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Comparison of Neural Network Methods for Forecasting Lumpy Demand
Mohammad R. Amin-naseri 1*, Bahman Rostami Tabar 2
1 Department of Industrial Engineering, Faculty of Engineering,Tarbiat Modares University, Tehran, Iran
2 Laboratoire de l'Intégration du Matériau au SystèmeUniversité Bordeaux1, Bordeaux, France
Abstract
Accurate demand forecasting of spare parts is a difficult task particularly for items with
lumpy pattern. This study investigates four different types of artificial neural networks
(ANNs) namely multilayer perceptron (MLP), generalized regression neural network
(GRNN), recurrent neural network (RNN), Time delay neural network (TDNN) and two
traditional methods for lumpy demand forecasting. Demand records for 30 types of spare
parts from Arak Petrochemical Company in Iran were used to evaluate the accuracy of
different methods. It was found that the ANNs provided suitable forecasts, and that estimation
using RNN yields better results than other neural networks.
Keyword: Forecasting; Neural network methods; Lumpy demand; Spare parts
Corresponding Author.
Tel: +982182880; Fax: +982188006544 Tarbiat Modares University, Ale Ahmad High Way, Tehran, Iran E-mail Address: [email protected]
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1. Introduction
Forecasting the future is a critical element of management decision making. The final
effectiveness of any decision depends upon the consequence of events following this
decision. The ability to forecast the uncontrollable aspects of these events earlier to making
the decision should permit an improved choice over that which would otherwise be made[1].
The need for forecasting is increasing as management attempts to decrease its dependence on
chance and becomes more scientific in dealing with its environment [2]. Statistical methods,
such as exponential smoothing and regression analysis, have been used by analysts in
forecasting demand for a number of decades. Many of these methods may perform poorly
when demand for an item is lumpy.
Lumpy demand patterns are characterized by intervals in which there is no demand and, for
periods with actual demand occurrences, by a large variation in demand levels [3]. The
problem of modeling the future consumption becomes especially difficult for lumpy patterns
which are common for the spare parts inventory systems. Forecasting the lumpy demand
requires special techniques in comparison with the smooth and continuous case, since the
assumptions for continuity and normal demand distribution do not hold [4]. Lumpy demand
patterns are very common, particularly in organizations that hold many spare parts. In the
aerospace, and automotive sectors, for example, organizations may have thousands or tens of
thousands of stock keeping units (SKUs) classified as intermittent or lumpy [5]. For instance,
lumpy demand has been observed in the automotive industry [5,6], in durable goods spare
parts [7], in aircraft maintenance service parts [8], in petrochemical industry [9,10] and in
telecommunication systems, large compressors, and textile machines [3].
Croston [11] implied that traditional forecasting methods such as single exponential
smoothing (SES) may lead to sub-optimal stocking decisions and proposed an alternative
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forecasting method. In proposed procedure, two forecasts for the mean demand-interval and
the mean demand-size have been done. The forecast for the demand per period is then
calculated as the ratio of the forecasts for demand size and demand interval. Modifications of
the original Croston's method were later proposed by several other authors. Willemain et al
[12] compared Croston’s method with exponential smoothing and concluded that Croston’s
method is robustly superior to exponential smoothing, although results with real data in some
cases show a more modest benefit. Johnston and Boylan [13] obtained similar results, but
further showed that Croston’s method is always better than exponential smoothing when the
average inter-arrival time between demands is greater than 1.25 review intervals. Sani and
Kingsman [14] compared various forecasting and inventory control methods on some long
series of low demand real data from a typical spare parts depot in the UK. They concluded
based on cost and service level, that the best forecasting method is the Croston’s method. An
important contribution is that by Syntetos and Boylan [15]. They show that Croston’s method
lead to a biased estimate of demand per unit of time. They also proposed a modified method
and demonstrated the improvement in a simulation experiment. Ghobbar and Friend[7]
compared various forecasting methods using real data of aircraft maintenance repair parts
from an airlines operator. The data is lumpy in nature and they showed that moving average,
Holt’s and Croston’s forecasting methods, are superior to other methods such as the
exponential smoothing. Willemain et al. [16] compared various forecasting methods using
large industrial data sets. They showed that the bootstrapping method produces more accurate
forecasts than both exponential smoothing and Croston’s method. In an attempt to develop a
forecasting procedure that can handle both fast moving and slow moving items, Levén and
Segerstedt [17] proposed a modification of Croston’s method which was thought to avoid the
bias indicated by Syntetos and Boylan[15]. Boylan and syntetos [18] reviewed modified
Croston procedure to intermittent demand forecasting proposed by Levén and Segerstedt [17].
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It has been found to be more biased than Croston's method, especially for highly intermittent
series. They assessed the accuracy of this method using simulated data and the mean square
error measure, and showed that Croston's method is generally more accurate than its
modification, particularly for strongly intermittent series. Eaves and Kingsman [19] compared
various forecasting methods using real data from the UK’s Royal Air Force. They showed
that the modified Croston’s method [11] by Syntetos and Boylan[15] is the best forecasting
method for spare parts inventory control. Syntetos et al. [20] analyzed a wider range of
intermittent demand patterns and made a categorization to guide the selection of forecasting
methods. They indicated that there are demand categories that are better used with the
original Croston’s method and there are others that go well with the Syntetos/Boylan
modification. In an attempt to further confirm the good performance of their modified
Croston’s method, Syntetos and Boylan [21] carried out a comparison of forecasting methods
including theirs and the original Croston’s method. A simulation exercise was carried out on
3,000 products from the automotive industry with “fast intermittent” demand.
It was shown that the modification is the most accurate estimator. Syntetos and Boylan [22]
evaluated the empirical stock control performance of the Syntetos-Boylan approximation
(SBA). They first discussed the nature of the empirical demand data set and specified the
stock control model to be used for experimentation purposes. Performance measures were
then selected to report customer service level and stock volume differences. The out-of
sample empirical comparison results demonstrate the superior stock control performance of
the SBA and enable insights to be gained into the empirical utility of the other estimators.
Hua et al. [9, 10] developed a hybrid approach for forecasting the intermittent demand of
spare parts. Their approaches provide a mechanism to integrate the demand autocorrelated
process and the relationship between explanatory variables and the nonzero demand of spare
parts during forecasting occurrences of nonzero demands over lead times. Nezih et al. [23]
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investigated an alternative method developed by Wright (1986) for data sets with missing
values and compared it to a Syntetos-Boylan approximation in a simulated environment.
A little work has been done in lumpy demand forecasting using neural network. Carmo and
Rodrigues [24] applied NN modeling on ten “irregularly spaced” time-series. They used a
Radial Basis Function (RBF) network. Gaussian basis function networks were shown to be
adequate models for short-term prediction of irregularly spaced time series, with the ability to
generate better predictive performance than alternative models, by taking into account
nonlinear correlations in the data. Gutierrez et al. [25] adopted the most widely used method,
a multi-layered perceptron (MLP) trained by a back-propagation (BP) algorithm.
Their research objective was to assess whether the NN-based approach is a superior
alternative to traditional approaches for modeling and forecasting lumpy demand.
In this research, generalized regression neural network (GRNN), Elman recurrent neural
network (RNN) and Time delay neural network (TDNN) have been applied to 30 types of
spare parts in Arak Petrochemical Company in Iran. Statistical accuracy measures showed
that our approaches produce more accurate forecasts than two traditional methods (Coston's
method and Syntetos-Boylan Approximation) and applied MLP to lumpy patterns by Guttirez
et al.
This paper is divided into four sections. In the next section, forecasting methods for
estimating the lumpy demand problem, accuracy measures and characteristics of data are
presented. In the third section, results of our studies are presented. Conclusions and future
studies are found in the fourth section.
2. Forecasting Methods
2.1 Croston 's method and Syntetos-Boylan Approximation
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Croston’s original method estimates the mean demand per period, by applying the
exponential smoothing method separately, to the intervals between nonzero demands and
their sizes. This method has become an industry standard, and has been adopted by many
companies and software vendors [15]. The method works as follows:
If a demand occurs, then forecast the mean demand-size using SES and forecast the mean
demand-interval using SES, and finally, forecast the mean demand per period as the ratio of
Size to Interval; Else, if there is no demand, then do not change any of the forecasts.
A disadvantage of the original Croston's method is that it is positively biased. Syntetos and
Boylan [15] noted this and proposed a modification. Syntetos and Boylan [21] correct
original method by multiplying the forecast for the demand per period with(1−α2 ), where is
the smoothing constant. Therefore forecast demand per period calculated as follow:
d t=(1−α2 ) st
nt
The selection of the smoothing constants will definitely affect the accuracy of forecasts [23].
The use of low exponential smoothing constant values in the range of 0.20-0.5 has been
considered realistic and recommended in the literature on lumpy demand [11, 13]. Syntetos
and Boylan [15] suggest that should be no more than 0.15. Eaves [26] choose values in the
range of 0.01- 0.1. Syntetos and Boylan [21] used four smoothing constant values of 0.05,
0.10, 0.15, and 0.20 in the single exponential smoothing, Croston's method and Syntetos-
Boylan approximation. Nezih et al. [23] used smoothing constant ranges from 0.05 to 0.45 for
Syntetos and Boylan method, also they used 0.1<α<0.9 and 0.1<β<0.9 with increments of
0.1 for their method. . In this study the smoothing constant value ranges from 0.05 to 0. 5
used in Croston's method and Syntetos-Boylan approximation.
6
)1(
2.2 Artificial neural network models
Hill et al. [27] demonstrated that statistical time-series methods need human interaction and
evaluation and can misjudge the functional form relating the independent and dependent
variables. These traditional methods can also fail to make necessary data transformations.
Traditional time-series methods may not sometimes capture the nonlinear pattern in data [27].
The estimation of artificial neural networks (ANN) or simply Neural network (NN), however,
can be automated. These models need to be recalibrated on all previous data. In spite of the
great deal of time and effort spent by the researchers in both conventional and soft computing
techniques for time series forecasting, the need of producing more and more accurate time
series forecasts has forced the researchers to develop innovative methods to model time
series. ANNs have been applied in many areas for time series forecasting and widely touted
as solving many forecasting and decision modeling problems. NN were introduced as
efficient tools of modeling and forecasting about two decades ago. A great deal of research
has been devoted to using neural networks techniques especially in business and financial
forecasting. One can find numerous ANN applications in a wide range of areas for time series
forecasting. Wong et al [28] reviewed bibliography of neural network application research in
business during the period of 1994-1998. Their extensive literature searches have identified a
total of 302 research articles. Production/operations, finance, marketing/distribution, and
information systems were found as the most popular application areas. Regarding the lumpy
demand case, most traditional forecasting methods assume a specific distribution of demand
time series or the time series of forecast error, while neither assumption is valid for that[8].
Artificial neural network modeling is a logical choice to overcome these limitations [25].
High capabilities of artificial neural networks in capturing non-linearity of lumpy demand
patterns can make it a promising choice in lumpy demand forecasting. A little work has been
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done in lumpy demand forecasting using neural network, and we are aware of only two works
by Carmo and Rodrigues [24] and Gutierrez et al. [25]. In the subsequent section, the
architectural of 4 types of neural network, Multi-layered perceptron(MLP), Generalized
Regression Neural Network(GRNN), Recurrent Neural Network(RNN) and Time Delay
Neural network(TDNN) have been illustrated.
2.2.1 Multilayer perceptron network(MLP)
There are many methods in the NN literature that can be used for flexible nonlinear modeling.
The most widely used method is multilayer perceptron (MLP) trained by back-propagation
(BP) algorithm. The standard MLP network consists of three components: an input layer, one
hidden layer and one output layer. A layer being groups of neurons (or processing units) that
share the same input-output connections. In conventional MLP network, the information (or
input signal) is passed forward. Thus the MLP network is also referred to as multilayer
feedforward network.
A major advantage of the MLP is that it is less complex than temporal neural networks such
as the time delay networks and the recurrent networks, and has the same nonlinear input–
output mapping capability. Furthermore, the MLP can be trained even by using the standard
Backpropagation algorithm [29]. Gutierrez et al [25] used three layers of MLP for lumpy
demand forecasting: one input layer for input variables, one hidden unit layer, and one output
layer. The MLP had three nodes in the hidden layer. One output unit was used in the output
layer. All the input nodes were fully connected to all the hidden nodes.
The hidden nodes were in turn connected to the output node. The input nodes represented two
variables: (1) the demand at the end of the immediately preceding period and (2) the number
of periods separating the last two nonzero demand transactions as of the end of the
immediately preceding period. The output node represented the predicted value of the
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demand transaction for the current period (figure 1). They used a learning rate value of 0.1
and momentum factor value of 0.9. Logistic sigmoid transfer function has been used and all
input patterns have been scaled between 0 and 1.
<<Insert figure 1 about here>>
To exploit the time structure in the data, the neural network must have access to this time
dimension. While MLPs are popular in many application areas, they are not well suited for
such temporal sequences processing due to the lack of time delay and/or feedback
connections necessary to provide a dynamic model.
2.2.2 Proposed models
2.2.2.1 Generalized regression neural network
Generalized Regression Neural Network (GRNN) proposed by Spetch [30] does not require
an iterative training procedure as in back propagation method. It approximates any arbitrary
function between input and output vectors, drawing the function estimate directly from the
training data. Furthermore, it is consistent; that is, as the training set size becomes large, the
estimation error approaches zero, with only mild restrictions on the function. The GRNN is
used for estimation of continuous variables, as in standard regression techniques. It is related
to the radial basis function network and is based on a standard statistical technique called
kernel regression [30].
The GRNN consists of four layers: input layer, pattern layer, summation layer and output
layer. The input units are in the first layer. The second layer has the pattern units and the
outputs of this layer are passed on to the summation units in the third layer. The final layer
covers the output units. Details of the GRNN are presented by Specht and Schematic diagram
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of GRNN architecture is presented in Figure 2. In GRNN approach, users only choose the
value of smoothing parameter σ and the number of input nodes in the input layer [30]. In this
research, wide ranges of the smoothing parameter σ, i.e. from 0 to 50, have been used and
based on extracted input variables from original data series, GRNN model was adapted. In
this model defined input variables were extracted from whole observations.
<<Insert figure 2 about here>>
In the GRNN model the input nodes include following items:
1. The demand at the end of the immediately preceding target period (lag1).
2. The number of consecutive period with no demand transaction immediately preceding
target period.
3. The number of periods separating the last two nonzero demand transactions as of the
end of the immediately preceding target period.
4. The mean of demand for four period immediately preceding target periods.
In order to identify the effective input variables in GRNN architectures, several combinations
of different input variables were applied into the network and the performance of networks
were evaluated using statistical measures, and then best input sets have been selected based
on these measures [31].
2.2.2.2 Recurrent neural network(RNN)
To exploit the time structure in the data, the neural network must have access to this time
dimension. One way of introducing ‘‘memory’’ in the MLP is by considering a feedback
connections. The RNN used in this study is the basic Elman-type RNN [32], also referred to
as the globally connected RNN. The network consists of four layers (Figure. 3): an input 10
layer, a hidden layer, a context layer and an output layer. Each input unit is connected to
every hidden unit, as is each context unit. Conversely, there are one-by-one downward
connections between the hidden nodes and the context units leading to an equal number of
hidden and context units. In fact, the downward connections allow the context units to store
the outputs of the hidden nodes (i.e. internal states) at each time step, and then the fully
distributed upward links feed them back as additional inputs. Therefore, the recurrent
connections allow the hidden units to recycle the information over multiple time steps and
thereby to discover temporal information contained in the sequential input and relevant to the
target function. Thus the RNN has an inherent dynamic (or adaptive) memory provided by the
context units in its recurrent connections [29]. The mathematical model of RNN is as follows:
a j1 (t )=F (∑
i=1
R
w j , i1 pi ( t )+∑
c=1
s1
w j ,cC ac
1 ( t−1 )+b j1)1≤ j≤ s1
ak2 (t )=G ¿
In which
R: the number of input nodes
P: input vector.
w1, wc: hidden layer weight's matrix for actual and recursive inputs, recursively.
w2: Output layer weight's matrix.
s2 , s1: The number of hidden and output nodes, respectively.
b1,b2: the bias vector for hidden and output layers.
a1,a2: the output vector for hidden and output layers.
11
)2(
)3(
F and G: transfer function in hidden and output layer, respectively.
<<Insert Figure 3 about here>>
In this network, following variables have been defined for input nodes in input layer:
1. The demand at the end of the immediately preceding target period (lag1).
2. The number of consecutive periods with demand transaction, immediately preceding
target period.
3. The number of consecutive period with no demand transaction, immediately
preceding target period.
4. The number of periods separating the last two non-zero demand transactions as of
the end of the immediately preceding target period.
5. The number of period between target period and first non-zero demand immediately
preceding target period.
6. The number of period between target period and first zero demand immediately
preceding target period.
7. The mean of demand for six periods immediately preceding target period.
8. The maximum of demand for six periods immediately preceding target period.
The input patterns should be normalized before being presented to the network. When
variables are loaded into a neural network, they must be scaled from their numeric range into
the numeric range that the neural network deals with efficiently. In this network tan-sigmoid
and satlins transfer functions, have been used in hidden and output layers, respectively. Thus,
according to following formula, all input patterns were scaled to range of (-1, 1):
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Si (scaled )=[ S i−min (S)max ( S )−min (S) ]×2−1
Where S is the time series of variable under consideration, Si is the value of observations and
Si)scaled) is equal to normalized values. To find the best number of neurons in the hidden
layer, range of 1 to 15 neurons have been examined and neurons with minimum errors were
selected. When an input is presented to the network the training algorithm (learning equation)
attempts to adjust the weights so that the desired output is produced. In this research back
propagation algorithm have been used as training algorithm with learning rate of 0.01. In
addition, network's parameter adjusting have been done using adaptive calibration algorithm.
Because of the limited historical data, this algorithm preferred to batch mode training for train
the network.
2.2.2.3 Time-Delayed Neural Network(TDNN)
Another way of introducing memory in the MLP is by replacing the neurons in the input layer
with a memory structure called a Tap Delay Line (TDL). Such type of MLP is called Time-
Delay feed forward neural network (TDNN) and also referred to as a pseudo-dynamic neural
network owing to the static memory structure as opposed to the adaptive memory structure
used in recurrent neural networks. To predict temporal patterns, an ANN requires two distinct
components: a memory and an associator. The memory holds the relevant past information,
and the associator uses the memory to predict future events. In this case the associator is
simply a static MLP network and the memory is generated by the tapped delay line by simply
holding past samples of the input signal as shown in figure 4 [33].
The mathematical model of TDNN is as follows:
13
)4(
)5(
a j1 ( t )=F (∑
d=0
D
∑i=1
R
w j ,i , d1 P i ,d+1 ( t )+b j
1)1 ≤ j ≤ s1
ak2 (t )=G(∑
j=1
s1
w k , j2 a j
1 (t )+bk2)1≤ k ≤ s2
In which
D: Time delay memory degree
R: the number of input nodes
P: input vector.
w1,wc: hidden layer weight's matrix for actual and recursive inputs, recursively.
w2: Output layer weight's matrix.
s2 , s1: The number of hidden and output nodes, respectively.
b1,b2: the bias vector for hidden and output layers.
a1,a2: the output vector for hidden and output layers.
F and G: transfer function in hidden and output layer, respectively.
<<Insert figure 4 about here>>
In this research we use a special type of TDNN, in which TDLs have been applied in input
layer. This network was named as input delay neural network (IDNN). In this network, the
input nodes are the same as elman recurrent neural network architecture. In hidden and output
layer tan-sig transfer function was used, therefore, all patterns were scaled to (-1, 1). To find
the best results, the range of 1 to 15 neurons in the hidden layer has been used. For training
14
)6(
the network, back propagation algorithm with learning rate of 0.01 was used. Same as the
RNN architecture we use an adaptive calibration algorithm for adjusting the network
parameters. In this study, tap delay line with length 2 to 5 have been used in input layer.
2.3 Data analysis
In this study real data sets of 30 types of spare parts demand in Arak Petrochemical enterprise
in Iran have been used. The data were gathered from Arak Petrochemical enterprise's
inventory control package, which include 79 monthly periods from 2001 to 2007. Average
inter-demand interval (ADI) and squared coefficient of variation (CV2) measures were used
to classify demand patterns into four categories. The demand patterns are classified into
lumpy demand category when the average inter-demand interval (ADI) is greater than 1.32
and squared coefficient of variation (CV2) is greater than 0.49. The ADI ranges from 1.7 to
7.4 month and CV2 ranges from 0.52 to 2. Table 1 summarizes the statistical characteristics
of the 30 types of spare parts. All 30 items under consideration satisfy criteria specified for
demand lumpiness. For Croston's method and Syntetos-Boylan approximation (SBA) data
series have been divided into three blocks: (i) initialization, (ii) calibration and (iii)
performance measurement. The ‘initialization block’ is used to initialize values required for
methods based on recursive formulae (such as the mean inter-demand interval for Croston’s
method). In the ‘calibration block’, the optimal smoothing constants are identified based on
mean square error (MSE). Finally, the optimal smoothing constants are used to update
forecasts in the ‘performance block’, in which performance statistics are calculated [34].
The lengths of these blocks are 12- 51-16. For neural network methods, data series were
divided into two blocks. (i) training set (ii) test set. From 79 monthly observations, 63
observations have been used for train the network, and five Methods tested using last 16
observations.
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<<Insert Table 1 about here>>
2.4 Statistical Forecast Accuracy Measures
Forecast-accuracy measures are critical guidelines for proper selection and implementation of
forecast models. Standard forecast error measures frequently appearing in the literature do not
provide fair evaluations for lumpy demand cases due to the large number of zero demand
periods. The conventional measures, such as MAPE and GMAE, have many problems when
the actual demand in the dominator of the calculation is zero or the forecast of demand equals
to actual one.
Hoover [35], Boylan and Syntetos [36], Willemain [37], and Hyndman [38] suggested
various metrics and discussed their strengths and weaknesses. Although it seems that no
single metric comes out clean, but in this research adjusted mean absolute error(A-MAPE),
mean absolute scaled error(MASE) and percentage best(PB) measures have been used.
MASE is a scale-free error measure which uses naïve forecasts as a benchmark. Let e t
indicate forecast error, et = dt – ft. The scaled error, q at time t is then calculated using (7) and
MASE is the average of absolute values of qt :
q t=e t
1n−1
∑i=2
n
(di−d i−1)
Hoover has been suggested three variations on the MAPE: the denominator-adjusted MAPE,
the symmetric MAPE, and the ratio of MAD to MEAN. In this study latest measure was used
as Adjusted MASE.
16
)7(
Adjusted MAPE ¿
∑i=1
n
|d t−f t|n
∑i=1
n
d t
n
The third error measure, the percentage best (PB), is the percentage of time periods one
method performs better than the other methods under consideration. This approach is robust
to large forecast errors and the results can be subjected to formal statistical tests .It is a useful
measure, although it does not quantify the degree of improvement in forecast error [39].
The mathematical expression for PB for method m is
PBm=((∑t=1
n
Bm, t)n
)×100
Where, for time period t, Bm,t=1if |dm,t−f m ,t| is the minimum of |dk ,t−f k ,t|for all methods k
under consideration and Bm,t=0 otherwise .
To compare the alternative methods under consideration, results were reported on these three
error measures, A-MAPE, MASE and PB.
3. Result and Discussion
In croston's method and the Syntetos-Boylan Approximation, each of 30 time series is divided
into three blocks including: initialization, calibration and performance measurement. The
value of smoothing constant with minimum error in calibration block was selected for using in
performance measurement block according to the Mean Squared Error (MSE) measure.
17
)8(
)9 (
Therefore the results of forecasting using two traditional forecast methods in performance
measurement blocks were compared to four neural network approaches.
In this paper, three types of neural networks such as generalized regression neural network,
elman recurrent neural network and time delay neural network have been used in modeling
lumpy demand. In addition, the results of these models were compared to forecasts of multi-
layered perceptron network applied by Gutierrez et al. [25] in lumpy demand forecasting.
Table 2 reports overall A-MAPEs for the methods under consideration. The performance of
RNN model is superior among all methods in general (a simple average of 30 A-MAPEs:
99.68 for RNN vs. 111.44 for IDNN, 126.07 for GRNN, 155.33 for MLP, 163.3 for SBA and
190.87 for Croston method). In calculation of this measure we considered all periods
including the periods with no transaction. In 27 and 24 of 30 items, RNN model produce
more accurate forecasts than SBA and MLP, respectively.
<<Insert Table 2 about here>>
Table 3 shows results of MASEs measure for all forecasting methods. In overall RNN
approach has a better performance than those other methods (a simple average of 30 MASEs:
0.72 for RNN vs. .83 for IDNN, 0.97 for GRNN, 1.14 for MLP, 1.22 for SBA and 1.39 for
CR), Also according to this measure in 29 and 27 of 30 items, RNN model outperforms to
SBA and MLP, respectively.
<<Insert Table 3 about here>>
Table 4 and 5 reports on model performance based on Percentage Best statistics. Because of
the superiority of SBA over the Croston's method and recurrent neural network to other
proposed neural network in subsequent analysis, the performance of RNN only compared to
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the SBA and MLP. RNN had the highest PB values for 25 series vs. SBA and 24 series vs.
MLP. These PB statistics further establishes the overall superiority of RNN model (averaging
68.71% vs. 31.29% for SBA and 65.67% vs. 34.33% for multi-layered perceptron proposed
by Gutierrez et al[25].
<<Insert Table 4 about here>>
<<Insert Table 5 about here>>
In summary, we found that our proposed neural network models to be superior to Croston's
method and Syntetos-Boylan Approximation and recently applied multi-layered perceptron
by Gutierrez et al(2007) in general, based on all three error measures i.e., Adjusted MAPE,
PB, and MASE.
4. Conclusions and Future Studies
Spare parts inventories frequently display lumpy demand streams, which are difficult to
forecast. Croston's method introduced in 1972 is often used as a benchmark when testing new
methods [23]. Little work has been done on the application of NN modeling in lumpy demand
forecasting. We are aware of only two studies, Carmo and Rodrigues [24] who applied NN
modeling on 10 demand "irregularly spaced" time series and recently Gutierrez et al [25]
adopted MLP on 24 demand time series. In this study, the generalized regression neural
network (GRNN), the elman recurrent neural network (RNN) and the time delay neural
network (TDNN) have been used in lumpy demand forecasting. Our study compares the
performance of GRNN, RNN and TDNN model to those using two traditional methods like
Croston's method and the Syntetos-Boylan approximation and recently suggested Multi-
Layered Perceptron neural network (MLP) by Gutierrez et al [25]. Using real data sets of 30
19
types of spare parts from Arak Petrochemical Company in Iran and three performance
measures, A-MAPE, PB and MASE, we show that our proposed approaches produce more
accurate forecasts. Moreover, this study showed that the elman recurrent neural network is the
most effective approach in lumpy demand forecasting.
In future studies we shall attempt to use other dynamic neural networks like time delay
models with TDL in hidden layer, as well as, hybrid of time delay networks and recurrent
neural network. Another interesting research is the use of genetic algorithm to obtain the
optimal neural network architecture. The possibility of combining neural network to another
one for forecasting the occurrence and then the quantity of demand may be useful in lumpy
demand forecasting.
Acknowledgements
The authors wish to thank Dr. A.A. Syntetos for his encouragement on this research.
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