A DEMAND FORECAST MODEL FOR SEMICONDUCTOR SPARE PARTS
Pao-Long Chang1), Ying-Chyi Chou2), Ming-Guang Huang3)
1)
National Chiao Tung University (paolong@cc.nctu.edu.tw)
2)
National Chiao Tung University (paolong@cc.nctu.edu.tw)
3)
National Chiao Tung University (paolong@cc.nctu.edu.tw)
Abstract
Most demand forecast models are based on the mean and variance of historical demand
data. The spare parts demand of semiconductor equipment is closely related to equipment
characteristics. Thus, this study recommends using a regression model with independent
variables, including machines quantity, average usage time of machines and whether their
design is modified to forecast spare parts demand. With this model, Applied Materials Taiwan
(AMT), a major semiconductor equipment manufacturer in Taiwan, is employed as an
example to demonstrate the estimation of parts demand quantity and demand size distribution
parameters.
Keywords: spare parts; demand forecast; regression model; semiconductor equipment.
1.
Introduction
The spare parts of semiconductor equipment differ from those of common equipment. A
semiconductor machine contains hundreds or thousands of parts for which the demand
characteristics vary significantly. Some parts are replaced less than 12 times a year, other
parts may require regular replacement and weekly purchases. Therefore, parts are divided
into fast movers and slow movers, and the classification criteria are determined based on
annual demand quantity.
Parts requirement rate is closely related to the life cycle of a machine. Generally, parts
are supplied by the manufacturer and are free of charge during the three to six months period
following machine installation. Moreover, until a customer officially accepts a machine,
repeated tests must be conducted. This process results in extreme parts consumption. During
the one-year warranty period, the equipment manufacturers provide parts that have to be
placed due to regular use. This also increases the demand for the parts. However, their
demand rate is lower during this period than during that of the installation period. Once a
warranty has expired, equipment owners must pay for these parts, and as a result, the demand
rate declines. Furthermore, improvements in technology accelerate the research and
development of semiconductor production process, which causes machines to be upgraded
frequently, thus shortening the replacement cycle of machine parts. Hence it may be
concluded that the demand for semiconductor equipment parts is very closely related to the
life cycle of the equipment.
The aim of this study is to establish a demand forecast model for semiconductor
equipment spare parts. Since machine technologies develop quickly in the semiconductor
industry, detailed information on machine failure distribution is not available. Therefore, a
regression model is developed to forecast parts demand which uses machines characteristics,
such as number of machines, average usage time of machines, and whether machines design
are modified, as the independent variables.
2.
Literature Review
There have been numerous papers discussing the fast and slow moving parts inventory
models. For slow moving spare parts, Vereecke and Verstraeten [13] have developed an
inventory management system based on the assumption that demand of spare parts follows a
Poisson distribution. Segerstedt [10] and Yeh [15]、[16] focus on the parts in intermittent
demand situation. They assumed that the three variables---the time between two consecutive
demands, the demand size and the lead time---are all Gamma distributed. Burgin [1]
proposed demand size during lead time is Gamma distributed if data are positively skewed.
For fast moving spare parts, Dilworth [4] proposed many inventory control systems using
normal distribution to approximate the demand during lead time. In addition, Vereecke and
Verstraeten [13], Silver et al. [11] also indicated that fast moving spare parts demand should
be normal distributed during the lead time.
From those inventory models for slow and fast moving spare parts, it is necessary to
predict the demand size during lead time. Accurate parameters of demand distribution
estimates are very important because inventory cost or the probability of shortage during the
lead time are functions of those parameters.
With regard to demand forecast model, Buzacott [2] and Jun [6] both use exponential
smoothing to estimate the demand. It requires only two pieces of data, the last forecast and
the observation of the latest period. It is claimed to be the method most frequently used for
forecasting low and intermittent demand. Croston [3] developed a method for forecasting in
intermittent demand situations which he showed the method has lower variance than the
exponential smoothing forecast. Willemain et al. [14] emphasize the key role of demand
forecasting in planning production, inventories and work force and economic lot sizing. They
conclude that Croston's method is robustly superior to exponential smoothing. Foote [5]
discussed the implementation of forecasting system for aircraft spare parts. He used ARIMA
forecasting lead time demand and average monthly demand. Researchers such as Tsay [12],
looking at outliers, level shifts and variance changes for ARIMA series. When forecasts using
judgment and different models are combined, a simple average method is shown by Kang [7]
to be the best. Lawrence et al. [8] also show that simple models and averaging of different
forecasts are likely to be most effective. Sani et al. [9] described several forecasting methods
for low demand items, including exponential smoothing method, moving average method and
other simple empirical methods.
The forecast methods, including single exponential smoothing, Croston's method,
ARIMA, moving average method and so on, are all based on the mean and variance of past
demand data. However, they are not suitable for the demand forecast of spare parts of
semiconductor equipment since the spare parts demand is highly related to equipment
characteristics. In general, demand of spare parts is increasing along with the machine
quantity and its usage time. Moreover, when equipment design changed, parts reliability will
be increased and hence the spare parts demand size is decreased. Therefore, one must
consider the above mentioned factors of equipment in building the demand forecast model for
spare parts.
3.
Description of the Model
From the experiences of semiconductor manufactures, demand of slow moving spare
parts will increase sharply with machine quantity and average machine usage. However, the
demand change rate of fast moving spare parts is much lower than that of slower moving
ones. Therefore, demand size functions of these parts are assumed to follow a second-degree
polynomial regression model and an exponential pattern, respectively. Machine design
modifications are also a key factor that affects demand size. Thus, the regression models with
independent variables, including machines quantity, average usage time and design
modifications is used to forecast future demand for spare parts. Furthermore, in these models,
parts are assumed to belong to the same type of machine. If the parts belong to various types
of machine, the regression model can be employed to forecast the parts demand size for each
individual type and then the sum of which will become the total demand size for the parts.
The demand forecast models of slow moving and fast moving spare parts are shown as
following:
slow moving：
log y S   S 0   S1 x1   S 2 x2   S 3 x3
(1)
fast moving：
y F   F 0   F1 x1   F 2 x 2   F 3 x3   F 4 x12   F 5 x 22   F 6 x32   F 7 x1 x 2   F 8 x1 x3   F 9 x 2 x3
where
y S ：demand units of slow moving parts per period.
y F ：demand units of fast moving parts per period.
x1 ：number of machines.
x 2 ：average usage time of machines.
( 2 )
x3 ：if x3 =1, it means the machine design is modified in last period; if x3 =0, it means the
machine design is the same as last period.
The first step for forecasting the parameters of regression model is to collect the
historical data, say {( y, i ; x1,i , x 2,i , x 3,i )}, i=1,2,…,n, where i denotes the ith period, and
y , i ; x1,i , x 2,i , x 3,i denote the sample data in ith period. Using the data and software package
SAS, we can obtain the estimates of parameters  S , j , j=0,1,2,3,  F , j , j=0,1, …,9 in the
regression models. The estimates of spare parts demand in next period (i.e. (n+1)th period)
and unknown population variance can be computed very easily as the following:
^
slow moving： log y S ,n 1   S 0   S1 x1,n1   S 2 x 2,n 1   S 3 x3,n1
(3)
^
fas t m ovi ng ：
y F ,n 1   F 0   F 1 x1,n 1   F 2 x 2,n 1   F 3 x 3,n 1   F 4 x12,n 1   F 5 x 22,n 1

 F 6 x 32,n 1
(4)
  F 7 x1,n 1 x 2,n 1   F 8 x1,n 1 x 3,n 1   F 9 x 2,n 1 x 3,n 1
^
n
 ( y ,i  y ,i )
S 2  i 1
(5)
n2
where

y,i ：the estimation of parts demand in ith period.
S 2 ：the
estimation of population variance.
In general, if the parts is a slow mover, the demand size follows a Gamma distribution,
say G (  ,  ), and its expected value and variance are  and  2 . Using the forecast data,
the estimation of parameters are:
^
^

( y ,n1 ) 2
S
2
,
S2
^
 
(6)
^
y , n 1
If the parts is a fast mover, the demand size generally follows a Normal distribution, say
N (  ,  2 ). The parameters are estimated by
^
^
  y , n  1 ,
^2
  S2
(7)
Of course, the parameters of other types of demand distribution can be estimated in a similar
way.
4.
Examples
According to the model described in section 3, we use real data from Applied Materials
Taiwan (AMT), a major semiconductor equipment manufacturer in Taiwan, to demonstrate the
applicability of the model. The parts in Case 1 is diamond disk which is a slow moving spare
parts. In Case 2, the parts is mass flow controller which is a fast moving spare parts. The data of
Case 1 parts was recorded from April 1997 to December 1998. During this period of time,
machines design has never been changed. The data of Case 2 parts was recorded from July
1995 to January 1998, in which machines design has been modified many times. In these two
cases, we assume the unit period is one month. With the monthly data of the demand size,
numbers of machines and average usage time of machines, the forward selection procedure of
variables from equation (1) and (2) leads to the model that contains the variables x1 and x 2 in
Case 1, and x1 , x 2 , x12 , x 22 , x 32 , x1 x2 , x1 x 3 , and x2 x3 in Case 2. The computing
algorithm is implemented in SAS language, and the parameter estimates are obtained as shown
in Table 1 and Table 2. Thus, the regression models are:
Case 1:
l o gy S  0.9 9 6 2 7 3 411.20 4 8 9 6 5 6
x1210.2 6 6 6 4 5 0x22 5
Case 2:
y F  60362  2848.62 x1  585.16 x2  33.67 x12  1.45 x22  1161.67 x32
 13.85 x1 x2  28.72 x1 x3  8.95 x2 x3
Table 1: The results of SAS for Case 1.
ParameterParameter
Standard Error T for H0:
Estimate
Parameter=0
Intercept
-0.996273412
0.39207224
-2.541
X1
1.048965621
0.10597206
9.899
X2
-0.266645025
0.02568270
-10.382
Prob > |T|
0.0199
0.0001
0.0001
Table 2: The result of SAS for Case 2.
ParameterParameter
Standard Error T for H0:
Estimate
Parameter=0
Intercept
60362 20791.54676886
2.903
X1
-2848.61841351 955.1683670268
-2.982
X2
585.16493494 193.4528764866
3.025
X1*X1
33.67032642 11.0045863474
3.060
X2*X2
1.44692250 0.4284677630
3.377
X3*X3 -1161.66773800 793.60004084.
-1.464
X1*X2
-13.84699236 4.4190635911
-3.133
X1*X3
28.72234434 18.1275061251
1.584
X2*X3
-8.953284
3.55606165
-2.518
Prob > |T|
0.0082
0.0069
0.0062
0.0057
0.0027
0.1574
0.0048
0.1274
0.0196
In Case 1, the data in current period is x1  8 , x2  26.82 , x3  0 . Since
dy S
1 dy S
  S 2  0.27 implying that
 0 . It means demand size tends to decrease
y S dx 2
dx 2
monotonously as the average usage time of machines is increased. This result indicates that
there are second sources for Case 1 parts which is indeed the situation in practice. Using the
regression model, the forecast demand size is 0.98 next month from equation (3), variance is
1.65 from equation (5). If the demand size of Case 1 parts is Gamma distributed, we can get
the parameters   0.58,   1.68 of Gamma distribution from equation (6).
In Case 2, x1  54
x 2  55 .83
x3  1 , and dy F   F 1   F 8  2  F 4 x1   F 7 x 2  43.42  0 ,
dx1
dy F
  F 2   F 9  2  F 2 x 2   F 7 x1  65167.99  0 . It reveals that spare parts demand quantity
dx 2
increases along with machine quantity and average usage time. This shows that Case 2 is a
normal parts. In the next period, the average usage time of spare parts becomes 56.83.
Suppose
the
machine
quantity
is
still
54,
then
y F ( x3  1)  y F ( x3  0)   F 3   F 6   F 8 x1   F 9 x2  109.42  0 . This means that parts demand quantity is
decreasing after machine design is modified. In fact, if machine design is not modified next
month, the parts demand forecast is 152.72 from equation (4); if machine design is modified,
the parts demand forecast will drop to 33.30.
5.
Conclusion
Most of the demand forecast methods are based on the past data of spare parts demand.
In semiconductor industry, since parts demand quantity is closely related to equipment
characteristics, therefore, this paper suggests a regression model with machines quantity,
average usage time and whether machines design is modified as independent variables to
forecast the spare parts demand. It is also demonstrated that the model can be used to
estimate the parameters of parts demand distribution.
6.
Acknowledgment
The authors would like to thank The National Science Council of the Republic of China
for financially supporting this research under Contract No. NSC 89-2213-E-009-042.
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