Differential Calculus of Fermatean Fuzzy Functions: Continuities, Derivatives, and Differentials

Zaoli Yang; Harish Garg; Xin Li

doi:10.2991/ijcis.d.201215.001

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Volume 14, Issue 1, 2021, Pages 282 - 294

Differential Calculus of Fermatean Fuzzy Functions: Continuities, Derivatives, and Differentials

Authors

Zaoli Yang¹, Harish Garg²^{, *}^,, Xin Li¹^,

¹College of Economics and Management, Beijing University of Technology, Beijing, China

²School of Mathematics, Thapar Institute of Engineering and Technology, Deemed University, Punjab, India

^*Corresponding author. Email: harishg58iitr@gmail.com

Corresponding Author

Harish Garg

Received 19 October 2020, Accepted 7 December 2020, Available Online 22 December 2020.

DOI: 10.2991/ijcis.d.201215.001 How to use a DOI?
Keywords: Fermatean fuzzy sets; Continuities; Derivatives; Differentials; Calculus
Abstract: Fermatean fuzzy sets are an effective way to handle uncertainty and vagueness by expanding the spatial scope of membership and nonmembership of the intuitionistic fuzzy set and the Pythagorean fuzzy set. However, existing studies only analyzed the discrete information and neglected the continuous state of Fermatean fuzzy sets. In this paper, we investigated the properties of continuous Fermatean fuzzy information by firstly proposing Fermatean fuzzy functions, then defining the subtraction and division operations of Fermatean fuzzy functions and discussing their properties. Further, we examined the continuity, derivatives, and differentials of Fermatean fuzzy functions. Effective approximate calculations regarding nonlinear problems in the Fermatean fuzzy environment were provided, and some examples were presented to verify the feasibility and effectiveness of approximate calculations using the Fermatean fuzzy functions.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Since Zadeh proposed the concept of fuzzy sets [1], many scholars have researched fuzzy set theory. For example, Atanassov [2] proposed the intuitionistic fuzzy sets (IFSs) to characterize uncertainty information according to the degree of membership and nonmembership, providing a basis for other scholars to develop some new fuzzy states of interval IFSs [3,4], intuitionistic 2-tuple linguistic sets [5], intuitionistic trapezoidal fuzzy sets [6,7], intuitionistic normal fuzzy sets [8], intuitionistic uncertain linguistic [9], triangular intuitionistic fuzzy numbers [10–12], linguistic interval-valued intuitionistic neutrosopic fuzzy sets [13], generalized intuitionistic fuzzy Einstein hybrid geometric fuzzy sets [14], interval type-2 fuzzy sets [15], among others. However, IFS and its extensions must meet the criterion that the sum of the degree of membership and nonmembership is less than 1, thereby restricting its application in some decision and information environments [16–18]. For instance, when decision-makers independently evaluate the degree of membership and nonmembership, the sum may be greater than 1 but their quadratic sum would not be greater than 1. To handle this problem, Yager proposed the Pythagorean fuzzy set (PFS) [19,20] to satisfy the quadratic sum of membership and nonmembership degree, i.e., not greater than 1, allowing us to easily infer that the PFS is more useful than IFS in depicting fuzzy information.

After this presentation, PFS-based multi-attribute decision-making (MADM) methods were conducted by many scholars. Peng and Yang [21] investigated the division and subtraction operations of PFS and developed some aggregation operators. Liang et al. [22] combined the TOPSIS and three-way decisions model to present a novel Pythagorean fuzzy decision-making method. Verma and Merigo [23] examined the similarity measures between PFSs. Liu et al. [24] proposed an interval-valued Pythagorean hesitant fuzzy best-worst MADM method to support the product selecting. Rahman and Abdullah [25] developed the family of induced generalized Einstein geometric aggregation operators under interval-valued Pythagorean fuzzy environment. Moreover, Hussian and Yang [26] developed a method for measuring the distance between PFSs using the Hausdorff metric theory. Muhammad et al. [27] developed the ELECTRE I-based MADM method under the Pythagorean fuzzy information. Similarly, Ren et al. [28] presented the TODIM-based MADM method based on the PFSs. Apart from above, Garg [29] established the generalized Einstein operations in PFSs. Yang et al. [30] developed some interval-valued Pythagorean fuzzy Frank power aggregation operators and analyzed its several limiting cases. Meanwhile, Garg [31] defined the neutrality operations in PFSs to aggregate the Pythagorean fuzzy information.

Although IFS and PFS facilitate the resolution of fuzzy decision problems, they still have obvious shortcomings, especially in extremely contradictory decision environments. PFS and IFS are unable to handle a situation where the sum of membership and nonmembership is greater than 1 and the sum of squares is still greater than 1, but the sum of three is less than 1 [32]. For these cases, Senapati and Yager [32] developed the novel concept of Fermatean fuzzy set (FFS), which satisfies the criterion that the sum of the third power of membership and nonmembership must be less than 1. Compared to IFS and PFS, FFS gains a stronger ability to describe uncertain information by expanding the spatial scope of membership and nonmembership. Based on FFS, Wang et al. [33] developed a hesitant Fermatean fuzzy multicriteria decision-making method using Archimedean Bonferroni mean operators, Senapati and Yager [34] proposed Fermatean fuzzy information weighted aggregation operators, and Liu et al. [35] developed a distance measure method for Fermatean fuzzy linguistic term sets. Furthermore, Liu et al. [36] defined a new concept of a Fermatean fuzzy linguistic set and Senapati and Yager [37] developed some new operations between Fermatean fuzzy numbers (FFNs).

However, previous research only focused on how to address the FFS-based decision-making problems with discrete information and did not consider the continuous states of FFS. In many decision-making situations, people need to make decisions in a continuous information environment, such as the diagnosis of the patient's condition, predict the weather and traffic condition, etc. For these issues, some authors have carried out some studies on the continuous fuzzy information. For instance, Lei and Xu [38], Lei et al. [39] defined the intuitionistic fuzzy function (IFFs) to depict the continuous intuitionistic fuzzy information. Based on that, some other scholars analyzed the properties of continuities, derivatives, and differential approximate calculations of IFFs [40,41]. Further, Gou et al. [42] proposed the concept of Pythagorean fuzzy function (PFF) and investigated the continuity and derivability of PFFs. On account of that the deficiencies of IFS and PFS also exist in the differential calculus of IFFs and PFFs, the Fermatean fuzzy function (FFF) and its properties need to be investigated. Therefore, inspired the idea by [38–42], in this work, we treat FFNs as variables to study their continuous states, instead of just treating them as constants. Based on that, we define the novel concept of FFF and examine the properties of continuous Fermatean fuzzy information, such as continuities, derivatives, and differentials. The main aim of this paper is to discuss continuity and calculus theories in the Fermatean fuzzy environment, which provide a method for dealing with nonlinear problems. The main continuations of this paper are summarized as follows:

Defined the novel concept of FFF.
Investigated the properties of continuities, derivatives, and differentials of FFF.
Established a concise decision application framework under continuous Fermatean fuzzy information.

The remaining content of this paper is organized as follows. Section 2 recalls some basic concepts of IFS, PFS, and FFS. The subtraction and division operations on FFNs are defined in Section 3. The definition of FFF and its continuous properties are described in Section 4. The derivative operations between FFFs are defined in Section 5. The differentials of FFFs are introduced in Section 6. And some numerical examples are given to illustrate the properties of FFS in Section 7. Finally, the conclusions are summarized in Section 8.

2. PRELIMINARIES

In this section, we briefly recall some basic concepts about IFS, PFS, and FFS.

Definition 2.1.

[2] With a nonempty set, the form of IFS was defined by Atanassov as

A=〈x,αAx,βAx〉|x∈X,(1)

where αAx and βAx designate the degree of membership and nonmembership of element x∈X to set A, respectively, thereby satisfying 0≤αAx≤1 and 0≤βAx≤1, and meeting the condition 0≤αAx+βAx≤1.

Definition 2.2.

[19] Let X be an ordinary set, with the form of PFS defined by Yager as

P=〈x,αPx,βPx〉|x∈X,(2)

where αPx and βPx designate the degree of membership and nonmembership of element x∈X to set P, respectively, thereby satisfying 0≤αPx≤1 and 0≤βPx≤1, and meeting the condition 0≤αP2x+βP2x≤1. The degree of indeterminacy is given as πPx=1−αPx2−βPx21/2.

Definition 2.3.

[32] Assuming that X is a universe of discourse, the form of FFS was defined by Senapati and Yager as

F=〈x,αFx,βFx〉|x∈X,(3)

where αPx and βPx designate the degree of membership and nonmembership of element x∈X to set F, respectively, thereby satisfying 0≤αFx≤1 and 0≤βFx≤1, and meeting the condition 0≤αF3x+βF3x≤1. For any FFS, F and x∈X, πFx=1−αFx3−βFx31/3 is identified as the degree of indeterminacy of x to F.

For convenience, we considered αF,βF to be a FFN, which was denoted as F=αF,βF. For simplicity, we considered the FFNs to be the components of the FFS.

The membership grades related to FFSs are herein referred to as Fermatean membership grades (FMGs).

Definition 2.4.

[32] Let F1=αF1,βF1, F2=αF2,βF2, and F=αF,βF be three FFNs, defined as

F1∩F2=minαF1,αF2,maxβF1,βF2,
F1∪F2=maxαF1,αF2,minβF1,βF2,
Fc=βF,αF.

Definition 2.5.

[32] Let F1=αF1,βF1, F2=αF2,βF2, and F=αF,βF be three FFNs and λ be a positive real number, defined as

F1⊕F2=αF13+αF23−αF13αF231/3,βF1βF2,
F1⊗F2=αF1αF2,βF13+βF23−βF13βF231/3,
λF=1−1−αF3λ1/3,βFλ,
Fλ=αFλ,1−1−βF3λ1/3.

Theorem 2.1.

[32] Let F1=αF1,βF1, F2=αF2,βF2, and F=αF,βF be three FFNs, and λ, λ1, and λ2 be three positive real numbers, therefore, the following operations are valid:

F1⊕F2=F2⊕F1,
F1⊗F2=F2⊗F1,
λF1⊕F2=λF1⊕λF2,
λ1+λ2F=λ1F⊕λ2F,
F1⊗F2λ=F1λ⊗F2λ,
F1λ1⊗F2λ2=Fλ1+λ2.

Definition 2.6.

[32] Let F=αF,βF be a FFN, therefore, the score function, S of F, is defined as

SF=αF3−βF3,(4)

and the accuracy function, H of F, is defined as

HF=αF3+βF3.(5)

Theorem 2.2.

For any FFN F=αF,βF, the suggested function score is SF∈−1,1, whereby the bigger the value of HF, the bigger the accuracy of the FFS F.

Theorem 2.3.

For any FFN F=αF,βF, the accuracy function is HF∈0,1, whereby the bigger the value of HF, the greater the accuracy of F.

According to Definition 2.3 and 2.6, πF3+HF=1 was identified.

Definition 2.7.

[32] Let F1=αF1,βF1 and F2=αF2,βF2 be any two FFNs, with SF1 and SF2 being the score functions of F1 and F2, respectively, and HF1 and HF2 being the accuracy functions of F1 and F2, respectively, then,

If SF1<SF2, then F1<F2,
If SF1>SF2, then F1>F2,
If SF1=SF2, then,
1. If HF1<HF2, then F1<F2,
2. If HF1>HF2, then F1>F2,
3. If HF1=HF2, then F1=F2.

Definition 2.8.

Let F1=αF1,βF1, F2=αF2,βF2, and F3=αF3,βF3 be three FFNs, therefore, the natural quasi-ordering can be defined as

F1≥F2 when and when αF1≥αF2 and βF1≤βF2,
F1≤F2 when and when αF1≤αF2 and βF1≥βF2.

3. SUBTRACTION AND DIVISION OPERATIONS ON FFNs

To further derive the derivative and differential of FFFs, we define the subtraction and division operations based on Definition 2.5 and Gao et al.'s study [43] as follows.

Definition 3.1.

Let F1=αF1,βF1 and F2=αF2,βF2 be two FFNs, therefore, the subtraction operation of FFNs was defined as

F2⊖F1=αF23−αF131−αF131/3,βF2βF1,(6)

which satisfies

0≤βF2βF1≤1−αF231/31−αF131/3≤1,(7)

and can be replaced by

αF2≥αF1, βF2≤βF1, βF2πF1≤βF1πF2.

Definition 3.2.

Let F1=αF1,βF1 and F2=αF2,βF2 be two FFNs, therefore, the division operation between FFNs was defined as follows:

F2⊘F1=αF2αF1,βF23−βF131−βF131/3,(8)

which satisfies

0≤αF2αF1≤1−βF231/31−βF131/3≤1.

Theorem 3.1.

Let F1=αF1,βF1 and F2=αF2,βF2 be two FFNs, then the result of the subtraction operation of the FFNs, i.e., F2⊖F1, is an FFN, and the result of the division operation of FFNs, i.e., F2⊘F1, is also an FFN.

The following two conditions required proof.

0≤α≤1,0≤β≤1,
0≤α3+β3≤1.

Since 0≤αF1≤1, 0≤αF2≤1, 0≤βF1≤1, and 0≤βF2≤1, and αF2≥αF1,βF2≤βF1,βF2πF1≤βF1πF2,

αF23−αF13≤1−αF13 was obtained.

Further,

0≤αF23−αF131−αF13≤1,

then,

0≤αF23−αF131−αF131/3≤1,

and 0≤βF2βF1≤1,

therefore, 0≤α≤1.

Similarly, 0≤β≤1 was obtained, satisfying condition (i).

Since βF2πF1≤βF1πF2,

βF21−αF13−βF131/3≤βF11−αF23−βF231/3 was obtained.

Further, βF23βF13≤1−αF231−αF13.

Moreover, 0≤αF23−αF131−αF131/33+βF2βF13≤αF23−αF131−αF13+1−αF231−αF13=1.

Therefore, condition (ii) was met, meaning that the result of the subtraction operation of F2⊖F1 was an FFN. Similarly, we proved that the result of the division subtraction operation for FFNs, i.e., F2⊘F1, was also an FFN.

Theorem 3.2.

If F2⊖F1 is an FFN, then αF2≥αF1 and βF1≥βF2 must exist. Similarly, if F2⊘F1 is an FFN, αF2≤αF1 and βF1≤βF2 must exist.

According to [43] and [44], a novel order relationship based on the addition and subtraction of FFNs is presented as follows.

Definition 3.3.

Let F1=αF1,βF1, F2=αF2,βF2, and F3=αF3,βF3 be three FFNs, thereby satisfying F2=F1⊕F3, therefore, F1 is less than or equal to F2 as denoted by F1⊴F2. Particularly, F1⊲F2 if F3≠0,1.

Definition 3.4.

Let F1=αF1,βF1 and F2=αF2,βF2 be two FFNs, with the set F consisting of all FFNs, therefore,

F⊕F1=F2⊕F1∈F|F2∈F,(9)

F⊖F1=F1⊖F2∈F|F2∈F,(10)

F⊗F1=F2⊖F1∈F|F2∈F,(11)

F⊘F1=F1⊘F2∈F|F2∈F.(12)

According to Definition 2.5 and 3.1, the partition relationships between the addition and subtraction regions and between the multiplication and division regions were analyzed, as shown in Figures 1–3.

Theorem 3.3.

Let F1=αF1,βF1, F2=αF2,βF2, and F3=αF3,βF3 be three FFNs, with the set F consisting of all FFNs. For any F3∈F⊕F1, F3=F1⊕F2 implies F2=F3⊖F1.

According Theorem 3.3 and Definition 2.5, we obtained

F1≤F3, βF33βF13≤1−αF331−αF13,

therefore,

βF33βF13+αF331−αF13≤11−αF13.(13)

Theorem 3.4.

Let F1=αF1,βF1, F2=αF2,βF2, F3=αF3,βF3, and F4=αF4,βF4 be four FFNs, with the set F consisting of all FFNs, therefore,

If F2∈F⊕F1 and F3∈F⊕F2, then,
1. F1⊕F2=F2⊕F1,
2. F1⊕F2⊖F3⊕F4=F1⊖F2⊕F2⊖F4,
3. F1⊖F2⊖F3⊖F4=F1⊖F3⊕F2⊖F4.
  
  Proof for (ii).
  F1⊕F2⊖F3⊕F4=αF13+αF23−αF13αF231/3,βF1βF2⊖αF33+αF43−αF33αF431/3,βF3βF4=αF13+αF23+αF33αF43−αF33−αF43−αF13αF231−αF33−αF43+αF33αF431/3,βF1βF2βF3βF4αF13+αF23+αF33αF43−αF33−αF43−αF13αF231−αF33−αF43+αF33αF431/3,αF13−αF331−αF331/3,βF1βF3⊕αF23−αF431−αF431/3,βF2βF4=F1⊖F2⊕F2⊖F4.
If F2∈F⊗F1 and F3∈F⊗F2, then,
1. F1⊗F2⊘F2⊗F3=F1⊘F3,
2. F1⊘F2⊘F3⊘F2=F1⊘F3.
For λ1>0,λ2>0,
1. λ1F1⊕F2=λ1F1⊕λ1F2,
2. λ1F1⊖F2=λ1F1⊖λ1F2,
3. λ1+λ2F1=λ1F1⊕λ2F1,
4. λ1−λ2F1=λ1F1⊖λ2F1.

Proof for (i).

λ1F1⊕F2=λ1αF13+αF23−αF13αF231/3,βF1βF2=1−1−αF13−αF23+αF13αF23λ11/3,βF1λ1βF2λ1=1−1−αF13λ1+1−1−αF23λ1−1−1−αF13λ11−1−αF23λ11/3,βF1λ1βF2λ1=λ1F1⊕λ1F2.

Proof for (iii).

λ1F1⊕λ2F1=1−1−αF3λ11/3,β3λ1⊕1−1−αF3λ21/3,β3λ2=1−1−αF3λ1+λ21/3,β3λ1+λ2=λ1+λ2F1.

The others could be proven in a similar way, therefore, the procedures were omitted from explanation.

4. FFF AND ITS CONTINUITY

In this section, we first propose a definition of FFF based on the basic information regarding the FFNs. Besides, we introduce the continuous information of FFF, which plays a fundamental role in calculus. Then, we discuss the calculus properties of FFFs.

Definition 4.1.

The continuous functions of the multivariable were defined as

hαF,βF:0,1×0,1↦0,1,(14)

and

yαF,βF:0,1×0,1↦0,1.(15)

In addition, we assumed that

0≤h3+y3≤1.(16)

Let

φF=hαF,βF,yαF,βF:=hF,yF,(17)

then the function φF is considered as an FFF in terms of h and y.

The function φ in Equation (13) was still observed to be an FFN. For the analysis below, we guaranteed that all basic operations could be closed in FFS, so that φF⊕φI, φF⊖φI, φF⊗φI, and φF⊘φI still made sense for any F,I∈F. Therefore, we introduced some subsets of F into the theorem.

According to Definition 2.5, Definitions 3.1, and 3.2, we obtained the following:

Theorem 4.1.

Let φF be an FFF, as introduced in Definition 4.1.

For a fixed F∈F, we defined
F⊕φF:=I∈F⊕F|0≤yIyF≤1−hI31−hF31/3≤1,
where F⊕F was taken from Equation (9). Therefore, for any φI∈F⊕φF, φI⊖φF was also an FFN.
For a fixed F∈F, we defined
F⊕φF:=I∈F⊕F|0≤hIhF≤1−yI31−yF31/3≤1,
where F⊕F was taken from Equation (11). Therefore, for any φI∈F⊕φF, φI⊘φF was also an FFN.

Based on the basic properties introduced in Definition 4.1 and Theorem 4.1, the calculus properties of FFFs can be discussed, starting with the continuity of FFFs and considering whether the following limit was suitable for FFFs regarding their operations.

limI⊖ℱ→0,1φI⊘φF=0,1, φI∈F⊕φF(18)

Equation (18) was described as follows:

Definition 4.2.

ε−δ rule Let φF be an FFF, as defined in Definition 4.1. For any fixed ε=αε,βε∈F, if δ=αδ,βδ exists, α and ε should be depended on, such that

φI∈F⊕φF|F⊕φF⊕ε

holds true for all FFN I, satisfying

I∈F⊕F|F⊕F⊕δ.

Therefore, φ is continuous at F, and φF is continuous in F on the condition that φF is continuous at every F.

Definition 4.3.

If Equation (18) is unsuitable for FFFs in their operations, then φF is called a discontinuous FFF at F∈F.

Theorem 4.2.

Assuming that φF and ϕF are both continuous FFFs, φF⊕ϕF, φF⊖ϕF, φF⊗ϕF, and φF⊘ϕF, as well as their compositions φϕF and ϕφF, are all continuous, as long as their operations make sense.

5. THE DERIVATIVE OPERATOR OF FFFs

The derivative operation is an essential concept in calculus, and the essence of the derivative is the local linear approximation of the function through limit operation, which can indicate the rate of change of a function value relative to its variable. Therefore, it can be used as an effective method to calculate the instantaneous rate of change in many fields, such as economics, physics, and medicine.

For this section, we modified classical calculus theories and introduced a definition for the derivative operations of FFFs. A derivate formula of FFF is proposed in this section, allowing us to further discuss some properties of the derivative, such as Chain's law and other operations.

Definition 5.1.

Let φF=hF,yF and φI=hI,yI exist, as given in Definition 4.1, and if the quantity

limI⊖ℱ→0,1φI⊘φFI⊖F

made sense, and it was still an FFN in terms of F, then φ allowed for a derivative at F, as denoted by

dφFdF=limI⊖ℱ→0,1φI⊘φFI⊖F.(19)

Theorem 5.1.

The uniqueness of the limit showed that if dφ(F)dF|F, then it was unique. For convenience, we used IF:=I⊘F.

Theorem 5.2.

Let FFF φF=hF,yF exist, as given in Definition 4.1, where it is continuous in F. Therefore, the sufficient and necessary condition on the existence of dφ(F)dF|F was

0≤1−αF3hF21−hF3αF2⋅∂hF∂αF≤βFyF∂yF∂βF≤1.(20)

In particular, if

∂hF∂βF=0=∂yF∂αF,(21)

then we obtained

dφFdFF=dhF,yFdF=1−αF3hF21−hF3αF2⋅∂hF∂αF1/3,1−βFyF∂yF∂βF1/3,(22)

which played an essential role in our research.

For proof, according to Equations (6) and (7), we obtained

dφFdFF=limI⊖F→0,1φI⊖φFI⊖F=limI⊖F→0,1hI,yI⊖hF,yFαI,βI⊖αF,βF=limI⊖F→0,1hI3−hF31−hF31−αF3αI3−αF31/3,yIyF3−βIβF31−βIβF31/3=limI⊖F→0,1I,limI⊖F→0,1II(23)

We first considered I and utilized the binomial expression formula

aq−bq=a−b∑i=0q−1aq−1−ibi,q∈N+,(24)

and the continuity assumption, thereby obtaining

limI⊖F→0,1hI3−hF3αI3−αF3=limI⊖F→0,1hI3−hF3αI3−αF3⋅∑i=02hF3−1−ihFi∑i=02αF3−1−iαFi=hF2αF2limI⊖F→0,1hI−hFαI−αF=hF2αF2limI⊖F→0,1hI−hαF,βIαI−αF+hαF,βI−hFβI−βFβI−βFαI−αF=hF2αF2∂hF∂αF+∂hF∂βFcosθ,F(25)

where cosθ,F:=limI⊖F→0,1βI−βFαI−αF denoted the tangential derivative at the point F=αF,βF.

According to Theorem 4.1, we obtained

hF≤hI and βF≤βI when β∈F⊕F∩F⊕φF,

therefore,

yFβF=limI⊖F→0,1hI−hFβI−βF≤0.(26)

Based on Equations (26) and (29), we derived

yFβF⋅cosθ,F≥0.

Thus, the last equal sign in (25) was reasonable.

Hence,

limI⊖F→0,1I=1−αF31−hF3⋅limI⊖F→0,1hI3−hF3αI3−αF31/3=1−αF31−hF3⋅hF2αF2⋅∂hF∂αF+∂hF∂βFcosθ,F1/3.(27)

Next, we considered II.

limI⊖F→0,1II=limI⊖F→0,1II31/3=limI⊖F→0,1yIyF3−βIβF31−βIβF31/331/3=limI⊖F→0,1yIyF3−βIβF31−βIβF31/3=1yF3limI⊖F→0,1yI3βF3−yF3βI3βF3−βI31/3=βF3y3αF,βFlimI⊖F→0,1yI3−y3αI,βFβF3−βI3+βF3yF3limI⊖F→0,1y3αI,βF−yF3βF3−βI3+11/3=−βFyF∂yαF,βF∂βF−βFyF∂yF∂αFcos−1θ,F+11/3=1−βFyF∂yF∂βF+∂yF∂αFβFyFcos−1θ,F1/3(28)

Thus, according to Equation (21), the above deduction led to

limI⊖F→0,1I,limI⊖F→0,1II=1−αF3hF21−hF3αF2⋅∂hF∂αF1/3,1−βFyF∂yF∂βF1/3,

which was equal to Equation (22).

Theorem 5.3.

Equation (20) was necessary to guarantee that dφ(F)dF|F was still an FFF.

Theorem 5.4.

We found that functions h and y were both single-variable dependent.

Theorem 5.5.

In terms of Theorem 3.2, I⊖F implies that αF≤αI and βF≥βI, therefore,

cosθ,F:=limI⊖F→0,1βI−βFαI−αF≤0.(29)

Definition 5.2.

The FFF φF=αF,βF admits derivative for all F∈F. Particularly, dφFdF=1,0.

For proof, according to Equation (17), we obtained

hF=αF andyF=βF.

For all F∈F,

1−αF31−hF3hF2αF2⋅∂hF∂αF=1−αF31−αF3=1, βFyF∂yF∂βF=1.

Therefore, Equations (20) and (21) were guaranteed.

Furthermore, from Equation (22), we obtained

dφFdF=1,0.(30)

The basic operations of the derivatives of FFFs are discussed in the following theorems.

Theorem 5.6.

Assuming the existence of FFF derivatives, then,

φ1F=h1αF,βF,y1αF,βF=h1,y1

and

φ2F=h2αF,βF,y2αF,βF=h2,y2

are derivable.

Then,

dφ1F⊕φ2FdF=1−αF3h121−h23αF2⋅∂h1∂αF+1−αF3h221−h23αF2⋅∂h2∂αF1/3,1−βFy1∂y1∂βF−βFy2∂y2∂βF1/3,(31)

dφ1F⊖φ2FdF=1−αF3h121−h23αF2⋅∂h1∂αF−1−αF3h221−h23αF2⋅∂h2∂αF1/3,1−βFy1∂y1∂βF+βFy2∂y2∂βF1/3,(32)

dφ1F⊗φ2FdF=1−αF3h12h23∂h1+h13h22∂h21−h13h23αF2∂αF1/3,1−βFy121−y23dy1+y221−y13dy2h13+h23−h13h23∂βF1/3, and(33)

=dφ1F⊖φ2FdF=1−αF3h12h23−h13αF2⋅h2∂h1−h1∂h2∂αF1/3,1−βFy121−y23dy1−y221−y13dy2h13−h231−h23∂βF1/3.(34)

For proof, regarding Equation (31),

Since dφ1F⊕φ2FdF=ddFh13+h23−h13h231/3,y1y2, then we obtained

d(h13+h23−h13h23)1/3=13(h13+h23−h13h23)1/2d(h13+h23−h13h23)(1−h23)h12dh1+(1−h3)h22dh2,

dy1y2=y2dy1+y1dy2,

and

1−h13+h23−h13h231/33=1−h131−h23.

According to Equation (22), we derived

dφ1F⊕φ2FdF=1−αF3h121−h23αF2⋅∂h1∂αF+1−αF3h221−h23αF2⋅∂h2∂αF1/3,1−βFy1∂y1∂βF−βFy2∂y2∂βF1/3.

This is Equation (31).

For Equation (22),

Since dφ1F⊖φ2FdF=ddFh13−h231−h231/3,y1y2, then we obtained

dh13−h231−h231/3=13h13−h231−h231/2dh13−h231−h23=h13−h231−h231/2h121−h23dh1−h221−h13dh21−h232,

dy1y2=y2dy1−y1dy2y22,

and

1−h13−h231−h231/33=1−h231−h13.

According to Equation (22), we obtained

dφ1F⊖φ2FdF=1−αF3h121−h23αF2⋅∂h1∂αF−1−αF3h221−h23αF2⋅∂h2∂αF1/3,1−βFy1∂y1∂βF+βFy2∂y2∂βF1/3.

This is Equation (32).

The proof for Equations (33) and (34) was similar to the proof for Equations (31) and (32), so was omitted from the explanation.

As a direct consequence, we obtained

dφ1F⊕CdF=ddFφ1F⊖C=ddFφ1F,

where C is a constant FFF.

In terms of Chain's law of derivatives, we obtained the following derivative operations for compound FFFs φϕF and ϕφF.

Theorem 5.7.

Let φϕF and ϕφF be two FFF compounds. Assuming that both the FFFs φF and ϕF exist as derivatives, we obtained

dφϕFdF=dφdϕ⊗dϕdF and dϕφFdF=dϕdφ⊗dφdF.

As the first instance of proof,

dφϕFdF=limI⊖F→0,1φϕF⊕ΔF⊖φϕFϕF⊕ΔF⊖ϕF⊗ϕF⊕ΔF⊖ϕFΔF=limI⊖F→0,1φϕF⊕ΔF⊖φϕFϕF⊕ΔF⊖ϕF⊗limI⊖F→0,1ϕF⊕ΔF⊖ϕFΔF=dφϕFdϕF⊗dϕFdF.

As the second instance of proof, let φϕF=hφhϕα,yφyϕβ=Hα,Yβ be an FFF compound, therefore, based on Equation (17), we obtained

dφϕFdF=1−α3H2α1−α3αα2⋅dHdα1/3,1−βYβ⋅dYdβ1/3=1−α3hφhϕα21−hφhϕα3α2⋅dhφhϕαdα1/3,1−βyφyϕβ⋅dyφyϕβdβ1/31−α3hφhϕα21−hφhϕα3α2⋅dhφhϕαdα1/3=1−hϕα3hφhϕα21−hφhϕα3hϕα2⋅dhφhϕαdhϕα⋅1−α3hϕα21−hϕα3α2⋅dhϕαdα1/3,1−yϕβyφyϕβ⋅dyφyϕβdβ⋅βyϕβ⋅dyϕβdβ1/3.1−hϕα3hφhϕα21−hφhϕα3hϕα2⋅dhφhϕαdhϕα⋅1−α3hϕα21−hϕα3α2⋅dhϕαdα1/3

Since,

1−hϕα3hφhϕα21−hφhϕα3hϕα2⋅dhφhϕαdhϕα⋅ 1−α3hϕα21−hϕα3α2⋅dhϕαdα1/3=1−hϕα3hφhϕα21−hφhϕα3hϕα2⋅dhφhϕαdhϕα1/3 ⋅1−α3hϕα21−hϕα3α2⋅dhϕαdα1/3

and

1−yϕβyφyϕβ⋅dyφyϕβdβ⋅βyϕβ⋅dyϕβdβ1/3=1−βyϕβ⋅dyϕβdβ1/33+1−yϕβyφyϕβ⋅dyφyϕβdβ1/33⋅−1−βyϕβ⋅dyϕβdβ1/33⋅1−yϕβyφyϕβ⋅dyφyϕβdβ1/331/3,

the proof of dφϕFdF=dφdϕ⊗dϕdF was completed.

6. FFF DIFFERENTIALS

In mathematics, a differential operator is an operator defined as a function of the differentiation operation. Differential operators encompass an effective method to estimate function changes based on the proportion of variable variation, which can be regarded as an effective way to linearly approximate nonlinearity problems on the condition that the changes are appropriate. Besides, the FFF differential can be used to estimate the approximate value. For example, decision-makers want to change the value or add some new values after calculation, but the workload of recalculation is too large in some decision-making environments to obtain the accurate results. For this situation, we can use differentiation to estimate the approximate value.

In this section, the intrinsic properties of differential operators in Fermatean fuzzy environment are studied, and the theories of FFFs are discussed. The definition of an FFF differential operator is first discussed, followed by the necessary assumptions needed to ensure an FFF is differentiable. We further propose a differential formula usually used in a variety of applications and numerical calculations.

Definition 6.1.

Let F⊕ΔF∈F⊕φF exist with F∈F, and let Δφ=φF⊕ΔF⊖φF be the difference in φ in light of ΔF. If there an FFN exists, i.e., F1,F2, which depends only on F, as denoted by

Δφ=F1+οαF⊕ΔF−αF,F2+οαF⊕ΔF−αF≈F1,F2,(35)

with οαF⊕ΔF−αF and οαF⊕ΔF−αF satisfying

limΔα→0,1οαF⊕ΔF−αFαF⊕ΔF−αF=0(36)

and

limΔα→0,1οβF⊕ΔF−βFβF⊕ΔF−βF=0,(37)

then the FFF φ is considered to be differential at F, as denoted by

dφ=F1,F2.

Theorem 6.1.

Let F⊕ΔF∈F⊕φF exist with F∈F, assuming the hypotheses in Theorem 5.6 hold true, then FFN φ is differentiable. In particular,

dφ=(F1,F2)=dφdF|F⊗ΔF(38)

As proof, according to Equation (6), we obtained

ΔF=αF⊕ΔF3−αF31−βF31/3,βF⊕ΔFβF.(39)

If the hypotheses in Theorem 5.6 hold true, namely, dφdF|F exists, then in terms of Theorem 5.6 and Equation (39), we obtained

dφdF|F⊗ΔF=1−αF3hF21−hF3αF2⋅∂hF∂αF1/3,1−βFyF∂yF∂βF1/3⊕ΔF=αF⊕ΔF3−αF3hF21−hF3αF2⋅∂hF∂αF1/3,1−βF⊕ΔF3−βF3yFβF2∂yF∂βF1/3.(40)

Based on Definition 3.2, we deduced that

φF⊕ΔF⊖φF=hF⊕ΔF3−hF31−hF31/3,yF⊕ΔFyF.(41)

Next, we compared Equations (40) with (41) to identify whether Definition 6.1 holds true. We explore Taylor's expansion theorem to handle the membership degree, as follows:

hF⊕ΔF3−hF31−hF3=∂hF3∂αFαF⊕ΔF−αF+οαF⊕ΔF−αF1−hF3=αF⊕ΔF3−αF3hF21−hF3αF2⋅∂hF∂αF+∂hF3∂αFοαF⊕ΔF−αF−αF⊕ΔF3−αF33αF2+οαF⊕ΔF−αF1−hF3=αF⊕ΔF3−αF3hF21−hF3αF2⋅∂hF∂αF+οαF⊕ΔF−αF,(42)

where οαF⊕ΔF−αF differed from line to line and satisfied Equation (36).

In a similar way, we derived

yF⊕ΔFhF=1−βF⊕ΔF3−βF3βF3∂yF∂βF+οβF⊕ΔF−βF.(43)

Based on Equations (39–43), we obtained

φF⊕ΔF⊖φF=hF⊕ΔF3−hF31−hF31/3,yF⊕ΔFyF=αF3−αF3hF21−hF3αF2⋅∂hF∂αF+οαF⊕ΔF−αF1/3, 1−βF⊕ΔF3−βF3yFβF2∂yF∂βF1/3=αF⊕ΔF3−αF3hF21−hF3αF2⋅∂hF∂αF1/3, 1−βF⊕ΔF3−βF3yFβF2∂yF∂βF+οβF⊕ΔF−βF1/3≈αF⊕ΔF3−αF3hF21−hF3αF2⋅∂hF∂αF1/3,1−βF⊕ΔF3−βF3yFβF2∂yF∂βF1/3=dφdF|F⊗ΔF.

Therefore, Equation (41) was regarded as Equation (40), i.e.,

Δφ=φF⊕ΔF⊖φF≈dφdF|F⊗ΔF:=F1,F2.(44)

By combining all of the above, including Equations (36) and (37) φ was deduced to be differentiable at α.

Theorem 6.2.

If φF is differentiable, then dφdF|F exists.

Based on [44], we proposed an approach approximating a nonlinear FFF in the following corollary:

Corollary 6.1.

Let F⊕ΔF∈F⊕φF and F∈F. If FFF φ was differentiable, then,

φF⊕ΔF≈φF⊕dφdF|F⊗αF⊕ΔF3−αF31−hF31/3,βF⊕ΔFβF.(45)

As proof, based on Definition 3.2, Equations (35), (38), and (40), we obtained

φF⊕ΔF=φF⊕φF⊕ΔF⊖φF≈φF⊕dφ(F)dF|F⊗ΔF=φF⊕dφ(F)dF|F⊗αF⊕ΔF3−αF31−αF31/3,βF⊕ΔFβF,,

which was equal to Equation (45). Therefore, the proof of Corollary 6.1 was completed.

Corollary 6.2.

If the FFF φh,y satisfied

∂2h∂α2=0=∂2y∂β2,(46)

then

φF⊕ΔF⊖φF=dφ(F)dF|F⊗αF⊕ΔF3−αF31−αF31/3,βF⊕ΔFβF.(47)

The proof of Corollary 6.2 was similar to the proof of Theorem 6.1, so was omitted from explanation.

When comparing Equation (35) with Equation (47), the sign “≈” in Equation (35) was replaced with “=” in Equation (47), on the condition that Equation (47) is true. Therefore, the differential dφ represents the value of the difference Δφ on the condition that FFF φF is linear.

7. NUMERICAL EXAMPLES AND APPLICATIONS

In order to illustrate the validity and accuracy of the differential approximate calculation formula in this paper, some numerical examples of the approximation of nonlinear FFF with Fermatean fuzzy continuous information is provided as follows:

Example 7.1.

Let ΔF=0.1,0.9,λ=0.2 and the form of FFF φF=hF,yF be shown as

φF=1−1−αF30.21/3,βF0.2.

Based on Definition 2.5 and Theorem 3.4, we derived

ΔφF=φF⊕ΔF⊖φF=0.2F⊕ΔF⊖0.2F=Δh,Δy=0.0585,0.9816.(48)

Based on Theorem 5.6 and Theorem 6.1, we obtained

dφ=dφdF⊗ΔF=1−αF31−αF30.2d1−1−αF30.2dαF31/3,1−βFβF0.2dβF0.2dβF1/3⊗ΔF=0.21/3,0.81/3⊗ΔF=0.0585,0.9791=dh,dy.(49)

According to the result of Equation (49), we can find the validity of the differential approximate calculation formula, then we can further conduct its application as follows:

Example 7.2.

Five FFN decision values were derived using the Fermatean method from five experts, i.e.,

F1=(0.6,0.46),F2=(0.93,0.52),F3=(0.92,0.46)F4=(0.65,0.43),F4=(0.72,0.82)

where the weight of their evaluation is ω=(0.25,0.2,0.15,0.25,0.15)T. According to Definition 2.5 and Fermatean fuzzy weighted average (FFWA) operator in [32], the aggregation value is

φF=FFWAF1,F2,F3,F4,F5=1−∏i=151−αi3wi1/3,∏i=15βiwi=0.8113,0.5055.(50)

In case some experts wanted to change their preference, we assumed that first expert wanted to change the value of F1 and give a new assessment, i.e., F˜1=0.79,0.42, then we can obtained

ΔF1=0.7070,0.9130,

and the aggregation value is

φF˜=FFWAF˜1,F2,F3,F4,F5=1−∏i=151−αi3wi1/3,∏i=15βiwi=0.8350,0.4942.(51)

For simplicity, based on Theorem 5.6, and Theorem 6.1, we assumed F˜1∈F⊕F1 and used Corollary 6.1, thereby obtaining the following approximated aggregation:

φF⊕ΔF=φF⊕φF⊕ΔF⊖φF≈φF⊕dφ(F1)dF1|F1⊗ΔF1=φF⊕dφ(F1)dF1|F1⊗F1,F˜1=0.8113,0.5055⊕0.4454,0.9797=0.8316,0.4952.(52)

In conclusion, there were tiny differences observed between the results of Equations (51) and (52). Hence, the differential approximate calculation formulae regarding FFFs were effective and feasible.

If the first expert changes the value of F1 and gives an extreme value, i.e., F1′=1,0, then the aggregation value is

φF′=FFWAF′1,F2,F3,F4,F5=1−∏i=151−αi3wi1/3,∏i=15βiwi=1,0.(53)

For simplicity, based on Theorem 5.6, and Theorem 6.1, we assumed F′1∈F⊕F1 and used Corollary 6.1 to obtain the following approximated aggregation:

φF⊕ΔF=φF⊕φF⊕ΔF⊖φF≈φF⊕dφ(F1)dF1|F1⊗ΔF1=φF⊕dφ(F1)dF1|F1⊗F1,F′1=0.8113,0.5055⊕0,0.75=0.8113,0.3791(54)

We find that there is a big difference between Equations (53) and (54). The reason is the FFWA operator cannot handle extreme cases. So, we further discuss the extreme value situation through the q-ROF interaction Maclaurin symmetry mean (q-ROFIWMSM) operator in [47].

According to the q-ROFIWMSM operator, we obtain the following aggregation value:

φF=q−ROFIWMSMF1,F2,F3,F4,F50.7387,0.8829.

If the first expert changes the value of F1 and gives an extreme value, i.e., F1′=1,0, then the aggregation value is

φF′=q−ROFIWMMF1′,F2,F3,F4,F5=0.8781,0.7204.(55)

For simplicity, based on Theorem 5.6, and Theorem 6.1, we assumed F1′∈F⊕F1 and used Corollary 6.1 to get the following approximated aggregation:

φF⊕ΔF=φF⊕φF⊕ΔF⊖φF≈φF⊕dφ(F1)dF1|F1⊗ΔF1=φF⊕dφ(F1)dF1|F1⊗F1,F1′=0.8699,0.7268.(56)

Obviously, there are little differences between Equations (55) and (56). Therefore, the differential approximate calculation formulae regarding FFFs are effective and feasible.

8. CONCLUSION

Considering the continuous Fermatean fuzzy information, we discussed the continuities, derivatives, and differentials of FFF. In it, we firstly proposed the subtraction and division operations between FFNs and discussed their properties, which laid the foundation for further discuss the derivatives and differentials. Then, we defined the concept of FFF and investigated its continuity that plays an important role in calculus. Moreover, we defined the derivatives and differentials of FFFs and examined the algebraic and compound operations of the derivatives of FFFs. Finally, we conducted some examples to verify the feasibility and effectiveness of the approximate calculation on FFFs. Compared to the existing studies, our method is nonlinear with its application in a more expansive range. Besides, the proposed method can address both continuous and discrete information.

In further studies, the elastic coefficients of FFFs and their relationships with the derivatives can be investigated. The inverse operations of the derivatives of the FFFs, such as indefinite integral and definite integral derivatives, can be further examined. Apart from those, the FFF-based decision-making method can be applied to complex uncertainty information decision-making environments, such as the selection of business partners in the supply chain, the prediction of traffic conditions, etc [48–50].

CONFLICTS OF INTEREST

We declare that there are no conflicts of interest regarding the publication of this paper.

AUTHORS' CONTRIBUTIONS

Zaoli Yang: Conceptualization, Funding acquisition, Supervision, Software, Resources, Data curation, Supervision, Writing - review & editing; Harish Garg: Conceptualization, Data curation, Supervision, Software, Resources, Writing - review & editing; Xin Li: Formal analysis, Investigation, Writing - review & editing.

ACKNOWLEDGMENTS

This work was supported in part by the Natural Science Foundation of China (No. 71704007), the Beijing Social Science Foundation of China (No. 18GLC082).

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 282 - 294
Publication Date: 2020/12/22
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.201215.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Zaoli Yang
AU  - Harish Garg
AU  - Xin Li
PY  - 2020
DA  - 2020/12/22
TI  - Differential Calculus of Fermatean Fuzzy Functions: Continuities, Derivatives, and Differentials
JO  - International Journal of Computational Intelligence Systems
SP  - 282
EP  - 294
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.201215.001
DO  - 10.2991/ijcis.d.201215.001
ID  - Yang2020
ER  -

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