# A Kind of Weakening Buffer Operators with Time-Varying Parameters and Their Applications

Authors: Pumei Gao; Jiefang Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Buffer operators can effectively weaken the impact disturbance of the system and restore the original appearance of the system. However, the buffer strength of the existing buffer operator is fixed, and the buffer strength cannot be adjusted. So the buffer effect for the system impact disturbance is not flexible enough, and the accuracy of the prediction model established is not high. In order to solve this problem, this paper proposes weakened buffer operators with time-varying parameters, analyzes the properties of the newly constructed buffer operator, and proves the superiority of the new buffer operators theoretically. Finally, the new buffer operators proposed in this paper are applied to the prediction of the number of employees in China’s air transport industry and the unsafe events in China’s civil aviation. The results show that the proposed buffer operators are better than the existing buffer operators, which further verify the validity and practicability of the proposed buffer operators.


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## Body

## 1. Introduction

In the quantitative prediction of the system, the model often loses its effectiveness due to the disturbance of external shock. This leads to inconsistencies between qualitative analysis and quantitative calculations. Professor Liu et al. made an in-depth study of this kind of problem and believed that the problem was not the model itself, but the shock disturbance of the system [1]. At this time, the system behavior data cannot correctly reflect the real change law of the system. Then, professor Liu first proposed the axiom system of buffering operators and constructed a class of effective buffering operators based on arithmetic mean, which has been widely used in the prediction modeling of shock disturbance systems [2, 3].Dang et al. constructed several practical buffer operators from different angles [4]. Dang and colleagues constructed a series of weakening buffer operators and studied their properties [4]. For the system whose development trend is too flat due to the existence of system shock disturbance, Xie and Liu built several enhanced buffer operators [5]. Qian and Dang constructed a new weakening buffer operator by using the average growth rate of the sequence and conducted an in-depth study on the adjustment of buffer strength [6]. Hu et al. built a class of enhanced buffer operators by using the monotony of functions, which added a new research direction for the construction of enhanced buffer operators [7]. Liu et al. constructed a new class of weakening buffer operators based on the relationship between the existing data and the latest data [8]. Liu and Li proposed a combination method of the same kind of buffer operators [9]. Wu et al. proposed a multivariable weakening buffer operator, which can meet the requirements of using the latest data [10]. Wang and He constructed a kind of full-information variable weight weakening buffer operator and strengthening buffer operator [11]. Guo et al. proposed a kind of bidirectional weakening buffer operator, which also considered the influence of new information on old information [13].Liao et al. used classical average weakening buffer operators to solve practical problems [14–16]. Gao proposed a new weakening buffer operator based on the exponential type and applied the weakening buffer operator to the short-term prediction of China’s energy demand [17]. Wang and Wang used a variable weight buffer operator to realize the dynamic processing of original load data [18]. When the permanent magnet fault-tolerant motor system was affected by an external shock, Yang used the enhanced buffer operator to process the data sequence [19]. Wan and Li introduced an exponential weakening buffer operator to eliminate the random interference of the original data series of surrounding rock displacement [20]. Zeng used a grey weakening buffer operator to model and analyze shale gas production in China and achieved a good modeling effect [21].Dang constructed the weakening buffer operator considering time variation [4]. Because the time factor is taken into account, the buffering effect can be improved effectively. Since this method was proposed, it has been widely used. However, because the buffering effect at each time point is fixed, the model is not flexible enough. This will lead to the problem of poor buffering effect for different time series.In view of this, we constructed a class of weakening buffer operators with time-varying parameters and analyzed the properties of the new buffer operators. The remainder of the paper is organized as follows: in Section2, we constructed a class of weakening buffer operators with time-varying parameters. In Section 3, we constructed an extended model for a class of weakening buffering operators with time-varying parameters. In Section 4, the newly established models are applied to air transport forecasting. The cases include the number of people engaged in China’s air transport industry and China’s civil aviation insecurity events.


## 2. Construction of a Class of Time-Varying Parameter Weakening Buffer Operators

Theorem 1 (see [2]).
Suppose thatX=x1,x2,...,xn is the sequence of system behavior, XD=x1d,x2d,⋅⋅⋅,xnd is its buffer sequence. Then, the following conclusions can be drawn.(1)
X is a monotonic growth sequence, and D is the weakening buffer operator. ⇔xk≤xkd,k=1,2,...,n.(2)
X is a monotonic descending sequence, and D is the weakening buffer operator.(1)⇔xk≥xkd,k=1,2,…,n.(3)
X is an oscillating sequence, and D is the weakening buffer operator. Then, the following conclusion holds:(2)max1≤k≤nxk≥max1≤k≤nxkd,min1≤k≤nxk≤min1≤k≤nxkd.Theorem 2.
SupposeX=x1,x2,…,xnxk>0 is the systemic behavioral data sequence, and XD1=x1d1,x2d2,⋯xndn is the buffer sequence. Then,(3)xkd1=kαxk+k+1αxk+1+…+nαxn∑i=kniα,k=1,2,...,n.IfX is the monotone increasing sequence or the monotone decreasing sequence or the stochastic vibration sequence, D1 is the weakening buffer operator.Proof.
It can be seen from the construction process ofD1 that D1 satisfies the buffer operator’s three axioms, so D1 is the buffer operation. The next part is to prove that D1 is a weakening buffer operator.(1)
IfX is the monotone increasing sequence, then(4)xkd1−xk=xkd1=kαxk+k+1αxk+1+…+nαxn∑i=kniα−xk>kαxk+k+1αxk+…+nαxk∑i=kniα−xk=xk∑i=kniα∑i=kniα−xk=xk−xk=0.Therefore,xkd1>xk.According to Theorem1, D1 is the weakening buffer operator. Also,(5)xkd1xk+1d1=kαxk+k+1αxk+1+…+nαxn/∑i=kniαk+1αxk+1+k+2αxk+2+…+nαxn/∑i=k+1niα=kαxk+k+1αxk+1+…+nαxn/∑i=kniαk+1αxk+1+k+2αxk+2…+nαxn+kαxk−kαxk/∑i=k+1niα+kα−kα=kαxk+k+1αxk+1+...+nαxnk+1αxk+1+k+2αxk+2+...+nαxn+kαxk−kαxk∑i=k+1niα+kα−kα∑i=kniα=1−1/1+∑i=k+1niα/kα1−1/1+k+1/kαxk+1/xk+k+2/kαxk+2/xk+...+n/kαxn/xk=1−1/1+k+1/kα+k+2/kα+...+n/kα1−1/1+k+1/kαxk+1/xk+k+2/kαxk+2/xk+...+n/kαxn/xk.Supposeq1=1+k+1/kα+k+2/kα+...+n/kα(6)q2=1+k+1kαxk+1xk+k+2kαxk+2xk+...+nkαxnxk,xkd1xk+1d1=1−1/q11−1/q1.BecauseX is the monotone increasing sequence,(7)xk<xk+1⇔xk+1xk>1.Therefore,q1<q2⇔1/q1>1/q2⇔1−1/q1<1−1/q2⇔1−1/q1/1−1/q2<1(8)xkd1xk+1d1<1⇔xkd1<xk+1d1.The buffered sequenceXD1 is still a monotone increasing sequence, and the monotone is the same as the original sequence.(2)
IfX is a monotone decreasing sequence, then(9)xkd1−xk=xkd1=kαxk+k+1αxk+1+...+nαxn∑i=kniα−xk,<kαxk+k+1αxk+...+nαxk∑i=kniα−xk=xk∑i=kniα∑i=kniα−xk=xk−xk=0.Therefore,xkd1<xk.According to Theorem1, D1 is the weakening buffer operator. Also,(10)xkd1xk+1d1=kαxk+k+1αxk+1+...+nαxn/∑i=kniαk+1αxk+1+k+2αxk+2...+nαxn/∑i=k+1niα=kαxk+k+1αxk+1+...+nαxn/∑i=kniαk+1αxk+1+k+2αxk+2+...+nαxn+kαxk−kαxk/∑i=k+1niα+kα−kα=1−1/q11−1/q2,q1=1+k+1kα+k+2kα+...+nkα,q2=1+k+1kαxk+1xk+k+2kαxk+2xk+...+nkαxnxk.BecauseX is a monotone decreasing sequence, therefore,(11)xk>xk+1⇔xk+1xk<1.Then,q1>q2⇔1/q1<1/q2⇔1−1/q1>1−1/q2⇔1−1/q1/1−1/q2>1(12)xkd1xk+1d1>1⇔xkd1>xk+1d1.The buffered sequenceXD1 is still a monotone decreasing sequence, and the monotone is the same as the original sequence.(3)
IfX is a stochastic vibration sequence, suppose that(13)xM=maxxk|k=1,2,…,n,xm,=minxk|k=1,2,…,n,xMd1−xM=MαxM+M+1αxM+1+...+nαxn∑i=Mniα−xM<MαxM+M+1αxM+...+nαxM∑i=Mniα−xM=Mα+M+1α+...+nα∑i=MniαsxM−xM.=xM−xM=0.
Therefore,(14)xMd1−xM<0.
Also,(15)xmd1−xm=mαxm+m+1αxm+1+...+nαxn∑i=mniα−xm>mαxm+m+1αxm+...+nαxm∑i=mniα−xm=mα+m+1α+...+nα∑i=mniαxm−xm=xm−xm=0.
Therefore,(16)xmd1−xm>0,xMd1<xM,xmd1>xm.
According to Theorem1, D1 is the weakening buffer operator.Theorem 3.
IfX=x1,x2,...,xnxk>0 is the systemic behavioral data sequence, XD1=x1d1,x2d1,⋅⋅⋅,xnd1 is its buffer sequence. Then,(17)xkd1=kαxk+k+1αxk+1+…+nαxn∑i=kniα,k=1,2,…,n.,xkd1=kαxk+k+1αxk+1+…+nαxn∑i=kniα=kα∑i=knkα,k+1α∑i=knkα,…,nα∑i=knkαxk,xk+1,…,xnT.Then, the weight sequence can be denoted as(18)λ=kα∑i=knkα,k+1α∑i=knkα,…,nα∑i=knkα.Then, it is not hard to get the following results:(1)
Ifα=0,λ=1/n−k+1,1/n−k+1,…,1/n−k+1, the new operator degenerates into an average weakening buffer operator. It can be seen that the weight of each data is equal at this time.(2)
Ifα=1,λ=k/n−k+1k+n/2,k+1/n−k+1k+n/2,…,n/n−k+1k+n/2 the new operator degenerates into a weighted weakening buffer operator. It can be seen that, at this time, the weight of the sequence data increases linearly with the change of the weight, and the weight of the new information is greater than that of the old information, which is in accordance with the “new information first principle” of the grey system theory.(3)
Ifα⟶+∞, λ=kα/kα+k+1α+...+nα,k+1α/kα+k+1α+...+nα,...,nα/kα+k+1α+...+nα(19)=11+k+1/kα+…+n/kα,1k/k+1α+1+…+n/k+1α,…,1k/nα+k+1/nα+…+n/nα=0,0,…,0,1.This is an extreme case where the most recent information has a weight of 1 and all other information has a weight of 0.At this point, the sequence becomes a constant sequence, which means that the system is no longer changing and remains stable.As can be seen from the expression ofλ, with the increase of α， λ1=1/1+k+1/kα+...+n/kα is going down, and λn=1/k/nα+k+1/nα+…+n/nα is increasing.This means that, with the increase ofα, the weight of the old information decreases while the weight of the new information increases. Therefore, a value α can be appropriately reduced if old information is to be emphasized, and a value α can be appropriately increased if new information is to be emphasized. For different system sequences, the optimal value α can be solved by the intelligent optimization algorithm.


## 3. Construction of an Extended Model for a Time-Varying Parameter Weakening Buffer Operator

Theorem 4.
IfX=x1,x2,…,xnxk>0 is the systemic behavioral data sequence, XD2=x1d1,x2d1,⋅⋅⋅,xnd1 is the buffer operator.(20)xkd2=kα+βxk+k+1α+βxk+1+…+nα+βxn∑i=kniα+β,k=1,2,…,n.Then, whenX is a monotonic growth sequence, a monotonic attenuation sequence and an oscillation sequence, D2, are all weakening buffer operators.Proof.
It can be seen from the construction process ofD2, D2 satisfies the three axioms of buffer operator, then D2 is a buffer operator, and D2 is proved to be a weakening buffer operator as follows.(1)
IfX is the monotone increasing sequence, then(21)xkd2−xk=kα+βxk+k+1α+βxk+1+...+nα+βxn∑i=kniα+β−xk>kα+βxk+k+1α+βxk+...+nα+βxk∑i=kniα+β−xk,=xk∑i=kniα+β∑i=kniα+β−xk=xk−xk=0,xkd2>xk.According to Theorem1, D2 is the weakening buffer operator. Also,(22)xkd2xk+1d2=kα+βxk+k+1α+βxk+1+...+nα+βxn/∑i=kniα+βk+1α+βxk+1+...+nα+βxn/∑i=kniα+β=kα+βxk+k+1α+βxk+1+...+nα+βxn/∑i=kniα+βk+1α+βxk+1+...+nα+βxn+kα+βxk−kα+βxk/∑i=k+1niα+β+kα+β−kα+β=kα+βxk+k+1α+βxk+1+...+nα+βxnk+1α+βxk+1+...+nα+βxn+kα+βxk−kα+βxk,∑i=k+1niα+β+kα+β−kα+β∑i=kniα+β=11−kα+βxk/kα+βxk+k+1α+βxk+1+...+nα+βxn1−kα+β∑i=kniα+β=11−1/1+k+1α+β/kα+βxk+1/xk+...+nα+β/kα+βxn/xk1−11+k+1α+β/kα+β+...+nα+β/kα+β.Supposeq1=1+k+1α+β/kα+β+…+nα+β/kα+β(23)q2=1+k+1α+βkα+βxk+1xk+…+nα+βkα+βxnxk.BecauseX is the monotone increasing sequence, then(24)xk<xk+1⇔xk+1xk>1,q1<q2⇔1q1>1q2⇔1−1q1<1−1q2⇔1−1/q11−1/q2<1,xkd2xk+1d2<1⇔xkd2<xk+1d2.The buffered sequenceXD2 is still monotone decreasing sequence, and the monotone is the same as the original sequence.(2)
IfX monotone decreasing sequence(25)xkd2−xk=kα+βxk+k+1α+βxk+1+...+nα+βxn∑i=kniα+β−xk<kα+βxk+k+1α+βxk+...+nα+βxk∑i=kniα+β−xk=xk∑i=kniα+β∑i=kniα+β−xk=xk−xk=0,xkd2<xk.According to Theorem1, D2 is the weakening buffer operator. Also,(26)xkd2xk+1d2=kα+βxk+k+1α+βxk+1+...+nα+βxn/∑i=kniα+βk+1α+βxk+1+...+nα+βxn/∑i=kniα+β=11−1/1+k+1α+βxk+1/kα+βxk+...+nα+βxn/kα+βxk1−11+k+1α+β/kα+β+...+nα+β/kα+β,q1=1+k+1α+βkα+β+...+nα+βkα+β,q2=1+k+1α+βxk+1kα+βxk+...+nα+βxnkα+βxk.BecauseX is monotone decreasing sequence, then(27)xk>xk+1⇔xk+1xk<1,q1>q2⇔1q1<1q2⇔1−1q1>1−1q2⇔1−1/q11−1/q2>1,xkd2xk+1d2>1⇔xkd2>xk+1d2.The buffered sequenceXD1 is still a monotone decreasing sequence, and the monotone is the same as the original sequence.(3)
IfX is a stochastic vibration sequence, suppose that(28)xM=maxxk|k=1,2,...,nxm=minxk|k=1,2,...,n,xMd2−xM=Mα+βxM+M+1α+βxM+1+...+nα+βxn∑i=Mniα+β−xM<Mα+βxk+M+1α+βxM+...+nα+βxM∑i=Mniα+β−xM=xM∑i=Mniα+β∑i=Mniα+β−xM=xM−xM=0.
Therefore,(29)xMd1<xM.
And(30)xmd2−xm=mα+βxm+m+1α+βxm+1+...+nα+βxn∑i=mniα+β−xm<mα+βxk+m+1α+βxm+...+nα+βxm∑i=mniα+β−xm=xm∑i=mniα+β∑i=mniα+β−xm=xm−xm=0,xmd2<xm.
Therefore,xMd2<xM,xmd2>xm.
Then, it can be known from Theorem1 that D2 is the weakening buffer operator. For the convenience of writing, the improved buffer operator is abbreviated as WAWBO-V (weighted average weakening buffer operator with variable parameters). Through the above derivation and analysis, it can be seen that the improved time-varying weighted buffer operator (D2) has better adaptability, and D1 can be regarded as a special case of D2, the parameter β=0.


## 4. Numerical Examples

### 4.1. Forecast of the Number of People Engaged in China’s Air Transport Industry

With the rapid development of the air transportation industry, a large number of advanced technology and advanced experience are widely used in air transportation. This situation calls for professional talent to be put into the air transport industry. The number of employees in China’s air transport industry from 2010 to 2018 is shown in Table1 and Figure 1.Table 1
The number of employees in China’s air transport industry from 2010 to 2018 (persons).
Time201020112012201320142015201620172018People272023335260376100494397507789553358595301624318645957Source: China statistical yearbook.Figure 1
The number of employees in the air transport industry from 2010 to 2018.As can be seen from Table1 and Figure 1, from 2011 to 2018, the annual growth rate of the number of employees in the air transport industry of CAAC was 12.18%, 31.45%, 2.71%, 8.97%, 7.58%, 4.87%, 3.47%, respectively, and the maximum reaching 31.45%. With the rapid growth of the number of practitioners, the growth rate of the sequence decreases year by year, and the sequence shows a typical trend of fast at first and then slow, which meets the application conditions of the buffer operator.Therefore, this paper takes the data from 2011 to 2016 as the modeling data and the data from 2017 to 2018 as the prediction data to test the accuracy of different modeling methods. The calculation results of different methods are shown in Table2. WAWBO (weighted average weakening buffer operator) is the weighted average weakening buffer operator. In WAWBO-V, the optimal parameter is obtained by a genetic algorithm. The optimal parameter is α=1,β=0.4.Table 2
Prediction results of different grey prediction models.
Original dataDGM(1,1)APEMAPEWAWBOAPEMAPEWAW BO-VAPEMAPE6243186438608.1617.215730603.744.586071401.990.8764595772688016.435930505.016243700.01APE (absolute percentage error) =100%xk−x^k/xk, MAPE (mean absolute percentage error) = 100%1/2∑k=1nxk−x^k/xk.In general, the mean relative error and simulation error are required to be as small as possible. The accuracy test grade is shown in Table3. It can be seen from the calculation results in Table 2 that the original sequence is directly used to build the grey prediction model due to the disturbance of impact disturbance to the system, the average error of the three-step prediction reached 17.21%. It can be seen that the prediction accuracy is level 4, and the error is too large, so it is not suitable for prediction. After using the weighted average weakening operator, the error is reduced to a certain extent, still reaching 4.58%, and the prediction accuracy is level 2. The new weakening buffer operator with time-varying parameters reduces the prediction error to 0.87%, and the prediction accuracy reaches the first level. The accuracy has been greatly improved, which further enriches and improves the theoretical system of buffer operator.Table 3
Accuracy test grade.
Accuracy classIndicator thresholdRelative errorFirst level0.01Second level0.05Three level0.10Four level0.20


### 4.2. Prediction of China’s Civil Aviation Insecurity Events

Civil aviation safety is the top priority of civil aviation work. Civil aviation unsafe events refer to civil aircraft accidents, signs of civil aircraft accidents, and general events that affect the flight safety of civil aircraft [22]. Causes of civil aviation insecurity events include crew errors, total flight time, mechanical causes, and weather reasons [23], which contain both deterministic and uncertain information. Therefore, civil aviation insecurity events can be regarded as a grey system. The number of unsafe incidents in Chinese civil aviation is shown in Table 4 and Figure 2 [24].Table 4
Number of unsafe incidents in Chinese civil aviation from 2008 to 2015.
year20082009201020112012201320142015Number of unsafe incidents128025644126670310367116971246913207Figure 2
The trend of unsafe incidents in CAAC from 2008 to 2015.Since 2008, the annual growth rate of unsafe incidents in China’s civil aviation is 100.31%, 60.92%, 62.46%, 54.66%, 12.83%, 6.60%, 5.92%, and the maximum reaching 100.31%. With the attention paid by the Civil Aviation Administration of China and various airlines to safety, the growth rate of the number of unsafe incidents decreases year by year, and the sequence shows a typical trend of fast in the beginning and slow in the end, which meets the application conditions of the weakening buffer operator.Therefore, this paper takes the data from 2008 to 2013 as the modeling data and the data from 2014 to 2015 as the prediction data. The calculation results of different methods are shown in Table5.Table 5
Prediction results of different grey prediction models.
Original dataDGM (1, 1)APEMAPEAWOAPEMAPEWAWBO-VAPEMAPE1246918056.1844.8167.6013631.029.3213.0912659.581.532.021320725144.4290.3915434.1316.8613538.622.51It can be seen from the calculation results in Table5 that when the grey prediction model was directly established for the original sequence, the average relative error of the predicted value reached 67.60%, which could not be applied to the prediction. After the traditional average weakening buffer operator is weakened, the prediction ability of the model has been improved to a certain extent, and the relative error of prediction is reduced to 13.09%, and the prediction error is four levels, which can be basically applied to the prediction. After the new buffer operator with time-varying parameters proposed in this paper is weakened, the prediction accuracy of the model reaches 2.02%, and the prediction error is level 2, which indicates that the buffer operator proposed in this paper can effectively weaken the impact disturbance of the system and restore the real law of the system, so that the established grey prediction model has a high prediction accuracy.


## 4.1. Forecast of the Number of People Engaged in China’s Air Transport Industry

With the rapid development of the air transportation industry, a large number of advanced technology and advanced experience are widely used in air transportation. This situation calls for professional talent to be put into the air transport industry. The number of employees in China’s air transport industry from 2010 to 2018 is shown in Table1 and Figure 1.Table 1
The number of employees in China’s air transport industry from 2010 to 2018 (persons).
Time201020112012201320142015201620172018People272023335260376100494397507789553358595301624318645957Source: China statistical yearbook.Figure 1
The number of employees in the air transport industry from 2010 to 2018.As can be seen from Table1 and Figure 1, from 2011 to 2018, the annual growth rate of the number of employees in the air transport industry of CAAC was 12.18%, 31.45%, 2.71%, 8.97%, 7.58%, 4.87%, 3.47%, respectively, and the maximum reaching 31.45%. With the rapid growth of the number of practitioners, the growth rate of the sequence decreases year by year, and the sequence shows a typical trend of fast at first and then slow, which meets the application conditions of the buffer operator.Therefore, this paper takes the data from 2011 to 2016 as the modeling data and the data from 2017 to 2018 as the prediction data to test the accuracy of different modeling methods. The calculation results of different methods are shown in Table2. WAWBO (weighted average weakening buffer operator) is the weighted average weakening buffer operator. In WAWBO-V, the optimal parameter is obtained by a genetic algorithm. The optimal parameter is α=1,β=0.4.Table 2
Prediction results of different grey prediction models.
Original dataDGM(1,1)APEMAPEWAWBOAPEMAPEWAW BO-VAPEMAPE6243186438608.1617.215730603.744.586071401.990.8764595772688016.435930505.016243700.01APE (absolute percentage error) =100%xk−x^k/xk, MAPE (mean absolute percentage error) = 100%1/2∑k=1nxk−x^k/xk.In general, the mean relative error and simulation error are required to be as small as possible. The accuracy test grade is shown in Table3. It can be seen from the calculation results in Table 2 that the original sequence is directly used to build the grey prediction model due to the disturbance of impact disturbance to the system, the average error of the three-step prediction reached 17.21%. It can be seen that the prediction accuracy is level 4, and the error is too large, so it is not suitable for prediction. After using the weighted average weakening operator, the error is reduced to a certain extent, still reaching 4.58%, and the prediction accuracy is level 2. The new weakening buffer operator with time-varying parameters reduces the prediction error to 0.87%, and the prediction accuracy reaches the first level. The accuracy has been greatly improved, which further enriches and improves the theoretical system of buffer operator.Table 3
Accuracy test grade.
Accuracy classIndicator thresholdRelative errorFirst level0.01Second level0.05Three level0.10Four level0.20


## 4.2. Prediction of China’s Civil Aviation Insecurity Events

Civil aviation safety is the top priority of civil aviation work. Civil aviation unsafe events refer to civil aircraft accidents, signs of civil aircraft accidents, and general events that affect the flight safety of civil aircraft [22]. Causes of civil aviation insecurity events include crew errors, total flight time, mechanical causes, and weather reasons [23], which contain both deterministic and uncertain information. Therefore, civil aviation insecurity events can be regarded as a grey system. The number of unsafe incidents in Chinese civil aviation is shown in Table 4 and Figure 2 [24].Table 4
Number of unsafe incidents in Chinese civil aviation from 2008 to 2015.
year20082009201020112012201320142015Number of unsafe incidents128025644126670310367116971246913207Figure 2
The trend of unsafe incidents in CAAC from 2008 to 2015.Since 2008, the annual growth rate of unsafe incidents in China’s civil aviation is 100.31%, 60.92%, 62.46%, 54.66%, 12.83%, 6.60%, 5.92%, and the maximum reaching 100.31%. With the attention paid by the Civil Aviation Administration of China and various airlines to safety, the growth rate of the number of unsafe incidents decreases year by year, and the sequence shows a typical trend of fast in the beginning and slow in the end, which meets the application conditions of the weakening buffer operator.Therefore, this paper takes the data from 2008 to 2013 as the modeling data and the data from 2014 to 2015 as the prediction data. The calculation results of different methods are shown in Table5.Table 5
Prediction results of different grey prediction models.
Original dataDGM (1, 1)APEMAPEAWOAPEMAPEWAWBO-VAPEMAPE1246918056.1844.8167.6013631.029.3213.0912659.581.532.021320725144.4290.3915434.1316.8613538.622.51It can be seen from the calculation results in Table5 that when the grey prediction model was directly established for the original sequence, the average relative error of the predicted value reached 67.60%, which could not be applied to the prediction. After the traditional average weakening buffer operator is weakened, the prediction ability of the model has been improved to a certain extent, and the relative error of prediction is reduced to 13.09%, and the prediction error is four levels, which can be basically applied to the prediction. After the new buffer operator with time-varying parameters proposed in this paper is weakened, the prediction accuracy of the model reaches 2.02%, and the prediction error is level 2, which indicates that the buffer operator proposed in this paper can effectively weaken the impact disturbance of the system and restore the real law of the system, so that the established grey prediction model has a high prediction accuracy.


## 5. Conclusion

On the basis of the existing research, this paper presents a kind of weakening buffer operator with time-varying parameters and makes a deep analysis of their properties. Through mathematical derivation, the rationality of the buffer operator proposed in this paper is proved. Compared with the existing buffer operator, the superiority of the buffer operator proposed in this paper is proved in theory. By using the intelligent optimization algorithm to calculate the value of the parameters, the buffer operator can get the optimal calculation effect. The proposed buffer operators are applied to the industry and civil aviation workers’ unsafe events forecast. The prediction accuracy can be greatly improved compared with the traditional grey forecasting model by the new weakening buffer operators. Also, satisfactory results can be obtained. The study of this paper shows that the buffer operator can effectively reduce the impact disturbance of the system and restore the real appearance of the system. The construction of different buffer operators can effectively promote the application and popularization of buffer operators. At the same time, it is necessary to choose a suitable buffer operator for different impact disturbance systems, so as to achieve a better buffer effect. The research in this paper expands the application range of buffer operator and further enriches and improves the theoretical system of buffer operator.



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*Source: 1106349-2022-07-12.xml*