Forecasting irregular demand for spare parts inventory
Dang Quang Vinh
Department of Industrial Engineering, Pusan National University, Busan 609-735, Korea [email protected]
Abstract. Accurate demand forecasting is one kind of basic approach in supply chain management, especially in spare parts area. The characteristic of demand for spare parts inventories is difficult to predict because of not only random demand but also a large proportion of zero values. Croston’s method is a widely used to predict this kind of demand. In this paper, we forecast the cumulative distribution of demand over a fixed lead time using the modification of Croston’s method. By applying exponentially weighted average to Croston’s method, we show that the new one produces better forecast of the distribution of demand during a fixed lead time.
Keyword: Croston’s method; exponential smoothing; irregular demand; spare part; service parts inventory.
1. Introduction
Inventory with irregular demands are quite popular in practice. Item with intermittent demand include spare parts, heavy machinery, and high-priced capital good. Data for such items is composed of time series of non-negative integer values where some values are zero.
Accurate forecasting of demand is one of the most important aspects in inventory management. However, the characteristic of spare parts makes this procedure especially difficult. Up to now, Croston’s method is the most widely used approach for irregular demand forecasting which involves exponential smoothing forecasts on the size of demand and the time periods between demands.
In this paper, we develop a modification of Croston’s method for forecasting the cumulative distribution of intermittent demand during a fixed lead time. Using a numerical experiment, we show that the new forecasting method is better than the original Croston’s method.
2. Related research
Croston’s method is the most popular approach in intermittent demand forecasting area. Croston (1972) stated that the assumptions of this method were the distribution
of inter-arrival times is IID; and demand sizes and inter-arrival times are mutually independent.
Other authors, including Johnston & Boylan (1996) have suggested a few modifications to Croston’s method that can provide improved forecast accuracy. One such modification is to use log transformations of both demands and inter-arrival times. Another modified method proposed that inter-arrival times are assumed to have an IID Geometric distribution. The other model is combination between the above modifications that use logarithms of the demands and Geometric distribution of inter-arrival times.
3. Forecasting methods
3.1. Exponential Smoothing
Exponential smoothing refers to a particular type of method applied to time series data, either to produces smoothed data or to forecast. It was also used to predict intermittent demand. The exponential smoothing forecast assumes that lead time demand (LTD) is a normally distributed sum of L IID random variables. Based on this assumption, exponential smoothing process was used to estimate the mean of the normal distribution as follows:
Let X(t) be the observed demand in period t, t = 1…T. Let M(t) be the estimate of mean demand per period. Let L be the fixed lead time over that forecasts are desired.
M(t) = αX(t) + (1-α)M(t-1) (1)
where α is a smoothing constant between 0 and 1. We estimated the mean of the L demands over the lead time as L.M(T).
3.2. Croston’s method
The Croston method is a forecasting approach that was developed to provide a more accurate estimate for products with intermittent demand.
The Croston method consists of two main steps. First, Croston method calculates the mean demand per period by separately applying exponential smoothing. Second, the mean interval between demands is calculated. This is then used in a form of the model to predict the future demand.
Let Y(t) be the estimate of the mean size of a nonzero demand, let P(t) be the estimate of the mean interval between nonzero demands, and let Q be the time interval since the last nonzero demand.
If X(t) = 0 then Y(t) = Y(t-1) P(t) = P(t-1) Q = Q + 1
Y(t) = X(t) + (1α - )Y(tα -1)
P(t) = Q + (1α - )P(tα -1) Q = 1
The estimate of mean demand per period
M(t) = Y(t)/P(t) (2)
Croston’s method assumes the lead time demand (LTD) follows normal distribution with mean
L.M
(T)
(3) 4. Modification of Croston’s Method
The modification of Croston’s Method estimates the mean demand per period by applying exponentially weighted average forecasting method for any first interval which the demand increases again from zero value and also taking account of this approach to that point of time separately when we consider the intervals between nonzero demands and their sizes.
In this circumstance, we apply exponentially weighted average method of two last past values because of some reason. Firstly, this kind of method will bring out the results that are better than the original one due to considering the effects of two last periods on next period’s the forecasting value. Secondly, we just consider two last periods because the irregular demand is not influenced much by past trends. Two is a reasonable number in order to apply in the exponentially weighted average forecasting method.
Using ratio of the mean square forecast error (MSE) from the modified method and the Croston’s method as a measure of the improvement, we can evaluate the efficiency of the modified one.
The modification of Croston’s Method works as follows: If X(t) = 0 then Y(t) = Y(t-1) P(t) = P(t-1) Q = Q + 1 Else Y(t) = X(t) + α α(1- )Y(tα -1) + (1- )α 2 Y(t-2) P(t) = Q + α α(1- )P(tα -1) + (1- )α 2 P(t-2) Q = 1 With α + (1α - ) + (1α - )α 2 = 1
The value of α that minimizes the MSE can be obtained through Solver in Excel commercial software.
lead time demand (LTD) can be achieved as follows (2) and (3), respectively. 5. Numerical Examples
Table 1 gives an example about the intermittent demand data. Table 1. Intermittent demand data
Month Demand Month Demand 1 0 13 0 2 0 14 0 3 19 15 3 4 0 16 0 5 0 17 0 6 0 18 19 7 4 19 0 8 18 20 0 9 17 21 0 10 0 22 5 11 0 23 4 12 0 24 5
Table 2 and 3 give the results of the original Croston’s method and the modified one that are computed based on the above data.
Table 2. The result of original Croston’s method
Forecast α 0.3 Month (t) Demand X(t) Y(t) P(t) Q M(t) (X(t) – M(t))2 1 0 0 0 1 0 0.0 2 0 0 0 2 0 0.0 3 19 5.7 0.6 1 9.5 90.3 4 0 5.7 0.6 2 9.5 90.3 5 0 5.7 0.6 3 9.5 90.3 6 0 5.7 0.6 4 9.5 90.3 7 4 5.2 1.6 1 3.2 0.6 8 18 9 1.4 1 6.3 136.9
10 0 11 1.3 2 8.8 76.8 11 0 11 1.3 3 8.8 76.8 12 0 11 1.3 4 8.8 76.8 13 0 11 1.3 5 8.8 76.8 14 0 11 1.3 6 8.8 76.8 15 3 8.9 2.7 1 3.3 0.1 16 0 8.9 2.7 2 3.3 10.8 17 0 8.9 2.7 3 3.3 10.8 18 19 12 2.8 1 4.3 217.2 19 0 12 2.8 2 4.3 18.2 20 0 12 2.8 3 4.3 18.2 21 0 12 2.8 4 4.3 18.2 22 5 9.8 3.2 1 3.1 3.5 23 4 8.1 2.5 1 3.2 0.6 24 5 7.2 2.1 1 3.5 2.3 MSE 52.1
Table 3. The result of modified model
Forecast α 0.3 Month (t) Demand X(t) Y(t) P(t) Q M(t) (X(t) – M(t))2 1 0 0 0 1 0 0 2 0 0 0 2 0 0 3 19 5.7 0.6 1 9.5 90.3 4 0 5.7 0.6 2 9.5 90.3 5 0 5.7 0.6 3 9.5 90.3 6 0 5.7 0.6 4 9.5 90.3 7 4 5.2 1.62 1 3.2 0.6 8 18 9.3 0.93 1 9.9 65.0 9 17 9.6 1.29 1 7.4 91.5 10 0 9.6 1.29 2 7.4 55.3 11 0 9.6 1.29 3 7.4 55.3 12 0 9.6 1.29 4 7.4 55.3 13 0 9.6 1.29 5 7.4 55.3
15 3 7.6 2.7 1 2.8 0.03 16 0 7.6 2.7 2 2.8 7.9 17 0 7.6 2.7 3 2.8 7.9 18 19 11 2.79 1 4 226.5 19 0 11 2.79 2 4 15.6 20 0 11 2.79 3 4 15.6 21 0 11 2.79 4 4 15.6 22 5 9.2 4.52 1 2 8.8 23 4 13 4.83 1 2.7 1.7 24 5 15 5.9 1 2.6 5.9 MSE 45.8
Fig. 1. The comparison between the Croston’s method and the modified one
Table 2 and 3 show that MSE of the modified Croston’s method is smaller than that of the original one with optimal value of α (0.3) , namely 45.8 compared with
out the better outcome as opposed to the original Croston’s method regarding to MSE criterion.
6. Conclusion
In this paper, we propose a new forecasting approach to deal with the intermittent demand problem. Traditional statistical forecasting methods such as exponential smoothing and moving average that work well with normal and smooth demands do not give the accurate results with intermittent data because they ignore the zero values in forecasting demand. In the contrast, the modified Croston’s model developed in this material takes the special role of zero values into account. It can be considered as one forward step of original approach by applying the exponentially weighted average to Croston’s method. Numerical experiments show that the proposed method give the better mean square error when we compare with the traditional one. For further studies, the more appropriate assumption of lead time demand’s distribution is expected to be stated so that the better results can be found.
References
Thomas R. Willemain, Charles N. Smart, Henry F. Schwarz, 2004. A new approach to forecast intermittent demand for service parts inventories. International Journal of Forecasting, 20, 375–387.
Matteo Kalchschmidt, Giulio Zotteri, Roberto Verganti, 2003. Inventory management in a multi-echelon spare parts supply chain. International Journal Production Economics, 81– 82, 397–413.
F.R. Johnston, J.E. Boylan, 1996. Forecasting intermittent demand: a comparative evaluation of Croston’s method. Comment. International Journal of Forecasting, 12, 297–298.
Charles N. Smart, 2002. Accurate intermittent demand forecasting for inventory planning: new technologies and dramatic results. International Conference Proceedings.
