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. 2019 Jun 26;21(7):628. doi: 10.3390/e21070628

Spherical Fuzzy Logarithmic Aggregation Operators Based on Entropy and Their Application in Decision Support Systems

Yun Jin 1, Shahzaib Ashraf 2, Saleem Abdullah 2,*
PMCID: PMC7515119  PMID: 33267343

Abstract

Keeping in view the importance of new defined and well growing spherical fuzzy sets, in this study, we proposed a novel method to handle the spherical fuzzy multi-criteria group decision-making (MCGDM) problems. Firstly, we presented some novel logarithmic operations of spherical fuzzy sets (SFSs). Then, we proposed series of novel logarithmic operators, namely spherical fuzzy weighted average operators and spherical fuzzy weighted geometric operators. We proposed the spherical fuzzy entropy to find the unknown weights information of the criteria. We study some of its desirable properties such as idempotency, boundary and monotonicity in detail. Finally, the detailed steps for the spherical fuzzy decision-making problems were developed, and a practical case was given to check the created approach and to illustrate its validity and superiority. Besides this, a systematic comparison analysis with other existent methods is conducted to reveal the advantages of our proposed method. Results indicate that the proposed method is suitable and effective for the decision process to evaluate their best alternative.

Keywords: spherical fuzzy sets, logarithmic spherical operational laws, logarithmic spherical aggregation operators, entropy, multi-criteria group decision making (MCGDM) problems

1. Introduction

The complication of a system is growing every day in real life and getting the finest option from the set of possible ones is difficult for the decision makers. To attain a single objective is difficult to summarize but not incredible. Many organizations found difficulties with setting motivations, goals and opinions’ complications. Thus, organizational decisions simultaneously include numerous objectives, whether they regard individuals or committees. This reflection suggests, according to criteria solved optionally, restricting each decision maker to attain an ideal solution-optimum under each criterion involved in practical problems. Consequently, the decision maker is more concentrated to establish more applicable and reliable techniques to find the best options.

To handle the ambiguity and uncertainty data in decision-making problems, the classical or crisp methods cannot be always effective. Thus, dealing with such uncertain situations, Zadeh [1] in 1965 presented the idea of the fuzzy set. Zadeh assigns membership grades to elements of a set in the interval [0,1] by offering the idea of fuzzy sets (FSs). Zadeh’s work in this direction is remarkable as many of the set theoretic properties of crisp cases were defined for fuzzy sets. Fuzzy set theory got the attention of researchers and found its applications in decision science [2], artificial intelligence [3], and medical diagnosis [4], and its enormous applications are discussed in [5].

After many applications of fuzzy set theory, Atanassov observed that there are many shortcomings in this theory and introduced the notion of intuitionistic fuzzy sets [6] to generalize the concept of Zadeh’s fuzzy set. In intuitionistic fuzzy sets, each element is expressed by an ordered pair, and each pair is characterized by membership and non-membership grades on the condition that the sum of their grades are less than or equal to 1. During the last few decades, the intuitionistic fuzzy sets (IFSs) are fruitful and broadly utilized by researchers to grasp the ambiguity and imprecision data. To cumulate all the executive of criteria for alternatives, aggregation operators play a vital role throughout the information merging procedure. Xu [7] presented a weighted averaging operator while Xu and Yager [8] developed a geometric aggregation operator for aggregating the different intuitionistic fuzzy numbers. Verma [9] in 2015 proposed the generalized Bonferroni mean operator and, in [10], Verma and Sharma proposed the measure of inaccuracy using intuitionistic fuzzy information. Deschrijver [11] developed the IFS representation of t-norms and t-conorms. In some decision theories, the decision makers deal with the situation of particular attributes where values of their summation of membership degrees exceeds 1. In such conditions, IFS has no ability to obtain any satisfactory result. To overcome this situation, Yager [12] developed the idea of a Pythagorean fuzzy set (PyFS) as a generalization of IFS, which satisfies the fact that the value of square summation of its membership degrees is less then or equal to 1. Clearly, PyFS is more flexible than IFS to deal with the imprecision and ambiguity in the practical multi-criteria decision-making (MCDM) problems. Zhang and Xu [13] established an extension of TOPSIS to MCDM with PyFS information. The error for the proof of distance measure in Zhang and Xu [13] has been pointed out by Yang et al. [14]. For MCDM problems in a Pythagorean fuzzy environment, Yager and Abbasov [15] developed a series of aggregation operators. Peng and Yang [16] explained their relationship among these aggregation operators and established the superiority and inferiority ranking (SIR) for the multi-criteria group decision-making (MCGDM) method. Using Einstein operation, Garg [17] generalized Pythagorean fuzzy information aggregation. Gou et al. [18] studied many Pythagorean fuzzy functions and investigated their fundamental properties such as continuity, derivatively, and differentiability in detail. Zhang [19] put forward a ranked qualitative flexible (QUALIFLEX) multi-criteria approach with the closeness index-based ranking methods for multi-criteria Pythagorean fuzzy decision analysis. Zeng et al. [20] explored a hybrid method for Pythagorean fuzzy MCDM. Zeng [21], applied the Pythagorean fuzzy probabilistic OWA (PFPOWA) operator for MAGDM problems. For more study, we refer to [22,23,24].

IFS theory and PyFS theory have been successfully applied in different areas, but there are situations that cannot be represented by it in real life, such as voting, we may face human opinions involving more answers of the type: yes, abstain, no and refusal (for example, in a democratic election station, the council issues’ 500 voting papers for a candidate. The voting results are divided into four groups accompanied with the number of papers that are vote for (251), abstain (99), vote against (120) and refusal of voting (30). Here, group abstain means that the voting paper is a white paper rejecting both agree and disagree for the candidate but still takes the vote, group refusal of voting is either invalid voting papers or did not take the vote. The candidate is successful because the number of support papers is over 50% (i.e., 250).

However, at least five people said later on in their blogs that they support the candidate in the last moment because they find that the support number seems larger than the against number. Such kind of examples (in which the number of abstains is a key factor and the group refusal of voting indeed exists) happened in reality and IFS and PyFS could not handle it). Thus, Cuong [25] proposed a new notion named picture fuzzy sets, which is an extension of fuzzy sets and intuitionistic fuzzy sets. Picture fuzzy sets give three membership degrees of an element named the positive membership degree, the neutral membership degree, and the negative membership degree, respectively. The picture fuzzy set solved the voting problem successfully, and is applied to clustering [26], fuzzy inference [27], and decision-making [28,29,30,31,32,33,34,35].

The neutrosophic set is another important generalizations of the classic set, fuzzy set, intuitionistic fuzzy set and picture fuzzy set to deal with uncertainties in decision-making problems. Many authors contributed in the decision-making theory using neutrosophic information. Ashraf [36] proposed the logarithmic hybrid aggregation operators for single value neutrosophic sets. Dragan et al. [37] proposed the novel approach for the selection of power generation technology using the combinative distance-based assessment (CODAS) method. Many decision-making approaches like [38,39] make important contributions using neutrosophic information.

The picture fuzzy set becomes more famous by introducing various kinds of aggregation operators. However, it has a shortcoming in that it is only valid for the environment whose sum of degrees is less than or equal to one. However, in day-to-day life, there are many situations where this condition is ruled out. For instance, if a person giving their preference in the form of positive, neutral and negative membership degrees towards a particular object is 0.7, 0.3 and 0.5, then clearly this situation is not handling with picture fuzzy set. In order to resolve it, Ashraf [40] proposed the notion of the spherical fuzzy set. For instance, corresponding to the above-considered example, we see that (0.7)2+(0.3)2+0.52=0.94 and hence a spherical fuzzy set (SFS) is an extension of the existing extensions of fuzzy set theories. In the spherical fuzzy set, each element can be written in the form of triplet component, and each pair is categorized by a positive membership degree, a neutral membership degree and a negative membership degree such that the sum of their square is less than or equal to one. Ashraf [41] proposed some series of spherical aggregation operators using t-norm and t-conorm and gave its applications to show the effectiveness of proposed operators. Ashraf in [42] proposed the GRA method based on a spherical linguistic fuzzy Choquet integral environment and gave its application. For more study, we refer to [43,44,45].

The logarithmic operations being good alternatives, compared with the algebraic operations, have the potential to offer similar smooth estimations as the algebraic operations. However, there is little investigation on logarithmic operations on the IFSs and PyFSs. Motivated by these ideas, we develop a spherical fuzzy MCDM method based on the logarithmic aggregation operators, with the logarithmic operations of the spherical fuzzy sets (SFSs) handling spherical fuzzy MCDM within SFSs.

Thus, the goal of this article is to propose the decision-making method for MCDM problems in which there exist the interrelationships among the criteria. The contributions of this study are:

  • We develop some novel logarithmic operations for spherical fuzzy sets, which can overcome the weaknesses of algebraic operations and capture the relationship between various SFSs.

  • We extend logarithmic operators to logarithmic spherical fuzzy operators, namely logarithmic spherical fuzzy weighted averaging (L-SFWA), logarithmic spherical fuzzy ordered weighted averaging (L-SFOWA), logarithmic spherical fuzzy hybrid weighted averaging (L-SFHWA), logarithmic spherical fuzzy weighted geometric (L-SFWG), logarithmic spherical fuzzy ordered weighted geometric (L-SFOWG) and logarithmic spherical fuzzy hybrid weighted geometric (L-SFHWG) to SFSs, which can overcome the algebraic operators’ drawbacks.

  • We develop the spherical fuzzy entropy for spherical fuzzy information, which can help to find the unknown weights information of the criteria.

  • We develop an algorithm to deal with multi-attribute decision-making problems using spherical fuzzy information.

  • To show the effectiveness and reliability of the proposed spherical fuzzy logarithmic aggregation operators, the application of the proposed operator in emerging technology enterprises is developed.

  • Results indicate that the proposed technique is more effective and gives more accurate output as compared to existing studies.

In order to attain the research goal that has been stated above, the organization of this article is offered as: Section 2 concentrates on some basic notions and operations of existing extensions of fuzzy set theories and also some discussion to propose the spherical fuzzy entropy. Section 3 presents some novel logarithmic operational laws of SFSs. Section 4 defines the logarithmic aggregation operators for SFNs and discusses its properties. Section 5 presents an approach for handling the spherical fuzzy MCDM problem based on the proposed logarithmic operators. Section 5.1 uses an application case to verify the novel method and Section 5.2 presents the comparison study about algebraic and logarithmic aggregation operators. Section 6 concludes the study.

2. Preliminaries

The concepts and basic operations of existing extensions of fuzzy sets are recalled in this section, and they are the foundation of this study.

Definition 1

([27]). A mapping T^:Θ×ΘΘ is said to be triangular-norm if each element T^ satisfies that:

(1) T^ is commutative, monotonic and associative,

(2) T^v*,1=v*, each v*T^,

where Θ=[0,1] is the unite interval.

Definition 2

([27]). A mapping S^:Θ×ΘΘ is said to be triangular-conorm if each element S^ satisfies that

(1) S^ is commutative, monotonic and associative,

(2) S^v*,0=v*, each v*S^,

where Θ=[0,1] is the unite interval.

As different norms are the curial elements for proposing aggregation operators in fuzzy set theory, here we enlist some basic norms operations for fuzzy sets in Figure 1.

Figure 1.

Figure 1

Basic Norms Operations.

Now, we enlist different types of norms with its generators too in Figure 2 and Figure 3.

Figure 2.

Figure 2

T-norm with its Generators.

Figure 3.

Figure 3

T-conorm with its Generators.

Definition 3

([15]). For a set ℜ, by a Pythagorean fuzzy set in ℜ, we mean a structure

ε=Pσrˇγ,Nσrˇγ|rˇγ,

in which Pσ:Θ and Nσ:Θ indicate that the positive and negative grades in ,Θ=0,1 are the unit intervals. In addition, the following condition satisfied by ρσ and Nσ is 0Pσ2rˇγ+Nσ2rˇγ1; for all rˇ. Then, ε is said to be a Pythagorean fuzzy set in ℜ.

Definition 4

([25]). For a set ℜ, by a picture fuzzy set in ℜ, we mean a structure

ε=Pσrˇγ,Iσrˇγ,Nσrˇγ|rˇγ,

in which Pσ:Θ,Iσ:Θ and Nσ:Θ indicate that the positive, neutral and negative grades in ℜ, Θ=0,1 are the unit intervals. In addition, the following condition satisfied by Pσ,Iσ and Nσ is 0Pσrˇγ+Iσrˇγ+Nσrˇγ1, for all rˇγ. Then, ε is said to be a picture fuzzy set in ℜ.

Definition 5

([40]). For a set ℜ, by a spherical fuzzy set in ℜ, we mean a structure

ε=Pσrˇγ,Iσrˇγ,Nσrˇγ|rˇγ,

in which Pσ:Θ,Iσ:Θ and Nσ:Θ indicate that the positive, neutral and negative grades in ℜ, Θ=0,1 are the unit intervals. In addition, the following condition satisfies by Pσ,Iσ and Nσ is 0Pσ2rˇγ+Iσ2rˇγ+Nσ2rˇγ1, for all rˇγ. Then, ε is said to be a spherical fuzzy set in ℜ.

χσrˇγ=1Pσ2rˇγ+Iσ2rˇγ+Nσ2rˇγ is said to be a refusal degree of rˇγ in ℜ, for SFS Pσrˇγ,Iσrˇγ,Nσrˇγ|rˇγ, which is triple components Pσrˇγ,Iσrˇγ,andNσrˇγ is said to SFN denoted by e=Pe,Ie,Ne, where Pe,Ie and Ne[0,1], with the condition that: 0Pe2+Ie2+Ne21.

Ashraf and Abdullah [40] proposed the basic operations of spherical fuzzy set as follows:

Definition 6.

For any two SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ and εq=Pξqrˇγ,Iξqrˇγ,Nξqrˇγ in ℜ. The union, intersection and compliment are proposed as:

(1) ερεqiffrˇγ,PξρrˇγPξqrˇγ,IξρrˇγIξqrˇγ and NξρrˇγNξqrˇγ;

(2) ερ=εqiffερεq and εqερ;

(3) ερεq=maxPξρ,Pξq,minIξρ,Iξq,minNξρ,Nξq;

(4) ερεq=minPξρ,Pξq,minIξρ,Iξq,maxNξρ,Nξq;

(5) ερ=Nξρ,Iξρ,Pξρ.

Definition 7.

For any two SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ and εq=Pξqrˇγ,Iξqrˇγ,Nξqrˇγ in ℜ and β0; then, the operations of SFNs are proposed as

(1) ερεq=Pξρ2+Pξq2Pξρ2·Pξq2,Iξρ·Iξq,Nξρ·Nξq;

(2) β·ερ=1(1Pξρ2)β,(Iξρ)β,(Nξρ)β;

(3) ερεq=Pξρ·Pξq,Iξρ·Iξq,Nξρ2+Nξq2Nξρ2·Nξq2;

(4) ερβ=(Pξρ)β,(Iξρ)β,1(1Nξρ2)β.

(5) βερ=β1ρξρ2,1β2Iξρ,1β2Nξρifβ0,1,1β1ρξρ2,11β2Iξρ,11β2Nξρifβ1.

Ashraf and Abdullah [40] introduced some properties based on Definition 7 as follows:

Definition 8.

For any three SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ, εq=Pξqrˇγ,Iξqrˇγ,Nξqrˇγ and εl=Pσlrˇγ,Iσlrˇγ,Nσlrˇγ in ℜ and β1,β20. Then,

(1) ερεq=εqερ;

(2) ερεq=εqερ;

(3) β1(ερεq)=β1ερβ1εq,β1>0;

(4) (ερεq)β1=ερεq,β1>0;

(5) β1ερβ2ερ=(β1+β2)ερ,β1>0,β2>0;

(6) ερερ=ερ,β1>0,β2>0;

(7) (ερεq)εl=ερ(εqεl);

(8) (ερεq)εl=ερ(εqεl).

Definition 9.

For any SFN, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ in ℜ. Then, score and accuracy values are defined as

(1) S˜(ερ)=13(2+PξρIξρNξρ)0,1

(2) A˜(ερ)=PξρNξρ0,1.

The score and accuracy values defined above suggest which SFN is greater than other SFNs. The comparison technique is defined in the next definition.

Definition 10.

For any SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2 in ℜ. Then, the comparison technique is proposed as

(1) If S˜(ε1)<S˜(ε2), then ε1<ε2,

(2) If S˜(ε1)>S˜(ε2), then ε1>ε2,

(3) If S˜(ε1)=S˜(ε2), then

(a) A˜(ε1)<A˜(ε2), then ε1<ε2,

(b) A˜(ε1)>A˜(ε2), then ε1>ε2,

(c) A˜(ε1)=A˜(ε2), then ε1ε2.

Ashraf and Abdullah [40] proposed aggregation operators for SFNs based on different norms:

Definition 11.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ. The structure of the spherical weighted averaging (SFWA) operator is

SFWAε1,ε2,,εn=ρ=1nβρερ,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1.

Definition 12.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ. The structure of the spherical order weighted averaging (SFOWA) operator is

SFOWAε1,ε2,,εn=ρ=1nβρεη(ρ),

where βρρ=1,2,,n are weight vectors with βρ0, ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ) consequently by total order εη(1)εη(2)εη(n).

Definition 13.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ. The structure of the spherical hybrid weighted averaging (SFHWA) operator is

SFHWAε1,ε2,,εn=ρ=1nβρεη(ρ)*,

where βρρ=1,2,,n are weight vectors with βρ0, ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ)*εη(ρ)*=nβρεη(ρ),ρN consequently by total order εη(1)*εη(2)*εη(n)*. In addition, the associated weights are ω=(ω1,ω2,,ωn) with ωρ0, Σρ=1nωρ=1.

Definition 14.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ. The structure of spherical weighted geometric (SFWG) operator is

SFWGε1,ε2,,εn=ρ=1nερβρ,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1.

Definition 15.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ. The structure of the spherical order weighted geometric (SFOWG) operator is

SFOWGε1,ε2,,εn=ρ=1nεη(ρ)βρ,

where βρρ=1,2,,n are weight vectors with βρ0, ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ) consequently by total order εη(1)εη(2)εη(n).

Definition 16.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ. The structure of spherical hybrid weighted geometric (SFHWG) operator is

SFHWGε1,ε2,,εn=ρ=1nεη(ρ)*βρ,

where βρρ=1,2,,n are weight vectors with βρ0, ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ)*εη(ρ)*=nβρεη(ρ),ρN consequently by total order εη(1)*εη(2)*εη(n)*. In addition, associated weights are ω=(ω1,ω2,,ωn) with ωρ0, Σρ=1nωρ=1.

3. Entropy

Basically, we familiarize the concept of entropy, when probability measures the discrimination of criteria being imposed on multi-attribute decision-making problems. Non-probabilistic entropy firstly approximated by De Luca and Termini [46] also presented some necessities to find intuitive comprehension of the degree of fuzziness. Many researchers are getting interest in this field and have done a lot of work such as Scmidt and Kacprzyk [47] proposing some axioms for distance between intuitionistic fuzzy sets and non-probabilistic entropy measure for them. In this section, we recall the concept of Shannon entropy, fuzzy entropy, entropy for Pythagorean fyzzy numbers and propose the entropy for spherical fuzzy numbers.

Definition 17

([48]). Let δηρρ1,2,,n be the set of n-complete probability distributions. Shannon entropy for δηρρ1,2,,n probability distribution is defined as

Esδ=ρ=1nδηρlogδηρ.

Definition 18

([49]). Let F be any fuzzy set in ℜ, fuzzy entropy for the set F, we mean a structure

Fδ=1nρ=1nPηρlogPηρ+1Pηρlog1Pηρ.

Definition 19

([49]). Let F be any fuzzy set in ℜ, and, for Pythagorean fuzzy entropy for the set F, we mean a structure

Pyq=1+1nρ=1nPilogPi+NilogNiq=1n1+1nρ=1nPilogPi+NilogNi.

Definition 20.

Let F be any fuzzy set in ℜ, spherical fuzzy entropy for the set F, and we mean a structure

γq=1+1nρ=1nPilogPi+IilogIi+NilogNiq=1n1+1nρ=1nPilogPi+IilogIi+NilogNi.

4. Spherical Fuzzy Logarithmic Operational Laws

Motivated by the novel concept of spherical fuzzy set, we introduced some novel logarithmic operational laws for SFNs. As real number system ogσ0 is meaningless and ogσ1 is not defined therefore, in our study, we take nonempty spherical fuzzy sets and σ1, where σ is any real number.

Definition 21.

For any SFN, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ in ℜ. The logarithmic spherical fuzzy number is defined as

ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγ|rˇγ

in which Pσ:Θ,Iσ:Θ and Nσ:Θ indicate that the positive, neutral and negative grades in ℜ, Θ=0,1 are the unit intervals. In addition, the following condition satisfies by Pσ,Iσ and Nσ is 0Pσ2rˇγ+Iσ2rˇγ+Nσ2rˇγ1 for all rˇγ. Therefore, the membership grade is

1ogσPξρrˇγ2:Θ,suchthat01ogσPξρrˇγ21,rˇγ,

the neutral grade is

ogσ1Iξρ2rˇγ:Θ,suchthat0ogσ1Iξρ2rˇγ1,rˇγ,

and the negative grade is

ogσ1Nξρ2rˇγ:Θ,suchthat0ogσ1Nξρ2rˇγ1,rˇγ.

Therefore,

ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγ|rˇγ0<σminPξρ,1Iξρ2,1Nξρ21,σ1

is SPN.

Definition 22.

For any SFN, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ in ℜ. If

ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγ0<σminPξρ,1Iξρ2,1Nξρ2<1,1og1σPξρrˇγ2,og1σ1Iξρ2rˇγ,og1σ1Nξρ2rˇγ0<1σminPξρ,1Iξρ2,1Nξρ2<1,σ1,

then the function ogσερ is known to be a logarithmic operator for a spherical fuzzy set, and its value is called logarithmic SFN (L-SFN). Here, we take ogσ0=0,σ>0,σ1.

Theorem 1.

For any SFN, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ in ℜ, then ogσερ is also a spherical fuzzy number.

Proof. 

Since any SFN ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ in , which means that Pσ:Θ, Iσ:Θ and Nσ:Θ indicate that the positive, neutral and negative grades in , Θ=0,1 are the unit intervals. In addition, the following condition satisfied by Pσ,Iσ and Nσ is 0Pσ2rˇγ+Iσ2rˇγ+Nσ2rˇγ1. The following two cases happen.

Case-1 When 0<σminPξρ,1Iξρ2,1Nξρ2<1, σ1 and since ogσερ is a decreasing function w.r.t σ. Thus, 0ogσPξρ,ogσ1Iξρ2,ogσ1Nξρ21 and hence 01ogσPξρrˇγ21, 0ogσ1Iξρ2rˇγ1, 0ogσ1Nξρ2rˇγ1 and 01ogσPξρrˇγ2+ogσ1Iξρ2rˇγ+ogσ1Nξρ2rˇγ1. Therefore, ogσερ is SFN.

Case-2 When σ>1, 0<1σ<1 and 1σminPξρ,1Iξρ2,1Nξρ2; similar to the above, we can find that ogσερ is SFN. Thus, the procedure is eliminated here. □

Example 1.

Suppose that, for any SFN, ερ=0.8,0.5,0.3 in ℜ with σ=0.4, then

ogσερ=1og0.40.82,og0.410.52,og0.410.32=0.969,0.156,0.051.

In addition, if σ=8, then it follows:

og1σερ=1og180.82,og1810.52,og1810.32=0.994,0.069,0.022.

Now, we give some discussion on the basic properties of the L-SFN.

Theorem 2.

For any SFN, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ in ℜ. If 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1 then

(1) σogσερ=ερ;

(2) ogσσερ=ερ.

Proof. 

(1) According to Definitions 7 and 22, we obtain

σogσερ=σ11ogσρξρ22,1σ2ogσ1Iξρ2,1σ2ogσ1Nξρ2=σ11ogσρξρ2,11Iξρ2,11Nξρ2=σogσρξρ,Iξρ,Nξρ=ρξρ,Iξρ,Nξρ=ερ.

(2) According to Definition 22, we obtain

ogσσερ=ogσσ1ρξρ2,1σ2Iξρ,1σ2Nξρ=1ogσσ1ρξρ22,ogσ11σ2Iξρ2,ogσ11σ2Nξρ2=11ρξρ2,ogσ11σ2Iξρ,ogσ11σ2Nξρ=ρξρ,Iξρ,Nξρ=ερ.

Definition 23.

For any two L-SFNs, ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγ and ogσεq=1ogσPξqrˇγ2,ogσ1Iξq2rˇγ,ogσ1Nξq2rˇγ in ℜ and β0, then the logarithmic operations of L-SFNs are proposed:

(1) ogσερogσεq=1ogσPξρrˇγ2·ogσPξqrˇγ2,ogσ1Iξρ2rˇγ·ogσ1Iξq2rˇγ,ogσ1Nξρ2rˇγ·ogσ1Nξq2rˇγ;

(2) β·ogσερ=1ogσPξρrˇγ2β,ogσ1Iξρ2rˇγβ,ogσ1Nξρ2rˇγβ;

(3) ogσερogσεq=1ogσPξρrˇγ2·1ogσPξqrˇγ2,11ogσ1Iξρ2rˇγ2·1ogσ1Iξq2rˇγ2,11ogσ1Nξρ2rˇγ2·1ogσ1Nξq2rˇγ2;

(4) ogσερβ=1ogσPξρrˇγ2β,11ogσ1Iξρ2rˇγ2β,11ogσ1Iξρ2rˇγ2β.

Theorem 3.

For any two L-SFNs, ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγρ=1,2 in ℜ, with 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1. Then,

(1) ogσε1ogσε2=ogσε2ogσε1,

(2)ogσε1ogσε2=ogσε2ogσε1.

Proof. 

This is straightforward from Definition 23, so the procedure is eliminated here. □

Theorem 4.

For any two L-SFNs, ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγρ=1,2,3 in ℜ, with 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1. Then,

(1) ogσε1ogσε2ogσε3=ogσε1ogσε2ogσε3,

(2)ogσε1ogσε2ogσε3=ogσε1ogσε2ogσε3.

Proof. 

This is straightforward from Definition 23, so the procedure is eliminated here. □

Theorem 5.

For any two L-SFNs, ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγρ=1,2 in ℜ, with 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1,β,β1,β2>0 be any real numbers. Then,

(1) βogσε1ogσε2=βogσε1βogσε2;

(2) ogσε1ogσε2β=ogσε1βogσε2β;

(3) β1ogσε1β2ogσε1=β1+β2ogσε1;

(4) ogσε1β1ogσε1β2=ogσε1β1+β2;

(5) ogσε1β1β2=ogσε1β1β2.

Proof. 

(1) Since, from Definition 23, we have

ogσε1ogσε2=1ogσPξ12·ogσPξ22,ogσ1Iξ12·ogσ1Iξ22,ogσ1Nξ12·ogσ1Nξ22,

for any real number β>0, we obtain

βogσε1ogσε2=1ogσPξ12·ogσPξ22β,ogσ1Iξ12·ogσ1Iξ22β,ogσ1Nξ12·ogσ1Nξ22β=1ogσPξ12β,ogσ1Iξ12β,ogσ1Nξ12β1ogσPξ22β,ogσ1Iξ22β,ogσ1Nξ22β=βogσε1βogσε2.

(2) Since, from Definition 23, we have

ogσε1ogσε2=1ogσPξ12·1ogσPξ22,11ogσ1Iξ122·1ogσ1Iξ222,11ogσ1Nξ122·1ogσ1Nξ222,

for any real number β>0, we obtain

ogσε1ogσε2β=1ogσPξ12β·1ogσPξ22β11ogσ1Iξ122β·1ogσ1Iξ222β11ogσ1Nξ122β·1ogσ1Nξ222β=ogσε1βogσε2β;

(3) and (4) are similarly as above, so the procedure is eliminated here.

(5) Since, from Definition 23, we have

ogσε1β1β2=1ogσPξ12β111ogσ1Iξ122β111ogσ1Nξ122β1β2=1ogσPξ12β1β211ogσ1Iξ122β1β211ogσ1Nξ122β1β2=ogσε1β1β2,

this is therefore proved. □

Definition 24.

For any L-SFN, ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγ in ℜ. Then, score and accuracy values are defined as

(1) S˜(ogσερ)=1ogσPξρrˇγ2ogσ1Iξρ2rˇγ2ogσ1Nξρ2rˇγ2,

(2) A˜(ogσερ)=1ogσPξρrˇγ2+ogσ1Nξρ2rˇγ2.

The score and accuracy values defined above suggest which L-SFN is greater than other L-SFNs. The comparison technique is defined in the next definition.

Definition 25.

For any L-SFN, ogσερ=1ogσPξρrˇγ2,ogσ1Iξρ2rˇγ,ogσ1Nξρ2rˇγρ=1,2 in ℜ. Then, the comparison technique is proposed as

(1) If S˜(ogσε1)<S˜(ogσε2), then ogσε1<ogσε2,

(2) If S˜(ogσε1)>S˜(ogσε2), then ogσε1>ogσε2,

(3) If S˜(ogσε1)=S˜(ogσε2), then

(a) A˜(ogσε1)<A˜(ogσε2), then ogσε1<ogσε2,

(b) A˜(ogσε1)>A˜(ogσε2), then ogσε1>ogσε2,

(c) A˜(ogσε1)=A˜(ogσε2), then ogσε1ogσε2.

5. Logarithmic Aggregation Operators for L-SFNs

Now, we propose novel spherical fuzzy logarithmic aggregation operators for L-SFNs based on defined spherical fuzzy logarithmic operations laws as follows:

5.1. Logarithmic Averaging Operators

Definition 26.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. The structure of logarithmic spherical weighted averaging (L-SFWA) operator is

LSFWAε1,ε2,,εn=ρ=1nβρogσρερ,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1.

Theorem 6.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. Then, by using logarithmic operations and Definition 26, LSFWA is defined as

LSFWAε1,ε2,,εn,=1ρ=1nogσρPξρ2βρ,ρ=1nogσρ1Iξρ2βρ,ρ=1nogσρ1Nξρ2βρ0<σρminPξρ,1Iξρ2,1Nξρ2<1,1ρ=1nog1σρPξρ2βρ,ρ=1nog1σρ1Iξρ2βρ,ρ=1nog1σρ1Nξρ2βρ0<1σρminPξρ,1Iξρ2,1Nξρ2<1,σ1,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1.

Proof. 

Using mathematical induction to prove Theorem 6, we therefore proceed as follows:

(a) For n=2, since

β1ogσ1ε1=1ogσ1Pξ12β1,ogσ11Iξ12β1,ogσ11Nξ12β1

and

β2ogσ2ε2=1ogσ2Pξ22β2,ogσ21Iξ22β2,ogσ21Nξ22β2.

Then,

LSFWAε1,ε2=β1ogσ1ε1β2ogσ2ε2=1ogσ1Pξ12β1,ogσ11Iξ12β1,ogσ11Nξ12β11ogσ2Pξ22β2,ogσ21Iξ22β2,ogσ21Nξ22β2=1ogσ1Pξ12β1·ogσ2Pξ22β2,ogσ11Iξ12β1·ogσ21Iξ22β2,ogσ11Nξ12β1·ogσ21Nξ22β2=1ρ=12ogσρPξρ2βρ,ρ=1nogσρ1Iξρ2βρ,ρ=1nogσρ1Nξρ2βρ.

(b) Now, Theorem 6 is true for n=k,

LSFWAε1,ε2,,εk=1ρ=1kogσρPξρ2βρ,ρ=1kogσρ1Iξρ2βρ,ρ=1kogσρ1Nξρ2βρ.

(c) Now, we prove that Theorem 6 for n=k+1, which is

LSFWAε1,ε2,,εk,εk+1=ρ=1kβρogσρερ+βk+1ogσk+1εk+1
LSFWAε1,ε2,,εk,εk+1=1ρ=1kogσρPξρ2βρ,ρ=1kogσρ1Iξρ2βρ,ρ=1kogσρ1Nξρ2βρ1ogσk+1Pξk+12βk+1,ogσk+11Iξk+12βk+1,ogσk+11Nξk+12βk+1=1ρ=1k+1ogσρPξρ2βρ,ρ=1k+1ogσρ1Iξρ2βρ,ρ=1k+1ogσρ1Nξρ2βρ.

Thus, Theorem 6 is true for n=z+1. Hence, it is satisfied for all n. Therefore,

LSFWAε1,ε2,,εn=1ρ=1nogσρPξρ2βρ,ρ=1nogσρ1Iξρ2βρ,ρ=1nogσρ1Nξρ2βρ.

In a similar way, if 0<1σρminρξρ,1Iξρ2,1Nξρ2<1, σ1, we can also obtain

LSFWAε1,ε2,,εn=1ρ=1nog1σρPξρ2βρ,ρ=1nog1σρ1Iξρ2βρ,ρ=1nog1σρ1Nξρ2βρ,

which completes the proof. □

Remark 1.

If σ1=σ2=σ3==σn=σ, that is 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1, then the LSFWA operator is reduced as follows:

LSFWAε1,ε2,,εn=1ρ=1nogσPξρ2βρ,ρ=1nogσ1Iξρ2βρ,ρ=1nogσ1Nξρ2βρ.

Properties: The LSFWA operator satisfies some properties that are listed below:

(1) Idempotency: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . Then, if the collection of SFNs ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n is identical,

LSFWAε1,ε2,,εn=ε.

(2) Boundedness: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . ερ=minρρξρ,maxρIξρ,maxρNξρ and ερ+=maxρPξρ,minρIξρ,minρNξρ ρ=1,2,,n in , therefore

ερLSFWAε1,ε2,,εnερ+.

(3) Monotonically: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in .If ερ ερ* for ρ=1,2,,n, then

LSFWAε1,ε2,,εnLSFWAε1*,ε2*,,εn*.

Definition 27.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. The structure of the logarithmic spherical ordered weighted averaging (L-SFOWA) operator is

LSFOWAε1,ε2,,εn=ρ=1nβρogσρεηρ,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ) consequently by total order εη(1)εη(2)εη(n).

Theorem 7.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. Then, by using logarithmic operations and Definition 27, LSFOWA is defined as

LSFOWAε1,ε2,,εn=1ρ=1nogσρPξηρ2βρ,ρ=1nogσρ1Iξηρ2βρ,ρ=1nogσρ1Nξηρ2βρ0<σρminPξρ,1Iξρ2,1Nξρ2<1,1ρ=1nog1σρPξηρ2βρ,ρ=1nog1σρ1Iξηρ2βρ,ρ=1nog1σρ1Nξηρ2βρ0<1σρminPξρ,1Iξρ2,1Nξρ2<1,σ1, (1)

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ) consequently by total order εη(1)εη(2)εη(n).

Proof. 

The proof is similar to Theorem 6. Thus, the procedure is eliminated here. □

Remark 2.

If σ1=σ2=σ3==σn=σ, that is, 0<σminρξρ,1Iξρ2,1Nξρ2<1,σ1, then the LSFOWA operator is reduced as follows:

LSFOWAε1,ε2,,εn=1ρ=1nogσPξηρ2βρ,ρ=1nogσ1Iξηρ2βρ,ρ=1nogσ1Nξηρ2βρ.

Properties: The LSFOWA operator satisfies some properties that are listed below:

(1) Idempotency: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in .Then, if a collection of SFNs ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n is identical, that is,

LSFOWAε1,ε2,,εn=ε.

(2) Boundedness: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . ερ=minρPξρ,maxρIξρ,maxρNξρ and ερ+=maxρPξρ,minρIξρ,minρNξρ ρ=1,2,,n in ; therefore,

ερLSFOWAε1,ε2,,εnερ+.

(3) Monotonically: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . If ερ ερ* for ρ=1,2,,n, then

LSFOWAε1,ε2,,εnLSFOWAε1*,ε2*,,εn*.

Definition 28.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. The structure of a logarithmic spherical hybrid weighted averaging (L-SFHWA) operator is

LSFHWAε1,ε2,,εn=ρ=1nβρogσρεηρ*,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ)*εη(ρ)*=nβρεη(ρ),ρN consequently by total order εη(1)*εη(2)*εη(n)*. In addition, associated weights are ω=(ω1,ω2,,ωn) with ωρ0, Σρ=1nωρ=1.

Theorem 8.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. Then, by using logarithmic operations and Definition 28, LSFHWA is defined as

LSFHWAε1,ε2,,εn,=1ρ=1nogσρPξηρ*2βρ,ρ=1nogσρ1Iξηρ*2βρ,ρ=1nogσρ1Nξηρ*2βρ0<σρminPξρ,1Iξρ2,1Nξρ2<1,1ρ=1nog1σρPξηρ*2βρ,ρ=1nog1σρ1Iξηρ*2βρ,ρ=1nog1σρ1Nξηρ*2βρ0<1σρminPξρ,1Iξρ2,1Nξρ2<1,σ1, (2)

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ)*εη(ρ)*=nβρεη(ρ),ρN consequently by total order εη(1)*εη(2)*εη(n)*. In addition, associated weights are ω=(ω1,ω2,,ωn) with ωρ0, Σρ=1nωρ=1.

Proof. 

This proof issimilar to Theorem 6, so the procedure is eliminated here. □

Remark 3.

If σ1=σ2=σ3==σn=σ, that is, 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1, then the LSFHWA operator reduces to

LSFHWAε1,ε2,,εn=1ρ=1nogσPξηρ*2βρ,ρ=1nogσ1Iξηρ*2βρ,ρ=1nogσ1Nξηρ*2βρ.

Properties: The LSFHWA operator satisfies some properties that are listed below:

(1) Idempotency: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . Then, if a collection of SFNs ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n are identical, that is,

LSFHWAε1,ε2,,εn=ε.

(2) Boundedness: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . ερ=minρPξρ,maxρIξρ,maxρNξρ and ερ+=maxρPξρ,minρIξρ,minρNξρ ρ=1,2,,n in ; therefore,

ερLSFHWAε1,ε2,,εnερ+.

(3) Monotonically: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in .If ερ ερ* for ρ=1,2,,n, then

LSFHWAε1,ε2,,εnLSFHWAε1*,ε2*,,εn*.

5.2. Logarithmic Geometric Operators

Definition 29.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. The structure of logarithmic spherical weighted geometric (L-SFWG) operator is

LSFWGε1,ε2,,εn=ρ=1nogσρερβρ,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1.

Theorem 9.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. Then, by using logarithmic operations and Definition 29, LSFWG is defined as

LSFWGε1,ε2,,εn=ρ=1n1ogσρPξρ2βρ1ρ=1n1ogσρ1Iξρ22βρ1ρ=1n1ogσρ1Nξρ22βρ0<σρminPξρ,1Iξρ2,1Nξρ2<1,ρ=1n1og1σρPξρ2βρ1ρ=1n1og1σρ1Iξρ22βρ1ρ=1n1og1σρ1Nξρ22βρ0<1σρminPξρ,1Iξρ2,1Nξρ2<1,σ1,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1.

Proof. 

Using mathematical induction to prove Theorem 9, we proceed as follows:

(a) For n=2, since

ogσ1ε1β1=1ogσ1Pξ12β111ogσ11Iξ122β111ogσ11Nξ122β1

and

ogσ2ε2β2=1ogσ2Pξ22β211ogσ21Iξ222β211ogσ21Nξ222β2,

then

LSFWGε1,ε2=ogσ1ε1β1ogσ2ε2β2=1ogσ1Pξ12β111ogσ11Iξ122β111ogσ11Nξ122β11ogσ2Pξ22β211ogσ21Iξ222β211ogσ21Nξ222β2=1ogσ1Pξ12β1·1ogσ2Pξ22β211ogσ11Iξ122β1·1ogσ21Iξ222β211ogσ11Nξ122β1·1ogσ21Nξ222β2=ρ=121ogσρPξρ2βρ1ρ=121ogσρ1Iξρ22βρ1ρ=121ogσρ1Nξρ22βρ.

(b) Now, Theorem 9 is true for n=k,

LSFWGε1,ε2,,εk=ρ=1k1ogσρPξρ2βρ1ρ=1k1ogσρ1Iξρ22βρ1ρ=1k1ogσρ1Nξρ22βρ.

(c) Now, we prove that Theorem 9 for n=k+1, that is,

LSFWGε1,ε2,,εk,εk+1=ρ=1kogσρερβρogσk+1εk+1βk+1
LSFWGε1,ε2,,εk,εk+1=ρ=1k1ogσρPξρ2βρ1ρ=1k1ogσρ1Iξρ22βρ1ρ=1k1ogσρ1Nξρ22βρ1ogσk+1Pξk+12βk+111ogσk+11Iξk+122βk+111ogσk+11Nξk+122βk+1=ρ=1k+11ogσρρξρ2βρ1ρ=1k+11ogσρ1Iξρ22βρ1ρ=1k+11ogσρ1Nξρ22βρ.

Thus, Theorem 9 is true for n=z+1. Hence, it is satisfied for all n. Therefore,

LSFWGε1,ε2,,εn=ρ=1n1ogσρPξρ2βρ1ρ=1n1ogσρ1Iξρ22βρ1ρ=1n1ogσρ1Nξρ22βρ.

In a similar way, if 0<1σρminρξρ,1Iξρ2,1Nξρ2<1, σ1, we can also obtain

LSFWGε1,ε2,,εn=ρ=1n1og1σρPξρ2βρ1ρ=1n1og1σρ1Iξρ22βρ1ρ=1n1og1σρ1Nξρ22βρ,

which completes the proof. □

Remark 4.

If σ1=σ2=σ3==σn=σ, that is, 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1, then the LSFWG operator reduces to

LSFWGε1,ε2,,εn=ρ=1n1ogσPξρ2βρ1ρ=1n1ogσ1Iξρ22βρ1ρ=1n1ogσ1Nξρ22βρ.

Properties: The LSFWG operator satisfies some properties that are listed below:

(1) Idempotency: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in .Then, if a collection of SFNs ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n are identical, that is,

LSFWGε1,ε2,,εn=ε.

(2) Boundedness: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . ερ=minρρξρ,maxρIξρ,maxρNξρ and ερ+=maxρPξρ,minρIξρ,minρNξρ ρ=1,2,,n in , therefore

ερLSFWGε1,ε2,,εnερ+.

(3) Monotonically: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in .If ερ ερ* for ρ=1,2,,n, then

LSFWGε1,ε2,,εnLSFWGε1*,ε2*,,εn*.

Definition 30.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. The structure of logarithmic spherical ordered weighted geometric (L-SFOWG) operator is

LSFOWGε1,ε2,,εn=ρ=1nogσρεηρβρ,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ) consequently by total order εη(1)εη(2)εη(n).

Theorem 10.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. Then, by using logarithmic operations and Definition 30, LSFOWG defined as

LSFOWGε1,ε2,,εn=ρ=1n1ogσρPξηρ2βρ1ρ=1n1ogσρ1Iξηρ22βρ1ρ=1n1ogσρ1Nξηρ22βρ0<σρminPξρ,1Iξρ2,1Nξρ2<1,ρ=1n1og1σρPξηρ2βρ1ρ=1n1og1σρ1Iξηρ22βρ1ρ=1n1og1σρ1Nξηρ22βρ0<1σρminPξρ,1Iξρ2,1Nξρ2<1,σ1,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ) consequently by total order εη(1)εη(2)εη(n).

Proof. 

This proof is similar to Theorem 9, so the procedure is eliminated here.□

Remark 5.

If σ1=σ2=σ3==σn=σ, that is, 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1, then LSFOWG operator reduces to

LSFOWGε1,ε2,,εn=ρ=1n1ogσPξηρ2βρ1ρ=1n1ogσ1Iξηρ22βρ1ρ=1n1ogσ1Nξηρ22βρ.

Properties: The LSFOWG operator satisfies some properties that are listed below:

(1) Idempotency: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in .Then, if the collection of SFNs ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n is identical, that is,

LSFOWGε1,ε2,,εn=ε.

(2) Boundedness: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . ερ=minρPξρ,maxρIξρ,maxρNξρ and ερ+=maxρPξρ,minρIξρ,minρNξρ ρ=1,2,,n in , therefore

ερLSFOWGε1,ε2,,εnερ+.

(3) Monotonically: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . If ερ ερ* for ρ=1,2,,n, then

LSFOWGε1,ε2,,εnLSFOWGε1*,ε2*,,εn*.

Definition 31.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminPξρ,1Iξρ2,1Nξρ2<1,σ1. The structure of logarithmic spherical hybrid weighted geometric (L-SFHWG) operator is

LSFHWGε1,ε2,,εn=ρ=1nogσρεηρ*βρ,

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ)*εη(ρ)*=nβρεη(ρ),ρN consequently by total order εη(1)*εη(2)*εη(n)*. In addition, associated weights are ω=(ω1,ω2,,ωn) with ωρ0, Σρ=1nωρ=1.

Theorem 11.

For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγρ=1,2,,n in ℜ, with 0<σρminρξρ,1Iξρ2,1Nξρ2<1,σ1. Then, by using logarithmic operations and Definition 31, LSFHWG is defined as

LSFHWGε1,ε2,,εn=ρ=1n1ogσρPξηρ*2βρ1ρ=1n1ogσρ1Iξηρ*22βρ1ρ=1n1ogσρ1Nξηρ*22βρ0<σρminPξρ,1Iξρ2,1Nξρ2<1,ρ=1n1og1σρPξηρ*2βρ1ρ=1n1og1σρ1Iξηρ*22βρ1ρ=1n1og1σρ1Nξηρ*22βρ0<1σρminPξρ,1Iξρ2,1Nξρ2<1,σ1, (3)

where βρρ=1,2,,n are weight vectors with βρ0 and ρ=1nβρ=1 and the ρth biggest weighted value is εη(ρ)*εη(ρ)*=nβρεη(ρ),ρN consequently by total order εη(1)*εη(2)*εη(n)*. In addition, the associated weights are ω=(ω1,ω2,,ωn) with ωρ0, Σρ=1nωρ=1.

Proof. 

This proof is similar to Theorem 9. Thus, the procedure is eliminated here. □

Remark 6.

If σ1=σ2=σ3==σn=σ, that is, 0<σminPξρ,1Iξρ2,1Nξρ2<1,σ1; then, LSFHWG operator reduces to

LSFHWGε1,ε2,,εn=ρ=1n1ogσPξηρ*2βρ1ρ=1n1ogσ1Iξηρ*22βρ1ρ=1n1ogσ1Nξηρ*22βρ.

Properties: The LSFHWG operator satisfies some properties that are listed below:

(1) Idempotency: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . Then, if collection of SFNs ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n are identical, that is,

LSFHWGε1,ε2,,εn=ε.

(2) Boundedness: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . ερ=minρPξρ,maxρIξρ,maxρNξρ and ερ+=maxρρξρ,minρIξρ,minρNξρ ρ=1,2,,n in , therefore

ερLSFHWGε1,ε2,,εnερ+.

(3) Monotonically: For any collection of SFNs, ερ=Pξρrˇγ,Iξρrˇγ,Nξρrˇγ ρ=1,2,,n in . If ερ ερ* for ρ=1,2,,n, then

LSFHWGε1,ε2,,εnLSFHWGε1*,ε2*,,εn*.

6. Proposed Technique for Solving Decision-Making Problems

In this section, we propose a new approach to decision-making based on the spherical fuzzy set. This approach will use data information provided by the decision problem only and does not need any additional information provided by decision makers, in order to avoid the effect of subjective information influencing the decision results. In the following, we will introduce a spherical fuzzy set decision-making matrix as indicated below.

  • Step 1:

    Let H=(h1,h2,,hm) be a distinct set of m probable alternatives and Y=(y1,y2,,yn) be a finite set of n criteria, where hi indicates the ith alternatives and yj indicates the jth criteria. Let D=(d1,d2,,dt) be a finite set of t experts, where dk indicates the kth expert. The expert dk supplies her appraisal of an alternative hi on an attribute yj as a SFNs (i=1,2,,m;j=1,2,,n). The experts’ information is represented by the spherical fuzzy set decision-making matrix Ds=Eiρsm×n. Assume that βρ(ρ=1,2,,m) is a weight vector of the attribute yj such that 0βρ1, ρ=1nβρ=1 and ψ=(ψ1,ψ2,,ψm) is the weight vector of the decision makers dk such that ψk1, k=1nψk=1.

    When we construct the spherical fuzzy decision-making matrices, Ds=Eiρsm×n for decisions. Basically, criteria have two types: one is benefit criteria and the other one is cost criteria. If the spherical fuzzy decision matrices have cost type criteria matrices, Ds=Eiρsm×n can be converted into the normalized spherical fuzzy decision matrices, rˇs=rˇiρsm×n, where rˇips=Eiρs,forbenefitcriteriaApE¯iρs,forcostcriteriaAp,j=1,2,,n, and E¯iρs is the complement of Eiρs. If all the criteria have the same type, then there is no need for normalization.

    Taking the decision information from the given matrix rˇk and using the SFWA/SFWG operator, the individual total spherical fuzzy preference value rˇik of the alternative hi is derived as follows:
    rˇik=LSFWA(εi1k,εi2k,,εink),(i=1,2,,m;k=1,2,,t),
    where β=(β1,β2,,βn)T is the weight vector of the attribute.
  • Step 2:

    In this step, we find the collective spherical information using a spherical fuzzy weighted averaging aggregation operator.

  • Step 3:
    In this step, we find the weights of each of the criteria by using the spherical fuzzy entropy:
    γq=1+1nρ=1nPilogPi+IilogIi+NilogNiq=1n1+1nρ=1nPilogPi+IilogIi+NilogNi.
  • Step 4:

    In this step, we calculate the aggregated information using all the logarithmic aggregation operators of spherical fuzzy sets.

  • Step 5:

    We find the score value S˜(ogσερ) and the accuracy value A˜(ogσερ) of the cumulative overall preference value hi (i=1,2,,m).

  • Step 6:

    By the definition, rank the alternatives hi (i=1,2,,m) and choose the best alternative that has the maximum score value.

The Algorithmic Steps are shown in Figure 4.

Figure 4.

Figure 4

Algorithmic Steps.

6.1. Numerical Example

Assume that there is a committee that selects five applicable emerging technology enterprises Hg(g=1,2,3,4,5), which are given as follows:

(1) Augmented Reality (H1),

(2) Personalized Medicine H2,

(3) Artificial Intelligence H3,

(4) Gene Drive H4,

(5) Quantum Computing H5

To assess the possible rising technology enterprises according to the four attributes, which are

(1) Advancement D1,

(2) market risk D2,

(3) financial investments D3 and

(4) progress of science and technology D4.

To avoid the conflict between them, the decision makers gives the weight β=(0.314,0.355,0.331)T. Construct the spherical fuzzy set decision making matrices as shown in Table 1, Table 2 and Table 3:

Table 1.

Emerging technology Eenterprises F1.

D1 D2 D3 D4
H1 0.84,0.34,0.40 0.43,0.39,0.78 0.67,0.50,0.30 0.31,0.21,0.71
H2 0.60,0.11,0.53 0.23,0.35,0.59 0.72,0.31,0.41 0.11,0.25,0.82
H3 0.79,0.19,0.39 0.11,0.21,0.91 0.71,0.41,0.13 0.34,0.25,0.51
H4 0.63,0.51,0.13 0.49,0.33,0.42 0.61,0.43,0.45 0.49,0.37,0.59
H5 0.57,0.36,0.29 0.50,0.15,0.60 0.70,0.32,0.40 0.33,0.44,0.65.

Table 2.

Emerging technology enterprises F2.

D1 D2 D3 D4
H1 0.61,0.15,0.53 0.16,0.35,0.62 0.61,0.35,0.47 0.55,0.17,0.74
H2 0.66,0.11,0.51 0.43,0.23,0.77 0.93,0.08,0.09 0.02,0.06,0.99
H3 0.88,0.09,0.07 0.05,0.06,0.89 0.56,0.17,0.44 0.43,0.13,0.61
H4 0.59,0.32,0.34 0.24,0.48,0.51 0.68,0.53,0.39 0.34,0.21,0.61
H5 0.71,0.31,0.24 0.35,0.41,0.69 0.73,0.44,0.21 0.22,0.49,0.74

Table 3.

Emerging technology enterprises F3.

D1 D2 D3 D4
H1 0.85,0.25.0.15 0.14,0.23,0.88 0.78,0.38,0.18 0.29,0.39,0.83
H2 0.94,0.04,0.07 0.39,0.19,0.61 0.63,0.18,0.35 0.48,0.49,0.56
H3 0.73,0.13,0.46 0.19,0.39,0.88 0.87,0.35,0.18 0.41,0.13,0.81
H4 0.82,0.12,0.43 0.55,0.21,0.63 0.53,0.33,0.47 0.46,0.23,0.51
H5 0.61,0.33,0.29 0.28,0.41,0.63 0.74,0.34,0.14 0.37,0.32,0.65

Since D1, D3 are benefit type criteria and D2, D4 is cost type criteria, we need to have normalized the decision matrices. Normalized decision matrices are shown in Table 4, Table 5 and Table 6:

Table 4.

Emerging technology enterprises rˇ1.

D1 D2 D3 D4
H1 0.84,0.34,0.40 0.78,0.39,0.43 0.67,0.50,0.30 0.71,0.21,0.31
H2 0.60,0.11,0.53 0.59,0.35,0.23 0.72,0.31,0.41 0.82,0.25,0.11
H3 0.79,0.19,0.39 0.91,0.21,0.11 0.71,0.41,0.13 0.51,0.25,0.34
H4 0.63,0.51,0.13 0.42,0.33,0.49 0.61,0.43,0.45 0.59,0.37,0.49
H5 0.57,0.36,0.29 0.60,0.15,0.50 0.70,0.32,0.40 0.65,0.44,0.33

Table 5.

Emerging technology enterprises rˇ2.

D1 D2 D3 D4
H1 0.61,0.15,0.53 0.62,0.35,0.16 0.61,0.35,0.47 0.74,0.17,0.55
H2 0.66,0.11,0.51 0.77,0.23,0.43 0.93,0.08,0.09 0.99,0.06,0.02
H3 0.88,0.09,0.07 0.89,0.06,0.05 0.56,0.17,0.44 0.61,0.13,0.43
H4 0.59,0.32,0.34 0.51,0.48,0.24 0.68,0.53,0.39 0.61,0.21,0.34
H5 0.71,0.31,0.24 0.69,0.41,0.35 0.73,0.44,0.21 0.74,0.49,0.22

Table 6.

Emerging technology enterprises rˇ3.

D1 D2 D3 D4
H1 0.85,0.25.0.15 0.88,0.23,0.14 0.78,0.38,0.18 0.83,0.39,0.29
H2 0.94,0.04,0.07 0.61,0.19,0.39 0.63,0.18,0.35 0.56,0.49,0.48
H3 0.73,0.13,0.46 0.88,0.39,0.19 0.87,0.35,0.18 0.81,0.13,0.41
H4 0.82,0.12,0.43 0.63,0.21,0.55 0.53,0.33,0.47 0.51,0.23,0.46
H5 0.61,0.33,0.29 0.63,0.41,0.28 0.74,0.34,0.14 0.65,0.32,0.37

Step 2: Use the SFWA operator to aggregate all the individual normalized spherical fuzzy decision matrices. The aggregated spherical fuzzy decision matrix is shown in Table 7.

Table 7.

Collective spherical fuzzy decision information matrix rˇ.

D1 D2 D3 D4
H1 0.788,0.229,0.319 0.785,0.315,0.208 0.696,0.402,0.297 0.767,0.239,0.371
H2 0.807,0.078,0.279 0.674,0.246,0.342 0.818,0.160,0.227 0.919,0.188,0.097
H3 0.814,0.128,0.223 0.893,0.165,0.099 0.748,0.284,0.223 0.677,0.159,0.393
H4 0.702,0.267,0.271 0.533,0.324,0.395 0.615,0.424,0.433 0.573,0.258,0.421
H5 0.639,0.331,0.271 0.644,0.298,0.363 0.724,0.365,0.224 0.685,0.411,0.296

Step 3: The entropy of each attribute can be calculated by the following equation:

γq=1+1nρ=1nPilogPi+IilogIi+NilogNiq=1n1+1nρ=1nPilogPi+IilogIi+NilogNi

and the calculated weights are

W=w1=0.256,w2=0.248,w3=0.245,w4=0.251.

Step 4: Now, we apply all the proposed logarithmic aggregation operators to collective spherical fuzzy information to find the aggregated information as follows:

Case-1: Using logarithmic spherical fuzzy weighted averaging aggregation operator, we obtained

graphic file with name entropy-21-00628-i001.jpg

Case-2: Using a logarithmic spherical fuzzy ordered weighted averaging aggregation operator, we obtained

graphic file with name entropy-21-00628-i002.jpg

Case-3: Using a logarithmic spherical fuzzy hybrid weighted averaging aggregation operator, we obtained

graphic file with name entropy-21-00628-i003.jpg

Case-4: Using a logarithmic spherical fuzzy weighted geometric aggregation operator, we obtained

graphic file with name entropy-21-00628-i004.jpg

Case-5: Using a logarithmic spherical fuzzy ordered weighted geometric aggregation operator, we obtained

graphic file with name entropy-21-00628-i005.jpg

Case-6: Using a logarithmic spherical fuzzy hybrid weighted geometric aggregation operator, we obtained

graphic file with name entropy-21-00628-i006.jpg

Step 5: We find the score value S˜(ogσερ) and the accuracy value A˜(ogσερ) of the cumulative overall preference value hi (i=1,2,3,4,5).

Case-1: Score of aggregated information for the L-SFWA Operator, and we obtained

graphic file with name entropy-21-00628-i007.jpg

Case-2: Score of aggregated information for the L-SFOWA Operator, and we obtained

graphic file with name entropy-21-00628-i008.jpg

Case-3: Score of aggregated information for the L-SFHWA Operator, and we obtained

graphic file with name entropy-21-00628-i009.jpg

Case-4: Score of aggregated information for the L-SFWG Operator, and we obtained

graphic file with name entropy-21-00628-i010.jpg

Case-5: Score of aggregated information for the L-SFOWG Operator, and we obtained

graphic file with name entropy-21-00628-i011.jpg

Case-6: Score of aggregated inf ormation for the L-SFHWG Operator, and we obtained

graphic file with name entropy-21-00628-i012.jpg

Step 6: We find the best (suitable) alternative that has the maximum score value from set of alternatives hi(i=1,2,3,4,5). Overall preference value and ranking of the alternatives are summarized in Table 8.

Table 8.

Overall preference value and ranking of the alternatives for σ=0.5>0.

S˜(ogσH1) S˜(ogσH2) S˜(ogσH3) S˜(ogσH4) S˜(ogσH5) Ranking
L-SFWA 0.985356 0.996515 0.993345 0.737170 0.920198 H2>H3>H1>H5>H4
L-SFOWA 0.985236 0.996558 0.993479 0.737233 0.921374 H2>H3>H1>H5>H4
L-SFHWA 0.999817 0.999998 0.999863 0.913067 0.996664 H2>H3>H1>H5>H4
L-SFWG 0.982645 0.988581 0.984383 0.545559 0.910307 H2>H3>H1>H5>H4
L-SFOWG 0.982461 0.988519 0.984477 0.544563 0.911435 H2>H1>H3>H5>H4
L-SFHWG 0.999914 0.999997 0.999657 0.91689 0.999278 H2>H1>H3>H5>H4

The ranking of the alternatives are shown in the Figure 5.

Figure 5.

Figure 5

Ranking of Alternatives.

6.2. Sensitivity Analysis and Comparison Discussion

In this section, we give the comparison analysis on how our proposed logarithmic aggregation operators are effective and reliable to aggregate the spherical fuzzy information. Ashraf [40,41] proposed the spherical aggregation operators to aggregate the spherical fuzzy informatio; in this part of our study, we give a comparison between proposed and novel logarithmic spherical fuzzy aggregation operators. We take the spherical fuzzy information from [40] as follows:

graphic file with name entropy-21-00628-i013.jpg

Now, we utilized a logarithmic spherical fuzzy weighted averaging operator to choose the best alternative as follows:

graphic file with name entropy-21-00628-i014.jpg

The ranking of the alternatives using Ashraf [40] information is shown in the Figure 6.

Figure 6.

Figure 6

Comparison Ranking.

The bast alternative is H2. The obtained result utilizing a logarithmic spherical fuzzy weighted averaging operator is the same as results shown by Ashraf [40]. Hence, this study proposed the novel logarithmic aggregation operators to aggregate the spherical fuzzy information. This study gives a more reliable technique to aggregate and to deal with uncertainties in decision-making problems using spherical fuzzy sets. Utilizing proposed spherical fuzzy logarithmic aggregation operators, we find the best alternative from a set of alternatives given by the decision maker. Hence, the proposed MCGDM technique based on spherical fuzzy logarithmic aggregation operators gives another technique to find the best alternative as an application in decision support systems.

7. Conclusions

In this paper, we have revealed a novel logarithmic operation of SFSs with the real base number σ. Additionally, we have analyzed their properties and relationships. In view of these logarithmic operations, we built up the weighted averaging and geometric aggregation operators named L-SFWA, L-SFOWA, L-SFHWA, L-SFWG, L-SFOWG and L-SFHWG. A spherical fuzzy MCDM problem with interactive criteria, an approach based on the proposed operators, was proposed. Finally, this method was applied to MCDM problems. From the decision results displayed in numerical examples, we can find that our created approach can overcome the drawbacks of existing algebraic aggregation operators. In the succeeding work, we shall combine the proposed operator with some novel fuzzy sets, such as type-2 fuzzy sets, neutrosophic sets, and so on. In addition, we may investigate our created approach in the field of different areas, such as personnel evaluation, medical artificial intelligence, energy management and supplier selection evaluation. In addition, we can develop more decision-making approaches like GRA, TODAM, TOPSIS, VIKOR and so on to deal with uncertainties in the form of spherical fuzzy information.

Acknowledgments

The authors would like to thank the editor in chief, associate editor and the anonymous referees for detailed and valuable comments that helped to improve this manuscript.

Author Contributions

Conceptualization, S.A. (Shahzaib Ashraf) and S.A. (Saleem Abdullah); methodology, S.A. (Shahzaib Ashraf); software, S.A. (Shahzaib Ashraf); validation, S.A. (Shahzaib Ashraf), S.A. (Saleem Abdullah) and Y.J.; formal analysis, S.A. (Saleem Abdullah) and Y.J.; investigation, S.A. (Shahzaib Ashraf); writing—original draft preparation, S.A.; writing—review and editing, S.A. (Shahzaib Ashraf); visualization, Y.J.; supervision, S.A. (Saleem Abdullah); funding acquisition, Y.J.

Funding

This paper is supported by Major Humanities and Social Sciences Research Projects in Zhejiang Universities (No. 2018QN058), the China Postdoctoral Science Foundation (No. 2018QN058), the Zhejiang Province Natural Science Foundation (No. LY18G010007) and the National Natural Science Foundation of China (No. 71761027).

Conflicts of Interest

The authors declare no conflict of interest.

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