Establishing inventory measurements using intermittent demand forecasting

Abstract

Demand forecasting is a fundamental element in industrial problems. Forecasts are crucial for accurately estimating intermittent demand to establish inventory measurements. The demand estimation by the Croston method gives less accurate values, which increases the standard deviation value. This increase indicates that the forecasted method is an inappropriate method of intermittent demand data because of the zero values. However, real data were adopted in an industrial sector for three years with constant lead-time. Furthermore, an integration of Bernoulli distribution and geometric distribution has been done to establish the new formulation, then extracting the mean equation and the standard deviation equation of intermittent demand during lead-time. Relying on it, the optimal quantity of safety stock and reorder levels have been obtained. Furthermore, the proposed modified forecasting method was evaluated based on the criteria of CV and the results that obtained gives a less ratio dispersion of data thus accurate results. These procedures are very important to the industrial sectors in drawing future inventory policies.

Keywords

Intermittent demand forecasting croston’s method reorder level and safety stock

1 Introduction

During the past fifty years, intermittent demand forecasting has received a huge considerable attention due to its complexity for many companies.

Since the early work of [1], many academic researchers have focused on the forecasting problem for fast moving items (i.e., [2 –6]). In contrast, forecasting for slow moving items (intermittent demand) has received less interest, even though these items are no less important than fast moving items and need inventory management. Intermittent demand control for repair parts inventories in many industries constitute more than 60% of inventory values [7].

Traditionally, the related literature addresses inventory management and demand forecasting as independent problems. The Creston method is the standard method to forecast intermittent demand. Intermittent demands are very essential in most manufacturers. Intermittent demands are characterized by their high impact and large number on companies’ operations. Therefore, companies try to analyse intermittent demand and estimate future demand. Nevertheless, it is difficult to develop an optimal forecasting method that addresses the intermittent demand [8].

Forecasting is a method of dealing with past experience to estimate the future. However, there are two types of forecasting methods: quantitative and qualitative methods [9, 10]. The appropriate forecasting method to address intermittent demand items is the Croston method (CR) [11]. In [12, 13] Croston introduced a method that divides the forecasts in two parts, namely, demand size and time between withdrawals or demand (time intervals) to obtain forecasts, which is better than simple exponential smoothing (SES).

In [13] Segerstedt and Lev, created a procedure to handle regular fast moving items and slow moving items. The suggested procedure implemented in a modern computerized ERP-system with a periodic review system to build a Croston forecasting procedure. The mean and the variance of the forecasted demand rate were fitted base on an Erlang distribution. They showed theoretically the Croston forecasting method is more accurate than ordinary exponential smoothing method for slow moving items. However, it was not clear that the Croston procedure with demand distributed Erlang outperforms ordinary exponential smoothing. Later [14], used four forecasting methods (Simple moving average-SMA, Single Exponential Smoothing-SES, Croston’s method and their developed Croston’s approach) in an automotive industry. They have proved that the root-mean-square errors are meeting the theoretical and practical requirements of slow items. They have proved that the root-mean-square errors are meeting the theoretical and practical requirements of slow items. Moreover, they showed by a simulation experiment the out-of-sample comparison results indicate superior performance of the new method. A few years later [15], criticized the work of [14]. The criticism was because of the ignored the damping effect on the bias of the probability that a demand occurs which lead to excessive compensation and a negative bias. Thus, bigger positive bias of the original method. Various approaches have been examined for such products, giving careful consideration to the requirement for inventory planning over a multi-period non-stationary lead-time [16]. In addition, optimizing the smoothing parameter dynamically by the Excel solver to reach the best performance of Croston’s method has been studied. However, the study proposed a new performance measure that is applies in the field of the irregular demand forecasting. The study concluded that the optimal parameter value leads best performance for each of the methods [17]. Later [18], identified the strongest and the weaknesses of the intermittent items replenishment system of the Army. The study applied on a conducted managerial ranks and those who working in the military logistics by using a survey. However, evaluation the ability of the intermittent items replenishment system and its capability to adopt a new replenishment system that is lean in peace time to save money and agile during war to increase reliability of equipment achieved by a certainty of supply. The results highlighted the strengths and the weaknesses of the system by adopting a dynamic lean-agile replenishment system. Furthermore, the best fit spare parts demand distribution analyzed by using the Kolmogorov Smirnov goodness-of-fit test to find the best-fitting distributions and compering their results with those in the literature [19]. In the same year, Type-2 fuzzy employed to deal with vagueness and unknown information in real life problems with ABC analysis for intermittent demand inventory classification to cope the decision making process under uncertainty condition. The study compared their results with the results of intuitionistic fuzzy AHP-ABC approach. The comparison showed that all the included items matched with the exception of few items [20].

Generally, the classical forecasting method used in inventory system is SES. However, the non-stationary fluctuation behaviour of demands made this method unsuitable for intermittent demand (slow moving), which gives biased estimates of distribution parameters and increases the value of variance (i.e., the demand estimates did not have the Best Linear unbiased estimation, BLUE properties) [21, 22]. In addition, the literature classified the intermittent demand forecasting under three main categories which they are, time-series, contextual and comparative studies [23]. In [24] Yang, et. al., proposed a new forecasting method of intermittent demand by modifying TSB (mTSB) model based on TSB model which applies the probability occurrence for both zero and nonzero demand estimation. The modified mTSB model achieves optimal results on RMASE and MASE among all comparison models

The most widely technique used in practice is exponential smoothing. However, this application of this technique to intermittent demand has a significant impediment. This impediment make the exponential smoothing is biased because of the positive-demand periods and the bias rely on average inter arrival time. Accordingly, estimating intermittent demand produces a saw tooth forecast pattern. That leads to over- estimate actual demand and increase this over- estimate during the intermittent demand interval [12, 23].

The real problem appears when the demands (intermittent demands) chain is not under the Bernoulli distribution condition. The reality of the intermittent demand impose different chains which is, the demand occurs for a while of periods (t) and then no-demand occurs. Based on that, it is a mixture of Bernoulli and Geometric distribution. Therefore, a new treatment and procedure is needed. The data considered in this paper were collected over three years (January 2017 to December 2019) with an average lead-time of three months. Therefore, the demand data are converted to (36 months) a base period of lead-time. In addition, only six items are selected because of the difficulty obtaining the data. The data were collected at an industrial sector. The remainder of this paper is organized as follows. The next section reviews the related literature. Sections 3 cover the theoretical background of the study. Sections 4, 5 and 6 describe the problem and research methodology. The data analysis is provided in section 7. Model evaluation and discussion and conclusions are presented in section 8 and 9.

2 Related literature review

The literature presented many studies focusing on different aspects of intermittent demand forecasting and inventory control, including demand during lead-time distribution [10 , 25–27], parameter revision frequencies [28, 29] demand forecasting models [10 , 30], classification of items [31, 32] and the selection of time bucket [33, 34].

In [12] Croston, illustrates that the exponential smoothing method on intermittent demand items creates high forecasting errors, and thus, overstock safety inventories are created. The study assumes an alternative method; this method works on separating the estimation of intervals between demands of the quantities demanded in each situation. However [35], proved that the mean interval between demands is greater than 1.25 time periods. They reached this conclusion by comparing the Croston Method and forecasting made through exponential smoothing.

In [36] Ghobbar and Friend, proved the forecasting method superiority by comparing three techniques in aircraft parts demand: the Croston Method, double exponential smoothing (Holts) and the weight moving average. In the same vein, similar results were provided by [37]. In [31] Eaves and Kingsman, used the Croston Method and SBA forecasting methods to evaluate spare parts demand in the British Air Force (RAF). The evaluation showed the superiority of the SBA to a certain service level.

Bootstrapping method has been used to develop forecasting models for intermittent demands considering autocorrelation and small demand variation to evaluate demand during lead-time distribution. Moreover, this method compared with the Croston Method and the results showed the bootstrapping method provides better results [38]. Latter, [39] showed the advantage of demand forecasting in petrochemical industry by using bootstrapping with a regression analysis.

Based on neural networks a forecasting model introduced to be superior to the Croston method, SBA and exponential smoothing. Furthermore, the developed model enhanced fuzzy neural network (EFNN) model by using fuzzy logic connection with the analytic hierarchy process (AHP) and a genetic algorithm and the model provides better results [40, 41]. In [22] Doszyn, Suggested a stochastic simulation system for an expert intermittent demand forecasting, the system was applied in a work clothes company. The proposed expert system was associated and compared with the common technique in the intermittent demand forecasting methods which is exponential smoothing such as SES, SBA, TSB and Croston’s method. A new measure of forecast “Absolute Mean Error” was suggested and used in the comparisons.

3 Theoretical background

In this section, the theoretical background of the study is elaborated; beginning with a discussion of the concept of forecasting method, followed by an explanation of the mean and standard deviation estimation

3.1 Forecasting method

The forecasting method is the widely used method used in production and inventory system. Forecasting is a set of processes completed in order to obtain an appropriate estimate of the future situation [42]. The forecasting system is a mathematical method used to forecast variables, which are very important to decision making on parameters such as demand size and demand inter arrival intervals [3]. The growing importance of using the forecasting method of inventory control is buoyed by the ability to know future directions of production and the sudden changes in demand size [43].

3.2 Exponential smoothing

Smoothing is defined as the disposal of the fluctuations and random disorders in the data series. The basic principle of exponential smoothing is to modify the previous observation in the time series to obtain an appropriate future forecasting. The idea of the exponential smoothing is forecasting for next periods based on the modified weights for the actual value of the current period, as well as the forecasted value of the current value. The weight that gives to the forecasted value is (1-α). Therefore, the mathematical relation of the observation forecasting in the period (t) is [1]: ${\hat{Z}}_{t} = α Z_{t} + (1 - α) {\hat{Z}}_{t - 1}$ (1)

where

${\hat{Z}}_{t} :$ the forecasted value of the period, t.

${\hat{Z}}_{t - 1} :$ the forecasted value of the period, t-1.

α: the smoothing constant, ranging between zero and 1, (0≤α≤1).

Let us compensate for the value of ${\hat{Z}}_{t - 1}$ in Equation (1) until ${\hat{Z}}_{t - k}$ ; we obtain

$\begin{matrix} {\hat{Z}}_{t} = α Z_{t} + α (1 - α) {\hat{Z}}_{t - 1} + α {(1 - α)}^{2} {\hat{Z}}_{t - 2} \\ + . . . + α {(1 - α)}^{k} {\hat{Z}}_{t - k} + . . . \end{matrix}$ (2)

Equation (2) can be written sequentially as: ${\hat{Z}}_{t} = α \sum_{k = 0}^{\infty} {(1 - α)}^{k} {\hat{Z}}_{t - k}$ (3)

We can observe from Equation (2) that the probability value of (1 - α) given to the previous quantities and this value increases whenever they are updated. The compensation for the weight (1 - α) ^k becomes zero whenever the value of k is increased. When using simple exponential smoothing, it is preferred that the initial forecast to value Z_t equals the average of the demand series [5]. This is due to the lack of availability of the data. In addition, the selection value of the smoothing constant α depends on the Mean Square Error (MSE), which makes the value of the MSE as small as possible [44, 45].

3.3 Estimating the standard deviation of demand

Always A forecasting error is a difference between the real value and the predicted value of a time series phenomenon [46]. Irrespective of which method is used to get the forecasting value of X, for a single period or more periods, the real value will remain different from the predicted value [47]. To make an effort to estimate this error, the forecasting error is used, where the forecasting error is the difference between the real value X_t and the forecasted value ${\hat{Z}}_{t}$ as shown in Equation (1) [48]. To ensure that the predictive values are equal to the real values, the differences between these two values must be equal to zero [49]. However, this is not reality because the value of e_t is unknown unless the value of ${\hat{Z}}_{t}$ is already known. It was found that the forecasting error is normally distributed [1]. To determine the parameters of this distribution, the mean, variance, and standard deviation must be known. The variance error for the forecasting model exponential smoothing, ES [44] is: $σ_{e}^{2} = \frac{2}{2 - σ} σ_{ɛ}^{2}$ (4)

where $σ_{ɛ}^{2}$ is the variance of the random variable. To calculate the variance error $σ_{e}^{2}$ from Equation (4), it must be specified that $σ_{ɛ}^{2}$ of the latter is also unknown. Therefore, we use the numerical relationship between mean absolute deviation (MAD) and standard deviation in order to estimate the value of $σ_{ɛ}^{2}$ [9, 50]. The original reason that MAD is estimated instead of σ or σ² was that this simplified the computations. It is acceptable to estimate σ or σ² directly, but most forecasting systems first evaluate MAD. It is obvious that MAD and σ in most cases give a very similar picture of the variations around the mean. It is also possible to relate them to each other. A common assumption is that the forecast errors are normally distributed [3]. Thus, it is easy to show that $σ_{t} = \sqrt{\frac{π}{2}} {MAD}_{t} = 1.25 {MAD}_{t}$ (5)

$where \sqrt{\frac{π}{2}} = 1.25;$ Therefore $MAD = sum of absolute error / number of error .$ (6)

Most facilities, manufacturers, and stakeholders prefer to use mean absolute deviation to obtain the standard deviation [51 –53]. The mean absolute deviation can fit the exponential smoothing method because it can help obtain the estimate of the mean absolute deviation by using an exponential smoothing method. Therefore: ${MAD}_{t + 1} = α | e_{t} | + (1 - α) {MAD}_{t}$ (7)

4 Problem description

The most common method used in practice is exponential smoothing. In spite of, this application of this technique to intermittent demand has a significant barrier. This barrier makes the exponential smoothing is biased because of the positive-demand periods and the bias based on average inter arrival time. Therefore, estimating intermittent demand produces a saw tooth forecast pattern. That leads to over- estimate actual demand and increase this over- estimate during the intermittent demand interval [23]. Intermittent demands inventory control is particularly challenging due to the nature of their demand, which is usually slow-moving, lumpy and erratic. The problem appears when the demands chain is not under the Bernoulli distribution condition. In Bernoulli distribution the demand chain occurrence and non-occurrence must be sequential. In other words, assume 1 represents a demand that occurs and zero represents no demand occurs. Thus, the chain can be represented as follows (1,0,1,0,1, ...) which describes the Bernoulli distribution. But the reality of the intermittent demand impose different chains which is, the demand occurs for a while of periods (t) and then no-demand occurs. For example, assume that the demands on an item are (1,0,0,0,1,1,0,0,1,1,1,0,0, ...). In this case, the forecasted method is not a suitable forecasting method, and it must be modifying. Therefore, to modify the forecasting method an integration of Bernoulli distribution with geometric distribution must be done to establish the new formulation then extracting the mean equation and the standard deviation equation of demand during lead-time, which plays the important role to obtain the considered inventory measurements.

5 Model description

This paper considers different items of intermittent demand with lead-time approximately equal to three months. The purpose of the model is to obtain the new mean and standard deviation of demand during lead-time based on the modified the forecasting method, which is later used to optimize the inventory measurements (i.e., the optimal reorder point, R and safety stock, S). Figure 1 shows the steps of the methodology framework that adopted in this research.

Fig. 1

Methodology framework procedures.

6 Methodology

Intermittent demands inventory control is particularly challenging due to the nature of their demand, which is usually slow-moving, lumpy and erratic. The Croston method is considered as a scientific method to solve inventory problems when the demands are intermittent due to the property of best nonlinear estimation of the distributions parameters [38]. The chains of intermittent demand usually have large amounts of zero values. The non-zero values of the chains are one (demand occur) or zero values (no demand occur). In this situation, it is unsuitable to use simple exponential smoothing method (SEM), due to the zero values, which leads to non-stationary fluctuations of the forecasting. Therefore, the forecasting method, which divides the chains into two groups, is used to determine the average period time between arrivals (demand occurrence) and the demand size. The demand size chain is built on demand values except zero values, which are not updated.

6.1 Data types and collection

This study includes secondary data about demand and lead-time data. The data were obtained from an industrial sector in Kurdistan region of Iraq for three years (January 2017 – December 2019) with constant lead-time. The data were processed to meet the requirements of the study. Clarifications of each type of data are provided in the following sections.

6.1.1 Demand data

Demand is the quantity that meets customers’ needs (item) per unit per month. The quantity of the demand are converted to (36 months) a base period of the lead-time. The objective for the demand data is to establish and estimate the demd parameters by using the appropriate forecasting method.

6.1.2 Lead-time data

The lead time is the time interval between the preparation of an order and its delivery to the depot and is calculated separately for each order [54]. These times or periods are subject to deterministic conditions with period equal to three months.

6.1.3 Probabilistic demand and lead-time

This study assume the demand size Z_t occurs independently and follows the normal distribution with the mean, μ, and variance, σ², and the occurred demand for any periodic review P, which is generated from a Bernoulli distribution, has a probability (1/P), where P is an integer value. Z_t from Equation (3) is identical independent distributions. Therefore, the expected value of ${\hat{Z}}_{t}$ can be obtaining from Equation (3) as: $\begin{matrix} E ({\hat{Z}}_{t}) = E (α \sum_{K = 0}^{\infty} {(1 - α)}^{k} {\hat{Z}}_{t - k}) \\ E ({\hat{Z}}_{t}) = α \sum_{K = 0}^{\infty} {(1 - α)}^{k} E ({\hat{Z}}_{t - k}) \\ E ({\hat{Z}}_{t}) = μ \end{matrix}$ (8)

The variance can be obtaining from Equation (3) as: $\begin{matrix} V ({\hat{Z}}_{t}) = V (α \sum_{K = 0}^{\infty} {(1 - α)}^{k} {\hat{Z}}_{t - k}) \\ V ({\hat{Z}}_{t}) = α \sum_{K = 0}^{\infty} {(1 - α)}^{2 k} V ({\hat{Z}}_{t - k}) \\ V ({\hat{Z}}_{t}) = \frac{α}{2 - α} σ^{2} \end{matrix}$ (9)

Now, we assume the model of demand occurrence is: ${\hat{Z}}_{t - k} = {\begin{matrix} 0 with probability (1 - \frac{1}{P}) (no demand occurs) \\ μ + e_{t} with probability (\frac{1}{P}) (demand occurs) \end{matrix}}$ (10)

where, $e_{t} \sim N (0, σ^{2}), E (e_{i} e_{k}) 0, \forall i \neq k$ (10)

Now, after substituting Equations (10) in (3), the expected value and the variance are obtained as $\begin{matrix} E ({\hat{Z}}_{t}) = E (α \sum_{K = 0}^{\infty} {(1 - α)}^{k} {\hat{Z}}_{t - k}) \\ E ({\hat{Z}}_{t}) = E (μ + e_{t}) \\ E ({\hat{Z}}_{t}) = μ + \frac{1}{P} \\ V ({\hat{Z}}_{t}) = V (α \sum_{K = 0}^{\infty} {(1 - α)}^{k} {\hat{Z}}_{t - k}) \\ V ({\hat{Z}}_{t}) = \frac{α}{2 - α} V (μ + e_{t}) \end{matrix}$ (11)

Using the mathematical expression, we obtain: $V ({\hat{Z}}_{t}) = \frac{α}{2 - α} {\frac{(P - 1)}{P^{2}} μ^{2} + \frac{1}{P} σ^{2}}$ (12)

The first limit of μ² represents the increase in demand quantity due to intermittent demand, therefore the pattern of demand will be updated as it will increase the variance. Moreover, the estimation of this process is biased, which leads to poor use of the exponential smoothing method.

6.6.1.1 Modifying forecasting method

As the process is mixed between SES and Bernoulli distribution, we assume the following relation: $Z_{t}^{*} = Y_{t} * Z_{t}$ (13)

Where,

Y_t: is Bernoulli random variable $Y_{t} = {\begin{matrix} 0 with probability (1 - \frac{1}{P}) (no demand occurs) \\ 1 with probability (\frac{1}{P}) (demand occurs) \end{matrix}}$ (14)

and $e_{t} \sim N (μ, σ^{2}), E {(Z_{t} - μ) (Y_{t} - \frac{1}{P})} = 0$

In this method, there are two cases of demand forecasting.

Demand occurs, $Z_{t}^{*} > 0$

The procedure in this case is mixing between Bernoulli distribution and SES method, but as mentioned earlier; the estimation in this method is biased. To update the demand, we add a new random variable, Q_t, which follows Geometric distribution. However, the initial forecasted value is equal to the mean of the demand series data (inter arrivals) that is used to extract the time period of demand and inters arrival to obtain unbiased estimation. ${\hat{q}}_{t} = α Q_{t} + (1 - α) {\hat{q}}_{t - 1}$ (15)

where,

${\hat{q}}_{t}$ : the estimated value of Q_t at the end of the time, t: $\begin{matrix} E ({\hat{q}}_{t}) = E (α \sum_{K = 0}^{\infty} {(1 - α)}^{k} Q_{t - k}) \\ E ({\hat{q}}_{t}) = P \end{matrix}$ (16) $\begin{matrix} V ({\hat{q}}_{t}) = V (α \sum_{K = 0}^{\infty} {(1 - α)}^{k} Q_{t - k}) \\ V ({\hat{Z}}_{t}) = \frac{α}{2 - α} V (Q_{t - k}) \\ V ({\hat{Z}}_{t}) = \frac{α}{2 - α} P^{2} {1 - \frac{1}{P}} \end{matrix}$ (17)

6.6.1.2 No demand occurs

In this case, the current forecasted value is equal to previous forecasted values due to the non-updated demand. ${\hat{q}}_{t} = {\hat{q}}_{t - 1}$ (18)

Based on that, the calculating procedure of demand estimation is: ${\hat{D}}_{t} = \frac{{\hat{Z}}_{t}}{{\hat{q}}_{t}}$ (19)

Therefore, to obtain the expected value and the variance, the procedure is as follows: $\begin{matrix} E ({\hat{D}}_{t}) = E (\frac{{\hat{Z}}_{t}}{{\hat{q}}_{t}}) = \frac{E ({\hat{Z}}_{t})}{E ({\hat{q}}_{t})} \\ E ({\hat{D}}_{t}) = \frac{μ}{P} \end{matrix}$ (20) $\begin{matrix} V ({\hat{D}}_{t}) = V (\frac{{\hat{Z}}_{t}}{{\hat{q}}_{t}}) \\ V ({\hat{D}}_{t}) = \frac{α}{2 - α} (\frac{1}{P^{2}} σ^{2} + \frac{1}{P^{2}} (1 - \frac{1}{P}) μ^{2}) \end{matrix}$ (21)

6.1.4 Optimizing the reorder point

To determine the reorder point, the R level, the service level, K of the inventory must be determined, which is extracted from the distribution tables. This service level, K depends on the demand during the lead-time distribution. The reorder point level consists of two dimensions, the mean of the demand during lead-time, μ_L and safety stock, S. The safety stock in turn depends on the standard deviation of demand during lead-time, σ_L, multiplied by the value of the service level [3]. The best service level adopted in most similar cases or in the literature had 95% credibility and 0.05% errors [55, 56]. $R = μ_{L} + S$ (22)

and $S = K σ_{L}$ (23) where the value of μ_L and σ_L can be obtained from $μ_{L} = μ_{\hat{D}} * L$ (24) $σ_{L} = σ_{\hat{D}} * \sqrt{L}$ (25) where $μ_{\hat{D}}$ and $σ_{\hat{D}}$ are the mean and standard deviation of the demand data respectively, which are extracted from the modified Croston method as in Equations (20) and (21), and L is the lead-time.

7 Analyses results and discussion

First, the exponential smoothing method has been applied and the demand estimation gives less accurate values, which leads to the increase of the standard deviation value. This increase indicates that the exponential smoothing method is an inappropriate method of intermittent demand data because of the zero values. Moreover, the forecasting values by using exponential smoothing method of intermittent demand data should be in the form of a saw tooth. Figure 2 shows that the fluctuations of the forecasted demand data are very high and non-stationary due to the instability of demand during a period of time when using the exponential smoothing method.

Fig. 2

Forecasted demand fluctuations using exponential smoothing method of first item.

When applying the modified forecasting method for the six items (item 1, item 2 and item 3,..., item 6), the demand estimation gives a more accurate value, which leads to a decrease of the standard deviation value. This dynamic indicates that the modified forecasting method is the appropriate method for the intermittent demand data. Table 1 exhibits how the modified forecasting method has been applied for the first item.

Table 1

Extracting the mean and standard deviation from first item using the modified forecasting method

T	Z_t	$Z_{t} = 0, {\hat{q}}_{t} = {\hat{q}}_{t - 1} :$	${\hat{Z}}_{t} = 0.1 Q_{t} + 0.9 {\hat{Z}}_{t - 1}$	${\hat{D}}_{t} = \frac{{\hat{Z}}_{t}}{{\hat{q}}_{t}}$	\|e_t\|	MAD_t	σ = 1.25MAD_t	Var( ${\hat{D}}_{t}$ )	σ _D
		$Z_{t} > 0, {\hat{q}}_{t} = . 1$
		$Q_{t} + . 9 {\hat{q}}_{t - 1}$
0		2.1818181	3.08333	1.413194444		1.460975	1.82621876	0.09380894	0.306282
1	1	1.96363636	2.875	1.46412037	0.46412037	1.36129	1.70161193	0.09488972	0.308042
2	0	1.96363636	2.875	1.46412037	1.46412037	1.371573	1.71446578	0.09548908	0.309013
3	0	1.96363636	2.875	1.46412037	1.46412037	1.380827	1.72603425	0.09603235	0.309891
4	3	2.06727272	2.8875	1.39676781	1.60323219	1.403068	1.75383485	0.09089338	0.301485
5	0	2.06727272	2.8875	1.39676781	1.39676781	1.402438	1.75304734	0.09085937	0.301429
6	0	2.06727272	2.8875	1.39676781	1.39676781	1.401871	1.75233858	0.09082877	0.301378
7	2	2.16054545	2.79875	1.295390474	0.704609526	1.332145	1.66518091	0.07870405	0.280542
8	0	2.16054545	2.79875	1.295390474	1.295390474	1.328469	1.66058663	0.07853177	0.280235
9	0	2.16054545	2.79875	1.295390474	1.295390474	1.325161	1.65645178	0.07837713	0.279959
10	4	2.24449090	2.918875	1.300461939	2.699538061	1.462599	1.82824886	0.0842738	0.290299
11	0	2.24449090	2.918875	1.300461939	1.300461939	1.446385	1.80798171	0.08350386	0.28897
12	0	2.24449090	2.918875	1.300461939	1.300461939	1.431793	1.78974128	0.08281826	0.287782
13	3	2.32004181	2.9269875	1.261609802	1.738390198	1.462453	1.82806593	0.08034049	0.283444
14	0	2.32004181	2.9269875	1.261609802	1.261609802	1.442368	1.80296056	0.07944914	0.281867
15	0	2.32004181	2.9269875	1.261609802	1.261609802	1.424293	1.78036573	0.07865746	0.280459
16	3	2.38803763	2.93428875	1.228744767	1.771255233	1.458989	1.82373606	0.07688451	0.277281
17	0	2.38803763	2.93428875	1.228744767	1.228744767	1.435964	1.79495555	0.07592331	0.275542
18	0	2.38803763	2.93428875	1.228744767	1.228744767	1.415242	1.76905309	0.07507131	0.273991
19	3	2.44923387	2.94085987	1.200726443	1.799273557	1.453646	1.81705698	0.07386793	0.271787
20	0	2.44923387	2.94085987	1.200726443	1.200726443	1.428354	1.78544208	0.07286866	0.269942
21	0	2.44923387	2.94085987	1.200726443	1.200726443	1.405591	1.75698868	0.07198432	0.268299
22	4	2.50431048	3.04677388	1.21661188	2.78338812	1.543371	1.92921333	0.07802925	0.279337
23	0	2.50431048	3.04677388	1.21661188	1.21661188	1.510695	1.88836848	0.07672069	0.276985
24	0	2.50431048	3.04677388	1.21661188	1.21661188	1.481286	1.85160811	0.07556692	0.274894
25	4	2.55387943	2.74209649	1.073698491	2.926301509	1.625788	2.03223499	0.07024393	0.265036
26	0	2.55387943	2.74209649	1.073698491	1.073698491	1.570579	1.96322380	0.06801892	0.260804
27	0	2.55387943	2.74209649	1.073698491	1.073698491	1.520891	1.90111373	0.06608213	0.257064
28	3	2.59849149	2.76788684	1.065189883	1.934810117	1.562283	1.95285362	0.06646226	0.257803
29	0	2.59849149	2.76788684	1.065189883	1.065189883	1.512574	1.89071700	0.06460066	0.254167
30	0	2.59849149	2.76788684	1.065189883	1.065189883	1.467835	1.83479403	0.06297668	0.250952
31	0	2.59849149	2.76788684	1.065189883	1.065189883	1.427571	1.78446368	0.06155679	0.248106
32	4	2.73864234	2.89109816	1.055668394	2.944331606	1.579247	1.97405848	0.06458327	0.254132
33	0	2.73864234	2.89109816	1.055668394	1.055668394	1.526889	1.90861118	0.06280008	0.250599
34	0	2.73864234	2.89109816	1.055668394	1.055668394	1.479767	1.84970861	0.0612466	0.247481
35	0	2.73864234	2.89109816	1.055668394	1.055668394	1.437357	1.79669630	0.05989011	0.244725
36	3	2.86477811	2.90198834	1.012988873	1.987011127	1.492322	1.86540306	0.05747105	0.239731
t + 1		2.57830029	2.90198834	1.125543192					0.239731
t + 2		2.32047026	2.90198834	1.250603547					0.239731
t + 3		2.08842324	2.90198834	1.389559496					0.239731

Table 1 shows that the forecasted mean and standard deviation based on the modified forecasting method is the sum of (t + 1, t + 2 and t + 3). Figure 3 shows that the fluctuations of the forecasted demand data are stationary during the period of time when using the modified forecasting method.

Fig. 3

Forecasted demand fluctuations using the modified forecasting method of first item.

Based on these results, Table 2 shows the extracted mean and standard deviation from the six items based on the modified forecasting method.

Table 2

Forecasted mean and standard deviation using the modified forecasting method

Items	$μ_{\hat{D}}$	$σ_{\hat{D}}$	Items	$μ_{\hat{D}}$	$σ_{\hat{D}}$
1	3.766	0.719	4	5.341	1.012
2	4.824	0.956	5	3.576	0.618
3	7.241	1.298	6	5.240	1.003

By applying Equations (24) and (25), we can extract the mean and standard deviation of demand during lead-time, which is later used to establish the safety stock and reorder level of the six items. Therefore, Table 3 shows the μ_L and σ_L for the six items.

Table 3

The mean and standard deviation of demand during lead-time

Items	μ _L	σ _L	Items	μ _L	σ _L
1	11.298	1.245	4	16.023	1.753
2	14.472	1.656	5	10.728	1.070
3	21.723	2.249	6	15.720	1.737

One of the necessary measures in the inventory system are the reorder level, R and safety stock, S. Optimal safety stock makes a balance between overstock and under-stock to avoid shortage. Referring to Equation (23) the safety factor, K, under a service level of % 95 (confidence level) was extracted. Thus, from Equation (22) the optimal R is established. Table 4 shows the optimal reorder level and safety stocks under a service level of % 95.

Table 4

Optimal R and S under service level % 95

Items	S	R	Items	S	R
1	2.04784	13.34584	4	2.883429	18.90643
2	2.72387	17.19588	5	1.759994	12.48799
3	3.69927	25.42228	6	2.857111	18.57711

8 Model evaluation

This section evaluates the modified forecasting method based on the criteria of coefficient of variation, CV. One of the important statistical criteria to measure the dispersion of the data is CV. Whenever the CV proportion is less, there is less dispersion of data and indicates the accuracy of the results. Therefore, the comparison has been done with [19] to show our results are better. However, Table 5 represents the evaluation based on the criteria of CV. Moreover, to prove the proposed method give an accurate standard deviation value of the demand during lead time, a comparison has been done with [24] to prove the extracted measures from the modified method gives accurate results. Table 6 represents the evaluation based on the criteria of standard deviation value of the demand during lead time.

Table 5
Results of data dispersion based on CV

Modified forecasting method Measures

Average, μ Standard deviation, SD CV Percentage %

Item 1 11.29 1.245 % 11

Item 2 14.47 1.656 % 11.43

Item 3 21.723 2.249 % 10.35

Item 4 16.023 1.753 % 9.97

Item 5 10.728 1.070 % 9.97

Item 6 15.720 1.737 % 11.04

Turrini &Meissner, 2018 3.83 3.06 % 79.89

Modified forecasting method	Measures
Item 1	11.29	1.245	% 11
Item 2	14.47	1.656	% 11.43
Item 3	21.723	2.249	% 10.35
Item 4	16.023	1.753	% 9.97
Item 5	10.728	1.070	% 9.97
Item 6	15.720	1.737	% 11.04
Turrini &Meissner, 2018	3.83	3.06	% 79.89

Table 6

Standard deviation value of the demand during lead time

Modified forecasting method	Standard deviation value of the demand during lead time
Item 1	1.245
Item 2	1.656
Item 3	2.249
Item 4	1.753
Item 5	1.070
Item 6	1.737
Yang et al., (2021)	3.052

From the results in Table 5, note that the proportion of the CV, in the modified forecasting method for all the six items are less dispersion with which has been compared. Based on this measure, the modified forecasting method gives a less ratio dispersion of data, which means that accurate results are obtained from the proposed modified forecasting method.∥From the results in Table 6, we can observe the standard deviation value of the demand during lead time for the six items are less than with which has been compared. As we know the standard deviation indicates the consistency of the model. Less standard deviation means more consistent. Thus, the modified forecasting method gives accurate results.

9 Discussion and conclusions

Time period with zero demand always challenging the accurate intermittent demand forecasting. Accordingly, this type of demand tends to be widespread, particularly in industries section such as heavy equipment.

In this paper, a forecasting method was modified to establish the mean and standard deviation of demand during lead-time. The estimation of the classical Croston method gives less accurate values, which leads to the increase of the standard deviation value. In contrast, the proposed modified forecasting method estimation gives more accurate values, which leads to a decrease in the standard deviation value. Therefore, the modified forecasting method is the appropriate method of intermittent demand data. Based on that, the optimal safety stock and reorder level have been determined. Furthermore, the proposed modified forecasting method were evaluated based on the criteria of CV and the results that obtained gives a less ratio dispersion of data thus accurate results. These procedures are very important to the industrial sectors in drawing future inventory policies.

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Modified forecasting method	Measures
	Average, μ	Standard deviation, SD	CV Percentage %
Item 1	11.29	1.245	% 11
Item 2	14.47	1.656	% 11.43
Item 3	21.723	2.249	% 10.35
Item 4	16.023	1.753	% 9.97
Item 5	10.728	1.070	% 9.97
Item 6	15.720	1.737	% 11.04
Turrini &Meissner, 2018	3.83	3.06	% 79.89