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Journal of Intelligent & Fuzzy Systems 31 (2016) 2911–2918 DOI:10.3233/JIFS-169174 IOS Press
Intermittent demand forecasting with fuzzy markov chain and multi aggregation prediction algorithm Ming Leia,∗ , Shalang Lia and Qian Tanb a Department
of Management Science and Technology, Guanghua School of Management, Peking University, Beijing, P.R. China b State Grid Corporation of China, Xicheng District, Beijing, P.R. China
Abstract. Demand forecasting is very important, both in academic and business practice. Intermittent demand forecasting has bothered managers and scholars for a long time. The zero demand point and extreme high values in the demand time series make it difficult to forecast. Under this condition, the fuzzy time series model performs well in eliminating the side effects of extreme data. To make our model more stable, we review the latest intermittent forecasting literature and combine the MultiAggregation Prediction Algorithm (MAPA) and the fuzzy Markov chain model to generate a new forecasting algorithm. In this paper, we use the material demand data of STATE GRID Corporation of China to test the forecasting accuracy of the new forecasting algorithm, as well as compare the results to the exponential smoothing (ES) model and the new forecasting algorithm. The results show that FMC-MAPA with an equal weight method in the time disaggregating process is the best forecasting method in this case. The forecasting ability of this method is more stable and robust in different fuzzy partition numbers and data adjustment than the ES and FMC model. We also study the impact of data adjustment on forecasting error. The results indicate that the unadjusted data have lower forecasting errors when compared to the linear trend adjustment, additive seasonal adjustment and the combination of these two adjustment methods. Keywords: Intermittent demand, forecasting, fuzzy Markov chain, multi-aggregation prediction algorithm
1. Introduction Demand forecasting is very important to material planning in supply chain management, and it is very popular in both the academic world and the industry. There are a large number of forecasting methods applied in business practices, among which the time series forecasting methods are most widely used. According to the review of the time series forecasting methods [4, 9], the most commonly ∗ Corresponding author. Ming Lei, Department of Management Science and Technology, Guanghua School of Management, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing, P.R. China. Tel./Fax: +86 10 62756243; E-mail:
[email protected].
used time series forecasting models are listed below: Exponential smoothing models, ARI-MA models, Seasonality State space and structural models and the Kalman filter, Nonlinear models, Long memory models, ARCH/GARCH models, Count data forecasting models, Combining models and so on. However, the conventional forecasting methods are not suitable for intermittent demand forecasting (intermittent demand means that some time periods have no demand, which makes the forecasting more difficult than normal and smooth demand). A stream of literature has discussed the intermittent demand forecasting methods in regards to the modification of conventional forecasting methods. These include the Croston model [2], bootstrapping method [3],
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M. Lei et al. / Intermittent demand forecasting with fuzzy markov chain and multi aggregation prediction algorithm
Kalman filter weighting function [7], modified Holt’s method [5] and so on. To the best of our knowledge, most intermit-tent demand forecasting literature only uses 0 to proxy the no demand point, which may have more information than 0. In this consideration, we apply the fuzzy time series theory to intermittent demand forecasting. The fuzzy time series forecasting theory is proposed by Song and Chissom [6], and this new time series forecasting method has been studied in many areas. The algorithm of this forecasting method consists of 4 steps: (1) define the universe of discourse U and partition the universe of discourse U. (2) define the fuzzy sets and fuzzify the time series data. (3) Model the fuzzy relationship. (4) de-fuzzify the forecasting results. The fuzzy time series forecasting methods can deal with the uncertainty and vagueness inherent in data. This method has been widely used in literature. Chen et al. [8] used the fuzzy time series and automatically generated weights of multi-factors to forecast the Taiwan Stock Index. This showed that the proposed method was better than other methods. Leite et al. [1] applies time varying systems and a fuzzy algorithm to forecast the rainfall problem. Shah and Mrinalini [17] used if-then based fuzzy rules to capture the trend prevailing in time series and applied these methods to university enrolment, sales and the GDP forecasting problem. Chen et al. [14] introduces the fuzzy time series forecasting methods in supply chain disruption forecasting. As a result, significant improvement has been shown. A great number of studies have proven that fuzzy time series forecasting methods outperform other conventional forecasting methods [15, 18, 24, 26, 27]. However, the application of fuzzy time series forecasting on intermittent demand is scarce. As far as we are concerned, only two research papers have done the relevant study. Kis¸i [21] applies fuzzy neural networks in intermittent demand forecasting and Chen [13] uses a similar approach to forecast the inventory of automobile spares parts. Those papers fill the gap in this area by introducing a fuzzy time series forecasting method to solve the intermittent demand forecasting problem. What’s more, we also incorporate the latest literature about intermittent demand forecasting in our research. In this paper, we combine the Multi Aggregation Prediction Algorithm (MAPA) method [20] and the fuzzy Markov chain (FMC) method (we will use MAPA-FMC in the rest of the paper). The merit of the FMC model is the use of fuzzification to eliminate the side effect of
the extreme data point, and the MAPA is effective in decreasing the volatility and intermittency of the time series. The results of this paper manifest that the combination of these two methods will generate more reliable forecasting results. The rest of this paper is organized as follows: Section 2 reviews the algorithm of the MAPA method, FMC forecasting model and the forecasting error measurement method. Section 3 introduces the algorithm of forecasting in this paper. Section 4 presents the descriptive statistics of data, as well as the forecasting results, and it compares the forecasting results of the FMC-MAPA model with the benchmark forecasting model. Section 5 has the conclusions and future study.
2. MAPA and fuzzy Markov chain model 2.1. MAPA for intermittent demand forecasting The main property of intermittent demand is the discontinuity of the time series. The MAPA method comes from the idea that less frequent, time series data will have lower intermittency. However, temporal aggregation will induce information loss, which is not proper for inventory and supply chain management. Nikolopoulos et al. [16] proposes an aggregation and disaggregation approach to overcome the information loss of temporal aggregation. However, the aggregation and disaggregation approach need to decide the best aggregation level, which is not an easy job. Kourentzes et al. [20] modified this research by taking multiple aggregation levels into consideration and combining the forecasting results of different levels, which is called the MAPA. This method has attracted the attention among scholars [11, 12, 19]. The MAPA method is shown below: Denote yj[k] as the temporal aggregation of time series yt (t = 1, 2, . . . , T ) where [k] means the aggregation level is k periods. Specifically, yj[1] = yt jk and generally yj[k] = 1k t=1+(j−1)k yt if n/k is integer, while if n/k is not integer, we will remove n − n/k × k from the original time series in order to form complete aggregation buckets. The next step is forecasting aggregated time series yj[k] (k = 1, 2, . . . , K) and obtaining forecasts of different [k] (h is foreaggregation levels denoted by yˆ n/k+h casting periods). Then we combine these forecasts to generate the final forecasting results. There are two combination methods in this paper [11]:
M. Lei et al. / Intermittent demand forecasting with fuzzy markov chain and multi aggregation prediction algorithm
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Unweighted mean: yˆ T +hk =
K 1 [k] yˆ n/k+h K
(1)
k=1
Weighted mean: yˆ T +hk =
K
[k] wk yˆ n/k+h
(2)
k=1
where w1 =
K+1 2K ,
wk =
Fig. 1. The membership function of fuzzy sets. 1 2K .
2.2. Fuzzy Markov chain forecasting method 2.2.1. Fuzzy time series In this section, we mainly describe the concept of fuzzy time series. In this paper, we use the definition of the first paper proposed by Song and Chissom [6]. Definition 1. [28] Let U be the universe of discourse and U = {u1 , u2 , . . . , un }. Then we define the fuzzy set A of the universe of discourse U as follows: A = fA (u1 )/u1 + fA (u2 )/u2 + . . . + fA (un )/un (3) Where fA is the membership function of the fuzzy set A and fA : U → [0, 1], fA (ui ) denotes the membership degree of ui (i = 1, 2, . . . , n) in the fuzzy set A. the symbol “+” means the operation union, not the summation. Definition 2. [6, 28]: Let Y (t) (t = 1, 2, 3 . . . , T ) be the original time series and Y ⊂ R. Denote Y (t) as the universe of discourse and {A1 , A2 , . . . , An } as fuzzy sets given Y (t). fAi (Y (t)) (i = 1, 2, . . . , n; t = 1, 2, . . . , T ) denotes the membership degree of t-th item of original time series on fuzzy set Ai . F (t) denotes the fuzzy time series on Y (t), which is given below: F (t) = [fA1 (Y (t)), fA2 (Y (t)), . . . , fAn (Y (t))] (4) 2.2.2. Fuzzy partition There is no restriction on the number of fuzzy sets. In this paper, we regard the partition number as variable parameter. We set n = 3, 4, 5, 6, 7, 8 and compare the effect of different partition number on forecasting error. The universe of discourse U is defined by U = [min(Y ) − y1 , min(Y ) + y2 ], where y1 and y2 are positive numbers [6]. In the intermittent demand setting, min(Y ) = 0 and no demand will be smaller than 0. Therefore, we set y1 = 0, y2 = std (Y ) (std means standard deviation).
To keep in line with existing literature about interval partition, we use the Fuzzy C-Means (FCM) clustering methods to decide the partition interval. FCM is aimed at minimizing the clustering loss function by iteration. We derive the cluster center by using the Matlab function “fcm”. After obtaining the center, we choose the middle point of the neighbor center point to separate the universe of discourse into n fuzzy sets. We define the membership function of the n fuzzy sets by the triangular function, which is shown in Fig. 1. 2.2.3. Fuzzy Markov chain forecasting method The Fuzzy Markov chain (FMC), which is widely studied in literature [10, 23, 25, 29], combines the fuzzy time series theory and Markov chain theory. The Markov chain model is applied in state transition and assumes that the present state only relates with the first lagged state. In our paper, each fuzzy set can be regarded as an individual state, and the degree of Y (t) on each fuzzy set can be regarded as the probability of Y (t) lines in each fuzzy set. Then, we model a fuzzy state transition matrix to describe the relation between F (t) and F (t − 1). The model is described as follows: Step 1. Compute the initial state probability At first, we denote ρt,i = Ai ∩ Y (t) / Y (t) as the probability of Y (t) lies in fuzzy set Ai where Y (t) = m i=1 fAi (Y (t)), Ai ∩ Y (t) = fAi (Y (t)). The initial state probability of Y lies in Ai as follows: T −1
Pi =
t=1
ρt,i
T −1
(5)
Step 2. Compute the transition matrix T −1
Pi,j =
t=1
ρt,i ρt+1,i
(T − 1)Pi
(6)
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Pi,j denotes the transition probability of fuzzy set Ai to fuzzy set Aj , and we denote P = [Pi,j ]m×m . Step 3. Forecasting At first we compute the membership degree of Y (T ) on all the fuzzy sets and get F (T ) according to Equation (4). Then using the following equation we obtain the fuzzy forecasting results of n + 1. F (T + 1) = F (T )P
(7)
Step 4. Defuzzification Defuzzification methods are different in many papers, among which Center of Area (COA) is the most widely used method. The detailed computation method can be seen in Equation (9). C Fˆ (T + 1) (8) C 1 Where C is the vector of fuzzy sets’ center. Fˆ (T + 1) is the forecasted vector of the membership degree of corresponding fuzzy sets. 1 is the vector of 1 (the same length as C). Y (T + 1) =
2.3. Forecasting error measurement There are a large number of forecasting error measurements, both in literature and business practice. As reviewed by Prestwich [23], the most commonly used measurements, mean error group (ME, MSE, RMSE, MAE, MdAE), is useful for comparing different methods. In this paper, we use RMSE to measure the forecasting error: T (yt − yˆ t ) RMSE =
t=1
(9) T Where yt (t = 1, 2, . . . , T ) is the original time series, yˆ t (t = 1, 2, . . . , T ) is the forecasting results. RMSE is Root-Mean Square Error.
3. Forecasting algorithm The forecasting algorithm is shown in Fig. 2, and the detailed steps are presented as follows: Step 1. Data preparation In most cases, time series data have trend and seasonality patterns. These patterns may have some effect on the forecasting results. In this paper, we will use four types of data adjustment methods: (1) no adjustment, (2) linear trend removed, (3) additive
Fig. 2. The forecasting algorithm.
seasonal effect removed, (4) linear trend removed and additive seasonal effect removed [22]. Step 2. Temporal aggregation K is the aggregation level as defined in Section 2. In this paper, we use four aggregations levels: K = 1, K = 2, K = 3, K = 4. Then we will forecast each aggregation level data separately. Step 3. Fuzzification At first, we set partition numbers, n = 3, 4, 5, 6, 7, 8 separately, and then derive fuzzy sets by the Fuzzy-C means clustering method. Finally, we will use Equation (4) to transform the ordinary time series data into the fuzzy time series data. Step 4. Markov chain forecasting Suppose we have T historical data and forecast the T + 1 point data. We will use Equation (6) to compute the Markov transition matrix first. Then we will use Equation (7) to derive the fuzzy time series forecasting results. Step 5. Defuzzification In this paper, we use COA method, which is described in Equation (8), to defuzzify the forecasted fuzzy time series. Step 6. Combining Finally, we will combine different, temporal aggregation level forecasting results together to generate
M. Lei et al. / Intermittent demand forecasting with fuzzy markov chain and multi aggregation prediction algorithm
one forecasting result. The two combination methods are described in Equations (1 and 2). We apply an exponential smoothing (ES) forecasting method as the benchmark model and we set the exponential smoothing factor equal to 0.3, 0.5 and 0.7, separately. Apart from the ES model, we also run the FMC model alone and compare the forecasting error with the new forecasting method proposed in this paper. To sum up, we will run 21 forecasting models here and 4 data adjustment methods. Therefore, the number of total forecasting results will be 84.
4. Data description and forecasting results 4.1. Date set STATE GRID Corporation of China needs to buy a large quantity of materials for power grid construction and overhaul every year. Our data comes from one province electric power company of STATE GRID, as shown in Fig. 31 . As we can see in the figure, the time series is volatile and includes some zero points, which make forecasting and inventory management very difficult. The time series is a monthly demand from April 2010 to March 2016. The highest demand is 6,286 units and the lowest demand is 0 unit. There are 4 zero points in this time series and many demand points are very close to zero. In this paper, our forecasting study
Fig. 3. Original time series. 1 The data used in this paper comes from the ERP system of STATE GRID Corporation and we got this data with the help of the staff in this company.
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begins in April 2014, and the demand data before this date are regarded only as a training set. 4.2. Forecasting results Table 1 shows the RMSE of all the forecasting methods. In the table, “FMC-MAPA1” means the combination of FMC model and temporal aggregation and disaggregation method with Equation (1). “FMC-MAPA2” means the combination of FMC model and temporal aggregation and disaggregation method with Equation (2). In the second column, a stands for an exponential smoothing factor, and n is the fuzzy partition number. There is no agreement on the determination of the partition number in literature. In general, when data has more noise, we may choose smaller n. While the data is less volatile, we may choose bigger n. In this paper, we choose n in the range of 3∼8 in each model and compare the differences among them. The unadjusted data is more volatile and the linear trend and seasonal trend adjusted data is also less volatile. The results showed that unadjusted data needs smaller n and trend adjusted data should use bigger n, which is consistent with the general rules of choosing a partition number n. From the forecasting results of unadjusted data, we may conclude that the FMC based methods perform better than the ES model. In the FMC model, the best partition number in the fuzzification process is 4, and the best forecasting method is FMC-MAPA1. This is because the RMSE of this method has the lowest RMSE and the RMSE is stable across different fuzzy partition numbers. From the data of column 4, column 5 and column 6, the linear adjustment will not decrease the RMSE of the original time series. As we can see in Fig. 4, the linear trend is very flat. Therefore, it has weak impact on the forecasting process. The table also showed that FMC-MAPA1 is the best forecasting method in the linear de-trend scenario. The seasonal adjustment may induce the increase of RMSE of all the forecasting methods. As shown in Fig. 4, the seasonal pattern is obvious before the 40th month. However, the seasonal pattern is changing after the 40th month, which will have negative impact on demand forecasting. It is worth noting that the FMC-MAPA2 generates better forecasting results than other forecasting methods, while the FMC-MAPA1 is more stable and robust through different fuzzy partition numbers and data adjustment. The best fuzzy partition number is between 4 and 6.
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M. Lei et al. / Intermittent demand forecasting with fuzzy markov chain and multi aggregation prediction algorithm Table 1 The RMSE of all the forecasting methods and data adjustment Methods ES
FMC
FMC-MAPA1
FMC-MAPA2
a = 0.3 a = 0.5 a = 0.7 n=3 n=4 n=5 n=6 n=7 n=8 n=3 n=4 n=5 n=6 n=7 n=8 n=3 n=4 n=5 n=6 n=7 n=8
Original Data
Linear De-trend
Additive De-season
Det-rend and De-season
334.05 324.59 308.90 297.37 278.80 391.32 464.03 482.21 539.25 282.75 269.93 286.93 295.08 300.89 320.54 289.49 272.01 331.61 365.11 386.40 405.78
334.30 324.96 309.50 342.90 302.35 366.25 414.00 425.95 535.24 295.25 289.08 288.27 286.30 294.65 305.91 317.57 302.21 320.41 339.29 382.35 397.41
593.89 563.45 545.18 913.39 560.99 514.85 504.75 505.66 514.00 592.91 526.47 511.60 513.59 516.50 517.16 737.25 541.15 511.25 506.47 508.58 506.63
593.99 563.62 545.50 952.63 598.25 567.51 512.77 517.43 525.58 596.23 531.36 514.06 511.25 498.70 501.57 757.35 560.93 519.60 509.71 505.08 505.28
Fig. 4. Original time series and linear trend and seasonal trend.
These results are similar with many previous studies [13, 15, 16, 20, 21, 24, 27]. In this paper, the demand is intermittent with many extreme data points. Conventional time series methods, such as the ES model, will be misled by these extreme data points. However, Fuzzification is good at getting rid of the negative impact of the extreme data and MAPA will capture both short term trend and long trend, which is effective in decreasing the volatility of the time series. Therefore, the forecasting ability is: FMC-MAPA1, FMC-MAPA2> FMC>ES. Kourentzes et al. [20] shows that MAPA2 will improve forecasting accuracy in some cases. This paper obtained the same results (column 5 and
Fig. 5. Original time series and forecasting result of unadjusted data.
column 6 in Table 1). However, we also find that the forecasting ability of the FMC-MAPA1 is more stable than the FMC-MAPA2. The reason for this result is that equal weight is neutral from a different temporal hierarch. Also, the time series pattern in this paper is not stable. Therefore, the neutral strategy will get more stable forecasting results, which implies that the FMC-MAPA1 is the best forecasting strategy when time series are very volatile and intermittent. Figure 5 shows the original data and forecasting results in the period between April 2014 and March 2016. The figure presents the best forecasting results of ES, FMC, FMC-MAPA1, FMC-MAPA2 methods
M. Lei et al. / Intermittent demand forecasting with fuzzy markov chain and multi aggregation prediction algorithm
in unadjusted data. As we can see in the figure, the ES method has obvious lag effects, which means the previous volatility will induce the fat tail in later forecasting results. However, these lag effects are smaller in the fuzzy Markov chain based forecasting model because the fuzzification process will eliminate the impact of extreme values and performs better at deriving the trend of the time series. 5. Conclusions and future study
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FMC-MAPA method. However, intermittent demand is very common in real business. More data sets from different areas should be tested in the future to verify the forecasting stability and robustness of the method. In this case, linear trend and seasonal trend adjustment will not contribute to more accurate forecasting results. The main reason is due to the pattern change in the time series. Future study may focus on the pattern detection and use different adjustment in different patterns.
5.1. Conclusions Acknowledgments This paper introduces a novel FMC-MAPA scheme for intermittent demand forecasting with extreme low and extreme high time series data. The forecasting results show that FMC methods will decrease the forecasting error compared to the ES model. However, the decreasing effect is not stable and is robust in different fuzzy partition numbers and data adjustment. The FMC-MAPA methods proposed in this paper will decrease the forecasting error further when compared to the FMC and ES. The FMC-MAPA1 method, which uses the Equation (1), is the best forecasting method in this case. It is worth noting that the forecasting ability of this method is more stable and robust in different fuzzy partition numbers and data adjustment. The merit of the FMC model is in using the fuzzification to eliminate the side effects of extreme data points. The MAPA is also effective in decreasing the volatility and intermittency of the time series. The results of this case prove that the combination of these two methods will generate more reliable forecasting results. Regarding data adjustment, the unadjusted data have lower forecasting error when compared to the three other adjustment methods. From the historical data of the time series. We may find that the linear trend and seasonal trend changed in different periods. In this condition, data adjustment will not effectively remove the gross linear trend and seasonal trend. As we can see, the linear trend is flat and the seasonal trend only fits the first half of the time series, not the later period of the time series. In this condition, data adjustment is bias in some parts of the time series. That is why the unadjusted data have better forecasting results. 5.2. Future study This paper only uses the material demand of the STATE GRID Corporation of China to test the
This paper is sponsored by the management consulting program of STATE GRID Corporation of China “The Big Data Study about Supply Chain Management of STATE GRID Corporation of China” and the number is 0711-15OTL02511025.
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