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


 Hugh Owens
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1 An inventory control system for sare arts at a refinery: An emirical comarison of different reorder oint methods Eric Porras a*, Rommert Dekker b a Instituto Tecnológico y de Estudios Sueriores de Monterrey, Camus Santa Fe, Mexico City, Mexico b Econometric Institute, Erasmus University Rotterdam, the Netherlands Abstract Inventory control of sare arts is essential to many organizations, since excess inventory leads to high holding costs and stock outs can have a great imact on oerations erformance. This aer rooses a methodology for effective sare arts inventory control, motivated by a case study at a large oil refinery. Different demand modeling techniques and inventory olicies are evaluated using real data. Keywords: Sare arts; inventory control; reorder oints; demand classes; service levels. Introduction Effective inventory management of sare arts is essential to many comanies, from caitalintensive manufacturers to service organizations, such as car manufacturers, chemical lants, telecom comanies and airlines. Different from workinrocess (WIP) and finished roduct inventories, which are driven by roduction rocesses and customer demands, sare arts are ket in stock to suort maintenance oerations and to rotect against equiment failures. Although this function is well understood by maintenance managers, many comanies face the challenge of keeing on stock large inventories of sares with excessive associated holding and obsolescence costs. Thus, effective cost analysis can be an imortant tool to evaluate the effects of stock control decisions related to sare arts. However, the difficulty in assessing good strategies for the management of sare arts lies in their secific nature, normally very slowmoving arts with highly stochastic and erratic demands. For examle, tyical industrial data sets comrise limited demand history with long streams of zero demand values and a few large demands (Willemain et al. (2004)). This makes the estimation of the lead time demand (LTD) distributions very difficult, which is essential to obtain the control arameters of most inventory olicies. Although different inventory models have been roosed in the literature to tackle this roblem (see next section), there is a lack of emirical testing of theoretical models with data from real industrial environments. * Corresonding author:
2 This aer concerns a study on sare arts at a maor oil refinery in the Netherlands, which consisted of two hases. In the first hase a case study was conducted with the comany, where the SAP system used as latform for its oerations was under examination. As a result, some imrovement measures were rovided and later imlemented by the comany. The second hase was focused on the analysis of the demand data rovided by the comany. This aer reorts on the findings related to this hase, where the obective was to erform an emirical comarison of different inventory models. The aforementioned comany kees stock of a large number of sare arts related to equiment used in its etrochemical rocesses. Although these stocks are essential for the continuity of its oerations, management was concerned with the savings oortunities at the rocess floor by having better inventory control of its sare arts, whose value was worth at the moment of the study more than 27 million euros. One maor difficulty of the study was the limited demand history available. By describing the case, we make general observations about the ractical asects of inventory control. Moreover, our aim is to comare various olicies with real demand data from the case to see which one is best under what circumstances. Common methods resented in the literature rather use given statistical demand distributions to assess the erformance of inventory models. Consequently, with our methodology we can better identify the real limitations of industrial data sets. The remainder of the aer is organized as follows: the next section resents briefly related literature. Section 3 includes the case study descrition. Next the methodology is exlained in Section 4. The comutation results are included in Section 5 and the final conclusions are resented in the last section. 2. Review of related literature One of the maor areas of inventory research over the ast decades is the one related to the management of sare arts inventories. Although theoretical models for slowmoving items are abundant in inventory literature since 965, case studies are few (for a comrehensive overview of recent literature on sare arts management see Kennedy et al. (2002)). In the arena of theoretical models, one of the most extensively studied inventory olicies is the socalled (S, S) model, a articular case of (s, S) models, with an underlying Poisson demand distribution (see Feeney and Sherbrooke (966)). Although well studied and suitable for slowmoving items, this tye of olicy requires continuous review of the inventory system. Moreover, the Poisson distribution assumes randomness of demand, with interarrival times between unit size demands following an exonential distribution. This distribution needs no information of demand other than the average demand, which is the solely arameter of the demand distribution. When transactions are larger than unit size, authors have roosed the use of comoundpoisson models (see Williams (984) and Silver et al. (97)). However, these models are difficult to aly in ractice because they need an assumtion on the comounding distribution. For examle Williams (984) develoed a method to identify soradic demand items, where three arameters are needed: one for the exonential distribution of interarrival times of demands, and two arameters of an underlying gamma distribution for the demand size.
3 Most of the emirical studies in sare arts literature are focused on testing forecasting methods for demand of slowmoving items rather than on imlementing inventory models. This is an imortant distinction since forecast methods are used to estimate oint forecast of the mean (like the moving average method) while for evaluating control arameters of inventory models (like the (s,q) model) one needs an estimation of the entire LTD distribution. Moreover, inventory models are used to meet secific customer service levels in the long run, while forecasting models aim to obtain accurate demand forecasts as determined by the mean average ercentage error (MAPE) or the mean square error (MSE). In this area of research Ghobbar and Friend (2003) resent a comarative study of 3 different forecasting methods for the management of sare arts in the aviation industry. They use the MAPE measurement alied to forecast errors to assess the accuracy of the different methods but no inventory models are included in the study. They confirm the sueriority of the weighted moving average and Croston s methods over exonential smoothing and seasonal regression models. Silver et al. (998) also advise on the use of Croston s method for roducts with intermittent and erratic demand. This method (Croston (972)) assumes that the LTD has a normal distribution, and estimates the mean demand er eriod by alying exonential smoothing searately to the intervals between nonzero demands and their sizes. Willemain et al. (2004) roose the use of a modified bootstra method to forecast intermittent demand of service arts, and they imlement the method on a large industrial data set. Croston s method and exonential smoothing are evaluated as well, yet like in the revious aer no attemt is made to imlement an inventory control model. They show that the modified bootstra method roduces more accurate forecasts (based on the MAPE measurement) than the exonential smoothing and the Croston s method. We use in the study the bootstra method roosed by Willemain and we comare it with the erformance of an emirical distribution model. We assess also the erformance of models based on Poisson and normal demand distributions. A similar research to the resent study is resented in Stribosch et al. (2000), where the erformance of two different (s, Q) models for sare arts in a roduction lant environment is examined. Unlike our method, they test the inventory models roosed using simulation where demands are generated from an Erlang distribution, whereas we assess the inventory models using the historical demand data for the items (around 8,000 items). Also related to our study, Gelders and van Looy (978) resented a case study carried out in a large etrochemical lant. They develoed different inventory models to control slow and fast moving items, which were clustered in classes using ABC analysis together with criticality and value considerations. As they had limited information on consumtion rates for slow movers, a Poisson underlying distribution was assumed to comare between existing ractices and the models roosed. We use in our study a similar aroach, but different to their study we estimate the LTD distribution using the methods mentioned above and we test the models with real demand data, rather than using simulation. Preventive maintenance (PM) is another imortant managerial issue that has been addressed in the literature (see Bridgman and MountCambell (993)). Information on PM can be used to better control inventories as it takes advantage of lanned demand, by correcting the effect of stochastic failures of equiment. Thus, models can erform more accurate as they are not erturbed further by PM demands.
4 3. The case study  system descrition The comany under study consists of a maor etrochemical comlex located in the Netherlands, which includes 60 different lants divided in chemicals manufacturing and oil refinering. The comlex dates from 930, and many new installations have been added since then. A large art of it however, stems from the 960s. The rocurement deartment offers service to all lants. There is one central warehouse owned by the comany. At the moment of the study (2000), there were in total 30 thousand catalogued materials, of which only 43 thousand were ket on stock at the site, with a total value of more than 27 million euros. There are 22 additional small decentral storages on site, containing fast moving materials that can be directly used if needed. No stock registration is done for these items and they are relenished on a batch basis. Therefore, we only need to consider a single stock echelon, being the warehouse as user of sares for equiment and not a roducer of arts. In total there are 80 thousand requests of material er year, both for nonstock and stock materials. Requests for materials ket on stock are sulied from available stock. If there is shortage of a material an emergency relenishment order is generated. Controlling 43 thousand materials reresents a difficult task, esecially because of the differences in tyes and consumtion atterns associated with them. It also requires efficient use of the manower available and of the information system at hand. Until 997 an inhouse develoed information system for inventory control was used by the comany. In 997 they moved to the information system SAP R/3, which is a comlete ERPsystem, but not secific for inventory control. Almost the whole demand history before 997 has been lost in the transfer to SAP. Within SAP, the comany alied the MM (materials management) module for the control of its sare arts. Since SAP evolved out of MRP systems for the manufacturing and assembly industry, the MM module is very much based on the MRP lanning hilosohy (see Heizer and Bender (200)). Demand is exressed by actual orders or by forecasts of demands. Next demand of end roducts is converted to demand for assemblies, comonents and arts. Stock control is erformed in SAP on a eriodic basis (socalled eriodic review). Items are ordered when the MRP run is made. The SAP user can set the aroriate time interval, e.g. daily, weekly or monthly. At the comany they run the MRP every week. The actual stock control within SAP occurs in terms of minmax levels (equivalently to (s, S) olicies), or MRPtye control based on lead times. Minor functionality is available in SAP to determine the minimum level s and the maximum level S. Safety stocks can be used to determine the reorder level s, and lot sizing methods are available to evaluate the difference between s and S. Before the roect 90% of the control levels were set manually, and afterwards some 70%. As a result still many relenishment orders were checked manually before sending them out. With resect to forecasting, several methods are available in SAP, like exonential smoothing and moving averages, both with trends and seasonality. It is however the intermittent nature of demand that makes the alication of these methods articularly difficult to sare arts. For the determination of safety stock levels, the normal loss model is available which aroximates the demand during the lead time with a normal distribution. This model works with the cycle service level as service level obective. However, no fill rate service levels can be defined within the MM module. A more striking asect of SAP is that within its functionality no continuous review models can be imlemented. Therefore the classical and much advised (S, S) model with Poisson distributed demand over the lead time cannot be alied.
5 3.. Data structure Statistical information for the consumtion of sare arts was available for 5 years (the last year only until August). The demand information was recorded in monthly eriods, so a total of 55 eriods of demand information was available for the study. One imortant limitation of the demand set was that it did not secify whether demands were due to failures or reventive maintenance activities. Different arts used by the comany are divided in two main categories: materials related to a iece of equiment and the ones not related to any articular equiment, like rotecting shoes, helmets, generalurosed electrical equiment and instrumentation. From the total of 43,000 materials in stock, 4,383 were sare arts, accounting for 80% of the total stock value. These sare arts are the focus of the resent study. The arts related to equiment are classified according to criticality codes, which are based on how unavailability affects the safety of the eole and environment, the cost of down time and the quality of the rocesses. Materials not related to equiment do not have such a criticality code. Accordingly, three criticality codes are used: High (H), Medium (M) and Low (L), which are defined as follows: High (H): Unavailability of these materials would result in exensive downtime or cause danger to the safety of the eole and the environment. Risk taken in the rocess of ordering and stocking cannot be ustified. Medium (M): Unavailability of these materials would result in significant loss of roduction, but does not endanger the safety of the eole or the environment. A calculated risk can be taken in the rocess of ordering and stocking. Low (L): Unavailability of these materials would not result in serious effects on the rocesses or on the safety of the eole and the environment. The revious classification is made on exert udgement and no quantitative methods are used to date. We would have liked to use these codes as related to stockout costs, but these were difficult to assess by the management. A further insection of the materials with criticality code leads to a more refined classification: materials that are uniquely installed in a articular iece of equiment (60% of the materials related to equiment), and materials which are related to more than one iece of equiment of different criticality codes. That means that there are sare arts that have combined criticality codes (H/M/L, M/L) deending on whether they are installed in multile iece of equiment of different criticality. The comany used these codes to decide on the stock levels of the different arts. Thus, items identified as highly critical should be on stock since they require high fill rates, low critical ones destock, and medium critical ones on stock deending on costeffective considerations. However, as no models are available in SAP that incororate criticality considerations, these levels were set mostly by exert udgement Classification of arts A more refined analysis of the sare arts data revealed that imortant differences among them existed not only in terms of criticality codes but also with resect to demand and rice. Therefore, we aimed at grouing them in different classes to see whether we should aly different stock control methods for different classes.
6 Below we describe the different classes considered in this study. Criticality classes Based on the criticality codes, the following criticality classes were defined for the sare arts: Criticality class : H 2: H/M/L or H/L 3: M 4: M/L 5: L 6: Not related to any articular iece of equiment For the current olicy used by the comany, we exect to observe that the service levels associated with high critical items are higher than the ones for low critical items (see discussion in Section 5.4). Demand classes The original data set consisted of more than 4,000 sare arts, for which we observed a high variability in demand atterns. For examle some arts had only 0/ demands while others exerienced either few large demands or no realization of demands during 5 years. For other arts we observed large negative demands due to returns. Thus, a classification was needed for the sare arts based on consumtion rates. For arts with total ositive demand over the fiveyear eriod and some demand values higher than, we identified from a histogram two main grous: arts with relatively high total demand and arts with low total demand. Although the boundary between these two grous was not clearly identified in the histogram of demand, from a Pareto analysis we could reasonably establish it in 60 units. We observe that 90% of the items had a demand below this value, and this contributed 25% of the total demand. At the same time, items with 0/ demands also had a total demand of less than 60. According to this, we established the following demand classes for the arts: Demand class : arts with only 0/ demands. 2: arts with negative total demand. 3: arts with no realizations of demand (all demands equal to zero). 4: arts with total demand larger than 0 but less than 60, and not only 0/ demands. 5: arts with total demand higher than 60. 6: arts with , 0, demands. Price classes For the sare arts in the data set, 5 different rice levels were identified in a histogram. Table shows the different rice classes for the sare arts. The arts recorded in SAP with a rice of 0 euros are items not owned by the materials deartment (rice class ). We observed rices as low as 0.0 euros for some arts (rice class 2) and the most exensive ones had a rice of 20,000 euros (rice class 5). Table. Price classes of arts Price class Price () in euros = 0 0< < 69 69< 22 >22 Sare arts (total = 4383) 0% 9% 33% 29% 8%
7 Combined classes Using the criticality, demand and rice classes, we include each item in a combined class defined by three digits. Accordingly, an item in class xyz corresonds to an item with demand class x, criticality class y and rice class z. This classification allows us to otimize the system er class rather than for individual items. That is, once a service level is defined for the combined class, the arameters for the different inventory olicies are evaluated for each item in the class. Then a simulation tool is used to evaluate the erformance of the selected model of each individual item using its demand data. Finally total costs are aggregated across all items in the class. In this way we aim at obtaining an otimization rule for each combined class considered in the study (see section 4) Considerations on item classes and anomalous observations The analysis of sare arts data is erformed for all combined classes incororating demand classes, 4, 5 and 6. Thus, all criticality and rice classes which combine those demand classes are considered for the evaluation of the inventory models, excet rice class, since items with rice zero do not have associated holding costs. Since for items with total negative demand a zero inventory olicy is otimal, we do not incororate in the analysis demand class 2 (4% of the arts). Moreover, although negative demands can be associated with returns due to reventive maintenance ractices (arts that were ordered but not actually installed) or with reaired arts that were brought back to the system, we did not have secific information in this resect. As for demand class 3 (2.2% of the arts), since items in this class have no demand realizations in five years, we leave it out of the analysis. From these considerations we are left with,984 items. Additionally, we observed items with an error in the criticality secification. These items accounted for 4.% of the total numbers of arts. After excluding these items, we were finally left with,790 sare arts for the analysis. We also identified in the data set one articular month for which a large number of very high demands was recorded. We considered that this was due to an administrative rebooking in the warehouse and thus we eliminated this month from the data set in our analysis Lead times The lead times for the sare arts were recorded in days. However the demand data set for the items was registered in months, without secification of the day within the month that a articular demand took lace. Therefore, for ease of imlementation in the simulation we rounded the lead times off to full months using 30 days er month. In this way an item with a lead time of 80 days was considered to have a lead time of 3 months. Observe that this conversion is also necessary for the estimation of the distribution of the lead time demand, since demand forecasts for the items are roduced in months. Although the rounding u of lead times was introduced to better coe with the demand data, by doing this we take a conservative aroach, in which the service levels achieved by the system in the simulation will be generally lower than they are in reality. Observe that lead time management is an imortant issue, esecially for older arts, as suliers may no longer be able to meet the original romised times.
8 3.2. Cost structure In general, three tyes of costs are associated with inventories: holding costs, ordering costs and stockout costs. Holding costs reresent the cost of caital tied u in the sare arts inventory. An annual fixed rate of 25% was used in the study. Ordering costs reresent the cost associated with lacing an order for a sare art, which includes the costs of telehone calls, insection and handling of the incoming items, aying the bill and registration of the arts. This cost is indeendent of the number of arts included in the order. An ordering cost of 36 euros was used in the study. Since our obective is to evaluate the otimal balance between service levels and holding costs, we consider stockout costs in a searate study included in the Aendix. 4. Methodology We use two aroaches for the otimization of the sare arts inventory system under consideration, namely an exante and an exost aroach. In the exost rocedure the same data set is used for both fitting and testing uroses. Oosite to this, the exante rocedure, once a distribution has been fitted to the data, uses an entirely different set for testing uroses. In this resect the exante aroach is more relevant from a scientific and ractical ersective, since in reality systems face future unknown demands (Silver et al. (998)). In order to achieve this, we divide the historical demand data into two sets, namely a fitting eriod and a testing eriod. The fitting eriod will be used to estimate the lead time demand distribution (LTD) which is used in turn to determine the inventory olicy arameters. The testing eriod is used to erform a simulation to evaluate the erformances of the inventory olicies selected and comare them with the erformance of the current one. We consider two tyes of service levels, the cycle service level (CSL) and the fill rate. The reason for using the exost aroach is that many industrial data sets are rather short for forecasting uroses, and this rocedure will give the advantage of using the whole data set to get a better icture of the real demand rocess. The erformance of both aroaches will be comared to assess the advantages of each one. We have to note however that in the data set there are many items with only one or two demand realizations, and therefore we exect highly variable results. One of the main issues we address is whether theoretical models can outerform stock analysts. In ractice this is difficult to assess because of lack of information (e.g. short demand data sets, little information on reventive maintenance ractices), as well as imlementation constraints. For instance real lead times of items are normally in days but in the models one may refer to use full eriods of time for ease of imlementation. Other ractical issues like the erformance of different methods to model the demand rocess are exlored as well in this study. In order to achieve this, below we give the demand modeling methods used and next we describe the inventory models considered. 4.. Modeling the lead time demand In inventory decision making, one needs to determine inventory control arameters, such as reorder oints and safety stocks. In order to do so, we need a secification of the lead time demand distribution. This is traditionally done by modeling lead time
9 demand using common robability distributions found in the literature, such as the normal distribution (Silver et al. (998)) or the Poisson distribution (see Schultz (987)). Other authors roose the use of models based on forecast techniques such as moving average or exonential smoothing (see Croston (972) and Silver et al. (998)). In order to coe better with real data sets and to give a more realistic icture of demand, authors have roosed the use of bootstra techniques (Bookbinder and Lordahl (989); Efron and Tibshirani (993); Willemain et al. (2004). As we are interested in the erformance of theoretical models using real data, we estimate the LTD distribution using the Willemain s bootstra method along with another novel rocedure using emirical data. In this way we estimate the distribution of demand over the lead time for each model, which is used in turn to evaluate the arameters of the inventory olicies selected. However, different to the methods found in the literature, our obective is to use directly the real demand values observed to assess the erformance of the olicies using the simulation tool. To kee the study tractable, we do not aly udating for the estimation of the LTD distribution. We also evaluate the erformance of the system using normal and Poisson distribution based models. Below we describe these methods. Willemain s Bootstra method (W) We imlemented the modified bootstra method resented in Willemain et al. (2004). This method, as comared to traditional bootstra techniques resented in the literature, has the advantage of caturing better the autocorrelations between demand realizations, esecially when dealing with intermittent demands with a high roortion of zero values. The method first evaluates the emirical transition robabilities between states of zero demand and states of ositive demand for the different items. Then using this information, a stream of zero and nonzero demands is randomly generated for a eriod of length equal to the lead time. The nonzero values are filled with demand values samle from the data set. In this way estimates of LTD for each item are obtained for a large number of realizations (000 in this study). This information is finally used to estimate the distribution of LTD. Willemain et al. (2004) alied this method to nine large industrial data sets of service arts inventories and comared it with the exonential smoothing method and the Croston s method. He concluded that the modified bootstra method gave the best erformance of all three methods. In Fig. we show a lot of an estimation of the LTD distribution using Willemain s method for an item corresonding to class 45. The item (labelled #74) has a lead time of 9 eriods and its demand data for the 55 eriods is as follows: in eriods 5 and 7 it observed ositive demands of unit size each, and in eriod 9 a demand of 2. The rest of the eriods no demands were observed. Notice that although only lead time demand values of and 3 were realized in the data set, the method is able to roduce a LTD estimation where many other lead time demand values are taken into account. Emirical distribution of lead time demand (E) We imlemented an emirical model to estimate the distribution of LTD. Different to the traditional bootstra method, we construct a histogram of demands over the lead time without samling. This method is new to the literature as no attemts have been made to use it for inventory control. Since demands are taken directly from the data set over fixed eriods of time equal to the lead time, this method also catures autocorrelations and fixed demand intervals due to reventive maintenance, and is far easier to imlement than the modified bootstra method described above. For
10 examle, consider an item with the following stream of demands over 40 eriods with a lead time of 3 eriods: Period Demand Assuming that the above sequence of demand values can occur at any oint in time after a future demand observation, then 4 is a ossible realization for a LTD value. Of course the same is true for and 2. In this case we require that the estimation method catures these ossibilities; that is of having a LTD of 4 with a ositive associated robability, and a LTD of 2 with a higher associated robability. As the emirical method will construct a cumulative distribution function (cdf) over a lead time of 3 consecutive eriods for the whole data set, the LTD values of 2 and 4 will be estimated with robabilities of 0.2 and 0.053, corresondingly. In Fig. 2 we show a lot of the LTD distribution (cdf) for this item. This examle was constructed for illustrative uroses. Item # 74 Willemain's CDF F(x) x Figure. Cumulative distribution function using Willemain s method Emirical cdf 00% 80% 60% 40% 20% 0% Lead time demand Figure 2. Emirical cumulative distribution of lead time demand Normal distribution (N) We imlement a normal based model assuming that demand for the arts follows a normal distribution. To this end, the average ( D ) and standard deviation (S.D.) of the observed eriod demand is evaluated to estimate the arameters µ LTD and σ LTD of a normal LTD distribution, as follows:
11 µ LTD = D L σ = S. D. L, LTD where L is the lead time of an item in full eriods of time (days, months, etc.), and with D and S.D. evaluated using the whole data set of demands, including zero and negative values. Thus, for integration over a normal LTD distribution we use common formulas found in the literature (e.g. Silver et al. (998) or Chora 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 recommended (Silver et al. (998)). Thus, we do not exect the normal based model to give better results than the others. However, we want to investigate its erformance as comared to the other models considered. To this end, when we evaluate reorder oints based on the normal LTD distribution to meet desired fill rates, the values obtained are rounded u to integer values, and negative values are set to zero. We require this since for our system reorder oints are defined as ositive integer values in accordance with the discrete demand for arts. In the case of negative values which are set to zero no comensation for the gain mass is alied in the integration of the normal LTD distribution. This causes only a minor distortion in our results as we normally look at high service levels which have associated ositive reorder oint values. We neither correct for the gain mass associated with the rounding off of fractional values, as this has also a minor effect. Accordingly, to evaluate a reorder oint s for a given fill rate level β, we use a similar rocedure as the Excel goal seek routine (see Chora and Meindl (2004)), utilizing the formula: σ UNLI ( z) 00 β % = LTD 00, Q where UNLI(z) is the unit normal loss integral associated with the unit normal variate z. Recall that z corresonds to a reorder oint s associated with a CSL value (see Silver et al. (998)). Thus the roduct in the denominator of the above formula gives the exected units short for a given reorder level s. As mentioned in Silver et al. (998), the revious formula underestimates the true fill rate if σ LTD is large relative to Q. Therefore, a correction should be made in the numerator substituting the term σ LTD UNLI(z) by σ LTD (UNLI (z) UNLI (z + Q/σ LTD )). By using the uncorrected formula, we obtained conservative values of the fill rates, which did not have a maor imact in the otimization of the system (see results in sections 5. and 5.2). Remark. Although in many situations the normality assumtion is not satisfied, this distribution has widely been used in ractice. This is due to the simlicity to evaluate reorder oints and other arameters based on the normal distribution. Poisson distribution (P) Silver et al. (998) suggest that the Poisson distribution is suitable to model demand of slow moving items. We use the Poisson distribution to estimate the LTD distribution for items in demand class. The reason is that demands for these items are of unit size, and hence the basic assumtion of the Poisson distribution is satisfied.
12 The only arameter of the Poisson distribution, the average rate of demand over the lead time, is estimated from the demand data for the different items. For other demand classes, a comoundpoisson based model would be more aroriate, but a number of secific assumtions need to be satisfied in the comounding in order to exect a good erformance of this model. Since we wanted to develo robust methods that alies equally well for different industrial data sets, we did not consider the comound Poisson for demand classes different from. The Poisson based model is comared to the normal, emirical and bootstra methods for demand class Inventory models We use an (s, nq) inventory olicy for the system, with the reorder oint s evaluated using the LTD distribution according to the modeling methods described above. Thus, when overshooting of the reorder oint s cannot be overcome by the lot size Q, an alternative lot size equal to nq is ordered, such that the inventory osition is brought above s, where n is an integer value. This is a common ractice in inventory management (see Silver et al. (998)). The lot size Q will be evaluated according to the economic order quantity (EOQ) using average annual demand. We round off the EOQ calculation according to Axsäter (2000), as follows:. Evaluate: m = EOQ 2. Set Q = m m + if m = 0 if m 0 and otherwise EOQ m m + EOQ When we use the Poisson distribution to estimate the LTD, we use Q =. This model is often referred to as (S, S) model, with s = S. Notice that for demand class, since average demand is generally low, the EOQ calculation is likely to roduce also a value of. For the classes under study the roosed olicies are comared to the current (min, max) olicy in terms of the selected service level and total costs. The aim of the roosed methodology is to otimize the system, that is, to establish which model and olicy erform best under which conditions (see Table 2). Table 2. Inventory models considered Model Parameters Demand Classes Current olicy (C) minmax (s, S), 4, 5, 6 Poisson based model (P) (S, S) Normal based model (N) (s N, nq), 4, 5, 6 Emirical based model (E) (s E, nq), 4, 5, 6 Willemain based model (W) (s W, nq), 4, 5, 6
13 4.2.. Handling of large demands Consider the situation in which an item with a relatively short lead time observed unusual high demands. For such an item the analysis becomes difficult as the associated reorder oint (say evaluated according to a normal LTD distribution) is likely to be overshot when using a simulation tool to assess the erformance of a given inventory model. To illustrate this we give in Table 3 the demand data and other relevant information for item # 307, which has a lead time of one month. For this item we consider that the demand value of 450 is an outlier, since this value is larger than the average of the rest of the ositive demands lus 0 times their standard deviation. So here the aim is to construct a tool that filters out this large demand. To this end, we imlement a demand filter in the following way: all demands larger than the average lus k standard deviations (evaluated using only ositive demands) are filtered out. For this articular item observe that k = 3 will not roduce the desired result and hence a value k = 2 is a better choice. In a similar way, for items belonging to demand classes 4 and 5 we found that actually k = 2 was the best selection for the demand filter. As a result of alying the mentioned filter, 8.7% of the ositive demand values for these two classes were filtered out. Table 3. Demand data for item # 307 Demand number of Total value occurrancens demand Total Average of demands > Stand. deviation of demands > Average of ositive demands < Stand. dev. of ositive dem. < Remark 2. Although for the otimization of the system we use the above filter of demands, we also assess the effect of having large demands included in the otimization rocess. We resent some numerical results in the Aendix. Remark on the imlementation of the exost and exante aroaches Due to the limitations inherent to the data set used in our simulation study, it may well be that not enough information is used for fitting uroses. Therefore, items with only zero or one ositive demand during the fitting eriod are excluded from the analysis. This consideration is used in both the exost and the exante aroaches. As for classes with few items (6 or less), they were excluded from the analysis in both the exante and the exost aroach. According to these considerations, of the original,790 items, a total of 8,494 were included in the exost aroach and 4,326 in the exante aroach (see analysis of results in sections 5. and 5.2). Remark on classification of items We include each item in a class according to the demand attern and other relevant information (criticality and rice). We do the classification using the whole data set in both the exante and the exost aroaches, since we want to assure that an item in a certain category will not exhibit demand values not corresonding to that category during the simulation in the testing eriod.
14 4.3. Otimization of the system: a decomosition aroach We focus on otimization of the system based on service levels, for which both the fill rate and the cycle service level are used. The utilization of service levels to set safety stocks is a referred method in industry as oosed to cost minimization. The reason for this is that the latter requires the evaluation of stock out costs which deend on down time enalties and other factors which are difficult to evaluate in ractice. Considering the size of the system related to the resent study (recall that originally it comrises some 4,000 items), we use a decomosition aroach for the otimization of the system. This tye of aroach is well known to the literature, where it is often used for the analysis of comlex systems, e.g. for the otimization of multiechelon inventory systems (see van der Heiden et al. (997)). In our case the basic idea is to otimize the system at grou level, defined by the classes described earlier, rather than doing it at item level. Thus, we evaluate the different inventory arameters for the items based on a single target service level for all items in the class. Notice that deviations of the realized service levels with resect to the target values are exected due to the discrete nature of inventory levels. Thus, the obective of the method is to find the right level of a target service level that otimizes the classes under study. We first introduce the following notation: : index for each item in class, =,, N, where N is the size of class. X: identifier for the model alied in the simulation, where X = current olicy (C), Poisson (P), Normal (N), Emirical (E) or Willemain (W). ( X ) c ( X ) soc (t) D : number of inventory cycles comleted by item during the testing eriod using model X. Note: an inventory cycle is defined between the lacing of an order and its arrival to the system, i.e. the inventory cycle over a lead time (also referred to as relenishment cycle). : number of inventory cycles for item with stockouts during the testing eriod using model X. : total number of units demanded of item during the testing eriod. ( X ) S : total number of units of item sulied from on hand stock over the testing eriod using model X. CSL : cycle service level achieved by the system using the model X. ( X ) ( X ) β : fill rate achieved by the system using model X. β : target fill rate for class. ES (s ): exected units short for item for a given reorder oint s, according to the corresonding LTD distribution. Now for each combined class (xyz) roceed as follows:. Evaluate the service levels (CSL and fill rate) achieved by the current system over the testing eriod according to the exost or the exante aroaches. These are the current service levels of class, which are defined by
15 CSL ( C) N = =, N = soc c ( C) ( C) N = ( C ) β =. N = S D ( C) ( t) 2. For each item, estimate the cumulative distribution of LTD over the fitting eriod, according to the model selected (see Section 4.). Use β = β together with the LTD distribution to estimate the corresonding arameters of the inventory olicy (i.e. reorder oint and lot size), in the following way: Calculate Q, the lot size, according to the EOQ formula and round it off as exlain in Section 4.2. If class = yz (demand class ) then set Q =. Chose the smallest s which satisfies a 00β % fill rate, i.e.: ES ( ) s 00 β % 00. Q 3. For each item, run the simulation over the testing eriod alying the (s, nq) or the (S, S) olicy as required, using the arameters selected in the revious oint. For each model selected, evaluate for class the realized fill rate (rβ ), and its total costs (TC ), according to N ( X ) S N ( X ) = ( X ) β =, TC N, target = TC, target, ( t) = D = r where TC,target is the total cost of item for the target service level selected, which comrises the holding and ordering costs. Comute the total savings achieved by the selected model with resect to the current system according to: (C) Total savings = TC TC ( model) ( current) ( model), target, target Within the classes we exect deviations of the realized service levels for the individual items. In order to assess the magnitude of those deviations, we evaluate the weighted variances of the realized fill rate as follows: N = ( X ) ( r ) X ) ( X ) ( X Var( rβ ) = c β β. ( N ( X ) c = ) 2 An estimation of the standard deviation of the target fill rate is thus given by 2
16 ) ˆ ( X ) ( X σ = Var( rβ ). 4. For each model selected check whether the realized service levels are better than the service levels of the current system. If the answer is yes, check if it achieves lower total costs. This will serve as a guide for otimization. 5. For classes with lower realized service levels or negative savings, construct exchange service levelcost curves for the different inventory olicies, in the following way: define a set of fixed target values for the fill rate (β ), say from % to 00% in ste sizes of %. Next evaluate the arameters of the selected inventory model as in oint 2 (excet for the current system). For each value of the target service level obtain the corresonding rβ, TC,target and total savings. Finally, identify the value of β, for which the corresonding realized fill rate equals or exceeds the current fill rate (within % recision), and that achieves ositive savings. In this way we otimize the system among the different classes. Remark 3. The revious method assumes that the system is otimized using the fill rate as otimization criterion, which lays the role of control arameter. However, it is ossible to imlement the above rocedure using the CSL for the evaluation of reorder oints in stes 2 and 5, in the classical way. Even in the case for which the otimizing criterion is the fill rate, one may decide to aly the CSL as control arameter due to limitations associated with the evaluation of reorder oints, and otimize the system with resect to the fill rate. In the Aendix we give some results using this method. Remark 4. When more than one order is outstanding for an item in the simulation, that is when 2 or more relenishments cycles overla, we associate a stockout with the immediately recedent cycle that has not finished yet, thus causing only that cycle to be a stockout cycle. This situation is not considered in inventory text books such as Silver et al. (998), thus confusion as to how to register stockout cycles may arise. To illustrate this consider the item #2248 (see Fig. 3.) with lot size of 2 units, lead time of 2 eriods and reorder oint of 2 units. In eriod 9 the inventory osition reaches the reorder oint, triggering the first relenishment cycle. This cycle ends at the beginning of eriod 2 when the system is relenished and causing the net stock to reach 4 units. This cycle ends u with no stockouts (thus coloured white). The second cycle starts in eriod 22 with net stock still ositive. Notice that before this cycle ends, a third cycle starts in eriod 23, where the net stock dros to . This causes the second cycle to be a stockout cycle (thus coloured grey). Similarly, the demand of 8 units in eriod 24 causes the third cycle to be a stockout cycle, triggering a fourth cycle. Here the question arises as to whether the fourth cycle should be counted as stockout cycle or not. Although these two last cycles overla we only consider the former to be a stockout cycle. Notice that actually the fourth cycle ends u with a ositive net stock. In this way and assuming no further demands after eriod 3, the item will achieve a CSL of 60%. Notice that if the fourth cycle was considered a stockout cycle the CSL calculation would give 40%.
17 Item # 2248 ROP = 2 Q = 2 L = 2 eriods Period Demands Net stock Inventory Position Cycles 3 4? Figure 3. Examle to illustrate the occurrence of stockout cycles Remark on the imlementation of the otimization methodology In the study we otimized the system based on the fill rate, since this service level is of more ractical imortance for sare arts. We investigated however what were the realized cycle service levels (CSL) achieved by the different classes under the models considered. We observed that in many cases for a articular target fill rate the corresonding CSL was way behind the current CSL. Therefore, in order to revent such low values of the CSL we imosed an extra condition in Ste 5 of the revious algorithm. Accordingly, for the identification of the otimal target fill rate we checked that the realized CSL was within 5% of the current CSL as long as the system still achieved ositive savings. For some classes this condition was difficult to satisfy exactly, and hence the realized CSL was below the 5% value. For the evaluation of the realized CSL of class for the different models we use the following formula: rcsl ( X ) N = =, N = soc c ( X ) ( X ) with corresonding variance given by Var( rcsl ( X ) ) = c N c = N ( X ) = ( X ) soc c ( X ) ( X ) 2 ( X ) ( rcsl ) 2 Accuracy of the LTD modeling methods Since we are interested in the erformance of the modeling methods alied to inventory olicies, a useful measure of the accuracy of the different models can be rovided by the average of the absolute difference between the target and the realized service levels. As we observed earlier, differences are exected due to the discreteness of the inventory levels. We call this measure MAD, defined by MAD N ( ) CSL = N i= rcsl α, for a target cycle service level α, and i.
18 N ) MADβ = rβ i β, for a target fill rate β. N ( i= 4.4. The simulation model We imlemented the roosed methodology in MatLab v.7.0. The system was simulated over the testing eriod using the different inventory models. Recall that demands for the items in each class were drawn directly from the data set and used as inut to the simulation. Thus, we were able to evaluate the erformance of the different olicies and models in a real inventory environment. This allowed us to cature the eculiarities of the system under study and to discern under which conditions which olicy erforms better. Since demands cannot occur with a higher frequency of one er month, we consider that the system is reviewed on a continuous basis, and therefore valid for the alication of the (s, nq) and (S, S) olicies. For the evaluation of inventory cycles and comutation of inventory related costs we use the classical inventory methodology. Thus, the (s, nq) olicy is imlemented in the classical way as exlained in Section 4.2. Considerations on starting stocks Since the system orders nq units whenever the inventory osition dros below the reorder oint (or Q when it is at the reorder oint), the maximum level that the system can have at any oint in time is given by its reorder oint s lus Q. Therefore, for comarison uroses it is fair to consider the starting stock of an item as its reorder level lus its lot size Q. For the current system, an equivalent choice is to set the starting stock of an item at its max level, since this can be considered as an orderuto level S with S = s + Q. This choice of starting stocks allows us to comare the erformance of the different inventory olicies considered. Note that for an infinite horizon, the CSL is indifferent to the choice of starting stock, whereas the fill rate is not. However for a fixed data set, the starting stock may also have an effect on the cycle service level of an item, since for higher starting stocks fewer inventory cycles will be comleted. As we use the same definition of starting stocks for all models, the total number of inventory cycles will be the same among them, excet for the current system. To see this notice that the number of inventory cycles for an item is indeendent of the choice of its reorder oint. Since for all models we use the same demand values and Q is fixed, it follows that cycles will be the same regardless of the model used. 5. Results from the simulation otimization After having alied the classification criteria exlained in Section 3, and using the methodology resented in the revious section, we erformed an otimization of the inventory system of sare arts. In order to have a clear icture of the whole rocess, we resent in Fig. 4 a flow chart diagram of the entire rocedure, from cleaning and classification of demand, rice and criticality data for sare arts until the otimization of the system itself.
19 START Data cleaning of sare arts: identification of secification errors and anomalous observations Exost otimization: whole data set for fitting and testing Selection of otimization aroach Exante otimization: data set slitted for fitting and testing Classification of arts based on demand, criticality and rice using whole data set Alication of the otimization rocess using the decomosition aroach method (section 4.3) Selection of relevant item classes for the otimization and exclusion of classes with few items Filtering of large demands for demand classes 4 and 5 Fitting to find LTD distribution of each item according to demand model (exclusion of items with not enough demand data for fitting) Evaluating realized service levels for each item class as a whole Setting target service levels for each item class Testing inventory model for each item inside the class using simulation with actual demand as inut Analysis of otimization results Figure 4. Flow chart diagram of the otimization study In tables 4 and 5 we give a summary of the numerical results obtained from the simulation otimization for the exost and exante aroaches, using as otimization criterion the fill rate (for a comlete set of results of the different item classes see tables 3 and 4 in the Aendix). We include the relevant classes under study, for which the realized CSL, the realized fill rate and the total savings are reorted for each model. We also include the total cost of the current system with the current achieved service levels (target CSL and target fill rate). For convenience, we use the following notation in the otimization results tables: Notation for otimization results tables tcsl: target cycle service level, that is, the cycle service level achieved by the current system (C) using the (min, max) olicy. t fill rate: target fill rate, that is, the fill rate achieved by the current system under the (min, max) olicy. rcsl: realized cycle service level, that is the cycle service level achieved by each of the four inventory models under consideration, these are: Normal (N), Poisson (P), Emirical (E) and Willemain model (W). r fill rate: realized fill rate, same observations as for rcsl. Savings (*) vs. (C): This is the difference between the holding and ordering costs of the current system with resect to (N), (P), (E) and (W).
20 Table 4. Summary of otimization results for the system under exost aroach Item class # of Current System (C) Normal Model (N) Poisson Model (P) Emirical Model (E) Willemain Model (W) items Tot. Costs* Service level Savings Service level Savings* Service level Savings Service level Savings Service level tcsl t fill rate (N) vs. (C) rcsl r fill rate (P) vs. (C) rcsl r fill rate (E) vs. (C) rcsl r fill rate (W) vs. (C) rcsl r fill rate Summary by demand class Demand class 2,226 3,693, % 95.6% 774, % 96.8% 703, % 96.8% 762, % 96.9% 75, % 96.6% 4 5,85 9,469, % 86.% 408,750 8.% 90.7% 202, % 9.7% 70, % 89.7% , % 85.7% 3, % 90.2% 5,4 37.4% 8.4% 7, % 8.7% 6 475,676, % 96.6% 92, % 97.3% 04, % 97.3% 6, % 97.5% Summary by criticality class Criticality class 493,704, % 88.3% 20, % 9.8% 3, % 92.9% 5, % 87.7% 6, % 88.7% , % 95.3% 37, % 97.%, % 99.8% 28,67 9.2% 97.5% 2, % 96.4% 3 3,90 9,65,00 90.% 89.8% 853, % 92.7% 648, % 97.5% 802, % 93.5% 743, % 92.2% 4, , % 92.9% 07, % 94.9% 20, % 97.0% 83, % 95.7% 54, % 94.2% , % 89.8% 6, % 93.4%, % 96.% 30, % 94.0% 54, % 92.9% 6 2,597 2,838, % 85.4% 227, % 90.6% 28, % 95.6% 5, % 89.9% 74, % 88.2% Summary by rice class Price class 2 2, , % 90.0% 79, % 97.0% 5, % 97.9% 66, % 95.7% 7, % 95.0% 3 3,266,608, % 88.5% 245, % 9.3% 25, % 98.2% 5, % 92.7% 73, % 9.4% 4 2,487 3,637, % 89.4% 200,40 90.% 90.7% 54, % 95.4% 7, % 90.4% 43, % 88.4% ,755, % 88.2% 682, % 9.5% 628, % 96.0% 700, % 89.9% 677, % 89.3% Total 8,494 5,525, % 89.2%,307, % 92.6% 703, % 96.8%,054, % 92.7% 964, % 9.4% * All cost and savings figures are in euros. Item class Table 5. Summary of otimization results for the system under exante aroach # of items Current System (C) Normal Model (N) Poisson Model (P) Emirical Model (E) Willemain Model (W) Costs* Service level Savings* Service level Savings Service level Savings Service level Savings Service level t fill r fill r fill r fill r fill tcsl rate (N) vs. (C) rcsl rate (P) vs. (C) rcsl rate (E) vs. (C) rcsl rate (W) vs. (C) rcsl rate Summary by demand class Demand class , % 95.7% 94, % 97.2% 63, % 96.3% 96, % 97.% 36,83 96.% 97.2% 4 2,873,975, % 87.3% 53, % 9.5% 0, % 9.% 8, % 89.6% , % 90.4%, % 92.9% 6, % 90.% 2,9 72.6% 90.8% , % 99.0% , % 97.6% 2, % 97.6% 3,0 96.6% 97.6% Summary by criticality class Criticality class 32 52, % 85.7% % 90.7% % 87.0% 4, % 9.5% 9, % 88.9% , % 96.5% 2, % 97.8%, % 00.0% 4, % 96.6% 8, % 96.9% 3,457,70, % 90.5% 5, % 9.7% 56, % 96.5% 63, % 9.% 8, % 89.9% , % 94.3% 29, % 96.% 4, % 97.7% 26, % 95.7% 22, % 94.8% , % 95.2% 6, % 95.2% 2, % 98.7% 2, % 95.6% 2, % 94.7% 6, , % 84.6% 58, % 9.4% 8, % 94.0% 28, % 90.5% 2, % 89.9% Summary by rice class Price class 2,372 44,420 8.% 90.5% 58, % 98.3% , % 95.2% 63, % 97.8% 58, % 98.3% 3, , % 88.7% 62, % 92.3%, % 98.8% 38, % 92.2% 2, % 9.2% 4,59 83, % 90.0% 9, % 88.% 0, % 94.5% 9,3 84.9% 86.6% 8, % 84.5% 5 47,526, % 90.2% 7,07 86.% 88.3% 53, % 96.3% % 87.4% 6, % 86.4% Total 4,326 2,878, % 89.7% 57, % 93.0% 63, % 96.3% 2,8 8.7% 92.3% 27, % 9.5% * All cost and savings figures are in euros 5.. Otimization results for the system as a whole From the results at the bottom of Table 4 we can see that for the aggregated values across all classes considered in the system, all the models outerformed the current system under the exost aroach, achieving lower total costs and higher fill rates, with overall CSL within 5%±% of the current CSL. For the different models under the exost aroach, we make the following observations: The normal model achieved the best overall erformance among the models, achieving the highest total savings, although for demand class the emirical and Willemain models erformed very similar with resect to the normal. For this class the Poisson model erformed the worst of all models. Total savings for the normal model were of 8.4% over a total current cost of 5.5 million euros for the 5year eriod, and with overall fill rate of 92.6% versus a current fill rate of 89.2%. The cycle service level achieved by this model was 85.%, slightly lower than the one of the current system. The emirical model outerformed Willemain s model in terms of total savings (.05 vs million euros), both achieving overall fill rates above 90% and similar overall cycle service levels. To exlain the difference in erformance of these models, we found that the Willemain model evaluates slightly higher reorder oints to achieve
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