#### Abstract

To deal with the forecasting with small samples in the supply chain, three grey models with fractional order accumulation are presented. Human judgment of future trends is incorporated into the order number of accumulation. The output of the proposed model will provide decision-makers in the supply chain with more forecasting information for short time periods. The results of practical real examples demonstrate that the model provides remarkable prediction performances compared with the traditional forecasting model.

#### 1. Introduction

The supply chain forecasting can be made more accurate when human judgements are incorporated into the forecast system [1]. The performance of purely quantitative forecasting method can be flawed when historical data is limited [2]. In certain cases, the number of available samples is so scarce that providing reliable estimates is a challenging problem [3]. In order to analyze and predict the small samples systems accurately, a large number of studies on supply chain forecasting using grey models and improved grey models have been reported [4–7]. The GM(1,1) (grey first-order differential equation model) forecasting model is formulated for solving limited time series data [8]. There is no strict hypothesis for the distribution of parent data. However, the reliability and validity of the GM(1,1) have never been discussed. First, without considering other causes when using limited time series data, the forecasting of the GM(1,1) is unreliable and provides insufficient information to a decision-maker. Second, human judgment cannot be incorporated into the forecasting systems. A grey prediction model with fractional-order accumulation is newly proposed and has better performance and more freedom compared with the traditional grey model [9]. Multivariable nonequidistance grey model with fractional order accumulation is discussed [10]. The grey model predictor design is modified by using optimal fractional-order accumulation calculus [11]. The air quality indicators in the Beijing-Tianjin-Hebei region are predicted by the grey prediction model with fractional-order accumulation [12]. Therefore, in This paper, three grey models with fractional-order accumulation are put forward. The output of the proposed model can obtain a general interval. This interval is not the grey number [13]. It will provide more forecasting information for a short time period. Most importantly, human judgment for future trend is incorporated into the order number of accumulation.

The rest of the paper proceeds as follows. Grey single variable forecasting models with fractional order accumulation are presented in Section 2. The multivariable grey model with fractional order accumulation is proposed in Section 3. Interval forecasting method of grey models for supply chain management is discussed in Section 4. Some conclusions are given in the last section.

#### 2. Grey Single Variable Forecasting Models with Fractional Order Accumulation

For the original data sequence , a new sequence can be generated by the fractional order accumulated generating operator (FAGO) as [14]. (1,1) model and grey exponential smoothing are given in the following way.

##### 2.1. Model

*Definition 1. *is referred to as the original form of the (1,1) model, where It is the traditional GM(1,1) model when . The ordinary least squares estimate sequence of the (1,1) model satisfieswhere

The solution of the whitenization equation for (1,1) is given by

The order inverse accumulated generating operator (IAGO) of is

The procedures of (1,1) can be summarized as follows.

*Step 1. *For the sequence from [4], 0.7-order accumulation sequence and in (2) are

*Step 2. *Substituting and into (2), we have the parameters . Then (1,1) can be represented as

*Step 3. *The 0.7-order accumulated generating operation values can be obtained by employing (4): The 2-order accumulated generating operation values is

The predicted values are , which are listed in Table 1. Mean absolute percentage error () compares the actual values with the forecasted values to evaluate the precision. The use of intelligent methods is a new trend in the grey models [15, 16]. In this paper, particle swarm optimization is adopted to find the optimal order which produces the minimum MAPE. The experiments are conducted in the MATLAB R2015b.

The results of Table 1 mean that (1,1) have better performance than the traditional GM(1,1). To investigate the feasibility of the (1,1) model in the supply chain, the following cases are given.

*The Supply Chain Performance Resilience Forecasting Example [4]*. The periodic resilience performance indicators of case firm are listed in Table 2. By (1,1), respectively, the forecasting values and MAPE of different models are given in Table 3.

In Table 3, the error analysis shows that (1,1) ensure the best fit of the data to achieve strong prediction capability. (1,1) can predict the supply chain resilience of an Indian electronics manufacturer.

*The Sales Volume of Printers Forecasting Example [20]*. Shih et al used the sales volume of printers from 2002 to 2007 as training data. The volume of 2008 is the testing data. Actual values are listed in Table 4 and the errors of eight models are shown in Table 5. The results show that the forecasting accuracy of the (1,1) is better than traditional GM(1,1).

*The LCD TV Demand Forecasting Example [19]*. Tsaur collected the LCD TV demand data from 2001 to 2006 as shown in Table 6. It was obvious that the collected data are limited time series data, and therefore the possible methods such as GM(1,1) and single exponential smoothing model were used for forecasting. The forecasting results are shown in Table 6. In Table 6, we find that the (1,1) has better forecasting performance of MAPE than the single exponential smoothing model.

##### 2.2. Grey Exponential Smoothing Model

FAGO is widely used in grey models for its ability to smooth the randomness of original data [9]. Through FAGO, the disorderly data may be converted into regular form. Actually, by using FAGO, the disorderly sequence may be converted into an approximately increased sequence. For example, , the -AGO sequence is The lines of these sequences are illustrated in Figures 1 and 2, respectively. Comparing the two lines, it is clear that the trend of sequence in Figure 2 is more obvious than the sequence in Figure 1.

An irregular and increased sequence can be predicted by using double exponential smoothing (GDES). Then we give the following definition.

*Definition 2 (see [21]). *For the original time series , -AGO is given in Definition 1. GDES follows the equations where and are the single and double exponential smoothing values for time , respectively. The forecasting form is .

If , GDES is the traditional double exponential smoothing. The process of calculating GDES can be summarized as follows.

*Step 1. *Set the order number and obtain the -FAGO sequence of according to Definition 1.

*Step 2. *Calculate the parameters ( and ) by using Definition 2.

*Step 3. *Compute the predictive value using , where is the out-of-sample size.

*Step 4. *Transform the prediction value back to the original sequence by means of IAGO [21].

In this paper, the predictive values are calculated by using different s. The that produces a small mean square error for the fitted values and shows an expected future growth is chosen. We assumed that and are equal to the initial historical values.

In this case, the data from [17] is the spare demand of a firm. The accuracy of demand forecast significantly affects the firm‘s sustainability and profitability. To compare the GDES with traditional double exponential smoothing, the smoothing constant of GDES and traditional double exponential smoothing is set to . The predictive values of two models are shown in Table 7.

As shown in Table 7, GDES can enhance the accuracy of small data forecasting.

#### 3. The Multivariable Grey Model with Fractional Order Accumulation

The existing multivariable grey models (GMC(1,N)) all used first-order accumulated generating operation sequence [22–26]. In this section, the fractional order accumulated generating operation sequence is introduced into the GMC(1,N) model.

*Definition 3. *It is assumed that is the sequence of system characteristic and ,N) are the sequences of relevant factors. Then is the (1,N) model, where is the order accumulation of , [14]. It is the traditional GMC(1,N) model when . Using ordinary least squares, the parameters of (1,N) model are estimated: where

Set ; when , the convolution integral of the right hand of (12) can also be discretised as where Then the order inverse accumulated generating operator of the predicted value is the same as (5).

For example, the customer perception indicators of engine product are listed in Table 8 from [18]. To obtain the sales volume forecasting value, the engine sales volume is . Energy efficiency, product safety, cost performance, and service promptness are , and , respectively. This case has limited data. Thus we can build (1,5). (1,5) has the smaller MAPE than the (1,5) with the other accumulated order numbers. Thus the results of (1,5) are given in Table 9.

By comparing the MAPE in Table 9, the example of demand forecasting for engine products demonstrated the applicability and validity of the (1,5).

Actually, is taken as the supply chain demand. ,N) are the sequences of relevant factors. (1,N) are suitable to predict the supply chain demand, because (1,N) can depict the model more precisely with new degrees of freedom and performances.

#### 4. Interval Forecasting of Grey Models for Supply Chain Management

Fractional derivatives accumulate the whole history of the system in weighted form and it is referred to as the memory effect. As we know, big samples forecasting models depend on statistical laws. In this section, the small samples forecasting models which depend on the memory effect are a new path. For example, in grey forecasting models denotes the weight of as 1. The larger of is, the larger the weight of old data is. The smaller of is, the smaller the weight of old data is. Reducing can reduce the weights of old data, which can put more emphasis on the newer data.

Although the grey models (including (1,1), , and (1,N)) have better forecasting performance, the input data is too limited and the forecasting values are point estimations which supply too limited information for a decision-maker. In this section, in order to obtain a better fitted model with smaller estimated errors and a smaller support forecasting interval, a novel fractional grey modelling mechanism is applied for solving limited time series data using the following steps.

Actually, for many forecasting cases in the supply chain, the accumulated order number often satisfied . If the future trend is similar to the newer data, the setting value of is smaller. If the future trend is similar to the order data, the setting value of is bigger. In the supply chain. for many forecasting cases with limited information, if it is difficult to judge the future trend, the forecasting value must be in a forecasting interval. The upper value and the lower value of forecasting interval can be obtained by different fractional grey models.

For the original data sequence , the forecasting values of (1,1) model (or and (1,N)) are denoted as and the forecasting values of (1,1) model (or and (1,N)) are denoted as For the same point , the smaller of and is set to the lower value of the forecasting interval. The bigger of and is set to the upper value of the forecasting interval. Thus we can obtain a grey forecasting interval. For example, the forecasting values of (1,1) model (or and (1,N)) are and the forecasting values of (1,1) model (or and (1,N)) are Therefore, the grey forecasting interval values are The above method is called the grey interval model (GIM). To investigate the feasibility of GIM in supply chain forecasting, seven cases are listed as follows.

*Case 1. *The data are from [27]. The data come from the sales volume of semiconductor components in a firm. To compare the proposed model with the other model, the data before November are the training set. The fitting values of different models are given in Table 10.

The predictive results (December) of different models are given in Table 11.

We can clearly see from the results given in Table 10 that (1,1) can obtain higher fitting accuracy. From the results given in Table 11, we can see that GIM can obtain shorter intervals including the actual value. But the simple linear regression derives a wide forecasting interval which provides too little information for the decision-maker.

*Case 2. *The data come from [28]. The data refer to the numbers of end-of-life vehicles in the West Anatolia. Forecasting the return flow of an end-of-life product is important for all decision levels of the reverse supply chain. In this paper, the predictive results of GIM are given in Table 12. The results of interval value can provide guidance to the managers and practitioners of recovery and recycling systems.

*Case 3. *The data come from [29]. In recent years, China’s high production of thin film transistor liquid crystal display (TFT-LCD) panels has led to intense competition in this industry, causing its supply to be greater than the actual demand. Under such circumstances, reducing inventory levels and inventory turnover is a critical issue faced by panel manufacturers. The production quantity is maintained at an appropriate balance point considering the total cost. To achieve this balance in production marketing coordination, an accurate short-term demand forecast is essential. Because the demand for TFT-LCD panels is affected tremendously by the global business cycle, whereas the business cycle has substantially changed in recent years, the use of numerous long-term historical observations does not satisfy the needs of the short-term forecasts. Therefore, this study applied data produced by a leading Taiwanese TFT-LCD panel manufacturer to verify the forecasting performance of GIM [29]. In this paper, the predictive results of GIM are given in Table 13. The actual value is in the forecasting interval. It demonstrates that GIM can provide more information with predictive values.

*Case 4. *The data come from [30]. They are the return quantity for a third party e-waste firm in Turkey. The predictive results are listed in Table 14.

The predictive results in Table 14 indicate that (1,1) and (1,1) can obtain small fitting errors. By GIM, the obtained forecasting interval indeed includes the actual value. Thus, we can conclude that GIM is effective for limited sample forecasting problems.

*Case 5. *Shih et al. used the sales volume of printers from 2002 to 2007 as the training data. The volume of 2008 is the testing data [20]. Actual values are listed in Table 4 and the results of GIM are shown in Table 15. The results show that the actual value of 2008 is in the forecasting interval. Thus, we can conclude that GIM is effective for the limited sample forecasting problem.

*Case 6. *Actual values are listed in Table 7 and the results of GIM are shown in Table 16. As shown in Table 16, the results show that the actual demand values are all in the forecasting interval. It also indicates that highly volatile demand can obtain volatile intervals. These results can provide more information for the supply chain manager. Thus, we can conclude that GIM is effective for limited sample forecasting problems.

*Case 7 (see [19]). *Tsaur collected the LCD TV demand data from 2001 to 2006 as shown in Table 6. The forecasting results are shown in Table 17. In Table 17, we find that the interval of GIM is shorter than that of the fuzzy autoregressive model.

#### 5. Conclusion

In theory, (1,1), , and (1,N) models are discussed in this paper, respectively. Their application scopes are listed in Table 18. The results show that these models outperform the traditional grey models in the prediction precision.

Using traditional grey models, the forecasting values of limited data are point estimations which supply too limited information for a decision-maker. It is easy to arouse suspicion for this kind of point estimations. Thus, a novel interval forecasting method is put forward. From the empirical results, we found that the forecasting capability of the GIM is quite encouraging, but those of the time series models are not. In addition, the interval of GIM is smaller than that of the simple linear regression and fuzzy autoregressive model. In the real world, the environment is uncertain and we can only use a limited amount of data to provide future forecasts for a short time period. In this situation, GIM is more satisfactory than the time series model.

In practice, GIM is summarized in Table 19. This paper contributes to the literature with an interval model for small samples forecasting in supply chain. It significantly improves small sample forecasting due to the interval result carrying more information. This paper demonstrates that grey modelling can be successfully applied to the forecasting problem in supply chain. Moreover, the proposed forecasting system can be used as a strategic tool for forecasting under uncertain conditions with a small amount of recent data.

For future research, in order to investigate the feasibility of the novel model in supply chain, it may be used for other real world cases for forecasting and the performances of the methods can be compared.

#### Data Availability

All the data are from the references in this paper.

#### Conflicts of Interest

The authors declare no conflicts of interest.

#### Acknowledgments

The relevant researches carried out in this paper are supported by the National Natural Science Foundation of China (No. 71871084, 71401051, and 71801085). We also acknowledge the Project funded by China Postdoctoral Science Foundation (2018M630562).