Comparison of Fuzzy Time Series Chen and Cheng to Forecast Indonesia Rice Productivity

How to cite this article: Sofhya, H., N. (2022). Comparison of Fuzzy Time Series Chen and Cheng to Forecast Indonesia Rice Productivity. Eduma : Mathematics Education Learning and Teaching, 11(1), 119 – 128. doi:http://dx.doi.org/10.24235/eduma.v11i1. 10936


INTRODUCTION
Rice is the main staple food of Indonesia. Based on Badan Pusat Statistik (BPS) data in 2019, around 90% of Indonesians consume rice as the main product of their carbohydrate needs. The 2020 population census conducted by BPS for Indonesia population in September 2020 was 270,20 million people (Badan Pusat Statistik, 2021). This means that Indonesia must have a large enough supply of rice. But based on BPS data 2021, it shows that the harvested area and productivity of rice in Indonesia in 2021 has decreased compared to the previous year (Badan Pusat Statistik, 2021). There are many factors that affect the decreased of rice productivity such as rainfall, fertilizer quality, seed quality and human resources. Indonesia's rice production is currently not able to supply the needs of domestic rice consumption (Ishaq et al., 2017). Therefore, the Indonesian government chose to import rice for supply the needs of domestic rice consumption. However, rice import policies need to be carried out with caution. Rice import policy will have an impact on price stability if carried out excessively. Therefore, it is necessary to estimate domestic rice productivity so that the government can make appropriate rice import policies. Prediction of domestic rice productivity can be obtained using the forecasting method.
Forecasting is the art or science of predicting future (Heizer & Render, 2015). The purpose of forecasting is to reduce uncertainty about the future. The right forecasting method is needed to be able to provide accurate forecasting results so that it can be used as a reference to make the right decisions regarding future policies. Data on the movement of the value of Indonesian rice productivity follows a time series that is summarized annually. The time series forecasting method is carried out using previous data to determine future data (Hanke, et al, 2005). New approaches to forecasting techniques continue to be developed to obtain accurate predictions. Fuzzy logic is proven to have a better performance in solving uncertainty problems (Sofhya, 2020). Forecasting time series data using fuzzy models is known as fuzzy time series.
Fuzzy time series was first introduced by Song & Chissom (1993), The superiority of fuzzy time series is to defines a fuzzy relation that was built by determining the logical relation of the training data. Fuzzy relation is formed by training data logical relation which involves fuzzy sets from universal partition. Universal partition division based on statistical distribution on each partition. The use of statistical distributions as a consideration for repartitioning and the use of data for compiling universal sets is still an open problem in the context of how to determine the optimal forecasting model to improve forecasting performance (Maria et al., 2019). As science develops, there are several models of fuzzy time series was proposed to obtain optimal forecasting results. Some of them are models proposed by Chen (1996), Singh (2007, and Cheng (2008). These models have different ways of defuzzification to get the forecast value. Chen proposes using the average in obtaining the forecast value. Then, Singh proposes using a supremum in obtaining the forecast value while cheng uses the weighting in obtaining the forecast value. The optimal forecasting model is determined based on the accuracy of the forecasting result. The mean absolute percentage error (MAPE) is one of the most popular measures of the forecast accuracy, it is recommended in the most textbooks (Kim & Kim, 2016). In some research, the forecasting methods are compared to obtain the best result determined based on the MAPE value. A forecasting method that produces a smaller MAPE value means a better method and more appropriate method to forecast the data. The research conducted by Arnita et al (2020) which compare three fuzzy time series model to forecast rainfall in Medan shows that Chen model is the best model compared to the other two models based on the MAPE value. Then, the research conducted by Tauryawati & Irawan (2014) which compare Fuzzy time series cheng model with Box-Jenkins method to forecast IHSG, shows that from the two methods, the cheng model produces more accurate forecasting value with smaller MAPE values. Both studies show that the Chen and Cheng models are among the best methods for data forecasting. Therefore, this research will discuss the comparison of forecasting using fuzzy time series Chen and Cheng models in forecasting rice productivity in Indonesia. In this research, the comparison based on the MAPE value. The forecasting method that produces the smaller MAPE value means that those method is more appropriate method to forecast Indonesia rice productivity.

Fuzzy Set
The definition of fuzzy set was first introduced by Zadeh in 1965. Fuzzy set is a pair of (A, ), where A is a set and is membership function : A→ |0,1|. For a finite set A={ 1, 2, … , }. Grade of membership of in (A, ) is ( ). Let ∈ , is not include of fuzzy set (A, ) when ( ) = 0. In other ways is fully included of fuzzy set (A, ) when ( ) = 1. Then is called member of fuzzy set (A, ) when 0 < ( ) < 1. When ( ) > 0, is the support of (A, ) and is kernel when ( ) = 0. There are many ways to represent fuzzy sets, fuzzy sets (A, ) can denoted by

Time Series
Time series is a set of ordered data observations in time. (Hanke & Wichren, 2005). There are four types of data time series, namely horizontal, trend, seasonal and cyclical. The horizontal data is an unexpected and characteristic event random, but the occurrence can affect time series data fluctuation. The trend data is a trend of direction data in the long term, it can increase or decrease. Seasonal data is fluctuations in data that occur periodically. Then the cyclical data is a fluctuation of data more than one year.

Fuzzy Time Series
Fuzzy Time Series is forecasting method that uses fuzzy principles. This method applies fuzy sets to time series analysis. Therefore, fuzzy time series captures past data then uses it to predict future data (Tauryawati & Irawan, 2014). In this method the time series data will be represented into a fuzzy sets. Then the relationship between the data will be defined. Let F(t -1) = Ai and F(t) = Aj. Relationship between two consecutive observations, F(t) and F(t -1), referred to as a fuzzy logical relationship (FLR), can be denoted by Ai→ Aj , where Ai is called the left-hand side (LHS) and Aj is the right hand side (RHS) of the FL. After determining the FLR, then the next stage is combine and group the FLR into Fuzzy Logic Relationship Group (FLRG). This FLRG is uses as the basis to determine the forecast value. Then in the finel stage, Defuzzyfication is carried out to obtain the forecast value. There are several ways to determine the forecast value in fuzzy time series, including the model proposed by Chen and Cheng. Calculating the forecast value from defuzzyfication using chen model can be obtained from calculating the mean of the median in each intervals. Let ( ) appropriate with 1, 2, … , and maximum membership value occurs at the interval 1 , 2 , . . , where the median is 1 , 2 , . . , . Then the equation to find the forecast value using Chen model is obtain as: On the other hand, Forecasting using cheng model is slightly different since the grouping of FLR because in the cheng model there is a weighting for each FLR. Then represent these weights into a weighted matrix that has been normalizes ] where is the median of each intervals the equation to find the forecast value using cheng model is obtain as

Mean Absolute Percentage Error
Mean Absolute Percentage Error (MAPE) is an error measurement that calculates the percentage deviation between the actual data and forecast value. MAPE calculated by using the average absolute error in each periode divided by the actual observed value in that period.
With ( ) is actual data and ( ) is forecast value. The forecasting method has good accuracy if the MAPE value less than 20% (Margi & Pandawa, 2015).

Population and Sample
The data used in this study is Indonesia rice productivity from 2001-2021 which wes obtained through Badan Pusat Statistik Indonesia.

Reseach Design
Forecasting method used in this research is fuzzy time series. Two fuzzy time series models will be compared, namely chen and cheng models. Basically the step of fuzzy time series is almost the same. The steps that will be carried out in this study that accordance with forecasting using fuzzy time series method are as follows: step 1 is determine the set of universes. At this stage, looking for the minimum and maximum values of actual data. Then, step 2 is specifies the number and range of classes, This research uses the Sturgess rule to determine the number of intervals divided.
Step 3 is determine the fuzzy set against the set universe This stage changes the set of universes that have been divided into a fuzzy sets. Fuzzy sets are formed with the size of the n*n matrix. After that, step 4 is fuzzyfication of historical data. This stage determines the membership value of each fuzzy set from historical data.
Step 5 is determine Fuzzy Logical Relationship (FLR). The relationship between two sequentially data, F(t) and F(t-1), becomes F(t-1) → F(t ).Next step is determine Fuzzy Logical Relationship Group (FLRG). The value of each relationship obtained will be combined and grouped known as FLRG (Fuzzy Logical Relationship Group). The way of grouping is from the same left side. The final steps of FTS method is defuzzyfication and calculating the forecast value. At this stage, the result of defuzzyfication data will be obtained the forecast value.In this study, two approach models will be used in determining forecast value of the fuzzy time series, that is the model proposed by chen and cheng. The next stage is determined the accuracy of the forecast result using MAPE value. The smaller MAPE value obtained, the more accurate forecast value is. At the end, compare the MAPE value from the forecasting result using chen and cheng models. Then determine which model is more appropriate to forecast Indonesia rice productivity.

RESULT AND DISCUSSION
The actual data used in this research is Indonesia rice productivity in 2001-2021 obtained from BPS Indonesia.

Data anlyzes
Indonesia rice productivity forecasting will be carried out using the chen and cheng fuzzy time series method as described above. Set of universes, the minimum and maximum values of the actual data to be studied are sought. Set U= [Umin, Umax] where Umin= Xmin-d1 and Umax=Xmax+d2, Xmin is minimum value of the actual data and Xmax is the maximum value of the actal data with d1, d2 is any numbers. Based on the Table 1  The next steps is fuzzification. This steps is changes the set of universes that have been divided and are still a set of crips numbers into a fuzzy set based on intervals. Fuzzy sets are formed with the size of the nxn matrix. The value of n is the value obtained from the results of the universe of discourse. we obtained five equal intervals so n=6. Therefore we get 6x6 matrix of fuzzy sets as follows: Table 2  Matrix of Fuzzy Sets  A1  A2  A3  A4  A5  A6  A1 1,0 0,5 0,0 0,0 0,0 0,0 A2 0,5 1,0 0,5 0,0 0,0 0,0 A3 0,0 0,5 1,0 0,5 0,0 0,0 A4 0,0 0,0 0,5 1,0 0,5 0,0 A5 0,0 0,0 0,0 0,5 1,0 0,5 A6 0,0 0,0 0,0 0,0 0,5 1,0 We can also represent the fuzzy set above in the following equention: After defining the fuzzy sets against the universes, then fuzzyfication the actual data. Determines the membership value of each fuzzy set from actual data. The results of fuzzyfication shown in the following Table 3. One of the important parts in the fuzzy time series method is determine the relation between sequential data. The relationship between sequential data that has been converted into fuzzy set is called a Fuzzy Logical Relationship (FLR). Determining the relationship is by rationalize the fuzzy logical relationship, for example, and there are F(i) = dan F(i + 1) = . The relationship between two consecutive observations, F(i) and F(i + 1), becomes F(i) → F(t + 1), is called the fuzzy logic relation, denoted by → , where is named with LHS (Left Hand Side) or current data and is named with RHS (Right Hand Side) or subsequent data. After getting the FLR, the next step is combine and group FLR. The value of each relationship obtained will be combined or commonly known as FLRG (Fuzzy Logical Relationship Group). The way of grouping is from the same left side. The next step is defuzzification to get the forecast value. At this stage there is a different ways to obtain the forecast value using chen or cheng model. Therefore we will find the forecast value using Chen model. As described above we need to find the middle value of each interval. Forecast value using chen model is obtained by calculate the mean of median based on the FLRG. Then the forecast value is obtained as in Table 7.  Table 7, we can get forcasting value of Indonesia rice productivity using chen model as shown Table 8.  The next step is to calculate forecast value using cheng model. To obtain forecasting value of cheng model we need to do the weighting. After the weighting is obtained, we can get the forecast value of Indonesia productivity rice using cheng model shown in Table 9   The last step in analyzing data is calculate the MAPE value to see the accuracy of forecasting method. A method that get smaller MAPE value means that method is better to forecast Indonesia prodictivity rice. The calculations results based on the forecast value, the MAPE value obtained using chen model is 18% and using cheng model is 12%. It means the MAPE value using cheng model is smaller than chen model.

Conclusion
Based on the result of data analysis, it was found that the MAPE value using cheng model is smaller than using chen model. It means that the cheng model is more appropriate used in forecasting data of Indonesia productivity rice. However, the chen and cheng models both give good forecasting result because the MAPE value is less than 20%.the chen and chengs models provides forecasting value that are quite accurate. So

C O M P A R I S O N C H A R T U S I N G C H E N G M O D E L
actual data forecasting value that this method can be used as a reference for government to forecast rice productivity in determining rice import policies

Implication
There are many fuzzy time series models to forecast data, so that for the further research more fuzzy time series models can be compared to give more accurate results in forecasting Indonesia rice productivity.