On the Performance of Standard Practices in Inventory Control when Demand is Frequently a Multiple of some Base Demand Quantity

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On the Performance of Standard Practices in Inventory Control

when Demand is Frequently a Multiple of some Base Demand

Quantity

W.D. Touber

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: prof. dr. R.H. Teunter

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On the Performance of Standard Practices in Inventory Control

when Demand is Frequently a Multiple of some Base Demand

Quantity

W.D. Touber, s1977482 Abstract

We investigate the performance of two standard practices in inventory control when demand is frequently a multiple of some base demand quantity (BDQ). Although such demand characteristics commonly arise in practice, they have not been studied before. Firstly, it is standard practice to model lead time demand by a normal distribution and estimate lead time demand variance on the basis of one period ahead forecast errors. This standard approach can cause poor inventory control performance as using the normal distribution is inappropriate under many demand conditions and the variance estimator tends to be biased. We explore the potential benefits of alternative approaches on the basis of theoretically generated demand data as well as demand data from two Dutch food manufacturers. More specifically, we consider three alternatives to the normal distribution among which a distribution specifically designed to model the demand characteristics under consideration. Results indicate that the alternative distributions can improve performance substantially. Moreover, we consider an alternative variance estimator which employs lead time forecast errors. However, results regarding both estimators are inconclusive, hence future research is needed. Secondly, judgmentally adjusting statistical forecasts is common practice. Still, the effect of these adjustments is ill-researched. We find that the adjustments made by a Dutch food manufacturer considerably enhance inventory control performance.

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List of Abbreviations

ADIDA Aggregate-Disaggregate Intermittent Demand Approach

BDQ Base Demand Quantity

CRO Croston’s method

CSL Cycle Service Level

CV Coefficient of Variation

EOQ Economic Order Quantity

FFC Final Forecast

MAE Mean Absolute Error

MSE Mean Squared Error

PB Percentage of times Better

pp Percentage point

RGRMSE Relative Geometric mean of RMSEs

RMSE Root Mean Squared Error

SBA Syntetos-Boylan Approximation

SES Single Exponential Smoothing

SFC Statistical Forecast

SKU Stock Keeping Unit

SMA Simple Moving Average

TSB Teunter, Syntetos and Babai method

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1 Introduction

Inventory control is a crucial problem for almost all organizations in every sector of the economy (Axs¨ater,

2006). One wishes to find the optimal trade-off between inventory investment and customer service. If customers cannot be served due to insufficient inventory you miss out on potential revenue. In the worst case customers take their business elsewhere and never return which can be catastrophic for the future of the company. Enhanced customer service can be attained by maintaining higher inventory levels, however the corresponding cost increase may trigger profit losses. This research considers a common approach to tackle this problem. In this one specifies appropriate service level targets and attempts to design an inventory control system that achieves these service targets at minimal costs.

Perfect customer service is often impossible to reach and if attainable, the associated inventory costs will likely be out of proportion. Consequently, service targets are below 100%, meaning that stock-outs will occur even when targets are met. In case demand cannot be met immediately due to depleted inventory, some customers might be willing to wait until the next replenishment epoch before receiving their order. On the other hand, if customers do not opt for backorders, unmet demand leads to lost sales. We make the assumption that all customers opt for backorders, hence lost sales never occur.

A main decision problem in inventory control is determining the timing and size of replenishment orders. After an order has arrived, the inventory levels should be able to cover demand up to the moment of the next replenishment to such an extent that service targets are met. The required inventory level can be defined as the sum of expected demand during this replenishment cycle and extra stock, called safety stock, maintained to mitigate demand uncertainty. Hence, in order to determine order quantities appropriately an estimation of demand regarding a certain future period is needed. In this study this future period concerns the lead time, i.e. the time from the moment an order is placed until it is available on shelf. To model lead time demand, usually a probability distribution is employed which is then referred to as a predictive demand distribution.

Predictive distributions are commonly fitted on the basis of two demand parameters, namely the demand mean and the demand variance. Next, we describe a widely applied procedure to obtain estimates of the lead time demand mean and variance. First, a forecasting method uses historic demand data to generate statistical forecasts concerning the periods that comprise the lead time. Judgmentally adjusting

these statistical forecasts is very common practice in industry (Fildes and Goodwin, 2007). Usually

managers revise statistical forecasts in an attempt to incorporate additional information that cannot be captured by the forecast models. The adjustments, however, do not necessarily improve forecasts, they may even damage forecast accuracy (Franses and Legerstee, 2010). Despite its practical relevance, just two studies empirically investigate the influence of judgmental revisions on inventory control performance (Syntetos, Nikolopoulos, Boylan, Fildes, and Goodwin, 2009; Syntetos, Nikolopoulos, and Boylan, 2010). The small number of studies and the corresponding mixed results indicate a need for complementary empirical research into the effect of judgmental adjustments to statistical forecasts.

The sum of the final, possibly judgmentally adjusted, forecasts is used as an estimate for the lead

time demand mean. As some textbooks (e.g. Silver, Pyke, and Peterson, 1998; Axs¨ater, 2006) suggest,

the variance of historic one period ahead forecast errors is used to represent the per-period demand variance. Multiplying the estimate (e.g. Mean Squared Error, MSE, or Mean Absolute Error, MAE) of this variance by the lead time provides an estimate of the lead time demand variance. We remark that this research considers variance estimation through MSE. Although this way the per-period forecast error in the demand mean is captured correctly, the correlation of forecast errors for different periods of the lead time as well as the error in the per-period demand variance is ignored. As a consequence, application of the aforementioned estimation procedure can lead to significant service target underachievements (Prak, Teunter, and Syntetos, 2017).

For many reasons, the continuous normal distribution is the dominant choice of probability distribution

for modeling demand, in particular for high demand items (Axs¨ater, 2006; Silver et al., 1998). Thus, fitting

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Therefore, using the resulting predictive distribution to make inventory decisions is referred to as the standard approach. A disadvantage of the standard approach is that its predictive distribution attaches a positive probability to the occurrence of negative demand. Although negative demand may be interpreted as items that are returned, this characteristic of the normal distribution is often inconvenient in practice. Additionally, in practice (high demand) items appear for which the demand characteristics and the normal distribution do not match. An interesting example is items for which demand frequently, but not always, equals a multiple of some Base Demand Quantity (BDQ). Such demand characteristics appear in practice since fixed order quantities are frequently administered, e.g. the well-known Economic Order Quantity (EOQ) derived by Harris (1913) in an attempt to minimize total ordering and holding costs. Fixed order quantities can be very convenient from a logistical perspective as well. A BDQ can, for example, correspond to a physical unit containing multiple trading units, e.g. a pallet that carries 100 items. Next, we list some reasons causing compound demand to be frequently, but not always, a multiple of some BDQ:

• One or more customers always or frequently order a multiple of same BDQ while other customers sporadically order quantities that are not a multiple of this BDQ;

• Customers, for whatever reason, occasionally deviate from their BDQ;

• When multiple customers frequently order multiples of distinct BDQs, the greatest common divisor of these BDQs can be considered as BDQ.

Because such demand features cannot be modeled by the normal distribution, application of the standard approach may cause poor inventory control performance. Surprisingly, to the best of our knowledge, this matter has not been explored before.

In this paper, we investigate the inventory control performance of standard practices in inventory control, namely applying the standard approach and judgmentally adjusting statistical forecasts, in a backorder system when demand is frequently, but not always, a multiple of some BDQ. Inventory control performance is measured in terms of the achieved service level and the required inventory investment. Two service measures are considered, namely the cycle service level (CSL), i.e. the probability of no stock-out during a replenishment cycle, and the volume fill rate (VFR), i.e. the proportion of demand that is satisfied directly from stock, also known as type 1 and type 2 service measure, respectively.

Moreover, we use the standard approach as benchmark and explore the potential benefits of alternative approaches which differ in one or two aspects from the standard approach. The first aspect concerns the estimation of the lead time demand variance. As mentioned earlier, the standard approach ignores the correlation of forecast errors as well as the error in the per-period demand variance which may cause poor inventory control performance. We consider an alternative estimation procedure proposed by Syntetos and Boylan (2006a) that directly estimates the lead time demand variance by the MSE regarding lead time forecast errors. Although, this way the correlation of forecast errors is intuitively corrected for, the error in the variance estimate is still ignored. Prak and Teunter (2018) present a general framework for dealing with estimation uncertainty, but this is beyond the scope of this research. Syntetos and Boylan (2006b) empirically compare the inventory control performance of the lead time MSE method and the standard approach variance estimation method. They model demand by a normal distribution, consider CSL, produce forecasts through single exponential smoothing (SES) and conclude that the lead time MSE method does not perform particularly well. To the best of our knowledge, we are the first to compare the inventory control performance of these methods in case of judgmentally adjusted forecasts and distinct predictive distributions for both CSL and VFR.

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selection rule of Adan et al. (1995) is applied as well. The last alternative employs a mixture of the first two alternative distributions. In this, the probability assigned to the BDQ distribution is used to reflect the probability with which demand is a multiple of the BDQ. Updating this probability periodically, pre-sumably renders this so-called general distribution particularly suitable for demand that is frequently a multiple of some BDQ.

The standard and alternative approaches are compared both theoretically and empirically. In the theoretical setting we simulate a stationary demand process involving demands that are frequently a multiple of some BDQ by sampling from a theoretical demand distribution. Forecasts are produced by a simple moving average (SMA) and a periodic review order-up-to-level inventory policy is implemented in order to evaluate inventory control performance. A sensitivity analysis is performed to gain insights into the effects of several demand, estimation and inventory control parameters on inventory control performance. The empirical analysis is similar but relies upon weekly historic demand series (ranging between week 1 2016 and week 52 2017) of 135 stock keeping units (SKUs) from two Dutch food manufacturers. These food manufacturers wish to remain anonymous and are hereafter referred to as food manufacturer A and food manufacturer B. Food manufacturer A provided us with historic statistical forecasts generated by their forecast tool as well as their final, possibly judgmentally adjusted, forecasts for 87 SKUs. This allows us to investigate the influence of judgmental adjustments to statistical forecasts on inventory control performance for each of the considered approaches. Furthermore, the demand data from both food manufacturers are used to evaluate the performance of the approaches under consideration. Here the method proposed by Croston (1972) (CRO) is used to generate forecasts.

The contribution of this work to the existing literature is threefold. Firstly, to the best of our knowledge, we introduce and explore demand characteristics that have not been described before in literature, namely demand that is frequently, but not always, a multiple of some BDQ. Secondly, we add to the literature on the application of discrete predictive distributions in inventory control. There has been little work in this regard, see Section 2.2, and research in this direction is urgently needed (Syntetos, Babai, Boylan, Kolassa, and Nikolopoulos, 2016). Finally, by investigating the effect of judgmental adjustments on forecasting and inventory control performance, we make an important contribution to the research field of judgmental

forecasting. This contribution is valuable because of the small number of available empirical studies

concerning this subject and the corresponding inconclusive findings, see Section 2.1.

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2 Related literature

This section presents an overview of the most relevant literature regarding judgmentally adjusting sta-tistical forecasts (Section 2.1) and the use of predictive distributions in inventory control (Section 2.2). Consequently, we show the relevance and innovativeness of this study.

2.1 Judgmentally adjusting statistical forecasts

Statistical forecasts can be reliable in certain contexts, however they may be bad at forecasting special

events of which the effects cannot be captured by the forecast model. Examples of such events are

(competitive) sales campaigns and introductions of new (competitive) products. In case experts have partial knowledge regarding such events, their judgmental adjustments to the statistical forecasts can improve forecast accuracy (Goodwin, 2002; Sanders and Ritzman, 2001). On the other hand, statistical forecasts are frequently, and typically unnecessarily, revised for the following reasons:

• A desire for a sense of ownership of the forecasting system (Goodwin, 2002; Onkal and Gonul, 2005). Due to a lack of expertise regarding forecasting (tools), the forecasting system can be perceived as a ‘black box’. Then, in order to gain a better understanding of how forecasts are derived, managers intervene into the system.

• The misinterpretation of noise (Goodwin, 2005; Harvey, 1995). The tendency of managers to rec-ognize patterns where there are not necessarily any patterns at all typically results in unnecessary small adjustments to statistical forecasts.

• Political pressures can favor adjusting statistical forecasts (Goodwin, 1996). For example, in sales functions, managers may inflate forecasts in an attempt to avoid stock-outs.

Although judgmentally adjusting statistical forecasts is a very common practice (Fildes and Goodwin, 2007), there is just a moderate number of empirical studies that compare the accuracy of statistical and judgmentally adjusted forecasts. Mathews and Diamantopoulos (1986, 1989, 1990, 1992) analyze a sample of more than 900 products from a major UK pharmaceutical manufacturer and retailer and conclude that judgmental adjustments to statistical forecasts enhance forecast accuracy.

Fildes, Goodwin, Lawrence, and Nikolopoulos (2009) collect more than 60,000 pairs of statistical

forecasts and judgmentally adjusted forecasts from four companies. For three of the four companies

(home cleaning, pharmaceutical and food & beverages) judgmental adjustments appeared to, on average, increase forecast accuracy. A detailed analysis revealed that relatively larger adjustments usually lead to more significant improvements in accuracy while smaller adjustments often damage accuracy. Moreover, positive adjustments are less likely to improve accuracy than negative adjustments. In addition, the positive adjustments were made in the wrong direction more frequently, indicating a bias towards optimism. In the fourth company (a major UK retailer), that was adjusting statistical forecasts less frequently, judgmental adjustments were actually found to reduce forecast accuracy.

Franses and Legerstee (2009, 2010, 2011a,b, 2013) explore a database containing statistical and judg-mentally adjusted forecasts from the pharmaceutical industry, across 35 countries and seven distinct prod-uct categories. The most important result regarding this series of publications was that Franses and Legerstee (2010) find that judgmentally adjusted forecasts are, at best, as good as statistical forecasts, but in most cases they are worse.

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Syntetos et al. (2009) consider CSL for intermittent demand forecasts and conclude that judgmental adjustments improve service levels at the expense of a modest stock increase. Syntetos et al. (2010) conduct similar research concerning fast moving items for both CSL and VFR. They find that a small improvement in forecasting accuracy due to judgmental adjustments may result in a far more significant inventory control performance improvement. Contrary to Syntetos et al. (2009), they find adjustments not only to improve achieved service levels but also to reduce inventory levels substantially. The effect on inventory control performance appeared to be similar for both service measures.

By investigating the effect of judgmental adjustments on forecasting and inventory control performance, we make an important contribution to the research field of judgmental forecasting. This contribution is especially valuable because of the small number of available empirical studies regarding this subject and the corresponding inconclusive findings. Moreover, these studies employ the normal distribution to model demand and use the lead time MSE method for variance estimation whereas we also facilitate other distributions and another variance estimation method.

2.2 Predictive distributions in inventory control

In this section we solely discuss distributions that can be fitted on the first two moments as this is common practice. Still, fitting procedures involving other moments have been proposed in literature, see e.g. Kottas and Lau (1980).

Burgin (1975) presents two criteria to be satisfied by a demand distribution: 1) the distribution can represent non-negative demand; 2) the shape of the density function changes from monotonic decreasing for low demand to a normal type distribution for high demand. The normal distribution satisfies neither of these criteria since its density function is symmetric and attaches a positive probability to negative

demand. Yet, the normal distribution may be appropriate for modeling high demand (Axs¨ater, 2006; Silver

et al., 1998) or demand over many periods (Syntetos and Boylan, 2006b). The latter can be explained by the central limit theorem. Certain researchers argue that the normal distribution is appropriate if the coefficient of variation (CV) of demand is low (e.g. Silver et al., 1998; Tadikamalla, 1984). Lau and Lau (2003) summarize the findings in literature regarding the appropriateness of the normal distribution and show cases where the use of the normal distribution can cause poor inventory control performance, even if the CV is low.

Syntetos, Babai, Lengu, and Altay (2011) use both the CV and the inter-demand interval to identify the appropriateness of several distributions. They compared the empirical fit of the normal, gamma, Poisson, negative binomial and stuttering Poisson distribution and conclude that the normal distribution should only be used if the inter-demand interval and CV are small. The Poisson distribution is to be used for low CVs and the gamma distribution for extreme CVs and inter-demand intervals. In the remaining cases the negative binomial or stuttering Poisson distribution is deemed appropriate.

Furthermore, distribution selection rules on the basis of the first two moments have been proposed, e.g.

by Axs¨ater (2006) and Adan et al. (1995). In this research we employ the procedure suggested by Adan

et al. (1995) which fits a negative binomial distribution, a Poisson distribution or a mixture of binomial

distributions, see Appendix A. Rossetti and ¨Unl¨u (2011) compare the inventory control performance of the

normal, lognormal, gamma, Poisson and negative binomial distribution as well as distribution selection

rules under a wide variety of demand conditions. Their theoretical results indicate that distribution

selection rules, among which the one that we employ, offer great potential for modeling demand.

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3 Methodological framework

This section presents the methodological framework that we use to investigate the problem at hand. First, in Section 3.1 the inventory policy that we implement is discussed. The inventory control performance measures are addressed in Section 3.2. Section 3.3 formally defines the four predictive distributions that we compare in this research. Then, Section 3.4 considers the forecasting models employed in this study. Next, it is explained in Section 3.5 how the distributional parameters are estimated on the basis of historic forecasts and demand observations. Finally, Section 3.6 demonstrates how to calculate order-up-to levels for each of the predictive distributions. Table 1 presents the notation used in this paper.

Table 1: Notation

Symbol Definition

α0, α∗ Achieved and target cycle service level, respectively β0, β∗ Achieved and target volume fill rate, respectively δ Exponential smoothing parameter

Dt,L Random variable representing cumulative demand over the L periods succeeding

period t

φ, Φ Standard normal probability density function and cumulative distribution function, respectively

G(z) Standard normal loss function, i.e. φ(z) − z(1 − Φ(z)) N (µ, σ2

) Continuous normal distribution with mean µ and variance σ2

fN, FN Probability density function and cumulative distribution function corresponding to

a normal distribution, respectively

D(µ, σ2₎ _{Discrete distribution fitted on the basis of mean µ and variance σ}2 _{according to the}

procedure proposed by Adan et al. (1995), see Appendix A

fD, FD Probability density function and cumulative distribution function corresponding to

D(µ, σ2_{), respectively}

fM, FM Probability density function and cumulative distribution function corresponding to

the discrete distribution that models the multiplicity of B, i.e. D(µ/B, σ2_/B2_),

respectively

t Time period index

B Base demand quantity (BDQ)

Ω The set of multiples of the BDQ, i.e. Ω = {kB|k ∈ N+}

T Review period

L ∈ N+

. Lead time

Qt Quantity ordered in period t

ILt Inventory level at the end of period t

IPt Inventory position at the end of period t

St Order-up-to-level determined at the end of period t

dt ∈ N. Demand in period t

ˆ

dt,k k period ahead forecast made at the end of period t

t Forecast error in period t, i.e. dt− ˆdt−1,1

ˆ

µt,L Estimate of the lead time demand mean made at the end of period t

M SEt Estimate of the per-period demand variance made at the end of period t

M SEt,L Estimate of the lead time demand variance made at the end of period t

ˆ

σ2t,L Estimate of the lead time demand variance made at the end of period t

ˆ

γt Estimate of the probability with which per-period demand is a multiple of B made

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3.1 Inventory control policy

A properly defined inventory control policy regulates the timing and size of replenishment orders. If service targets have been specified, the inventory policy should attempt to achieve these targets while keeping inventory levels as low as possible. We consider a widely applied inventory model that is relatively simple and close to optimal (Sani, 1995), namely a periodic order-up-to-level (T, S) model. Before we can explain the mechanics of this policy we should distinguish between the inventory level (IL) and the inventory position (IP). The former equals the stock on hand minus backorders. Then, the inventory position is obtained by adding the outstanding orders to the inventory level.

When administering the aforementioned (T, S) policy, the inventory position is reviewed periodically. Here, the review period T , i.e. the time interval between reviews, is constant. The food manufacturers that provided us with demand data collect data on a weekly basis, so that the inventory review period can be considered to be one week (T = 1). At a review an order is placed that raises the inventory position to the order-up-to-level S. The lead time L, which must be integer since T = 1, specifies the number of weeks from the moment the order is placed until it is available on shelf. For example, in production, L often includes order preparation time, transit time, production time and the time required for quality inspections.

In reality one usually faces supply uncertainty, e.g. varying lead times and order quantities. Yet, for the sake of simplicity, we ignore this source of variation and assume that an order of Q units placed now, will lead to an inventory level increase of precisely Q units exactly L time periods later. On the other hand, demand uncertainty is not ignored. Future demand is estimated periodically and S is set accordingly. The order of events within an arbitrary period t is as follows:

1. The order placed L time units ago, i.e. Qt−L, is received;

2. Random demand dtoccurs;

3. To reflect inventory costs, determine inventory level ILtat the end of the period:

ILt= ILt−1+ Qt−L− dt;

4. Given an estimation of future demand, determine Stand order Qtthat raises the inventory position

(= IPt−1− dt) to St, hence

Qt= max{St− IPt−1+ dt, 0},

and IP is updated as follows:

IPt= IPt−1− dt+ Qt.

Thus, the order-up-to level S should cover demand during L subsequent periods to such an extent that

service targets are met while keeping inventory costs at a minimum. However, as future demand is

unknown, one requires an estimation of demand for this future period in order to determine S appropriately. In this study this estimation takes the form of a predictive distribution. Section 3.3 presents the four predictive distributions that we explore and Section 3.6 demonstrates how S is determined for each of these predictive demand distributions.

3.2 Inventory control performance measures

On the basis of ILt values regarding n periods, the inventory control measures can be calculated. The

achieved CSL α0, i.e. the fraction of replenishment cycles in which no stock-out occurred, is calculated as:

α0=

Pn

t=11{ILt≥0}

n ,

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VFR β0, i.e. the proportion of demand that is satisfied directly from stock or the proportion of demand that is not backordered, is found as follows:

β0= Total demand directly satisfied

Total demand , = Pn t=1min{dt, ILt−1+ Qt−L} Pn t=1dt .

The average inventory on hand at the end of the period, used as a measure of inventory investment, equals:

IL =

Pn

t=1max{ILt, 0}

n .

3.3 Predictive demand distributions

This section introduces the four lead time demand distributions explored in this study. All four distribu-tions are fitted directly on the basis of lead time parameters. Their formal definidistribu-tions are presented in Definition 1.

The standard approach fits a continuous normal distribution with lead time demand mean µL and

lead time demand variance σ2

L. The first alternative to the normal distribution that we explore is a

discrete distribution. More specifically, given µLand σL2 we fit a negative binomial distribution, a Poisson

distribution or a mixture of binomial distributions according to the procedure proposed by Adan et al. (1995) as explained in Appendix A. The normal as well as the discrete distribution might be inappropriate in case demand is frequently a multiple of some BDQ. Then, depending on the frequency with which demand is a multiple of B, enhanced inventory control performance might be achieved by an alternative approach that (incorrectly) assumes that demand is always a multiple of B. We implement this approach

by fitting a discrete distribution that models the multiplicity of B. We use µL/B and σL2/B

2_{to represent}

the lead time mean and variance regarding this multiplicity, respectively. The distribution that models the multiplicity of B is then obtained on the basis of these demand parameters by applying the fitting procedure of Adan et al. (1995). Multiplying the resulting distribution by B provides the lead time demand distribution which is referred to as the BDQ distribution.

However, our focus is on demand that is just frequently, and not always, a multiple of B. Hence, we consider a so-called general distribution which takes into account the frequency with which demand is a multiple of B. To this end, a mixture of the discrete and BDQ distribution is employed. We define γ as the probability with which per-period demand is a multiple of B and raise it to the power of L, i.e.

γL_{, to obtain an approximation of the probability with which lead time demand is a multiple of B}1_{. The}

probability γLis attached to the BDQ distribution since it is used to model demands that are a multiple of

B. As a consequence, the general distribution is equal to the discrete distribution with probability 1 − γL_.

Since the demand characteristics concerning demands that are a multiple of B and demands that are not may differ significantly, it is appropriate to separately estimate the parameters concerning the

distributions that comprise the general distribution. This means that the lead time parameters µ1,Land

σ2

1,Lof the BDQ distribution should solely be based on historic demands that are a multiple of B. Then,

the remaining demands ought to be used for estimating the lead time parameters µ2,L and σ2,L2 of the

discrete distribution. Unfortunately, in the empirical analysis the true value of B is not known with certainty which complicates splitting historic demands observations into demands that are a multiple of B and demands that are not. Moreover, the statistical and final forecasts provided by food manufacturer A cannot be split. To overcome these issues, in the empirical analysis we simply use the same parameter

estimates for both distributions, i.e. we use µ1,L = µ2,L = µL and σ1,L2 = σ

2 2,L = σ

2

L. Yet, in the

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theoretical analysis B is known and we will investigate whether estimating parameters separately does affect inventory control performance.

We realize that the specification of the general distribution is not exact. One should actually consider the convolution of the per-period mixture distributions concerning the periods that comprise the lead time. However, due to time constraints we were unable to investigate whether doing so improves inventory control performance. Still, the performance of applying the general distribution for L = 1 may lead to a hypothesis with respect to the performance of the exact distribution for L > 1. For example, if our results for L = 1 indicate poor inventory control performance, we expect similar conclusions for the exact distribution for L > 1.

At first it might seem redundant to consider the discrete and BDQ distributions next to the general distribution as the former two distributions are special cases of the latter. Yet, we include all three distribu-tions for performance evaluation purposes. For instance, if employing the general distribution outperforms using the normal distribution, it does not necessarily mean that this is due to its capability of adequately modeling the demand characteristics under consideration. It might be the case that implementing the discrete and/or BDQ distribution outperforms applying the normal distribution as well. Then, we should look at the performance differences between these distributions and the general distribution in order to evaluate the contribution (in terms of inventory control performance) of the general distribution.

A restriction of the discrete, BDQ and general distribution, is that the mean should be greater than zero. Although a zero mean for non-negative demand is unrealistic, we should specify how to deal with this case. Therefore, the order-up-to-level is simply set to zero when the lead time demand mean is

zero. The case σ_L2 = 0 is treated differently as well because it represents the situation where demand is

assumed to be equal to µLwith certainty. Then, fitting a demand distribution becomes redundant as the

order-up-to-level can be determined directly on the basis of the service target.

Definition 1 Let the base demand quantity be given by B, then we define four probability distributions used to model lead time demand:

• Normal: Lead time demand is modeled by a continuous normal distribution with mean µL and

variance σ2

L, i.e. it is assumed that DL∼ N (µL, σL2).

• Discrete: Lead time demand is modeled by a discrete distribution with mean µL and variance σ2L,

denoted by D(µL, σL2), and fit according to the procedure proposed by Adan et al. (1995), see Appendix

A.

• BDQ: It is assumed that demand is always a multiple of B. To model the multiplicity of B we fit

a discrete distribution with mean µL/B and variance σ2L/B

2 _{according to the procedure proposed by}

Adan et al. (1995), see Appendix A. Hence, the BDQ distribution provides the following assumption:

DL∼ B · D(µL/B, σ2L/B2).

• General: It is assumed that lead time demand is equal to a multiple of B with probability γL_{. To}

model this type of demand a mixture of the discrete and BDQ distribution is used. As explained

in this section, the parameters µ1,L and σ21,L of the BDQ distribution and the parameters µ2,L and

σ2

2,L of the discrete distribution can be estimated separately. The general distribution provides the

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3.4 Forecasting

The mean and variance of the lead time distributions are estimated on the basis of forecasts and forecast errors, respectively. The corresponding calculations are addressed in Section 3.5. This study employs forecasts provided by food manufacturer A as well as statistical forecasts we generate ourselves. The former forecasts are discussed in Section 5.1 while the latter forecasts are treated in this section.

We generate forecasts using two widely applied statistical forecasting methods, namely simple moving average (SMA) and the method proposed by Croston (1972) (CRO). As the name already reveals, SMA gives the arithmetic mean of the M most recent demand observations, i.e.:

ˆ dt,k = 1 M M X i=1 dt−i+1 k = 1, 2, . . . .

In case of stationary demand SMA is an unbiased forecasting method and provides the minimum variance unbiased estimator for the per-period demand mean if M covers all past demand observations. Yet, in reality one usually observes that the mean of a demand process only is constant during a limited period of time. One does not know if the demand mean is constant and if so, for how long. This clarifies the frequent use of SMA in practice and why it performs well in forecasting competitions (Ali and Boylan, 2011, 2012; Gardner, 1990, 2006).

Forecasts produced by SMA are prone to bias in case of intermittent demand (also called irregular or sporadic demand), i.e. when demand is frequently equal to zero. More specifically, after observing a positive demand, the forecast will be biased high whereas it is biased low just before a positive demand. To overcome these problems Croston (1972) proposed a method that separately forecasts the non-zero

demand size (Yt) and the time interval between non-zero demands (It) using exponential smoothing. If a

non-zero demand occurs (dt> 0), the estimates ˆYtand ˆItare updated as follows:

ˆ

Yt= δ1dt+ (1 − δ1) ˆYt−1,

ˆ

It= δ2It+ (1 − δ2) ˆIt−1.

Here It represents the number of periods since the preceding non-zero demand. Although using distinct

smoothing parameters 0 < δi< 1 (i = 1, 2) may lead to better forecasting accuracy, we will simply apply

δ = δ1 = δ2 for reasons of convenience. In periods with dt = 0 the estimates are not updated, hence

ˆ

Yt= ˆYt−1and ˆIt= ˆIt−1. The demand forecast made at the end of period t is given by:

ˆ dt,k = ˆ Yt ˆ It k = 1, 2, . . .

The updating procedure is initialized as follows. The first forecast for the demand size is set equal to the average non-zero demand size regarding the preceding demand observations. Dividing the number of preceding non-zero demands by the total number of preceding demand observations provides an initial estimate of time interval.

As opposed to SMA, CRO is actually a biased estimator for a level demand process, see e.g. Teunter and Sani (2009). Syntetos and Boylan (2005) proposed the so-called Syntetos-Boylan Approximation (SBA) that knows a smaller bias than CRO. Other well-known intermittent demand forecasting meth-ods are the Teunter, Syntetos and Babai (TSB) method (Teunter, Syntetos, and Babai, 2011) and the aggregate-disaggregate intermittent demand approach (ADIDA) proposed by Nikolopoulos, Syntetos, Boy-lan, Petropoulos, and Assimakopoulos (2011). Nevertheless, since we focus on standard practices we will solely consider CRO as intermittent demand forecasting method as it is the common choice in practice.

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3.5 Parameter estimation

This section explains how the parameters µt,L, σt,L2 and γtare estimated at the end of period t. The lead

time demand mean is estimated by the sum of the one up to L period ahead forecasts. In case of SMA or

CRO, which produce forecasts that are the same for all future periods, ˆµt,Lcan be obtained by multiplying

the one period ahead forecast by the lead time. However, the forecasts provided by food manufacturer A may vary across future periods. Therefore, we present the following general formula that applies to all forecasts: ˆ µt,L= L X k=1 ˆ dt,k.

The lead time demand variance is estimated on the basis of past forecast errors. In this study we consider two methods to do so. Their formal definitions have been presented in Definition 2.

Definition 2 We define two methods for estimating the lead time demand variance ˆσ2_t,L at the end of

period t using the M preceding demand observations:

• Per-period MSE: The one-period-ahead forecast error, t = dt− ˆdt−1,1, is used to calculate the

per-period MSE: M SEt= 1 M t X k=t−M +1 2_k. (1)

If an initial per-period MSE, which can be obtained using (1), is available, the per-period MSE can be updated periodically through exponential smoothing:

M SEt= δ · 2t+ (1 − δ) · M SEt−1. (2)

Regardless of how the per-period MSE is obtained, an estimate of the lead time variance is obtained via multiplying the per-period MSE by the lead time:

ˆ

σ_t,L2 = L · M SEt.

• Lead time MSE: The lead time forecast error, t,L =Pt_k=t−L+1dt− ˆµt−L,L, is used to calculate

the lead time MSE:

M SEt,L= 1 M − L + 1 t X k=t−M +L 2_k,L. (3)

If an initial lead time MSE, which can be obtained using (3), is available, the per-period MSE can be updated periodically through exponential smoothing:

M SEt,L = δ · 2t,L+ (1 − δ) · M SEt−1,L. (4)

The lead time MSE, regardless of how it is computed, is directly employed as an estimate of the lead time demand variance:

ˆ

σt,L2 = M SEt,L.

The standard approach uses the MSE concerning one-period ahead forecast errors as an estimate of the per-period demand variance. Then, an estimate of the lead time demand variance is obtained through

multiplying the per-period MSE by the lead time. This so-called per-period MSE method, however,

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In the theoretical analysis (1) and (3) are applied in order to control for estimation accuracy via M . On the other hand, in the empirical analysis these equations will solely be used to provide initial estimates. After initialization (2) and (4) are employed.

Wat remains is the estimation of γ, i.e. the probability with which per-period demand is a multiple of

B. Let γ = {kB|k ∈ N+_{} be the set of multiples of B. Then, at the end of period t an estimate of γ on}

the basis of the M preceding demand observations is obtained as follows:

ˆ

γt=

PM

k=11γ(dt−k+1)

M ,

where 1γ(dt) is an indicator function that is 1 if dt∈ γ and 0 otherwise.

3.6 Order-up-to-level calculation

On the basis of the assumed lead time demand distribution the smallest order-up-to level S is determined that is expected to satisfy the specified service target. Logically, this calculation depends on the practiced service measure. We will analyze two service measures, namely the cycle service level α and the volume

fill rate β, also referred to as type 1 and type 2 service measure, respectively. We define α∗ and β∗ as

the target cycle service level and target volume fill rate, respectively. Subsequently, the order-up-to level

calculation given α∗ and β∗ for all four lead time demand distributions is explained.

3.6.1 Cycle service level

The cycle service level α gives the probability of no stock-out during a replenishment cycle, thus for an arbitrary period t it holds that:

αt= P (Dt,L ≤ St),

where Dt,L is a random variable representing cumulative demand over the L periods subsequent to period

t. As explained in Section 3.4, for each of the four considered demand distributions the parameters are updated at the end of each period. As a consequence, S has to be updated as well. Next, we demonstrate

for each of the lead time demand distributions how St, the order-up-to-level at the end of period t, is

determined. Since all calculations concern the same period, the index t is dropped.

If lead time demand is modeled by a normal distribution it is assumed that DL∼ N (ˆµL, ˆσ2L). Let FN

represent the corresponding cumulative distribution function. Then, the order-up-to-level is given by: SN ormal= dF_N−1(α∗)e,

= dˆµL+ Φ−1(α∗)ˆσL2e,

here Φ denotes the standard normal distribution function and dxe is the ceiling function which maps x to the least integer greater than or equal to x. The ceiling function is applied because we consider integer demand.

In order to determine the order-up-to-level for the discrete, BDQ and general distribution we first derive functions α(S) that relate the expected achieved CSL to S. The estimated discrete distribution

yields DL∼ D(ˆµL, ˆσL2). Now, let FD be the corresponding cumulative distribution function, then we find

for the discrete distribution the following function:

αDiscrete(S) = FD(S).

Similarly, if we denote by FM the cumulative distribution function corresponding to the distribution that

models the multiplicity of B, i.e. D(ˆµL/B, ˆσ2L/B

2_{), the function for the BDQ distribution reads:}

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where bxc is the floor function which maps x to the greatest integer less than or equal to x. The calculation

for the general distribution is less straightforward. First we define F1 and F2 as the cumulative

distri-bution associated with D(ˆµ1,L/B, ˆσ21,L/B

2_{) and D(ˆ}_µ

2,L, ˆσ2,L2 ), respectively. Consequently, the cumulative

distribution function corresponding to the general distribution is given by:

FG(x) = ˆγL· F1(bx/Bc) + (1 − ˆγL) · F2(x), (5)

and the achieved CSL function for the general distribution is: αGeneral(S) = FG(S).

Finally, the order-up-to-level for distribution X ∈ {Discrete, BDQ, General} is found as follows:

SX= arg min

S∈N

{αX_{(S) ≥ α}∗_}.

3.6.2 Volume fill rate

The volume fill rate service measure β gives the proportion of demand that can be met from stock immediately or the fraction of demand within a period that is not backordered:

β = 1 − Expected Backorder

Expected per-period demand.

For a certain period t we then have that:

βt= 1 − E[max{D

t,L− St, 0}]

Expected per-period demand.

We employ ˆµt,L/L as an estimate of expected per-period demand. Subsequently, we address the calculation

of the order-up-to-level Stat the end of period t given a target volume fill rate β∗for each of the predictive

distributions. To this end, for each of these distributions first the function that relates the order-up-to-level to the expected achieved VFR is derived. Since all computations in the remainder of this section concern the same period, the index t is dropped.

In case lead time demand is modeled by a normal distribution, the standard normal loss function G, which is defined as:

G(z) =

Z ∞

z

(x − z)φ(x)dx = φ(z) − z(1 − Φ(z)),

is used to calculate the order-up-to-level. Since ˆσLG(S− ˆσˆLµL) gives the expected backorder, the function

that relates S to the expected achieved VFR reads:

βN ormal(S) = 1 − ˆσLG( S− ˆµL ˆ σL ) ˆ µL/L .

Unfortunately, the calculations involving discrete distributions are less elegant. If lead time demand is modeled by a discrete distribution, the expected achieved VFR as a function of S can be expressed as:

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In similar fashion, the following function for the BDQ distribution can be derived: βBDQ(S) = 1 − ˆ µL− BPbS/Bcx=0 xfM(x) − S 1 − FM(bS/Bc) ˆ µL/L ,

where b·c is the floor function. In order to obtain the expected achieved volume fill rate function for

the general distribution we define the lead time demand mean ˆµG_L according to this distribution. Denote

by ˆµ1,L and ˆµ2,Lthe estimates of the lead time demand mean regarding demands that are a multiple of

B and demands that are not, respectively. Then, the lead time demand mean according to the general distribution equals:

ˆ

µG_L= ˆγL· ˆµ1,L+ (1 − ˆγL) · ˆµ2,L.

Subsequently, using (5) we find the following function:

βGeneral(S) = 1 − ˆ µG_L− ˆγLBPbS/Bc x=0 xf1(x) − (1 − ˆγL)P S x=0xf2(x) − S 1 − FG(S) ˆ µG_L/L .

Finally, the order-up-to-level for distribution X ∈ {N ormal, Discrete, BDQ, General} is found as follows:

SX = arg min

S∈N

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4 Theoretical analysis

In the theoretical setting we consider a stationary demand process, that is, demands dt(t = 1, 2, . . .) are

independently and identically distributed (i.i.d.) with mean µ and variance σ2_{: d}

t i.i.d.

∼ (µ, σ2_{). Section 4.1}

presents the theoretical distribution employed to simulate a stationary demand process involving demands that are frequently a multiple of some BDQ. Next, in Section 4.2 we discuss the mechanics of the simulation model that we employ to evaluate the inventory control performance of the approaches under consideration. Finally, Section 4.3 addresses the simulation results.

4.1 Demand process

In order to simulate a demand process featuring demands that are frequently a multiple of some BDQ, we employ a mixture distribution similar to the general distribution:

dt∼

(

B · D(µ1/B, σ21/B2) w.p. θ,

D(µ2, σ22) w.p. 1 − θ.

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The mean of this mixture distribution is given by:

µ = θ · µ1+ (1 − θ) · µ2, (7)

and the variance equals (see Appendix C):

σ2= θ · σ₁2+ (1 − θ) · σ2₂+ θ(1 − θ)(µ1− µ2)2. (8)

By varying the parameters B, µ1, µ2, σ12, σ22and θ of the theoretical distribution a wide range of demand

processes can be simulated.

4.2 Simulation model

In each period the order of events is as explained in Section 3.1. The demands are sampled from a theoretical distribution, namely the one presented in (6). Since the simulated demand process is stationary, increasing the number of observations on the basis of which demand parameters are estimated likely improves estimation accuracy. Hence, by using SMA as forecasting method and estimating both the lead time demand variance and γ using M demand observations, M can be used to control for estimation

accuracy. This way, the first forecast is produced at the end of period M and the first estimate of σ2

L

is obtained at the end of period 2M . Consequently, the initialization period covers 2M periods. The n periods following the initialization period provide the inventory control performance measures according to the calculations presented in Section 3.2. Preliminary tests revealed that using n = 1, 000, 000 for CSL and n = 100, 000 for VFR leads to acceptably stable results. This means that the variation with respect to the results of multiple simulations (with equal control parameter settings) is considered negligible.

The main purpose of this simulation study is to gain insights into the effects of several demand, estimation and inventory control parameters on the inventory control performance of the approaches under consideration. This is accomplished by performing a sensitivity analysis. More specifically, base values for the control parameters are specified and subsequently the value of each control parameter is varied while keeping the remaining control parameters fixed. A disadvantage of this procedure is that interaction effects regarding different control parameters cannot be identified.

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demand characteristics concerning demands that are a multiple of B and demands that are not. The influence of the difference in mean and standard deviation are examined simultaneously. A difference of ∆ between the means of the distributions comprising the theoretical distribution, see (6), is reached by

setting µ1= µ +1₂∆ and µ2= µ −1₂∆. Here µ = 180, i.e. the base value. Similarly, we set σ1= σ +1₂∆

and σ2 = σ −1₂∆ where σ represents the base value of 120. This way, because θ = 0.5 (base value), the

mean of the demand process remains constant while the variance increases by θ(1 − θ)∆2_{, see (7) and (8).}

It is important to note that even for ∆ = 0 the mean and standard deviation concerning demands that are a multiple of B and demands that are not, are unequal. This can be explained as follows. The discrete

distribution associated with µ1 and σ1, see (6), attaches a positive probability to the occurrence of zero

demand. Consequently, the difference in demand characteristics arises because zero is not considered a multiple of B.

When addressing the effect of ∆, we consider two approaches for obtaining estimates of the lead time demand mean and variance regarding the general distribution. The first approach, which is also applied during all other simulations, uses the preceding M (=10) demand observations to produce one estimate of

the lead time demand mean and one estimate of the lead time demand variance, i.e. ˆµ1,L = ˆµ2,L= ˆµL

and ˆσ2

1,L= ˆσ2,L2 = ˆσL2. The second approach called ”Split” makes a distinction between demands that are

a multiple of B and demands that are not. Historic demands are split into demands that are a multiple

of B and demands that are not, hence its name. Subsequently, ˆµ1,L and ˆσ21,L are based on the last 5

observations from the former subsample while ˆµ2,L and ˆσ2,L2 are based on the last 5 observations from

the latter subsample. As such, both parameter estimation approaches rely on 10 demand observations. Since the demand characteristics regarding demands that are a multiple of B and demands that are not differ for all considered values of ∆, we expect that separately estimating demand parameters enhances the inventory control performance results of the general distribution.

The mean and standard deviation have been expressed in terms of B. As a consequence, the CV remains constant when varying B. As mentioned above, M is used to control for estimation accuracy. Thus, by varying M we can explore the influence of estimation accuracy on inventory control performance. The relevant inventory control parameters are the target service levels and lead time L. Note that T is also an inventory control parameter, however we assume it to be one, see Section 3.1. Solely one service level target is considered because we expect the effects of the other control parameters to be similar for other service targets. The empirical analysis will look at both service targets of 0.90 and 0.95. The two lead time demand variance estimation methods, see Definition 2, are only compared when varying L since they are identical for L = 1.

Table 2: Control parameters values

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4.3 Simulation output

The simulation results are presented graphically in this section. In the figures displaying achieved service levels, the service target has been depicted by a dashed line. Because the inventory control performance appeared not to be affected by B, the graphs showing the results for various values of B have been omitted. The remaining results for CSL and VFR are discussed in Section 4.3.1 and Section 4.3.2, respectively.

4.3.1 Cycle service level

Figure 1a and 1b display for various values of θ the achieved CSLs and average inventory levels, respectively. We observe that for each of the distributional approaches the achieved CSL gets closer to the target when θ increases, that is when demand is more frequently a multiple of B. The normal distribution is outperformed significantly by the alternative distributions for all considered values of θ. The achieved CSLs of the alternative distributions are very similar for θ = 0.5. For θ ≤ 0.5 the discrete and general distribution achieve almost identical CSLs. Yet, as the general distribution knows smaller average inventory levels it attains superior inventory control performance. On the other hand, when θ ≥ 0.5 the achieved CSLs concerning the BDQ and general distribution visually coincide. Here, the BDQ distribution requires smaller average inventory levels meaning that it outperforms the other approaches. Hence, there appears to exist a threshold value for θ below which the general distribution is best to be used and above which one should employ the BDQ distribution. Investigating whether a distribution selection rule on the basis of such a threshold can improve inventory control performance is left for future research.

The achieved CSLs and average inventory levels for different values of M are depicted in Figure 1c and 1d, respectively. As M increases, thus when estimation accuracy improves, the inventory control performance, both in terms of achieved service level and average inventory level, improves for all four distributions. The effect of enhanced estimation accuracy seems to be similar for each of the approaches since the lines appear to be parallel.

0.0 0.2 0.4 0.6 0.8 1.0 0.90 0.92 0.94 0.96 θ Achie v ed CSL

Normal Discrete BDQ General

(a) Effect θ on achieved CSL

0.0 0.2 0.4 0.6 0.8 1.0 200 220 240 260 θ In v entor y le v el

(b) Effect θ on inventory level

5 10 15 20 25 30 35 40 0.90 0.92 0.94 0.96 M Achie v ed CSL

(c) Effect M on achieved CSL 5 10 15 20 25 30 35 40 200 220 240 260 M In v entor y le v el

(d) Effect M on inventory level

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Next, we explore the inventory control performance of the two lead time demand variance estimation methods, see Definition 2, for various values of L. The achieved CSLs and average inventory levels for the standard method, which uses the per-period MSE, are presented in Figure 2a and 2c, respectively. The second method directly estimates the lead time demand variance by the lead time MSE. Figure 2b and 2d respectively depict the achieved CSLs and average inventory levels for this method. Recall from Section 3.5 that both methods coincide for L = 1. For L > 1 we observe that the difference in achieved CSLs increases with L in favor of the lead time MSE method. This may be explained by the fact that the per-period method ignores the correlation of forecast errors. The performance of the per-period MSE method in combination with each of the distributions decreases with L, however the extent of this decrease diminishes slightly with L. The degradation is largest for the BDQ distribution for small values

of L, which is likely due to the declining probability γL _{with which lead time demand is a multiple of B.}

In combination with the lead time MSE method, the decline in inventory control performance of all four distributions seems to stop for L ≥ 5.

1 2 3 4 5 6 0.90 0.94 L Achie v ed CSL

(a) Effect L on achieved CSL (per-period MSE)

1 2 3 4 5 6 0.90 0.94 L Achie v ed CSL

(b) Effect L on achieved CSL (lead time MSE)

1 2 3 4 5 6 200 400 600 L In v entor y le v el

(c) Effect L on inventory level (per-period MSE)

1 2 3 4 5 6 200 400 600 L In v entor y le v el

(d) Effect L on inventory level (lead time MSE)

Figure 2: Achieved cycle service levels and average inventory levels regarding the normal, discrete, BDQ and general distribution. Various values of L are considered for both the lead time variance estimate based on the per-period MSE and the lead time MSE. SMA is used as forecasting method and the service target equals 0.95. Results based on 1,000,000 periods.

Figure 3 shows the inventory control performance measures for several values of µ and σ. When µ increases, ceteris paribus, the CV decreases. This likely causes the improvement in inventory control performance for all distributions. Some authors argue that the normal distribution should only be applied when the CV is small, see Section 2.2. This may explain why, compared to the alternative distributions, the normal distribution benefits more significantly from the increase in µ. The achieved CSLs for the alternative distributions are very similar for all considered values of µ. However, the BDQ distribution is superior as it knows the smallest average inventory levels.

Opposed to µ, an increase in σ, ceteris paribus, leads to a higher CV. This presumably clarifies the substantial decrease in inventory control performance for all distributions. Especially the normal

distribution performs poorly when demand variability is high. The alternative distributions perform

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2B 3B 4B 5B 6B 7B 0.90 0.92 0.94 0.96 µ Achie v ed CSL

(a) Effect µ on achieved CSL

2B 3B 4B 5B 6B 7B 200 220 240 260 µ In v entor y le v el

(b) Effect µ on inventory level

1B 1.5B 2B 2.5B 3B 3.5B 0.90 0.92 0.94 0.96 σ Achie v ed CSL

(c) Effect σ on achieved CSL 1B 1.5B 2B 2.5B 3B 3.5B 100 200 300 400 500 σ In v entor y le v el

(d) Effect σ on inventory level

Figure 3: Achieved cycle service levels and average inventory levels regarding the normal, discrete, BDQ and general distribution for various values of µ and σ. SMA is used as forecasting method and the service target equals 0.95. Results based on 1,000,000 periods.

Contradicting expectations, the general distribution does not yield superior inventory control perfor-mance in a substantial number of the considered cases. This may be explained by the fact that the general distribution is not exact as discussed in Section 3.3. In addition, estimating more parameters on the basis of a limited amount of data may lead to less accurate estimates and consequently less adequate order-up-to-levels. But probably the main reason is the difference in demand characteristics concerning demands that are a multiple of B and demands that are not. As explained in Section 4.2 such a difference is present even for ∆ = 0. Therefore, we expect that applying the Split variant of the general distribution yields superior inventory control performance.

Figure 4 shows the results for several values of ∆. As expected, we observe that Split improves the inventory control performance of the general distribution and that this gain in performance increases

with ∆. For ∆ = 0 we find that opposed to the standard variant, the Split variant of the general

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0 20 40 60 80 0.90 0.92 0.94 0.96 ∆ Achie v ed CSL

Normal Discrete BDQ General Split

(a) Effect ∆ on achieved CSL

0 20 40 60 80 200 240 280 ∆ In v entor y le v el

(b) Effect ∆ on inventory level

Figure 4: Achieved cycle service levels and average inventory levels regarding the normal, discrete, BDQ, general distribution fitted on the basis of one mean estimate and one variance estimate and the general distribution employing two mean estimates and two variance estimates (Split). Various values of ∆ are considered. SMA is used as forecasting method and the service target equals 0.95. Results based on 1,000,000 periods.

4.3.2 Volume fill rate

Figure 5a and 5b display for various values of θ the achieved VFRs and average inventory levels, respec-tively. We observe that for all distributions the achieved VFRs get closer to the target when θ increases, hence when demand is more frequently a multiple of the BDQ. Like for the CSL case, we find that the normal distribution, compared to the alternative distributions, performs very poorly. Of the considered distributions, the discrete and general distributions achieve best inventory control performance for θ = 0. Contradicting expectations, the discrete distribution outperforms the other distributions for all other val-ues of θ. The gain in inventory control performance accomplished by employing the discrete distribution instead of the general distribution grows with θ. Thus, the distributions that attempt to take into account the fact that demand is frequently a multiple of B, do not perform particularly well in case of VFR, not even for θ = 1.

The achieved VFRs and average inventory levels for different values of M are depicted in Figure 5c and 5d, respectively. The findings are comparable to those for CSL, i.e. improved estimation accuracy enhances the inventory control performance, both in terms of achieved service level and average inventory level, similarly for all four distributions.

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0.0 0.2 0.4 0.6 0.8 1.0 0.90 0.92 0.94 0.96 θ Achie v ed VFR

(a) Effect θ on achieved VFR

0.0 0.2 0.4 0.6 0.8 1.0 140 160 180 200 220 θ In v entor y le v el

(b) Effect θ on inventory level

5 10 15 20 25 30 35 40 0.90 0.92 0.94 0.96 M Achie v ed VFR

(c) Effect M on achieved VFR 5 10 15 20 25 30 35 40 140 180 220 M In v entor y le v el

(d) Effect M on inventory level

Figure 5: Achieved volume fill rates and average inventory levels regarding the standard, discrete, BDQ and general demand distributions for various values of θ and M . SMA is used as forecasting method and the service target equals 0.95. Results based on 100,000 periods.

1 2 3 4 5 6 0.84 0.88 0.92 0.96 L Achie v ed VFR

(a) Effect L on achieved VFR (per-period MSE)

1 2 3 4 5 6 0.84 0.88 0.92 0.96 L Achie v ed VFR

(b) Effect L on achieved VFR (lead time MSE)

1 2 3 4 5 6 200 400 600 800 L In v entor y le v el

(c) Effect L on inventory level (per-period MSE)

1 2 3 4 5 6 200 400 600 800 L In v entor y le v el

(d) Effect L on inventory level (lead time MSE)

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Figure 7 shows the inventory control performance measures for several values of µ and σ. Increasing µ and σ know opposite effects on the inventory control performance of each of the considered distributions. This can be explained using the CV. When µ increases or σ decreases, ceteris paribus, the CV decreases. The achieved inventory control performance measures for all distributions converge when the CV dimin-ishes. Compared to the alternative distributions, the impact of the CV on the performance of the normal distribution is large. 2B 3B 4B 5B 6B 7B 0.88 0.92 0.96 µ Achie v ed VFR

(a) Effect µ on achieved VFR

2B 3B 4B 5B 6B 7B 100 150 200 250 µ In v entor y le v el

(b) Effect µ on inventory level

1B 1.5B 2B 2.5B 3B 3.5B 0.86 0.90 0.94 0.98 σ Achie v ed VFR

(c) Effect σ on achieved VFR 1B 1.5B 2B 2.5B 3B 3.5B 100 300 500 σ In v entor y le v el

(d) Effect σ on inventory level

Figure 7: Achieved volume fill rates and average inventory levels regarding the standard, discrete, BDQ and general approach demand distributions for various values of µ and σ. SMA is used as forecasting method and the service target equals 0.95. Results based on 100,000 periods.

The general distribution does not yield superior inventory control performance in a nearly all considered cases. Like for CSL, we expect this to be a result of not taking into account the difference in demand characteristics concerning demands that are a multiple of B and demands that are not.

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0 20 40 60 80 0.90 0.92 0.94 0.96 ∆ Achie v ed VFR

(a) Effect ∆ on achieved VFR

0 20 40 60 80 140 180 220 260 ∆ In v entor y le v el

(b) Effect ∆ on inventory level

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5 Empirical analysis

In this section we consider data provided by two Dutch food manufacturers. These data are described in Section 5.1. Next, in Section 5.2 we address the judgmental adjustments and their influence on forecasting performance. Finally, in Section 5.3 we evaluate the inventory control performance of the approaches under consideration for statistical and final forecasts provided by food manufacturer A as well as forecasts generated by CRO.

5.1 Data description

The empirical analysis can mainly be partitioned into two parts which employ distinct datasets. The first part investigates the effect of judgmental adjustments to statistical forecasts on forecasting and inventory control performance. To this end, we employ demand data series, historical statistical forecasts and final, possibly judgmentally adjusted, forecasts regarding 87 SKUs of food manufacturer A. These data concern 52 consecutive weeks from week 1 2017 up to and including week 52 2017. The demand data series characteristics are summarized in Table 3. The descriptive statistics have been rounded to two decimal places. The statistics regarding the inter-demand interval reveals that we deal with both fast moving SKUs and SKUs with an intermittent demand pattern. Moreover, the variability of demand varies significantly across SKUs.

Table 3: Characteristics of the demand data series provided by food manufacturer A.

87 SKUs Demand Demand size Inter-demand interval

Mean St. dev. CV Mean St. dev. CV Mean St. dev. CV

Min. 9.58 5.37 0.32 9.76 5.25 0.11 1.00 0.00 0.00

25th percentile 48.88 47.95 0.58 83.06 34.05 0.42 1.02 0.14 0.14

Median 143.42 108.89 0.76 165.73 97.10 0.54 1.09 0.33 0.31

75th percentile 334.11 242.44 1.17 438.71 245.53 0.67 1.36 0.63 0.47

Max. 992.94 1017.06 3.03 1622.72 1002.21 2.14 9.25 4.79 0.72

CV2_{of demand sizes ranges from 0.01 to 4.59 (25th percentile=0.26, median=0.45, 75th percentile=0.81).}

In addition, Table 4 lists several statistics related to the demand characteristics that we focus on in this study, namely demand that is frequently a multiple of some BDQ. The BDQs that we employ for the SKUs of food manufacturer A range from 3 to 200. ”% Multiple” gives the percentage of demands within a demand series that is a multiple of the associated BDQ. For each demand series, regarding the demands that are a multiple of B we have calculated the mean, standard deviation and CV with respect to the multiplicity of B.

Table 4: BDQ related characteristics of the demand data series provided by food manufacturer A.

87 SKUs Multiplicity

B % Multiple Mean St. dev. CV

Min. 3 27.45 1.03 0.19 0.01

25th percentile 11 59.47 2.05 1.03 0.38

Median 60 85.71 4.83 2.13 0.52

75th percentile 80 93.81 9.47 4.38 0.67

Max. 200 98.08 84.64 81.43 2.16

On the Performance of Standard Practices in Inventory Control when Demand is Frequently a Multiple of some Base Demand Quantity

On the Performance of Standard Practices in Inventory Control

when Demand is Frequently a Multiple of some Base Demand

Quantity

W.D. Touber

On the Performance of Standard Practices in Inventory Control

when Demand is Frequently a Multiple of some Base Demand

Quantity

List of Abbreviations

Contents

1

Introduction

2

Related literature

2.1

Judgmentally adjusting statistical forecasts

2.2

Predictive distributions in inventory control

3

Methodological framework

3.1

Inventory control policy

3.2

Inventory control performance measures

3.3

Predictive demand distributions

3.4

Forecasting

3.5

Parameter estimation

3.6

Order-up-to-level calculation

4

Theoretical analysis

4.1

Demand process

4.2

Simulation model

4.3

Simulation output

5

Empirical analysis

5.1

Data description