an inventory control system for spare parts at a refiner

8/4/2019 An Inventory Control System for Spare Parts at a Refiner

http://slidepdf.com/reader/full/an-inventory-control-system-for-spare-parts-at-a-refiner 1/32

Production, Manufacturing and Logistics

An inventory control system for spare parts at a refinery:An empirical comparison of different re-order point methods

Eric Porras a,*, Rommert Dekker b

a Instituto Tecnolo gico y de Estudios Superiores de Monterrey, Campus Santa Fe, 01389 Mexico City, Mexicob Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands

Received 5 August 2005; accepted 9 November 2006

Abstract

Inventory control of spare parts is essential to many organizations, since excess inventory leads to high holding costsand stock outs can have a great impact on operations performance. This paper compares different re-order point methodsfor effective spare parts inventory control, motivated by a case study at a large oil refinery. Different demand modelingtechniques and inventory policies are evaluated using real data.Ó 2006 Elsevier B.V. All rights reserved.

Keywords: Spare parts; Inventory control; Re-order points; Demand classes; Service levels

1. Introduction

Effective inventory management of spare parts isessential to many companies, from capital-intensivemanufacturers to service organizations, such as carmanufacturers, chemical plants, telecom companiesand airlines. Different from work-in-process (WIP)and finished product inventories, which are driven

by production processes and customer demands,spare parts are kept in stock to support maintenanceoperations and to protect against equipment fail-ures. Although this function is well understoodby maintenance managers, many companies facethe challenge of keeping on stock large inventoriesof spares with excessive associated holding and

obsolescence costs. Thus, effective cost analysis canbe an important tool to evaluate the effects of stockcontrol decisions related to spare parts. However,the difficulty in assessing good strategies for themanagement of spare parts lies in their specific nat-ure, normally very slow-moving parts with highlystochastic and erratic demands. For example, typi-cal industrial data sets comprise limited demand his-

tory with long streams of zero demand values and afew large demands (Willemain et al., 2004). Thismakes the estimation of the lead time demand(LTD) distributions very difficult, which is essentialto obtain the control parameters of most inventorypolicies. Although different inventory models havebeen proposed in the literature to tackle this prob-lem (see next section), there is a lack of empiricaltesting of theoretical models with data from realindustrial environments.

0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2006.11.008

* Corresponding author. Tel.: +52 55 91778000.E-mail address: [email protected] (E. Porras).

European Journal of Operational Research xxx (2007) xxx–xxx

www.elsevier.com/locate/ejor

ARTICLE IN PRESS

Please cite this article in press as: Porras, E., Dekker, R., An inventory control system for spare parts at a refinery: ...,Eur. J. Oper. Res. (2007), doi:10.1016/j.ejor.2006.11.008

http://-/?-

mailto:[email protected]

mailto:[email protected]

http://-/?-



This paper concerns a study on spare parts at amajor oil refinery in the Netherlands, which con-sisted of two phases. In the first phase a case studywas conducted with the company with the idea tooptimize the use of the SAP system. As a result,

some improvement measures were provided andlater implemented by the company. The secondphase was focused on the analysis of the demanddata provided by the company. This paper reportson the findings related to this phase, where theobjective was to perform an empirical comparisonof different inventory models.

The aforementioned company keeps stock of alarge number of spare parts related to equipmentused in its petrochemical processes. Although thesestocks are essential for the continuity of its opera-tions, management was concerned with the savings

opportunities at the process floor by having betterinventory control of its spare parts, whose valuewas worth at the moment of the study more than27 million euros. One major difficulty of the studywas the limited demand history available.

By describing the case, we make general observa-tions about the practical aspects of inventory con-trol. Moreover, our aim is to compare variouspolicies with real demand data from the case tosee which one is best under what circumstances.Common methods presented in the literature rather

use given statistical demand distributions to assessthe performance of inventory models. Conse-quently, with our methodology we can better iden-tify the real limitations of industrial data sets.

The remainder of the paper is organized as fol-lows: the next section presents briefly related litera-ture. Section 3 includes the case study description.Next the methodology is explained in Section 4.The computation results are included in Section 5and the final conclusions are presented in the lastsection.

2. Review of related literature

One of the major areas of inventory research overthe past decades is the one related to the manage-ment of spare parts inventories. Although theoreti-cal models for slow-moving items are abundant ininventory literature since 1965, case studies arefew (for a comprehensive overview of recent litera-ture on spare parts management see Kennedyet al. (2002)). These studies are generally concen-

trated on the mathematical optimization of inven-

tory models but unlike our work they do notvalidate them using empirical data.

In the arena of theoretical models, one of themost extensively studied inventory policies is theso-called (S À 1, S ) model, a particular case of

(s,

S ) models, with an underlying Poisson demanddistribution (see Feeney and Sherbrooke, 1966).

Although well studied and suitable for slow-movingitems, this type of policy requires continuous reviewof the inventory system. Moreover, the Poisson dis-tribution assumes randomness of demand, withinterarrival times between unit size demands follow-ing an exponential distribution. This distributionneeds no information of demand other than theaverage demand, which is the sole parameter of the demand distribution. When transactions are lar-ger than unit size, authors have proposed the use of

compound-Poisson models (see Williams, 1984; Sil-ver et al., 1971). However, these models are moredifficult to apply in practice because they need anassumption on the compounding distribution. Forexample Williams (1984) developed a method toidentify sporadic demand items, where three param-eters are needed: one for the exponential distribu-tion of interarrival times of demands, and twoparameters of an underlying gamma distributionfor the demand size.

Most of the empirical studies in spare parts liter-

ature are focused on testing forecasting methods fordemand of slow-moving items rather than on imple-menting inventory models. This is an important dis-tinction since forecast methods are used to estimatepoint forecast of the mean (like the moving averagemethod) while for evaluating control parameters of inventory models (like the (s, Q) model) one needsan estimation of the entire LTD distribution. More-over, inventory models are used to meet specific cus-tomer service levels in the long run, whileforecasting models aim to obtain accurate demandforecasts as determined by the mean average per-centage error (MAPE) or the mean square error(MSE). In this area of research Ghobbar and Friend(2003) present a comparative study of 13 differentforecasting methods for the management of spareparts in the aviation industry. They use the MAPEmeasurement applied to forecast errors to assessthe accuracy of the different methods but no inven-tory models are included in the study. They confirmthe superiority of the weighted moving average andCroston’s methods over exponential smoothing andseasonal regression models. Silver et al. (1998) also

advise on the use of Croston’s method for products

2 E. Porras, R. Dekker / European Journal of Operational Research xxx (2007) xxx–xxx

ARTICLE IN PRESS


http://-/?-

http://-/?-

http://-/?-

http://-/?-

http://-/?-

http://-/?-



with intermittent and erratic demand. This method(Croston, 1972) assumes that the LTD has a normaldistribution, and estimates the mean demand perperiod by applying exponential smoothing sepa-rately to the intervals between non-zero demands

and their sizes.Willemain et al. (2004) propose the use of a mod-ified bootstrap method to forecast intermittentdemand of service parts, and they implement themethod on a large industrial data set. Croston’smethod and exponential smoothing are evaluatedas well, yet like in the previous paper no attemptis made to implement an inventory control model.They show that the modified bootstrap method pro-duces more accurate forecasts (based on the MAPEmeasurement) than the exponential smoothing andthe Croston’s method. We use in the study the boot-

strap method proposed by Willemain and we com-pare it with the performance of an empiricaldistribution model. We assess also the performanceof models based on Poisson and normal demanddistributions.

A similar research to the present study is pre-sented in Strijbosch et al. (2000), where the perfor-mance of two different (s, Q) models for spareparts in a production plant environment is exam-ined. Unlike our method, they test the inventorymodels proposed using simulation where demands

are generated from an Erlang distribution, whereaswe assess the inventory models using the historicaldemand data for the items (around 8000 items).Also related to our study, Gelders and van Looy(1978) presented a case study carried out in a largepetrochemical plant. They developed differentinventory models to control slow and fast movingitems, which were clustered in classes using ABCanalysis together with criticality and value consider-ations. As they had limited information on con-sumption rates for slow movers, a Poissonunderlying distribution was assumed to comparebetween existing practices and the models proposed.We use in our study a similar approach, but differ-ent to their study we estimate the LTD distributionusing the methods mentioned above and we test themodels with real demand data, rather than usingsimulation.

3. The case study

In this section we describe the problem setting,the demand data, the classification criteria for the

parts and the costs.

3.1. Problem description

The company under study consists of a majorpetrochemical complex located in the Netherlands,which includes 60 different plants divided in chemi-

cals manufacturing and oil refining. The complexdates from 1930, and many new installations havebeen added since then. A large part of it, however,stems from the 1960s. The procurement departmentoffers service to all plants. There is one central ware-house owned by the company. At the moment of thestudy (2000), there were in total 130 thousand cata-logued materials, of which only 43 thousand werekept on stock at the site, with a total value of morethan 27 million euros. There are 22 additional smallde-central storages on site, containing fast movingmaterials that can be directly used if needed. No

stock registration is done for these items and theyare replenished on a batch basis. Therefore, we onlyneed to consider a single stock echelon, being thewarehouse as user of spares for equipment andnot a producer of parts. In total there are 180 thou-sand requests of material per year, both for non-stock and stock materials. Requests for materialskept on stock are supplied from available stock. If there is shortage of a material an emergency replen-ishment order is generated.

Controlling 43 thousand materials represents a

difficult task, especially because of the differencesin types and consumption patterns associated withthem. It also requires efficient use of the manpoweravailable and of the information system at hand.Until 1997 an in-house developed informationsystem for inventory control was used by the com-pany. In 1997 they moved to the information sys-tem SAP R/3, which is a complete ERP-system,but not specific for inventory control. Almost thewhole demand history before 1997 has been lostin the transfer to SAP. Within SAP, the companyapplied the MM (materials management) modulefor the control of its spare parts. Since SAPevolved out of MRP systems for the manufactur-ing and assembly industry, the MM module is verymuch based on the MRP planning philosophy (seeHeizer and Bender, 2001). Demand is expressed byactual orders or by forecasts of demands. Nextdemand of end products is converted to demandfor assemblies, components and parts. Stock con-trol is performed in SAP on a periodic basis (so-called periodic review). Items are ordered whenthe MRP run is made. The SAP user can set the

appropriate time interval, e.g. daily, weekly or

E. Porras, R. Dekker / European Journal of Operational Research xxx (2007) xxx–xxx 3

ARTICLE IN PRESS


http://-/?-

http://-/?-

http://-/?-

http://-/?-



monthly. At the company they run the MRP everyweek.

The actual stock control within SAP occurs interms of min–max levels (equivalently to (s, S ) poli-cies), or MRP-type control based on lead times.

Minor functionality is available in SAP to deter-mine the minimum level s and the maximum levelS . Safety stocks can be used to determine the re-order level s, and lot sizing methods are availableto evaluate the difference between s and S . Beforethe project 90% of the control levels were set manu-ally, and afterwards some 70%. As a result stillmany replenishment orders were checked manuallybefore sending them out.

With respect to forecasting, several methods areavailable in SAP, like exponential smoothing andmoving averages, both with trends and seasonality.

It is however the intermittent nature of demand thatmakes the application of these methods particularlydifficult to spare parts. For the determination of safety stock levels, the normal loss model is avail-able which approximates the demand during thelead time with a normal distribution. This modelworks with the cycle service level as service levelobjective. However, no fill rate service levels canbe defined within the MM module. A more strikingaspect of SAP is that within its functionality no con-tinuous review models can be implemented. There-

fore the classical and much advised (S À 1, S )model with Poisson distributed demand over thelead time cannot be applied.

3.2. Data structure

Statistical information for the consumption of spare parts was available for 5 years (the last yearonly until August). The demand information wasrecorded in monthly periods, so a total of 55 periodsof demand information was available for the study.One important limitation of the demand set wasthat it did not specify whether demands were dueto failures or preventive maintenance activities.

Different parts used by the company are dividedin two main categories: materials related to a pieceof equipment and the ones not related to anyparticular equipment, like protecting shoes, helmets,general-purposed electrical equipment and instru-mentation. From the total of 43,000 materials instock, 14,383 were spare parts, accounting for 80%of the total stock value. These spare parts are the

focus of the present study. The parts related to

equipment are classified according to criticalitycodes as follows:

• High (H): Unavailability of these materialswould result in expensive downtime or cause dan-

ger to the safety of the people and the environ-ment. Risk taken in the process of ordering andstocking cannot be justified.

• Medium (M): Unavailability of these materialswould result in significant loss of production,but does not endanger the safety of the peopleor the environment. A calculated risk can betaken in the process of ordering and stocking.

• Low (L): Unavailability of these materials wouldnot result in serious effects on the processes or onthe safety of the people and the environment.

The previous classification is made on expert judgment and no quantitative methods are used todate. A further inspection of the materials with crit-icality code leads to a more refined classification:materials that are uniquely installed in a particularpiece of equipment (60% of the materials related toequipment), and materials which are related to morethan one piece of equipment of different criticalitycodes. That means that there are spare parts thathave combined criticality codes (H/M/L, M/L)depending on whether they are installed in multiple

piece of equipment of different criticality. The com-pany used these codes to decide on the stock levelsof the different parts. Thus, items identified as highlycritical should be on stock since they require high fillrates, low critical ones destock, and medium criticalones on stock depending on cost-effective consider-ations. However, as no models are available inSAP that incorporate criticality considerations,these levels were set mostly by expert judgment.

3.2.1. Classification of parts

A more refined analysis of the spare parts datarevealed that important differences among themexisted not only in terms of criticality codes but alsowith respect to demand and price. Therefore, weaimed at grouping them in different classes to seewhether we should apply different stock controlmethods for different classes.

Below we describe the different classes consideredin this study.

3.2.1.1. Criticality classes. Based on the criticalitycodes, the following criticality classes were defined

for the spare parts:


ARTICLE IN PRESS




Criticality class 1: HCriticality class 2: H/M/L or H/LCriticality class 3: MCriticality class 4: M/LCriticality class 5: L

Criticality class 6: Not related to any particularpiece of equipment

For the current policy used by the company, weexpect to observe that the service levels associatedwith high critical items are higher than the onesfor low critical items (see discussion in Section 5.4).

3.2.1.2. Demand classes. The original data set con-sisted of more than 14,000 spare parts, for whichwe observed a high variability in demand patterns.For example some parts had only 0/1 demandswhile others experienced either few large demandsor no realization of demands during 5 years. Forother parts we observed large negative demandsdue to returns. Thus, a classification was neededfor the spare parts based on consumption rates.For parts with total positive demand over the five-year period and some demand values higher than1, we identified from a histogram two main groups:parts with relatively high total demand and partswith low total demand. Although the boundarybetween these two groups was not clearly identified

in the histogram of demand, from a Pareto analysiswe could reasonably establish it in 60 units. Thisalso implies that at least in one month demand isat least 2. We observe that 90% of the items had ademand below this value, and this contributed25% of the total demand. At the same time, itemswith 0/1 demands also had a total demand of lessthan 60. According to this, we established the fol-lowing demand classes for the parts:

Demand class 1: parts with only 0/1 demands.

Demand class 2: parts with total demand largerthan 0 but less than 60, and not only 0/1demands.Demand class 3: parts with total demand higherthan 60.Demand class 4: parts with À1, 0, 1 demands.

Parts with negative total demand or parts with norealizations of demand (all demands equal to zero)were excluded from the analysis.

3.2.1.3. Price classes. For the spare parts in the data

set, five different price levels were identified in a his-togram. Table 1 shows the different price classes forthe spare parts. The parts recorded in SAP with aprice of 0 euros are items not owned by the materi-als department (price class 1). We observed prices aslow as 0.01 euros for some parts (price class 2) andthe most expensive ones had a price of 20,000 euros(price class 5).

3.2.1.4. Combined classes. Using the criticality,demand and price classes, we include each item ina combined class defined by three digits. Accord-ingly, an item in class ‘‘xyz’’ corresponds to an itemwith demand class x, criticality class y and priceclass z. This classification allows us to optimizethe system per class rather than for individual items.That is, once a service level is defined for the com-bined class, the parameters for the different inven-tory policies are evaluated for each item in theclass. Then a simulation tool is used to evaluatethe performance of the selected model of each indi-vidual item using its demand data. Finally totalcosts are aggregated across all items in the class.

In this way we aim at obtaining an optimization rulefor each combined class considered in the study (seeSection 4).

3.2.2. Considerations on item classes and anomalous

observations

The analysis of spare parts data is performed forall combined classes incorporating demand classes1, 2, 3 and 4. Thus, all criticality and price classeswhich combine those demand classes are consideredfor the evaluation of the inventory models, except

price class 1, since items with price zero do not haveassociated holding costs. Since for items with totalnegative demand a zero inventory policy is optimal,we do not incorporate them in the analysis (4% of the parts). Moreover, although negative demandscan be associated with returns due to preventive

Table 1Price classes of parts

Price class 1 2 3 4 5

Price ( p) in euros p = 0 0 < p6 13.6 13.6 < p6 169 169 < p6 2112 p > 2112

Spare parts (total = 14 383) 10% 19% 33% 29% 8%


ARTICLE IN PRESS




maintenance practices (parts that were ordered butnot actually installed) or with repaired parts thatwere brought back to the system, we did not havespecific information in this respect. We also leaveout of the analysis items with no demand realiza-

tions in five years (2.2% of the parts). From theseconsiderations we are left with 11,984 items. Addi-tionally, we observed items with an error in the crit-icality specification. These items accounted for 4.1%of the total numbers of parts. After excluding theseitems, we were finally left with 11,790 spare parts forthe analysis. We also identified in the data set oneparticular month for which a large number of veryhigh demands was recorded. We considered thatthis was due to an administrative rebooking in thewarehouse and thus we eliminated this month fromthe data set in our analysis.

3.2.3. Lead times

The lead times for the spare parts were recordedin days. However the demand data set for the itemswas registered in months, without specification of the day within the month that a particular demandtook place. Therefore, for ease of implementation inthe simulation we rounded the lead times off to fullmonths using 30 days per month. In this way anitem with a lead time of 80 days was considered tohave a lead time of 3 months. Observe that this con-

version is also necessary for the estimation of thedistribution of the lead time demand, since demandforecasts for the items are produced in months.Although the rounding up of lead times was intro-duced to better cope with the demand data, bydoing this we take a conservative approach, inwhich the service levels achieved by the system inthe simulation will be generally lower than theyare in reality.

3.3. Cost structure

In general, three types of costs are associatedwith inventories: holding costs, ordering costs andstockout costs. Holding costs represent the cost of capital tied up in the spare parts inventory. Anannual fixed rate of 25% was used in the study.Ordering costs represent the cost associated withplacing an order for a spare part, which includesthe costs of telephone calls, inspection and handlingof the incoming items, paying the bill and registra-tion of the parts. This cost is independent of thenumber of parts included in the order. An ordering

cost of 36 euros was used in the study. Since our

objective is to evaluate the optimal balance betweenservice levels and holding costs we do not incorpo-rate in this study stockout costs.

4. Methodology

We use two approaches for the optimization of the spare parts inventory system under consider-ation, namely an ex-ante and an ex-post approach.In the ex-post procedure the same data set is usedfor both fitting and testing purposes. Opposite tothis, the ex-ante procedure, once a distribution hasbeen fitted to the data, uses an entirely differentset for testing purposes. In this respect the ex-ante

approach is more appropriate from a scientific andpractical perspective, since in reality systems facefuture ‘‘unknown’’ demands (Silver et al., 1998).

In order to achieve this, we divide the historicaldemand data into two sets, namely a fitting periodand a testing period. The fitting period will be usedto estimate the lead time demand distribution(LTD) which is used in turn to determine the inven-tory policy parameters. The testing period is used toperform a simulation to evaluate the performancesof the inventory policies selected and compare themwith the performance of the current one. We con-sider two types of service levels, the cycle servicelevel (CSL) and the fill rate. The reason for using

the ex-post approach is that many industrial datasets are rather short for forecasting purposes andthere is a lot of non-stationarity because of intro-duction or outphasing of parts. This procedure willgive the advantage of using the whole data set to geta ‘‘better picture’’ of the real demand process. Theperformance of both approaches will be comparedto assess the validation conclusions made by eachone. We have to note however that in the data setthere are many items with only one or two demandrealizations, and therefore we expect highly variableresults.

One of the main issues we address is whether the-oretical models can outperform stock analysts. Inpractice this is difficult to assess because of lack of information (e.g. short demand data sets, littleinformation on preventive maintenance practices),as well as implementation constraints. For instancereal lead times of items are normally in days but inthe models one may prefer to use full periods of timefor ease of implementation. Other practical issueslike the performance of different methods to modelthe demand process are explored as well in this

study. In order to achieve this, below we give the


ARTICLE IN PRESS




demand modeling methods used and next wedescribe the inventory models considered.

4.1. Modeling the lead time demand

In inventory decision making, one needs to deter-mine inventory control parameters, such as re-orderpoints and safety stocks. In order to do so, we needa specification of the lead time demand distribution.This is traditionally done by modeling the lead timedemand using common probability distributionsfound in the literature, such as the normal distribu-tion (Silver et al., 1998) or the Poisson distribution(see Schultz, 1987). Other authors propose the useof models based on forecast techniques such asmoving average or exponential smoothing (see Cro-ston, 1972; Silver et al., 1998). In order to cope bet-

ter with real data sets and to give a more realisticpicture of demand, authors have proposed the useof bootstrap techniques (Bookbinder and Lordahl,1989; Efron and Tibshirani, 1993; Willemain et al.,2004). As we are interested in the performance of theoretical models using real data, we estimate theLTD distribution using the Willemain’s bootstrapmethod along with another novel procedure usingempirical data. In this way we estimate the distribu-tion of demand over the lead time for each model,which is used in turn to evaluate the parameters of

the inventory policies selected. However, differentto the methods found in the literature, our objectiveis to use directly the real demand values observed toassess the performance of the policies using the sim-ulation tool. To keep the study tractable, we do notapply updating for the estimation of the LTD distri-bution. We also evaluate the performance of the sys-tem using normal and Poisson distribution basedmodels. Below we describe these methods.

4.1.1. Willemain’s bootstrap method (W)

We implemented the modified bootstrap methodpresented in Willemain et al. (2004). This method,as compared to traditional bootstrap techniques pre-sented in the literature, has the advantage of captur-ing better the autocorrelations between demandrealizations, especially when dealing with intermit-tent demands with a high proportion of zero values.The method first evaluates the empirical transitionprobabilities between states of zero demand andstates of positive demand for the different items.Then using this information, a stream of zero andnon-zero demands is randomly generated for a per-

iod of length equal to the lead time. The non-zero val-

ues are filled with demand values sample from thedata set. In this way estimates of LTD for each itemare obtained for a large number of realizations (1000in this study). This information is finally used to esti-mate the distribution of LTD. Willemain et al. (2004)

applied this method to nine large industrial data setsof service parts inventories and compared it with theexponential smoothing method and the Croston’smethod. He concluded that the modified bootstrapmethod gave the best performance of all three meth-ods. In Fig. 1, we show a plot of an estimation of theLTD distribution (CDF) using Willemain’s methodfor an item corresponding to class 215. The item(labelled #741) has a lead time of nine periods andits demand data for the 55 periods is as follows: inperiods 5 and 17 it observed positive demands of unitsize each, and in period 9 a demand of 2. The rest of

the periods no demands were observed. Notice thatalthough only lead time demand values of 0, 1, 2and 3 were realized in the data set, the method is ableto produce a LTD estimation where many other leadtime demand values are taken into account.

Once the LTD distribution is obtained, we canuse it to determine a re-order point to achieve agiven fill rate b as follows:

From the CDF, F (x), of LTD obtain the list of possible re-order point values s, by setting s = x,where x are the lead time demand values, and their

corresponding probabilities f (x). Now choose thesmallest s satisfying:

100b% 6 1 À ESð sÞQ

Â 100;

0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

1

x

F ( x )

Item# 741 Willemain's CDF

Fig. 1. Cumulative distribution function using Willemain’s

method.


ARTICLE IN PRESS


http://-/?-

http://-/?-

http://-/?-

http://-/?-



where Q is a pre-determined lot size and ES(s) is theexpected units short for a given re-order point s,which is evaluated as follows:

ESð sÞ ¼X xj x> s

ð x À sÞ f ð xÞ:

Note: The term 1 À ES(s)/Q is the calculated fill ratefor a given s.

We use the above procedure to evaluate a re-order point s for item #741 for a given fill rate of 82%. This item has a price of 4172 euros. Using thisinformation and the procedure describe in Section4.2 we obtain Q = 1. Now following the above pro-cedure we have: for s = 3 we obtain ES(s) = 0.254with associated fill rate of 74.6%, and for s = 4 weobtain ES(s) = 0.136 with associated fill rate of 86.4%. Thus, in this case s = 4 is the re-order point

value sought.

4.1.2. Empirical distribution of lead time demand (E)

We implemented an empirical model to estimatethe distribution of LTD. Differently from the tradi-tional bootstrap method, we construct a histogramof demands over the lead time without sampling.This method is new to the literature as no attemptshave been made to use it for inventory control.Since demands are taken directly from the data setover fixed periods of time equal to the lead time, this

method also captures autocorrelations and fixeddemand intervals due to preventive maintenance,and is far easier to implement than the modifiedbootstrap method described above. Using theempirical model, we build once more the CDF forthe LTD of item #741 which is depicted in Fig. 2.As the empirical model uses only the demand valuesfrom the demand data set to construct the CDF,fewer lead time demand values are produced in this

CDF than in the corresponding one using the Wille-main’s method (see Fig. 1). Using a similar proce-dure as the one described for the Willemain’sbootstrap method, we can use the empirical CDFto determine a re-order point for a given fill rate.

As an illustration of this, using the empirical CDFof item #741 we obtain for s = 1 and s = 2 calcu-lated fill rates of 68.1% and 87.2%. Thus, for a givenfill rate of 82% we determine a re-order point values = 2.

4.1.3. Normal distribution (N)

We implement a normal based model assumingthat demand for the parts follows a normal distribu-tion. To this end, the average ( DÞ and standard devi-ation (SD) of the observed period demand isevaluated to estimate the parameters lLTD and rLTDof a normal LTD distribution, as follows:

lLTD ¼ D Á LrLTD ¼ SD Á

ffiffiffi L

p ;

where L is the lead time of an item in full periods of time (days, months, etc.), and with D and SD eval-uated using the whole data set of demands, includ-ing zero and negative values. Thus, for integrationover a normal LTD distribution we use commonformulas found in the literature (e.g. Silver et al.,

1998 or Chopra and Meindl, 2004).The normal distribution is not generally advised

for modeling the demand of slow-moving items,for which a Poisson distribution is better recom-mended (Silver et al., 1998). Thus, we do not expectthe normal based model to give better results thanthe others. However, we want to investigate its per-formance as compared to the other models consid-ered. To this end, when we evaluate re-orderpoints based on the normal LTD distribution tomeet desired fill rates, the values obtained arerounded up to integer values, and negative valuesare set to zero. We require this since for our systemre-order points are defined as positive integer valuesin accordance with the discrete demand for parts. Inthe case of negative values which are set to zero nocompensation for the gain mass is applied in theintegration of the normal LTD distribution. Thiscauses only a minor distortion in our results as wenormally look at high service levels which haveassociated positive re-order point values. We neithercorrect for the gain mass associated with the round-ing off of fractional values, as this has also a minor

effect. Accordingly, to determine a re-order point s

Item # 741: Empirical CDF

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4

Lead time demand

Fig. 2. Empirical cumulative distribution of lead time demand.


ARTICLE IN PRESS




for a given fill rate b, we use a similar procedure asthe Excel ‘‘goal seek’’ routine (see Chopra andMeindl, 2004), by first finding a z-value satisfying:

100b% 6 1 À rLTD Á UNLIð z ÞQ

Â 100;

where UNLI(z) is the unit normal loss integral asso-ciated with the unit normal variate z. Then s followsfrom s = lLTD + z Æ rLTD, rounded up to the nearestinteger. Recall that z corresponds to a re-orderpoint s associated with a CSL value (see Silveret al., 1998). Thus the product in the denominatorof the above formula gives the expected units shortfor a given re-order level s and the whole right-handterm gives the calculated fill rate. As mentioned inSilver et al. (1998), the previous formula underesti-mates the true fill rate if rLTD is large relative to

Q. Therefore, a correction should be made in thenumerator substituting the term rLTD Æ UNLI(z)by rLTD(UNLI(z) À UNLI(z + Q/rLTD)). By usingthe uncorrected formula, we obtained conservativevalues of the fill rates, which did not have a majorimpact in the optimization of the system (see resultsin Sections 5.1 and 5.2).

Using the data for item #741 we obtain lLTD =0.654 and rLTD = 0.975. Now, applying the aboveprocedure for a given fill rate of 82% we have:

For s = 1: CSL = 63.8%, z = 0.354, UNLI(z) =

0.247, calculated fill rate = 75.9%.For s = 2: CSL = 91.6%, z = 1.379, UNLI(z) =

0.038, calculated fill rate = 96.3%.Thus, s = 2 is the re-order point to be selected.

Remark 1. Although in many situations the nor-mality assumption is not satisfied, this distributionhas widely been used in practice and it is available inmany packages. This is due to the simplicity toevaluate re-order points and other parameters basedon the normal distribution.

4.1.4. Poisson distribution (P)

Silver et al. (1998) suggest that the Poisson distri-bution is suitable to model demand of slow-movingitems. We use the Poisson distribution to estimatethe LTD distribution for items in demand class 1.The reason is that demands for these items are of unit size, and hence the basic assumption of thePoisson distribution is satisfied. The only parameterof the Poisson distribution, the average rate of demand over the lead time, is estimated from thedemand data for the different items. For other

demand classes, a compound-Poisson based model

would be more appropriate, but it is not clearbeforehand which batch distribution should be usedas often very specific values are observed (e.g. item15 has demands of 30, 80 and 120 on a total of 792 installed). The Poisson-based model is com-pared to the normal, empirical and bootstrap meth-ods for demand class 1.

In order to determine a re-order point for a givenfill rate using the Poisson model, we can apply asimilar procedure as the one described above forthe Willemain’s or the empirical method, substitut-ing f (x) for the Poisson probabilities (with parame-ter kLTD ¼ D Á LÞ and using s = x as the set of positive integers 1,2, . . . For the Poisson model weuse a (s, S )-type policy with s = S À 1 (see Section4.2). Applying the above procedure to the data of item #741 and a given fill rate of 82% we obtain:

For s = 0: calculated fill rate = 34.5%.

For s = 1: calculated fill rate = 82.6%.For s = 2: calculated fill rate = 96.6%.

Thus, the re-order point value selected is s = 1.Note: As for item #741 we observed demand val-

ues higher than one, this item belongs to a demandclass where the Poisson model is not applied. There-fore, the above calculation is presented for illustra-tive purposes (see Table 2 in Section 4.2).

4.2. Inventory models

We use an (s, nQ) inventory policy for the system,with the re-order point s evaluated using the LTDdistribution according to the modeling methodsdescribed above. Thus, when overshooting of there-order point s cannot be overcome by the lot sizeQ, an alternative lot size equal to nQ is ordered,such that the inventory position is brought aboves, where n is an integer value. This is a commonpractice in inventory management (see Silver et al.,1998). The lot size Q will be evaluated according

to the economic order quantity (EOQ) using aver-

Table 2Inventory models considered

Model Parameters Demand classes

Current policy (C ) min–max (s, S ) 1, 2, 3, 4Poisson based model (P ) (S À 1, S ) 1Normal based model (N ) (sN , nQ) 1, 2, 3, 4

Empirical based model (E ) (sE , nQ) 1, 2, 3, 4Willemain based model (W ) (sW , nQ) 1, 2, 3, 4


ARTICLE IN PRESS




age annual demand. We round off the EOQ calcula-tion according to Axsater (2000), as follows:

1. Evaluate: m ¼ bEOQc

2. Set Q ¼1 if m ¼ 0

m if m 6¼ 0 and

EOQ

m 6

m

þ1

EOQm þ 1 otherwise

8<:When we use the Poisson distribution to estimatethe LTD, we use Q = 1. This model is often referredto as (S À 1, S ) model, with s = S À 1. As averagedemand is generally low in this case, the EOQ calcu-lation also yields a value of 1.

For the classes under study the proposed policiesare compared to the current (min, max) policy interms of the selected service level and total costs.The aim of the proposed methodology is to optimize

the system, that is, to establish which model andpolicy perform best under which conditions (seeTable 2).

4.2.1. Handling of large demands

Consider the situation in which an item with arelatively short lead time observed ‘‘unusual’’ highdemands. For such an item the analysis becomesdifficult as the associated re-order point (say evalu-ated according to a normal LTD distribution) islikely to be overshot when using a simulation tool

to assess the performance of a given inventorymodel. To illustrate this we give in Table 3 thedemand data and other relevant information foritem #1307, which has a lead time of one month.For this item we consider that the demand valueof 450 is an outlier, since this value is larger than

the average of the rest of the positive demands plus10 times their standard deviation. So here the aim isto construct a tool that filters out this large demand.To this end, we implement a demand filter in the fol-lowing way: all demands larger than the average

plusk

standard deviations (evaluated using onlypositive demands) are filtered out. For this particu-lar item observe that k = 3 will not produce thedesired result and hence a value k = 2 is a betterchoice. In a similar way, for items belonging todemand classes 2 and 3 we found that actuallyk = 2 was the best selection for the demand filter.As a result of applying the mentioned filter, 8.7%of the positive demand values for these two classeswere filtered out.

Remark 2. Although for the optimization of the

system we used the above filter of demands, we alsoassessed the effect of having large demands includedin the optimization process, for which we obtainedsimilar results (see Porras, 2005).

Remark 3. In the inventory analysis of all fourmodels we neglect the so-called undershoot. Thisis the amount by which the inventory positionsdrops below the re-order point. If the demand sizeis always one, the re-order level is always exactlyreached, but if the order size is large, there can besubstantial drop below the re-order level. When cal-culating the leadtime demand distribution, oneshould take the undershoot distribution thereforeinto account. We did not include it in our analysisbecause we removed most large demands.

4.2.1.1. Remark on the implementation of the ex-post

and ex-ante approaches. Due to the limitationsinherent to the data set used in our simulationstudy, it may well be that not enough informationis available for fitting purposes. Therefore, itemswith only zero or one positive demand during thefitting period are excluded from the analysis. Thisconsideration is used in both the ex-post and theex-ante approaches. As for classes with few items(6 or less), they were excluded from the analysis inboth the ex-ante and the ex-post approach. Accord-ing to these considerations, of the original 11,790items, a total of 8494 were included in the ex-post

approach and 4326 in the ex-ante approach (seeanalysis of results in Sections 5.1 and 5.2).

4.2.1.2. Remark on classification of items. We

include each item in a class according to the demand

Table 3Demand data for item #1307

Demand value Number of occurrences

Totaldemand

0 47 087 1 87100 4 400120 1 120150 1 150450 1 450Total 55 1207

Average of demands > 0 150.9Standard deviation of

demands > 0122.4

Average of positivedemands < 450

108.1

Standard deviation of positive

demands < 450

20.8


ARTICLE IN PRESS




pattern and other relevant information (criticalityand price). We do the classification using the wholedata set in both the ex-ante and the ex-post

approaches, since we want to assure that an itemin a certain category will not exhibit demand values

not corresponding to that category during the simu-lation in the testing period.

4.3. Optimization of the system: A decomposition

approach

We focus on optimization of the system based onservice levels, for which both the fill rate and thecycle service level are used. The utilization of servicelevels to set safety stocks is a preferred method inindustry as opposed to cost minimization. The rea-

son for this is that the latter requires the evaluationof stock out costs which depend on down time pen-alties and other factors which are difficult to evalu-ate in practice.

Considering the size of the system related to thepresent study (recall that originally it comprisessome 14,000 items), we use a decompositionapproach for the optimization of the system. Thistype of approach is well known to the literature,where it is often used for the analysis of complexsystems, e.g. for the optimization of multi-echeloninventory systems (see van der Heijden et al.,1997). In our case the basic idea is to optimize thesystem at group level, defined by the classesdescribed earlier, rather than doing it at item level.Thus, we evaluate the different inventory parame-ters for the items based on a single target servicelevel for all items in the class. Notice that deviationsof the realized service levels with respect to the tar-get values are expected due to the discrete nature of inventory levels. Thus, the objective of the methodis to find the right level of a target service level thatoptimizes the classes under study.

We first introduce the following notation:

j index for each item j in class p, j = 1, . . . ,N p, where N p is the size of class p

X identifier for the model applied in the simu-lation, where X = current policy (C ), Pois-son (P ), Normal (N ), Empirical (E ) orWillemain (W ).

cð X Þ j number of inventory cycles completed by

item j during the testing period using modelX . Note: an inventory cycle is defined be-tween the placing of an order and its arrival

to the system, i.e. the inventory cycle over a

lead time (also referred to as replenishmentcycle)

socð X Þ j number of inventory cycles for item j with

stockouts during the testing period usingmodel X

D

ðt Þ j total number of units demanded of item

j

during the testing periodS

ð X Þ j total number of units of item j supplied

from on hand stock over the testing periodusing model X

CSLð X Þ p cycle service level achieved by the system

using the model X

bð X Þ p fill rate achieved by the system using model

X

b p target fill rate for class p

ES j (s j ) expected units short for item j for a givenre-order point s j , according to the corre-

sponding LTD distribution

Now for each combined class p(xyz) we proceedas follows:

1. Evaluate the service levels (CSL and fill rate)achieved by the current system over the testingperiod according to the ex-post or the ex-ante

approaches. These are the current service levelsof class p, which are defined by

CSLðC Þ p ¼ 1 À

P N p j¼1socð

C Þ jP N p

j¼1cðC Þ j

; bðC Þ p ¼

P N p j¼1S ð

C Þ jP N p

j¼1 Dðt Þ j

:

2. For each item j , estimate the cumulative distribu-tion of LTD over the fitting period, according tothe model selected (see Section 4.1). Use b p ¼ bðC Þ

p

together with the LTD distribution to estimatethe corresponding parameters of the inventorypolicy (i.e. re-order point and lot size), in the fol-lowing way:

• Calculate Q j , the lot size, according to theEOQ formula and round it off as explain inSection 4.2. If class p = 1 yz (demand class 1)then set Q = 1.

• Chose the smallest s j which satisfies a 100b p%fill rate, i.e.:

100b p % 6 1 À ES jð s jÞQ j

!Â 100:

3. For each item j , run the simulation over the test-

ing period applying the (s, nQ) or the (S À 1, S )


ARTICLE IN PRESS


http://-/?-

http://-/?-

http://-/?-

http://-/?-



policy as required, using the parameters selectedin the previous point. For each model selected,evaluate for class p the realized fill rate (rb p),and its total costs (TC p), according to

r bð X

Þ p ¼ P N p j

¼1S

ð X Þ jP N p

j¼1 Dðt Þ j

; TCð X

Þ p ;target ¼ X N p j¼1

TC j;target;

where TC j ,target is the total cost of item j for thetarget service level selected, which comprises theholding and ordering costs. Compute the totalsavings achieved by the selected model with re-spect to the current system according to:

Total savingsðmodelÞ p ¼ TCðcurrentÞ

p ;target À TCðmodelÞ p ;target :

Within the classes we expect deviations of therealized service levels for the individual items.

In order to assess the magnitude of those devia-tions, we evaluate the weighted variances of therealized fill rate as follows:

Varðr bð X Þ p Þ ¼ 1P N p

j¼1cð X Þ j

X N p j¼1

cð X Þ j bð X Þ2

j À ðr bð X Þ p Þ2

:

An estimation of the standard deviation of thetarget fill rate is thus given by

rð X Þ p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVarðr bð X Þ

p Þq

:

4. For each model selected check whether the real-ized service levels are better than the service levelsof the current system. If the answer is yes, calcu-late total costs.

5. For classes with lower realized service levels ornegative savings, construct exchange servicelevel-cost curves for the different inventory poli-cies, in the following way: define a set of fixed tar-get values for the fill rate (b p), say from 1% to100% in step sizes of 1%. Next evaluate theparameters of the selected inventory model as inpoint 2 (except for the current system). For each

value of the target service level obtain the corre-sponding rb p, TC p,target and total savings. Finally,identify the value of b p, for which the correspond-ing realized fill rate equals or exceeds the currentfill rate (within 1% precision), and that has thelargest positive savings. In this way we optimizethe system among the different classes.

Remark 4. The previous method assumes that thesystem is optimized using the fill rate as optimiza-

tion criterion, which plays the role of control

parameter. However, it is also possible to implementthe above procedure using the CSL for the evalu-ation of re-order points in steps 2 and 5, in theclassical way. Even in the case for which the trueoptimizing criterion is the fill rate, one may decide

to apply the CSL as control parameter due tolimitations associated with the software available,which may not allow the evaluation of the fill rate.

Remark 5. When more than one order is outstand-ing for an item in the simulation, that is when 2 ormore replenishments cycles overlap, we associate astockout with the immediately precedent cycle thathas not finished yet, thus causing only that cycle tobe a stockout cycle. This situation is not consideredin inventory text books such as Silver et al. (1998),

thus confusion as to how to register stockout cyclesmay arise. To illustrate this consider the item#2248 (see Fig. 3) with lot size of 2 units, lead timeof 2 periods and re-order point of 2 units. In period19 the inventory position reaches the re-order point,triggering the first replenishment cycle. This cycleends at the beginning of period 21 when the systemis replenished and causing the net stock to reach 4units. This cycle ends up with no stockouts (thus col-oured white). The second cycle starts in period 22with net stock still positive. Notice that before thiscycle ends, a third cycle starts in period 23, wherethe net stock drops to À1. This causes the secondcycle to be a stockout cycle (thus coloured grey). Sim-ilarly, the demand of 8 units in period 24 causes thethird cycle to be a stockout cycle, triggering a fourthcycle. Here the question arises as to whether thefourth cycle should be counted as stockout cycle ornot. Although these two last cycles overlap we onlyconsider the former to be a stockout cycle (one ordersafter the demand occurrence). Notice that actuallythe fourth cycle ends up with a positive net stock.In this way and assuming no further demands after

period 31, the item will achieve a CSL of 60%. Noticethat if the fourth cycle was considered a stockoutcycle the CSL calculation would give 40%.

4.3.1. Remark on the implementation of the

optimization methodology

In the study we optimized the system based onthe fill rate, since this service level is of more practi-cal importance for spare parts. We investigatedhowever the realized cycle service levels (CSL)achieved by the different classes under the models

considered. We observed that in many cases for a


ARTICLE IN PRESS




particular target fill rate the corresponding CSL wasway below the current CSL. Therefore, in order toprevent such low values of the CSL we imposedan extra condition in Step 5 of the previous algo-rithm. Accordingly, for the identification of the

optimal target fill rate we checked that the realizedCSL was within 5% of the current CSL as long asthe system still achieved positive savings. In somecases (especially for the empirical and Willemain’smodels) this condition was difficult to satisfy exactlyfor particular classes (see e.g. classes 213, 214 and215 in Table 11), and hence the realized CSL wasbelow the 5% value. For the evaluation of the real-ized CSL of class p for the different models we usethe following formula:

rCSLð X Þ p ¼ 1 ÀP N p

j¼1soc

ð X Þ jP N p

j¼1cð X Þ j

with corresponding variance given by

VarðrCSLð X Þ p Þ ¼ 1P N p

j¼1cð X Þ j

X N p j¼1

cð X Þ j 1 À soc

ð X Þ j

cð X Þ j

!2

À rCSLð X Þ p

2

:

4.3.2. Accuracy of the LTD modeling methodsSince we are interested in the performance of the

modeling methods applied to inventory policies, auseful measure of the accuracy of the different mod-els can be provided by the average of the absolutedifference between the target and the realized servicelevels. As we observed earlier, differences areexpected due to the discreteness of the inventory lev-els. We call this measure MAD, defined by

MADð p ÞCSL

¼1

N p X N p

i¼1 jrCSLi

Àa p

j

for a target cycle service level a p ; and

MADð p Þb ¼ 1

N p

X N p i¼1

jr bi À b p j

for a target fill rate b p :

4.4. The simulation model

We implemented the proposed methodology inMatLab v.7.0. The system was simulated over thetesting period using the different inventory models.Recall that demands for the items in each class weredrawn directly from the data set and used as inputto the simulation. Thus, we were able to evaluatethe performance of the different policies and modelsin a real inventory environment. This allowed us to

capture the peculiarities of the system under studyand to discern under which conditions which policyperforms better.

Since demands cannot occur with a higher fre-quency of one per month, we consider that the sys-tem is reviewed on a continuous basis, and thereforevalid for the application of the (s, nQ) and (S À 1, S )policies.

For the evaluation of inventory cycles and com-putation of inventory related costs we use the classi-cal inventory methodology. Thus, the (s, nQ) policyis implemented in the classical way as explained inSection 4.2.

4.4.1. Considerations on starting stocks

Since the system orders nQ units whenever theinventory position drops below the re-order point(or Q when it is at the re-order point), the maximumlevel that the system can have at any point in time isgiven by its re-order point s plus Q. Therefore, forcomparison purposes it is fair to consider the start-ing stock of an item as its re-order level s plus its lotsize Q. For the current system, an equivalent choice

is to set the starting stock of an item at its max level,

Item # 2248

ROP = 2

Q = 2

L = 2 periods

Period 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Demands 0 1 0 0 0 0 0 1 0 0 2 3 8 0 1 0 1 0 0 0

Net stock 4 3 3 3 3 3 3 2 2 4 2 -1 -7 -4 3 3 2 2 4 4

Inventory Position 4 3 3 3 3 3 3 4 4 4 4 4 4 4 3 3 4 4 4 41 2 5

Cycles 3

4 ?

Fig. 3. Example to illustrate the occurrence of stockout cycles.


ARTICLE IN PRESS




since this can be considered as an order-up-to levelS with S = s + Q. This choice of starting stocksallows us to compare the performance of the differ-ent inventory policies considered. Note that for aninfinite horizon, the CSL is indifferent to the choice

of starting stock, whereas thefill rate

is not. How-ever for a fixed data set, the starting stock may alsohave an effect on the cycle service level of an item,since for higher starting stocks fewer inventorycycles will be completed. As we use the same defini-tion of starting stocks for all models, the total num-ber of inventory cycles will be the same amongthem, except for the current system. To see thisnotice that the number of inventory cycles for anitem is independent of the choice of its re-orderpoint. Since for all models we use the same demandvalues and Q is fixed, it follows that the number of

cycles will be the same regardless of the model used.

5. Results from the simulation optimization

After having applied the classification criteriaexplained in Section 3, and using the methodologypresented in the previous section, we performed anoptimization of the inventory system of spare parts.In order to have a clear picture of the whole process,we present in Fig. 4 a flow chart diagram of the

entire procedure, from cleaning and classificationof demand, price and criticality data for spare partsuntil the optimization of the system itself.

In Tables 4 and 5, we give a summary of thenumerical results obtained from the simulation opti-

mization for theex-post

andex-ante

approaches,using as optimization criterion the fill rate (for acomplete set of results of the different item classessee Tables 11 and 12 in the Appendix). We includethe relevant classes under study, for which the real-ized CSL, the realized fill rate and the total savingsare reported for each model. We also include thetotal cost of the current system with the currentachieved service levels (target CSL and target fillrate). For convenience, we use the following nota-tion in the optimization results tables:

tCSL target cycle service level, that is, the cycleservice level achieved by the current system(C ) using the (min, max) policy

t fill rate target fill rate, that is, the fill rate achieved bythe current system under the (min, max) policy

rCSL realized cycle service level, that is the cycleservice level achieved by each of the fourinventory models under consideration,these are: Normal (N ), Poisson (P ), Empir-ical (E ) and Willemain model (W )

Fig. 4. Flow chart diagram of the optimization study.


ARTICLE IN PRESS


http://-/?-

http://-/?-



Table 4Summary of optimization results for the system under ex-post approach

Itemclass

# of items

Current system (C ) Normal model (N ) Poisson model (P ) Empirical model (E )

Totalcostsa

Service level (%) Savings Service level (%) Savingsa Service level (%) Savings Service l

tCSL t fill rate (N ) vs.(C )

rCSL r fill rate (P ) vs.(C )

rCSL r fill rate (E ) vs.(C )

rCSL

Summary by demand class

1 2226 3,693,400 95.6 95.6 774,220 96.2 96.8 703,250 96.8 96.8 762,330 96.3 2 5185 9,469,700 84.3 86.1 408,750 81.1 90.7 202,860 80.0 3 608 685,920 77.2 85.7 31,370 70.0 90.2 À15,411 37.4 4 475 1,676,100 96.5 96.6 92,713 96.3 97.3 104,950 96.3

Summary by criticality class

1 493 1,704,700 88.9 88.3 20,428 87.3 91.8 3541 92.9 92.9 À5074 76.1 2 447 277,250 94.1 95.3 37,882 95.0 97.1 1215 99.8 99.8 28,617 91.2 3 3190 9,165,100 90.1 89.8 853,480 88.4 92.7 648,930 97.5 97.5 802,150 86.9 4 1395 905,370 92.6 92.9 107,170 90.7 94.9 20,420 97.2 97.0 83,448 90.0 5 372 634,320 90.8 89.8 61,002 90.4 93.4 1060 96.1 96.1 30,051 88.6 6 2597 2,838,300 79.5 85.4 227,090 75.3 90.6 28,077 95.7 95.6 115,530 70.8

Summary by price class

2 2223 524,350 83.2 90.0 179,030 79.4 97.0 À5415 97.9 97.9 166,870 75.1 3 3266 1,608,000 87.9 88.5 245,180 84.7 91.3 25,627 98.2 98.2 115,240 83.3 4 2487 3,637,300 90.2 89.4 200,410 90.1 90.7 54,393 95.5 95.4 71,791 86.7 5 518 9,755,400 90.4 88.2 682,440 88.4 91.5 628,640 96.0 96.0 700,820 83.2 Total 8494 15,525,060 87.5 89.2 1,307,055 85.1 92.6 703,246 96.8 96.8 1,054,723 82.2

a All cost and savings figures are in euros.

P l e a s e c i t e t h i s ar t i c l e i n pr e s s a s : P or r a s ,E . ,D e k k e r ,R . ,

Ani n v e n t or y c on t r ol s y s t e m

f or s p ar e

p ar t s a t ar e fi n e r y : . . . ,

E ur . J . O p e r .R e s . ( 2 0 0 7 ) , d oi : 1 0 .1 0 1 6 / j . e j or .2 0 0 6 .1 1 . 0 0 8



Table 5Summary of optimization results for the system under ex-ante approach

Itemclass

# of items


Totalcostsa

Service level (%) Savingsa Service level (%) Savings Service level (%) Savings Service

tCSL t fill rate (N ) vs.(C )

rCSL r fill rate (P ) vs.(C )

rCSL r fill rate (E ) vs.(C )

rCSL


1 800 504,650 95.6 95.7 94,010 96.5 97.2 63,575 96.3 96.3 96,409 96.2 2 2873 1,975,100 84.8 87.3 53,384 81.3 91.5 10,900 79.1 3 475 152,750 76.3 90.4 11,873 80.1 92.9 16,379 67.2 4 178 246,130 98.9 99.0 À1816 96.6 97.6 À2571 96.6

Summary by criticality class1 132 152,500 86.5 85.7 À540 80.7 90.7 À468 87.0 87.0 À14,322 87.1 2 301 81,914 94.8 96.5 12,877 97.2 97.8 1496 100.0 100.0 14,668 85.1 3 1457 1,701,000 90.0 90.5 51,385 85.6 91.7 56,230 96.5 96.5 63,253 83.1 4 818 212,440 93.3 94.3 29,042 93.4 96.1 À4,659 97.7 97.7 26,538 92.6 5 159 84,877 94.8 95.2 6130 92.5 95.2 2398 98.7 98.7 2132 92.5 6 1459 645,930 76.3 84.6 58,558 75.6 91.4 8578 94.0 94.0 28,848 71.8


2 1372 144,420 81.1 90.5 58,581 82.9 98.3 À1919 95.2 95.2 63,142 77.6 3 1648 377,020 87.5 88.7 62,667 83.9 92.3 1126 98.8 98.8 38,379 82.6 4 1159 831,190 90.7 90.0 19,185 87.4 88.1 10,508 94.5 94.5 19,131 84.9 5 147 1,526,000 90.6 90.2 17,017 86.1 88.3 53,860 96.3 96.3 466 83.4 Total 4326 2,878,665 86.4 89.7 157,452 84.6 93.0 63,575 96.3 96.3 121,118 81.7

a All cost and savings figures are in euros.

P l e a s e c i t e t h i s ar t i c l e

i n pr e s s a s : P or r a s ,E . ,D e k k e r ,R . ,Ani n v e n t or y c on t r ol s y s t e m

f or s p ar e p ar t s a t ar e fi n e r y : . . . ,




r fill rate realized fill rate, same observations as forrCSL

Savings (*) vs. (C ) Difference between the holdingand ordering costs of the current systemwith respect to (N ), (P ), (E ) and (W )

5.1. Optimization results for the system as a whole

From the results at the bottom of Table 4 we cansee that for the aggregated values across all classesconsidered in the system, all the models outper-formed the current system under the ex-post

approach, achieving lower total costs and higher fillrates, with overall CSL within 5 ± 1% of the currentCSL. For the different models under the ex-post

approach, we make the following observations:

The normal model achieved the best overall per-formance among the models, achieving the highesttotal savings, although for demand class 1 theempirical and Willemain models performed verysimilar with respect to the normal. For this classthe Poisson model performed the worst of all mod-els. Total savings for the normal model were of 8.4%over a total current cost of 15.5 million euros for the5-year period, and with overall fill rate of 92.6% ver-sus a current fill rate of 89.2%. The cycle servicelevel achieved by this model was 85.1%, slightly

lower than the one of the current system.The empirical model outperformed Willemain’s

model in terms of total savings (1.05 vs. 0.96 millioneuros), both achieving overall fill rates above 90%and similar overall cycle service levels. To explainthe difference in performance of these models, wefound that the Willemain model evaluates slightlyhigher re-order points to achieve the same servicelevel as the ones of the empirical model. Thisbecause the Willemain method produces a largerrange of different lead time demand values, whereasthe empirical method uses the exact values observedin the data set to construct the LTD distribution.Therefore, larger deviations are expected in theLTD distribution based on the Willemain modeland expensive individual items are likely to have ahigher effect with respect to holding costs. In thisrespect, observe that the difference in total savingsbetween the two models comes from few classeswith expensive items (e.g. classes 135, 235, 435),whereas for other classes both models performedvery similar or the Willemain model performed bet-ter (see Table 11 in the appendix). On the other

hand the normal model uses a continuous distribu-

tion of LTD, which is smoother than the empiricalor the Willemain’s one, making it less expensive tomatch or improve the target service levels.

As for the ex-ante approach, we have the follow-ing conclusions (see bottom of Tables 5 and 12 of

the appendix):As the ex-ante approach uses a reduced data setfor fitting purposes, the optimization of the systembecame more difficult than when using the ex-post

approach, which could better match or improvethe target service levels at lower total costs. Recallthat fewer items are used under this approach sincewe only consider items with at least two positivedemands during the fitting period. Despite theselimitations, the results obtained from ex-ante

approach confirmed the general conclusions derivedunder the ex-post approach. First of all observe that

the current system was outperformed by all modelsalthough with lower savings.

Second, under this approach again the normalmodel achieved the best performance among allmodels, with total savings of 5.5% with respect tothe current system and overall fill rate of 93% (overa current one of 89.7%).

Next, as in the ex-post approach, the empiricalmodel achieved higher savings than the Willemainmodel, both with realized fill rates above 90%. Wecan see that again the difference comes from the very

same classes as the ones discussed above for the ex- post approach.

Finally, for demand class 1, the Poisson modelperformed the worst of all models, which is the sameresult found under the ex-post approach.

5.2. Optimization results by demand, criticality and

price classes

From the optimization results aggregated bydemand, criticality and price classes presented inTables 4 and 5, we can see the effect of the differentmethods on particular classes. As we had moreinformation available for the individual classesunder the ex-post approach, we give the followingobservations under this approach, unless otherwiseindicated.

All the models outperformed the current systemfor the different aggregated classes with positive sav-ings and higher fill rates, except for demand class 3,where the empirical and the Willemain models hadhigher total costs and lower fill rates; criticality class1, where the empirical model yielded higher total

costs; and price class 2, where the Poisson model


ARTICLE IN PRESS


http://-/?-

http://-/?-

http://-/?-

http://-/?-



achieved also higher total costs. For the latter two,however, the models achieved the same or higher fillrates with respect to the current system within 1%precision. In the ex-ante approach there are fewmore classes where the models could not outper-

form the current system (demand class 4, criticalityclass 3 and price class 5, with varied results for thedifferent models).

From the optimization results by demand classes,we can see that, according to what one wouldexpect, for all models and for the current systemthe achieved service levels (CSLs and fill rates)decreased as demand for the items increased. Forlow demand items (demand classes 1 and 4), allmodels achieved high CSLs and fill rates, with sav-ings ranging from 6.2% to 20% with respect to thecurrent system. However for high demand items

(demand classes 2 and 3) the normal model achievedmuch better results than the other models, with totalcombined savings of 4.3%, as compared to the cor-responding savings of 1.8% and 1.5% of the empir-ical and Willemain models.

From the results aggregated by criticality classeswe conclude that for low to medium criticality allmodels performed well. However for high criticalitems the normal model performed much betterthan the others. For high critical items under theex-ante approach, observe that the current system

was outperformed by the models but at the expenseof higher costs. For example the normal modelachieved a fill rate 5% higher than the one of thecurrent system but with 0.35% higher costs. (For amore specific examination of the relation betweencriticality and service levels see analysis below.)

Finally, from the results by price classes, we cansee that the models performed similarly, except forprice classes 2 and 3 where the Poisson modelachieved negative or low total savings.

5.3. A closer look at a class with high differences:

explaining the results

We want to give in this section some specificresults for one class where we observed large differ-ences between the models. By doing this we want toexplain the differences in the general results dis-cussed in the previous sections. Accordingly, wechose class 215 as an illustrative example (see opti-mization results for this class in Table 11 of theAppendix). Notice that for this class the normalmodel was the only one achieving positive savings.

The empirical model yielded lower total costs than

the Willemain model, though it achieved a lower fillrate than the current system.

In Table 6, we present the different steps in theoptimization of four typical items inside class 215,with price per item of 20,000 euros. These items

are representative of the 15 most expensive itemsin this class, with similar lead times and totaldemands, where the total number of items was 39.In Table 6 the following additional results are pre-sented for each item and model, along with the real-ized CSL, the realized fill rate and the total costs:

(s, S ) the (min, max) parameters of the currentsystem

s the re-order point corresponding to a targetfill rate (t fill rate)

#c total number of inventory cycles completed

by the item#soc number of stock out cycles for the given re-

order point#D total number of units demanded#S total number of units supplied for the given

re-order point.

After running an optimization procedure forthese items, we obtained the results shown at thebottom of Table 6. As we can see from these results,the normal model is the only one for which a target

fill rate can be identified for which the combinedperformance of all the items produced an optimizedresult with respect to the current system (a target fillrate of 76%). For the empirical model, since we hadfewer available re-order points, this match is moredifficult to obtain for a particular target fill rate.On the other hand, the Willemain model producesa greater range of re-order points as compared tothe normal or the empirical model. However the tar-get fill rate values for the different re-order points donot allow an optimal combination for a specificoptimal target fill rate.

The previous analysis explains the high variabil-ity that we observe in the optimized results for someof the classes presented in Tables 4 and 5. Althoughthis is a drawback of the optimization proceduredue to the decomposition approach, it allows a trac-table analysis of large scale systems like the onerelated to the present study. Since the method usesthe same target parameters for group of items, thestock control is less cumbersome than in the caseof parameters set at an individual level. An alterna-tive approach is discussed in Section 5.8, where we

optimize class 215 using an item level optimization.


ARTICLE IN PRESS


http://-/?-

http://-/?-



Table 6Optimization results for some items in class 215

Item # Current system (C ) Normal model (N ) Empirical model (E ) W

(s, S ) #c #soc #D #S Total

costs

CSL

(%)

fill

rate(%)

t fill

rate(%)

s #c #soc #D #S Tot.

costs

rCSL

(%)

r fill

rate(%)

t fill

rate(%)

s #c #soc #D #S Tot.

costs

rCSL

(%)

r fill

rate(%)

t

ra(%

870

(L = 1)

(1, 2) 2 0 3 3 44,655 100 100 1 0 2 1 3 2 22,155 50 67 1 0 2 1 3 2 22,155 50 67

86 1 2 0 3 3 44,655 100 100 95 1 2 0 3 3 44,655 100 100 9

99 2 2 0 3 3 67,572 100 100 9

9

9

10

11216

(L = 6)

(2, 2) 2 0 4 4 44,239 100 100 1 0 2 2 4 2 26,322 0 50 1 0 2 2 4 2 26,322 0 50

43 1 2 0 4 4 44,239 100 100 53 2 2 0 4 4 67,155 100 100 3

83 2 2 0 4 4 67,155 100 100 5

97 3 2 0 4 4 90,072 100 100 7

100 5 2 0 4 4 135,910 100 100 8

9

9

9

10

11186

(L = 3)

(2, 2) 2 0 4 4 49,239 100 100 1 0 2 2 4 2 28,822 0 50 1 0 2 2 4 2 28,822 0 50

62 1 2 0 4 4 49,239 100 100 78 2 2 0 4 4 72,155 100 100 6

95 2 2 0 4 4 72,155 100 100 7

100 3 2 0 4 4 95,072 100 100 8

9

9

10

5982

(L = 8)

(2, 2) 2 1 4 2 40,072 50 50 1 0 2 1 4 1 23,405 50 25 1 0 2 1 4 1 23,405 50 25

32 1 2 1 4 2 40,072 50 50 34 2 2 1 4 3 62,572 50 75

76 2 2 1 4 3 62,572 50 75 96 4 2 0 4 4 107,990 100 100 4

94 3 2 0 4 4 85,072 100 100 6

99 4 2 0 4 4 107,990 100 100 8

100 5 2 0 4 4 130,910 100 100 99

9

10

Opt. results 8 1 15 13 178,205 8 7.5 86.7 7 6 8 2 15 13 178,205 7 5.0 86.7 95 8 1 15 14 246,537 8 7.5 93.3 7



f or s p ar e





5.4. Service level vs. class structure

Although service levels would usually be reflectedby different price or criticality levels, we could notobserve for the current system (ex-post results) aconsistent pattern in this respect. In order to havea better picture of this result, we present in Fig. 5the aggregated current service levels in a two-entrymatrix for price and criticality. We would expectthat as price increases for the same criticality level

the corresponding service levels would decrease.On the other hand, for the same price, we expectthat as criticality increases the service level alsoincreases. However we observe that often expensiveitems have higher associated service levels than

cheap ones, and low critical items have higher ser-vice levels than high critical items. For some othercases we could observe a more consistent behaviorwithin the line of expectation, but a general conclu-sion in this respect is far from obvious. We concludethat the inconsistencies observed are maybe theresult of a wrong judgment on the part of the stockanalyst when deciding on the re-order points for theitems. Also rounding up the lead times could havean effect, since by doing this we are generally evalu-ating worst service levels than the ones experiencedby the system in reality.

5.5. Optimization curves: Cost vs. service level

In Fig. 6, we show a couple of typical optimiza-tion curves for a particular class. The upper plotcorresponds to the exchange curve between targetand realized fill rates. The current policy is repre-sented by a horizontal line. The lower picture showsthe total holding and ordering costs as a function of

91.1%93.3%89.3%93.2%

90.4%89.6%91.4%94.4%

96.6%91.4%87.8%95.8%

Current fill rates

91.7%93.4%92.2%86.6%

91.8%89.3%89.8%91.4%

96.2%91.7%90.1%95.0%

Current CSLs

p>2112169<p<211213.6<p<169p<13.6

91.1%93.3%89.3%93.2%High

90.4%89.6%91.4%94.4%Medium

96.6%91.4%87.8%95.8%Low

Current fill rates

91.7%93.4%92.2%86.6%High

91.8%89.3%89.8%91.4%Medium

96.2%91.7%90.1%95.0%Low

Current CSLs

p>2112169<p<211213.6<p<169p<13.6Criticality

Price in

euros

Criticality

increases

Price increases

Fig. 5. Double-entry matrices for CSLs and fill rates of thecurrent system.

0 10 20 30 40 50 60 70 80 90 10060

70

80

90

100Realized fill rate vs. Taget fill rate

Target fill rate

R e a l i z e d f i l l r a t e

60 65 70 75 80 85 90 95 1000

2

4

6

8

10x 10

5 Total costs vs. Realized fill rate

Realized fill rate

T o t a l C o s t s

Current policy

Current policy

(E)

(E)

(N)

(N)

(W)

(W)

Fig. 6. Optimization curves for class 233 (ex-post analysis).


ARTICLE IN PRESS




the realized fill rate for the current system and forthe normal, empirical and Willemain models. Obvi-ously the current policy is represented by a singlepoint. The plots correspond to class 233 (733 itemswith total individual demands larger than 1 and

total demand lower than 60, medium criticalityand 13.6 < price 6 169 euros), and the precision of the graphs is 1%.

As we can see from the lower plot, the point asso-ciated to the current system lies above the curves forthe three models. Therefore we can achieve higherfill rates with positive savings for the three models(see numerical results for this class in Table 11 of the Appendix). Notice that the normal curve is lesssteep than the ones for the empirical and Willemainmodels in the optimization region (around a real-ized fill rate of 90%), therefore achieving higher sav-

ings for this class. This behavior of the normalmodel illustrates the better performance of thismodel over the others for some relevant classes, aswe observed the same effect discussed here. Observealso that for this class the Willemain curve isabove the empirical one, which illustrates the bet-ter performance of the latter with respect to theformer. Finally, notice that the normal modelneeded a higher target fill rate to achieve its optimal

realized fill rate than the other models. As before,we observed the same effects in other relevant clas-ses. From this we can conclude that the empiricaland Willemain models are more accurate in termsof predictability of the service levels. Although in

this case the three models yielded a ‘‘good’’ perfor-mance over the current policy, for other classes theoptimization was more difficult, as we showed inSection 5.3.

In Fig. 7, we present the corresponding optimiza-tion curves for class 135 (demand class 1, mediumcriticality and price >169 euros). Here we can com-pare the performance of the Poisson model withrespect to other models. As we can see from thelower plot the performance curves for the four mod-els are very similar around the optimization region(realized fill rate of 95–97%), all of them clearly out-

performing the current policy. From the upper plotwe can see again that the ‘‘predictability’’ of the nor-mal model is the worst of all, achieving higher fillrates for lower target values than the other models.Notice that for high fill rates the Poisson model isthe most accurate in this respect. Observe howeverthat in terms of the optimization performance thisdoes not have a great impact for the normal orthe other models.

93 94 95 96 97 98 99 100

1

0

2

3

4x 106

Current policy(E)

(P)

(N)

(W)

Current policy

(E)

(N)

(W)

(P)

0 10 20 30 40 50 60 70 80 90 100

92

90

94

96

98

100Realized fill rate vs. Taget fill rate

Target fill rate

R e a l i z e d f i l l r a t e

Total costs vs. Realized fill rate

Realized fill rate

T o t a l C o s t s

Fig. 7. Optimization curves for class 135 (ex-post analysis).


ARTICLE IN PRESS


http://-/?-

http://-/?-



5.6. Accuracy of the models and variability within the

classes

As the number of inventory cycles completed byeach item inside a given class may vary greatly, we

expect variations inside the classes between the tar-get and the realized fill rates for the individual items.In order to give an estimation of the accuracy of themodels, in Table 7 we present the average and max-imum values of the fill rate MADs, aggregated forthe different demand classes. For the calculationthe current fill rates were used as target values underthe ex-post approach. As we can see from theresults, the MAD calculation associated with highdemand classes yielded higher average and maxi-mum values. From the average results we can con-clude that the accuracy of the models varies

between 4% and 17% as given by the MADs, whichis due to the low number of demand occurrences.

Another measure of variability is the one pro-vided by the weighted variances of the realized ser-vice levels, as defined in Section 4.3. This measure ismore related to the performance of the system thanthe MAD calculation, as it assesses the variability of the realized service levels for a given set of parame-ters within a class. In Table 8, we give the values

of the variances aggregated for the demand classesfor the different models. As we can see from theresults the service level variances associated withhigh demand classes are higher than for low demandclasses. In general, for the same demand class the

variances of realized CSLs and fill rates are withinsimilar ranges for the different models.

5.7. Individual items

Although the optimization of the system has sofar been performed at class level, where target ser-vice levels are the same for all items inside the class,it is interesting to look at the difference between themodels for individual items. With this idea we showin Fig. 8 a number of interesting plots for two typ-ical items belonging to classes 334 and 124. These

plots are analogous to the ones shown in Figs. 5and 6 but at an item level. The data for these itemsis given in Table 9. As we can see from the plots thecurves of item #1835 are smoother than the ones of item #7559. Since demands for the latter are of unitsize, the steps in realized service levels and associ-ated costs are bigger for a given target service level.For some classes these discrete ‘‘jumps’’ in realizedservice levels for the items made the optimization

Table 7

Average and maximum MADs for target fill rates

Demandclass

Normal model (N ) Poisson model (P ) Empirical model (E ) Willemain model (W )

Average(%)

Maximum(%)

Average(%)

Maximum(%)

Average(%)

Maximum(%)

Average(%)

Maximum(%)

1 5.5 15.2 5.8 15.4 5.5 15.2 5.8 15.22 12.3 18.2 13.2 20.9 14.0 23.03 12.9 24.6 17.6 36.3 15.3 24.64 4.3 8.8 4.4 8.5 4.4 8.5

Table 8Variances of realized fill rates and CSLs

Demandclass

Normal model (N ) Poisson model (P ) Empirical model (E ) Willemain model (W )

Average(%)

Maximum(%)

Average(%)

Maximum(%)

Average(%)

Maximum(%)

Average(%)

Maximum(%)

Variance of CSLs1 1.2 3.8 0.9 3.2 1.2 3.8 1.3 3.82 6.9 20.2 9.0 17.8 9.9 20.7 9.9 20.73 5.3 12.8 9.1 14.5 8.0 11.34 1.0 3.0 1.0 2.5 0.9 2.5

Variance of fill rates1 1.1 4.2 0.9 3.2 1.0 4.2 1.2 4.22 4.6 10.5 2.2 6.2 4.6 9.73 8.8 35.2 6.4 24.3 4.2 18.2

4 0.6 3.2 0.6 3.2 0.6 3.2


ARTICLE IN PRESS




very difficult, as it was the case for class 124. As foritem #1835 the normal and Willemain modelsyielded very smooth curves in spite of the fewobserved demands. For the class where this itembelongs this proved to be advantageous for thesetwo models with respect to the empirical model.

Note that there seems to be some regularity in thedemands, which could be due to preventive mainte-nance, but the data do not provide informationabout it.

5.8. Optimization using a marginal approach (item

level optimization)

As we saw in the analysis presented in Section5.3, the optimization method based on the decom-position approach may not find an optimal solutionfor classes where items have large differences in

holding costs. This because we use a single target fill

rate for all items inside the class. As this effectdepends on the ratio of most expensive items tothe cheapest ones, we try to minimize it by cluster-ing the items in the price classes defined for thestudy. However, as we saw in the optimizationresults, for some classes we could not outperformthe current policy using the proposed methodology.In order to overcome this problem, one should opti-mize the different classes using a marginal approach.To this end, the optimal re-order point for each itemshould be found independently using a measure of the contribution of the item to the overall fill rateof the class per monetary unit invested. As the pro-cedure considers each item individually, for largescale systems the associated computation effortmay be large and therefore the implementation of such a method is not feasible. Hence, for these casesan approach of the type presented in this study may

be more suitable.

(N)

(E)

(W)

Item #1835, Class 334

0 50 1000

50

100Realized fill rate vs. Taget fill rate Realized fill rate vs. Taget fill rate

target fill rate

0 50 100target fill rate

r f i l l r a t e

r f i l l r a t e

r fill rate

r fill rate

r fill rate

0 50 1000

50

100

target fill rate

0 50 100

0 50 1000 50 100

target fill rate

r f i l l r a t e

0

50

100

r f i l l r a t e

20 40 60 80 1000

1

2x 104

104

104

x

x

Tot. Costs vs. r fill rate

T o t a l c o s t s

T o t a l c o s t s

T o t a l c

o s t s

T o t a l c o s t s

T o t a l c o s t s

T o t a l c o s t s

T o t a l c o s t s

0

2

4

0

1

2

3

Item #7559, Class 124

(N)

(E)

(W)

(P)

90

95

100


r f i l l r a

t e

90

95

100


r f i l l r a t e

90

95

100


r f i l l r a t e

90

95

100

90 95 1000

2000

4000Total Costs vs. Realized fill rate

r fill rate

90 95 100r fill rate



0

1000

2000

0

5000

1000

500

1500

Fig. 8. Plots for typical items.

Table 9Data for items depicted in Fig. 8

Item #1835, Class 334L = 31 days (1 month)Price = 236 eurosDemands 10 24 76Period 12 49 52

Item #7559, Class 124L = 44 days (2 months)Price = 485 eurosDemands 1 1 1 1 1 1 1 1 1 1 1Period 6 9 12 13 18 26 29 39 47 49 55


ARTICLE IN PRESS




Accordingly, we present in this section an alter-native optimization approach for the system basedon the marginal approach described above, usingthe fill rate as the optimization criterion. This proce-dure is similar to the one presented in Muckstadt

(2005), where a Poisson model is used to optimizethe system subject to a budget constraint. First weintroduce the following notation:

D j average annual demand of item j

s j (b) re-order point of item j associated with a fillrate b

b j (s j ) expected fill rate of item j corresponding toa re-order point s j as given by the LTD dis-tribution (see Step 2 of method in Section4.3)

s(k ) the k th set of given re-order points for a

group of items, say [s1, s2, . . . , sn]E (b p(s(k ))) expected fill rate for class p associated

with the k th set of re-order points for theitems inside the class, given by

E ðb p ð sðk ÞÞÞ ¼Pn

j¼1b jð sðk Þ j Þ Á D jPn

j¼1 D j

:

Note: For illustrative purposes, we implemented thismethod for some of the classes for which we couldnot outperform the current policy using the original

methodology.

5.8.1. Optimization method using expected marginal

contributions to the class fill rate

1. For each class p, say with n items, consider an ini-tial list s(0) of available re-order points as givenby the corresponding LTD distribution of eachitem, say for b = 0%:

sð0Þ ¼ ½ sð0Þ1 ; s

ð0Þ2 ; . . . ; sð0Þ

n �:

2. For each item j evaluate the expected marginalcontribution to the fill rate of the class per incre-mental euro invested if sjðk Þ is increased to s

ðk þ1Þ j ,

where sðk þ1Þ j is the next available re-order point

given by the LTD distribution of the item (inthe case of the normal model this is equivalentto increase the re-order point by one unit). Thatis, evaluate

Dð sðk þ1Þ j Þ ¼ D jðb jð sðk þ1Þ

j Þ À b jð sðk Þ j ÞÞh jð sðk þ1Þ

j À sðk Þ j Þ

;

where h j is the annual unit holding cost of item j . T a b l e 1 0

O p t i m i z a t i o n r e s u l t s f o r s e l e c t e d c l a s s e s u

s i n g t h e m a r g i n a l a p p r o a c h

I t e m

c l a s s

C u r r e n t s y s t e m ( C )

N o r m a l m o d e l ( N )

P o i s s o n m o d e l ( P )

E

m p i r i c a l m o d e l ( E )

W i l l e m a i n

m o d e l ( W )

T o t .

c o s t s

S e r v i c e l e v e l

( % )

M

e t h o d

S a v i n g s

S e r v i c e l e v e

l ( % )

S a v i n g s

S e r v i c e l e v e l ( % ) S

a v i n g s

S e r v i c e l e v e l ( % )

S a v i n g s

S e r v i c e l e v e l ( % )

C S L

fi l l

r a t e

( N ) v s .

( C )

r C S L

r fi l l

r a t e

( P ) v s .

( C )

r C S L

r fi l l

r a t e

( E ) v s .

( C )

r C S L

r fi l l

r a t e

( W ) v s .

( C )

r C S L

r fi l l

r a t e

1 6

5

3 1 6 , 2

8 0

9 5 . 4

9 5 . 4

M

a r g i n a l

2 3 2

9 5 . 4

9 5 . 4

2 3 2

9 5 . 4

3

9 5 . 4

3

À

3 7 0 6

9 5 . 4

9 5 . 4

À 8 0 8

9 6 . 1

9 6 . 1

D

e c o m p o s i t i o n

À 5 9

9 5 . 4

9 5 . 4

À 5 9

9 5 . 4

0

9 5 . 4

0

À 5 9

9 5 . 4

9 5 . 4

À 2 6 8 7

9 6 . 1

9 6 . 1

2 1

5

8 8 0 , 4

1 0

8 3 . 2

8 1 . 3

M

a r g i n a l

5 2 , 2

2 2

7 8 . 8

8 8 . 8

À

9 4 3 6

7 2 . 1

8 2 . 1

À 1 7 0 5

6 2 . 5

8 1 . 3

D


1 9 , 2

3 4

7 8 . 8

8 7 . 9

À

7 5 5 8

5 3 . 8

7 7 . 2

À 8 0 2 7

6 9 . 2

8 4 . 8

3 3

3

1 0 1 , 2

0 0

7 4 . 1

7 5 . 6

M

a r g i n a l

1 7 , 8

1 5

6 8 . 2

7 5 . 0

2

9 , 9

3 9

5 8 . 5

7 2 . 9

2 9 , 8

0 5

6 7 . 2

7 4 . 4

D


À 2 6 5 8

6 0 . 4

7 8 . 6

À

1 6 7 3

4 4 . 9

6 7 . 6

À 3 6 9 7

4 5 . 5

7 3 . 6


ARTICLE IN PRESS




3. Increase the re-order point of the item with thehighest value of Dð sðk þ1Þ

j Þ, say item l , to get anew list of re-order points, given by

sðk þ1Þ ¼ ½ sðk Þ1 ; sðk Þ2 ; . . . ; s

ðk þ1Þl ; . . . ; sðk Þn �:

4. Simulate the system with the list of re-orderpoints given by s(k +1) and evaluate the realizedCSL, the realized fill rate, the total costs andthe total savings for the class.

5. Compare with the current policy as in step 4 of the previous optimization method (Section 4.3).If the realized service levels for the class arebelow the current ones, repeat steps (2)–(4) untilthe service levels are within acceptable levels oruntil the total costs for the class exceed the cur-rent ones.

In Table 10, we present the results for the optimi-zation of classes 165, 415 and 533 using the abovemethod. We also include the results of the optimiza-tion using the decomposition approach of Section4.3. As we can see from the results, for all three clas-ses the models yielded better results with respect tothe previous optimization method, except in thecase of the empirical model for classes 165 and415, which achieved higher total costs than before.Notice that as we are looking at the optimal individ-ual re-order points, the new method can take betteradvantage of the LTD distributions modeledaccording to the different methods. However, asfor the previous optimization method, some prob-lems may arise as we are limited by the range of re-order point values associated with the LTD dis-tribution. This effect is more clearly manifested inthe case of the empirical model, as we observed inthe results. Recall that the LTD distribution in thecase of the empirical model is determined using onlyavailable demand values from the demand data, andthus the range of re-order points is lower than for

Willemain’s or the Normal models.

6. Conclusions

In this study we present a methodology forthe empirical test of different inventory modelswith actual data for spare parts, using two differ-ent approaches, namely an ex-post and an ex-ante

approach. Although in practice the ex-ante

approach is of more relevance than the ex-post,as information for spare parts is always limited it

is also more difficult to implement. As a result,

for the ex-post approach we showed that the cur-rent stock control practices can be improved byusing the theoretical models considered, howeverusing the ex-ante approach the results were lessconclusive. Consequently, total savings of up to

6.4% in holding costs can be achieved by havinga better inventory control. We have to note how-ever, that using rounded-up values for the leadtimes generally hurts the realized service levels. Amore refined study would require the use of moreaccurate values for the lead times. Of course thiswould also require more specific information aboutthe demand data, which is often in practice difficultto obtain.

According to our study, we can derive the follow-ing general conclusions on the inventory control of spare parts:

1. We observed erratic high demands partiallycaused by preventive maintenance, which needsto be identified for a better stock control of theparts.

2. Spare parts present low to very low demandvalues (in some cases no demand observationswere observed during 5 years). With so limiteddata the application of theoretical models istroublesome.

3. Predictability of theoretical models is an impor-

tant issue, as it measures the expected servicelevel of a policy with given parameters when con-fronted with real data. We observed large differ-ences between target and realized service levels,with maximal values of up to 40%.

4. ERP packages as SAP R/3 do not provide theright tools for the control of spare parts, sincethey only include the cycle service level (CSL)as service measure. In this case, a more appropri-ate measure is provided by the fill rate.

5. A theoretical analysis of the type presented inthis study may be hampered by non-stationarityof the demand data, as lead time demand pat-terns may change over time.

6. The information provided on the demand sizes isan important issue, as they are often not relatedto the number of items installed. As a conse-quence of this, compound-Poisson models aredifficult to use.

7. Inventory models can save money and improveservice levels, even in the ex-ante approach.

For the models under study we have the follow-

ing general conclusions:


ARTICLE IN PRESS




252 28 5593 90.0 91.6 2870 92.9 99.6 3194 85.7253 76 34,290 86.0 80.9 4197 83.2 90.1 À2058 84.2 254 80 147,890 86.7 85.7 12,366 86.0 87.8 À3589 79.7 255 14 210,480 93.2 94.2 34,719 93.0 95.7 24,706 93.0262 590 112,880 68.7 84.2 60,467 64.5 95.9 55,350 73.9263 798 310,520 76.4 80.7 51,941 72.5 86.3 23,856 72.0264 314 649,760 83.8 83.7 58,369 84.1 84.8 7215 82.8265 41 587,700 82.4 79.5 À3078 75.7 84.6 À1471 70.6

313 11 83,779 76.4 88.2

À3102 71.7 88.4

À7057 35.8

322 25 15,743 66.3 90.2 1761 92.7 99.0 5519 63.6323 6 8629 86.4 54.9 À333 68.0 59.8 À7221 28.0 332 144 82,834 80.2 89.7 14,346 86.5 97.8 15,984 78.8333 43 101,200 74.1 75.6 À2658 60.4 78.6 À1673 44.9 334 9 56,697 69.4 75.0 6403 69.0 76.6 À620 71.4 342 35 16,182 86.3 93.2 4249 89.1 96.2 4494 89.1343 9 105,480 75.8 86.4 À9147 78.3 85.8 À12,775 30.4 362 287 134,440 77.0 86.9 À624 58.9 90.4 À11,618 6.4 363 39 80,942 71.4 70.5 20,476 67.6 72.5 À443 37.5

413 14 2161 96.9 96.9 À73 95.8 96.9 À73 95.8 414 28 43,355 97.1 97.2 747 98.4 98.6 2347 96.8415 17 129,460 100.0 100.0 À312 94.3 95.2 À399 97.1 423 8 3446 100.0 100.0 1025 100.0 100.0 1203 100.0

424 11 15,405 98.0 98.0 À1087 97.7 98.0 À723 95.3 433 52 14,778 98.1 98.2 À95 98.2 98.8 417 98.2434 91 117,410 95.7 95.9 4430 96.3 97.0 2773 96.3435 69 788,730 93.2 93.3 86,946 92.3 94.8 88,500 93.0443 36 9741 97.8 98.5 2750 100.0 100.0 2780 100.0444 39 47,642 97.7 97.8 3602 98.7 98.9 4792 98.7445 6 32,143 91.7 92.3 À6158 100.0 100.0 1316 100.0454 26 34,420 95.3 95.5 À497 94.7 95.5 À497 94.7 455 7 42,668 100.0 100.0 2534 100.0 100.0 2534 100.0464 30 40,800 97.5 97.6 À1099 96.2 97.6 À21 96.2 465 41 353,910 96.6 95.9 0 93.6 95.9 0 93.6

Total 8494 15,525,060 87.5 89.2 1,307,055 85.1 92.6 703,246 96.8 96.8 1,054,723 82.2


1 2226 3,693,400 95.6 95.6 774,220 96.2 96.8 703,250 96.8 96.8 762,330 96.3 2 5185 9,469,700 84.3 86.1 408,750 81.1 90.7 202,860 80.03 608 685,920 77.2 85.7 31,370 70.0 90.2 À15,411 37.4 4 475 1,676,100 96.5 96.6 92,713 96.3 97.3 104,950 96.3



f or s p ar e





Table 11 (continued )

Itemclass

# of items


Tot. costs(euros)

Service level (%) Savings Service level (%) Savings Service level (%) Savings Service lev

tCSL t fillrate

(N ) vs.(C )

rCSL r fillrate

(P ) vs.(C )

rCSL r fillrate

(E ) vs.(C )

rCSL


1 493 1,704,700 88.9 88.3 20,428 87.3 91.8 3541 92.9 92.9 À5074 76.1

2 447 277,250 94.1 95.3 37,882 95.0 97.1 1215 99.8 99.8 28,617 91.2 3 3190 9,165,100 90.1 89.8 853,480 88.4 92.7 648,930 97.5 97.5 802,150 86.9 4 1395 905,370 92.6 92.9 107,170 90.7 94.9 20,420 97.2 97.0 83,448 90.0 5 372 634,320 90.8 89.8 61,002 90.4 93.4 1060 96.1 96.1 30,051 88.6 6 2597 2,838,300 79.5 85.4 227,090 75.3 90.6 28,077 95.7 95.6 115,530 70.8


2 2223 524,350 83.2 90.0 179,030 79.4 97.0 À5415 97.9 97.9 166,870 75.1 3 3266 1,608,000 87.9 88.5 245,180 84.7 91.3 25,627 98.2 98.2 115,240 83.3 4 2487 3,637,300 90.2 89.4 200,410 90.1 90.7 54,393 95.5 95.4 71,791 86.7 5 518 9,755,400 90.4 88.2 682,440 88.4 91.5 628,640 96.0 96.0 700,820 83.2







Table 12 (continued )

Itemclass

# of items

Current system (C ) Normal model (N ) Poisson model (P ) Empirical model (E

Tot. costs(euros)

Service level(%)

Savings Service level(%)

Savings Service level(%)

Savings Service(%)

tCSL t fillrate

(N ) vs.(C )

rCSL r fillrate

(P ) vs.(C )

rCSL r fillrate

(E ) vs.(C )

rCSL

252 14 655 92.3 95.3 415 100.0 100.0 415 100.0 253 40 8238 91.3 89.6 2235 88.9 94.8 841 88.9 254 41 27,507 93.8 96.0 À553 85.3 89.4 À1875 85.3 255 2 9770 100.0 100.0 0 100.0 100.0 À1765 100.0 262 343 34,861 64.3 80.4 20,357 63.6 98.8 19,640 61.4 263 460 94,819 75.3 80.5 17,881 71.6 87.9 3795 71.4 264 198 195,970 86.9 84.9 À711 82.8 81.5 À1157 79.3 265 16 127,180 79.3 75.0 À2150 67.9 71.4 À10,862 64.3

313 7 7632 89.5 97.1 À443 73.7 93.6 À337 57.9 322 24 6481 75.0 92.2 965 94.7 98.4 2333 73.7 323 5 4542 88.9 54.2 1493 75.0 33.4 1436 87.5 332 109 19,019 78.8 89.7 1637 84.1 96.8 2970 78.0 333 29 28,546 78.5 92.2 À865 64.1 87.0 À448 40.6 334 6 19,485 80.0 79.6 À712 56.3 56.0 683 62.5342 26 5832 86.8 90.7 2369 80.0 91.5 2386 80.0

343 7 9027 84.4 96.6 À730 100.0 100.0 À2246 40.0 362 235 35,786 72.9 93.5 8166 82.7 96.8 10,127 69.2 363 27 16,405 75.8 67.8 À9 48.1 63.1 À525 23.1

414 4 1258 100.0 100.0 89 100.0 100.0 89 100.0 423 5 1055 100.0 100.0 290 100.0 100.0 290 100.0 424 4 3432 100.0 100.0 À732 77.8 81.8 À732 77.8 433 22 3387 100.0 100.0 740 100.0 100.0 674 100.0 434 42 30,132 98.8 98.9 À343 93.8 95.5 À310 93.8 435 15 108,440 100.0 100.0 À8518 92.3 96.0 À9666 92.3 443 14 1819 100.0 100.0 681 100.0 100.0 681 100.0 444 27 15,882 97.0 97.1 2040 98.0 98.5 2298 98.0 445 1 3025 100.0 100.0 0 100.0 100.0 0 100.0 454 14 9630 95.7 95.8

À252 94.1 95.8

À85 94.1

455 3 12,883 100.0 100.0 1152 100.0 100.0 1152 100.0 464 15 11,295 100.0 100.0 1189 100.0 100.0 1189 100.0 465 12 43,899 100.0 100.0 1849 100.0 100.0 1849 100.0

Total 4326 2,878,665 86.4 89.7 157,452 84.6 93.0 63,575 96.3 96.3 121,118 81.7








1 800 504,650 95.6 95.7 94,010 96.5 97.2 63,575 96.3 96.3 96,409 96.2 2 2873 1,975,100 84.8 87.3 53,384 81.3 91.5 10,900 79.1 3 475 152,750 76.3 90.4 11,873 80.1 92.9 16,379 67.2 4 178 246,130 98.9 99.0 À1816 96.6 97.6 À2571 96.6


1 132 152,500 86.5 85.7 À540 80.7 90.7 À468 87.0 87.0 À14,322 87.1 2 301 81,914 94.8 96.5 12,877 97.2 97.8 1496 100.0 100.0 14,668 85.1 3 1457 1,701,000 90.0 90.5 51,385 85.6 91.7 56,230 96.5 96.5 63,253 83.1

4 818 212,440 93.3 94.3 29,042 93.4 96.1 À4659 97.7 97.7 26,538 92.6 5 159 84,877 94.8 95.2 6130 92.5 95.2 2398 98.7 98.7 2132 92.5 6 1459 645,930 76.3 84.6 58,558 75.6 91.4 8578 94.0 94.0 28,848 71.8


2 1372 144,420 81.1 90.5 58,581 82.9 98.3 À1919 95.2 95.2 63,142 77.6 3 1648 377,020 87.5 88.7 62,667 83.9 92.3 1126 98.8 98.8 38,379 82.6 4 1159 831,190 90.7 90.0 19,185 87.4 88.1 10,508 94.5 94.5 19,131 84.9 5 147 1,526,000 90.6 90.2 17,017 86.1 88.3 53,860 96.3 96.3 466 83.4



f or s p ar e





1. Contrary to expectation, the normal model per-formed very well in the optimization, being theone that achieved the best overall performancein both the ex-post and the ex-ante approaches,achieving total savings of 8.4% (about 1.3 million

euros) over the current system. This in spite of the fact that the demand process for spare partsdoes not generally follow a normal distribution.We observed that the normal LTD distributionis less sensitive to changes in service level values,and thus more conservative re-order points areadvised when using this distribution as com-pared to Poisson, Willemain or the empiricalLTD distributions. Yet the normal model givesbad predictions on the actual fill rate to beachieved.

2. The Poisson model performed equally or worse

than the others for items with 0–1 demands. Wepresume that the Poisson assumption may notbe appropriate, as this model seems to overstockin several cases. This could be caused by preven-tive maintenance or by items with wear-out.

3. The empirical model yielded a better overallperformance than the Willemain model, withtotal savings of 1.05 and 0.96 million euros,correspondingly.

4. All models outperformed the current system withsavings and service levels about the same order.

This can be seen as a validation of the implemen-tation of the different models.

5. For the different criticality, demand and priceclasses considered, all the models performed sim-ilarly, with only some particular classes where thenormal model yielded higher savings. In thisrespect, we did not observe a differentiated effectof the use of the different models applied to thevarious kinds of classes.

All previous conclusions were corroborated bythe ex-ante approach.

Acknowledgement

The authors are thankful to Anne Rijneveld fordoing the case study and providing the data and fur-thermore to Rutger de Mare and Jeroen Hazeu fortheir support in the implementation of the methodsused in this paper.

Appendix

References

Axsater, S., 2000. Inventory Control. Kluwer Academic Pub-lisher, Boston, MA.

Bookbinder, J.H., Lordahl, A.E., 1989. Estimation of inventoryre-order levels using the bootstrap statistical procedure. IIE

Transactions 21, 302–312.Chopra, S., Meindl, P., 2004. Supply Chain Management:Strategy, Planning and Operations, second ed. Prentice-HallInc., NJ.

Croston, J.D., 1972. Forecasting and stock control for intermit-tent demands. Operational Research Quarterly 23, 289– 303.

Efron, B., Tibshirani, R.J., 1993. An Introduction to theBootstrap. Chapman and Hall, New York.

Feeney, G.J., Sherbrooke, C.C., 1966. The (s À 1, s) inventorypolicy under compound Poisson demand. Management Sci-ence 12, 391–411.

Gelders, L.F., van Looy, P.M., 1978. An inventory policy forslow and fast movers in a petrochemical plant: A case study.

Journal of the Operational Research Society 29 (9), 867–874.Ghobbar, A.A., Friend, C.H., 2003. Evaluation of forecasting

methods for intermittent parts demand in the field of aviation:A predictive model. Computers and Operations Research 30,2097–2114.

Heizer, J., Bender, B., 2001. Operations Management, 6th ed.Prentice-Hall, New Jersey.

Kennedy, W.J., Patterson, J.W., Fredendall, L.D., 2002. Anoverview of recent literature on spare parts inventories.International Journal of Production Economics 76, 201– 215.

Muckstadt, J., 2005. Analysis and algorithms for service partssupply chains, Springer Series in Operations Research andFinancial Engineering.

Porras, E., 2005. Inventory Theory in Practice: Joint Replenish-ments and Spare Parts Control, Tinbergen Institute ResearchSeries No. 363, Erasmus University Rotterdam, theNetherlands.

Schultz, C.R., 1987. Forecasting and inventory control forsporadic demand under periodic review. Journal of theOperational Research Society 37, 303–308.

Silver, E.A., Ho, Chung-Mei, Deemer, R.L., 1971. Cost mini-mising inventory control of items having a special type of erratic demand pattern. INFOR 9, pp. 198–219.

Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Manage-ment and Production Planning and Scheduling, third ed. JohnWiley & Sons, NY.

Strijbosch, L.W.G., Heuts, R.M.J., van der Schoot, E.H.M.,2000. A combined forecast-inventory control procedure forspare parts. Journal of the Operational Research Society 51,1184–1192.

van der Heijden, M.C., Diks, E.B., de Kok, A.G., 1997.Allocation policies in general multi-echelon distributionsystems with (R, S ) order-up-to policies. International Journalof Production Economics 38, 225–244.

Willemain, T.R., Smart, C.N., Schwarz, H.F., 2004. A newapproach to forecasting intermittent demand for service partsinventories. International Journal of Forecasting 20, 375– 387.

Williams, T.M., 1984. Stock control with sporadic and slow-moving demand Journal of the Operational Research Society


ARTICLE IN PRESS

an inventory control system for spare parts at a refiner

Documents