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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.

The supply chain forecasting can be made more accurate when human judgements are incorporated into the forecast system [

The rest of the paper proceeds as follows. Grey single variable forecasting models with fractional order accumulation are presented in Section

For the original data sequence

The solution of the whitenization equation

The

The procedures of

For the sequence

Substituting

The 0.7-order accumulated generating operation values

The predicted values

The predictive values of different models.

Time | resilience performance | GM(1,1) | |
---|---|---|---|

1 | 1763 | 1763 | 1763 |

2 | 2376 | 2441 | 2384 |

3 | 2763 | 2685 | 2740 |

4 | 2983 | 2954 | 3002 |

5 | 3216 | 3250 | 3211 |

MAPE | 2.0 | 0.4 |

The results of Table

The periodic resilience performance indicators of a firm [

Time | RPI 1 | RPI 2 | RPI 3 | RPI 4 | RPI 5 |
---|---|---|---|---|---|

1 | 2386 | 1345 | 674 | 6784 | 1763 |

2 | 2399 | 1876 | 567 | 5647 | 2376 |

3 | 3453 | 2345 | 453 | 4563 | 2763 |

4 | 3645 | 2784 | 363 | 3743 | 2983 |

5 | 4064 | 3764 | 278 | 3345 | 3216 |

The error analysis of different models.

RPI 1 | RPI 2 | RPI 3 | RPI 4 | RPI 5 | |
---|---|---|---|---|---|

MAPE of GM(1,1) | 6 | 3 | 1 | 3 | 2 |

MAPE of | 4.1 | 2.3 | 0.4 | 1.2 | 0.4 |

In Table

Actual data of the sales volume of registered printers in Taiwan by year.

Year | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 |
---|---|---|---|---|---|---|---|

Sales volume | 1047957 | 1138440 | 1091705 | 1083829 | 981984 | 838822 | 820161 |

The forecasting values and MAPE of different models.

Forecasting technique | Forecasting value for 2008 | Actual value | MAPE |
---|---|---|---|

Moving average | 910403 | 11.0 | |

Exponential smoothing | 854168 | 4.1 | |

Quadratic trend | 646699 | 21.2 | |

Cubic trend | 665629 | 820161 | 18.8 |

Non-linear trend | 876998 | 6.9 | |

ANP | 830797 | 1.3 | |

| 817142 | 0.4 | |

GM(1,1) | 836018 | 1.9 |

The predictive values of different models (units:ten thousand).

Year | LCD TV demand | single exponential model | AR | |
---|---|---|---|---|

2001 | 81 | 59 | 81 | |

2002 | 150 | 151 | 95 | 197 |

2003 | 393 | 149 | 183 | 423 |

2004 | 970 | 567 | 505 | 898 |

2005 | 2018 | 1086 | 1092 | 1903 |

2006 | 4300 | 2682 | 2551 | 4026 |

MAPE | 37.6 | 41.1 | 10.9 |

FAGO is widely used in grey models for its ability to smooth the randomness of original data [

The line of

The line of

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

For the original time series

If

Set the order number

Calculate the parameters (

Compute the predictive value using

Transform the prediction value back to the original sequence by means of IAGO [

In this paper, the predictive values are calculated by using different

In this case, the data from [

The predictive values of different models.

Year | Actual demand [ | GDES | traditional double exponential smoothing |
---|---|---|---|

2006 | 16 | 16 | 16 |

2007 | 3 | 3 | -2 |

2008 | 7 | 9 | 4 |

2009 | 2 | 2 | -1 |

2010 | 4 | 4 | 3 |

2011 | 5 | 6 | 5 |

MAPE | 8.5 | 62.4 |

As shown in Table

The existing multivariable grey models (GMC(1,N)) all used first-order accumulated generating operation sequence [

It is assumed that

Set

For example, the customer perception indicators of engine product are listed in Table

The customer perception indicators of engine product [

Year | sales volume | energy efficiency | product safety | cost performance | service promptness |
---|---|---|---|---|---|

2005 | 850.15 | 8.2112 | 1 | 1 | 1 |

2006 | 972.18 | 8.5027 | 1.0326 | 1.0786 | 1.0405 |

2007 | 1078.47 | 8.2711 | 1.0352 | 1.1029 | 1.007 |

2008 | 1069.28 | 8.0806 | 1.0521 | 1.0624 | 0.9818 |

2009 | 1259.26 | 8.1915 | 1.0846 | 1.0277 | 1.0028 |

2010 | 1454.51 | 9.0849 | 1.0911 | 1.0705 | 1.1229 |

2011 | 1413.92 | 8.7934 | 1.0404 | 0.9908 | 1.0796 |

2012 | 1672.11 | 8.9042 | 1.0612 | 0.9769 | 1.0992 |

2013 | 1948.23 | 9.3961 | 1.1289 | 1.0243 | 1.1704 |

2014 | 2063.35 | 9.938 | 1.2214 | 0.9711 | 1.3701 |

The predictive values of different models.

Year | sales volume | Quadratic polynomial model [ | |
---|---|---|---|

2005 | 850.15 | 850.15 | 850.15 |

2006 | 972.18 | 861.7971 | 965.79 |

2007 | 1078.47 | 929.5540 | 1055.96 |

2008 | 1069.28 | 1058.6918 | 1144.18 |

2009 | 1259.26 | 1103.4097 | 1264.41 |

2010 | 1454.51 | 1285.5118 | 1399.63 |

2011 | 1413.92 | 1525.6792 | 1555.71 |

2012 | 1672.11 | 1567.0815 | 1743.53 |

2013 | 1948.23 | 1786.3352 | 1934.37 |

2014 | 2063.35 | 2095.0247 | 2100.82 |

MAPE | 8.24 | 3.0 |

By comparing the MAPE in Table

Actually,

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,

Although the grey models (including

Actually, for many forecasting cases in the supply chain, the accumulated order number often satisfied

For the original data sequence

The data are from [

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

We can clearly see from the results given in Table

The fitting values of different models.

Month | sales volume | | |
---|---|---|---|

1 | 103148 | 103148 | 103148 |

2 | 119066 | 119269 | 124066 |

3 | 137468 | 139992 | 143842 |

4 | 155892 | 162736 | 164238 |

5 | 186370 | 187063 | 186099 |

6 | 216230 | 212887 | 209924 |

7 | 249776 | 240225 | 236108 |

8 | 266829 | 269135 | 265023 |

9 | 294475 | 299698 | 297046 |

10 | 334595 | 332004 | 332578 |

11 | 365988 | 366158 | 372051 |

MAPE | 1.4 | 2.4 |

The predictive results of different models.

model | sales volume | predictive values | interval value |
---|---|---|---|

simple linear regression (Xue 2015) | 414032 | 436402 | [ |

GIM | 414032 | [ |

The data come from [

The predictive results of number of end-of-life vehicles.

Year | number of end-of-life vehicles | interval value |
---|---|---|

2008 | 1977 | [ |

2009 | 2455 | [ |

2010 | 2818 | [ |

2011 | 2266 | [ |

2012 | 2241 | [ |

2013 | 2265 | [ |

The data come from [

The predictive results of demand of TFT-LCD panels.

Months | Actual value | | | interval value |
---|---|---|---|---|

4 | 1.718 | 1.718 | 1.718 | [ |

5 | 1.728 | 1.714 | 1.715 | [ |

6 | 1.714 | 1.735 | 1.740 | [ |

7 | 1.753 | 1.746 | 1.741 | [ |

MAPE | 0.6 | 0.7 | ||

8 | 1.742 | 1.748 | 1.729 | [ |

The data come from [

The predictive results in Table

The predictive results of return quantity for third party.

Time | Actual value | | | interval value |
---|---|---|---|---|

1 | 536.4 | 536.4 | 536.4 | [ |

2 | 538.61 | 538.2 | 535.8 | [ |

3 | 569.35 | 569.6 | 554 | [ |

4 | 613 | 612.8 | 565.3 | [ |

MAPE | 0.03 | 2.7 | ||

5 | 633.32 | 664.6 | 572.6 | [ |

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 [

The forecasting interval value of the proposed model.

Forecasting technique | Actual value | interval value |
---|---|---|

GIM | 820161 | [ |

Actual values are listed in Table

The forecasting interval value of the proposed model.

Year | Actual demand [ | | | interval value |
---|---|---|---|---|

2006 | 16 | 16 | 16 | |

2007 | 3 | -1 | 4 | [-1, |

2008 | 7 | 5 | 9 | [ |

2009 | 2 | 0 | 2 | [ |

2010 | 4 | 3 | 4 | [ |

2011 | 5 | 5 | 6 | [ |

Tsaur collected the LCD TV demand data from 2001 to 2006 as shown in Table

The predictive results of different models (units:ten thousand).

Year | LCD TV demand | the proposed model | fuzzy autoregressive model [ |
---|---|---|---|

2001 | 81 | ||

2002 | 150 | [ | [ |

2003 | 393 | [ | [ |

2004 | 970 | [ | [ |

2005 | 2018 | [ | [ |

2006 | 4300 | [ | [ |

In theory,

The application scopes summary.

Situation | Method |
---|---|

| exponential trend |

| volatile trend |

| multi-variable limited data |

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

The proposed model summary.

Situation Judgement | Method |
---|---|

the future trend is similar to the newer data | |

the future trend is similar to the order data | |

difficult to judge the future trend | interval forecasting |

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.

All the data are from the references in this paper.

The authors declare no conflicts of interest.

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).