Industrial Engineering and Computer Sciences Division (G2I)

EXTENDED BETA-BINOMIAL MODEL FOR DEMAND FORECASTING OF MULTPLE SLOW-MOVING ITEMS

WITH LOW CONSUMPTION AND SHORT REQUESTS HISTORY

A. DOLGUI, M. PASHKEVICH

Septembre 2005

RESEARCH REPORT

2005-500-012

EXTENDED BETA-BINOMIAL MODEL FOR DEMAND FORECASTING OF MULTIPLE SLOW-MOVING ITEMS WITH LOW CONSUMPTION AND SHORT REQUESTS HISTORY

Alexandre Dolgui and Maxim Pashkevich

Centre for Industrial Engineering and Computer Sciences Ecole Nationale Supérieure des Mines de Saint-Etienne

158, Cours Fauriel, 42023 Saint Etienne, France e-mails: dolgui@emse.fr, pma@omegasoftware.com

Abstract. The paper considers the problem of modeling the lead-time demand for the multiple slow-moving inventory items in the case when the available demand history is very short, and a large percentage of items has only zero records. The Bayesian approach is used to overcome the mentioned problems with the past demand data: it is proposed to use the beta-binomial model to predict the lead-time demand probability distribution for each item. Further, an extension of this model is developed that allows accounting for the prior information regarding the maximum ex-pected probability of demand per period. Parameter estimation and Bayesian forecasting routines are derived for the new model. The efficiency and practical significance of the obtained results is proved by the simulation study.

1. Introduction Slow-moving stock keeping units are common for the spare parts inventory systems, and

are usually held to avoid very high costs that incur from the item unavailability when re-quested. They are particularly important for the military inventory systems, when not being able to replace one service part can result in the materiel unit not being ready to operate. A crucial issue for the effective management of the slow-moving inventory items is the lead-time demand forecasting (Nahmias, 2005).

Due to the extremely low consumption rate, the slow-moving spares are very inflexible with respect to overstocking, so overestimating the demand for these items can result in extra storage costs and losses due to the items becoming obsolete. As a result, trivial solutions like keeping extra safety stock for all items of this type are not satisfactory, and accurate demand forecasting needs to be performed to ensure effective inventory investments.

However, the available historical demand records are often very short and have a large number of zeros, with some records having only null values over the observation interval. For this reason, the inventory management of such items is performed on a group basis, when multiple stock keeping units are used to develop the “population-averaged” lead-time demand probability distribution. This distribution is then used for each item in the group when decid-ing on the replenishment order size and the re-order level.

While being acceptable in practice, the traditional method has clear disadvantage: due to the averaging the demand patterns, it underestimates the demand for the items with larger consumption, and overestimates it for the ones with lower requests count. It can be demon-strated by the following simple artificial example: for a group of SKUs, let the first half have the total demand over the observation interval equal to 2, and suppose that the second half has only zero demand records. The common approach will finally treat all these items based on a probability demand distribution with the mean equal to 1. But it is obvious that distinguishing between the two groups would lead to more accurate forecasts.

To develop intuition about the practical significance of the problem, consider the fol-lowing real-life example for the UK Royal Air Force (RAF) inventory system (Eaves and Kingsman, 2004) that is one of the largest and most diverse inventories in the world. At the

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beginning of 2000, RAF kept about 684,000 consumable line items resulting in approximately 145 million SKUs and a total value of stock 1.2 billion pounds. From the overall number of line items, about 8.5% accounted for 90% of the annual demand value (fast-moving items). Besides, 37.3% of the SKUs had less than 10 demand transactions, and 40.5% had zero de-mand over the observation period of six years. On total, about 60% of the SKUs could be classified as slow-moving with extremely low consumption, for which the demand per period (one month) is binary, and the maximum lead-time demand is finite and small (3-6 months). It is clear that accurate demand forecasting for the items of the considered type is very important for an inventory system like RAF.

This paper proposes a novel Bayesian-based methodology for estimating the demand distribution of multiple slow-moving items in the case of extremely low demand and short requests history. An extension of the beta-binomial probability distribution is proposed that allows flexible Bayesian forecasting for the SKUs with zero demand records relying on the demand for the similar items. Parameter estimation routines and forecasting technique are also presented, as well as the asymptotic analysis of the obtained theoretical results. The efficiency of the proposed approach is proved by a simulation study.

The reminder of the paper is organized as follows. Section 2 presents the review of the literature related to the paper topic. Section 3 deals with the problem statement. Section 4 de-scribes the usage of the classical beta-binomial model to solve the considered problem. An extension of this model that allows accounting for the prior information for the maximum probability of demand per period is presented in Section 5. The parameter estimation and Bayes forecasting issues for the proposed model are considered in Sections 6 and 7. Finally, Section 8 presents an overview of the simulation model used to prove the practical signifi-cance of the developed techniques, and discusses the computational results. We end with con-clusions in Section 9.

2. Related works Inventory control is a fundamental part of the supply chain management that is aimed at

optimal balancing service levels against investment over a very large assortment of the stock-keeping units (SKUs) and uncertainties (Graves et al., 1993). Since inventory investment in-volves up to 15-20% of the gross domestic product in the developed countries, and holding costs are on average 25-35% of the inventory value (Nahmias, 2005), employing correct in-ventory policies significantly influences the overall profit of an enterprise involved. This fact is supported by a number of successful implementations of the state-of-the-art inventory man-agement techniques (Bartezzaghi et al., 1999; Dolgui and Ould Louly, 2002; Kennedy et al., 2002; Ould Louly and Dolgui, 2004; Willemain et al., 2004; Zhang, 2004).

Accurate demand forecasting is a crucial issue for successful inventory management (Silver et al., 1998). According to the traditionally used approach that is based on the eco-nomic order quantities (Lewis, 1975), the demand distribution during the lead time is used to determine the order quantity and reorder points, and whereas the former requires only the av-erage demand, the latter involves the entire distribution over the interval between the genera-tion of a replenishment order and its arrival. Therefore, developing the accurate demand dis-tribution is critical for achieving a realistic estimate of the actual global service level, or a total fill rate in the case of multiple SKUs (Hopp and Spearman, 1995).

Intermittent demand analysis is a challenge for inventory control due to a specific na-ture of the underlying demand process. Being also called “irregular” or “erratic”, this kind of random demand is characterized by infrequent transactions and variable demand sizes when a transaction occur (Silver, 1981). Common examples of stock keeping units with intermittent demand include spare (service) parts, especially in the automobile and aircraft industry, and

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high-priced capital goods like heavy machinery. As noted in (Willemain et al., 2004), the in-termittent nature of demand makes the forecasting problem especially difficult.

The intermittent demand patterns can be divided into two categories: lumpy demand, when the requests can be made for more than a single unit during a transaction, and slow-moving patterns, when a request is made for a single unit in the most of the cases. Both of the categories create challenges while forecasting the demand distribution.

For the lumpy case, the major problem is modeling the demand magnitude during the transaction, since in this case the variability of demand is large relative to the mean. This can be coursed by a large number of small customers with approximately stationary behavior, and a few large customers that infrequently place orders with large magnitude (Silver, 1970). Be-sides, the lumpy demand can be caused by the correlation between the customer requests (Bartezzaghi et al., 1999) or the bull-whip effect in the multi-echelon system, when small variations in demand are magnified along the supply chain (Zhang, 2004).

For the slow-moving demand, the major problems are (i) the lack of past records for giv-ing reliable estimates of historic consumption, and (ii) zero consumption over a long period that would normally be more than adequate for analysis (Mitchell, 1962). Forecasting tech-niques developed for the smooth and continuous demand are not applicable for the intermit-tent demand due to inappropriateness of the assumptions of continuity and normal demand distribution. A number of alternative techniques that relax these assumptions were developed for forecasting the intermittent demand, the most widely used is the Croston’s method that applies the exponential smoothing separately to the intervals between nonzero demands and their sizes (Croston, 1972). A number of modifications of this approach with various im-provements were also proposed (Rao, 1973; Segerstedt, 1994; Willemain et al., 1994; Johns-ton and Boylan, 1996). Another way of handling the intermittent demand is based on the Pois-son probability distribution (Ward, 1978; Williams, 1984; Mitchell et al., 1983; Van Ness and Stevenson, 1983; Bagchi, 1987; Schultz, 1987; Watson, 1987; Dunsmuir and Snyder, 1989). Recently, the bootstrap methods (Efron and Tibshirani, 1998) became a popular tool for mod-eling the intermittent demand (Bookbinder and Lordahl, 1989; Wang and Rao, 1992; Kim et al., 1995; Willemain et al., 2004).

When some of the inventory items under consideration have only zero records in the past requests history, the problem of demand forecasting becomes even more complex (Mitchell, 1962), since the common techniques like the Croston’s method and its extensions can not be applied to estimate the probability distribution of demand. To solve this problem in real life, similar SKUs are grouped together, and the aggregated data is used to develop a population-averaged demand distribution (Eaves and Kingsman, 2004). It is clear that this approach has a number of drawbacks, the major one is that all the items from the group are treated in the same way with no respect to their specific requests.

Another challenge is the short demand history that is available to model the lead-time consumption. The Bayesian paradigm was historically used to overcome this difficulty with the past observations (Graves et al., 1993). First proposed by Scarf (1959) and Silver (1965), it was then successfully applied to various cases of the inventory management problems (Smith and Vemuganti, 1969; Brown and Rogers, 1973; Azoury, 1985; Popovic, 1987; Brad-ford and Sugrue, 1990; Hill, 1997, 1999; Aronis et al., 2004). Bayesian approach assumes that one has a prior information about the demand distribution, which is updated using the observed demand values to obtain the posterior distribution to be used for forecasting. How-ever, choosing an appropriate prior is also a problem, and often the informative or uniform priors are used to start with. An alternative approach, which is also employed in this paper, is to use the data for a group of related SKUs to estimate the prior distribution (Bradford and Sugrue, 1997).

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As mentioned by Graves et al. (1993), nearly all research work in inventory manage-ment related to demand forecasting was done for a single inventory item. This is justified by the fact in most of the cases, the real-life problems that involve handling of multiple SKUs can be separated or reduced to the single-item settings. However, as follows from the argu-ments presented above, when the demand history is short and a significant percentage of items has zero request records, the inventory management policy must deal with a group of items, and the mentioned reduction is not possible. Thus, this paper considers multiple inventory items that are managed to maximize the fill rate over all of them.

The demand probability distribution for the slow-moving SKUs is worth a separate dis-cussion. As noted by Hill (1999), much of the work in the Bayesian inventory modeling has been done assuming a Poisson demand distribution with a gamma prior that results in the negative binomial posterior. However, in the case of extremely low demand, the upper bound of the lead-time demand is finite and small, from our experience often 3-5 in real inventory management systems, and thus the binomial distribution is a better way to describe the de-mand data (the asymptotic relation between the binomial and Poison distributions is well known). Since a natural and conjugate prior for the binomial case is the beta distribution, the resulting demand distribution is the beta-binomial. This mixture probability distribution was successfully applied to a number of inventory management problems, see, for example, (Pet-rovic et al., 1989; Grange, 1998).

In this paper, an extension of the beta-binomial probability distribution is proposed, which is aimed at taking into account an additional information regarding the extremely low demand probability for multiple slow-moving SKUs. The following sections present the prob-lem formulation, followed by the theoretical and simulation results obtained.

3. Problem description It is assumed that the inventory control is performed over a group of the SKUs with the

correlated demand, with k being the group size. The items in the group can be, for example, spare parts from the same assembly unit, or the items that are historically known to have the correlated demands. Let n denote the number of periods for which the demand B data is avail-able, where B = (bij) is a binary k×n-matrix, and bij = 1 if, for the item i in the period j, the demand has occurred, and bij = 0 otherwise. The demand is assumed to be binary due to the specific properties of the SKUs under consideration: we suppose that it is always possible to select the period length so that not more than one request occurs in each period.

We also suppose that the availability of an item is crucial for the system, and a stock-out leads to high expenses (for instance, a piece of equipment not being able to operate). Since all items are slow-moving with some of them having zero demand records over the whole obser-vation period, it is assumed that the replenishment ordering costs are insignificant with re-spect to the inventory storage costs, and the size of the replenishment order is always one. Thus, the probability distribution of the lead-time demand, which must be calculated using the past demand data B, is used to decide on the re-order levels Li, i = 1, 2, …, k, for each SKU given the target fill rate. To estimate the parameters of a model for the lead-time demand, last n periods are always used.

In this paper, we utilize the inventory optimization approach (Hopp and Spear-man, 1995) that is aimed at satisfying the target fill rate (actual percent of demand met by the system) over the whole group of k items. The ultimate goal is to find the unknown re-order levels {Li} from the following constraint optimization problem:

FD

DLDtsLc k

i i

k

i iiik

iiiLL k

≥⋅≤

∑∑∑

=

=

=1

1

1},,{ }{

}{}{..,:min

1 E

EPK

, (1)

5

where Di denotes the random demand for the item i over the lead-time of m periods, ci is the unit holding cost for this item, and F is the target fill rate (for the comprehension convince, the notation used in the paper here and later is summarized in Table 1). The problem (1) is NP-hard, with k decision variables in the general case.

To solve the optimization problem (1), one must know the probability distributions of the random demands Di, i = 1, 2, …, k. These distributions must be assessed using the binary k×n-matrix B that contains the historical demand records, where k is large (hundreds or thou-sands), and n is small (5-10, for example). The traditional approach is to estimate the average request probability as

)(1 1

nkbpk

i

n

jijbm ⋅= ∑∑

= =

, (2)

and to assign the binomial distribution Bi(pBM, m) to all SKUs in the group. Afterwards, the problem (1) is solved for one “aggregated item” and takes the following simple form:

min: L s. t. FLD ≥≤ }{P , (3)

that is a usual service-level problem for a single inventory item, and L is the re-order level used for all SKUs in the group. This approach will be called the binomial model (BM) of the lead-time demand from now on in the paper.

This paper concentrates on accurate estimation of the probability distributions for {Di} and considers the problem of Bayesian forecasting of these distributions taking into account the individual requests history for each item. A number of arguments can be presented to ad-vocate this approach.

Table 1 General notation used in the paper

Symbols Interpretation Variables

k Number of related inventory items in the managed group n Number of periods with available historical demand data m Lead-time length B = (bij) Binary k×n-matrix with historical demand records bij Equals to one if for item i in period j demand has occurred; zero otherwise si Total number of requests for inventory item i in the past history (bi1, , bi2, …, bin) pi Unknown probability of request per period for inventory item i pBM Population-averaged probability of request per period π Extended beta-binomial model parameter, expert’s estimate for maximum of {pi} Li Unknown re-order level for inventory item i ci Unit holding cost for inventory item i Di Random lead-time demand for inventory item i DA Random population-averaged lead-time demand F Target fill rate B(α, β) Beta-function with parameters α, β Statistical notation

P{.} Probability of random event E{.} Mathematical expectation of random variable V{.} Variance of random variable L{.} Probability distribution law Bi(m, p) Binomial distribution with parameters m, p B(α, β) Beta distribution with parameters α, β θ̂ Statistical estimate of a parameter θ

Abbreviations

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BM Binomial model BBM Beta-binomial model EBBM Extended beta-binomial model

Firstly, it allows accounting for the correlation between the demands for the related items, while still distinguishing them. For example, in the case of spare parts, elements of the same assembly unit turn to wear out dependently, and if several items were already replaced, the probability of replacement for other elements in the unit goes up (Dolgui and Ould Louly, 2002; Ould Louly and Dolgui, 2004; Dolgui et al., 2005).

Secondly, the Bayesian technique allows solving the data availability problem of the real-life industrial data sets being “short and wide” (Gardner, 1990; Flores et al., 1993; Wil-lemain et al., 2004). The problem of having few observations (n) for each SKU is overcomed by utilizing the large number of items in a group (k).

Thirdly, the SKUs are forecasted now based on the demand for the related items and its own requests history, thus accounting for a particular item individuality. Therefore, the items are divided into classes based on their past requests, and each class gets its own lead-time demand distribution, as opposed to the binomial model with one aggregated item. The prob-lem (1) is being solved with respect to each item type; the number of decision variables re-duces to the maximum of n + 1, that, from one hand, makes the exhaustive search applicable, and, from the other hand, should lead to a more flexible solution of (1) with comparison to the BM case.

A separate note should be made on assessing the quality of the forecasting. Since the demand records are short and might contain a lot of zeroes, the performance of the forecasting system must be evaluated over the group of items under review, not for each item. This is consistent with the approach utilized in the recent paper of Willemain et al. (2004), as well as with the practical considerations: in real life, the average performance of the system is more important than its accuracy for a particular item (Hopp and Spearman, 1995). Thus, the simu-lation results presented at the end of the paper rely on the average system performance meas-ure, when the developed probability distributions for {Di} are used to solve the problem (1).

Let us now present on overview of the forecasting concept employed.

4. Beta-binomial model for slow-moving items with low demand and short history This section proposes a framework for the solution of the considered problem relying on

the ideas from the longitudinal statistical data analysis (Diggle et al., 2002) that were success-fully used to solve a number of problems with the similar data structure in other application areas; a review can be found in (Pashkevich and Kharin, 2004), (Pashkevich and Dol-gui, 2005). The authors would like to point out that the beta-binomial distribution, that serve as a foundation in the approach proposed, is often justified in the inventory management in a slightly different manner. It is usually supposed that a demand distribution for a single inven-tory item is of this kind because the “request probabilities” of the consumers have the beta distribution, and the number of consumers is limited and small (otherwise, the negative bino-mial distribution is used). In this paper, the beta-binomial distribution arises due to the het-erogeneity within the group of SKUs, that is similar to the approach proposed in (Bradford and Sugrue, 1997) for the mixed Poisson distribution.

The beta-binomial model (Collet, 2002), applied to the lead-time demand forecasting problem under consideration, is formulated as follows.

The first assumption is that the probability pi of an item i being ordered during the over-all review interval is invariant with respect to period number j:

A1. P{bij = 1} = pi, ∀ j = 1, 2, …, n.

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This assumption of the stationary demand over the review period is realistic from the practical point of view, since in our case the consumption rate is low, and the number of periods with available data n is small. The latter can be interpreted as the overall review period being neg-ligibly small when compared to the demand dynamics.

The second assumption introduces the heterogeneity among the SKUs by supposing that the demand probabilities {pi} are drawn from the same statistical distribution with the com-mon parameters that will be called the prior distribution from now on. Since the probability belongs to the interval [0; 1], this distribution is supposed to be beta with the parameters α, β:

A2. L{pi} = B(α, β), i = 1, 2, …, k.

The beta distribution is very flexible and can take a variety of shapes depending on the values of the parameters α, β (Collet, 2002). The probability density function is expressed as

),()1(),|( 11 βα−=βα −β−α Byyyfip , (4)

and, for example, if α and β are both less or equal to 1, the distribution will be U- or J-shaped. This shape represents a polarized distribution where some items have small response prob-abilities and others have large response probabilities, but few SKUs are in between. On the other hand, if α and β are both large, the distribution will resemble a spike so that all items have more or less the same response probability. Values of α and β just a little larger than 1 make the beta distribution look like an inverted U or like the central part of the normal curve. If α and β are both equal to one, the distribution becomes uniform. If one of the parameters is larger then 1 and the second is smaller, the probability density function has L-shape. Being also a conjugate prior for the binomial distribution, the beta distribution is capable to flexibly describe the diversity of the demand probabilities between the items within the group, provid-ing also computationally simple forecasting expressions.

The third assumption is that the random variables {pi} are independently drawn from the beta distribution, and thus are independent in total:

A3. Probabilities p1, p2, …, pk are i.i.d. random variables.

This implies that initially, the demand probabilities for the SKUs in the group do not influ-ence each other, and correlation shows itself with time, for example when the items involved in the same assembly unit start to wear out together.

Under the assumptions A1-A3, and the lead time being equal to m periods, the popula-tion-averaged lead-time demand probability distribution for the group is the beta-binomial with the parameters m, α, β (Collet, 2002):

mxB

xmxBxm

xDA ,,1,0,),(

),(}{ K=βα

−+β+α⋅

==P , (5)

where DA denotes the population-averaged lead-time demand. The major advantage of the Bayesian approach to forecasting is that it allows to adjust

the population-averaged probability distribution (5) using the observations specific to a par-ticular item. Since the beta distribution is a conjugate prior for the binomial model, the poste-rior distribution of the demand probability pi for the item i is also beta, but with the shifted parameters: ∑ =

=−+β+α=n

j ijiiiii bssnssp1

),,(}|{ BL , (6)

where si is the total demand over the observed period for the i-th SKU. Hence, the forecast of the lead-time demand Di for the item i has also the beta-binomial distribution, but with the parameters modified according to (6):

8

mxsnsB

xsnmxsBxm

sxDii

iiii ,,1,0,

)ˆ,ˆ()ˆ,ˆ(}|{ K=

−+β+α−−++β++α

⋅

==P , (7)

where the hats denote the estimates of the corresponding parameters. Finally, the expression (7) can used to calculate the re-order level and the replenishment order size.

Estimation of the model parameters is usually performed using the explicit expressions based on the method of moments (MM) or the iterative numerical estimators based on the maximum likelihood (ML). Denote by s , v the mean and variance of the sample {s1, s2, …, sk}, then the MM-estimators are expressed as (Johnson et al., 1995)

,)1//(

)/(ˆnnssv

ssvsnMM −+

−−=α .

)1//()()/(ˆ

nnssvsnsvsn

MM −+−−−

=β (8)

The maximum likelihood estimation involves maximization of the log-likelihood function

∑=

βα

βα−+β+α

⋅

=βα

k

i

ii

i BsnsB

sn

lmax1, ),(

),(ln),(: , (9)

that can be reduced to the solution of the following system of non-linear equations:

,01

0

11

0=

+β+α−

+α ∑∑−

=

−−

=

n

l

n

l

l

lA

lA ,0

1

1

0

1

0

1 =+β+α

−−−+β∑ ∑

−

=

−

=

−n

l

n

l

l

lA

lnB (10)

where ,21 nill fffA +++= ++ K ,10 ll fffB K++= and f0, f1, …, fn are the frequencies for the sample {s1, s2, …, sk}. Applying the Newthon-Raphson method for the numerical solution of (10) results in the following linear system for iterative computing of the estimates approximation:

∑−

=

−

=

−+++ +β+α

β−β+α−α−+α

α−α=1

0

1

02

11121 )(

)()(

)(n

l

n

l rrrrrr

r

lrrr l

Al

AF ∑ , (11)

∑ ∑−

=

−

=

−+++ +β+α

β−β+α−α−−−+β

β−β=1

0

1

02

11121 )(

)()1(

)(n

l

n

l rrrrrr

r

lrrr l

Aln

BG , (12)

where r is the iteration number, and Fr, Gr denote the corresponding residuals. The initial ap-proximations are usually found as the MM-estimates based on (8). A more detailed review of the parameter estimation techniques for the beta-binomial distribution, including some ad-vanced approaches, can be found in (Tripathi et al., 1994).

For the formulated above beta-binomial model (BBM) of the lead-time demand, it is proposed to divide all SKUs into (S + 1) sub-groups based on the previous requests {si}, S = max(s1, s2, …, sk), and to perform the inventory management based on S + 1 aggregated items, assigning the same service level to all items in a sub-group. Therefore, the problem (1) that balances the fill rate and the holding costs takes the following form:

FDr

DLDrtsLrmin S

x xcx

S

x xxxc

xS

xx

cxLL S

≥⋅

⋅≤⋅⋅

∑∑∑

=

=

=0

0

0}~,,~{ }~{

}~{}~~{..,~:

0 E

EPK

, (13)

where , and I(.) is the unit function that takes the value of one if the ar-

gument condition holds, and zero otherwise. The variables ∑ =

=⋅=k

i iicx xsIcr

1)(

xx DL ~,~ denote the re-order level and the random lead-time demand for the group x. Since in our case the past demand history n is very short, the problem (13) is efficiently solved using a simple branch-and-bound algo-

9

rithm with the following upper bound for an incomplete solution }~,,~,~{ , q < S, used to cut the unpromising branches:

21 qLLL K

Fx<

}

…

si

Posterdistributi(probab

0

1

S ,

ch d.

Dr

DrDLDrS

x xcx

S

qxcx

q

x xxxcx

⋅

⋅+⋅≤⋅

∑∑∑

=

+==

0

10

}~{

~{}~{}~~{

E

EEP. (14)

As a result, the proposed procedure of modeling the demand distribution for the group of slow-moving items with low and correlated demand can be outlined as follows.

(i). Obtain the historical demand data B for the group of items with the correlated demand.

(ii). Estimate the parameters α, β using the (k×n) binary matrix B and the expressions (8).

(iii). Obtain the Bayes posterior lead-time probability distribution (7) for each SKU based on the total demand si observed for it.

The graphical interpretation of the presented procedure is given in Figure 1.

An additional advantage of the proposed approach is its computational simplicity (ex-plicit expressions) that makes realistic its implementation for a large inventory control system. Even if one decides to use the ML-estimators for the parameters α, β, the Newthon-Raphson numerical procedure (11), (12) is known to converge quickly, with about five iterations being enough to stabilize the estimates up to three decimal places (Johnson et al., 1995). Another important issue is that the concept of adjusting the population-averaged probability distribu-tion using the past demand history for a particular item and its advantages are intuitively ap-pealing and clear for inventory managers in industry.

5. Extension of the model for the case of extremely low demand Although the classical beta-binomial model, which was suggested in the previous sec-

tion to handle the demand for the slow-moving SKUs, provides a reasonable framework to solve the problems with the short and zero demand histories, it does not utilize the fact that the demand is very low for all items. This paper proposes an extension of the beta-binomial model that allows taking this information into account. First, a corresponding probability dis-tribution is derived. After that, parameter estimation issues are considered. Subsequently, the Bayesian forecasting procedure for the new model that ensures mean square optimal predic-tion is developed. Asymptotic properties of the developed estimation and forecasting routines are also analyzed.

SKU #1

SKU #2

SKU #k

… … …

… … …

t = 1 2 n … …

0 1

p

f Prior probabilitydistribution

The data for all SKUs is used to estimate the prior parameters

Binary matrix with historical demand

ior on

ility)

Posteriordistribution(demand)

After estimating the parameters of the prior distribution for the demand probabilitythe SKUs are divided into S + 1 classes based on the past total demand si, with eaclass having its own posterior probability distribution of the lead-time deman

Fig 1. An overview of the proposed framework for forecasting the lead-time demand for the slow- moving inventory items with short past demand history and a large percentage of zero records

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The following extension of the assumption A2 is proposed: suppose that there exists an expert estimate π ∈ (0; 1] of the maximum demand probability per period for the considered group of SKUs, and this probability follows a special case of the generalized beta distribution:

1

11

),()(),,|( −β+α

−β−α

π⋅βα−π

=πβαB

yyyf gpi

. (15)

The parameter π is assumed to be small due to the specific properties of the considered inven-tory items. In the following theorem, we develop the population-averaged lead-time demand probability distribution under the prior (15), and then obtain the expressions for its mathe-matical expectation and variance. Afterwards, the asymptotic properties of the obtained distri-bution are explored.

Theorem 1. The population-averaged lead-time demand probability distribution for the extended beta-binomial model with the prior distribution (15) can be expressed as a weighted sum of the shifted beta-binomial probabilities:

, (16) mxmlPmwxDm

xlxlA ,,1,0,),,|(),(}{ 0 K=βα⋅π== ∑

=

P

where the weights wxl are computed as

, (17)

≤<

≤≤π−π⋅

=π

−

.,0

,0,)1(),(mxl

lxxl

mwxlx

xl

and P0(l | m, α, β) denotes the beta-binomial probability with the corresponding parameters:

),(

),(),,|(0 βα−+β+α

⋅

=βα

BlmlB

lm

mlP . (18)

Proof. Using the conditional probability approach, the probability of the lead-time de-mand being equal to x can be expressed as:

dyB

yyyyxm

yfyxDxD xmxgpAA i 1

11

00 ),()()1(),,|(}|{}{ −β+α

−β−απ−

π

π⋅βα−π

⋅−

=πβα⋅=== ∫∫PP . (19)

Introducing the transformation of variable y = π⋅z and employing the obvious identity zzz ⋅π−+−=⋅π− )1()1(1 allows applying the Binomial theorem, and (19) simplifies to

.

),()1()1(

),()1()1()1()(}{

1

0

11

1

0

11

)(

∫ ∑

∫

=

−−+β−+α−

−β−α−

βα−

⋅π−π

−−

=

=βα

−⋅⋅π−+−⋅π

==

dzB

zzlmxm

xm

dzB

zzzzzxm

xD

m

xl

lnlxlx

xmxAP

(20)

Using the property of the binomial coefficients that is easily proved from their definition

(21)

⋅

=

−−

⋅

xl

lm

lmxm

xm

leads to

. (22) ),(

)1()1(}{1

0

11

∫ ∑=

−−+β−+α−

βα−

⋅π−π

=== dz

Bzz

lm

xl

xDm

xl

lnlxlx

AP

11

Simplifying this expression by (i) changing the integration and summation order, (ii) taking into account the notation (17), and (iii) reminding the derivation of the beta-binomial prob-abilities (Skellam, 1948) concludes the proof of the theorem.

Remark. It is easy to check that the probabilities (16) sum to one by noting that the weights (17) satisfy the property ∑ that follows from the Binomial theorem. 1

0=

=

l

x xlw

Corollary 1. The mathematical expectation and variance of the developed probability distribution (16) are calculated as

β+α

α⋅π−π+

+β+αβ+α+β+ααβ

⋅π=β+α

α⋅π=

mmmDmD AA )1()1()(

)(}{,}{ 22VE . (23)

Proof. To obtain the expression for the mean value, the iterative expectation formula is employed using the fact that the distribution of DA conditional on p is binomial:

)(}|{}{ }{}{ β+απα⋅=== mmppDD AA EEEE . (24)

Here, the subscript for p is dropped since the variables are identically distributed, and the property of the general beta distribution )(}{ β+αα⋅π=pE is used.

For the variance, the following property is used: }{}{ |}{|}{}{ ηξ+ηξ=ξ VEEVV , where ξ and η are arbitrary random variables. Thus,

.

)1)(()1(

)1()(

|}{|}{}{

22

22

)}1({}{}{}{

+β+αβ+α+αα

π−β+α

απ+

+β+αβ+ααβ

π=

=+=+= −

mmm

pDpDD pmpmpAAA EVVEEVV (25)

After simplification, the latter expression reduces to the given above formula.

Corollary 2. The 2nd uncentered moment of the probability distribution (16) is found as

β+α

α⋅π+

+β+αβ+α+αα⋅−

⋅π=mnnDA )1)((

)1()1(}{ 22E . (26)

Proof. The corollary statement immediately follows from the following moments prop-erty: . }{}{}{ 22

AAA DDD EVE +=

Corollary 3. For the values of π close to one, the probabilities of the developed distri-bution (16) can be approximated by the following asymptotic expansion:

),1()1(}{}{ 0 π−+∆⋅π−−=== oxDxD AA PP (27)

(28)

<=⋅<+=⋅+−=⋅

=∆;,}{;,}1{)1(}{

0

00

mxxDrmxxDrxDr

A

AA

PPP

where P0{.} denotes the probability for the classical beta prior defined by (5).

Proof. First, let us linearize the weight coefficients (17) with respect to π. One can show that the matrix W = (wxl) satisfies the asymptotic expansion )1()1( π−+π−⋅+= π oWIW , where Wπ is a two-diagonal matrix such that Wjj = – j, Wj j + 1 = j. Afterwards, plugging in this expansion to the expressions for the probabilities (16) leads to the expansion (27).

The results presented in this section allow describing the lead-time demand via an ex-pended beta-binomial probability distribution that incorporates the expert’s estimate of the maximum demand probability π. Computationally fast expressions for the elements of the

12

probability row when π is close to one were also developed. The following section deals with the problem of parameter estimation for the proposed model.

6. Parameter estimation for the proposed model To estimate the parameters of the proposed extended beta-binomial model, two ap-

proaches, that are traditional in the mathematical statistics, are employed. We first develop the explicit estimators based on the method of moments. The corresponding estimates can be then used as the initial approximation for the maximum likelihood estimators, which are based on the numerical optimization routines. In all cases, the parameter π is assumed to be known. The raw data for the parameter estimation is the binary k×n-matrix B presented in the suffi-cient form {s1, s2, …, sk}, see expression (6).

Theorem 2. The method of moments estimators for the parameters α, β of the probabil-ity distribution (16) with the known parameter π can be expressed as

,,,1)1(

)()1(,)(,nvv

nss

ssvssnss

=′=′−′−′−′

′−π′−=λ

π′−π⋅λ

=βπ

′⋅λ=α (29)

where s and v are the mean and variance of the sample {s1, s2, …, sk} defined by (6).

Proof. The method of moments estimators are the solutions of the system of equations

vnnnsn=

β+αα

⋅π−π++β+αβ+α

+β+ααβ⋅π=

β+αα

⋅π )1()1()(

)(, 22 . (30)

Expressing )( β+αα , )( β+αβ from the first equation allows solving the second one for (α + β), that leads to the expressions (29).

To obtain the maximum likelihood estimates, the following optimization problem must be solved:

∏ ∑= =

βα

βα−+β+α

⋅

⋅π=βα

k

i

n

slls

i

i BlnlB

ln

wLmaximize1, ),(

),()(),(: (31)

This problem is solved using a modification of the steepest descent method. The following lemma provides the expressions that are necessary for the implementation of the correspond-ing numerical algorithm.

Lemma 1. The partial derivatives of the log-likelihood function are computed as

)( (.)ln(.) Ll =

,11),(0 0

1

0

1

0∑ ∑ ∑∑

= =

−

=

−

=

ν⋅

+β+α

−+α

⋅π⋅=α∂

∂ n

r

n

jrj

n

j

j

lrjr jj

nwfl (32)

,11),(0 0

1

0

1

0∑ ∑ ∑∑

= =

−

=

−−

=

ν⋅

+β+α

−+β

⋅π⋅=β∂

∂ n

r

n

jrj

n

j

jn

lrjr jj

nwfl (33)

),(

),(),(

),(),(1

0 βα−+β+α

⋅

⋅

βα

−+β+α⋅

⋅π=ν

−

=∑ B

jnjBjn

BtntB

tn

nwm

trtrj , (34)

where {fr} are the frequencies of the sample {s1, s2, …, sk} defined by (6).

13

Proof. The expressions are derived by explicit differentiation taking into account Theorem 1.

The comparative simulation results has indicated that the ML-estimators perform better on a long run that is consistent with the statistical estimation theory. Thus, it is proposed to obtain the initial approximation via the method of moments, and then apply the numerical optimization algorithm to compute the maximum likelihood estimates of the parameters α, β.

7. Bayes forecasting for the proposed model Once the model parameters are estimated, the empirical Bayes forecasting approach

(Winkler, 2003) is applied to obtain the individual probability distribution of the lead-time demand for each inventory item. In the following theorem, we develop the posterior distribu-tion of the demand probability under the assumption of the prior (15).

Theorem 3. For the prior (15), the posterior distribution of the demand probability can be represented as a the weighted sum of the shifted generalized beta distributions:

∑=

−−+β−+α

−+β+α−

⋅πω=n

l

lnl

lsigp lnlB

yynsyhii

0

11

),()1(),()|( , (35)

the corresponding mean-square-error optimal point forecast is computed as

∑= +β+α

+απ⋅πω=

n

llsi

gi n

lnspi

0

)(),()(ˆ , (36)

and the weights have the following form:

. (37) ),,|(),(),,|(),(),( 0

1

00 βα⋅π⋅

βα⋅π=πω

−

=∑ nlPnwnjPnwn rl

n

jrjrl

Proof. From the Bayes formula, the posterior probability density function of the de-mand probability pi can be found as

1

11

),()()1()|( −β+α

−β−α−

π⋅βα−π

⋅−

∝

Byyyy

sn

syh snsi

gpi

, (38)

where the symbol ∝ stands for ”proportional”. Following the same approach as in the proof of Theorem 1, the expression (38) can be rewritten as

∑=

−+β+α

−−+β−+α

π⋅βα−

⋅π∝

m

sln

lnl

lsigp

i

ii Byy

ln

nwsyh 1

11

),()1(),()|( . (39)

Fixing the denominator in the fraction to have the correct normalization coefficient for the probability density function of type (15) and reminding the notation (18), one gets

. (40) ∑=

π−+β+α⋅βα⋅π∝m

sl

gplsi

gp

i

iiilnlyfn|lPnwsyh ),,|(),,(),()|( 0

Normalizing the probability density function from this expression leads to (35). The expression (36) is derived from (35) by applying the mathematical expectation op-

erator and employing the expression for the generalized beta distribution probability mean. The optimality in the sense of minimum mean square error follows from the Bayesian proper-ties of the obtained point forecast.

14

Corollary 4. The mean square error of the forecast (36) is calculated as follows:

∑ ∑∑= =

+

+−+

=+

+

β+αβα

π⋅

+β+α

+αππω−

β+αα⋅π

=εn

r

n

rjn

jnj

rj

n

rlrl

gi j

nnw

nlnp

0][

])[(][2

]2[

]2[22

)(),()(),(

)()ˆ( , (41)

where the notation x[y+] stands for the incomplete factorial )1()1( −+⋅⋅+⋅ yxxx K .

Proof. The mean square forecast error of the arbitrary predictor in this case can be expressed as (Pashkevich and Kharin, 2004)

(.)ˆ ip

. (42) ∑=

=⋅⋅⋅−+=εn

lA

giiiii lDlplplppp

0

22 }{)(ˆ)(ˆ2)(ˆ}{)ˆ( )( PE

Plugging into the latter expression the Bayesian forecast (36) and applying the properties of the generalized beta and beta-binomial probability distributions leads to (41).

Corollary 5. For the values of π close to one, the following asymptotic relation between the developed predictor (36) and the predicator for the classical beta-binomial model

)()()(ˆ 0 nssp iii +β+α+α= holds:

1

)1()(,)(),1()1(1)1(1)(ˆ)(ˆ 0

−++β+α−β+α

=ξ+α

α−β+α=γπ−+

π−⋅ξ−π−⋅γ−

π⋅=i

i

i

iiii

gi sn

nss

nsospsp . (43)

Proof. Using the linearization of the weights (wrl) obtained in the proof of Corollary 3:

)1()1()1(),1()1(1 1 π−+π−⋅+=π−+π−⋅−= + orworw rrrr , (44)

the number of summation items the in expressions (36), (37) can be reduced down to two. As a result, the following asymptotic ratio holds for the weights (ωrl):

)1()1()1(

)()(1 π−+−−+β⋅+

+α⋅−=

ωω + o

rnrrrn

rr

rr . (45)

Using this property when simplifying (36) leads to the expressions (43).

Once the probability distribution of the demand probability pi is know, the next step is to develop the posterior lead-time demand probability distribution that adjusts the population-averaged consumption for a particular item with respect to its demand history. In the follow-ing theorem, we develop expressions for the corresponding probability row, and the mean and variance of the random lead-time demand.

Theorem 4. For the proposed extended beta-binomial model based on the prior (15), the probability distribution of the lead-time demand Di for the inventory item i is expressed as a weighted sum of the lead-time demand probability distributions for the extended beta-binomial model (16) with the shifted parameters, where the weights are given by the expression (37):

. (46) mxrnrmlPmwnsxDn

sr

m

xlxlrsii

i

i,,1,0,),,|(),(),(}|{ 0 K=−+β+α⋅π⋅πω== ∑ ∑

= =

P

and the mean demand and its variance are computed as

∑= +β+α

+α⋅πω⋅π=

n

srrsii

i

i nrnmsD ),(}|{E , (47)

15

)1(,}|{)(

}|{ ]2[2]2[

]2[]2[2 −⋅=−

β+α

α⋅π+

β+αα

⋅πω= −+

+−

=∑ mmmsDEmmsDV ii

n

srrsii i

. (48)

Proof. To develop the posterior lead-time demand probability distribution, remind that . Then, for the distribution of p),(}|{ iii pmpD BiL = i given by (35) one gets

, (49) ),(}{,}|{}|{}{ }{}{ rnrppxDpxDxD ri

rii

n

srrsiii

i

i−+β+α==⋅ω==== ∑

=

BLPEPEP

where the additive property of the mathematical expectation is used. Noting that the expecta-tions of the conditional probabilities in the last sum are the developed in Theorem 1 extended beta-binomial probabilities for x successes with the parameters m, α + r, β + n – r leads to the theorem statement (46). The formula (47) is derived by finding the mathematical expectation of (46) using the expression for the mean of the beta-binomial probability distribution and the properties of the weights (ωsr).

To derive the expression for the variance, we employ the same approach as in (49) to compute (relying on the result of Corollary 2), and then get the formula (48) as

. }|{ 2

ii sDE}{2

ADE − }{}{ 2AA DD EV =

The obtained result allows estimating the replenishment order sizes and the reorder points for the group of slow-moving SKUs with extremely low demand and the past records characterized by a large percentage of zeros and short observation period length. Its major advantage over the classical beta-binomial model is the additional parameter π, which allows specifying the prior distribution of the demand probability more exactly by assuming a smaller right border for its possible values. As proved by the simulation results presented at the end of the paper, this natural assumption leads to a significant performance increase over the classical model.

8. Simulation study To assess the efficiency of the proposed stochastic model for the lead-time demand, a

simulation study was performed that compared the developed extended beta-binomial model (EBBM), the classical beta-binomial model (BBM), and the straightforward binomial model (BM). Initially, it was also planned to include into the simulation a simple model that treats each SKU in a group separately when performing the inventory control. However, this model has performed so poorly that its results were not included to the paper.

A separate remark should be made on what we consider as a benchmark for comparison in our study. A traditional approach is to compare the developed techniques with the Cros-ton’s method when dealing with the slow-moving items. However, this paper explores a spe-cial case when the average demand is very low, and a considerable percentage of the stock keeping units can have zero demands records. This makes the classical Croston’s approach inapplicable, since it can not forecast the future demand based on zero observations only. Thus, we consider the BM, which treats all items based on the average demand per period probability, as a benchmark model, that is consistent with the literature review given in the beginning of the paper.

In the following subsections, we first describe the simulation model used, and then pre-sent the simulation results and their analysis that confirm the superiority of the proposed model over the known ones.

16

Table 2 Simulation model notation

Symbols Interpretation

t Period index Tmax Maximum number of periods, items life cycle length (simulation model parameter) Vi(t) Actual stock available for inventory item i at period t Ri(t) Replenishment order that arrives for inventory item i at the beginning of period t di(t) Random binary demand generated for inventory item i and period t f(t) Total not satisfied demand at period t for all SKUs FT Average not satisfied demand over the simulation run cT Average holding cost per period over the simulation run I(C) Unit function, takes value of one if condition C holds, and zero otherwise

8.1. Simulation model

This subsection describes the simulation model that was used to compare the mentioned above stochastic models of the lead-time demand. To remind the notation, n is the number of periods for which the historical data is available, m is the lead time, and Di is the random lead-time demand for the item i (see Table 1 for the general notation description). A general work-flow for one iteration of the simulation model of the inventory system is presented at the next page (simulation model iteration algorithm). The corresponding notation is given in Table 2. The steps 7-15 are performed for each of the three models (BM, BBM, EBBM) for the same values of the demand d, and the variables {V, L, R, f, fT, cT} are individual for each model. The graphical representation of the simulation iteration workflow is given in Figure 2.

Generate demand di(t) for i = 1, 2, …, k (t = 1, 2, …, Tmax)

t > n ? noyes

B(i,t) ← di(t), ∀ i

t = n ? no

yes

Estimate model parameters based on B and calculate lead-time demand

probability distribution

Find re-order levels {Li} by solving cost / fill-rate optimization problem

Define initial stock Vi(n) ← Li, replenishment orders: Ri(n + m) ← 1

Add arrived replenishment orders to stock: )()1()( tRtVtV iii +−←

Calculate not satisfied demand:

∑ =>⋅←

k

i iii tVtItdtf d1

)( )()()()(

Estimate model parameters based on Band calculate lead-time demand

probability distribution

Find re-order levels {Li} by solving cost / fill-rate optimization problem

Define new replenishment orders:

≤+←+ ∑

−

+=τi

m

tiii LtRtVImtR

1

1)()()(

Decrease stock: )()()( tdtVtV iii −← and define B as )( jntdb iij +−=

t ← t + 1

Fig 2. Simulation model for inventory control system

17

Simulation model iteration 1. Generate the demand for the first n periods and store it in the binary k×n-

matrix B = (bit), where bit is the demand for the item i in the period t. 2. Estimate parameters for each model of the lead-time demand:

BM: )(1 1

nkbp k

i

n

t itbm ⋅= ∑ ∑= =;

BBM: using the MM-estimators defined by expression (8); EBBM: using the MM-estimators developed in Theorem 2.

3. Calculate the lead-time probability distribution for each model: BM: Bi(m, pbm); BBM: based on the expression (7); EBBM: employing the result of Theorem 4.

4. For each model, find the re-order level {Li} as the solution of the following constraint optimization problem:

FD

DLDtsLmin k

i i

k

i iik

iiLL k

≥⋅≤

∑∑∑

=

=

=1

1

1},,{ }{

}{}{..,:

1 E

EPK

, (50)

where F is the target fill rate (the user-defined parameter ), and the holding cost per period for each SKU is assumed to be the same (without loss of gen-erality). Discussion of the solution of the problem (50) for each of the consid-ered models was presented earlier, in Sections 3 and 4.

5. Initialize stock V to the re-order levels and place the replenishment orders R:

kimtRLtV iii ,,2,1,1)(,)( K=←+← . (51)

6. for t = n + 1 to Tmax (the user-defined parameter) do 7. Add replenishment orders that arrived:

kitRtVtV iii ,,2,1),()1()( K=+−← . 8. Generate demand {di(t)} for the current period. 9. Calculate not satisfied demand: ∑ =

>⋅←k

i iii tVtItdtf d1

)( )()()()( .

10. Decrease the actual stock: V )()()( tdtVt iii −← . 11. Define the matrix B using the last n periods: )( jntdb iij +−← . 12. Estimate the model parameters as described in Step 2. 13. Find the re-order level {Li} as outlined in Step 3. 14. Define new replenishment orders:

. (52)

≤+←+ ∑

−

+=τi

m

tiii LtRtVImtR

1

1)()()(

15. end_ 16. Calculate the iteration statistics:

• percentage of demand not met: ∑∑∑= ==

←k

i

T

nti

T

ntT

maxmax

tdtff1

)()( ; (53)

• average holding cost per period: )( 1)(1

+−← ∑∑= =

nTtdc max

k

i

T

ntiT

max

. (54)

18

It should be noted that a number of papers that deal with modeling the lead-time de-mand probability distributions judge on the efficiency of the proposed techniques by compar-ing the actual and modeled probability distributions for the test periods. However, the right tail of the distribution is what actually impacts the inventory system performance (since the mean values are usually close for all models). One can easily check that distributions, which are quite different, can lead to the same re-order level values for a given service level. Con-sider the following simple example: lead time m is 5, and suppose that after applying different modeling approaches one gets: (i) the binomial distribution with m and p = 0.1, and (ii) the beta-binomial distribution with the parameters m, α = 1, β = 9:

Table 3 Example to support the chosen efficiency comparison method

x 0 1 2 3 4 5 PBM(x) 0.5905 0.3280 0.0729 0.0081 0.0005 0.0000 PBBM(x) 0.6429 0.2473 0.0824 0.0225 0.0045 0.0005

If the service level is fixed at 0.95, both models set the re-order level to 2, also the probability rows are quite different. For this reason, we consider simulating the inventory control system operation based on different models to be a more realistic approach, since it can reveal if the more accurate distribution approximation has any actual practical significance.

The following subsection presents the simulation results obtained using the described above simulation model for the inventory control system.

8.2. Simulation results

To compare the performance of the proposed extended beta-binomial model, the classi-cal beta-binomial model and the straightforward binomial model of the lead-time demand, there were selected three demand patterns given in Figure 3. The corresponding population-averaged lead-time demand probability distributions for each pattern are illustrated in Fig-ure 4. These demand shapes where selected since they should be hard for simple methods to deal with; they correspond to the situation when the considered group of items can be roughly subdivided into two parts, with one having extremely low demand probabilities, and another sub-group having similar non-negative probabilities of demand. The major difference between the considered shapes is their skewness.

The histograms in Figure 4 reveal that for the 1st pattern, the population-averaged lead-time demand probability distribution is multimodal, and thus will create problems for the classical BM. Also the 2nd and 3rd patterns are unimodal, the third one shows clear over-dispersion with respect to the binomial distribution.

0 0.2 0.4 0.6 0.8 1 0

2

4

6

8

10 f

p

α = 0.025

β = 0.025

π = 0.5

E = 0.25

V = 0.06

# 1

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100 f

α = 0.5

β = 0.025

π = 0.2

E = 0.19

V = 0.001

# 2

p

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10 f

p

α = 0.1

β = 0.5

π = 0.6

E = 0.1

V = 0.03

# 3

Fig 3. Experimental design: patterns for probability of demand density function;

p – demand probability; f – probability density function; E – mathematical expectation, V – variance

19

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5 # 1

x

P E = 1.5

V = 2.91

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4# 2

x

PE = 1.14

V = 0.96

0 1 2 3 4 5 6 0

0.2

0.4

0.6

0.8# 3

x

PE = 0.6

V = 1.48

Fig 4. Histograms for population-averaged lead time demand probability distribution

for considered demand shapes (m = 6): x – number of successes, P - probability

For each of the demand patterns, the experiment was designed as follows. It was as-sumed that the group consists of k = 1000 slow-moving items, n = 6 periods are available as the historical demand data, the lead time m was taking values from {1, 2, …, 6}, and the fill rate was varying from 0.95 to 0.99 with the step 0.01. The product life cycle length Tmax was set to 36, and 500 iterations were performed for each set of the parameter values. The period can be thought of as one month, with half of the year available to estimate the model parame-ters, and the item life cycle length being 3 years. The efficiency of each model was measured using the average actual fill rate being equal or higher then the target one, see expression (53), and the obtained average holding cost per period, see formula (54).

To develop intuition about the selected simulation settings, consider the following dis-tribution of the total number of requests over the 36 periods for the 1st demand pattern, lead time m = 6, and target fill rate F = 0.99:

Table 4 Typical distribution of the total number of request over the whole observation period in the simulation study

s 0 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 25 27 28 29f 461 8 6 6 2 2 4 4 6 6 4 7 12 19 24 50 32 74 64 64 46 47 21 14 12 4 0 0 0 1

As follows from this table, 46% of the SKUs had zero requests over the whole observa-tion period, and 49% of the items had from 10 to 25 requests over 36 months. Thus, the distri-bution of the total demand is consistent with the one described in the problem description.

The first part of the simulation results is given in Table 5, which presents the average holding cost per period for the considered models of the lead-time demand for three selected patterns. The general conclusion that follows from the figures in the table is that the EBBM ensures the lowest holding cost, the BM provides the highest cost, and the holding cost for the BBM is in between. The analysis of the results for each pattern follows below.

For the 1st demand pattern, the average gain for the developed EBBM vs. the BBM and BM for the considered lead times and target fill rates is 10% and 32% respectively. It means that relying on the extended beta-binomial model instead of the classical binomial one can lead to holding about 1/3 less stock while achieving the same target fill rate. At the same time, using the theoretical results obtained in this paper instead of the usual beta-binomial model can cut the holding costs by about ten percent. Both of these figures prove the practical sig-nificance of the proposed in the paper extended beta-binomial model.

The results for the 2nd and 3rd demand patterns support the formulated above thesis. The average gains of the EBBM over the known models are 1%, 8% and 6%, 30% respectively, that is also a significant holding cost decrease. Besides, as follows for the average gain fig-ures, the EBBM performs the best with respect to the other considered models for the demand pattern #1, when there exists a clear separation between the items with the very low demand probability, and the ones with the probability close to a non-zero value π. The reason why the

20

Table 5 Average holding cost per period for the considered models of the lead-time demand (in 1000’s)

BM BBM EBBM m\F 0.95 0.96 0.97 0.98 0.99 0.95 0.96 0.97 0.98 0.99 0.95 0.96 0.97 0.98 0.99

Pattern #1 1 2.00 2.00 2.00 2.00 2.00 1.53 1.54 1.57 1.64 1.65 1.47 1.47 1.48 1.48 1.60 2 2.52 2.53 2.53 2.54 2.76 1.82 1.86 1.97 2.12 2.23 1.61 1.67 1.86 1.92 2.00 3 2.76 2.76 2.76 2.76 3.12 2.06 2.18 2.32 2.34 2.79 1.99 2.11 2.20 2.35 2.50 4 3.01 3.08 3.08 3.14 3.57 2.33 2.43 2.61 2.80 3.11 2.27 2.42 2.48 2.56 2.83 5 3.09 3.29 3.30 3.30 3.38 2.57 2.58 3.07 3.23 3.31 2.34 2.40 2.41 2.76 3.01 6 3.15 3.30 3.84 3.88 3.88 2.78 2.90 3.24 3.39 3.64 2.38 2.63 2.61 2.86 3.28

Pattern #2 1 1.90 1.92 2.00 2.00 2.00 1.80 1.83 1.91 1.93 1.98 1.80 1.82 1.91 1.91 1.98 2 2.00 2.00 2.47 2.64 2.64 1.90 1.92 2.27 2.54 2.63 1.88 1.91 2.24 2.51 2.62 3 2.47 2.47 2.64 2.82 2.82 2.36 2.41 2.46 2.55 2.64 2.36 2.39 2.46 2.54 2.63 4 2.64 2.64 2.82 3.14 3.46 2.38 2.50 2.55 2.92 3.33 2.37 2.47 2.54 2.89 3.23 5 3.05 3.14 3.14 3.30 3.47 2.55 2.80 3.04 3.14 3.27 2.52 2.80 2.98 3.09 3.21 6 3.14 3.30 3.30 3.30 4.14 2.90 3.00 3.04 3.19 3.81 2.86 2.94 2.99 3.14 3.66

Pattern #3 1 1.81 1.81 1.90 1.90 2.00 1.20 1.24 1.24 1.33 1.33 1.20 1.24 1.24 1.33 1.33 2 1.90 1.90 2.00 2.00 2.53 1.35 1.43 1.45 1.55 2.16 1.27 1.43 1.43 1.45 1.66 3 2.00 2.00 2.23 2.64 2.72 1.51 1.59 1.94 2.26 2.41 1.48 1.57 1.86 1.95 2.09 4 2.52 2.63 2.63 2.81 2.81 1.72 2.14 2.28 2.40 2.57 1.60 1.87 2.25 2.38 2.47 5 2.55 2.55 2.55 2.70 2.97 2.18 2.26 2.36 2.55 3.19 2.00 2.19 2.35 2.53 2.58 6 2.63 2.70 2.72 2.72 3.55 2.38 2.42 2.50 2.81 3.20 2.21 2.30 2.47 2.60 3.01

EBBM and BBM perform approximately the same for the 2nd demand pattern is that the skewed to the left U-shaped demand distribution is easier approximated by the classical beta distribution, and the right tails for the probabilities of the lead-time demand are close for both models.

The second part of the simulation results is presented in Table 6, that gives the actual fill rate for the BM for the considered demand patterns, lead times, and the target fill rates. For the BBM and EBBM, the average actual fill rate was above the target level in all runs, and thus we do not show the corresponding figures in the table. Table 6 reveals that while hitting the target in the most of the cases, the BM can lead to very significant deteriorating of the actual fill rate with respect to the target one. For example, for the 3rd demand pattern, lead time 6 and target fill rate 0.95, the BM allows satisfying only 48% of the demand that is less than a half, and is absolutely unacceptable. A similar situation is for the 1st demand pattern, although in this case the considered model performs better. At the same time, for the 2nd de-mand pattern the BM performance is good.

An explanation of the difference in the BM’s performance lies in the properties of the population-averaged lead-time demand probability distribution. Looking back at Figure 4 re-minds that the 1st and 3rd demand patterns correspond to the multimodal and overdispersed Table 6 Actual fill rate of the beta-binomial model for the considered demand patterns

Pattern #1 Pattern #2 Pattern #3 m\F 0.95 0.96 0.97 0.98 0.99 0.95 0.96 0.97 0.98 0.99 0.95 0.96 0.97 0.98 0.99

1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.84 0.85 0.90 1.00 1.00 4 0.96 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.76 0.90 0.92 0.92 0.93 5 0.96 0.96 0.96 0.96 0.99 1.00 1.00 1.00 1.00 1.00 0.74 0.74 0.74 0.75 0.84 6 0.83 0.87 0.97 0.98 0.98 1.00 1.00 1.00 1.00 1.00 0.48 0.48 0.48 0.50 0.80

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0 1 2 3 4 5 6 0 200 400 600 800

1000

0 1 2 3 4 5 60

200

400

600

800

1000

0 1 2 3 4 5 6 0

200

400

600

800

1000Binomial

model Beta-binomial

model Extended

beta-binomial model

RL

Fr

RL

Fr Fr

Fig 5. Typical distribution of the re-order levels for the considered models (demand pattern # 1, m = 6, F = 0.95): RL – re-order level, Fr – frequency

cases, and thus are badly handled by the binomial model. The 2nd pattern is easily approxi-mated by the binomial distribution, and as a result, the corresponding lead-time demand model performs quite good both the for average stock holding cost (Table 5) and the actual fill rate (Table 6).

An intuitively appealing visualization of the considered models performance is given in Fig 5, which presents histograms for the typical re-order levels for the demand pattern #1 for the given model parameter values. The BM sets the re-order level to 3 to all SKUs. At the same time, the BBM and EBBM differentiate between the items and assign a low re-order level to about half of them (1 for the BBM, and 0 for the EBBM), and high levels (4-6) to the others. As a result, the BM keeps too much stock for one group of the items and thus leads to high holding cost (1.32 times worse then the EBBM and 1.13 times worse then the BBM). For another group of the SKUs, the BM underestimates the demand that leads to only 83% of the demand actually met. At the same time, the EBBM manages to more precisely reveal the de-mand heterogeneity and provides 1.17 times lower average holding costs than the BBM, while both models meet the target fill rate.

A separate computational study was performed to investigate the relation between the holding stock cost gain and the lead-time. For the demand pattern #1 under the assumption that nine months of historical data (n = 9) are available for models identification, and the tar-get fill rate is fixed at 0.99, the lead time was varied from 1 to 9 with step 1. Figure 6 illus-trates how the average holding stock cost gain of the EBBM vs. BBM and BM changes as the lead time increase; as expected, both curves increase with m. One can notice that for n = 9, the BM performs worth in comparison with the EBBM than for n = 6 (see Table 5), and, on the contrary, the gain with respect to BBM is smaller. The reason for this is that having more his-torical data allows the EBBM and BBM more precisely define the inventory items heteroge-neity, while the BM still relies on the population-averaged approach. On the other hand, the same reason leads to the decrease of the difference between the EBBM and BBM. Hence, the proposed model provides most of it advantages for the case with the short historical records.

1 2 3 4 5 6

1

1.1

1.2

1.3

1.4

1.5

1.6

Binomial model

Beta-binomial model

m

g

Extended beta-binomial model0.9

Fig 6. Gain in stock holding expenses of the developed extended beta-binomial model with respect to the binomial and beta-binomial models (demand pattern # 1, n = 9, F = 0.99): g – number of times decreased

22

9. Conclusions The paper considers the problem of modeling the lead-time demand for the multiple

slow-moving inventory items in the case when the available demand history is very short, and a large percentage of items has only zero records. To overcome the mentioned problems with the historical demand data, the Bayesian forecasting approach is employed to predict the lead-time demand probability distribution. Since the demand can be considered binary, and the past history records are short, we propose to use the beta-binomial model that is well-known to successfully solve problems with the similar data structure in the statistical longitudinal data analysis area. Further, an extension of this model is developed that allows accounting for the prior information regarding the maximum expected probability of demand per period. Pa-rameter estimation and Bayesian forecasting routines are derived for the new model. The effi-ciency and practical significance of the obtained results is proved by the simulation study.

The following conclusions follow from the conducted in the paper simulation study.

(i) The binomial model, that relies on the population-averaged demand probability per pe-riod, tends to overestimate the inventory needed to achieve the target fill rate, and is not reliable with respect to the actual target fill rate. For the inventory system settings considered in the paper, the BM can overestimate the inventory by near 30% or lead to only about 50% of the demand actually met, that is unacceptable for the systems with high reliability requirements.

(ii) The beta-binomial model, which is proposed as a solution for the problem of the data sets being “short and wide”, demonstrates stable and robust performance, ensures lower holding costs when compared to the BM, and guarantees the actual fill rate being con-sistent with the target. For the considered demand patterns, the BBM ensures the gain in inventory costs savings up to 20%, being at the same time reliable with respect to the fill rate.

(iii) The developed in the paper extended beta-binomial model, that allows limiting the de-mand per period by introducing an additional parameter π, ensures even lower holding costs then the BBM, and also provides the actual fill rate consistent with the target. The simulation study has indicated that the EBBM can lead to decreasing the holding ex-penses by about 10% and 30% when compared to the BBM and BM, while the compu-tational complexity with regards to the beta-binomial model is not significant.

As a result, the paper contribution can be viewed as follows. The practical aspect of the obtained results is the advance in the lead-time demand forecasting for the multiple slow-moving inventory items with extremely low demand and short historical records. The meth-odological aspect of the paper is the application of the statistical longitudinal data analysis techniques in the inventory management. Finally, the theoretical aspect deals with the pro-posed extension of the beta-binomial distribution, and development of the parameter estima-tion and Bayes forecasting routines for it.

Future work will deal with evaluating the performance of the proposed technique using real-life industrial datasets.

10. Acknowledgements This research was supported by the European Union grant INTAS YSF 03-55-869.

23

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