Reliability Evaluation Based on Uncertain Bayesian rule

CHUNXIAO ZHANG

Tianjin Key Laboratory for Civil Aircraft Airworthiness and Maintenance, Tianjin, CΗΙΝΑ

and College of Science, Civil Aviation University of China, Tianjin 300300, CΗΙΝΑ

YUANYUAN WANG

College of Science, Civil Aviation University of China, Tianjin 300300, CΗΙΝΑ

Abstract: This paper focuses on the reliability evaluation of a one-unit system based on uncertain Bayesian rule, in

which the unit’s lifetime is assumed to be an uncertain variable. Considering two types of the posterior uncertainty

distribution of the lifetime, the Bayesian estimation method of uncertainty parameter is first proposed. Then

reliability evaluation is carried out by calculating uncertainty reliability R(T)with a specific time Tand mean

time between failure MT BF . Finally, some numerical examples are conducted to illustrate the application of the

new method.

Key-Words: Uncertainty Bayesian rule, Parameter estimation, Reliability evaluation, MT BF

Received: September 22, 2022. Revised: November 9, 2022. Accepted: December 4, 2022. Published: January 5, 2023.

1 Introduction

Reliability evaluation is an essential aspect of re-

liability research, which assesses the ability of elec-

tronic equipment or systems to realize their functions

under specific conditions based on the life distribution

function. Its method mainly uses mathematical statis-

tics theory to analyze the specific distribution of the

lifetime and evaluates the reliability of systems by es-

timating the distribution parameters. Commonly, re-

liability is characterized by reliability function, mean

time between failures (MT BF ), and other indicators,

[1], [2], [3] and [4].

Bayes method has become an important means

of reliability evaluation, which can improve the ac-

curacy of parameter estimation by introducing the

prior engineering knowledge of systems into reliabil-

ity evaluation in the form of the prior distribution, [5],

[6]. First, the prior distribution of the assumed pa-

rameters and the likelihood function associated with

the observed data are obtained. Second, the posterior

distribution of the parameter is deduced. Finally, the

analytical solution or approximate solution of the pa-

rameter is determined.

In the Bayesian framework, the lifetime of the sys-

tem is regarded as an unknown parameter, which is

generally estimated using the expectation algorithm.

In [7] Breipohl et al. indicated the application of

Bayesian theory in making typical reliability deci-

sions via decision theory. Tillman et al. in [8] re-

viewed some reliability problems using Bayesian in-

ference. In [9] Sharma et al. analyzed various en-

gineering systems to their reliability characteristics

and studied the Bayesian analysis of system avail-

ability. Ando, T. in [10] proposed a Bayesian pre-

diction information criterion to estimate the posterior

mean of the expected log-likelihood of the predic-

tion distribution. In [11] Guo et al. investigated the

Bayesian melding method (BMM) for system reliabil-

ity analysis by effectively integrating various avail-

able sources of expert knowledge and data at both

subsystem and system levels. In [12] Lu and L. pro-

posed a Bayesian approach for evaluating the system

structure based on estimating the multiplicative or ad-

ditive discrepancy between the system and compo-

nent test data under the assumed structure while quan-

tifying the uncertainty. The real systems are mostly

uncertain random systems that are affected by both

aleatory and epistemic uncertainties. It is of great

significance to study effective reliability evaluation

methods in various fields. Song et al. in [13] pro-

posed a system reliability evaluation method based on

Bayesian theory and multi-source information fusion.

In [14] Alharbi et al. proposed a fuzzy Bayesian pro-

cedure to estimate the unknown parameters and fuzzy

reliability function and applied it to compare estima-

tors of cancer data set.

As mentioned in previous literature, the lifetime of

the system was usually assumed as a stochastic vari-

able. In practical cases, it is known that the estimated

distribution is not close enough to the frequency in

the world of eternal change. Therefore, the classical

method is not available and it should be treated as an

uncertainty distribution, [15],[16]. To deal with this

kind of problem, uncertainty theory was founded by

Liu in [17] and refined by Liu in [18]. It has become

a branch of axiomatic mathematics for modeling hu-

man uncertainty.

At present, the uncertainty theory has been fur-

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ther developed and popularized. It has become a

mathematical branch of modeling epistemic uncer-

tainties under small data sizes or no data and has been

introduced to the field of reliability. Zhang et al.

in [19],[20] developed some system belief reliability

formulas for different systems configurations. Zhang

et al. in [21] considered the structure component’s

failure time as an uncertain variable because of the ab-

sence of historical data. In recent years, based on un-

certainty theory, how to use limited failure time data

to obtain reliability distribution has become the focus

of scholars. For example, Z. et al. in [22] developed

a new method called the graduation formula to con-

struct belief reliability distribution with limited obser-

vations. In [23] Kang presented the lifetime model

and reliability evaluations based on uncertainty the-

ory. In [24] Lio and Kang gave a method to update

a prior uncertainty distribution to a posterior uncer-

tainty distribution based on the likelihood function

and observation data in the sense of uncertainty the-

ory.

In practical engineering, since system reliability

testing is widely costly and with high reliability, the

sample size of the system is normally very small,

and the problem of non-failure frequently occurs. In

addition, they have few historical operating data of

their lifetime. The above issues can lead to a lack

of knowledge in evaluating system reliability. Es-

pecially under the small sample size, the probability

theory based on large samples is not appropriate any-

more. However, using the system reliability in un-

certain Bayesian rule has not been investigated in the

literature. Therefore, this paper will propose a new

method to present the lifetime distribution and relia-

bility evaluations based on uncertain Bayesian rule.

The remainder of this paper is organized as fol-

lows: Section 2 is a preliminary basic knowledge

about uncertainty theory. In Section 3 and Section 4,

the definitions and theorems for calculating parameter

values and evaluating the reliability from two special

uncertainty distributions are provided. Some numeri-

cal examples with reliability evaluation are conducted

to illustrate the application of the new method in Sec-

tion 5. Finally, a concise conclusion is made in Sec-

tion6.

2 Preliminary

This section introduces some fundamental defini-

tions and theorems of uncertainty theory.

Definition 1. (Liu, [17]) An uncertain variable is

a measurable function ξfrom the uncertainty space

(Γ,L,M)to the set of real numbers such that {ξ∈

B}is an event for any Borel set Bof real numbers.

Definition 2. (Liu, [17]) The uncertainty distribution

Φof an uncertain variable ξis defined by

Φ(x) = M{ξ≤x}

for any real number x.

Definition 3. (Liu, [18]) Let ξbe an uncertain vari-

able with regular uncertainty distribution Φ(x). Then

the inverse function Φ−1(α)is called the inverse un-

certainty distribution of ξ.

An uncertain variable ξis called linear if it has a

linear uncertainty distribution

Φ(x) = 





0, if x ≤a

x−a

b−a, if a < x ≤b

1, if x > b

denoted by L(a, b)where a and b are real numbers

with a < b and the inverse uncertainty distribution of

linear uncertain variable L(a, b)is

Φ−1(α) = (1 −α)a+αb.

An uncertain variable ξis called normal if it has a

normal uncertainty distribution

Φ(x) = (1 + exp(π(e−x)

√3σ))−1, x ∈R

denoted by N(e, σ)where eand σare real numbers

with σ > 0and the inverse uncertainty distribution of

normal uncertain variable N(e, σ)is

Φ−1(α) = e+√3σ

πln α

1−α.

Expected value is the average value of uncertain

variable in the sense of uncertain measure. It is an

important feature of distribution and reflects the av-

erage value of the uncertain variable.

Theorem 1. (Liu, [17]) Let ξbe an uncertain vari-

able with uncertainty distribution Φ. Then

E[ξ] = +∞

(1 −Φ(x))dx −0

−∞

Φ(x)dx

Remark 1. (Liu, [18]) Let ξbe an uncertain variable

with regular uncertainty distribution Φ. Then

E[ξ] = 1

Φ−1(α)dα

Theorem 2. (Lio and Liu, [26], Likelihood Func-

tion) Suppose η1,η2,. . .,ηnare iid uncertain vari-

ables with uncertainty distribution F(y|θ)where θ

is an unknown parameter, and have observed values

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y1,y2,. . .,yn, respectively. If F(y|θ)is differentiable

at y1,y2,. . .,yn, then the likelihood function associ-

ated with y1,y2,. . .,ynis

L(s|y1, y2, . . . , yn) =



i=1

F′(yi|s).(1)

Definition 4. (Lio, [24])Suppose ξis an uncertain

variable with prior uncertainty distribution Φ(x), and

η1,η2,. . .,ηnare iid uncertain variables from a popu-

lation with uncertainty distribution F(y|ξ). Suppose

Φ′(x)and F′(y|ξ)can be obtained, and η1,η2,. . .,ηn

have observed values y1,y2,. . .,yn, respectively. Then

the posterior uncertainty distribution is defined by

Ψ(x|y1, y2, . . . , yn)

=x

−∞ L(s|y1, y2, . . . , yn)∧Φ′(x)ds

+∞

−∞ L(s|y1, y2, . . . , yn)∧Φ′(x)ds

=x

−∞



i=1

F′(yi|s)∧Φ′(x)ds

+∞

−∞



i=1

F′(yi|s)∧Φ′(x)ds

(2)

It is clear that if

+∞

−∞



i=1

F′(yi|s)Φ′(x)ds = 0

then the posterior uncertainty distribution defined by

Eq. (2) is a continuous monotone increasing function

satisfying

0≤Ψ(x|y1, y2, . . . , yn)≤1,

Ψ(x|y1, y2, . . . , yn)= 0,

Ψ(x|y1, y2, . . . , yn)= 1.

It was proved by Peng and Iwamura, [25], and Liu and

Lio, [26], that Eq.(2) is indeed an uncertainty distri-

bution.

3 Uncertain Bayesian parameter

estimation

In this section, we first present an estimation

method of uncertainty parameters based on the pos-

terior uncertainty distribution.

Definition 5. (Uncertain Posterior Expected Estima-

tion) Suppose ξis an uncertain variable with the pos-

terior uncertainty distribution Ψ(x|y1, y2, . . . , yn)

and y1, y2, . . . , ymare observed values, respectively.

If the inverse uncertainty distribution Ψ−1(α)exists,

then the posterior expected estimation of ξis

ξE=E(ξ|y1, y2, . . . , yn)

=+∞

(1 −Ψ(x|y1, y2, . . . , yn))dx

=1

Ψ(0)

Ψ−1(α)dα.

(3)

To present the lifetime distribution of systems in

the uncertainty theory, we consider two simple and

commonly use situations.

Lemma 1. (Lio, [24]) Suppose ξis an uncertain vari-

able with linear prior uncertainty distribution L(a, b),

and η1, η2, . . . , ηnare iid uncertain variables from a

population with linear uncertainty distribution L(ξ−

c, ξ +d), c,d≥0 and observed values y1, y2, . . . , ym,

respectively. If it is assumed that



i=1

(yi−d)∨a≤



i=1

(yi+c)∧b,

then the posterior uncertainty distribution is



i=1

(yi−d)∨a,



i=1

(yi+c)∧b).(4)

Theorem 3. Suppose an uncertain variable

ξhas the posterior uncertainty distribution

Ψ(x|y1, y2, . . . , yn)and y1, y2, . . . , ymare ob-

served values, when the prior distribution L(a, b)

is assumed to be linear and the data are observed

from a population whose distribution function

L(ξ−c, ξ +d), c,d≥0 is also linear. If the inverse

uncertainty distribution Ψ−1(α)exists, then the

posterior expected estimation is

ξE=

(



i=1

(yi−d)∨a)+(



i=1

(yi+c)∧b)

2.(5)

Proof. It follows from Definition 5 and Lemma 1 that

the posterior expected estimation is

ξE=1

Ψ−1(α)dα

=1

((



i=1

(yi−d)∨a)−(



i=1

(yi+c)∧b))α

+ (



i=1

(yi+c)∧b)dα

(



i=1

(yi−d)∨a)+(



i=1

(yi+c)∧b)

The Theorem is proved.

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According to Eq.(5), then the uncertainty distribu-

tion Fof the unit can be obtained.

F(y|ˆ

ξE) = 









0, if y ≤ˆ

ξE−c

y−(ˆ

ξE−c)

d+c, if ˆ

ξE−c < y ≤ˆ

ξE+d

1, if y > ˆ

ξE+d

(6)

Lemma 2. (Lio, [24]) Suppose ξis an uncertain

variable with normal prior uncertainty distribution

N(e, σ), and η1, η2, . . . , ηnare iid uncertain vari-

ables from a population with normal uncertainty dis-

tribution N(ξ, σ)and observed values y1, y2, . . . , ym,

respectively. Then the posterior uncertainty distribu-

tion is

Ψ(x|y1, y2, . . . , yn)

=ΦM(x)

ΦM((m+M)/2)+1−Φm((m+M)/2) , if x ≤m+M

ΦM((m+M)/2)+Φm(x)−Φm((m+M)/2)

ΦM((m+M)/2)+1−Φm((m+M)/2) , if x > m+M

(7)

where ΦMand Φmare the uncertainty distributions

N(M, σ)and N(m, σ), respectively, and



i=1

yi∧e, m =



i=1

yi∧e.

Theorem 4. Suppose an uncertain variable

ξhas the posterior uncertainty distribution

Ψ(x|y1, y2, . . . , yn)and y1, y2, . . . , ymare ob-

served values, when the prior distribution N(e, σ)is

assumed to be normal and the data are observed from

a population whose distribution function N(ξ, σ)is

also normal. If the inverse uncertainty distribution

Ψ−1(α)exists, then the posterior expected estimation

ξE=m+M

2(8)

where



i=1

yi∧e, m =



i=1

yi∧e.

Proof. It follows from Definition 5 and Lemma 2 that

the posterior expected estimation is

ξE=1

Ψ−1(α)dα

=1

(m+M

2−√3σ

πln α

1−α)dα

=m+M

The Theorem is verified.

According to Eq.(8), then the uncertainty distribu-

tion Fof the unit can be obtained.

F(y|ˆ

ξE) = (1 + exp(π(ˆ

ξE−y)

√3σ))−1(9)

4 Uncertain Bayesian reliability

evaluation

In this section, we consider uncertainty reliability

evaluation, which is defined as the measure that it will

perform a required function under stated conditions

for a stated period. To assess the system reliability,

two indicators the MT BF and R(T)for uncertainty

assessment are first defined.

Definition 6. (Mean Time Between Failure)Suppose

the failure free time η(η≥0) is a nonnegative uncer-

tain variable, and the uncertainty distribution of the

unit is F(y)≡ M{η⩽y}. The MTBF of the unit is

defined as

MT BF =+∞

0M{η > y}dy

=+∞

(1 −F(y))dy

=1

F(0)

F−1(α)dα

(10)

where F−1is the inverse uncertainty distributions of

In practical application, the uncertainty measure

of failure free time over a certain value Tis also an

important index, which represent the unceratinty re-

liability R(T)that a system will perform a required

function at the specific time Tunder stated operating

conditions using the uncertainty theory, expressed as,

R(T) = M{y > T }= 1 −F(T).(11)

Theorem 5. Suppose the failure free time ηis a non-

negative uncertain variable, and the uncertainty dis-

tribution of the unit F(y|ˆ

ξE)is L(ˆ

ξE−c, ˆ

ξE+d).

If the inverse uncertainty distribution F−1(α)exists,

the MT BF of the unit and uncertainty reliability R

are

MT BF =1

(1 −α)(ˆ

ξE−c) + α(ˆ

ξE+d)dα

(12)

R(T) = 









1, if y ≤ˆ

ξE−c

(ˆ

ξE+d)−T

d+c, if ˆ

ξE−c < y ≤ˆ

ξE+d

0, if y > ˆ

ξE+d

(13)

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Proof. It follows from Definition 6 and

F(y|ˆ

ξE) = 









0, if y ≤ˆ

ξE−c

y−(ˆ

ξE−c)

d+c, if ˆ

ξE−c < y ≤ˆ

ξE+d

1, if y > ˆ

ξE+d

that the MT BF of the unit is

MT BF =+∞

(1 −F(y|ˆ

ξE))dy

=1

F−1(α)dα

=1

(1 −α)(ˆ

ξE−c) + α(ˆ

ξE+d)dα

where F−1is the inverse uncertainty distributions

of F, and uncertainty reliability Ris

R(T) = M{y > T }

= 1 −F(T|ˆ

ξE)

=









1, if y ≤ˆ

ξE−c

(ˆ

ξE+d)−T

d+c, if ˆ

ξE−c < y ≤ˆ

ξE+d

0, if y > ˆ

ξE+d

The theorem is proved.

Theorem 6. Suppose the failure free time ηis a non-

negative uncertain variable, and the uncertainty dis-

tribution of the unit F(y|ˆ

ξE)is N(ˆ

ξE, σ). If the

inverse uncertainty distribution F−1(α)exists, the

MT BF of the unit and uncertainty reliability Rare

MT BF =1

(1+exp(πˆ

ξE

√3σ))−1

(ˆ

ξE+√3σ

πln α

1−α)dα

(14)

R(T) = 1 −(1 + exp(π(ˆ

ξE−T)

√3σ))−1.(15)

Proof. It follows from Definition 6 and

F(y|ˆ

ξE) = 1 + exp(π(ˆ

ξE−y)

√3σ)−1

that the MT BF of the unit is

MT BF =+∞

(1 −F(y|ˆ

ξE))dy

=1

(1+exp(πˆ

ξE

√3σ))−1

F−1(α)dα

=1

(1+exp(πˆ

ξE

√3σ))−1

(ˆ

ξE+√3σ

πln α

1−α)dα

where F−1is the inverse uncertainty distributions

of F, and uncertainty reliability Ris

R(T) = M{y > T }

= 1 −F(T|ˆ

ξE)

= 1 −(1 + exp(π(ˆ

ξE−T)

√3σ))−1

The theorem is verified.

5 Numerical examples

This section will provided some examples to illus-

trate the application of the new method.

Example 1. Suppose ξis an uncertain variable with

linear prior uncertainty distribution L(1510,1550),

and η1, η2, η3are iid uncertain variables from a pop-

ulation with linear uncertainty distribution L(ξ−

10, ξ + 20) and observed values y1= 1520, y2=

1530, y3= 1540, respectively. Then it follows from

Lemma 1 that the posterior uncertainty distribution is

L(1520,1530), i.e.

Ψ(x|1520,1530,1540)

=





0, if x ≤1520

x−1520

10 , if 1520 < x ≤1530

1, if x > 1530

Then, using the observation y1, y2, y3, the unknown

parameter can be estimated.

ξE=E(ξ) = 1525

According to Eq.(6), the uncertainty distribution of

the unit and its derivative F’can be obtained(see Fig-

ure 1).

F(y|ˆ

ξE) = 





0, if y ≤1515

y−1515

30 , if 1515 < y ≤1545

1, if y > 1545

F′(y|ˆ

ξE) = y

30 , if 1515 < y ≤1545

0, otherwise

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By substituting the estimation results into Eq.(12) and

Eq.(13) , the MT BF and R(1520) can be obtained.

As shown in Figure 2, it can be seen that the un-

certainty reliability of the unit decreases with the in-

crease in the expected time of failure.

MT BF =1

1515 + 30αdα = 1530

R(1520) = 1 −F(1520) ≈83.3%

Figure 1: Function F’ in Example 1

Figure 2: Uncertainty reliability R in Example 1

Example 2. Suppose ξis an uncertain variable with

normal prior uncertainty distribution N(1540,3),

and η1, η2are iid uncertain variables from a popula-

tion with normal uncertainty distribution N(ξ, 3) and

observed values y1= 1510, y2= 1550, respectively.

Then it follows from Lemma 2 that the posterior un-

certainty distribution is

Ψ(x|1510,1550)

=Φ2(x)

Φ2(1530)+1−Φm(1530) , if x ≤1530

Φ2(1530)+Φ2(x)−Φ1(1530)

Φ2(1530)+1−Φ1(1530) , if x > 1530

where Φ2and Φ1are the uncertainty distributions

N(1550,3) and N(1510,3). Then, using the obser-

vation y1, y2, the unknown parameter can be esti-

mated.

ξE=E(ξ) = 1530

According to Eq.(9), the uncertainty distribution of

the unit and its derivative F’can be obtained (see

Figure 3).

F(y|ˆ

ξE) = (1 + exp(π(1530 −y)

3√3))−1

F′(y|ˆ

ξE) =

3√3exp(π(1530−y)

3√3)

(1 + exp(π(1530−y)

3√3))2

By substituting the estimation results into Eq.(14) and

Eq.(15), the M T BF and R(1520) can be obtained.

As shown in Figure 4, it can be seen that the un-

certainty reliability of the unit decreases with the in-

crease in the expected time of failure.

MT BF =1

(1+exp(1530π

3√3))−1

(1530 + 3√3

πln α

1−α)dα

= 1530

R(1520) = 1 −(1 + exp(π(1530 −1520)

3√3))−1

≈99.76%

Example 3. Suppose ξis an uncertain variable with

normal prior uncertainty distribution N(1540,3),

and η1, η2are iid uncertain variables from a popula-

tion with linear uncertainty distribution L(ξ−20, ξ +

10) and observed values y1= 1530, y2= 1540, re-

spectively. Then it follows from Definition 4 that the

posterior uncertainty distribution is

Ψ(x|1530,1540)

=









Φ(x)−0.0024

0.3886 , if 1530 < x ≤1538.5

(x−1538.5)/30

0.3886 , if 1538.5< x ≤1541.5

Φ(x)−0.7124

0.3886 , if 1541.5< x ≤1550

where Φis the uncertainty distributions N(1540,3).

Then, using the observation y1, y2, the unknown pa-

rameter can be estimated.

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Figure 3: Function F’ in Example 2

Figure 4: Uncertainty reliability R in Example 2

ξE=E(ξ) = 1543.38

Moreover, the uncertainty distribution of the unit and

its derivative F’can be obtained(see Figure 6).

F(y|ˆ

ξE) = 





0, if y ≤1523.38

y−1523.38

30 , if 1523.38 < y ≤1553.38

1, if y > 1553.38

F′(y|ˆ

ξE) = y

30 , if 1523.38 < y ≤1553.38

0, otherwise

By substituting the estimation results into Eq.(10) and

Eq.(11) , the M T BF and R(1520) can be obtained.

As shown in Figure 5, it can be seen that the un-

certainty reliability of the unit decreases with the in-

crease in the expected time of failure.

MT BF =+∞

(1 −F(y|ˆ

ξE))dy

=1

F−1(α)dα

=1

(1 −α)(ˆ

ξE−c) + α

=1

1523.38 + 30αdα

= 1538.38

where F−1is the inverse uncertainty distributions of

R(1520) = 1 −F(1520) ≈88.74%

Figure 5: Function F’ in Example 3

6 Conclusion

This paper studied the reliability evaluation of the

system based on uncertain Bayesian rule, and the fol-

lowing conclusions can be drawn:

(i)Bayesian estimation method of uncertainty pa-

rameter was first proposed. There was not enough

data to obtain the probability distribution of the life-

time, so the stochastic method did not apply to our re-

search. Thus, the lifetime of the unit was regarded as

an uncertain variable, where two types of the uncer-

tainty distribution (linear and normal) were consid-

ered. It can be extended to Bayesian parameter test-

ing and decision-making, providing a basic method

for the research of uncertain statistical inference.

(ii)Based on the method of uncertain Bayesian

parameter estimation, reliability evaluation was de-

rived by calculating two reliability indexes (MT BF

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.7

Chunxiao Zhang, Yuanyuan Wang

E-ISSN: 2224-2880

Volume 22, 2023

Figure 6: Uncertainty reliability R in Example 3

and R(T)). Finally, Some numerical examples were

given to illustrate the application of the new method.

This method can be applied to reliability evaluation

in the engineering field, mainly aiming at the short-

age of failure data and considering the reliability of

experts. It can also be applied to project evaluation

and decision-making in the economic field.

For future works, the uncertain hypothesis testing

and decision-making for the unknown uncertainty pa-

rameters in uncertain Bayesian statistics will be stud-

ied. When the operational data is fully obtained, some

random lifetime can be considered, such as Weibull

distribution.

Contribution of individual authors to

the creation of a scientific article

Chunxiao Zhang proposed the idea of the method

and checked the correctness of the manuscript.

Yuanyuan Wang gave the method and wrote the

article.

Acknowledgements

This work is supported by the Open Fund of Civil

Aviation University of China for Provincial and Min-

isterial Scientific Research Institutions under Grant

No.TKLAM202201.

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Chunxiao Zhang, Yuanyuan Wang

E-ISSN: 2224-2880

Volume 22, 2023

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.7

Chunxiao Zhang, Yuanyuan Wang

E-ISSN: 2224-2880

Volume 22, 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.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

This work is supported by the Open Fund of Civil

Aviation University of China for Provincial and Min-

isterial Scientific Research Institutions under Grant

No.TKLAM202201.