and repair can also require kits which contain a collection of parts usedtogether. 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 dont cost much releative to the xedcost of ordering a batch.

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The probability that no more than 3 60W light bulbs will fail during thenext half year is then:

Pr ( S (0.5) 3) = e 4 0 .5(4 0.5)0

0! +

(4 0.5)1

1! +

(4 0.5)2

2! +

(4 0.5)3

3!= 0 .1353 [1 + 2 + 2 + 1 .3333]= 0 .86

or 86%. This means that the probability that 3 light bulbs will be sufficient forone half years 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 outof the system. Following Figure 2, we see that if a part fails, it is replaced bya good part from the service parts inventory, if available (otherwise the systemwaits until a good part becomes available). The failed part is then sent to therepair queue. When its turn comes, it is repaired and then placed in the goodparts inventory.

Sometimes there are restrictions on whether a part can be repaired. Amongthese restrictions are physical restrictions (e.g. the part may be too severelydamaged to be repaired) and policy restrictions (e.g. maximum number of three repairs is allowed for a specic part). In such cases, the cycle will be open ,meaning that material leaks out through disposal at a testing stage and getsresupplied through additional acquisitions. Figure 5 illustrates such a cycle.

Modelling the repair process

If part failures are memoryless with an annual rate of per year, then thearrivals into the repair queue are also memoryless. The repair lead-time isdened as the time spent by the failed part to reach the repair-depot, plus thetime spent in queue at the repair depot, plus the time spent in actual repairprocess, plus the time to return the xed part to the warehouse. It is typicallyassumed that the repair-lead times for parts are independent and identicallyditributed with a mean of L years. Thus L years is the average time from thepart 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 intransit to the repair shop, parts waiting in the repair queue, and parts in actualrepair 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 randomvariable with mean equal to L. The probability of j units being due fromrepair at the same time is then:

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Figure 5: Open cycle for repairables

Pr ( R = j ) = e L(L )j

j ! (3)

The above result is remarkable because the actual shape of the repair lead-time distribution does not matter. Only the mean of the repair lead-time isimportant to measure the repair pipeline. Although in practice repair lead-times 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 justied when there exists ample capacityto meet demand for repair.

Note that if not all parts of a specic type are repairable (i.,e. a percentageof failed parts are scrapped), then the part repair leadtime is a weighted averageof the repair and new buy procurement lead times.

2.2 Measuring Customer Service

Many alternative measures of customer service can be dened. We will considerhere two main types of service measures that are considered especially relevantfor service parts management. These two types dene part and system serviceperformance.

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Part service

Service metrics at an part level measure (and specify) the service level of partsindependently of each other. This is the traditional service approach for trackingcustomer service in inventory systems, and it applies primarily when the partis a nished product. Part service targets are typically specied for groups of parts, where the groups are made up of similar products. Examples includeretailers and mail order companies where a group could be a product family,e.g. mens dress shirts.

The specic service measures that are most often used at an part level are:

Probability of stockout This measures the probability that a stockout willoccur during a specied time interval at a location when either a customer

demands an part or the system generates a replenishment order to be lledfrom that location. It does not reect the number of replenishments peryear or the size of the stockout. While stockout probability is very easyto compute and to understand, in terms of the underlying mathematics,it does not relate well to the nal customers perception of service quality,especially when it is measured at an internal location (e.g. the centralwarehouse).

Fill rate This measures the expected value of the percentage of demand thatcan be fullled directly from available inventory, i.e. off-the-shelf. Thismetric is most often used by supply chain managers when they refer toservice level. It is dened by the (average) ratio of the quantity shippedto the quantity demanded.

Backorders Fill rate only considers if an part is immediately available from in-ventory, i.e. at the point in time when the demand for the part is realized- it does not reect how long the customer has to wait in case the partis not available. For spare parts this is an important issue, since delayin receiving a part needed for a system repair contributes to the unavail-ability of the system for customer use (downtime). The backorder ratespecically measures the average number of parts that systems/customers are waiting to receive . If, for example, one has no backorders half of theyear, a backorder of 1 unit a quarter of the year, and a backorder of 2 unitsfor the remaining quarter of the year, the average number of backordersfor the year is 0.75. The problem with the backorder rate is that it isnot very intuitive, and hence it is not easy for customers to relate it totheir overall service level needs. Another issue is that in some environ-ments excess demands are not backordered at the point where they enterthe inventory system. Rather shortages may be transferred to emergencybackup locations (either laterally to other forward locations or verticallyto a central warehouse).

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Response time/backorder delay This is the average amount of time that acustomer has to wait for their order to be fullled. If the part is availableoff the shelf, at the point where demand is realized then there is no (orminimal) delay. If the demand order cannot be lled at that location andthe demand is either backordered or transferred, then the customer willhave to wait until the part arrives.

It should be clear that for a given part, all of the part service metrics in-troduced here are inter-related and are determined by the interaction betweenthe level of inventory of the part, the process that generates demands and anydelays or lags in the system.

System service

From a customer/system user perspective, individual part service levels are notthat relevant. Consider for example a hot dog stand - having a really highservice 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 forservice parts since the purpose of the after sales service supply chain is to keep asystem up and running. As we shall see system service metrics are based on somecombination 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 serviceparts management include:

Average ll rate When measuring ll rate at a system level, one considersthe percentage of parts that are required for repair that are available from

on the shelf inventory. The system ll rate is then the weighted averageof the ll rates for the parts that supports the system, with the weightsbeing the relative demand rates for each part per year.

Total (System) Backorders The system wide backorders is simply the sumof expected backorders of all parts that are used to support the system.

System availability As we already have noted, the purpose of service parts isto keep systems up and running - in other words as available as possible.It can be proved that the average availability of a system is determinedby the total number of backorders. The resulting availability metric is ameasure that is both intuitive and directly reects 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 theservice supply chain organization. It also should be closely related to theintuitive understanding of service as perceived by the end user customer.

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It should reect the consequences that a stock out has on customer serviceand 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 canbe of enormous cost).

The expectations and requirements for availability (many people cannottolerate being without their TV for very long).

Inventory position

We dene inventory position for a specic 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 transitminus the number of units backordered. Let s be the target number of unitsfor the inventory position for a specic part type. If there currently is a systemthat 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 caseof one or more systems waiting for a specic part, (i.e. backorder) it means thatmore than s units are currently being repaired, or waiting in the repair queueor are in transit. We shall refer to the total number of such units to be thenumber of units in repair and denote it by R. If one system is waiting for aspecic part, then the number of units in repair of this part is R = s + 1, if twosystems are waiting for the type of part then R = s + 2, and so on.

Figures 6 and 7 illustrates how part failures, repair completions and thetarget inventory position drive the number of units in repair.

Probability of No Stockout

Following the previous discussion, the probability of lling a demand for a partis equal to 1 - Probability of a stockout and can be expressed as:

P NS (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 thesespare parts by i, so when a variable has subscript i, it means that we are talkingabout a specic service part out of the total of n service parts.

System ll rate probability is dened as:

P NS (s 1 , s 2 , . . . , s n ) = 1

ni =1 i

n

i =1 i Pr ( R i s i 1) (5)

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Figure 6: Inventory Dynamics a. S=2

Figure 7: Inventory Dynamics b. S=3

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where the probability of not stocking out for each part is weighted by its relativecontribution to the total demand for all parts 3 .

Backorders

As has been noted, the expected number of backorders for an part can be ex-pressed as:

BO (s) = Pr( R = s + 1) + 2 Pr( R = s + 2) + 3 Pr( R = s + 3) . . .

=

j = s

( j s)Pr ( R = j ) (6)

To obtain the expected or average number of backorders, the different real-izations of backorder values are weighted by their respective probability. So theexpected number of backorders is 1 times the probability of having 1 numberof 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 numberof backorders when the inventory stocking level is equal to zero units. This isgiven by the simple expression:

BO (0) = L (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 positionis equal to s i . The system-wide expected backorders is then:

SBO (s 1 , s 2 , . . . , s n ) =n

i =1BO i (s i ) (8)

System availability

Let K be the number of systems being supported, i.e. installed and being usedby customers. A simple estimate of the availability due to service parts is then:

A = K SBO

K (9)

If for example, there are 100 systems with an average number of systembackorders 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.

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measure is not completely accurate, but it is a reasonably good approximationfor most practical cases.

To summarize, both part and system service metrics are based on manage-ments selection of a vector of target inventory positions s1 , s 2 ,...,s n . The valuesfor each metric are also affected by the probability distribution for each partat each location (Poisson with mean i and demand lead time Li ). Variationin s i leads to variation in service metric levels for part i alone. System servicemetrics on the other hand, are a function of all of the target stocking levels,s1 , 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 maximumsystem availability. In other words, we want to spend the next dollar on in-ventory 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 AvailabilityS. T.Total Inventory Investment Inventory Budgetor,

Minimize Inventory BudgetS.T.Availability Target Availability

In practice, these optimization problems can be complicated by the existenceof additional service constraints (by location, system type, customer etc.) andadditional resource constraints (local budget, cash ow, etc.).

The following summarizes the procedure of system optimization:

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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 dollarinvested, and increase the inventory position of this part by one unit.For this part, recalculate the two measures of expected backorders andreduction in backorders per dollar invested caused by increasing the in-ventory level of this part by one unit.5. Continue to increase the inventory position of the most promising can-didate 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 i (s i ) be dened as the reduction in expected backorders for part i when increasing the inventory position from si to s i + 1 units . We then have:

i (s i ) = BO i (s i + 1) BO i (s i ) (10)

By writing this out long-hand, we get:

i (s i ) = Pr ( R i = s i + 2) + 2 Pr( 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:

i (s i ) = Pr ( R i = s i + 1) Pr ( R i = si + 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 expectedback orders resulting from increasing the inventory position by one unit for parti.

The measure we are interested in, however, is reduction in expected back orders per dollar invested in increasing inventory by one unit . This expressionfor part i is given by:

i (s i )ci (13)

We have now dened all the components required to optimize target in-ventory levels for a system of service parts at a single stocking location. In a

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subsequent chapter we will introduce an extension of the model and solutionmethodology to cover multiple locations and echelons 4 .

To clarify the approach, lets consider an example.

2.3.1 An example

The Medini School of Business is an American business school who recentlyaquired new telephone switching boards for all ve departments from an Eu-ropean producer. It is important to have high availability for the telephonesystem, and Medini needs to secure this. The appropriate goal of availabilityis set to 99.8%, which translates into one out of 500 days when the system isdown on average.

There are only three different parts that are critical in terms of failure tothe telephone system. The motherboard, denoted MOM, is the most expensiveof these at a unit cost $9000. The parallell port handling unit, denoted PAP,is a lot cheaper at $50. And nally, 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 returnedto Europe for repair. The lead time for this is 3 months. If a FUS fails, a newone can be ordered with a lead time of 1 1/2 month (the FUS is not worthrepairing). From the experience of the producer, on average 0.115 MOMs and0.265 PAPs are needed to be replaced per switching board per year. FUSs failat the rate of 0.5 per year per switching board.

We summarize the critical information for service parts management for thisexample in the following table (note that the demand rates are multiplied by 5which is the number of switching boards to be supported):

part# L cMOM 0.575 0.25 9000PAP 1.325 0.25 50FUS 2.5 0.125 1

Analysis of the historical failure data indicates that a memoryless failure rateassumption is reasonable and so we can model the number of units in repair asbeing distributed according to a Poisson distribution.

4 Note that this optimization process is bared on marginal analysis. It is sometimes referredto 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/orindentures) the greedy method is not guaranteed to generate an optimal solution and istherefore a huristic (approximate) method.

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Part approach

We start by nding the inventory positions necessary to achieve the desired(target) level of 99.8% availability using an approach where each part is treatedindependently. To do this, we rst need to gure out the maximum number of backorders that are associated with a system availability level of 99.8%. Let xdenote the (unknown) expected system backorders. Based on (9), x must satisfythe following equation:

5 x5

= 0 .998

By solving for x, we get:

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 theseparts 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. Thekey 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 thedetails for MOM.

First, based on (7), we nd the expected number of backorders for an inven-tory position of 0 units:

BO MOM (0) = 0 .575 0.25 = 0.14375

When nding the expected backorders for inventory position of 1, we adjustthe expected number of backorders for inventory position of 0 by MOM (0). The s are computed here directly, but they also can be easily evaluated using builtin Excel functions. We have:

BO MOM (1) = BOMOM (0) + MOM (0)= 0 .14375 + (Pr ( RMOM 0) 1)

= 0 .14375 + e 0 . 143750.143750

0! 1

= 0 .00985425In this calculation, note that 0 .143750 = 1 and that 0! = 1. We continue in

the same way, to compute the expected backorder for MOM when the targetinventory position is equal to 2:

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BO MOM (2) = BOMOM (1) + MOM (1)= 0 .00985425 + (Pr ( RMOM 1) 1)

= 0 .00985425 + e 0 . 143750.143750

0! +

0.143751

1! 1

= 0 .00046098

We see that this value is less than the target backorder rate of 0.0034 and sowe can stop the procedure for MOM. We conclude that for a target inventorylevel of 2 units, the expected number of backorders for MOM is less than themaximum target based on equal allocation of system backorders. This meansthat we should choose an inventory position of 2 for MOM in order for this partto support the system goal of availability.

The same procedure is carried out for PAP and FUS. The results are givenin the following table with the expected backorders for the different appropriateinventory positions indicated. A target inventory position of 3 is required forboth PAP and FUS in order to reduce the expected backorders to a value below0.0034.

part BO (0) BO (1) BO (2) BO (3)MOM 0.14375 0.00985425 0.00046098 0.000016328PAP 0.33125 0.04927564 0.00514727 0.00041215FUS 0.3125 0.04411563 0.00436114 0.00033007SYSTEM 0.7875 0.10324552 0.00996939 0.00075854

Note that this results in expected system backorders of:

SBO (2, 3, 3) = BOMOM (2) + BO P AP (3) + BO F US (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

5 = 0 .999759356

Thus with an inventory investment of $18153 (2 $9000+3 $50+3 $1), weget an estimated system availability of 99.98% when using stock levels based onthe part approach 5 .

5 Since we care here about whether the total expected backorders is lower than the target

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System approach

With the system approach we look at backorders for all parts simultaneouslywhile also taking cost into account. The calculations of , BO and A follow thesame procedure as for the part approach, and hence we will not describe thedetails here.

We start with inventory positions of 0 for all parts, and calculate (s)/c foreach of the parts.

part s (s)/c AMOM 0 -0.0000148773PAP 0 -0.0056394872FUS 0 -0.2683843711

SYS 0.8425From the table, we clearly see that FUS is the part that will improve expectedbackorders per dollar invested, i.e. it provides the greatest bang per buck, abackorder reduction of 0.268 per dollar of additional inventory.

part s (s)/c AMOM 0 -0.0000148773PAP 0 -0.0056394872FUS 1 -0.0397544870SYS 0.89617687

In the next iteration, FUS is still the most promise candidate for increasingthe inventory position.


At the next step, we see that PAP has become the most promising candidatefor getting an additional unit in inventory.


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.

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FUS has again become the most promising candidate.


Now PAP is the candidate for a unit increase in inventory position.

part s (s)/c AMOM 0 -0.0000148773PAP 2 -0.0000947025FUS 3 -0.0003098781

SYS 0.97015453Now FUS is the candidate for a unit increase in inventory position.


Now PAP is the candidate for a unit increase in inventory position.


We go back to FUS as the candidate for a unit increase in inventory position.


Finally, MOM is the candidate for a unit increase in inventory position, andgets its rst unit.

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part s (s)/c A

MOM 1 -0.0000010437PAP 3 -0.0000077091FUS 5 -0.0000009903SYS 0.99794651

Here, PAP is the candidate.


With this increase in PAPs inventory position, we have arrived at the desiredtarget of 99.8% availability. Using the systems procedure we have generated theoptimal stocking policy (lowest inventory investment) that achieves an avail-abiltiy 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 in-ventory investment required by the part approach ($18,53). This is of course aconstructed example, so the savings in real life may not be that high. Empiricalanalysis indicates, however, that the system approach often leads to signicantsavings.

Also note that the improvement to availability was rather large at the be-ginning of the procedure and smaller at the end. This is because the rst unitsadded have a greater impact since they are more likely to be needed. As moreunits are added the marginal benet (reduction in backorder rate per dollarinvested in inventory) decreases as we add stock to protect against shortageswith lower probability, (i.e. based on seeing many failures during one lead time).This indicates that investment in parts inventory exhibits decreasing returns toscale.

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2.4 Another example

We will consider here another simple example of the system approach. Considera situation where we have 10 systems, that are supported by two different ser-vice parts indexed by 1 and 2 with a target availability of at least 99.7%. Thenecessary data are given in the following table:

part# L c1 1.5 0.15 2002 2 0.1 50

When analyzing this system from a system perspective, we start by evaluat-ing the performance associated with having no service parts in inventory. Theexpected 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. Wethen need to nd the part that provides the highest reduction in backorders per dollar invested in an additional unit . We rst nd the reduction in expectedbackorders by increasing the inventory by one unit for both parts. For part 1we have:

1 (0) = Pr( R 1 0) 1

= e 1 .5 0 . 15(1.5 0.15)0

0! 1

= 0.2014838

And similarly for part 2:

2 (0) = Pr( R 2 0) 1

= e 2 0 . 1(2 0.1)0

0! 1

= 0.1812692

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But we are interested in the reduction per dollar invested in inventory . This isobtained by dividing by the unit cost:

1 (0)c1

= 0.2014838

200 = 0.001007419

and 2 (0)

c2=

0.181269250

= 0.003625384

This can be summarized in the following table:

part# s (s)/c1 0 -0.0010074192 0 -0.003625384

We see that part 2 has the highest reduction in backorders per dollar in-vested, and hence we choose to increase its inventory from 0 to 1 units. Thiswill result in an inventory investment of $50. The system backorders woulddecrease to:

SBO (0, 1) = SBO (0, 0) + 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 thatwe need to make another round of adding a unit of inventory of the part thathas the highest marginal reduction in backorders per dollar invested. Since wehave added a unit of part 2, we need to update 2 . We get:

2 (1) = Pr( R 2 1) 1

= e 2 0 . 1(2 0.1)0

0! +

(2 0.1)1

1! 1

= 0.0175231

Dividing this by the unit cost of $50 gives:

2 (1)c2

= 0.0175231

50 = 0.000350462

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We then get a new table for choosing the candidate for the additional inventoryincrease:

part# s (s)/c1 0 -0.0010074192 1 -0.000350462

Now we see that part 1 is the best candidate for an additional unit in inven-tory. By adding this, our inventory investment amounts to $250 (one of eachpart). The system backorders are updated to:

SBO (1, 1) = SBO (0, 1) + 1 (0)= 0 .2437308+ ( 0.2014838)

= 0 .042247This 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 thereforeconduct another round of adding a unit of inventory, where we need to updatethe necessary values for part 1 (which was our most recent candidate). We thenhave:

1 (1) = Pr( R 1 1) 1

= e 1 .5 0 . 15 (1.5 0.15)0

0! + (1.5 0.15)

1

1! 1

= 0.02181763

and 1 (1)

c1=

0.02181763200

= 0.000109088

The updated table then becomes:

part# s (s)/c1 1 -0.0001090882 1 -0.000350462

part 2 is now the candidate for adding an additional unit (meaning an in-crease in inventory position from 1 unit to 2 units). The updated system back-orders then becomes:

SBO (1, 2) = SBO (1, 1) + 2 (1)

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= 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 andtwo units of part 2 with a total inventory investment of $300.

These calculations can be easily done in Excel. For example 1 (1) is calcu-lated by entering:

POISSON (1, 1.5 0.15,TRUE ) 1

in a cell.

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References

If you are interested in learning more about this area, the following readingsare suggested.

J. J. Chamberlain, J. Nunes, Service Parts Management: A Real Life SuccessStory, 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 Researchin 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: De-livering 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 Lo-gistics. Interfaces, Vol. 20, No. 1, January-February 1990, pp. 65-82.

M. A. Cohen, Y. Wang and Y-S. Zheng Identifying Opportunities for Improv-ing 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 Prots After theSale, Supply Chain Management Review, Jan.-Feb. 2003, 42-49.

V. Deshpande, M.A. Cohen and K. Donohue A Threshold Inventory RationingPolicy for Service Differentiated Demand Classes, Management Science,Volume 49, No. 6, 2003.

V. Deshpande, M. A. Cohen, and K. Donohue, An Empirical Study of ServiceDifferentiation for Weapon System Service Parts. Operations Research,Volume 51, No. 4, 2003.

C. C. Sherbrooke. Optimal Inventory Modeling of Systems: Multi-EchelonTechniques . Kluwer Academic Publishers; 2nd edition, 2004.

R. Wise and P. Baumgartner, Go Downstream: The New Prot Imperativein Manufacturing, Harvard Business Review, Sept.-Oct. 1999, 133-141.

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repairable service parts inventory

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