The forecast demand models for slow and fast moving spare ...



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：

[pic] (1)

fast moving：

[pic] (2)

where

[pic]：demand units of slow moving parts per period.

[pic]：demand units of fast moving parts per period.

[pic]：number of machines.

[pic]：average usage time of machines.

[pic]：if [pic]=1, it means the machine design is modified in last period; if [pic]=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 {([pic];[pic], [pic], [pic])}, i=1,2,…,n, where i denotes the ith period, and [pic];[pic], [pic], [pic] denote the sample data in ith period. Using the data and software package SAS, we can obtain the estimates of parameters [pic], j=0,1,2,3, [pic], 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：[pic] (3)

fast moving ：[pic] (4)

[pic] (5)

where

[pic]：the estimation of parts demand in ith period.

[pic]：the estimation of population variance.

In general, if the parts is a slow mover, the demand size follows a Gamma distribution, say G ([pic],[pic]), and its expected value and variance are [pic] and [pic]. Using the forecast data, the estimation of parameters are:

[pic], [pic] (6)

If the parts is a fast mover, the demand size generally follows a Normal distribution, say N ([pic], [pic]). The parameters are estimated by

[pic][pic], [pic][pic] (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 [pic]and[pic]in Case 1, and [pic], [pic], [pic], [pic], [pic], [pic], [pic], and [pic] 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:

[pic]

Case 2:

[pic]

Table 1: The results of SAS for Case 1.

|Parameter |Parameter Estimate | Standard Error | T for H0: | Prob > |T| |

| | | |Parameter=0 | |

|Intercept | -0.996273412 | 0.39207224 | -2.541 | 0.0199 |

|X1 | 1.048965621 | 0.10597206 | 9.899 | 0.0001 |

|X2 | -0.266645025 | 0.02568270 | -10.382 | 0.0001 |

Table 2: The result of SAS for Case 2.

|Parameter |Parameter Estimate | Standard Error | T for H0: | Prob > |T| |

| | | |Parameter=0 | |

|Intercept | 60362 |20791.54676886 | 2.903 | 0.0082 |

|X1 |-2848.61841351 |955.1683670268 | -2.982 | 0.0069 |

|X2 | 585.16493494 |193.4528764866 | 3.025 | 0.0062 |

|X1*X1 | 33.67032642 |11.0045863474 | 3.060 | 0.0057 |

|X2*X2 | 1.44692250 | 0.4284677630 | 3.377 | 0.0027 |

|X3*X3 |-1161.66773800 | 793.60004084. | -1.464 | 0.1574 |

|X1*X2 | -13.84699236 | 4.4190635911 | -3.133 | 0.0048 |

|X1*X3 | 28.72234434 | 18.1275061251 | 1.584 | 0.1274 |

|X2*X3 | -8.953284 | 3.55606165 | -2.518 | 0.0196 |

In Case 1, the data in current period is [pic], [pic], [pic]. Since [pic] implying that [pic]. It means demand size tends to decrease 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 [pic]0.58, [pic]1.68 of Gamma distribution from equation (6).

In Case 2,[pic] [pic] [pic], and [pic], [pic]. It reveals that spare parts demand quantity 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 [pic]. 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.

Reference

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