Optimization of GMr (1,1) Model and Its Application in Forecast the

Number of Tourist Visits to Quang Ninh Province

VAN VIEN VU1, VAN THANH PHAN2,3*

1Faculty of Tourism, Ha Long University,

258 Bach Dang street, Nam Khe ward, Uong Bi city, Quang Ninh,

VIETNAM

2Faculty of Digital Economics and E-Commerce,

Vietnam-Korea University of Information and Communication Technology,

470 Tran Dai Nghia Street, Ngu Hanh Son ward, Da Nang city,

VIETNAM

3Department of Science Technology and International Affairs,

Quang Binh University,

312 Ly Thuong Kiet Street, Bac Ly ward, Dong Hoi city, Quang Binh,

VIETNAM

*Corresponding Author

Abstract: - Currently, many researchers pay more attention to improving the accuracy of the Grey forecasting

model. One of tendency is focused on the modification of the accumulated generating operation. In 2015, some

scholars used the r-fractional order accumulation to improve the accuracy. However, With the desire of users to

have a set of forecasting tools as accurate as possible. This paper based on the flexibility parameter of r-

accumulated generation operation proposed the systematic approach by optimizing the number of r for

improving the precision. To verify the performance in advance of the proposed approach, three case examples

were used, the simulation results demonstrated that the proposed systematic approach provides very remarkable

predictive performance with the accuracy performance of the proposed approach being higher than other

models in comparison. Furthermore, the real case in forecasting the number of tourism visits to Quang Ninh

was also conducted to compare the performance of models. The empirical results show that the proposed model

will get a higher accuracy performance with the lowest MAPE =19.722%. This result offers valuable insights

for Quang Ninh policymakers in building and developing policies regarding tourism industry management in

the future.

Key-Words: - GMr (1,1), fractional order accumulation, optimization, systematic approach, accuracy, number

of tourists, Quang Ninh province.

5HFHLYHG0D\5HYLVHG1RYHPEHU$FFHSWHG'HFHPEHU3XEOLVKHG'HFHPEHU

1 Introduction

The grey forecasting model is an important part of

the grey system theory. It is a time series forecasting

model. It was proposed by Prof. Deng in the early

1980s, [1], [2]. Among the grey models’ series, grey

model GM (1,1) is the basic model. It is constructed

by the first-order differential equation and one

variable with the non-negative original sequence.

Because of easy simulation and higher accuracy

compared with other time series forecasting models

in the uncertain information and the limited data, [3],

[4], [5], So, the GM (1,1) has been widely and

successfully applied to various fields such as

tourism, [6], [7], transportation, [8], [9], and, [10],

financial and economic, [11], [12], [13], integrated

circuit industry, [14], [15], [16], [17], energy

industry, [18], [19], [20], and so forth.

Some scholars concentrate on the improvement

of the GM (1,1) model with different perspectives.

For an instant, Wang and Lin used optimal

methodology to improve the background values,

[20], [21]. Hsu and Wang used different methods to

modify the development coefficient and the grey

input coefficient, [17], [22]. Some scholars focused

on modifying the residual error to improve the

performance of the GM (1,1) model, [7], and, [13].

In addition, some research proposed hybrid models

based on the combination of the GM (1,1) and other

methods like the grey econometric model, [5], the

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Van Vien Vu, Van Thanh Phan

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grey Markov model, [23], [24], the grey fuzzy

model, [20], and so on. Despite its improvement in

prediction accuracy, the prediction accuracy of

these models is not always satisfactory. Especially,

in an environment with the data are highly

fluctuating, or with lots of noise.

In recent years, Wu developed the grey model

with fractional order accumulation (abbreviated as

GMr (1,1)) to deal with fluctuating data, [25], [26].

This model is based on two fundamental principles

which are the principle of information differences

and the principle of new information priority.

Therefore, this model can solve the shortcomings of

previous models. For the GMr (1,1), the changing

initial condition affects the simulative value of GMr

(1,1) and the monotonicity and convexity of the

simulative value are uncertain when the actual value

is nonnegative increased. With the advantages of

this model, the GMr (1,1) has been widely applied in

the community, [27], [28]. In addition, with the

desire to improve the predictive accuracy, this study

proposed a new effective systematic approach based

on the mathematical algorithm of GMr (1,1) to

enhance the predictive performance. Based on three

case studies in research paper, [25], and the real

case in forecasting the number of tourists visit to

Quang Ninh province, the stimulation results

indicated that the proposed systematic approach can

significantly enhance the precision of the GMr (1,1)

model.

This paper is organized as follows. In section 2,

the main concept of the GM (1,1) with fractional

order accumulation is briefly introduced and

proposed a new systematic approach aims to

improve the accuracy precision of GMr (1,1).

Section 3 demonstrates the proposed approach has

better performances in several numerical examples

by compared with GM (1,1), and GMr (1,1). Section

4 illustrates the application of the proposed

approach in forecasting the number of tourism visits

to Quang Ninh province, Finally, the conclusions

are made in Section 5.

2Methodology and Proposed

Approach

Following the perspective of improving the GM (1,1)

model by modifying the background value, many

scholars indicated that the growth trend of the

original data sequence has a great influence on the

accuracy of prediction because the GM (1,1) model

is a kind of homogeneous exponential growth model.

If the original data sequence is smooth, the closer to

the exponential growth it is, the higher the

prediction precision the model can produce.

Therefore, Wu and partners proposed the r-order

accumulated generating operation to reduce the

fluctuation of the original data sequence, [25]. All

procedures for modifying the r-order accumulated

generating operation of GM (1,1) are given in

section 2.1

2.1 The GMr (1,1) Model

According to the research paper of Wu and partners,

[25], the overall process of the GMr (1,1) is as

follows:

Step1: Preparing the non-negative original data

sequence X0

 

)(),...,(),...,2(),1( )0()0()0()0()0( nxkxxxX 

4n

(1)

Where n is the sample size of the data

Step 2: Construct a new series data Xr by using

r-order accumulated generating operation (r-AGO)

( ) ( ) ( ) ( )

(0) (0) (0) (0)

1 2 1

( (1), (2),... ( ))

(1), (2),..., ( ),..., ( )

1 ....

0 1 ...

... ... ... ... ...

0 0 ... 1

0 0 ... 0 1

r r r r

r r n r n

r n r n

X x x x n

x x x k x n

C C C



   



   













(2)

Step 3: Building the GM (1,1) model by

establishing the r-order differential equation with

one variable is expressed as:

bkazkx rr  )()( )(

(

.,...,3,2)),1()((5.0)( )()( nkkkxkz rr 

) (3)

Where

)(

)1( kz

is the average value of consecutive

data.

So, whiteness equation is the following:

bkaz

kdx r

r )(

)(

)( )(

)(

(4)

The ordinary least square estimate of

and

can be obtained by:

YAAA

aTT 1

)( 















(5)

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Where



















)1()(

.............

)2()3(

)1()2(

)()(

nxnx

(6)

and

















1)(

..........

.........

1)3(

1)2(

)(

(7)

Step 4: The solution of the whitenization

equation (4) can be expressed as follows

)1()1()(

ˆ)1()()( akarr ee

xkx 











 

(8)

Step 5: The simulations and forecasting value

can be obtained by applying the r-order inverse

accumulated generating operator (r-IAGO).

Therefore, the fitted and predicted sequence is given

)0(

as:

(0) (0) (0) (0)

( ) ( ) ( ) ( )

1 1 1 2

ˆˆ ˆ ˆ

( (1), (2),... ( ))

ˆ ˆ ˆ ˆ

(1), (2),..., ( ),..., ( )

1 .... ( 1)

0 1 ... ( 1)

... ... ... ...

0 0 ...

0 0 ... 1

r r r r

r r n

X x x x n

x x x k x n































(9)

2.2 Proposed a Systematic Approach to

Improve the Predictive Accuracy of

GMr (1,1) by the r-order Accumulation

Optimization

Based on the mathematical algorithm of the GMr

(1,1), this study proposed a systematic approach to

improve the predictive accuracy of GMr (1,1) by r-

order accumulation optimization. The overall

procedure of the proposed approach is illustrated in

Figure 1:

Fig. 1: The procedure of the proposed approach

Step 4: Estimate the forecasted value of

)(

ˆr

(Eq. 8)

Step 5: Get the predicted value

)0(

from

)(

ˆr

by r-IAGO (Eq. 9)

Step 3: Construct GMr (1,1) model by the first-order different equation (Eq. 3, 4, 5, 6, 7)

Step 2: Construct time series data Xr(k) from X0by using r-AGO (Eq. 2)

Step 1: Input original time series data X0 (Eq. 1)

Step 6: Optimal the parameter r following equation

%100

)(

)(Min

1)0(

)0()0( 

















kkx

kxkx

(10)

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2.3 Evaluation Indicators Indexes

In order to evaluate forecast results, the present

study applied the MAPE (Mean Absolute

Percentages Error) performance indicator which is

expressed as follows:

%100

)(

)(1

2)0(

)0()0( 







kkx

kxkx

MAPE

(11)

Where

)(

)0( kx

and

)(

)0(

ˆkx

are actual and

forecasting values at time k, respectively, and n is

the total number of predictions.

3 Verification of the Proposed

Approach

In order to illustrate the effectiveness of the

proposed model, three cases in the research paper,

[25], are given below:

3.1 Electricity Consumption Forecasting

The examples from papers, [18], and, [25], provide

the sample data. As the same setting situation in

Wu’s paper, [25], the data from 2000 to 2003 (in-

sample data) are utilized to forecast the value of

2004 to 2007, the results are listed in Table 1. For

the electricity consumption in Russia, the three

compared models yielded lower MAPE in out-of-

sample (2004 to 2007). But the proposed approach

provides the best among them with a MAPE is 1.42.

This implies that the proposed approach has a strong

forecasting performance. Furthermore, Figure 2

illustrates the relative between MAPE and the r-

order accumulated generating operators.

Table 1. The actual, predicted values and MAPE

index of different grey models

Year

Actual

value

GM (1,1)

GM0.01(1,1)

Proposed

approach

with r=0.009

2004

55,516

54,800

55,294

55,329

2005

55,898

55,431

56,278

56,379

2006

58,600

56,069

57,308

57,510

2007

60,281

56,714

58,386

58,726

MAPE

(%)

3.10

1.61

1.42

For the case of forecasting the electricity

consumption in Vietnam, as can be seen from Table

2, from a forecasting viewpoint, the proposed

approach model is the best forecasting model

compared the GM (1,1) and GM0.05 (1,1). The

average MAPE from the years 2004 to 2007 of the

proposed approach is 0.6. This means that the

proposed approach can significantly enhance the

precision of the grey forecasting model. More

specific are shown in Figure 3.

Fig. 2: The effectiveness between MAPE and r-

order

Table 2. The actual, predicted values and three error

indicators of different grey models

Year

Actual

value

GM (1,1)

GM0.05(1,1)

Proposed

approach

with r=0.037

2004

3,437

3,477

3,457

3,464

2005

3,967

4,042

3,971

3,994

2006

4,630

4,699

4,537

4,587

2007

5,256

5,462

5,163

5,255

MAPE

(%)

2.16

1.13

0.6

Fig. 3: The impact of r-order to MAPE

3.2 The Power Load of Hubei Province in

China Forecasting

The sample from the paper, [25], is applied here to

compare the precision, the history load data of the

Hubei electric power network in China from the

years 1996 to 2007 are used to verify the forecasting

results. The power load for the year 2006 and 2007

are predicted based on the data before the year

2006. The performance of three compared models is

shown in Table 3. As can be seen from Table 3, the

MAPE of the proposed approach and the GM0.01

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(1,1) are the same. It means that the r = 0.01 in the

GMr (1,1) model is the optimal solution of the

proposed approach (Figure 4).

Table 3. The fitted values and three error indicators

of different models

Year

Actual

value

(104 kW)

(1,1)

GM0.01(1,1)

Proposed

approach

with r=0.01

1996

425.38

1997

440.26

402.20

433.94

1998

457.24

434.89

448.05

1999

457.38

470.23

467.27

2000

503.02

508.44

492.49

2001

526.02

549.77

525.11

2002

561.96

594.44

567.06

2003

629.20

642.75

620.83

2004

700.21

694.99

689.63

2005

788.15

751.47

777.57

MAPE

(%)

3.53

1.29

2006

876.76

812.54

889.88

2007

989.23

878.58

1033.26

MAPE

(%)

9.26

2.97

Fig. 4: The effectiveness between MAPE and r-

order

3.3 Computer Industry Output Value

Forecasting in Hsinchu Science Park

The example from a reference in, [25], provides the

sample data, the data from the years 2001 to 2007

are used to construct three forecasting models.

Actual values and simulative values of these

compared models are presented in Table 4. As can

be seen from Table 4, the proposed approach with

r=0.75 yielded the lowest MAPE in sample data.

This indicates that the proposed approach can

improve the fitted error of the GM (1,1) with

fractional order accumulation. More visualization of

r- order accumulation optimization was shown in

Figure 5.

Table 4. The fitted values and three error indicators

of different models.

Year

Actual

value

GM (1,1)

GM0.8(1,1)

Proposed

approach

with r=0.75

2001

1,610.71

2002

1,245.28

1,363.88

1,342.76

1,340.37

2003

1,347.71

1,274.91

1,280.32

1,279.60

2004

1,382.45

1,191.73

1,204.58

1,205.14

2005

1,018.45

1,113.99

1,122.64

1,123.34

2006

1,014.96

1,041.31

1,040.18

1,040.47

2007

949.46

973.38

960.34

960.12

MAPE

(%)

6.17

5.65

5.64

Fig. 5: The relative between MAPE and r-order

4 Application of Improved GMr (1,1)

in Forecast the Number of Tourist

Visits to Quang Ninh Province

In order to help the policymaker have more efficient

tools in formulating sustainable tourism industry

development, this paper uses three models which

are GM (1,1), GMr (1,1) and optimization of GMr

(1,1) model to forecast the number of tourism visits

to Quang Ninh province. Then we compare the

accuracy performance of these models to find out

the best model to suggest for forecasting in this

case.

4.1 Data Collection

The number of tourism visited to Quang Ninh

province from 2011- 2021 were obtained from the

report of Department of Tourism, Quang Ninh

province, [29], and are shown in Table 5.

Table 5 shows that the number of tourist visits

to Quang Ninh province significantly dropped from

14,005,090 people in 2019 to 8,843,000 people in

2020 and continues to fall to 4,830,000 people in

2021. The main cause of this decrease is due to the

COVID-19 pandemic. Therefore, the number of

tourists coming to Quang Ninh has decreased.

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Table 5. The total number of tourists visit to Quang

Ninh province during period time 2011 to 2022.

Year

Total number of tourists

2011

6,459,000

2012

7,000,000

2013

7,500,000

2014

7,500,000

2015

8,600,000

2016

8,350,000

2017

9,800,000

2018

12,246,000

2019

14,005,090

2020

8,843,000

2021

4,830,000

2022

11,589,000

4.2 Forecasting Models: Testing and Results

In this case, three forecasting models which are

traditional GM (1,1), GMr (1,1) and the proposed

approach were used to forecast the number of

tourism visits to Quang Ninh province. Then we

compare these models to find the best model based

on the MAPE index. For the best model strongly

suggest applying in this case.

4.2.1 The GM (1,1) Model

The GM (1,1) model is the basic grey forecasting

model. It is constructed by the first-order

differential model with one input variable. The

overall modeling algorithm of the GM (1,1)

forecasting model was shown in [21]. Based on the

mathematical algorithm of GM (1,1) and the

historical data in this case, the parameters were

identified as a = -0.02834 and b = 7,590,603.794.

The function of GM (1,1) for the number of tourists

forecasting is as follows:























 

02834.0

794.7590603

02834.0

794.7590603

6459000)(

ˆ02834.0)1( k

ekx

with k=1,2,.....n

All forecasted values and the average MAPE

index of the GM (1,1) model are recorded in Table

6. Table 6 shows the performance accuracy of the

forecasted value of the number of tourists coming to

Quang Ninh by using GM (1,1) with an average of

MAPE = 20.427 %.

Table 6. Actual and forecasted value of

forecasting models

Actual value

Forecasted value

GM (1,1)

GM1.01 (1,1)

GM1.029 (1,1)

6,459,000

7,000,000

7,699,208.566

7,646,065.689

7,542,921.217

7,500,000

7,920,571.674

7,817,696.142

7,620,177.084

7,500,000

8,148,299.283

8,010,307.708

7,747,608.097

8,600,000

8,382,574.382

8,215,457.149

7,899,337.94

8,350,000

8,623,585.221

8,430,446.6

8,066,989.065

9,800,000

8,871,525.462

8,654,122.258

8,246,792.598

12,246,000

9,126,594.334

8,885,938.281

8,436,771.447

14,005,090

9,388,996.796

9,125,640.063

8,635,795.999

8,843,000

9,658,943.699

9,373,131.559

8,843,190.817

4,830,000

9,936,651.956

9,628,411.584

9,058,546.502

11,589,000

10,222,344.72

9,891,540.14

9,281,620.43

MAPE (%)

20.427

19.969

19.722

4.2.2 The GMr (1,1) Model

Follow by the mathematical algorithm of GMr (1,1)

was detailed in section 2 and the real data of the

tourist number visits to Quang Ninh province. The

r-order accumulated generating operation,

coefficient parameters a and b are defined as: r=

1.01, a= -0.0276, b= 7,604,573.817. The function of

GMr (1,1) for the number of tourists forecasting is

as follows:























 

0276.0

871.7604573

0276.0

871.7604573

6459000)(

ˆ0276.0)01.1()01.1( k

ekx

with k=1,2,.....n

All forecasted values and the average MAPE

index of the GM1.01 (1,1) model are recorded in

Table 6. Table 6 showed the accuracy performance

of GM1.01 (1,1) is higher than GM (1,1) with the

MAPE index decreasing from 20.437 % to 19.969

4.2.3 Optimization of GMr (1,1) Model

As the same calculation of parameter of GMr (1,1)

model and based on the Eq.(10) we can get the

value of parameters are r= 1.029, a= -0.0263, b=

7,630,772.821.

The function of optimization of GMr (1,1) for the

number of tourist forecasting is as follows:























 

0263.0

821.7630772

0263.0

821.7630772

6459000)(

ˆ0263.0)029.1()029.1( k

ekx

with k=1,2,.....n

All forecasted values and the average MAPE

index of the optimization of GMr (1,1) model with

r=0.1029 are recorded in Table 6. Table 6 shows

the performance accuracy of the forecasted value of

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the number of tourists coming to Quang Ninh using

GM1.029 (1,1) with the MAPE =19.722 %.

4.3 Implication

Based on the empirical analysis, Table 6 represents

that the optimization of GM1.029 (1,1) is better than

the other compared models with the lowest MAPE

index. Therefore, the optimization of GM1.029 (1,1)

was strongly suggested for estimation in this case.

The forecasted value of tourists coming to Quang

Ninh province updated to the years 2023 and 2024

is shown in Table 7.

Table 7. Forecasted value of the number of tourists

coming to Quang Ninh by optimization of GM1.029

(1,1)

Year

Forecasted

2023

9,512,280.386

2024

9,750,470.664

Table 7 shows that the forecasted values in

2023 and 2024 will be over 9,512,000 and

9,750,000 people, respectively. This figure indicates

that the number of tourist visits to Quang Ninh will

slightly decrease compared to the year 2022, but

will significantly increase in the year 2024. This

empirical result will provide a basic scenario for the

policymaker to make a good decision in the future

regarding tourism industry management and

building the developing strategy.

5 Conclusions

Nowadays, the accuracy of forecasting toolkits is

becoming more and more important in supporting

policymakers in making the right decisions in the

future. To fill in this gap, this paper provides a new

systematic approach based on the optimization of r-

accumulated generation operation in (Eq. 10)

aiming to improve the predictive accuracy of GMr

(1,1). Via simulation in three cases and the real case

in forecasting the number of tourists visiting Quang

Ninh, the empirical results demonstrate that the

proposed systematic approach can significantly

improve the precision of the GMr (1,1) in all cases.

In the future direction, the proposed model can be

applied to forecast the performance of other

industries.

Acknowledgments:

The research paper is financial supported by the

Quang Binh University (CS:20.2023).

References:

[1] Deng J. L. (1986). Grey prediction and

decision. Huazhong University of Science and

Technology, Wuhan, China.

[2] Deng J. L. (2002). Solution of grey

differential equation for GM (1, 1| τ, r) in

matrix train. Journal of Grey System, vol. 13,

pp. 105-110.

[3] Deng J. L. (1982). Control problems of grey

systems. Systems & Control Letters, vol. 1,

no. 5, pp. 288-294.

[4] Lin Y and Liu S. (2004). A historical

introduction to grey systems theory. 2004

IEEE International Conference on Systems,

Man and Cybernetics, vol. 3, pp. 2403-2408.

[5] Liu S., Jeffrey F and Yang Y. (2012). A brief

introduction to grey systems theory. Grey

Systems: Theory and Application, no. 2, pp.

89-104.

[6] Huang Y. L and Lee Y. H. (2011). Accurately

forecasting model for the stochastic volatility

data in tourism demand. Modern Economy,

vol. 2, no. 5, pp. 823-829.

[7] Phan P. V. (2015). Enhanced Accuracy of

Grey Forecasting Model Case by tourism

industry in Vietnam. Hue University Journal

of Science: Economics and Development, vol.

113, no. 14, pp.157-167.

[8] Jiang F and Lei K. (2009). Grey prediction of

port cargo throughput based on GM (1,1, a)

model. Logistics Technology, vol. 9, pp. 68-

70.

[9] Guo, Z. J., Song X. Q and Ye J. (2005). A

Verhulst model on time series error corrected

for port throughput forecasting. Journal of the

Eastern Asia Society for Transportation

Studies, vol. 6, pp. 881-891.

[10] Lu I. J., Lewis C and Lin S. J. (2009). The

forecast of motor vehicle, energy demand and

CO2 emission from Taiwan's road transportation

sector. Energy Policy, vol. 37, no. 8, pp. 2952-

2961.

[11] Kayacan E., Ulutas B and Kaynak O.(2010).

Grey system theory-based models in time

series prediction. Expert Systems with

Applications, vol. 37, pp. 1784-1789.

[12] Askari M and Askari H. (2011). Time series

grey system prediction-based models: gold

price forecasting. Trends in Applied Sciences

Research, vol. 6, pp. 1287-1292.

[13] Wang C. N and Phan V. T (2014). Enhancing

the accurate of grey prediction for GDP

growth rate in Vietnam. 2014 IEEE

International Symposium on Computer,

Consumer and Control, pp. 1137-1139.

WSEAS TRANSACTIONS on BUSINESS and ECONOMICS

DOI: 10.37394/23207.2023.20.235

Van Vien Vu, Van Thanh Phan

E-ISSN: 2224-2899

2779

Volume 20, 2023

[14] Tsai L.C and Yu Y. S. (2004). Forecast of the

output value of Taiwan's IC industry using the

grey forecasting model. International Journal

of Computer Applications in Technology, vol.

19, no. 1, pp. 23- 27.

[15] Hsu L. C. (2003). Applying the grey

prediction model to the global integrated

circuit industry. Technological Forecasting &

Social Change, vol. 70, pp. 563-574.

[16] Hsu L. C., (2010). A genetic algorithm based

nonlinear grey Bernoulli model for output

forecasting in integrated circuit industry.

Expert Systems with Applications, vol. 37, no.

6, pp. 4318-4323.

[17] Hsu L. C. (2011). Using improved grey

forecasting models to forecast the output of

opto-electronics industry. Expert Systems with

Applications, vol. 38, no. 11, pp. 13879-

13885.

[18] Li D. C., Chang C. J., Chen C. C and Chen W. C.

(2012). Forecasting short-term electricity

consumption using the adaptive grey-based

approach-an Asian case. Omega, vol. 40, pp. 767-

773.

[19] Kang J and Zhao H. (2012). Application of

improved grey model in long-term load

forecasting of power engineering. Systems

Engineering Procedia, vol. 3, pp. 85-91.

[20] Lin Y. H., Chiu C. C., Lee P. C and Lin Y. J.

(2012). Applying fuzzy grey modification

model on inﬂow forecasting. Engineering

Applications of Artificial Intelligence, vol. 25,

no. 4, pp. 734-743.

[21] Wang Z. X., Dang Y. G and Liu S. F. (2007).

The optimization of background value in GM

(1,1) model. Journal of Grey System, vol. 10,

no. 2, pp. 69-74.

[22] Wang C. H and Hsu L. C. (2008). Using

genetic algorithms grey theory to forecast

high technology industrial output. Applied

Mathematics and Computation, vol. 195, pp.

256-263.

[23] Dong S., Chi K., Zhang Q. Y and Zhang X. D

(2012). The application of a grey Markov

model to forecasting annual maximum water

levels at hydrological stations. Journal of

Ocean University of China, vol. 11, no. 1, pp.

13-17.

[24] Hsu Y. T., Liu M. C., Yeh J and Hung H. F.

(2009). Forecasting the turning time of stock

market based on Markov-Fourier grey model.

Expert Systems with Applications, vol. 36, no.

4, pp. 8597-8603.

[25] Wu L., Liu S., Fang Z and Xu H. (2015).

Properties of the GM (1,1) with fractional

order accumulation. Applied Mathematics and

Computation, vol. 252, pp. 287-293.

[26] Wu L., Liu S., Yao L.,Yan S and Liu D.

(2013). Grey system model with the fractional

order accumulation, Communications in

Nonlinear Science and Numerical Simulation,

vol. 18, no. 7, pp. 1775-1785.

[27] Wu L. (2016). Using fractional GM (1,1)

model to predict the life of complex

equipment. Grey Systems: Theory and

Application, vol. 6, no. 1, pp. 32-40.

[28] Shili F., Wu L., Yan L and Zhigeng F. (2013).

Using fractional GM (1,1) model to predict

maintenance cost of weapon system, In

Proceedings of IEEE International

Conference on Grey systems and Intelligent

Services (GSIS), pp. 177-181.

[29] The report of Department of Tourism, Quang

Ninh province 2023, [Online].

https://bvhttdl.gov.vn/quang-ninh-phuc-hoi-

va-phat-trien-ben-vung-nganh-du-lich-

20230811100611591.htm (Accessed Date: 20,

August, 2023).

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution. All

author have read and agree to the published version

of the manuscript.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

The research paper is financial supported by the

Quang Binh University, Vietnam (CS:20.2023).

Conflict of Interest

The authors declare no conflicts of interest.

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WSEAS TRANSACTIONS on BUSINESS and ECONOMICS

DOI: 10.37394/23207.2023.20.235

Van Vien Vu, Van Thanh Phan

E-ISSN: 2224-2899

2780

Volume 20, 2023