An inventory control system for spare parts at a

refinery: An empirical comparison of different re-

order point methods

Eric Porrasa*, Rommert Dekker

b

a Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Santa Fe, Mexico City, Mexico

b Econometric Institute, Erasmus University Rotterdam, the Netherlands

Abstract

Inventory control of spare parts is essential to many organizations, since excess inventory leads to high

holding costs and stock outs can have a great impact on operations performance. This paper proposes a

methodology for effective spare parts inventory control, motivated by a case study at a large oil

refinery. Different demand modeling techniques and inventory policies are evaluated using real data.

Keywords: Spare parts; inventory control; re-order points; demand classes; service levels

1. Introduction

Effective inventory management of spare parts is essential to many companies, from

capital-intensive manufacturers to service organizations, such as car manufacturers,

chemical plants, telecom companies and 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 maintenance operations

and to protect against equipment failures. Although this function is well understood

by maintenance managers, many companies face the challenge of keeping on stock

large inventories of spares with excessive associated holding and obsolescence costs.

Thus, effective cost analysis can be an important tool to evaluate the effects of stock

control decisions related to spare parts. However, the difficulty in assessing good

strategies for the management of spare parts lies in their specific nature, normally

very slow-moving parts with highly stochastic and erratic demands. For example,

typical industrial data sets comprise 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 parameters of most inventory policies. Although

different inventory models have been proposed in the literature to tackle this problem

(see next section), there is a lack of empirical testing of theoretical models with data

from real industrial environments.

* Corresponding author: eric.porras@itesm.mx

This paper concerns a study on spare parts at a major oil refinery in the

Netherlands, which consisted of two phases. In the first phase a case study was

conducted with the company, where the SAP system used as platform for its

operations was under examination. As a result, some improvement measures were

provided and later implemented by the company. The second phase was focused on

the analysis of the demand data provided by the company. This paper reports on the

findings related to this phase, where the objective was to perform an empirical

comparison of different inventory models.

The aforementioned company keeps stock of a large number of spare parts related

to equipment used in its petrochemical processes. Although these stocks are essential

for the continuity of its operations, management was concerned with the savings

opportunities at the process floor by having better inventory control of its spare parts,

whose value was worth at the moment of the study more than 27 million euros. One

major difficulty of the study was the limited demand history available.

By describing the case, we make general observations about the practical aspects

of inventory control. Moreover, our aim is to compare various policies with real

demand data from the case to see which one is best under what circumstances.

Common methods presented in the literature rather use given statistical demand

distributions to assess the performance of inventory models. Consequently, with our

methodology we can better identify the real limitations of industrial data sets.

The remainder of the paper is organized as follows: the next section presents

briefly related literature. Section 3 includes the case study description. Next the

methodology is explained in Section 4. The computation results are included in

Section 5 and the final conclusions are presented in the last section.

2. Review of related literature

One of the major areas of inventory research over the past decades is the one related

to the management of spare parts inventories. Although theoretical models for slow-

moving items are abundant in inventory literature since 1965, case studies are few (for

a comprehensive overview of recent literature on spare parts management see

Kennedy et al. (2002)).

In the arena of theoretical models, one of the most extensively studied inventory

policies is the so-called (S-1, S) model, a particular case of (s, S) models, with an

underlying Poisson demand distribution (see Feeney and Sherbrooke (1966)).

Although well studied and suitable for slow-moving items, this type of policy requires

continuous review of the inventory system. Moreover, the Poisson distribution

assumes randomness of demand, with interarrival times between unit size demands

following an exponential distribution. This distribution needs no information of

demand other than the average demand, which is the solely parameter of the demand

distribution. When transactions are larger than unit size, authors have proposed the

use of compound-Poisson models (see Williams (1984) and Silver et al. (1971)).

However, these models are difficult to apply in practice because they need an

assumption on the compounding distribution. For example Williams (1984) developed

a method to identify sporadic demand items, where three parameters are needed: one

for the exponential distribution of interarrival times of demands, and two parameters

of an underlying gamma distribution for the demand size.

Most of the empirical studies in spare parts literature are focused on testing

forecasting methods for demand of slow-moving items rather than on implementing

inventory models. This is an important distinction since forecast methods are used to

estimate point forecast of the mean (like the moving average method) while for

evaluating control parameters of inventory models (like the (s,Q) model) one needs an

estimation of the entire LTD distribution. Moreover, inventory models are used to

meet specific customer service levels in the long run, while forecasting models aim to

obtain accurate demand forecasts as determined by the mean average percentage error

(MAPE) or the mean square error (MSE). In this area of research Ghobbar and Friend

(2003) present a comparative study of 13 different forecasting methods for the

management of spare parts in the aviation industry. They use the MAPE measurement

applied to forecast errors to assess the accuracy of the different methods but no

inventory models are included in the study. They confirm the superiority of the

weighted moving average and Croston’s methods over exponential smoothing and

seasonal regression models. Silver et al. (1998) also advise on the use of Croston’s

method for products with intermittent and erratic demand. This method (Croston

(1972)) assumes that the LTD has a normal distribution, and estimates the mean

demand per period by applying exponential smoothing separately to the intervals

between nonzero demands and their sizes.

Willemain et al. (2004) propose the use of a modified bootstrap method to forecast

intermittent demand of service parts, and they implement the method on a large

industrial data set. Croston’s method and exponential smoothing are evaluated as well,

yet like in the previous paper no attempt is made to implement an inventory control

model. They show that the modified bootstrap method produces more accurate

forecasts (based on the MAPE measurement) than the exponential smoothing and the

Croston’s method. We use in the study the bootstrap method proposed by Willemain

and we compare it with the performance of an empirical distribution model. We

assess also the performance of models based on Poisson and normal demand

distributions.

A similar research to the present study is presented in Strijbosch et al. (2000),

where the performance of two different (s, Q) models for spare parts in a production

plant environment is examined. Unlike our method, they test the inventory models

proposed 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 (1978)

presented a case study carried out in a large petrochemical plant. They developed

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 consumption rates for slow

movers, a Poisson underlying distribution was assumed to compare between existing

practices and the models proposed. We use in our study a similar approach, 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 important managerial issue that has been

addressed in the literature (see Bridgman and Mount-Campbell (1993)). Information

on PM can be used to better control inventories as it takes advantage of planned

demand, by correcting the effect of stochastic failures of equipment. Thus, models can

perform more accurate as they are not perturbed further by PM demands.

3. The case study - system description

The company under study consists of a major petrochemical complex located in the

Netherlands, which includes 60 different plants divided in chemicals manufacturing

and oil refinering. The complex dates from 1930, and many new installations have

been added since then. A large part of it however, stems from the 1960s. The

procurement department offers service to all plants. There is one central warehouse

owned by the company. At the moment of the study (2000), there were in total 130

thousand catalogued materials, of which only 43 thousand were kept on stock at the

site, with a total value of more than 27 million euros. There are 22 additional small

de-central 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 replenished on a

batch basis. Therefore, we only need to consider a single stock echelon, being the

warehouse as user of spares for equipment and not a producer of parts. In total there

are 180 thousand requests of material per year, both for non-stock and stock materials.

Requests for materials kept on stock are supplied from available stock. If there is

shortage of a material an emergency replenishment order is generated.

Controlling 43 thousand materials represents a difficult task, especially because of

the differences in types and consumption patterns associated with them. It also

requires efficient use of the manpower available and of the information system at

hand. Until 1997 an in-house developed information system for inventory control was

used by the company. In 1997 they moved to the information system SAP R/3, which

is a complete ERP-system, but not specific for inventory control. Almost the whole

demand history before 1997 has been lost in the transfer to SAP. Within SAP, the

company applied the MM (materials management) module for the control of its spare

parts. Since SAP evolved out of MRP systems for the manufacturing and assembly

industry, the MM module is very much based on the MRP planning philosophy (see

Heizer and Bender (2001)). Demand is expressed by actual orders or by forecasts of

demands. Next demand of end products is converted to demand for assemblies,

components and parts. Stock control is performed in SAP on a periodic basis (so-

called periodic review). Items are ordered when the MRP run is made. The SAP user

can set the appropriate time interval, e.g. daily, weekly or monthly. At the company

they run the MRP every week.

The actual stock control within SAP occurs in terms of min-max levels

(equivalently to (s, S) policies), or MRP-type 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 project

90% of the control levels were set manually, and afterwards some 70%. As a result

still many replenishment orders were checked manually before sending them out.

With respect to forecasting, several methods are available in SAP, like exponential

smoothing and moving averages, both with trends and seasonality. It is however the

intermittent nature of demand that makes the application of these methods particularly

difficult to spare parts. For the determination of safety stock levels, the normal loss

model is available which approximates the demand during the lead time with a normal

distribution. This model works with the cycle service level as service level objective.

However, no fill rate service levels can be defined within the MM module. A more

striking aspect of SAP is that within its functionality no continuous review models

can be implemented. Therefore the classical and much advised (S-1, S) model with

Poisson distributed demand over the lead time cannot be applied.

3.1. Data structure

Statistical information for the consumption of spare parts was available for 5 years

(the last year only until August). The demand information was recorded in monthly

periods, so a total of 55 periods of demand information was available for the study.

One important limitation of the demand set was that it did not specify whether

demands were due to failures or preventive maintenance activities.

Different parts used by the company are divided in two main categories: materials

related to a piece of equipment and the ones not related to any particular equipment,

like protecting shoes, helmets, general-purposed electrical equipment and

instrumentation. From the total of 43,000 materials in stock, 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 criticality

codes, which are based on how unavailability affects the safety of the people and

environment, the cost of down time and the quality of the processes. Materials not

related to equipment 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 expensive

downtime or cause danger to the safety of the people and the environment.

Risk taken in the process of ordering and stocking cannot be justified.

• Medium (M): Unavailability of these materials would result in significant loss

of production, but does not endanger the safety of the people or the

environment. A calculated risk can be taken in the process of ordering and

stocking.

• Low (L): Unavailability of these materials would not result in serious effects

on the processes or on the safety of the people and the environment.

The previous classification is made on expert judgement 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 inspection of the materials with criticality code leads to a more refined

classification: materials that are uniquely installed in a particular piece of equipment

(60% of the materials related to equipment), and materials which are related to more

than one piece of equipment of different criticality codes. That means that there are

spare parts that have combined criticality codes (H/M/L, M/L) depending on whether

they are installed in multiple piece of equipment of different criticality. The company

used these codes to decide on the stock levels of the different parts. 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 depending on cost-effective

considerations. However, as no models are available in SAP that incorporate

criticality considerations, these levels were set mostly by expert judgement.

3.1.1. Classification of parts

A more refined analysis of the spare parts data revealed that important differences

among them existed not only in terms of criticality codes but also with respect to

demand and price. Therefore, we aimed at grouping them in different classes to see

whether we should apply different stock control methods for different classes.

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

spare parts:

Criticality class 1: H

“ 2: H/M/L or H/L

“ 3: M

“ 4: M/L

“ 5: L

“ 6: Not related to any particular piece of equipment

For the current policy used by the company, we expect 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 14,000 spare parts, for which we

observed a high variability in demand patterns. For example some parts had only 0/1

demands while others experienced either few large demands or no realization of

demands during 5 years. For other parts we observed large negative demands due to

returns. Thus, a classification was needed for the spare parts based on consumption

rates. For parts with total positive demand over the five-year period and some demand

values higher than 1, we identified from a histogram two main groups: parts with

relatively high total demand and parts with low total demand. Although the boundary

between these two groups 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 0/1 demands also had a total demand of less

than 60. According to this, we established the following demand classes for the parts:

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

“ 2: parts with negative total demand.

“ 3: parts with no realizations of demand (all demands equal to zero).

“ 4: parts with total demand larger than 0 but less than 60, and not only

0/1 demands.

“ 5: parts with total demand higher than 60.

“ 6: parts with -1, 0, 1 demands.

Price classes

For the spare parts in the data set, 5 different price levels were identified in a

histogram. Table 1 shows the different price classes for the spare parts. The parts

recorded in SAP with a price of 0 euros are items not owned by the materials

department (price class 1). We observed prices as low as 0.01 euros for some parts

(price class 2) and the most expensive ones had a price of 20,000 euros (price class 5).

Table 1. Price classes of parts Price class 1 2 3 4 5

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

Spare parts (total = 14383) 10% 19% 33% 29% 8%

Combined classes

Using the criticality, demand and price classes, we include each item in a combined

class defined by three digits. Accordingly, an item in class “xyz” corresponds to an

item with demand class x, criticality class y and price class z. This classification

allows us to optimize the system per class rather than for individual items. That is,

once a service level is defined for the combined class, the parameters for the different

inventory policies are evaluated for each item in the class. Then a simulation tool is

used to evaluate the performance 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 optimization rule for each combined class considered in

the study (see section 4).

3.1.2. Considerations on item classes and anomalous observations

The analysis of spare parts data is performed for all combined classes incorporating

demand classes 1, 4, 5 and 6. Thus, all criticality and price classes which combine

those demand classes are considered for the evaluation of the inventory models,

except price class 1, since items with price zero do not have associated holding costs.

Since for items with total negative demand a zero inventory policy is optimal, we do

not incorporate in the analysis demand class 2 (4% of the parts). Moreover, although

negative demands can be associated with returns due to preventive maintenance

practices (parts that were ordered but not actually installed) or with repaired parts that

were brought back to the system, we did not have specific information in this respect.

As for demand class 3 (2.2% of the parts), 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 11,984 items. Additionally, we observed items with an error in the

criticality specification. These items accounted for 4.1% of the total numbers of parts.

After excluding these items, we were finally left with 11,790 spare parts for the

analysis. We also identified in the data set one particular 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.

3.1.3. Lead times

The lead times for the spare parts were recorded in days. However the demand data

set for the items was registered in months, without specification of the day within the

month that a particular demand took place. Therefore, for ease of implementation in

the simulation we rounded the lead times off to full months using 30 days per 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

produced in months. Although the rounding up of lead times was introduced to better

cope with the demand data, by doing this we take a conservative approach, 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 important issue,

especially for older parts, as suppliers may no longer be able to meet the original

promised times.

3.2. Cost structure

In general, three types of costs are associated with inventories: holding costs, ordering

costs and stockout costs. Holding costs represent the cost of capital tied up in the

spare parts inventory. An annual fixed rate of 25% was used in the study. Ordering

costs represent the cost associated with placing an order for a spare part, which

includes the costs of telephone calls, inspection and handling of the incoming items,

paying the bill and registration of the parts. This cost is independent of the number 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 between service levels and holding

costs, we consider stockout costs in a separate study included in the Appendix.

4. Methodology

We use two approaches for the optimization of the spare parts inventory system under

consideration, namely an ex-ante and an ex-post approach. In the ex-post procedure

the same data set is used for both fitting and testing purposes. Opposite to this, the ex-

ante procedure, once a distribution has been fitted to the data, uses an entirely

different set for testing purposes. In this respect the ex-ante approach is more relevant

from a scientific and practical perspective, since in reality systems face future

“unknown” demands (Silver et al. (1998)). In order to achieve this, we divide the

historical demand data into two sets, namely a fitting period and a testing period. The

fitting period will be used to estimate the lead time demand distribution (LTD) which

is used in turn to determine the inventory policy parameters. The testing period is used

to perform a simulation to evaluate the performances of the inventory policies

selected and compare them with the performance of the current one. We consider two

types of service levels, the cycle service level (CSL) and the fill rate. The reason for

using the ex-post approach is that many industrial data sets are rather short for

forecasting purposes, and this procedure will give the advantage of using the whole

data set to get a “better picture” of the real demand process. The performance of both

approaches will be compared 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 expect highly variable results.

One of the main issues we address is whether theoretical models can outperform

stock analysts. In practice this is difficult to assess because of lack of information (e.g.

short demand data sets, little information on preventive maintenance practices), as

well as implementation constraints. For instance real lead times of items are normally

in days but in the models one may prefer to use full periods of time for ease of

implementation. Other practical issues like the performance of different methods to

model the demand process are explored 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.

4.1. Modeling the lead time demand

In inventory decision making, one needs to determine inventory control parameters,

such as reorder points and safety stocks. In order to do so, we need a specification of

the lead time demand distribution. This is traditionally done by modeling lead time

demand using common probability distributions found in the literature, such as the

normal distribution (Silver et al. (1998)) or the Poisson distribution (see Schultz

(1987)). Other authors propose the use of models based on forecast techniques such as

moving average or exponential smoothing (see Croston (1972) and Silver et al.

(1998)). In order to cope better with real data sets and to give a more realistic picture

of demand, authors have proposed the use of 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 the

LTD distribution using the Willemain’s bootstrap method along with another novel

procedure using empirical 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 parameters of

the inventory policies selected. However, different to the methods found in the

literature, our objective is to use directly the real demand values observed to assess

the performance of the policies using the simulation tool. To keep the study tractable,

we do not apply updating for the estimation of the LTD distribution. We also evaluate

the performance of the system using normal and Poisson distribution based models.

Below we describe these methods.

Willemain’s Bootstrap method (W)

We implemented the modified bootstrap method presented in Willemain et al. (2004).

This method, as compared to traditional bootstrap techniques presented in the

literature, has the advantage of capturing better the autocorrelations between demand

realizations, especially when dealing with intermittent demands with a high

proportion of zero values. The method first evaluates the empirical transition

probabilities between states of zero demand and states of positive demand for the

different items. Then using this information, a stream of zero and non-zero demands is

randomly generated for a period of length equal to the lead time. The non-zero values

are filled with demand values sample from the data set. In this way estimates of LTD

for each item are obtained for a large number of realizations (1000 in this study). This

information is finally used to estimate the distribution of LTD. Willemain et al. (2004)

applied this method to nine large industrial data sets of service parts inventories and

compared it with the exponential smoothing method and the Croston’s method. He

concluded that the modified bootstrap method gave the best performance of all three

methods. In Fig. 1 we show a plot of an estimation of the LTD distribution using

Willemain’s method for an item corresponding to class 415. The item (labelled #741)

has a lead time of 9 periods and its demand data for the 55 periods is as follows: in

periods 5 and 17 it observed positive demands of unit size each, and in period 9 a

demand of 2. The rest of the periods no demands were observed. Notice that although

only lead time demand values of 1 and 3 were realized in the data set, the method is

able to produce a LTD estimation where many other lead time demand values are

taken into account.

Empirical distribution of lead time demand (E)

We implemented an empirical model to estimate the distribution of LTD. Different to

the traditional bootstrap method, we construct a histogram of demands over the lead

time without sampling. This method is new to the literature as no attempts have been

made to use it for inventory control. Since demands are taken directly from the data

set over fixed periods of time equal to the lead time, this method also captures

autocorrelations and fixed demand intervals due to preventive maintenance, and is far

easier to implement than the modified bootstrap method described above. For

example, consider an item with the following stream of demands over 40 periods with

a lead time of 3 periods:

Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Demand 0 0 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

Assuming that the above sequence of demand values can occur at any point in time

after a future demand observation, then 4 is a possible realization for a LTD value. Of

course the same is true for 1 and 2. In this case we require that the estimation method

captures these possibilities; that is of having a LTD of 4 with a positive associated

probability, and a LTD of 2 with a higher associated probability. As the empirical

method will construct a cumulative distribution function (cdf) over a lead time of 3

consecutive periods for the whole data set, the LTD values of 2 and 4 will be

estimated with probabilities of 0.21 and 0.053, correspondingly. In Fig. 2 we show a

plot of the LTD distribution (cdf) for this item. This example was constructed for

illustrative purposes.

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

Figure 1. Cumulative distribution function using Willemain’s method

Empirical cdf

0%

20%

40%

60%

80%

100%

0 1 2 3 4

Lead time demand

Figure 2. Empirical cumulative distribution of lead time demand

Normal distribution (N)

We implement a normal based model assuming that demand for the parts follows a

normal distribution. To this end, the average (D ) and standard deviation (S.D.) of the

observed period demand is evaluated to estimate the parameters µLTD and σLTD of a

normal LTD distribution, as follows:

LDLTD ⋅=µ

LDSLTD ⋅= ..σ ,

where L is the lead time of an item in full periods 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. (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 recommended (Silver et al.

(1998)). Thus, we do not expect the normal based model to give better results than the

others. However, we want to investigate its performance as compared to the other

models considered. To this end, when we evaluate reorder points based on the normal

LTD distribution to meet desired fill rates, the values obtained are rounded up to

integer values, and negative values are set to zero. We require this since for our

system reorder points are defined as positive integer values in accordance with the

discrete demand for parts. In the case of negative values which are set to zero no

compensation for the gain mass is applied 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 positive re-order point 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 re-order point s for a given fill rate

level β, we use a similar procedure as the Excel “goal seek” routine (see Chopra and

Meindl (2004)), utilizing the formula:

100)(

1%100 ×

⋅−=

Q

zUNLILTDσβ ,

where UNLI(z) is the unit normal loss integral associated with the unit normal variate

z. Recall that z corresponds to a re-order point s associated with a CSL value (see

Silver et al. (1998)). Thus the product in the denominator of the above formula gives

the expected units short for a given re-order level s. As mentioned in Silver et al.

(1998), the previous formula underestimates the true fill rate if σLTD is large relative to

Q. Therefore, a correction should be made in the numerator substituting the term

σLTD⋅UNLI(z) by σLTD(UNLI (z) – UNLI (z + Q/σLTD)). By using the uncorrected

formula, we obtained conservative values of the fill rates, which did not have a major

impact in the optimization of the system (see results in sections 5.1 and 5.2).

Remark 1. Although in many situations the normality assumption is not satisfied, this

distribution has widely been used in practice. This is due to the simplicity to evaluate

reorder points and other parameters based on the normal distribution.

Poisson distribution (P)

Silver et al. (1998) 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 1. The reason is that demands for these items

are of unit size, and hence the basic assumption of the Poisson distribution is satisfied.

The only parameter 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 compound-Poisson based model would be more appropriate, but a number

of specific assumptions need to be satisfied in the compounding in order to expect a

good performance of this model. Since we wanted to develop robust methods that

applies equally well for different industrial data sets, we did not consider the

compound Poisson for demand classes different from 1. The Poisson based model is

compared to the normal, empirical and bootstrap methods for demand class 1.

4.2. Inventory models

We use an (s, nQ) inventory policy for the system, with the reorder point s evaluated

using the LTD distribution according to the modeling methods described above. Thus,

when overshooting of the reorder point s cannot be overcome by the lot size Q, an

alternative lot size equal to nQ is ordered, such that the inventory position is brought

above s, where n is an integer value. This is a common practice in inventory

management (see Silver et al. (1998)). 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:

1. Evaluate: m = EOQ

2. Set

+

+≤≠

=

=

otherwisem

EOQ

m

m

EOQandmifm

mif

Q

1

10

01

When we use the Poisson distribution to estimate the LTD, we use Q = 1. This model

is often referred to as (S-1, S) model, with s = S-1. Notice that for demand class 1,

since average demand is generally low, the EOQ calculation is likely to produce also a

value of 1.

For the classes under study the proposed policies are compared to the current (min,

max) policy in terms 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 and

policy perform best under which conditions (see Table 2).

Table 2. Inventory models considered

Model Parameters Demand Classes

Current policy (C) min-max (s, S) 1, 4, 5, 6

Poisson based model (P) (S-1, S) 1

Normal based model (N) (sN, nQ) 1, 4, 5, 6

Empirical based model (E) (sE, nQ) 1, 4, 5, 6

Willemain based model (W) (sW, nQ) 1, 4, 5, 6

4.2.1. 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 point (say evaluated according to a normal LTD distribution) is

likely to be overshot when using a simulation tool to assess the performance of a

given inventory model. To illustrate this we give in Table 3 the demand data and other

relevant information for item # 1307, 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 positive demands plus 10 times their standard

deviation. So here the aim is to construct a tool that filters out this large demand. To

this end, we implement a demand filter in the following way: all demands larger than

the average plus k standard deviations (evaluated using only positive demands) are

filtered out. For this particular item observe that k = 3 will not produce 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 applying the mentioned filter, 8.7% of the positive

demand values for these two classes were filtered out.

Table 3. Demand data for item # 1307 Demand number of Total

value occurrancens demand

0 47 0

87 1 87

100 4 400

120 1 120

150 1 150

450 1 450

Total 55 1207

Average of demands > 0 150.9

Stand. deviation of demands > 0 122.4

Average of positive demands < 450 108.1

Stand. dev. of positive dem. < 450 20.8

Remark 2. Although for the optimization of the system we use the above filter of

demands, we also assess the effect of having large demands included in the

optimization process. We present some numerical results in the Appendix.

Remark on the implementation of the ex-post and ex-ante approaches

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 purposes. Therefore, items with only

zero or one positive demand during the fitting period are excluded from the analysis.

This consideration is used in both the ex-post and the ex-ante approaches. As for

classes with few items (6 or less), they were excluded from the analysis in both the

ex-ante and the ex-post approach. According to these considerations, of the original

11,790 items, a total of 8,494 were included in the ex-post approach and 4,326 in the

ex-ante approach (see analysis of results in sections 5.1 and 5.2).

Remark on classification of items

We include each item in a class according to the demand pattern and other relevant

information (criticality and price). We do the classification using the whole data set in

both the ex-ante and the ex-post approaches, since we want to assure that an item in a

certain category will not exhibit demand values not corresponding to that category

during the simulation in the testing period.

4.3. Optimization of the system: a decomposition approach

We focus on optimization 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 preferred method in industry as opposed to cost minimization. The

reason for this is that the latter requires the evaluation of stock out costs which depend

on down time penalties and other factors which are difficult to evaluate in practice.

Considering the size of the system related to the present study (recall that

originally it comprises some 14,000 items), we use a decomposition approach for the

optimization of the system. This type of approach is well known to the literature,

where it is often used for the analysis of complex systems, e.g. for the optimization of

multi-echelon inventory systems (see van der Heijden et al. (1997)). In our case the

basic idea is to optimize the system at group level, defined by the classes described

earlier, rather than doing it at item level. Thus, we evaluate the different inventory

parameters 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 respect to the target values

are expected due to the discrete nature of inventory levels. Thus, the objective of the

method is to find the right level of a target service level that optimizes the classes

under study.

We first introduce the following notation:

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

X: identifier for the model applied in the simulation, where X = current policy

(C), Poisson (P), Normal (N), Empirical (E) or Willemain (W). )(X

jc : number of inventory cycles completed by item j during the testing period

using model X. Note: an inventory cycle is defined between the placing of an

order and its arrival to the system, i.e. the inventory cycle over a lead time

(also referred to as replenishment cycle). )(X

jsoc : number of inventory cycles for item j with stockouts during the testing period

using model X. )(t

jD : total number of units demanded of item j during the testing period.

)(X

jS : total number of units of item j supplied from on hand stock over the testing

period using model X. )(X

pCSL : cycle service level achieved by the system using the model X.

)(X

pβ : fill rate achieved by the system using model X.

βp: target fill rate for class p.

ESj(sj): expected units short for item j for a given reorder point sj, according to the

corresponding LTD distribution.

Now for each combined class p (xyz) proceed as follows:

1. Evaluate the service levels (CSL and fill rate) achieved by the current system over

the testing period according to the ex-post or the ex-ante approaches. These are

the current service levels of class p, which are defined by

∑

∑

=

=−=

p

p

N

j

C

j

N

j

C

j

C

p

c

soc

CSL

1

)(

1

)(

)( 1 ,

∑

∑

=

==

p

p

N

j

t

j

N

j

C

j

C

p

D

S

1

)(

1

)(

)(β .

2. For each item j, estimate the cumulative distribution of LTD over the fitting

period, according to the model selected (see Section 4.1). Use βp =)(C

pβ together

with the LTD distribution to estimate the corresponding parameters of the

inventory policy (i.e. reorder point and lot size), in the following way:

• Calculate Qj, the lot size, according to the EOQ formula and round it off as

explain in Section 4.2. If class p = 1yz (demand class 1) then set Q = 1.

• Chose the smallest sj which satisfies a 100βp % fill rate, i.e.:

100)(

1%100 ×

−≤

j

jj

pQ

sESβ .

3. For each item j, run the simulation over the testing period applying the (s, nQ) or

the (S-1, S) policy as required, using the parameters selected in the previous point.

For each model selected, evaluate for class p the realized fill rate (rβp), and its

total costs (TCp), according to

∑

∑

=

==

p

p

N

j

t

j

N

j

X

j

X

p

D

S

r

1

)(

1

)(

)(β , ∑=

=pN

j

targetj

X

targetp TCTC1

,

)(

, ,

where TCj,target is the total cost of item j for the target service level selected, which

comprises the holding and ordering costs. Compute the total savings achieved by

the selected model with respect to the current system according to:

)(

,

)(

,

)( model

targetp

current

targetp

model

p TCTCsavingsTotal −=

Within the classes we expect 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:

( )∑∑

=

=

−=p

p

N

j

X

p

X

j

X

jN

j

X

j

X

p rc

c

rVar1

2)(2)()(

1

)(

)( 1)( βββ .

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

)(ˆ )()( X

p

X

p rVar βσ = .

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 optimization.

5. For classes with lower realized service levels or negative savings, construct

exchange service level-cost curves for the different inventory policies, in the

following way: define a set of fixed target values for the fill rate (βp), say from 1%

to 100% in step sizes of 1%. Next evaluate the parameters of the selected

inventory model as in point 2 (except for the current system). For each value of

the target service level obtain the corresponding rβp, TCp,target and total savings.

Finally, identify the value of βp, for which the corresponding realized fill rate

equals or exceeds the current fill rate (within 1% precision), and that achieves

positive savings. In this way we optimize the system among the different classes.

Remark 3. The previous method assumes that the system is optimized using the fill

rate as optimization criterion, which plays the role of control parameter. However, it

is possible to implement the above procedure using the CSL for the evaluation of

reorder points in steps 2 and 5, in the classical way. Even in the case for which the

optimizing criterion is the fill rate, one may decide to apply the CSL as control

parameter due to limitations associated with the evaluation of re-order points, and

optimize the system with respect to the fill rate. In the Appendix 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 replenishments cycles overlap, we associate a stockout with the

immediately precedent 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. (1998), 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 periods and reorder point of 2 units. In period 19 the inventory position reaches the

reorder point, triggering the first replenishment cycle. This cycle ends at the

beginning of period 21 when the system is replenished and causing the net stock to

reach 4 units. This cycle ends up with no stockouts (thus coloured white). The second

cycle starts in period 22 with net stock still positive. Notice that before this cycle

ends, a third cycle starts in period 23, where the net stock drops to -1. This causes the

second cycle to be a stockout cycle (thus coloured grey). Similarly, the demand of 8

units in period 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 overlap we only consider the

former to be a stockout cycle. Notice that actually the 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%. Notice that if the fourth cycle was considered a

stockout cycle the CSL calculation would give 40%.

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 4

1 2 5

Cycles 3

4 ? Figure 3. Example to illustrate the occurrence of stockout cycles

Remark on the implementation of the optimization methodology

In the study we optimized the system based on the fill rate, since this service level is

of more practical importance for spare parts. 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 particular target fill rate the

corresponding CSL was way behind the current CSL. Therefore, in order to prevent

such low values of the CSL we imposed an extra condition in Step 5 of the previous

algorithm. Accordingly, for the identification of the optimal target fill rate we checked

that the realized CSL was within 5% of the current CSL as long as the system still

achieved positive 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 p for the different models we use the following formula:

∑

∑

=

=−=

p

p

N

j

X

j

N

j

X

j

X

p

c

soc

rCSL

1

)(

1

)(

)( 1 ,

with corresponding variance given by

( )2)(

2

1)(

)(

)(

1

)(

)( 11

)( X

p

N

jX

j

X

jX

jN

j

X

j

X

p rCSLc

socc

c

rCSLVarp

p

−

−= ∑

∑=

=

.

Accuracy of the LTD modeling methods

Since we are interested in the performance of the modeling methods applied to

inventory policies, a useful measure of the accuracy of the different models can be

provided by the average of the absolute difference between the target and the realized

service levels. As we observed earlier, differences are expected due to the discreteness

of the inventory levels. We call this measure MAD, defined by

∑=

−=pN

i

pi

p

p

CSL rCSLN

MAD1

)( 1α , for a target cycle service level αp, and

∑=

−=pN

i

pi

p

p rN

MAD1

)( 1βββ , for a target fill rate βp.

4.4. The simulation model

We implemented the proposed methodology in MatLab v.7.0. The system was

simulated over the testing period using the different inventory models. Recall that

demands for the items in each class were drawn directly from the data set and used as

input to the simulation. Thus, we were able to evaluate the performance of the

different policies and models in a real inventory environment. This allowed us to

capture the peculiarities of the system under study and to discern under which

conditions which policy performs better.

Since demands cannot occur with a higher frequency of one per month, we

consider that the system is reviewed on a continuous basis, and therefore valid for the

application of the (s, nQ) and (S-1, S) policies.

For the evaluation of inventory cycles and computation of inventory related costs

we use the classical inventory methodology. Thus, the (s, nQ) policy is implemented

in the classical way as explained in Section 4.2.

Considerations on starting stocks

Since the system orders nQ units whenever the inventory position drops below the

reorder point (or Q when it is at the reorder point), the maximum level that the system

can have at any point in time is given by its reorder point s plus Q. Therefore, for

comparison purposes it is fair to consider the starting stock of an item as its reorder

level plus 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 order-up-

to level S with S = s + Q. This choice of starting stocks allows us to compare the

performance of the different inventory policies 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 completed. As we use the same definition of starting stocks for all models, the

total number of inventory cycles will be the same among them, except for the current

system. To see this notice that the number of inventory cycles for an item is

independent of the choice of its reorder point. 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 optimization

After having applied the classification criteria explained in Section 3, and using the

methodology presented in the previous section, we performed an optimization 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

classification of demand, price and criticality data for spare parts until the

optimization of the system itself.

Figure 4. Flow chart diagram of the optimization study

In tables 4 and 5 we give a summary of the numerical results obtained from the

simulation optimization for the ex-post and ex-ante approaches, using as optimization

criterion the fill rate (for a complete set of results of the different item classes see

tables 13 and 14 in the Appendix). We include the relevant classes under study, for

which the realized CSL, the realized fill rate and the total savings are reported 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 optimization results tables:

Notation for optimization results tables

tCSL: target cycle service level, that is, the cycle service level achieved by

the current system (C) using the (min, max) policy.

t fill rate: target fill rate, that is, the fill rate achieved by the current system under

the (min, max) policy.

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), Empirical (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 respect to (N), (P), (E) and (W).

Classification of parts based on demand,

criticality and price using whole data set

Data cleaning of spare parts: identification of specification errors

and anomalous observations

Selection of relevant item classes

for the optimization and exclusion of

classes with few items

Selection of optimization

approach

Ex-ante optimization: data set splitted for

fitting and testing

Ex-post optimization: whole data set

for fitting and testing

Filtering of large demands for demand classes 4 and 5

Application of the optimization process using the decomposition approach

method (section 4.3)

Analysis of optimization results

Fitting to find LTD distribution of each item according to demand model

(exclusion of items with not enough demand data for fitting)

Testing inventory model for each item inside the class using simulation with actual

demand as input

START

Setting target service

levels for each item class

Evaluating realized service levels for each item class

as a whole

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

Tot. Costs* Savings Savings* Savings Savings

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

1 2,226 3,693,400 95.6% 95.6% 774,220 96.2% 96.8% 703,250 96.8% 96.8% 762,330 96.3% 96.9% 751,500 96.0% 96.6%

4 5,185 9,469,700 84.3% 86.1% 408,750 81.1% 90.7% 202,860 80.0% 91.7% 170,000 79.1% 89.7%

5 608 685,920 77.2% 85.7% 31,370 70.0% 90.2% -15,411 37.4% 81.4% -17,797 37.8% 81.7%

6 475 1,676,100 96.5% 96.6% 92,713 96.3% 97.3% 104,950 96.3% 97.3% 61,126 96.7% 97.5%

Summary by criticality class

Criticality class

1 493 1,704,700 88.9% 88.3% 20,428 87.3% 91.8% 3,541 92.9% 92.9% -5,074 76.1% 87.7% 16,583 79.5% 88.7%

2 447 277,250 94.1% 95.3% 37,882 95.0% 97.1% 1,215 99.8% 99.8% 28,617 91.2% 97.5% 21,805 89.8% 96.4%

3 3,190 9,165,100 90.1% 89.8% 853,480 88.4% 92.7% 648,930 97.5% 97.5% 802,150 86.9% 93.5% 743,200 86.6% 92.2%

4 1,395 905,370 92.6% 92.9% 107,170 90.7% 94.9% 20,420 97.2% 97.0% 83,448 90.0% 95.7% 54,293 89.9% 94.2%

5 372 634,320 90.8% 89.8% 61,002 90.4% 93.4% 1,060 96.1% 96.1% 30,051 88.6% 94.0% 54,554 88.0% 92.9%

6 2,597 2,838,300 79.5% 85.4% 227,090 75.3% 90.6% 28,077 95.7% 95.6% 115,530 70.8% 89.9% 74,396 68.8% 88.2%

Summary by price class

Price class

2 2,223 524,350 83.2% 90.0% 179,030 79.4% 97.0% -5,415 97.9% 97.9% 166,870 75.1% 95.7% 171,200 73.7% 95.0%

3 3,266 1,608,000 87.9% 88.5% 245,180 84.7% 91.3% 25,627 98.2% 98.2% 115,240 83.3% 92.7% 73,012 83.3% 91.4%

4 2,487 3,637,300 90.2% 89.4% 200,410 90.1% 90.7% 54,393 95.5% 95.4% 71,791 86.7% 90.4% 43,397 85.6% 88.4%

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% 89.9% 677,220 84.0% 89.3%

Total 8,494 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% 92.7% 964,832 81.5% 91.4%

* All cost and savings figures are in euros.

Item class

# of

items

Current System (C)

Service level Service level

Normal Model (N) Poisson Model (P)

Service level Service level Service level

Empirical Model (E) Willemain Model (W)

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

Costs* Savings* Savings Savings Savings

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

1 800 504,650 95.6% 95.7% 94,010 96.5% 97.2% 63,575 96.3% 96.3% 96,409 96.2% 97.1% 36,831 96.1% 97.2%

4 2,873 1,975,100 84.8% 87.3% 53,384 81.3% 91.5% 10,900 79.1% 91.1% -18,623 79.6% 89.6%

5 475 152,750 76.3% 90.4% 11,873 80.1% 92.9% 16,379 67.2% 90.1% 12,119 72.6% 90.8%

6 178 246,130 98.9% 99.0% -1,816 96.6% 97.6% -2,571 96.6% 97.6% -3,110 96.6% 97.6%

Summary by criticality class

Criticality class

1 132 152,500 86.5% 85.7% -540 80.7% 90.7% -468 87.0% 87.0% -14,322 87.1% 91.5% -9,752 80.3% 88.9%

2 301 81,914 94.8% 96.5% 12,877 97.2% 97.8% 1,496 100.0% 100.0% 14,668 85.1% 96.6% 8,338 94.6% 96.9%

3 1,457 1,701,000 90.0% 90.5% 51,385 85.6% 91.7% 56,230 96.5% 96.5% 63,253 83.1% 91.1% -18,236 82.7% 89.9%

4 818 212,440 93.3% 94.3% 29,042 93.4% 96.1% -4,659 97.7% 97.7% 26,538 92.6% 95.7% 22,704 91.7% 94.8%

5 159 84,877 94.8% 95.2% 6,130 92.5% 95.2% 2,398 98.7% 98.7% 2,132 92.5% 95.6% 2,947 90.8% 94.7%

6 1,459 645,930 76.3% 84.6% 58,558 75.6% 91.4% 8,578 94.0% 94.0% 28,848 71.8% 90.5% 21,215 74.2% 89.9%

Summary by price class

Price class

2 1,372 144,420 81.1% 90.5% 58,581 82.9% 98.3% -1,919 95.2% 95.2% 63,142 77.6% 97.8% 58,886 84.3% 98.3%

3 1,648 377,020 87.5% 88.7% 62,667 83.9% 92.3% 1,126 98.8% 98.8% 38,379 82.6% 92.2% 21,502 80.9% 91.2%

4 1,159 831,190 90.7% 90.0% 19,185 87.4% 88.1% 10,508 94.5% 94.5% 19,131 84.9% 86.6% 8,416 83.0% 84.5%

5 147 1,526,000 90.6% 90.2% 17,017 86.1% 88.3% 53,860 96.3% 96.3% 466 83.4% 87.4% -61,587 82.0% 86.4%

Total 4,326 2,878,665 86.4% 89.7% 157,452 84.6% 93.0% 63,575 96.3% 96.3% 121,118 81.7% 92.3% 27,217 82.6% 91.5%

* All cost and savings figures are in euros

Item

class

# of

items

Current System (C)

Service level Service level

Normal Model (N) Poisson Model (P)

Service level Service level Service level

Empirical Model (E) Willemain Model (W)

5.1. Optimization 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 outperformed the current

system under the ex-post approach, achieving lower total costs and higher fill rates,

with overall CSL within 5%±1% of the current CSL. For the different models under

the ex-post approach, we make the following observations:

The normal model achieved the best overall performance among the models,

achieving the highest total savings, although for demand class 1 the empirical and

Willemain models performed very similar with respect to the normal. For this class

the Poisson model performed the worst of all models. Total savings for the normal

model were of 8.4% over a total current cost of 15.5 million euros for the 5-year

period, 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 85.1%, 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 million euros), both achieving overall fill rates above 90% and similar

overall cycle service levels. To explain the difference in performance of these models,

we found that the Willemain model evaluates slightly higher reorder points to achieve

the same service level as the ones of the empirical model. This because the Willemain

method produces a larger range of different lead time demand values, whereas the

empirical method uses the exact values observed in the data set to construct the LTD

distribution. Therefore, larger deviations are expected in the LTD distribution based

on the Willemain model and expensive individual items are likely to have a higher

effect with respect to holding costs. In this respect, observe that the difference in total

savings between the two models comes from few classes with expensive items (e.g.

classes 135, 435, 635), whereas for other classes both models performed very similar

or the Willemain model performed better (see Table 13 in the Appendix). On the other

hand the normal model uses a continuous distribution of LTD, which is smoother than

the empirical or the Willemain’s one, making it less expensive to match or improve

the target service levels.

As for the ex-ante approach, we have the following conclusions (see bottom of

Table 5 and Table 14 of the Appendix):

As the ex-ante approach uses a reduced data set for fitting purposes, the

optimization of the system became more difficult than when using the ex-post

approach, which could better match or improve the target service levels at lower total

costs. Recall that fewer items are used under this approach since we only consider

items with at least two positive demands during the fitting period. Despite these

limitations, the results obtained from ex-ante approach confirmed the general

conclusions derived under the ex-post approach. First of all observe that the current

system was outperformed by all models although with lower savings.

Second, under this approach again the normal model achieved the best

performance among all models, with total savings of 5.5% with respect to the current

system and overall fill rate of 93% (over a current one of 89.7%).

Next, as in the ex-post approach, the empirical model achieved higher savings than

the Willemain model, both with realized fill rates above 90%. We can 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 model performed the worst of all models,

which is the same result found under the ex-post approach.

5.2. Optimization results by demand, criticality and price classes

From the optimization results aggregated by demand, criticality and price classes

presented in tables 4 and 5, we can see the effect of the different methods on

particular classes. As we had more information available for the individual classes

under the ex-post approach, we give the following observations under this approach,

unless otherwise indicated.

All the models outperformed the current system for the different aggregated classes

with positive savings and higher fill rates, except for demand class 5, where the

empirical and the Willemain models had higher total costs and lower fill rates;

criticality class 1, where the empirical model yielded higher total costs; and price

class 2, where the Poisson model achieved also higher total costs. For the latter two,

however, the models achieved the same or higher fill rates with respect to the current

system within 1% precision. In the ex-ante approach there are few more classes where

the models could not outperform the current system (demand class 6, criticality class

3 and price class 5, with varied results for the different models).

From the optimization results by demand classes, we can see that, according to

what one would expect, for all models and for the current system the achieved service

levels (CSLs and fill rates) decreased as demand for the items increased. For low

demand items (demand classes 1 and 6), all models achieved high CSLs and fill rates,

with savings ranging from 6.2% to 20% with respect to the current system. However

for high demand items (demand classes 4 and 5) the normal model achieved much

better results than the other models, with total combined savings of 4.3%, as

compared to the corresponding savings of 1.8% and 1.5% of the empirical and

Willemain models.

From the results aggregated by criticality classes we conclude that for low to

medium criticality all models performed well. However for high critical items the

normal model performed much better than the others. For high critical items under the

ex-ante approach, observe that the current system was outperformed by the models

but at the expense of higher costs. For example the normal model achieved a fill rate

5% higher than the one of the current system but with 0.35% higher costs. (For a more

specific examination of the relation between criticality and service levels see analysis

below).

Finally, from the results by price classes, we can see that the models performed

similarly, except for price classes 2 and 3 where the Poisson model achieved 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 specific results for one class where we observed

large differences between the models. By doing this we want to explain the

differences in the general results discussed in the previous sections. Accordingly, we

chose class 415 as an illustrative example (see optimization results for this class in

Table 13 of the Appendix). Notice that for this class the normal model was the only

one achieving positive savings. The empirical model yielded lower total costs than the

Willemain model, however it achieved a lower fill rate than the current system.

In Table 6 we present the different steps in the optimization of 4 typical items

inside class 415, with price per item of 20,000 euros. These items are representative of

the 15 most expensive items in this class, with similar lead times and total demands,

where the total number of items was 39. In Table 6 the following additional results are

presented for each item and model, along with the realized CSL, the realized fill rate

and the total costs:

(s, S): the (min, max) parameters of the current system

s: the reorder point corresponding to a target fill rate (t fill rate)

#c: total number of inventory cycles completed by the item

#soc: number of stock out cycles for the given a reorder point

#D: total number of units demanded

#S: total number of units supplied for the given a reorder point.

After running an optimization procedure for these items, we obtained the results

shown at the bottom 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 combined

performance of all the items produced an optimized result with respect to the current

system (a target fill rate of 76%). For the empirical model, since we had fewer

available reorder points, this match is more difficult to obtain for a particular target

fill rate. On the other hand, the Willemain model produces a greater range of reorder

points as compared to the normal or the empirical model. However the regions of

target fill rates for the different reorder points does not allow an optimal combination

for a specific optimal target fill rate.

The previous analysis explains the high variability that we observe in the optimized

results for some of the classes presented in tables 4 and 5. Although this is a drawback

of the optimization procedure due to the decomposition approach, it allows a tractable

analysis of large scale systems like the one related to the present study. Since the

method uses the same target parameters for group of items, the stock control is less

cumbersome than in the case of parameters set at an individual level. An alternative

approach is discussed in section 5.8, where we optimize class 415 using an item level

optimization.

5.4. Service level vs. class structure

Although service levels would usually be reflected by different price or criticality

levels, we could not observe for the current system (ex-post results) a consistent

pattern in this respect. In order to have a better picture of this result, we present in Fig.

5 the aggregated current service levels in a two-entry matrix for price and criticality.

We would expect that as price increases for the same criticality level the

corresponding service levels would decrease. On the other hand, for the same price,

we expect that as criticality increases the service level also increases. However we

observe that often expensive items have higher associated service levels than cheap

ones, and low critical items have higher service levels than high critical items. For

some other cases we could observe a more consistent behavior within the line of

expectation, but a general conclusion in this respect is far from obvious. We conclude

that the inconsistencies observed are maybe the result of a wrong judgment on the part

of the stock analyst when deciding on the reorder points for the items. Also the effect

of rounding up the lead times could have an effect, since by doing this we are

generally evaluating worst service levels than the ones experienced by the system in

reality.

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

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

Figure 5. Double-entry matrices for CSLs and fill rates of the current system

Table 6. Optimization results for some items in class 415 Current system (C)

(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 fill rate s #c #soc #D #S Tot. costs rCSL r fill rate

870 (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% 1% 0 2 1 3 2 22,155 50% 67%

(L=1) 86% 1 2 0 3 3 44,655 100% 100% 95% 1 2 0 3 3 44,655 100% 100% 90% 1 2 0 3 3 44,655 100% 100%

99% 2 2 0 3 3 67,572 100% 100% 94% 2 2 0 3 3 67,572 100% 100%

97% 3 2 0 3 3 90,489 100% 100%

99% 4 2 0 3 3 113,410 100% 100%

100% 6 2 0 3 3 159,240 100% 100%

11216 (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% 1% 0 2 2 4 2 26,322 0% 50%

(L=6) 43% 1 2 0 4 4 44,239 100% 100% 53% 2 2 0 4 4 67,155 100% 100% 31% 1 2 0 4 4 44,239 100% 100%

83% 2 2 0 4 4 67,155 100% 100% 52% 2 2 0 4 4 67,155 100% 100%

97% 3 2 0 4 4 90,072 100% 100% 71% 3 2 0 4 4 90,072 100% 100%

100% 5 2 0 4 4 135,910 100% 100% 85% 4 2 0 4 4 112,990 100% 100%

93% 5 2 0 4 4 135,910 100% 100%

97% 6 2 0 4 4 158,820 100% 100%

99% 7 2 0 4 4 181,740 100% 100%

100% 12 2 0 4 4 296,320 100% 100%

11186 (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% 1% 0 2 2 4 2 28,822 0% 50%

(L=3) 62% 1 2 0 4 4 49,239 100% 100% 78% 2 2 0 4 4 72,155 100% 100% 65% 1 2 0 4 4 49,239 100% 100%

95% 2 2 0 4 4 72,155 100% 100% 76% 2 2 0 4 4 72,155 100% 100%

100% 3 2 0 4 4 95,072 100% 100% 87% 3 2 0 4 4 95,072 100% 100%

95% 4 2 0 4 4 117,990 100% 100%

99% 5 2 0 4 4 140,910 100% 100%

100% 7 2 0 4 4 186,740 100% 100%

5982 (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% 1% 0 2 1 4 1 23,405 50% 25%

(L=8) 32% 1 2 1 4 2 40,072 50% 50% 34% 2 2 1 4 3 62,572 50% 75% 14% 1 2 1 4 2 40,072 50% 50%

76% 2 2 1 4 3 62,572 50% 75% 96% 4 2 0 4 4 107,990 100% 100% 40% 2 2 1 4 3 62,572 50% 75%

94% 3 2 0 4 4 85,072 100% 100% 63% 3 2 0 4 4 85,072 100% 100%

99% 4 2 0 4 4 107,990 100% 100% 80% 4 2 0 4 4 107,990 100% 100%

100% 5 2 0 4 4 130,910 100% 100% 91% 5 2 0 4 4 130,910 100% 100%

96% 6 2 0 4 4 153,820 100% 100%

99% 7 2 0 4 4 176,740 100% 100%

100% 10 2 0 4 4 245,490 100% 100%

Opt. results 8 1 15 13 178,205 87.5% 86.7% 76% 8 2 15 13 178,205 75.0% 86.7% 95% 8 1 15 14 246,537 87.5% 93.3% 70% 8 1 15 14 223,621 87.5% 93.3%

Willemain's Model (W)

Item #

Normal Model (N) Empirical Model (E)

25

5.5. Optimization curves: cost vs. service level

In Fig. 6 we show a couple of typical optimization curves for a particular class. The

upper plot corresponds to the exchange curve between target and realized fill rates.

The current policy is represented by a horizontal line. The lower picture shows the

total holding and ordering costs as a function of the realized fill rate for the current

system and for the normal, empirical and Willemain models. Obviously the current

policy is represented by a single point. The plots correspond to class 433 (demand

class 4, medium criticality and 13.6 < price ≤ 169 euros), and the precision of the graphs is 1%.

Figure 6. Optimization curves for class 433.

As we can see from the lower plot, the point associated to the current system lies

above the curves for the three models. Therefore we can achieve higher fill rates with

positive savings for the three models (see numerical results for this class in Table 13

of the Appendix). Notice that the normal curve is less steep than the ones for the

empirical and Willemain models in the optimization region (around a realized fill rate

of 90%), therefore achieving higher savings for this class. This behavior of the normal

model illustrates the better performance of this model over the others for some

relevant classes, which had the overall impact discussed previously. Observe also that

for this class the Willemain curve is above the empirical one, which illustrates the

0 10 20 30 40 50 60 70 80 90 10060

70

80

90

100Class 433: Realized fill rate vs. Target fill rate

target fill rate

Realized fill rate

60 65 70 75 80 85 90 95 1000

2

4

6

8

10x 10

5 Class 433: Total costs vs. Realized fill rate

realized fill rate

Total Costs

Current policy

(E)

(N)

(W)

Current policy

(E) (N)

(W)

26

better performance of the latter with respect to the former. Finally notice that the

normal model needed a higher target fill rate to achieve its optimal realized fill rate

than the other models. From this we can conclude that the empirical and Willemain

models are more accurate in terms of predictability of the service levels. Although in

this case the three models yielded a “good” performance over the current policy, for

other classes the optimization was more difficult, as we showed in Section 5.3.

In Fig. 7 we present the corresponding optimization curves for class 135 (demand

class 1, medium criticality and price > 169 euros). Here we can appreciate the

performance of the Poisson model with respect to the other models. As we can see

from the lower plot the performance curves for the four models are very similar

around the optimization region (realized fill rate of 95%~97%), all of them clearly

outperforming the current policy. From the upper plot we can see again that the

“predictability” of the normal model is the worst of all, achieving higher fill rates for

lower target values than the other models. Notice that for higher fill rates the Poisson

model is the more accurate in this respect. Observe however that in terms of the

optimization performance this does not have a great impact for the normal or the other

models.

Figure 7. Optimization curves for class 135.

0 10 20 30 40 50 60 70 80 90 10090

92

94

96

98

100Class 135: Realized fill rate vs. Target fill rate

target fill rate

realiz

ed fill rate

93 94 95 96 97 98 99 1000

1

2

3

4x 10

6 Class 135: Total Costs vs. Realized fill rate

realized fill rate

Total Costs

Current policy (E)

(P)

(N)

(W)

Current policy

(E)

(N)

(W)

(P)

27

5.6. Accuracy of the models and variability within the classes

As the number of inventory cycles completed by each item inside a given class may

vary greatly, we expect variations inside the classes between the target and the

realized fill rates for the individual items. In order to give an estimation of the

accuracy of the models, in Table 7 we present the average and maximum values of the

fill rate MADs, aggregated for the different demand classes. For the calculation the

current fill rates were used as target values under the ex-post approach. As we can see

from the results, the MAD calculation associated with high demand classes yielded

higher average and maximum values. From the average results we can conclude that

the accuracy of the models varies between 4% and 17% as given by the MADs, which

for the type of system that we are dealing with is acceptable.

Table 7. Average and maximum MADs for target fill rates

Demand class

Average Maximum Average Maximum Average Maximum Average Maximum

1 5.5% 15.2% 5.8% 15.4% 5.5% 15.2% 5.8% 15.2%

4 12.3% 18.2% 13.2% 20.9% 14.0% 23.0%

5 12.9% 24.6% 17.6% 36.3% 15.3% 24.6%

6 4.3% 8.8% 4.4% 8.5% 4.4% 8.5%

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

Another measure of variability is the one provided by the weighted variances of the

realized service levels, as defined in section 4.3. This measure is more related to the

performance of the system than the MAD calculation, as it assesses the variability of

the realized service levels for a given set of parameters within a class. In Table 8 we

give the values of the variances aggregated for the demand classes for the different

models. As we can see from the results the service level variances associated with

high demand classes are higher than for low demand classes. In general, for the same

demand class the variances of realized CSLs and fill rates are within similar ranges

for the different models. Again taking into account the stochastic nature of the system

under study we consider that these values are within acceptable levels.

Table 8. Variances of realized fill rates and CSLs

Demand

class Average Maximum Average Maximum Average Maximum Average Maximum

Variance of CSLs

1 1.2% 3.8% 0.9% 3.2% 1.2% 3.8% 1.3% 3.8%

4 6.9% 20.2% 9.0% 17.8% 9.9% 20.7% 9.9% 20.7%

5 5.3% 12.8% 9.1% 14.5% 8.0% 11.3%6 1.0% 3.0% 1.0% 2.5% 0.9% 2.5%

Variance of fill rates

1 1.1% 4.2% 0.9% 3.2% 1.0% 4.2% 1.2% 4.2%

4 4.6% 10.5% 2.2% 6.2% 4.6% 9.7%

5 8.8% 35.2% 6.4% 24.3% 4.2% 18.2%

6 0.6% 3.2% 0.6% 3.2% 0.6% 3.2%

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

28

5.7. Individual items

Although the optimization of the system has so far been performed at class level,

where target service levels are the same for all items inside the class, it is interesting

to look at the difference between the models for individual items. With this idea we

show in Fig. 8 a number of interesting plots for two typical items belonging to classes

534 and 124. These plots are analogous to the ones shown in Figs. 5 and 6 but at an

item level. The data for these items is given in Table 9. As we can see from the plots

the curves of item #1835 are smoother than the ones of item #7559. Since demands

for the latter are of unit size, the steps in realized service levels and associated costs

are bigger for a given target service level. For some classes these discrete “jumps” in

realized service levels for the items made the optimization very difficult, as it was the

case for class 124. As for item #1835 the normal and Willemain models yielded very

smooth curves in spite of the few observed demands. For the class where this item

belongs this proved to be advantageous for these two models with respect to the

empirical model.

Table 9. Data for items depicted in Fig. 8. Item # 1835, Class 534

L = 31 days (1 month)

Price = 236 euros

Demands 10 24 76

Period 12 49 52

Item # 7559, Class 124

L = 44 days (2 months)

Price = 485 euros

Demands 1 1 1 1 1 1 1 1 1 1 1

Period 6 9 12 13 18 26 29 39 47 49 55

5.8. Optimization using a marginal approach (item level optimization)

As we saw in the analysis presented in Section 5.3, the optimization method based on

the decomposition approach may not find an optimal solution for 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 effect depends on the ratio of most expensive items

to the cheapest ones, we try to minimize it by clustering the items in the price classes

defined for the study. However, as we saw in the optimization results, for some

classes we could not outperform the current policy using the proposed methodology.

In order to overcome this problem, one should optimize the different classes using a

marginal approach. To this end, the optimal re-order point for each item should be

found independently using a measure of the contribution of the item to the overall fill

rate of the class per monetary unit invested. As the procedure considers each item

individually, for large scale systems the associated computation effort may be large

and therefore the implementation of such a method is not feasible. Hence, for these

cases an approach of the type presented in this study may be more suitable.

Accordingly, we present in this section an alternative optimization approach for the

system based on the marginal approach described above, using the fill rate as the

optimization criterion. This procedure is similar to the one presented in Muckstadt

(2005), where a Poisson model is used to optimized the system subject to a budget

constraint. First we introduce the following notation:

(N)

(E)

(W)

Item #1835, Class 534

0 50 1000

50

100Realized fill rate vs. Target fill rate

target fill rate

r fill rate

0 50 1000

50

100

target fill rate

r fill rate

0 50 1000

50

100

target fill rate

r fill rate

20 40 60 80 1000

1

2x 10

4 Tot. Costs vs. r fill rate

r fill rate

Total costs

0 50 1000

2

4x 10

4

r fill rate

Total costs

0 50 1000

1

2

3x 10

4

r fill rate

Total costs

(N)

(E)

(W)

Item #1835, Class 534

0 50 1000

50

100Realized fill rate vs. Target fill rate

target fill rate

r fill rate

0 50 1000

50

100

target fill rate

r fill rate

0 50 1000

50

100

target fill rate

r fill rate

20 40 60 80 1000

1

2x 10

4 Tot. Costs vs. r fill rate

r fill rate

Total costs

0 50 1000

2

4x 10

4

r fill rate

Total costs

0 50 1000

1

2

3x 10

4

r fill rate

Total costs

Item #7559, Class 124

(N)

(E)

(W)

(P)

0 50 10090

95

100Realized fill rate vs. Target fill rate

target fill rate

r fill rate

0 50 10090

95

100

target fill rate

r fill rate

0 50 10090

95

100

target fill rate

r fill rate

0 50 10090

95

100

target fill rate

r fill rate

90 95 1000

2000

4000Total Costs vs. Realized fill rate

r fill rate

Total costs

90 95 1000

1000

2000

r fill rate

Total costs

90 95 1000

5000

r fill rate

Total costs

90 95 100500

1000

1500

r fill rate

Total costs

Item #7559, Class 124

(N)

(E)

(W)

(P)

0 50 10090

95

100Realized fill rate vs. Target fill rate

target fill rate

r fill rate

0 50 10090

95

100

target fill rate

r fill rate

0 50 10090

95

100

target fill rate

r fill rate

0 50 10090

95

100

target fill rate

r fill rate

90 95 1000

2000

4000Total Costs vs. Realized fill rate

r fill rate

Total costs

90 95 1000

1000

2000

r fill rate

Total costs

90 95 1000

5000

r fill rate

Total costs

90 95 100500

1000

1500

r fill rate

Total costs

Figure 8. Plots for typical items.

30

jD : average annual demand of item j.

sj( β ) : reorder point of item j associated with a fill rate β.

)( jj sβ : expected fill rate of item j corresponding to a reorder point sj as given

by the LTD distribution (see Step 2 of method in Section 4.3).

s(k) : the k

th set of given reorder points for a group of items, say [s1, s2,…, sn]

( ))( )(k

p sE β : expected fill rate for class p associated with the kth set of reorder points

for the items inside the class, given by:

( )∑

∑

=

=

⋅

=n

j

j

n

j

j

k

jj

k

p

D

Ds

sE

1

1

)(

)(

)(

)(

β

β .

Note: For illustrative purposes, we implemented this method for some of the classes

for which we could not outperform the current policy using the original methodology.

Optimization method using expected marginal contributions to the class fill rate

1. For each class p, say with n items, consider an initial list s(0) of available reorder

points as given by the corresponding LTD distribution of each item, say for β = 0%:

],...,,[)0()0(

2

)0(

1

)0(

nssss = .

2. For each item j evaluate the expected marginal contribution to the fill rate of the

class per incremental euro invested if sj(k) is increased to sj

(k+1), where sj

(k+1) is the next

available reorder point given by the LTD distribution of the item (in the case of the

normal model this is equivalent to increase the reorder point by one unit). That is,

evaluate

)(

))()(()(

)()1(

)()1(

)1(

k

j

k

jj

k

jj

k

jjjk

jssh

ssDs

−

−=∆

+

+

+ββ

,

where hj is the annual unit holding cost of item j.

3. Increase the reorder point of the item with the highest value of )( )1( +∆ k

js , say item l,

to get a new list of reorder points, given by

],...,,...,,[)()1()(

2

)(

1

)1( k

n

k

l

kkk sssss ++ = .

4. Simulate the system with the list of reorder points given by s(k+1)

and evaluate the

realized CSL, the realized fill rate, the total costs and the 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 are below the current ones,

31

repeat steps (2)-(4) until the service levels are within acceptable levels or until the

total costs for the class exceed the current ones.

In Table 10 we present the results for the optimization of classes 165, 415 and 533

using the above method. We also include the results of the optimization using the

decomposition approach of Section 4.3. As we can see from the results, for all three

classes the models yielded better results with respect to the previous optimization

method, except in the case of the empirical model for classes 165 and 415, which

achieved higher total costs than before. Notice that as we are looking at the optimal

individual reorder points, the new method can take better advantage of the LTD

distributions modeled according to the different methods. However, as for the

previous optimization method, some problems may arise as we are limited by the

range of re-order point values associated with the LTD distribution. This effect is

manifested more clearly in the case of the empirical model, as we observed in the

results.

Table 10. Optimization results for selected classes using the marginal approach

Tot. costs Method Savings Savings Savings Savings

CSL 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

165 316,280 95.4% 95.4% Marginal 232 95.4% 95.4% 232 95.43% 95.43% -3,706 95.4% 95.4% -808 96.1% 96.1%

Decomposition -59 95.4% 95.4% -59 95.40% 95.40% -59 95.4% 95.4% -2,687 96.1% 96.1%

415 880,410 83.2% 81.3% Marginal 52,222 78.8% 88.8% -9,436 72.1% 82.1% -1,705 62.5% 81.3%

Decomposition 19,234 78.8% 87.9% -7,558 53.8% 77.2% -8,027 69.2% 84.8%

533 101,200 74.1% 75.6% Marginal 17,815 68.2% 75.0% 29,939 58.5% 72.9% 29,805 67.2% 74.4%

Decomposition -2,658 60.4% 78.6% -1,673 44.9% 67.6% -3,697 45.5% 73.6%

Item

Class

Current System (C) Normal Model (N) Poisson Model (P) Empirical Model (E) Willemain Model (W)

Service level Service level Service level Service level Service level

6. Conclusions

In this study we present a methodology for the empirical test of different inventory

models with actual data for spare parts, using two different 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

current stock control practices can be improved by using the theoretical models

considered, however using the ex-ante approach the results were less conclusive.

Consequently, total savings of up to 6.4% in holding costs can be achieved by having

a better inventory control. We have to note however, that using rounded-up values for

the lead times generally hurts the realized service levels. A more refined study would

require the use of more accurate values for the lead times. Of course this would also

require more specific information about the demand data, which is often in practice

difficult to obtain.

According to our study, we can derive the following general conclusions on the

inventory control of spare parts:

• We observed erratic high demands partially caused by preventive

maintenance, which needs to be identified for a better stock control of the

parts.

• Spare parts present low to very low demand values (in some cases no demand

observations were observed during 5 years). With so limited data the

application of theoretical models is troublesome.

32

• Predictability of theoretical models is an important issue, as it measures the

expected service level of a policy with given parameters when confronted with

real data. We observed large differences between target and realized service

levels, with maximal values of up to 40%.

• Inventory control packages as SAP R/3 do not provide the right tools for the

control of spare parts, since they only include the cycle service level (CSL) as

service measure. In this case, a more appropriate measure is provided by the

fill rate.

• A theoretical analysis of the type presented in this thesis may be hampered by

non-stationarity of the demand data, as lead time demand patterns may change

over time.

• The information provided on the demand sizes is an important issue, as they

are often not related to the number of items installed. As a consequence of

this, compound-Poisson models are difficult to use.

• Inventory models can save money and improve service levels, even in the ex-

ante approach.

For the models under study we have the following general conclusions:

• Contrary to expectation, the normal model performed very well in the

optimization, being the one that achieved the best overall performance in 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 parts does not generally follow a normal

distribution. We observed that the normal LTD distribution is less sensitive to

changes in service level values, and thus more conservative reorder points are

advised when using this distribution as compared to Poisson, Willeman or the

empirical LTD distributions.

• The Poisson model performed equally or worse than the others for items with

0-1 demands. We consider that the Poisson assumption may not be correct, as

this model seems to overstock in several cases.

• The empirical model yielded a better overall performance than the Willemain

model, with total savings of 1.05 and 0.96 million euros, correspondingly.

• All models outperformed the current system with savings and service levels

about the same order. This can be seen as a validation of the implementation

of the different models.

• For the different criticality, demand and price classes considered, all the

models performed similarly, with only some particular classes where the

normal model yielded higher savings. In this respect we did not observe a

differentiated effect of the use of the different models applied to the various

kinds of classes.

• All the previous conclusions we corroborated by the ex-ante approach.

Acknowledgements

The authors are thankful to Rutger de Mare and Jeroen Hazeu for their support in the

implementation of the methods used in this paper. The research reported in this paper

has been sponsored by the Transumo research program.

33

Appendix

Optimization with stockout costs included

Stockout costs are more difficult to assess in practice than holding or ordering costs,

as many things may happen in a stockout situation. Yet they are central for initial

stocking decisions. These costs were defined by the HQ of the company as a penalty

(in euros) for not having the part available from stock when needed. Accordingly, a

different penalty was assumed for the items depending on their criticality classes, as

follows:

High critical items: 12,500 euros per part short per day

Medium critical items: 2,500 euros per part short per day

Low critical items: 50 per stock out occasion

The previous figures are set by the management based on expert judgment.

Although these costs may occur in practice, when the system experiences shortages of

critical items a number of actions can be taken in order to avoid high stockout costs,

like emergency orders or the use of parts from a nearby stock site. We want to give in

this section an estimation of the effect of stockout costs in the optimization process.

In Table 11 we present the results corresponding to the optimization of class 533

(high demand items, medium criticality and 13.6 < price ≤ 169 euros), using the normal model, with total costs evaluated with and without stockout costs. Notice that

using stockout costs in the optimization leads us to select high values of the

optimization control parameter since this achieves the highest savings. In this example

the system would be optimized for a target fill rate of 100% (99.99%) that achieves a

realized fill rate of 97.4% and with total savings as indicated in Table 11 (last row of

the table). On the other hand, the optimization procedure without stockout costs

would produce optimized results for a target fill rate of 95%, as indicated in bold

numbers in Table 11. In both cases we outperform the current system; however, the

holding costs associated with the former case are much higher. Of course in this case

we aim to increase the service level as much as possible in order to avoid the high

stockout costs experienced by the current system. The question remains as whether

these high values for stockout costs are realistic, which was the main reason for not

including them in our previous analysis.

Table 11. Optimization results for class 533

Total costs

(without

stockout costs)

Total costs

(with stockout

costs) CSL fill rate

Target fill rates

(optimization

step)

Savings

without

stockout costs

Savings with

stockout

costs rCSL r fill rate

101,200 256,750,000 74.1% 75.6% 90% 15,049 3,165,000 51.9% 70.1%

101,200 256,750,000 74.1% 75.6% 91% 12,830 14,263,000 51.9% 71.3%

101,200 256,750,000 74.1% 75.6% 92% 10,621 25,061,000 53.5% 72.5%

101,200 256,750,000 74.1% 75.6% 93% 7,702 38,333,000 57.2% 73.9%

101,200 256,750,000 74.1% 75.6% 94% 4,846 52,130,000 58.3% 75.3%

101,200 256,750,000 74.1% 75.6% 95% 1,689 67,052,000 59.4% 76.8%

101,200 256,750,000 74.1% 75.6% 96% -2,658 84,522,000 60.4% 78.6%

101,200 256,750,000 74.1% 75.6% 97% -7,395 104,240,000 64.7% 80.7%

101,200 256,750,000 74.1% 75.6% 98% -14,354 127,640,000 67.4% 83.3%

101,200 256,750,000 74.1% 75.6% 99% -25,138 160,920,000 73.8% 87.1%

101,200 256,750,000 74.1% 75.6% 99.99% -84,397 240,590,000 93.0% 97.4%

Current Method (C) Normal Model (N)

34

Effects of the filtering process

In this section we want to investigate in more detail the effect of applying the large

demand filter described in Section 4.2.1. To appreciate better this effect we also

performed the optimization of the system without the filter, that is, with all demands

included. The results are summarized in Table 12. As we can see from the results the

system achieved similar fill rates and cycle service levels, but with associated lower

savings.

Table 12. Summary of optimization results without demand filter for high demands

Costs Savings Savings Savings Savings

tCSL tPFR (N) vs. (C) rCSL rPFR (P) vs. (C) rCSL rPFR (E) vs. (C) rCSL rPFR (W) vs. (C) rCSL rPFR

Summary by demand class

Demand class

1 2,226 3,693,400 95.6% 95.6% 774,220 96.2% 96.8% 703,250 96.8% 96.8% 762,330 96.3% 96.9% 770,490 96.0% 96.7%

4 5,185 9,473,100 83.9% 85.2% 401,760 81.3% 90.3% 187,840 79.8% 91.3% 84,126 77.8% 89.2%

5 608 688,220 74.0% 82.5% 26,913 67.3% 88.7% -24,994 34.7% 79.7% -33,776 38.3% 81.0%

6 475 1,676,100 96.5% 96.6% 92,713 96.3% 97.3% 104,950 96.3% 97.3% 81,393 96.5% 97.4%

Summary by criticality class

Criticality class

1 493 1,704,700 88.8% 87.8% 23,423 86.2% 91.0% 3,541 92.9% 92.9% -5,308 76.2% 87.3% -5,903 79.0% 88.2%

2 447 277,760 93.5% 94.6% 39,920 92.6% 96.5% 1,215 99.8% 99.8% 26,134 90.4% 97.3% 20,699 90.3% 96.2%

3 3,190 9,165,200 89.8% 89.5% 845,780 88.3% 92.6% 648,930 97.5% 97.5% 797,880 86.5% 93.4% 707,530 85.5% 92.2%

4 1,395 906,320 92.0% 91.9% 108,840 90.2% 94.5% 20,420 97.2% 97.0% 79,618 90.8% 95.4% 52,615 89.4% 94.1%

5 372 634,320 90.8% 89.8% 61,002 90.4% 93.4% 1,060 96.1% 96.1% 30,051 88.6% 94.0% 46,633 86.9% 92.7%

6 2,597 2,842,500 78.7% 84.0% 216,650 76.0% 89.9% 28,077 95.7% 95.6% 101,750 70.0% 89.1% 80,660 68.2% 87.4%

Summary by price class

Price class

2 2,223 529,470 82.0% 88.6% 172,090 79.9% 96.8% -5,415 97.9% 97.9% 160,830 74.8% 95.3% 163,790 71.0% 94.6%

3 3,266 1,609,500 87.5% 87.7% 241,060 84.2% 90.7% 25,627 98.2% 98.2% 95,707 83.2% 92.4% 64,196 83.3% 91.1%

4 2,487 3,636,500 90.0% 89.1% 200,020 89.9% 90.3% 54,393 95.5% 95.4% 72,772 86.2% 89.9% 50,236 85.6% 88.2%

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% 89.9% 624,010 83.8% 89.3%

Total 8,494 15,530,701 87.0% 88.4% 1,295,608 85.0% 92.2% 703,246 96.8% 96.8% 1,030,127 81.8% 92.3% 902,236 80.8% 91.0%

Poisson Model (P) Empirical Model (E) Willemain Model (W)

Service level Service level Service level Service level Service level

# of

items

Current System (C) Normal Model (N)

Optimization using the CSL as control parameter

Although we optimize the system using the fill rate as target service level, which is of

more relevance in industrial applications, the implementation of the optimization

method using this measure as control parameter is more complicated than using the

CSL. This is because of the calculation of the expected units short in a replenishment

cycle in the evaluation of the reorder points (step 2 of the optimization method given

in Section 4.3). Taking into account this and the fact that in many industrial systems

like SAP the preferred measure of service level is the CSL (and some times the only

one available), we want to give in this section evidence as to what extent this affects

the optimization process. Accordingly, we performed an optimization of the whole

system using the CSL to evaluate the reorder points although the system was

optimized against the current fill rate as before. For this we used the ex-post approach.

We give the full list of results in Table 15.

As we can see from the results, the different classes yielded higher variation in

terms of service levels and fill rates, especially for the empirical and Willemain

models, for which the maximum MADs were of 45%. Moreover, although the normal

and Poisson models achieved higher total savings than before, the corresponding

empirical and Willemain models yielded much lower savings. We conclude that lower

results can be derived in these cases, as we are not using the same measure for both

the evaluation of parameters and the optimization criterion as we did before.

35

Table 13. Optimization results for the system using the ex-post approach

Tot. Costs Savings Savings Savings Savings

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

113 47 9,194 94.3% 94.4% 1,683 93.6% 96.5% 157 95.8% 95.8% 1,683 93.6% 96.5% 1,683 93.6% 96.5%

114 60 50,336 87.6% 87.8% 2,671 89.3% 90.0% 2,498 89.4% 89.4% 2,671 89.3% 90.0% 2,671 89.3% 90.0%

115 23 278,720 92.0% 92.0% 886 96.0% 96.0% 886 96.0% 96.0% 886 96.0% 96.0% 19,924 94.0% 94.0%

122 17 2,712 100.0% 100.0% 2,101 100.0% 100.0% -122 100.0% 100.0% 2,101 100.0% 100.0% 2,101 100.0% 100.0%

123 43 12,908 100.0% 100.0% 3,045 100.0% 100.0% 2,175 100.0% 100.0% 3,667 100.0% 100.0% 2,711 100.0% 100.0%

124 36 44,260 98.9% 98.9% -313 99.4% 99.4% -838 99.4% 99.4% -313 99.4% 99.4% -551 99.4% 99.4%

132 109 11,882 99.6% 99.7% 9,341 100.0% 100.0% -1,857 97.8% 97.8% 9,341 100.0% 100.0% 9,341 100.0% 100.0%

133 356 93,357 96.6% 96.8% 26,254 96.2% 97.9% 11,838 97.9% 97.9% 29,178 95.9% 97.9% 27,010 96.0% 97.9%

134 461 413,830 94.8% 94.8% 10,447 96.2% 96.3% 5,392 97.3% 97.3% 13,046 96.9% 96.9% 14,412 95.7% 95.8%

135 77 1,821,500 94.1% 94.2% 633,560 95.9% 95.9% 633,560 95.9% 95.9% 613,930 95.9% 95.9% 599,000 95.9% 95.9%

142 66 7,245 100.0% 100.0% 5,307 100.0% 100.0% -2,667 98.5% 98.5% 5,307 100.0% 100.0% 5,307 100.0% 100.0%

143 165 44,549 97.5% 97.8% 13,592 97.5% 98.9% 4,016 99.4% 99.4% 14,796 97.5% 98.9% 13,326 97.5% 98.9%

144 159 141,200 93.6% 92.3% 21,488 93.8% 93.5% 20,364 94.5% 94.0% 21,643 94.0% 93.8% 17,418 93.3% 93.5%

145 9 35,389 96.2% 96.2% -1,294 96.2% 96.2% -1,294 96.2% 96.2% -4,027 100.0% 100.0% -4,228 96.2% 96.2%

152 17 1,622 100.0% 100.0% 1,136 100.0% 100.0% -245 100.0% 100.0% 1,136 100.0% 100.0% 1,136 100.0% 100.0%

153 39 10,519 94.2% 94.8% 4,108 90.9% 96.1% 1,735 96.1% 96.1% 4,108 90.9% 96.1% 4,108 90.9% 96.1%

154 76 69,500 93.1% 93.1% 4,021 94.8% 94.8% 4,021 94.8% 94.8% 4,968 94.4% 94.4% 795 94.0% 94.0%

155 9 77,338 95.5% 95.5% -4,452 100.0% 100.0% -4,452 100.0% 100.0% -4,452 100.0% 100.0% -1,532 95.5% 95.5%

162 49 5,880 100.0% 100.0% 4,442 100.0% 100.0% -524 95.9% 95.9% 4,442 100.0% 100.0% 4,442 100.0% 100.0%

163 157 40,285 98.1% 98.1% 12,549 98.5% 99.2% 5,705 98.8% 98.8% 11,052 98.9% 99.4% 11,659 98.5% 99.2%

164 204 204,900 92.1% 92.0% 23,708 93.4% 93.3% 22,956 93.3% 93.2% 27,226 93.1% 93.0% 23,452 93.2% 93.2%

165 47 316,280 95.4% 95.4% -59 95.4% 95.4% -59 95.4% 95.4% -59 95.4% 95.4% -2,687 96.1% 96.1%

412 49 6,942 84.8% 84.4% 3,018 89.3% 99.0% 3,043 82.1% 98.5% 2,896 85.7% 98.6%

413 120 47,285 88.8% 86.2% -1,028 85.7% 89.7% -201 61.5% 82.6% -1,118 69.4% 84.3%

414 85 173,060 86.5% 85.9% -3,296 80.4% 86.3% -416 70.2% 80.1% -340 70.2% 80.1%

415 39 880,410 83.2% 81.3% 19,234 78.8% 87.9% -7,558 53.8% 77.2% -8,027 69.2% 84.8%

422 112 19,791 95.3% 98.2% 11,836 96.9% 99.7% 12,777 93.8% 99.9% 12,778 90.6% 99.3%

423 129 63,598 94.8% 93.5% 15,038 91.6% 95.9% 12,245 91.0% 97.4% 7,650 91.0% 96.8%

424 60 90,761 92.3% 92.0% 4,808 93.8% 93.0% -636 89.0% 93.2% -2,696 84.5% 88.2%

432 411 61,073 88.6% 89.2% 34,385 84.7% 98.3% 31,649 91.5% 99.6% 30,899 92.0% 99.3%

433 730 366,720 87.9% 87.6% 69,228 83.4% 89.0% 19,864 85.3% 93.8% 9,795 83.7% 90.9%

434 519 1,034,500 87.0% 83.9% 40,986 85.0% 85.2% -7,146 73.8% 83.8% -6,714 74.4% 81.0%

435 119 4,200,600 84.1% 75.9% -80,091 78.7% 83.5% -13,092 65.0% 79.7% -25,708 65.4% 77.3%

442 284 39,537 93.4% 94.8% 24,393 88.9% 99.2% 24,153 88.9% 99.5% 23,351 93.3% 99.6%

443 388 164,630 90.9% 90.2% 35,732 86.5% 91.9% 21,898 87.1% 94.6% 19,096 86.2% 91.5%

444 199 261,640 87.2% 87.6% 12,656 87.4% 89.2% -929 83.2% 88.5% -2,833 79.7% 85.0%

452 28 5,593 90.0% 91.6% 2,870 92.9% 99.6% 3,194 85.7% 97.5% 3,041 92.9% 98.3%

453 76 34,290 86.0% 80.9% 4,197 83.2% 90.1% -2,058 84.2% 93.3% -1,198 81.5% 92.1%

454 80 147,890 86.7% 85.7% 12,366 86.0% 87.8% -3,589 79.7% 88.6% -1,147 77.7% 85.7%

455 14 210,480 93.2% 94.2% 34,719 93.0% 95.7% 24,706 93.0% 95.7% 47,314 93.0% 95.7%

462 590 112,880 68.7% 84.2% 60,467 64.5% 95.9% 55,350 73.9% 98.2% 59,157 66.1% 96.3%

463 798 310,520 76.4% 80.7% 51,941 72.5% 86.3% 23,856 72.0% 87.9% 11,960 72.6% 86.2%

464 314 649,760 83.8% 83.7% 58,369 84.1% 84.8% 7,215 82.8% 86.0% -7,557 79.1% 80.1%

465 41 587,700 82.4% 79.5% -3,078 75.7% 84.6% -1,471 70.6% 82.5% -597 65.4% 77.1%

513 11 83,779 76.4% 88.2% -3,102 71.7% 88.4% -7,057 35.8% 71.4% -1,469 34.0% 74.1%

522 25 15,743 66.3% 90.2% 1,761 92.7% 99.0% 5,519 63.6% 92.1% 5,203 63.6% 91.4%

523 6 8,629 86.4% 54.9% -333 68.0% 59.8% -7,221 28.0% 82.8% -5,964 28.0% 80.7%

532 144 82,834 80.2% 89.7% 14,346 86.5% 97.8% 15,984 78.8% 97.2% 15,239 80.3% 97.5%

533 43 101,200 74.1% 75.6% -2,658 60.4% 78.6% -1,673 44.9% 67.6% -3,697 45.5% 73.6%

534 9 56,697 69.4% 75.0% 6,403 69.0% 76.6% -620 71.4% 77.1% 3,462 66.7% 77.5%

542 35 16,182 86.3% 93.2% 4,249 89.1% 96.2% 4,494 89.1% 98.2% 4,482 82.8% 97.3%

543 9 105,480 75.8% 86.4% -9,147 78.3% 85.8% -12,775 30.4% 76.0% -25,708 34.8% 70.5%

562 287 134,440 77.0% 86.9% -624 58.9% 90.4% -11,618 6.4% 74.2% -8,168 6.4% 73.7%

563 39 80,942 71.4% 70.5% 20,476 67.6% 72.5% -443 37.5% 73.7% -1,176 42.6% 77.0%

613 14 2,161 96.9% 96.9% -73 95.8% 96.9% -73 95.8% 96.9% -73 95.8% 96.9%

614 28 43,355 97.1% 97.2% 747 98.4% 98.6% 2,347 96.8% 97.2% 747 98.4% 98.6%

615 17 129,460 100.0% 100.0% -312 94.3% 95.2% -399 97.1% 97.6% -312 94.3% 95.2%

623 8 3,446 100.0% 100.0% 1,025 100.0% 100.0% 1,203 100.0% 100.0% 1,296 100.0% 100.0%

624 11 15,405 98.0% 98.0% -1,087 97.7% 98.0% -723 95.3% 96.0% -723 95.3% 96.0%

633 52 14,778 98.1% 98.2% -95 98.2% 98.8% 417 98.2% 98.8% 871 99.1% 99.4%

634 91 117,410 95.7% 95.9% 4,430 96.3% 97.0% 2,773 96.3% 97.0% 587 96.3% 97.0%

635 69 788,730 93.2% 93.3% 86,946 92.3% 94.8% 88,500 93.0% 95.2% 68,706 93.0% 95.2%

643 36 9,741 97.8% 98.5% 2,750 100.0% 100.0% 2,780 100.0% 100.0% 2,250 100.0% 100.0%

644 39 47,642 97.7% 97.8% 3,602 98.7% 98.9% 4,792 98.7% 98.9% 4,009 98.7% 98.9%

645 6 32,143 91.7% 92.3% -6,158 100.0% 100.0% 1,316 100.0% 100.0% -2,178 100.0% 100.0%

654 26 34,420 95.3% 95.5% -497 94.7% 95.5% -497 94.7% 95.5% -497 94.7% 95.5%

655 7 42,668 100.0% 100.0% 2,534 100.0% 100.0% 2,534 100.0% 100.0% 2,534 100.0% 100.0%

664 30 40,800 97.5% 97.6% -1,099 96.2% 97.6% -21 96.2% 97.6% -1,099 96.2% 97.6%

665 41 353,910 96.6% 95.9% 0 93.6% 95.9% 0 93.6% 95.9% -14,991 96.2% 97.2%

Total 8,494 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% 92.7% 964,832 81.5% 91.4%

Summary by demand class

Dem. class

1 2,226 3,693,400 95.6% 95.6% 774,220 96.2% 96.8% 703,250 96.8% 96.8% 762,330 96.3% 96.9% 751,500 96.0% 96.6%

4 5,185 9,469,700 84.3% 86.1% 408,750 81.1% 90.7% 202,860 80.0% 91.7% 170,000 79.1% 89.7%

5 608 685,920 77.2% 85.7% 31,370 70.0% 90.2% -15,411 37.4% 81.4% -17,797 37.8% 81.7%

6 475 1,676,100 96.5% 96.6% 92,713 96.3% 97.3% 104,950 96.3% 97.3% 61,126 96.7% 97.5%

Summary by criticality class

Crit. class

1 493 1,704,700 88.9% 88.3% 20,428 87.3% 91.8% 3,541 92.9% 92.9% -5,074 76.1% 87.7% 16,583 79.5% 88.7%

2 447 277,250 94.1% 95.3% 37,882 95.0% 97.1% 1,215 99.8% 99.8% 28,617 91.2% 97.5% 21,805 89.8% 96.4%

3 3,190 9,165,100 90.1% 89.8% 853,480 88.4% 92.7% 648,930 97.5% 97.5% 802,150 86.9% 93.5% 743,200 86.6% 92.2%

4 1,395 905,370 92.6% 92.9% 107,170 90.7% 94.9% 20,420 97.2% 97.0% 83,448 90.0% 95.7% 54,293 89.9% 94.2%

5 372 634,320 90.8% 89.8% 61,002 90.4% 93.4% 1,060 96.1% 96.1% 30,051 88.6% 94.0% 54,554 88.0% 92.9%

6 2,597 2,838,300 79.5% 85.4% 227,090 75.3% 90.6% 28,077 95.7% 95.6% 115,530 70.8% 89.9% 74,396 68.8% 88.2%

Summary by price class

Price class

2 2,223 524,350 83.2% 90.0% 179,030 79.4% 97.0% -5,415 97.9% 97.9% 166,870 75.1% 95.7% 171,200 73.7% 95.0%

3 3,266 1,608,000 87.9% 88.5% 245,180 84.7% 91.3% 25,627 98.2% 98.2% 115,240 83.3% 92.7% 73,012 83.3% 91.4%

4 2,487 3,637,300 90.2% 89.4% 200,410 90.1% 90.7% 54,393 95.5% 95.4% 71,791 86.7% 90.4% 43,397 85.6% 88.4%

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% 89.9% 677,220 84.0% 89.3%

Item

class

# of

items

Current System (C)

Service level Service level

Normal Model (N) Poisson Model (P)

Service level Service level Service level

Empirical Model (E) Willemain Model (W)

36

Table 14. Optimization results for the system using the ex-ante approach

Costs Savings Savings Savings Savings

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

113 10 1,148 95.5% 95.5% -7 100.0% 100.0% -49 95.5% 95.5% -7 100.0% 100.0% -56 100.0% 100.0%

114 18 6,686 81.1% 81.6% 513 90.6% 92.1% -419 81.6% 81.6% 513 90.6% 92.1% -555 93.8% 94.7%

115 1 1,030 100.0% 100.0% 0 100.0% 100.0% 0 100.0% 100.0% 0 100.0% 100.0% 0 100.0% 100.0%

122 8 675 100.0% 100.0% 559 100.0% 100.0% -14 100.0% 100.0% 559 100.0% 100.0% 559 100.0% 100.0%

123 26 3,435 100.0% 100.0% 1,938 100.0% 100.0% -44 100.0% 100.0% 1,938 100.0% 100.0% 1,938 100.0% 100.0%

124 20 10,834 98.4% 98.5% 562 100.0% 100.0% 1,555 100.0% 100.0% 1,597 100.0% 100.0% -167 98.0% 98.5%

132 38 1,800 100.0% 100.0% 1,387 100.0% 100.0% -682 97.0% 97.0% 1,387 100.0% 100.0% 1,387 100.0% 100.0%

133 126 17,510 96.7% 96.9% 6,618 95.4% 98.4% 686 98.0% 98.0% 6,826 94.3% 98.0% 6,239 94.3% 98.0%

134 160 64,218 94.0% 94.0% 11,686 94.8% 95.1% 7,638 95.1% 95.1% 10,819 94.8% 95.1% 6,328 95.9% 96.1%

135 20 251,940 93.1% 93.5% 48,588 96.8% 96.8% 48,588 96.8% 96.8% 57,403 96.8% 96.8% -489 93.5% 93.5%

142 29 1,619 100.0% 100.0% 1,161 100.0% 100.0% -1,115 95.9% 95.9% 1,161 100.0% 100.0% 1,161 100.0% 100.0%

143 73 10,227 97.3% 97.7% 4,001 98.0% 99.4% -453 100.0% 100.0% 4,086 98.0% 99.4% 3,594 95.9% 98.8%

144 70 27,042 95.1% 95.2% 4,291 96.3% 96.6% -686 95.9% 95.9% 4,291 96.3% 96.6% 2,496 96.3% 96.6%

145 3 4,573 83.3% 83.3% -2,405 100.0% 100.0% -2,405 100.0% 100.0% -2,405 100.0% 100.0% -1,334 83.3% 83.3%

152 3 141 100.0% 100.0% 83 100.0% 100.0% -20 100.0% 100.0% 83 100.0% 100.0% 83 100.0% 100.0%

153 18 1,993 100.0% 100.0% 852 100.0% 100.0% 0 100.0% 100.0% 852 100.0% 100.0% 852 100.0% 100.0%

154 23 9,840 97.4% 97.4% -667 97.2% 97.4% -448 97.4% 97.4% -352 97.2% 97.4% -1,707 97.2% 97.4%

155 1 4,221 100.0% 100.0% 2,867 100.0% 100.0% 2,867 100.0% 100.0% 2,867 100.0% 100.0% 2,867 100.0% 100.0%

162 15 1,221 100.0% 100.0% 943 100.0% 100.0% -87 85.7% 85.7% 943 100.0% 100.0% 943 100.0% 100.0%

163 39 6,267 97.3% 97.6% 3,090 100.0% 100.0% 985 98.8% 98.8% 3,090 100.0% 100.0% 3,090 100.0% 100.0%

164 85 40,389 91.5% 91.5% 3,142 92.5% 93.2% 2,869 93.2% 93.2% 5,058 91.8% 92.6% 4,791 91.8% 92.6%

165 14 37,848 91.4% 91.4% 4,810 94.3% 94.3% 4,810 94.3% 94.3% -4,299 91.4% 91.4% 4,810 94.3% 94.3%

412 17 1,476 80.8% 84.3% 669 66.7% 98.6% 761 100.0% 100.0% 416 100.0% 100.0%

413 41 8,479 85.9% 84.1% 695 81.4% 91.9% 349 86.0% 93.6% -29 69.8% 89.4%

414 24 19,921 88.4% 86.7% -559 72.7% 78.8% -43 84.1% 81.4% -220 75.0% 73.5%

415 10 104,870 85.7% 74.4% -1,497 78.6% 82.1% -15,647 71.4% 87.2% -9,334 64.3% 82.1%

422 82 6,297 97.0% 99.2% 3,800 100.0% 100.0% 3,954 66.7% 98.7% 4,202 100.0% 100.0%

423 90 20,815 96.0% 96.6% 1,536 97.0% 99.1% 5,053 92.9% 97.8% 459 94.9% 98.0%

424 37 24,348 92.6% 93.9% 2,466 93.5% 96.4% -1,760 90.9% 93.9% -423 87.0% 92.1%

432 231 16,018 90.7% 91.6% 8,233 82.8% 97.7% 8,330 82.8% 97.7% 8,471 89.7% 98.0%

433 363 80,282 89.5% 88.3% 10,956 84.8% 91.6% 3,097 84.6% 93.1% -26 78.6% 90.1%

434 247 243,860 86.5% 83.1% -344 77.4% 78.5% -353 69.7% 75.6% -1,523 68.1% 72.2%

435 49 816,360 87.8% 90.1% -27,678 78.6% 82.1% -18,160 73.8% 78.4% -38,595 71.4% 77.2%

442 198 12,536 93.6% 95.5% 7,838 100.0% 100.0% 8,096 100.0% 100.0% 8,340 100.0% 100.0%

443 246 51,392 92.8% 92.3% 11,714 88.1% 94.4% 8,931 88.5% 94.4% 5,584 88.5% 93.5%

444 124 69,464 89.7% 92.2% -1,919 88.1% 89.9% -740 85.0% 87.1% -627 80.8% 83.7%

452 14 655 92.3% 95.3% 415 100.0% 100.0% 415 100.0% 100.0% 415 100.0% 100.0%

453 40 8,238 91.3% 89.6% 2,235 88.9% 94.8% 841 88.9% 96.4% 858 88.9% 95.3%

454 41 27,507 93.8% 96.0% -553 85.3% 89.4% -1,875 85.3% 89.4% -940 78.7% 86.8%

455 2 9,770 100.0% 100.0% 0 100.0% 100.0% -1,765 100.0% 100.0% 0 100.0% 100.0%

462 343 34,861 64.3% 80.4% 20,357 63.6% 98.8% 19,640 61.4% 98.8% 19,118 68.2% 99.1%

463 460 94,819 75.3% 80.5% 17,881 71.6% 87.9% 3,795 71.4% 88.1% -759 71.6% 87.6%

464 198 195,970 86.9% 84.9% -711 82.8% 81.5% -1,157 79.3% 78.0% -1,627 74.9% 73.3%

465 16 127,180 79.3% 75.0% -2,150 67.9% 71.4% -10,862 64.3% 73.8% -12,382 67.9% 76.2%

513 7 7,632 89.5% 97.1% -443 73.7% 93.6% -337 57.9% 78.7% -63 57.9% 82.9%

522 24 6,481 75.0% 92.2% 965 94.7% 98.4% 2,333 73.7% 95.2% 1,898 78.9% 97.9%

523 5 4,542 88.9% 54.2% 1,493 75.0% 33.4% 1,436 87.5% 35.6% 400 87.5% 37.3%

532 109 19,019 78.8% 89.7% 1,637 84.1% 96.8% 2,970 78.0% 95.5% 1,641 81.7% 96.0%

533 29 28,546 78.5% 92.2% -865 64.1% 87.0% -448 40.6% 74.2% -239 42.2% 76.1%

534 6 19,485 80.0% 79.6% -712 56.3% 56.0% 683 62.5% 62.4% -38 62.5% 49.7%

542 26 5,832 86.8% 90.7% 2,369 80.0% 91.5% 2,386 80.0% 92.3% 1,578 80.0% 93.6%

543 7 9,027 84.4% 96.6% -730 100.0% 100.0% -2,246 40.0% 95.4% -1,670 40.0% 95.4%

562 235 35,786 72.9% 93.5% 8,166 82.7% 96.8% 10,127 69.2% 95.3% 8,676 77.9% 95.9%

563 27 16,405 75.8% 67.8% -9 48.1% 63.1% -525 23.1% 50.9% -64 21.2% 51.2%

614 4 1,258 100.0% 100.0% 89 100.0% 100.0% 89 100.0% 100.0% 89 100.0% 100.0%

623 5 1,055 100.0% 100.0% 290 100.0% 100.0% 290 100.0% 100.0% 203 100.0% 100.0%

624 4 3,432 100.0% 100.0% -732 77.8% 81.8% -732 77.8% 81.8% -732 77.8% 81.8%

633 22 3,387 100.0% 100.0% 740 100.0% 100.0% 674 100.0% 100.0% 609 100.0% 100.0%

634 42 30,132 98.8% 98.9% -343 93.8% 95.5% -310 93.8% 95.5% -287 93.8% 95.5%

635 15 108,440 100.0% 100.0% -8,518 92.3% 96.0% -9,666 92.3% 96.0% -1,713 92.3% 96.0%

643 14 1,819 100.0% 100.0% 681 100.0% 100.0% 681 100.0% 100.0% 583 100.0% 100.0%

644 27 15,882 97.0% 97.1% 2,040 98.0% 98.5% 2,298 98.0% 98.5% 3,000 98.0% 98.5%

645 1 3,025 100.0% 100.0% 0 100.0% 100.0% 0 100.0% 100.0% 0 100.0% 100.0%

654 14 9,630 95.7% 95.8% -252 94.1% 95.8% -85 94.1% 95.8% -632 94.1% 95.8%

655 3 12,883 100.0% 100.0% 1,152 100.0% 100.0% 1,152 100.0% 100.0% 1,152 100.0% 100.0%

664 15 11,295 100.0% 100.0% 1,189 100.0% 100.0% 1,189 100.0% 100.0% 1,189 100.0% 100.0%

665 12 43,899 100.0% 100.0% 1,849 100.0% 100.0% 1,849 100.0% 100.0% -6,570 100.0% 100.0%

Total 4,326 2,878,665 86.4% 89.7% 157,452 84.6% 93.0% 63,575 96.3% 96.3% 121,118 81.7% 92.3% 27,217 82.6% 91.5%

Summary by demand class

Dem. class

1 800 504,650 95.6% 95.7% 94,010 96.5% 97.2% 63,575 96.3% 96.3% 96,409 96.2% 97.1% 36,831 96.1% 97.2%

4 2,873 1,975,100 84.8% 87.3% 53,384 81.3% 91.5% 10,900 79.1% 91.1% -18,623 79.6% 89.6%

5 475 152,750 76.3% 90.4% 11,873 80.1% 92.9% 16,379 67.2% 90.1% 12,119 72.6% 90.8%

6 178 246,130 98.9% 99.0% -1,816 96.6% 97.6% -2,571 96.6% 97.6% -3,110 96.6% 97.6%

Summary by criticality class

Crit. class

1 132 152,500 86.5% 85.7% -540 80.7% 90.7% -468 87.0% 87.0% -14,322 87.1% 91.5% -9,752 80.3% 88.9%

2 301 81,914 94.8% 96.5% 12,877 97.2% 97.8% 1,496 100.0% 100.0% 14,668 85.1% 96.6% 8,338 94.6% 96.9%

3 1,457 1,701,000 90.0% 90.5% 51,385 85.6% 91.7% 56,230 96.5% 96.5% 63,253 83.1% 91.1% -18,236 82.7% 89.9%

4 818 212,440 93.3% 94.3% 29,042 93.4% 96.1% -4,659 97.7% 97.7% 26,538 92.6% 95.7% 22,704 91.7% 94.8%

5 159 84,877 94.8% 95.2% 6,130 92.5% 95.2% 2,398 98.7% 98.7% 2,132 92.5% 95.6% 2,947 90.8% 94.7%

6 1,459 645,930 76.3% 84.6% 58,558 75.6% 91.4% 8,578 94.0% 94.0% 28,848 71.8% 90.5% 21,215 74.2% 89.9%

Summary by price class

Price class

2 1,372 144,420 81.1% 90.5% 58,581 82.9% 98.3% -1,919 95.2% 95.2% 63,142 77.6% 97.8% 58,886 84.3% 98.3%

3 1,648 377,020 87.5% 88.7% 62,667 83.9% 92.3% 1,126 98.8% 98.8% 38,379 82.6% 92.2% 21,502 80.9% 91.2%

4 1,159 831,190 90.7% 90.0% 19,185 87.4% 88.1% 10,508 94.5% 94.5% 19,131 84.9% 86.6% 8,416 83.0% 84.5%

5 147 1,526,000 90.6% 90.2% 17,017 86.1% 88.3% 53,860 96.3% 96.3% 466 83.4% 87.4% -61,587 82.0% 86.4%

Item

class

# of

items

Current System (C)

Service level Service level

Normal Model (N) Poisson Model (P)

Service level Service level Service level

Empirical Model (E) Willemain Model (W)

37

Table 15. Optimization results using CSL for the evaluation of reorder points

Tot. Costs Savings Savings Savings Savings

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

113 47 9,194 94.3% 94.4% 1,683 93.6% 96.5% 232 95.8% 95.8% 1,683 93.6% 96.5% 1,683 93.6% 96.5%

114 60 50,336 87.6% 87.8% 2,671 89.3% 90.0% 2,498 89.4% 89.4% 2,671 89.3% 90.0% 2,671 89.3% 90.0%

115 23 278,720 92.0% 92.0% 886 96.0% 96.0% 886 96.0% 96.0% -12,043 92.0% 92.0% 19,924 94.0% 94.0%

122 17 2,712 100.0% 100.0% 2,101 100.0% 100.0% -14 98.6% 98.6% 2,101 100.0% 100.0% 2,101 100.0% 100.0%

123 43 12,908 100.0% 100.0% 5,860 100.0% 100.0% 2,424 100.0% 100.0% 5,809 100.0% 100.0% 5,860 100.0% 100.0%

124 36 44,260 98.9% 98.9% -744 99.4% 99.4% -1,070 99.4% 99.4% -744 99.4% 99.4% -2,450 99.4% 99.4%

132 109 11,882 99.6% 99.7% 9,341 100.0% 100.0% -1,857 97.8% 97.8% 9,341 100.0% 100.0% 9,341 100.0% 100.0%

133 356 93,357 96.6% 96.8% 28,734 97.1% 98.5% 13,417 98.1% 98.1% 29,301 96.4% 98.1% 29,135 96.0% 97.9%

134 461 413,830 94.8% 94.8% 38,016 95.8% 95.8% 36,030 95.9% 95.8% 20,305 96.1% 96.0% 7,523 95.6% 95.6%

135 77 1,821,500 94.1% 94.2% 633,560 95.9% 95.9% 633,560 95.9% 95.9% 588,220 95.4% 95.4% 565,660 95.4% 95.4%

142 66 7,245 100.0% 100.0% 5,307 100.0% 100.0% -2,667 98.5% 98.5% 5,307 100.0% 100.0% 5,307 100.0% 100.0%

143 165 44,549 97.5% 97.8% 15,873 97.8% 99.0% 5,367 99.4% 99.4% 15,938 97.5% 98.9% 15,234 97.5% 98.9%

144 159 141,200 93.6% 92.3% 23,032 93.6% 93.3% 22,951 93.8% 93.3% 17,382 94.5% 93.5% 13,436 93.8% 93.7%

145 9 35,389 96.2% 96.2% -1,294 96.2% 96.2% -1,294 96.2% 96.2% -1,294 96.2% 96.2% -4,228 96.2% 96.2%

152 17 1,622 100.0% 100.0% 1,136 100.0% 100.0% -245 100.0% 100.0% 1,136 100.0% 100.0% 1,136 100.0% 100.0%

153 39 10,519 94.2% 94.8% 4,108 90.9% 96.1% 1,384 97.4% 97.4% 4,108 90.9% 96.1% 4,108 90.9% 96.1%

154 76 69,500 93.1% 93.1% 4,669 94.4% 94.4% 1,734 94.8% 94.8% 3,279 94.4% 94.4% -794 94.0% 94.0%

155 9 77,338 95.5% 95.5% -4,452 100.0% 100.0% -4,452 100.0% 100.0% -1,431 95.5% 95.5% -6,669 95.5% 95.5%

162 49 5,880 100.0% 100.0% 4,442 100.0% 100.0% -524 95.9% 95.9% 4,442 100.0% 100.0% 4,442 100.0% 100.0%

163 157 40,285 98.1% 98.1% 11,599 98.5% 99.2% 6,884 99.4% 99.4% 11,125 98.5% 99.2% 9,535 98.5% 99.2%

164 204 204,900 92.1% 92.0% 24,092 93.4% 93.5% 22,956 93.3% 93.2% 25,633 93.2% 93.0% 16,257 93.2% 93.0%

165 47 316,280 95.4% 95.4% -59 95.4% 95.4% -59 95.4% 95.4% -59 95.4% 95.4% -2,687 96.1% 96.1%

412 49 6,942 84.8% 84.4% 3,256 82.1% 98.4% 2,749 96.4% 99.9% 2,786 85.7% 98.9%

413 120 47,285 88.8% 86.2% 2,361 85.7% 87.2% -29 81.7% 85.8% -1,711 81.7% 86.5%

414 85 173,060 86.5% 85.9% -116 80.4% 85.7% -1,148 76.1% 81.0% -3,078 71.0% 80.6%

415 39 880,410 83.2% 81.3% -1,320 83.7% 79.5% -24,392 56.7% 78.6% -15,091 67.3% 81.7%

422 112 19,791 95.3% 98.2% 12,616 93.8% 99.3% 12,207 96.9% 99.8% 11,960 96.9% 99.6%

423 129 63,598 94.8% 93.5% 16,765 90.7% 95.0% 13,364 91.9% 96.1% 9,207 90.1% 94.5%

424 60 90,761 92.3% 92.0% 5,100 93.8% 93.0% -6,763 92.8% 92.3% -4,283 90.0% 88.7%

432 411 61,073 88.6% 89.2% 34,062 84.7% 98.1% 32,008 86.4% 98.1% 31,816 84.1% 97.7%

433 730 366,720 87.9% 87.6% 73,304 84.3% 88.6% 46,570 86.9% 90.6% 17,438 86.0% 89.3%

434 519 1,034,500 87.0% 83.9% 33,793 88.0% 85.4% -72,393 84.0% 86.2% -716 77.3% 79.7%

435 119 4,200,600 84.1% 75.9% 156,640 82.5% 77.1% -15,798 70.3% 78.6% -175,050 66.6% 76.4%

442 284 39,537 93.4% 94.8% 24,087 90.0% 99.2% 22,146 92.2% 99.6% 21,868 91.1% 99.2%

443 388 164,630 90.9% 90.2% 37,771 86.2% 91.4% 25,445 88.9% 92.8% 11,690 88.5% 92.3%

444 199 261,640 87.2% 87.6% 15,595 88.4% 88.7% 12,149 83.4% 87.2% -17,899 80.5% 84.7%

452 28 5,593 90.0% 91.6% 3,191 85.7% 98.7% 3,323 92.9% 98.1% 3,188 92.9% 98.3%

453 76 34,290 86.0% 80.9% 5,632 82.6% 89.1% -2,838 87.5% 92.3% 229 81.5% 89.9%

454 80 147,890 86.7% 85.7% 20,452 86.7% 86.8% -11,397 87.7% 90.4% -7,669 81.7% 84.8%

455 14 210,480 93.2% 94.2% 34,719 93.0% 95.7% 14,379 93.0% 95.7% 46,598 93.0% 95.7%

462 590 112,880 68.7% 84.2% 61,099 64.5% 95.1% 60,174 70.2% 95.1% 61,188 66.1% 94.2%

463 798 310,520 76.4% 80.7% 67,051 71.6% 84.3% 55,955 73.3% 85.0% 39,930 71.8% 84.6%

464 314 649,760 83.8% 83.7% 75,246 84.9% 84.7% 5,799 86.1% 85.8% -10,437 81.4% 81.1%

465 41 587,700 82.4% 79.5% -8,477 77.9% 80.4% -517 71.3% 77.7% -48,382 69.1% 76.5%

513 11 83,779 76.4% 88.2% 4,088 77.4% 87.6% 1,980 79.2% 87.8% 1,628 81.1% 89.2%

522 25 15,743 66.3% 90.2% 5,058 78.2% 91.2% 4,436 81.8% 94.8% 4,317 80.0% 93.9%

523 6 8,629 86.4% 54.9% -93 68.0% 58.9% -11,190 100.0% 100.0% -11,943 100.0% 100.0%

532 144 82,834 80.2% 89.7% 27,918 77.7% 92.5% 28,156 78.5% 92.3% 27,157 79.9% 92.8%

533 43 101,200 74.1% 75.6% -1,498 69.5% 80.1% -21,328 84.5% 85.5% -8,670 76.5% 80.1%

534 9 56,697 69.4% 75.0% 7,323 69.0% 76.3% -13,351 88.1% 88.6% 3,676 78.6% 77.5%

542 35 16,182 86.3% 93.2% 3,128 96.9% 98.2% 4,638 84.4% 94.8% 4,750 87.5% 95.5%

543 9 105,480 75.8% 86.4% 761 91.3% 85.8% -26,280 95.7% 97.4% -1,741 95.7% 90.8%

562 287 134,440 77.0% 86.9% 17,054 73.8% 89.8% -36,155 78.5% 94.7% 5,151 75.9% 91.4%

563 39 80,942 71.4% 70.5% 24,480 71.0% 71.7% 20,900 83.5% 73.2% 18,809 84.7% 73.6%

613 14 2,161 96.9% 96.9% 22 100.0% 100.0% 22 100.0% 100.0% 22 100.0% 100.0%

614 28 43,355 97.1% 97.2% -2,536 96.8% 97.2% -548 96.8% 97.2% 747 98.4% 98.6%

615 17 129,460 100.0% 100.0% -3,518 97.1% 97.6% -312 94.3% 95.2% -312 94.3% 95.2%

623 8 3,446 100.0% 100.0% 1,622 100.0% 100.0% 1,622 100.0% 100.0% 1,622 100.0% 100.0%

624 11 15,405 98.0% 98.0% -2,747 97.7% 98.0% -723 95.3% 96.0% -723 95.3% 96.0%

633 52 14,778 98.1% 98.2% 1,719 99.1% 99.4% -109 100.0% 100.0% 900 100.0% 100.0%

634 91 117,410 95.7% 95.9% 2,018 96.6% 97.2% 393 96.6% 97.2% 4,236 96.6% 97.2%

635 69 788,730 93.2% 93.3% 88,500 93.0% 95.2% 55,137 93.0% 95.2% 65,668 93.0% 95.2%

643 36 9,741 97.8% 98.5% 2,207 100.0% 100.0% 1,976 100.0% 100.0% 1,003 100.0% 100.0%

644 39 47,642 97.7% 97.8% 2,592 98.7% 98.9% 1,842 98.7% 98.9% 1,903 98.7% 98.9%

645 6 32,143 91.7% 92.3% -2,540 100.0% 100.0% 1,316 100.0% 100.0% 1,316 100.0% 100.0%

654 26 34,420 95.3% 95.5% -497 94.7% 95.5% -497 94.7% 95.5% -497 94.7% 95.5%

655 7 42,668 100.0% 100.0% 2,534 100.0% 100.0% 2,534 100.0% 100.0% 2,534 100.0% 100.0%

664 30 40,800 97.5% 97.6% 2,616 96.2% 97.6% -21 96.2% 97.6% -796 96.2% 97.6%

665 41 353,910 96.6% 95.9% -2,396 93.6% 96.6% -7,852 93.6% 95.9% -3,442 92.3% 95.2%

Total 8,494 15,525,060 87.5% 89.2% 1,661,202 85.8% 92.0% 738,141 96.6% 96.5% 909,788 86.6% 92.6% 787,423 84.9% 91.4%

Summary by demand class

Dem. class

1 2,226 3,693,400 95.6% 95.6% 810,560 96.3% 96.8% 738,140 96.6% 96.5% 732,220 96.2% 96.7% 696,520 96.0% 96.6%

4 5,185 9,469,700 84.3% 86.1% 672,830 81.5% 89.8% 170,990 82.4% 90.4% -26,412 79.7% 88.7%

5 608 685,920 77.2% 85.7% 88,218 75.9% 88.5% -48,193 80.3% 92.0% 43,134 79.0% 90.0%

6 475 1,676,100 96.5% 96.6% 89,597 96.7% 97.6% 54,779 96.6% 97.5% 74,181 96.6% 97.5%

Summary by criticality class

Crit. class

1 493 1,704,700 88.9% 88.3% 7,476 87.2% 90.4% 3,616 92.9% 92.9% -29,368 84.5% 89.1% 9,269 83.7% 89.5%

2 447 277,250 94.1% 95.3% 45,539 93.2% 96.3% 1,340 99.5% 99.5% 20,120 94.7% 97.3% 15,668 93.7% 96.3%

3 3,190 9,165,100 90.1% 89.8% 1,133,400 89.0% 92.2% 681,150 96.9% 96.9% 686,460 89.0% 92.9% 578,120 87.1% 91.3%

4 1,395 905,370 92.6% 92.9% 126,520 91.3% 94.7% 24,358 96.9% 96.7% 80,565 91.6% 95.0% 52,640 90.8% 94.4%

5 372 634,320 90.8% 89.8% 71,491 89.8% 92.8% -1,579 96.5% 96.5% 12,596 91.4% 94.1% 42,164 88.8% 92.3%

6 2,597 2,838,300 79.5% 85.4% 276,750 76.8% 89.6% 29,256 95.9% 95.8% 139,420 79.4% 90.5% 89,569 77.1% 89.2%

Summary by price class

Price class

2 2,223 524,350 83.2% 90.0% 213,800 80.4% 96.2% -5,307 97.8% 97.8% 156,010 83.6% 96.9% 196,510 81.5% 96.2%

3 3,266 1,608,000 87.9% 88.5% 304,050 85.0% 90.6% 29,707 98.5% 98.5% 174,030 86.7% 91.6% 143,970 85.7% 90.9%

4 2,487 3,637,300 90.2% 89.4% 250,570 90.9% 90.6% 85,099 94.7% 94.6% -18,132 89.8% 90.7% 1,106 87.0% 88.2%

5 518 9,755,400 90.4% 88.2% 892,780 90.0% 89.3% 628,640 96.0% 96.0% 597,890 84.2% 88.9% 445,840 84.0% 88.6%

Service level Service level

Empirical Model (E) Willemain Model (W)

Service level

Normal Model (N) Poisson Model (P)

Service level

Item

Class

# of

items

Current System (C)

Service level

38

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