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Symmetric Intuitionistic Fuzzy Weighted Mean Operators Based on Weighted Archimedean t-Norms and t-Conorms for Multi-Criteria Decision Making

Article type: Research Article

Authors: Ma, Zhen Ming; | Yang, Wei

Affiliations: School of Mathematics and Statistics, Linyi University, Linyi 276005, China

Correspondence: [*] Corresponding author.

Keywords: multi-criteria decision making, intuitionistic fuzzy set, weighted Archimedean t-norm and t-conorm, symmetric intuitionistic fuzzy weighted mean operator

DOI: 10.15388/20-INFOR390

Journal: Informatica, vol. 31, no. 1, pp. 89-112, 2020

Received February 2019
|
Accepted September 2019
|
Published: 2020
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Abstract

Using different operational laws on membership and non-membership information, various intuitionistic fuzzy aggregation operators based on Archimedean t-norm and t-conorm or their special cases have been extensively investigated for multi-criteria decision making. In spite of this, they are not suitable for some practical cases. In this paper, symmetric intuitionistic fuzzy weighted mean operators w.r.t. general weighted Archimedean t-norms and t-conorms are introduced to deal neutrally or fairly with membership and non-membership information to meet the need of decision makers in some cases. The relationship among the proposed operators and the existing ones is discussed. Particularly, using the parameters in the aggregation operators, the attitude whether the decision maker is optimistic, pessimistic or impartial is reflected. At last, an example is given to show the behaviour of the proposed operators for multi-criteria decision making under intuitionistic fuzzy environment.

1Introduction

Since the introduction of fuzzy sets by Zadeh (1965), various generalizations of fuzzy sets have been provided, such as intuitionistic fuzzy sets introduced by Atanassov (1986), abbreviated here as A-IFS (the reasons for this are presented in Dubois et al., 2005), grey set (Deng, 1989), vague set (Gau and Buehrer, 1993), interval-valued fuzzy set (Sambuc, 1975; Zadeh, 1975), and so on. Deschrijver and Kerre (2003, 2007) investigated the relationships among some extensions of fuzzy set theory, and proved that A-IFS, grey set, vague set and interval-valued fuzzy set are equivalent. As we know, it is constructed through the membership degree, the non-membership degree and the hesitancy degree, which can describe the uncertainty and fuzziness more objectively than the usual fuzzy set. Thus A-IFS has attracted more and more attention from researchers and has been used to solve many problems, especially the multi-criteria decision making (MCDM) problems.

An effective method to deal with the MCDM problem under intuitionistic fuzzy environment is to calculate the aggregation values of the alternatives. At present, plenty of aggregation operators in fuzzy environment have been extended to fit different situations in intuitionistic fuzzy case. All kinds of mean operators, such as quasi-arithmetic means (Hardy et al., 1934), (generalized) Bonferroni mean (BM, GBM) (Bonferroni, 1950; Yager, 2009) and (induced) ordered weighted averaging (OWA, IOWA) operators (Yager, 1988; Yager and Filev, 1999), are hot topics in aggregation, and a lot of related work has been done. With respect to the operations defined for intuitionistic fuzzy numbers (IFNs) based on algebraic product t-norm, probabilistic sum t-conorm and OWA operator, Xu and Yager (2006), Xu (2007) generalized the weighted geometric averaging operator to the intuitionistic fuzzy weighted geometric/averaging (IFWG/IFWA) operator, the intuitionistic fuzzy ordered weighted geometric/averaging (IFO WG/IFOWA) operator and the intuitionistic fuzzy hybrid geometric/averaging (IFHG/IFHA) operator and applied them to the MCDM problem under intuitionistic fuzzy environment. Although the IFHA (IFHG) operator generalized both the IFWA (IFWG) and IFOWA (IFOWG) operators by weighting the given importance and the ordered position of the arguments, there is a flaw pointed out by Liao and Xu (2014) that these hybrid aggregation operators do not satisfy some desirable properties, such as boundedness and idempotency; the developed operators not only can weigh both the arguments and their ordered positions simultaneously, but also have some desirable properties, such as idempotency, boundedness, and monotonicity. Based on the generalized OWA operator proposed by Yager (2004b), Li (2010) and Zhao et al. (2010) introduced the generalized IFWA, generalized IFOWA, and generalized IFHA operators, and applied them to multiple attribute decision making with intuitionistic fuzzy information. Using the operations defined in Xu (2007), Xu and Yager (2011) investigated the BM under intuitionistic fuzzy environments, developed an intuitionistic fuzzy BM (IFBM) and discussed its variety of special cases. Then, they applied the weighted IFBM to multicriteria decision making. Considering that the algebraic product and Einstein t-norms are two prototypical examples of the class of strict Archimedean t-norms (Klement et al., 2000), Wang and Liu (2011, 2012) proposed the intuitionistic fuzzy Einstein weighted geometric/averaging (IFEWG/IFEWA) operator and the intuitionistic fuzzy Einstein ordered weighted geometric/averaging (IFEOWG/IFEOWA) operator. By using Archimedean t-norm and t-conorm, Xia et al. (2012) defined the Archimedean t-norm and t-conorm based intuitionistic fuzzy weighted averaging (ATS-IFWA) operator and the Archimedean t-norm and t-conorm based intuitionistic fuzzy geometric (ATS-IFWG) operator to provide more choices for the decision makers by these parameterized t-norms and t-conorms. By extending the quasi-arithmetic ordered weighted averaging operator to different intuitionistic fuzzy situations, Yang and Chen (2012) introduced three kinds of new operators: the quasi-IFOWA operator, the quasi-intuitionistic fuzzy Choquet ordered averaging operator and the quasi-IFOWA operator based on the Dempster–Shafer belief structure. Tan et al. (2013) provided a critical analysis of Yang and Chen’s operations to elicit their disadvantages, and associating with operations in Xia et al. (2012), proposed a new quasi-IFOWA operator based on Archimedean t-norm and t-conorm to overcome these faults, and obtained some consistent conclusions. Beliakov et al. (2011) declared that the IFWA operator is not consistent with the limiting case of ordinary fuzzy sets, which is undesirable, and proposed a new construction method for the IFWA operator based on the Łukasiewicz t-norm, which is consistent with operations on ordinary fuzzy sets. Similarly to IFBM, Beliakov and James (2013) presented two alternative methods to extend the generalized Bonferroni mean to intuitionistic fuzzy sets. Particularly, they constructed a general form of intuitionistic fuzzy aggregation operators by pairing the usual aggregation operators and their duals. Since the above intuitionistic fuzzy aggregation operators are using different aggregation operators on membership and non-membership information, it was pointed out in Xia and Xu (2012) that it is necessary to develop some neutral aggregation operators in order to be neutral in some cases and to be treated fairly. For example, it is pointed out in Xu and Ma (2019), Yang et al. (2019) that when aggregating some individual intuitionistic fuzzy preference relations into a collective one, such operators are necessary. Based on algebraic product t-norm, new intuitionistic fuzy aggregation operators, which treat the membership and non-membership information fairly, were defined (Liao and Xu, 2015). Furthermore, due to the absence of parameters in these t-norms, the existing neutral aggregation operators can not provide more choices for the decision makers. Motivated by the idea of Beliakov and James (2013), Calvo and Mesiar (2003), Tan et al. (2013), Xia and Xu (2012), Xia et al. (2012), in this paper, some new intuitionistic fuzzy aggregation operators based on weighted Archimedean t-norm and t-conorm, which fairly treat membership and non-membership information and provide more choices for the decision maker, are developed.

To do so, the remainder of this paper is organized as follows: The basic concepts of weighted Archimedean t-norms and t-conorms and intuitionistic fuzzy sets are introduced in Section 2. In Sections 3 and 4, symmetric intuitionistic fuzzy weighted mean operators w.r.t. weighted Archimedean t-norms and t-conorms are defined, in which using parameters the attitude whether the decision maker is optimistic, pessimistic or impartial is reflected and the relationship among the proposed operators and the existing ones is discussed. Section 5 provides an example to illustrate the behaviour of the proposed operators. In the final section, our research is concluded.

2Preliminaries

To make the presentation self-contained, in what follows, we review some basic concepts.

2.1Weighted Archimedean t-Norms and t-Conorms

Definition 2.1

Definition 2.1(See Klement et al., 2000).

A triangular norm (t-norm) is a binary operation T on the unit interval [0,1][TeX:] $ [0,1]$, T:[0,1]2⟶[0,1][TeX:] $ T:{[0,1]^{2}}\longrightarrow [0,1]$, such that for all x,y,z∈[0,1][TeX:] $ x,y,z\in [0,1]$:

  • (T1) T(x,T(y,z))=T(T(x,y),z)[TeX:] $ T(x,T(y,z))=T(T(x,y),z)$,

  • (T2) T(x,y)=T(y,x)[TeX:] $ T(x,y)=T(y,x)$,

  • (T3) if x⩽y[TeX:] $ x\leqslant y$, then T(x,z)⩽T(y,z)[TeX:] $ T(x,z)\leqslant T(y,z)$,

  • (T4) T(x,1)=x[TeX:] $ T(x,1)=x$.

Definition 2.2

Definition 2.2(See Klement et al., 2000).

A triangular conorm (t-conorm) is a binary operation S on the unit interval [0,1][TeX:] $ [0,1]$, S:[0,1]2⟶[0,1][TeX:] $ S:{[0,1]^{2}}\longrightarrow [0,1]$, which, for all x,y,z∈[0,1][TeX:] $ x,y,z\in [0,1]$, satisfies (T1)–(T3) and

  • (S4) S(x,0)=x[TeX:] $ S(x,0)=x$ for all x∈[0,1][TeX:] $ x\in [0,1]$.

Definition 2.3

Definition 2.3(See Klement et al., 2000).

A t-norm T is called an Archimedean t-norm if it is continuous and T(x,x)<x[TeX:] $ T(x,x)<x$ for all x∈(0,1)[TeX:] $ x\in (0,1)$. An Archimedean t-norm is called a strictly Archimedean t-norm if it is strictly increasing in each variable for x,y∈(0,1)[TeX:] $ x,y\in (0,1)$.

Definition 2.4

Definition 2.4(See Klement et al., 2000).

A t-conorm S is called an Archimedean t-conorm if it is continuous and S(x,x)>x[TeX:] $ S(x,x)>x$ for all x∈(0,1)[TeX:] $ x\in (0,1)$. An Archimedean t-conorm is called a strictly Archimedean t-conorm if it is strictly increasing in each variable for x,y∈(0,1)[TeX:] $ x,y\in (0,1)$.

It is well known (see Klement et al., 2000) that a strict Archimedean t-norm can be expressed via its additive generator g as follows: T(x,y)=g−1(g(x)+g(y))[TeX:] $ T(x,y)={g^{-1}}(g(x)+g(y))$, and the same applies to its dual t-conorm, S(x,y)=h−1(h(x)+h(y)),[TeX:] $ S(x,y)={h^{-1}}(h(x)+h(y)),$ with h(x)=g(1−x)[TeX:] $ h(x)=g(1-x)$. That is, S(x,y)=1−g−1(g(1−x)+g(1−y))[TeX:] $ S(x,y)=1-{g^{-1}}(g(1-x)+g(1-y))$. If not otherwise specified, we remind that an additive generator of a continuous Archimedean t-norm is a strictly decreasing function g:[0,1]→[0,∞][TeX:] $ g:[0,1]\to [0,\infty ]$ such that g(1)=0[TeX:] $ g(1)=0$ in the following parts. For nilpotent operations the inverse changes to the pseudo-inverse.

For a given weight vector ω=(ω1,ω2,…,ωn)⊤[TeX:] $ \omega ={({\omega _{1}},{\omega _{2}},\dots ,{\omega _{n}})^{\top }}$ of x=(x1,x2,…,xn)[TeX:] $ x=({x_{1}},{x_{2}},\dots ,{x_{n}})$ where ωj∈[0,∞)[TeX:] $ {\omega _{j}}\in [0,\infty )$ is the weight of xj[TeX:] $ {x_{j}}$ ( j=1,2,…,n[TeX:] $ j=1,2,\dots ,n$), we denote the weighted t-norm aggregation operator as Tω(x)[TeX:] $ {T_{\omega }}(x)$. Let T be a continuous Archimedean t-norm with an additive generator g (Yager, 2004a), and we define the weighted aggregation as:

Tω,g(x)=g−1(∑j=1nωjg(xj)),Sω,g(x)=1−g−1(∑j=1nωjg(1−xj)).[TeX:] \[ {T_{\omega ,g}}(x)={g^{-1}}\bigg({\sum \limits_{j=1}^{n}}{\omega _{j}}g({x_{j}})\bigg),\hspace{2em}{S_{\omega ,g}}(x)=1-{g^{-1}}\bigg({\sum \limits_{j=1}^{n}}{\omega _{j}}g(1-{x_{j}})\bigg).\]
If we assign specific forms to the function g, then some weighted Archimedean t-norm from the well-known Archimedean t-norms (Klement et al., 2000) can be obtained:

Let gγSS(t)=1−tγγ[TeX:] $ {g_{\gamma }^{SS}}(t)=\frac{1-{t^{\gamma }}}{\gamma }$, γ≠0[TeX:] $ \gamma \ne 0$, then the weighted Schweizer–Sklar t-norm is provided as follows:

Tω,gγSS(x)=(∑j=1nωjxjγ)1γ.[TeX:] \[ {T_{\omega ,{g_{\gamma }^{SS}}}}(x)={\Bigg(\hspace{0.1667em}{\sum \limits_{j=1}^{n}}{\omega _{j}}{x_{j}^{\gamma }}\Bigg)^{\frac{1}{\gamma }}}.\]
Particularly, if γ=1[TeX:] $ \gamma =1$ and ω=(1,1)[TeX:] $ \omega =(1,1)$, then Schweizer–Sklar t-norm reduces to the Łukasiewicz t-norm.

Furthermore, let A=H(T,S)[TeX:] $ A=H(T,S)$ be a composed aggregation operator based on a continuous t-norm T, a continuous t-conorm S and a binary aggregation operator H, Calvo and Mesiar (2003) introduced weighted t-norms based aggregation operator Aω:[0,1]dimω→[0,1][TeX:] $ {A_{\omega }}:{[0,1]^{\dim \hspace{2.5pt}\omega }}\to [0,1]$ as Aω=H(Tω,Sω)[TeX:] $ {A_{\omega }}=H({T_{\omega }},{S_{\omega }})$, i.e. Aω(x)=H(Tω(x),Sω(x))[TeX:] $ {A_{\omega }}(x)=H({T_{\omega }}(x),{S_{\omega }}(x))$.

2.2Intuitionistic Fuzzy Sets

Definition 2.5

Definition 2.5(See Atanassov, 1986).

Let X be a given universe. An intuitionistic fuzzy set (IFS) A in X is defined as follows:

A={x,μA(x),νA(x)|x∈X},[TeX:] \[ A=\big\{x,{\mu _{A}}(x),{\nu _{A}}(x)\hspace{0.1667em}\big|\hspace{0.1667em}x\in X\big\},\]
μA(x),νA(x)∈[0,1][TeX:] $ {\mu _{A}}(x),{\nu _{A}}(x)\in [0,1]$ indicate the amounts of guaranteed membership and non-membership of x in A, respectively, and satisfy μA(x)+νA(x)⩽1[TeX:] $ {\mu _{A}}(x)+{\nu _{A}}(x)\leqslant 1$.

We recall for an intuitionistic fuzzy set the membership grade of x in A which is represented as a pair (μA(x),νA(x))[TeX:] $ ({\mu _{A}}(x),{\nu _{A}}(x))$ is called a intuitionistic fuzzy number (IFN) (Xu, 2007) and the set of all IFNs is denoted as IFN[TeX:] $ \mathcal{IFN}$. Here, the expression πA(x)=1−μA(x)−νA(x)[TeX:] $ {\pi _{A}}(x)=1-{\mu _{A}}(x)-{\nu _{A}}(x)$ is called the hesitancy of x. The IFN α=(μα,να)[TeX:] $ \alpha =({\mu _{\alpha }},{\nu _{\alpha }})$ has a physical interpretation, for example, if α=(0.3,0.2)[TeX:] $ \alpha =(0.3,0.2)$, then it can be interpreted as “the vote for resolution is 3 in favour, 2 against, and 5 abstentions (Gau and Buehrer, 1993). The following partial order ⩽ on IFN[TeX:] $ \mathcal{IFN}$, which is defined for α=(μα,να)[TeX:] $ \alpha =({\mu _{\alpha }},{\nu _{\alpha }})$ and β=(μβ,νβ)[TeX:] $ \beta =({\mu _{\beta }},{\nu _{\beta }})$ as β⩽α[TeX:] $ \beta \leqslant \alpha $ if and only if μβ⩽μα[TeX:] $ {\mu _{\beta }}\leqslant {\mu _{\alpha }}$ and να⩽νβ[TeX:] $ {\nu _{\alpha }}\leqslant {\nu _{\beta }}$. For an IFN α, a score function s (Chen and Tan, 1994), which is defined as the difference of membership and non-membership function, can be denoted as: s(α)=μα−να[TeX:] $ s(\alpha )={\mu _{\alpha }}-{\nu _{\alpha }}$, where s(α)∈[−1,1][TeX:] $ s(\alpha )\in [-1,1]$. The larger the score s(α)[TeX:] $ s(\alpha )$ is, the greater the IFN α is. To make the comparison method more discriminatory, an accuracy function h (Hong and Choi, 2000) is defined as follows: h(α)=μα+να,[TeX:] $ h(\alpha )={\mu _{\alpha }}+{\nu _{\alpha }},$ where h(α)∈[0,1][TeX:] $ h(\alpha )\in [0,1]$. When the scores are the same, the larger the accuracy h(α)[TeX:] $ h(\alpha )$ is, the greater the IFN α is. However, it is obvious that h(α)+πα=1[TeX:] $ h(\alpha )+{\pi _{\alpha }}=1$.

Definition 2.6

Definition 2.6(See Xu, 2007).

Let α,β[TeX:] $ \alpha ,\beta $ be two IFNs. Then, we have

  • (1) If s(β)>s(α)[TeX:] $ s(\beta )>s(\alpha )$, then β is bigger than α, i.e. β≻α[TeX:] $ \beta \succ \alpha $.

  • (2) If s(α)=s(β)[TeX:] $ s(\alpha )=s(\beta )$:

    • (a) if h(β)>h(α)[TeX:] $ h(\beta )>h(\alpha )$, then β is bigger than α, i.e. β≻α[TeX:] $ \beta \succ \alpha $;

    • (b) if h(α)=h(β)[TeX:] $ h(\alpha )=h(\beta )$, i.e. α=β[TeX:] $ \alpha =\beta $.

Definition 2.7

Definition 2.7(See Beliakov et al., 2011; Xia et al., 2012).

Let αi=(μαi,ναi)[TeX:] $ {\alpha _{i}}=({\mu _{{\alpha _{i}}}},{\nu _{{\alpha _{i}}}})$ (i=1,2)[TeX:] $ (i=1,2)$ and α=(μα,να)[TeX:] $ \alpha =({\mu _{\alpha }},{\nu _{\alpha }})$ be three IFNs, then we have

  • (1) α1⊕α2=(1−g−1(g(1−μα1)+g(1−μα2)),g−1(g(να1)+g(να2)))[TeX:] $ {\alpha _{1}}\oplus {\alpha _{2}}=(1-{g^{-1}}(g(1-{\mu _{{\alpha _{1}}}})+g(1-{\mu _{{\alpha _{2}}}})),{g^{-1}}(g({\nu _{{\alpha _{1}}}})+g({\nu _{{\alpha _{2}}}})))$;

  • (2) α1⊗α2=(g−1(g(μα1)+g(μα2)),1−g−1(g(1−να1)+g(1−να2)))[TeX:] $ {\alpha _{1}}\otimes {\alpha _{2}}=({g^{-1}}(g({\mu _{{\alpha _{1}}}})+g({\mu _{{\alpha _{2}}}})),1-{g^{-1}}(g(1-{\nu _{{\alpha _{1}}}})+g(1-{\nu _{{\alpha _{2}}}})))$;

  • (3) λα=(1−g−1(λg(1−μα)),g−1(λg(να)))[TeX:] $ \lambda \alpha =(1-{g^{-1}}(\lambda g(1-{\mu _{\alpha }})),{g^{-1}}(\lambda g({\nu _{\alpha }})))$, λ>0[TeX:] $ \lambda >0$;

  • (4) αλ=(g−1(λg(μα)),1−g−1(λg(1−να)))[TeX:] $ {\alpha ^{\lambda }}=({g^{-1}}(\lambda g({\mu _{\alpha }})),1-{g^{-1}}(\lambda g(1-{\nu _{\alpha }})))$, λ>0[TeX:] $ \lambda >0$;

  • (3) αc=(να,μα)[TeX:] $ {\alpha ^{c}}=({\nu _{\alpha }},{\mu _{\alpha }})$.

For convenience, if not otherwise specified, we always denote

α=(α1,α2,…,αn),μα=(μα1,μα2,…,μαn),να=(να1,να2,…,ναn),1−μα=(1−μα1,1−μα2,…,1−μαn),1−να=(1−να1,1−να2,…,1−ναn),[TeX:] \[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle \alpha & \displaystyle =& \displaystyle ({\alpha _{1}},{\alpha _{2}},\dots ,{\alpha _{n}}),\\ {} \displaystyle {\mu _{\alpha }}& \displaystyle =& \displaystyle ({\mu _{{\alpha _{1}}}},{\mu _{{\alpha _{2}}}},\dots ,{\mu _{{\alpha _{n}}}}),\hspace{1em}{\nu _{\alpha }}=({\nu _{{\alpha _{1}}}},{\nu _{{\alpha _{2}}}},\dots ,{\nu _{{\alpha _{n}}}}),\\ {} \displaystyle 1-{\mu _{\alpha }}& \displaystyle =& \displaystyle (1-{\mu _{{\alpha _{1}}}},1-{\mu _{{\alpha _{2}}}},\dots ,1-{\mu _{{\alpha _{n}}}}),\\ {} \displaystyle 1-{\nu _{\alpha }}& \displaystyle =& \displaystyle (1-{\nu _{{\alpha _{1}}}},1-{\nu _{{\alpha _{2}}}},\dots ,1-{\nu _{{\alpha _{n}}}}),\end{array}\]
for IFNs αj[TeX:] $ {\alpha _{j}}$ ( j=1,2,…,n[TeX:] $ j=1,2,\dots ,n$).

Definition 2.8

Definition 2.8(See Xia et al., 2012).

Let αj[TeX:] $ {\alpha _{j}}$ ( j=1,2,…,n[TeX:] $ j=1,2,\dots ,n$) be IFNs and Tω,g[TeX:] $ {T_{\omega ,g}}$ be a weighted Archimedean t-norm with an additive generator g and IFWATω,g:IFNn→IFN[TeX:] $ {\text{IFWA}\hspace{2.5pt}^{{T_{\omega ,g}}}}:{\mathcal{IFN}^{n}}\to \mathcal{IFN}$, if

IFWATω,g(α)=⨁j=1nωjαj=(1−Tω,g(1−μα),Tω,g(να)),[TeX:] \[ {\text{IFWA}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )={\underset{j=1}{\overset{n}{\bigoplus }}}{\omega _{j}}{\alpha _{j}}=\big(1-{T_{\omega ,g}}(1-{\mu _{\alpha }}),{T_{\omega ,g}}({\nu _{\alpha }})\big),\]
then IFWA Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ is called an intuitionistic fuzzy weighted averaging (IFWA Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$) operator of dimension n w.r.t. Tω,g[TeX:] $ {T_{\omega ,g}}$.

Definition 2.9

Definition 2.9(See Xia et al., 2012).

Let αj[TeX:] $ {\alpha _{j}}$ ( j=1,2,…,n[TeX:] $ j=1,2,\dots ,n$) be IFNs and Tω,g[TeX:] $ {T_{\omega ,g}}$ be a weighted Archimedean t-norm with an additive generator g and IFWMTω,g:IFNn→IFN[TeX:] $ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}:{\mathcal{IFN}^{n}}\to \mathcal{IFN}$, if

IFWMTω,g(α)=⨂j=1nαjωj=(Tω,g(μα),1−Tω,g(1−να)),[TeX:] \[ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )={\underset{j=1}{\overset{n}{\bigotimes }}}{\alpha _{j}^{{\omega _{j}}}}=\big({T_{\omega ,g}}({\mu _{\alpha }}),1-{T_{\omega ,g}}(1-{\nu _{\alpha }})\big),\]
then IFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ is called an intuitionistic fuzzy weighted mean (IFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$) operator of dimension n w.r.t. Tω,g[TeX:] $ {T_{\omega ,g}}$.

However, if we assign g(t)[TeX:] $ g(t)$ to gγH(t)=ln(γ+(1−γ)tt)[TeX:] $ {g_{\gamma }^{H}}(t)=\ln (\frac{\gamma +(1-\gamma )t}{t})$, gγSS(t)=1−tγγ[TeX:] $ {g_{\gamma }^{SS}}(t)=\frac{1-{t^{\gamma }}}{\gamma }$ and gγD(t)=(1−tt)γ[TeX:] $ {g_{\gamma }^{D}}(t)={(\frac{1-t}{t})^{\gamma }}$, respectively, then the following families of IFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ operators are obtained:

(1)
IFWMTω,gγH(α)=(Tω,gγH(μαj),1−Tω,gγH(1−ναj)),[TeX:] \[ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{H}}}}}}(\alpha )=\big({T_{\omega ,{g_{\gamma }^{H}}}}({\mu _{{\alpha _{j}}}}),1-{T_{\omega ,{g_{\gamma }^{H}}}}(1-{\nu _{{\alpha _{j}}}})\big),\]
(2)
IFWMTω,gγSS(α)=(Tω,gγSS(μαj),1−Tω,gγSS(1−ναj)),[TeX:] \[ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}(\alpha )=\big({T_{\omega ,{g_{\gamma }^{SS}}}}({\mu _{{\alpha _{j}}}}),1-{T_{\omega ,{g_{\gamma }^{SS}}}}(1-{\nu _{{\alpha _{j}}}})\big),\]
(3)
IFWMTω,gγD(α)=(Tω,gγD(μαj),1−Tω,gγD(1−ναj)).[TeX:] \[ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{D}}}}}}(\alpha )=\big({T_{\omega ,{g_{\gamma }^{D}}}}({\mu _{{\alpha _{j}}}}),1-{T_{\omega ,{g_{\gamma }^{D}}}}(1-{\nu _{{\alpha _{j}}}})\big).\]
Note that IFWMTω,gγH[TeX:] $ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{H}}}}}}$ has been investigated in Xia et al. (2012). In particular, if γ=1[TeX:] $ \gamma =1$, then
IFWMTω,g1H(α)=(∏j=1nμαjωj,1−∏j=1n(1−ναj)ωj),IFWMTω,g1SS(α)=(∑j=1nωjμαj,∑j=1nωjναj),IFWMTω,g1D(α)=((∑j=1nωjμαj−1)−1,1−(∑j=1nωj(1−ναj)−1)−1),[TeX:] \[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{1}^{H}}}}}}(\alpha )& \displaystyle =& \displaystyle \Bigg({\prod \limits_{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}},1-{\prod \limits_{j=1}^{n}}{(1-{\nu _{{\alpha _{j}}}})^{{\omega _{j}}}}\Bigg),\\ {} \displaystyle {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{1}^{SS}}}}}}(\alpha )& \displaystyle =& \displaystyle \Bigg({\sum \nolimits_{j=1}^{n}}{\omega _{j}}{\mu _{{\alpha _{j}}}},{\sum \limits_{j=1}^{n}}{\omega _{j}}{\nu _{{\alpha _{j}}}}\Bigg),\\ {} \displaystyle {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{1}^{D}}}}}}(\alpha )& \displaystyle =& \displaystyle \Bigg({\Bigg(\hspace{0.1667em}{\sum \limits_{j=1}^{n}}{\omega _{j}}{\mu _{{\alpha _{j}}}^{-1}}\Bigg)^{-1}},1-{\Bigg(\hspace{0.1667em}{\sum \limits_{j=1}^{n}}{\omega _{j}}{(1-{\nu _{{\alpha _{j}}}})^{-1}}\Bigg)^{-1}}\Bigg),\end{array}\]
IFWMTω,g1H=IFWGω[TeX:] $ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{1}^{H}}}}}}={\text{IFWG}\hspace{2.5pt}_{\omega }}$ defined by Xu (2007), IFWMTω,g1SS=IFWMω[TeX:] $ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{1}^{SS}}}}}}={\text{IFWM}\hspace{2.5pt}_{\omega }}$ defined by Beliakov et al. (2011) and IFWMTω,g1D[TeX:] $ {\text{IFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{1}^{D}}}}}}$ are natural generalizations of fuzzy weighted geometric mean, arithmetic mean and Harmonic mean, respectively.

However, Deschrijver and Kerre (2008) provided a natural extension of an aggregation function to the environment of interval-valued fuzzy set, and Beliakov and James (2013) gave the definition for A-IFS representation as follows:

Definition 2.10

Definition 2.10(See Beliakov and James, 2013).

Given an aggregation function agg:[0,1]n→[0,1][TeX:] $ agg:{[0,1]^{n}}\to [0,1]$, the natural extension of the aggregation function agg[TeX:] $ agg$ is given by Agg:IFNn→IFN[TeX:] $ Agg:{\mathcal{IFN}^{n}}\to \mathcal{IFN}$, Agg(α)=(agg(μα),1−agg(1−να))[TeX:] $ Agg(\alpha )=(agg({\mu _{\alpha }}),1-agg(1-{\nu _{\alpha }}))$.

3Symmetric Intuitionistic Fuzzy Weighted Mean Operators w.r.t. Weighted Archimedean t-Norms and t-Conorms

Considering the work of Beliakov and James (2013), Calvo and Mesiar (2003), Tan et al. (2013), Xia and Xu (2012), Xia et al. (2012), in this section we develop some symmetric intuitionistic fuzzy weighted mean operators w.r.t. weighted Archimedean t-norms and t-conorms to fairly treat membership and non-membership information and provide more choices for the decision maker by considering his/her attitude with parameters.

3.1Symmetric Intuitionistic Fuzzy Weighted Mean Operators w.r.t. Weighted Archimedean t-Norms

Lemma 3.1.

Let αj[TeX:] $ {\alpha _{j}}$ (j=1,2,…,n)[TeX:] $ (j=1,2,\dots ,n)$ be a collection of IFNs and Tω,g[TeX:] $ {T_{\omega ,g}}$ be a weighted Archimedean t-norm with an additive generator g. Then

(4)
(Tω,g(μα)Tω,g(μα)+Tω,g(1−μα),Tω,g(να)Tω,g(να)+Tω,g(1−να))[TeX:] \[ \bigg(\frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})},\frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}(1-{\nu _{\alpha }})}\bigg)\]
is an IFN.

Proof.

Since αj(j=1,2,…,n)[TeX:] $ {\alpha _{j}}(j=1,2,\dots ,n)$ are IFNs, it holds that μαj+ναj⩽1[TeX:] $ {\mu _{{\alpha _{j}}}}+{\nu _{{\alpha _{j}}}}\leqslant 1$. The antitonicity of g leads that Tω,g(να)⩽Tω,g(1−μα)[TeX:] $ {T_{\omega ,g}}({\nu _{\alpha }})\leqslant {T_{\omega ,g}}(1-{\mu _{\alpha }})$. Thus we have Tω,g(μα)Tω,g(μα)+Tω,g(1−μα)⩽Tω,g(μα)Tω,g(μα)+Tω,g(να)[TeX:] $ \frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})}\leqslant \frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}({\nu _{\alpha }})}$. Similarly, it holds that Tω,g(να)Tω,g(να)+Tω,g(1−να)⩽Tω,g(να)Tω,g(να)+Tω,g(μα)[TeX:] $ \frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}(1-{\nu _{\alpha }})}\leqslant \frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}({\mu _{\alpha }})}$. Then it yields immediately that Tω,g(μα)Tω,g(μα)+Tω,g(1−μα)+Tω,g(να)Tω,g(να)+Tω,g(1−να)⩽1[TeX:] $ \frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})}+\frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}(1-{\nu _{\alpha }})}\leqslant 1$, i.e. Eq. (4) is an IFN. □

Definition 3.2.

Let αj[TeX:] $ {\alpha _{j}}$ ( j=1,2,…,n[TeX:] $ j=1,2,\dots ,n$) be a collection of IFNs, Tω,g[TeX:] $ {T_{\omega ,g}}$ be a weighted Archimedean t-norm and SIFWMTω,g:IFNn→IFN[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}:{\mathcal{IFN}^{n}}\to \mathcal{IFN}$, if

(5)
SIFWMTω,g(α)=Tω,g(μα)Tω,g(μα)+Tω,g(1−μα),Tω,g(να)Tω,g(να)+Tω,g(1−να),[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )=\left(\frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})},\frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}(1-{\nu _{\alpha }})}\right),\]
then SIFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ is called a symmetric intuitionistic fuzzy weighted mean (SIFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$) operator of dimension n w.r.t. Tω,g[TeX:] $ {T_{\omega ,g}}$.

Especially, ναj=1−μαj[TeX:] $ {\nu _{{\alpha _{j}}}}=1-{\mu _{{\alpha _{j}}}}$ for all j=1,2,…,n[TeX:] $ j=1,2,\dots ,n$, that is, all αj[TeX:] $ {\alpha _{j}}$ are reduced to μαj[TeX:] $ {\mu _{{\alpha _{j}}}}$, respectively, then Eq. (5) has the following form:

SIFWMTω,g(α)=Tω,g(μα)Tω,g(μα)+Tω,g(1−μα),1−Tω,g(μα)Tω,g(μα)+Tω,g(1−μα),[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )=\left(\frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})},1-\frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})}\right),\]
which becomes both a symmetric sum operator of dimension n (Beliakov et al., 2007) and a weighted t-norm-based aggregation operator with H(x,y)=x1+x−y[TeX:] $ H(x,y)=\frac{x}{1+x-y}$ (Calvo and Mesiar, 2003) to aggregate fuzzy information.

Proposition 3.3.

Let αj,βj[TeX:] $ {\alpha _{j}},{\beta _{j}}$ (j=1,2,…,n)[TeX:] $ (j=1,2,\dots ,n)$ be two collections of IFNs with α=(α1,α2,…,αn)[TeX:] $ \alpha =({\alpha _{1}},{\alpha _{2}},\dots ,{\alpha _{n}})$, β=(β1,β2,…,βn)[TeX:] $ \beta =({\beta _{1}},{\beta _{2}},\dots ,{\beta _{n}})$, respectively, and Tω,g[TeX:] $ {T_{\omega ,g}}$ be a weighted Archimedean t-norm with an additive generator g.

  • (1) If all αj[TeX:] $ {\alpha _{j}}$ are equal, i.e. αj=δ=(μδ,νδ)[TeX:] $ {\alpha _{j}}=\delta =({\mu _{\delta }},{\nu _{\delta }})$, for all j, then SIFWMTω,g(α)=δ[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )=\delta $;

  • (2) If αj⩽βj[TeX:] $ {\alpha _{j}}\leqslant {\beta _{j}}$ for all j, then SIFWMTω,g(α)⩽SIFWMTω,g(β)[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )\leqslant {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\beta )$;

  • (3) Let α−=(min(μα),max(να))[TeX:] $ {\alpha ^{-}}=(\min ({\mu _{\alpha }}),\max ({\nu _{\alpha }}))$ and α+=(max(μα),min(ναj))[TeX:] $ {\alpha ^{+}}=(\max ({\mu _{\alpha }}),\min ({\nu _{{\alpha _{j}}}}))$, then α−⩽SIFWMTω,g(α)⩽α+[TeX:] $ {\alpha ^{-}}\leqslant {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )\leqslant {\alpha ^{+}}$.

Proof.

(1) By Definition 3.2, it holds that

SIFWMTω,g(α)=(Tω,g(μδ)Tω,g(μδ)+Tω,g(1−μδ),1−Tω,g(μδ)Tω,g(μδ)+Tω,g(1−μδ))=(μδ,νδ)=δ,[TeX:] \[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )& \displaystyle =& \displaystyle \bigg(\frac{{T_{\omega ,g}}({\mu _{\delta }})}{{T_{\omega ,g}}({\mu _{\delta }})+{T_{\omega ,g}}(1-{\mu _{\delta }})},1-\frac{{T_{\omega ,g}}({\mu _{\delta }})}{{T_{\omega ,g}}({\mu _{\delta }})+{T_{\omega ,g}}(1-{\mu _{\delta }})}\bigg)\\ {} & \displaystyle =& \displaystyle ({\mu _{\delta }},{\nu _{\delta }})=\delta ,\end{array}\]
and hence SIFWMTω,g(α)=δ[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )=\delta $.

(2) Since αj⩽βj[TeX:] $ {\alpha _{j}}\leqslant {\beta _{j}}$ for all j, i.e. μαj⩽μβj[TeX:] $ {\mu _{{\alpha _{j}}}}\leqslant {\mu _{{\beta _{j}}}}$ and νβj⩽ναj[TeX:] $ {\nu _{{\beta _{j}}}}\leqslant {\nu _{{\alpha _{j}}}}$, we have Tω,g(μα)⩽Tω,g(μβ)[TeX:] $ {T_{\omega ,g}}({\mu _{\alpha }})\leqslant {T_{\omega ,g}}({\mu _{\beta }})$, Tω,g(1−μβ)⩽Tω,g(1−μα)[TeX:] $ {T_{\omega ,g}}(1-{\mu _{\beta }})\leqslant {T_{\omega ,g}}(1-{\mu _{\alpha }})$, Tω,g(1−μα)⩽Tω,g(1−μβ)[TeX:] $ {T_{\omega ,g}}(1-{\mu _{\alpha }})\leqslant {T_{\omega ,g}}(1-{\mu _{\beta }})$ and Tω,g(μβ)⩽Tω,g(μα)[TeX:] $ {T_{\omega ,g}}({\mu _{\beta }})\leqslant {T_{\omega ,g}}({\mu _{\alpha }})$, and hence

Tω,g(μα)Tω,g(μα)+Tω,g(1−μα)=11+Tω,g(1−μα)Tω,g(μα)⩽11+Tω,g(1−μβ)Tω,g(μβ)=Tω,g(μβ)Tω,g(μβ)+Tω,g(1−μβ).[TeX:] \[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle \frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})}& \displaystyle =& \displaystyle \frac{1}{1+\frac{{T_{\omega ,g}}(1-{\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})}}\\ {} & \displaystyle \leqslant & \displaystyle \frac{1}{1+\frac{{T_{\omega ,g}}(1-{\mu _{\beta }})}{{T_{\omega ,g}}({\mu _{\beta }})}}=\frac{{T_{\omega ,g}}({\mu _{\beta }})}{{T_{\omega ,g}}({\mu _{\beta }})+{T_{\omega ,g}}(1-{\mu _{\beta }})}.\end{array}\]
Similarly, it holds that Tω,g(νβ)Tω,g(νβ)+Tω,g(1−νβ)⩽Tω,g(να)Tω,g(να)+Tω,g(1−να)[TeX:] $ \frac{{T_{\omega ,g}}({\nu _{\beta }})}{{T_{\omega ,g}}({\nu _{\beta }})+{T_{\omega ,g}}(1-{\nu _{\beta }})}\leqslant \frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}(1-{\nu _{\alpha }})}$. Thus we obtain SIFWMTω,g(α)⩽SIFWMTω,g(β)[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )\leqslant {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\beta )$.

(3) Since minj{μαj}⩽μαj⩽maxj{μαj}[TeX:] $ {\min _{j}}\{{\mu _{{\alpha _{j}}}}\}\leqslant {\mu _{{\alpha _{j}}}}\leqslant {\max _{j}}\{{\mu _{{\alpha _{j}}}}\}$ and minj{ναj}⩽ναj⩽maxj{ναj}[TeX:] $ {\min _{j}}\{{\nu _{{\alpha _{j}}}}\}\leqslant {\nu _{{\alpha _{j}}}}\leqslant {\max _{j}}\{{\nu _{{\alpha _{j}}}}\}$, it follows from (1) that

minj{μαj}⩽Tω,g(μα)Tω,g(μα)+Tω,g(1−μα),1−Tω,g(μα)Tω,g(μα)+Tω,g(1−μα)⩽maxj{μαj},minj{ναj}⩽Tω,g(να)Tω,g(να)+Tω,g(1−να),1−Tω,g(να)Tω,g(να)+Tω,g(1−να)⩽maxj{ναj},[TeX:] \[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle \underset{j}{\min }\{{\mu _{{\alpha _{j}}}}\}& \displaystyle \leqslant & \displaystyle \left(\frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})},1-\frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})}\right)\\ {} & \displaystyle \leqslant & \displaystyle \underset{j}{\max }\{{\mu _{{\alpha _{j}}}}\},\\ {} \displaystyle \underset{j}{\min }\{{\nu _{{\alpha _{j}}}}\}& \displaystyle \leqslant & \displaystyle \left(\frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}(1-{\nu _{\alpha }})},1-\frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}(1-{\nu _{\alpha }})}\right)\\ {} & \displaystyle \leqslant & \displaystyle \underset{j}{\max }\{{\nu _{{\alpha _{j}}}}\},\end{array}\]
thus we have α−⩽SIFWMTω,g(α)⩽α+[TeX:] $ {\alpha ^{-}}\leqslant {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,g}}}}(\alpha )\leqslant {\alpha ^{+}}$. □

When the additive generator g is assigned to different forms, some specific intuitionistic fuzzy aggregation operators can be provided as follows:

  • (1) If g(t)=gγH(t)[TeX:] $ g(t)={g_{\gamma }^{H}}(t)$, then the SIFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ operator is reduced to the following form:

    (6)
    SIFWMTω,gγH(α)=Tω,gγH(μα)Tω,gγH(μα)+Tω,gγH(1−μα),Tω,gγH(να)Tω,gγH(να)+Tω,gγH(1−να).[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{H}}}}}}(\alpha )=\left(\hspace{-0.1667em}\frac{{T_{\omega ,{g_{\gamma }^{H}}}}({\mu _{\alpha }})}{{T_{\omega ,{g_{\gamma }^{H}}}}({\mu _{\alpha }})+{T_{\omega ,{g_{\gamma }^{H}}}}(1-{\mu _{\alpha }})},\frac{{T_{\omega ,{g_{\gamma }^{H}}}}({\nu _{\alpha }})}{{T_{\omega ,{g_{\gamma }^{H}}}}({\nu _{\alpha }})+{T_{\omega ,{g_{\gamma }^{H}}}}(1-{\nu _{\alpha }})}\hspace{-0.1667em}\right).\]
    Especially, if γ=1[TeX:] $ \gamma =1$, i.e. gH1(t)=−ln(t)[TeX:] $ {g^{{H_{1}}}}(t)=-\ln (t)$, then the SIFWMTω,g1H[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{1}^{H}}}}}}$ operator with Tω,g1H(x)=∏j=1nxjωj[TeX:] $ {T_{\omega ,{g_{1}^{H}}}}(x)={\textstyle\prod _{j=1}^{n}}{x_{j}^{{\omega _{j}}}}$ is called a symmetric intuitionistic fuzzy weighted geometric (SIFWG) operator defined by Xia and Xu (2012) ; if γ=2[TeX:] $ \gamma =2$, i.e. gH2(t)=ln(2−tt)[TeX:] $ {g^{{H_{2}}}}(t)=\ln (\frac{2-t}{t})$, then the SIFWMTω,g2H[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{2}^{H}}}}}}$ operator with Tω,g2H(x)=2∏j=1n(2xj−1)ωj+1[TeX:] $ {T_{\omega ,{g_{2}^{H}}}}(x)=\frac{2}{{\textstyle\prod _{j=1}^{n}}{(\frac{2}{{x_{j}}}-1)^{{\omega _{j}}}}+1}$ is the symmetric form of IFWG operator based on the Einstein t-norm defined by Wang and Liu (2012) ; if γ→∞[TeX:] $ \gamma \to \infty $, then it holds that
    limγ→∞Tω,gγH(x)=limγ→∞γ∏j=1n(γxj+1−γ)ωj+γ−1=limγ→∞(∏j=1n(γxj+1−γ)ωj∏j=1nγωj+γ−1γ)−1=limγ→∞(∏j=1n(1xj+1−γγ)ωj+γ−1γ)−1=(∏j=1n(1xj−1)ωj+1)−1,[TeX:] \[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle \underset{\gamma \to \infty }{\lim }{T_{\omega ,{g_{\gamma }^{H}}}}(x)& \displaystyle =& \displaystyle \underset{\gamma \to \infty }{\lim }\frac{\gamma }{{\textstyle\textstyle\prod _{j=1}^{n}}{(\frac{\gamma }{{x_{j}}}+1-\gamma )^{{\omega _{j}}}}+\gamma -1}\\ {} & \displaystyle =& \displaystyle \underset{\gamma \to \infty }{\lim }{\bigg(\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{(\frac{\gamma }{{x_{j}}}+1-\gamma )^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{\gamma ^{{\omega _{j}}}}}+\frac{\gamma -1}{\gamma }\bigg)^{-1}}\\ {} & \displaystyle =& \displaystyle \underset{\gamma \to \infty }{\lim }{\bigg({\prod \limits_{j=1}^{n}}{\bigg(\frac{1}{{x_{j}}}+\frac{1-\gamma }{\gamma }\bigg)^{{\omega _{j}}}}+\frac{\gamma -1}{\gamma }\bigg)^{-1}}\\ {} & \displaystyle =& \displaystyle {\bigg({\prod \limits_{j=1}^{n}}{\bigg(\frac{1}{{x_{j}}}-1\bigg)^{{\omega _{j}}}}+1\bigg)^{-1}},\end{array}\]
    thus we have
    (7)
    SIFWMTω,g∞H(α)=∏j=1nμαjωj∏j=1n(1−μαj)ωj+∏j=1nμαjωj,∏j=1nναjωj∏j=1n(1−ναj)ωj+∏j=1nναjωj,[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\infty }^{H}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\mu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}},\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\nu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}\right),\end{array}\]
    if γ→0[TeX:] $ \gamma \to 0$, in a similar way, we have
    (8)
    SIFWMTω,g0H(α)=(1∑j=1nωjμαj,1∑j=1nωjναj).[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{0}^{H}}}}}}(\alpha )=\bigg(\frac{1}{{\textstyle\textstyle\sum _{j=1}^{n}}\frac{{\omega _{j}}}{{\mu _{{\alpha _{j}}}}}},\frac{1}{{\textstyle\textstyle\sum _{j=1}^{n}}\frac{{\omega _{j}}}{{\nu _{{\alpha _{j}}}}}}\bigg).\]

  • (2) If g(t)=gγSS(t)[TeX:] $ g(t)={g_{\gamma }^{SS}}(t)$, then the SIFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ operator is reduced to the following form:

    (9)
    SIFWMTω,gγSS(α)=Tω,gγSS(μα)Tω,gγSS(μα)+Tω,gγSS(1−μα),Tω,gγSS(να)Tω,gγSS(να)+Tω,gγSS(1−να).[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{T_{\omega ,{g_{\gamma }^{SS}}}}({\mu _{\alpha }})}{{T_{\omega ,{g_{\gamma }^{SS}}}}({\mu _{\alpha }})+{T_{\omega ,{g_{\gamma }^{SS}}}}(1-{\mu _{\alpha }})},\frac{{T_{\omega ,{g_{\gamma }^{SS}}}}({\nu _{\alpha }})}{{T_{\omega ,{g_{\gamma }^{SS}}}}({\nu _{\alpha }})+{T_{\omega ,{g_{\gamma }^{SS}}}}(1\hspace{-0.1667em}-\hspace{-0.1667em}{\nu _{\alpha }})}\right).\end{array}\]
    Particularly, if γ→∞[TeX:] $ \gamma \to \infty $, we assume that xk=max{x1,x2,…,xn}[TeX:] $ {x_{k}}=\max \{{x_{1}},{x_{2}},\dots ,{x_{n}}\}$, k∈{1,2,…,n}[TeX:] $ k\in \{1,2,\dots ,n\}$, then it follows from L’Hôpital’s rule that
    limγ→∞Tω,gγSS(x)=limγ→∞(∑j=1nωjxjγ)1γ=limγ→∞eln(∑j=1nωjxjγ)γ=limγ→∞e∑j=1nωjxjγlnxj∑j=1nωjxjγ=elimγ→∞∑j=1nωjxjγlnxj∑j=1nωjxjγ=elimγ→∞∑j=1nωj(xjxk)γlnxj∑j=1nωj(xjxk)γ=eωklnxkωk=xk,[TeX:] \[\begin{array}{l}\displaystyle \underset{\gamma \to \infty }{\lim }{T_{\omega ,{g_{\gamma }^{SS}}}}(x)=\underset{\gamma \to \infty }{\lim }{\Big({\sum \limits_{j=1}^{n}}{\omega _{j}}{x_{j}^{\gamma }}\Big)^{\frac{1}{\gamma }}}=\underset{\gamma \to \infty }{\lim }{e^{\frac{\ln \Big({\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{x_{j}^{\gamma }}\Big)}{\gamma }}}\\ {} \displaystyle \hspace{1em}=\underset{\gamma \to \infty }{\lim }{e^{\frac{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{x_{j}^{\gamma }}\ln {x_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{x_{j}^{\gamma }}}}}={e^{{\lim \nolimits_{\gamma \to \infty }}\frac{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{x_{j}^{\gamma }}\ln {x_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{x_{j}^{\gamma }}}}}={e^{{\lim \nolimits_{\gamma \to \infty }}\frac{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{(\frac{{x_{j}}}{{x_{k}}})^{\gamma }}\ln {x_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{(\frac{{x_{j}}}{{x_{k}}})^{\gamma }}}}}\\ {} \displaystyle \hspace{1em}={e^{\frac{{\omega _{k}}\ln {x_{k}}}{{\omega _{k}}}}}={x_{k}},\end{array}\]
    thus we get
    (10)
    SIFWMTω,g∞SS(α)=max(μα)max(μα)+max(1−μα),max(να)max(να)+max(1−να).[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\infty }^{SS}}}}}}(\alpha )=\left(\frac{\max ({\mu _{\alpha }})}{\max ({\mu _{\alpha }})+\max (1-{\mu _{\alpha }})},\frac{\max ({\nu _{\alpha }})}{\max ({\nu _{\alpha }})+\max (1-{\nu _{\alpha }})}\right).\]
    Similarly, if γ→−∞[TeX:] $ \gamma \to -\infty $, then we have
    (11)
    SIFWMTω,g−∞SS(α)=min(μα)min(μα)+min(1−μα),min(να)min(να)+min(1−να),[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{-\infty }^{SS}}}}}}(\alpha )=\left(\frac{\min ({\mu _{\alpha }})}{\min ({\mu _{\alpha }})+\min (1-{\mu _{\alpha }})},\frac{\min ({\nu _{\alpha }})}{\min ({\nu _{\alpha }})+\min (1-{\nu _{\alpha }})}\right),\]
    and if γ→0[TeX:] $ \gamma \to 0$, then we get
    (12)
    SIFWMTω,g0SS(α)=∏j=1nμαjωj∏j=1n(1−μαj)ωj+∏j=1nμαjωj,∏j=1nναjωj∏j=1n(1−ναj)ωj+∏j=1nναjωj.[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{0}^{SS}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\mu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}},\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\nu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}\right).\end{array}\]

  • (3) If g(t)=gγD(t)[TeX:] $ g(t)={g_{\gamma }^{D}}(t)$, then the SIFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ operator is reduced to the following case:

    (13)
    SIFWMTω,gγD(α)=Tω,gγD(μα)Tω,gγD(μα)+Tω,gγD(1−μα),Tω,gγD(να)Tω,gγD(να)+Tω,gγD(1−να).[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{D}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{T_{\omega ,{g_{\gamma }^{D}}}}({\mu _{\alpha }})}{{T_{\omega ,{g_{\gamma }^{D}}}}({\mu _{\alpha }})+{T_{\omega ,{g_{\gamma }^{D}}}}(1-{\mu _{\alpha }})},\frac{{T_{\omega ,{g_{\gamma }^{D}}}}({\nu _{\alpha }})}{{T_{\omega ,{g_{\gamma }^{D}}}}({\nu _{\alpha }})+{T_{\omega ,{g_{\gamma }^{D}}}}(1-{\nu _{\alpha }})}\right).\end{array}\]
    In particular, if γ=1[TeX:] $ \gamma =1$, then the SIFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ operator is reduced to the following case:
    SIFWMTω,g1D(α)=Tω,g1D(μα)Tω,g1D(μα)+Tω,g1D(1−μα),Tω,g1D(να)Tω,g1D(να)+Tω,g1D(1−να),[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{1}^{D}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{T_{\omega ,{g_{1}^{D}}}}({\mu _{\alpha }})}{{T_{\omega ,{g_{1}^{D}}}}({\mu _{\alpha }})+{T_{\omega ,{g_{1}^{D}}}}(1-{\mu _{\alpha }})},\frac{{T_{\omega ,{g_{1}^{D}}}}({\nu _{\alpha }})}{{T_{\omega ,{g_{1}^{D}}}}({\nu _{\alpha }})+{T_{\omega ,{g_{1}^{D}}}}(1-{\nu _{\alpha }})}\right),\end{array}\]
    where Tω,g1D(x)=(∑j=1nωjxj−1)−1[TeX:] $ {T_{\omega ,{g_{1}^{D}}}}(x)={({\textstyle\sum _{j=1}^{n}}{\omega _{j}}{x_{j}^{-1}})^{-1}}$. Now, we consider the case for γ→∞[TeX:] $ \gamma \to \infty $. Notice that
    limγ→∞(xj−1−1)γ=∞,xj<12,1,xj=12,0,xj>12.[TeX:] \[ \underset{\gamma \to \infty }{\lim }{\big({x_{j}^{-1}}-1\big)^{\gamma }}=\left\{\begin{array}{l@{\hskip4.0pt}l}\infty ,\hspace{1em}& {x_{j}}<\frac{1}{2},\\ {} 1,\hspace{1em}& {x_{j}}=\frac{1}{2},\\ {} 0,\hspace{1em}& {x_{j}}>\frac{1}{2}.\end{array}\right.\]
    For limγ→∞(∑j=1nωj(xj−1−1)γ)1γ[TeX:] $ {\lim \nolimits_{\gamma \to \infty }}{({\textstyle\sum _{j=1}^{n}}{\omega _{j}}{({x_{j}^{-1}}-1)^{\gamma }})^{\frac{1}{\gamma }}}$, taking xk=min{x1,x2,…,xn}[TeX:] $ {x_{k}}=\min \{{x_{1}},{x_{2}},\dots ,{x_{n}}\}$ with k∈{1,…,n}[TeX:] $ k\in \{1,\dots ,n\}$, then we have the following three cases:

    Case 1. xk<12[TeX:] $ {x_{k}}<\frac{1}{2}$, i.e. xk−1−1>1[TeX:] $ {x_{k}^{-1}}-1>1$. Then it follows from L’Hôpital’s rule that

    limγ→∞(∑j=1nωj(xj−1−1)γ)1γ=limγ→∞eln(∑j=1nωj(xj−1−1)γ)γ=limγ→∞e∑j=1nωj(xj−1−1)γln(xj−1−1)∑j=1nωj(xj−1−1)γ=limγ→∞e∑j=1nωj(xj−1−1xk−1−1)γln(xj−1−1)∑j=1nωj(xj−1−1xk−1−1)γ=eln(xk−1−1)=xk−1−1.[TeX:] \[\begin{array}{l}\displaystyle \underset{\gamma \to \infty }{\lim }{\Bigg({\sum \limits_{j=1}^{n}}{\omega _{j}}{({x_{j}^{-1}}-1)^{\gamma }}\Bigg)^{\frac{1}{\gamma }}}=\underset{\gamma \to \infty }{\lim }{e^{\frac{\ln ({\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{({x_{j}^{-1}}-1)^{\gamma }})}{\gamma }}}\\ {} \displaystyle \hspace{1em}=\underset{\gamma \to \infty }{\lim }{e^{\frac{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{({x_{j}^{-1}}-1)^{\gamma }}\ln ({x_{j}^{-1}}-1)}{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{({x_{j}^{-1}}-1)^{\gamma }}}}}=\underset{\gamma \to \infty }{\lim }{e^{\frac{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{(\frac{{x_{j}^{-1}}-1}{{x_{k}^{-1}}-1})^{\gamma }}\ln ({x_{j}^{-1}}-1)}{{\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{(\frac{{x_{j}^{-1}}-1}{{x_{k}^{-1}}-1})^{\gamma }}}}}\\ {} \displaystyle \hspace{1em}={e^{\ln ({x_{k}^{-1}}-1)}}={x_{k}^{-1}}-1.\end{array}\]
    Thus limγ→∞Tω,gγD(x)=xk[TeX:] $ {\lim \nolimits_{\gamma \to \infty }}{T_{\omega ,{g_{\gamma }^{D}}}}(x)={x_{k}}$.

    Case 2. xk=12[TeX:] $ {x_{k}}=\frac{1}{2}$, i.e. xk−1−1=1[TeX:] $ {x_{k}^{-1}}-1=1$. Then it follows from L’Hôpital’s rule that

    limγ→∞(∑j=1nωj(xj−1−1)γ)1γ=limγ→∞eln(∑j=1nωj(xj−1−1)γ)γ=1,[TeX:] \[ \underset{\gamma \to \infty }{\lim }{\Bigg({\sum \limits_{j=1}^{n}}{\omega _{j}}{\big({x_{j}^{-1}}-1\big)^{\gamma }}\Bigg)^{\frac{1}{\gamma }}}=\underset{\gamma \to \infty }{\lim }{e^{\frac{\ln ({\textstyle\textstyle\sum _{j=1}^{n}}{\omega _{j}}{({x_{j}^{-1}}-1)^{\gamma }})}{\gamma }}}=1,\]
    thus limγ→∞Tω,gγD(x)=12[TeX:] $ {\lim \nolimits_{\gamma \to \infty }}{T_{\omega ,{g_{\gamma }^{D}}}}(x)=\frac{1}{2}$.

    Case 3. xk>12[TeX:] $ {x_{k}}>\frac{1}{2}$, i.e. xk−1−1<1[TeX:] $ {x_{k}^{-1}}-1<1$. Then it is similar to Case 1 that

    limγ→∞(∑j=1nωj(xj−1−1)γ)1γ=xk−1−1,[TeX:] \[ \underset{\gamma \to \infty }{\lim }{\Bigg({\sum \limits_{j=1}^{n}}{\omega _{j}}{\big({x_{j}^{-1}}-1\big)^{\gamma }}\Bigg)^{\frac{1}{\gamma }}}={x_{k}^{-1}}-1,\]
    thus limγ→∞Tω,gγD(x)=xk[TeX:] $ {\lim \nolimits_{\gamma \to \infty }}{T_{\omega ,{g_{\gamma }^{D}}}}(x)={x_{k}}$.

    All in all, we have limγ→∞Tω,gγD(x)=xk[TeX:] $ {\lim \nolimits_{\gamma \to \infty }}{T_{\omega ,{g_{\gamma }^{D}}}}(x)={x_{k}}$, which yields that

    (14)
    SIFWMTω,g∞D(α)=min(μα)min(μα)+min(1−μα),min(να)min(να)+min(1−να).[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\infty }^{D}}}}}}(\alpha )=\left(\frac{\min ({\mu _{\alpha }})}{\min ({\mu _{\alpha }})+\min (1-{\mu _{\alpha }})},\frac{\min ({\nu _{\alpha }})}{\min ({\nu _{\alpha }})+\min (1-{\nu _{\alpha }})}\right).\]
    Similarly, if γ→0[TeX:] $ \gamma \to 0$, then we obtain
    (15)
    SIFWMTω,g0D(α)=∏j=1nμαjωj∏j=1n(1−μαj)ωj+∏j=1nμαjωj,∏j=1nναjωj∏j=1n(1−ναj)ωj+∏j=1nναjωj.[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{0}^{D}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\mu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}},\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\nu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}\right).\end{array}\]

3.2Symmetric Intuitionistic Fuzzy Weighted Mean Operators w.r.t. Weighted Archimedean t-Conorms

Lemma 3.4.

Let αj[TeX:] $ {\alpha _{j}}$ (j=1,2,…,n)[TeX:] $ (j=1,2,\dots ,n)$ be a collection of IFNs and Sω,g[TeX:] $ {S_{\omega ,g}}$ be the dual of a weighted Archimedean t-norm Tω,g[TeX:] $ {T_{\omega ,g}}$ with an additive generator g w.r.t. standard negation n(x)=1−x[TeX:] $ n(x)=1-x$. Then

(16)
(Sω,g(μα)Sω,g(μα)+Sω,g(1−μα),Sω,g(να)Sω,g(να)+Sω,g(1−να))[TeX:] \[ \bigg(\frac{{S_{\omega ,g}}({\mu _{\alpha }})}{{S_{\omega ,g}}({\mu _{\alpha }})+{S_{\omega ,g}}(1-{\mu _{\alpha }})},\frac{{S_{\omega ,g}}({\nu _{\alpha }})}{{S_{\omega ,g}}({\nu _{\alpha }})+{S_{\omega ,g}}(1-{\nu _{\alpha }})}\bigg)\]
is an IFN.

Proof.

Since αj[TeX:] $ {\alpha _{j}}$ (j=1,2,…,n)[TeX:] $ (j=1,2,\dots ,n)$ are IFNs, we have μαj+ναj⩽1[TeX:] $ {\mu _{{\alpha _{j}}}}+{\nu _{{\alpha _{j}}}}\leqslant 1$, i.e. ναj⩽1−μαj[TeX:] $ {\nu _{{\alpha _{j}}}}\leqslant 1-{\mu _{{\alpha _{j}}}}$. The antitonicity of g leads that ωjg(1−μαj)⩽ωjg(ναj)[TeX:] $ {\omega _{j}}g(1-{\mu _{{\alpha _{j}}}})\leqslant {\omega _{j}}g({\nu _{{\alpha _{j}}}})$, and hence Sω,g(να)⩽Sω,g(1−μα)[TeX:] $ {S_{\omega ,g}}({\nu _{\alpha }})\leqslant {S_{\omega ,g}}(1-{\mu _{\alpha }})$. Thus we get Sω,g(μα)Sω,g(μα)+Sω,g(1−μα)⩽Sω,g(μα)Sω,g(μα)+Sω,g(να)[TeX:] $ \frac{{S_{\omega ,g}}({\mu _{\alpha }})}{{S_{\omega ,g}}({\mu _{\alpha }})+{S_{\omega ,g}}(1-{\mu _{\alpha }})}\leqslant \frac{{S_{\omega ,g}}({\mu _{\alpha }})}{{S_{\omega ,g}}({\mu _{\alpha }})+{S_{\omega ,g}}({\nu _{\alpha }})}$. Similarly, Sω,g(να)Sω,g(να)+Sω,g(1−να)⩽Sω,g(να)Sω,g(να)+Sω,g(μα)[TeX:] $ \frac{{S_{\omega ,g}}({\nu _{\alpha }})}{{S_{\omega ,g}}({\nu _{\alpha }})+{S_{\omega ,g}}(1-{\nu _{\alpha }})}\leqslant \frac{{S_{\omega ,g}}({\nu _{\alpha }})}{{S_{\omega ,g}}({\nu _{\alpha }})+{S_{\omega ,g}}({\mu _{\alpha }})}$. Thus it yields that Sω,g(μα)Sω,g(μα)+Sω,g(1−μα)+Sω,g(να)Sω,g(να)+Sω,g(1−να)⩽1[TeX:] $ \frac{{S_{\omega ,g}}({\mu _{\alpha }})}{{S_{\omega ,g}}({\mu _{\alpha }})+{S_{\omega ,g}}(1-{\mu _{\alpha }})}+\frac{{S_{\omega ,g}}({\nu _{\alpha }})}{{S_{\omega ,g}}({\nu _{\alpha }})+{S_{\omega ,g}}(1-{\nu _{\alpha }})}\leqslant 1$, that is, (16) is an IFN. □

Definition 3.5.

Let αj[TeX:] $ {\alpha _{j}}$ ( j=1,2,…,n[TeX:] $ j=1,2,\dots ,n$) be a collection of IFNs, Sω,g[TeX:] $ {S_{\omega ,g}}$ be the dual of a weighted Archimedean t-norm Tω,g[TeX:] $ {T_{\omega ,g}}$ with an additive generator g w.r.t. standard negation n(x)=1−x[TeX:] $ n(x)=1-x$ and SIFWMSω,g:IFNn→IFN[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,g}}}}:{\mathcal{IFN}^{n}}\to \mathcal{IFN}$, if

(17)
SIFWMSω,g(α)=Sω,g(μα)Sω,g(μα)+Sω,g(1−μα),Sω,g(να)Sω,g(να)+Sω,g(1−να),[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,g}}}}(\alpha )=\left(\frac{{S_{\omega ,g}}({\mu _{\alpha }})}{{S_{\omega ,g}}({\mu _{\alpha }})+{S_{\omega ,g}}(1-{\mu _{\alpha }})},\frac{{S_{\omega ,g}}({\nu _{\alpha }})}{{S_{\omega ,g}}({\nu _{\alpha }})+{S_{\omega ,g}}(1-{\nu _{\alpha }})}\right),\]
then SIFWM Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$ is called a symmetric intuitionistic fuzzy weighted mean (SIFWM Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$) operator of dimension n w.r.t. Sω,g[TeX:] $ {S_{\omega ,g}}$.

Especially, if ναj=1−μαj[TeX:] $ {\nu _{{\alpha _{j}}}}=1-{\mu _{{\alpha _{j}}}}$ for all j=1,2,…,n[TeX:] $ j=1,2,\dots ,n$, that is, all αj[TeX:] $ {\alpha _{j}}$ are reduced to μαj[TeX:] $ {\mu _{{\alpha _{j}}}}$, respectively, then Eq. (17) is reduced to the following form:

SIFWMSω,g(α)=Sω,g(μα)Sω,g(1−μα)+Sω,g(μα),1−Sω,g(μα)Sω,g(1−μα)+Sω,g(μα),[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,g}}}}(\alpha )=\left(\frac{{S_{\omega ,g}}({\mu _{\alpha }})}{{S_{\omega ,g}}(1-{\mu _{\alpha }})+{S_{\omega ,g}}({\mu _{\alpha }})},1-\frac{{S_{\omega ,g}}({\mu _{\alpha }})}{{S_{\omega ,g}}(1-{\mu _{\alpha }})+{S_{\omega ,g}}({\mu _{\alpha }})}\right),\]
which becomes both a symmetric sum operator of dimension n (Beliakov et al., 2007) and a weighted t-norm-based aggregation operator with H(x,y)=y1−x+y[TeX:] $ H(x,y)=\frac{y}{1-x+y}$ (Calvo and Mesiar, 2003) to aggregate fuzzy information.

Proposition 3.6.

Let αj,βj[TeX:] $ {\alpha _{j}},{\beta _{j}}$ (j=1,2,…,n)[TeX:] $ (j=1,2,\dots ,n)$ be two collections of IFNs with α=(α1,α2,…,αn)[TeX:] $ \alpha =({\alpha _{1}},{\alpha _{2}},\dots ,{\alpha _{n}})$, β=(β1,β2,…,βn)[TeX:] $ \beta =({\beta _{1}},{\beta _{2}},\dots ,{\beta _{n}})$, respectively, and Sω,g[TeX:] $ {S_{\omega ,g}}$ be the dual of a weighted Archimedean t-norm Tω,g[TeX:] $ {T_{\omega ,g}}$ with an additive generator g w.r.t. standard negation n(x)=1−x[TeX:] $ n(x)=1-x$.

  • (1) If all αj[TeX:] $ {\alpha _{j}}$ are equal, i.e. αj=δ=(μδ,νδ)[TeX:] $ {\alpha _{j}}=\delta =({\mu _{\delta }},{\nu _{\delta }})$, for all j, then SIFWMSω,g(α)=δ[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,g}}}}(\alpha )=\delta $;

  • (2) If αj⩽βj[TeX:] $ {\alpha _{j}}\leqslant {\beta _{j}}$ for all j, then SIFWMSω,g(α)⩽SIFWMSω,g(β)[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,g}}}}(\alpha )\leqslant {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,g}}}}(\beta )$;

  • (3) Let α−=(min(μα),max(να))[TeX:] $ {\alpha ^{-}}=(\min ({\mu _{\alpha }}),\max ({\nu _{\alpha }}))$ and α+=(max(μα),min(ναj))[TeX:] $ {\alpha ^{+}}=(\max ({\mu _{\alpha }}),\min ({\nu _{{\alpha _{j}}}}))$, then α−⩽SIFWMSω,g(α)⩽α+[TeX:] $ {\alpha ^{-}}\leqslant {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,g}}}}(\alpha )\leqslant {\alpha ^{+}}$.

Proof.

It can be proved in a similar way as in Proposition 3.3. □

Next, we assign the additive generator g to different forms, some specific symmetric intuitionistic fuzzy aggregation operators can be obtained as follows:

  • 1. If g(t)=gγH(1−t)[TeX:] $ g(t)={g_{\gamma }^{H}}(1-t)$, then the SIFWM Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$ operator is reduced to the following form:

    (18)
    SIFWMSω,gγH(α)=Sω,gγH(μα)Sω,gγH(1−μα)+Sω,gγH(μα),Sω,gγH(να)Sω,gγH(1−να)+Sω,gγH(να),[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{\gamma }^{H}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{S_{\omega ,{g_{\gamma }^{H}}}}({\mu _{\alpha }})}{{S_{\omega ,{g_{\gamma }^{H}}}}(1-{\mu _{\alpha }})+{S_{\omega ,{g_{\gamma }^{H}}}}({\mu _{\alpha }})},\frac{{S_{\omega ,{g_{\gamma }^{H}}}}({\nu _{\alpha }})}{{S_{\omega ,{g_{\gamma }^{H}}}}(1-{\nu _{\alpha }})+{S_{\omega ,{g_{\gamma }^{H}}}}({\nu _{\alpha }})}\right),\end{array}\]
    where Sω,gγH(x)=∏j=1n(γ1−xj+1−γ)ωj−1∏j=1n(γ1−xj+1−γ)ωj+γ−1[TeX:] $ {S_{\omega ,{g_{\gamma }^{H}}}}(x)=\frac{{\textstyle\prod _{j=1}^{n}}{(\frac{\gamma }{1-{x_{j}}}+1-\gamma )^{{\omega _{j}}}}-1}{{\textstyle\prod _{j=1}^{n}}{(\frac{\gamma }{1-{x_{j}}}+1-\gamma )^{{\omega _{j}}}}+\gamma -1}$, γ>0[TeX:] $ \gamma >0$.

    Especially, if γ=1[TeX:] $ \gamma =1$, i.e. g1H(t)=−ln(1−t)[TeX:] $ {g_{1}^{H}}(t)=-\ln (1-t)$, then the SIFWMSω,g1H[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{1}^{H}}}}}}$ operator with Sω,g1H(x)=1−∏j=1n(1−xj)ωj[TeX:] $ {S_{\omega ,{g_{1}^{H}}}}(x)=1-{\textstyle\prod _{j=1}^{n}}{(1-{x_{j}})^{{\omega _{j}}}}$ is the symmetric form of intuitionistic fuzzy weighted averaging (IFWA) operator defined by Xu (2007) ; if γ=2[TeX:] $ \gamma =2$, i.e. g2H(t)=ln(1+t1−t)[TeX:] $ {g_{2}^{H}}(t)=\ln (\frac{1+t}{1-t})$, then the SIFWMSω,g2H[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{2}^{H}}}}}}$ operator with Sω,g2H(x)=∏j=1n(1+xj1−xj)ωj−1∏j=1n(1+xj1−xj)ωj+1[TeX:] $ {S_{\omega ,{g_{2}^{H}}}}(x)=\frac{{\textstyle\prod _{j=1}^{n}}{(\frac{1+{x_{j}}}{1-{x_{j}}})^{{\omega _{j}}}}-1}{{\textstyle\prod _{j=1}^{n}}{(\frac{1+{x_{j}}}{1-{x_{j}}})^{{\omega _{j}}}}+1}$ is the symmetric form of IFWG operator based on Einstein t-norm and t-conorm defined by Wang and Liu (2012) ; if γ→∞[TeX:] $ \gamma \to \infty $, then it is similar to the proof of limγ→∞Tω,gγH(x)[TeX:] $ {\lim \nolimits_{\gamma \to \infty }}{T_{\omega ,{g_{\gamma }^{H}}}}(x)$ that

    limγ→∞Sω,gγH(x)=∏j=1nxjωj∏j=1nxjωj+∏j=1n(1−xj)ωj,[TeX:] \[ \underset{\gamma \to \infty }{\lim }{S_{\omega ,{g_{\gamma }^{H}}}}(x)=\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{x_{j}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{x_{j}^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{x_{j}})^{{\omega _{j}}}}},\]
    and hence
    (19)
    SIFWMSω,g∞H(α)=∏j=1nμαjωj∏j=1n(1−μαj)ωj+∏j=1nμαjωj,∏j=1nναjωj∏j=1n(1−ναj)ωj+∏j=1nναjωj.[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{\infty }^{H}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\mu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}},\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\nu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}\right).\end{array}\]

  • 2. If g(t)=gγSS(1−t)[TeX:] $ g(t)={g_{\gamma }^{SS}}(1-t)$, then the SIFWM Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$ operator is reduced to the following form:

    (20)
    SIFWMSω,gγSS(α)=Sω,gγSS(μα)Sω,gγSS(μα)+Sω,gγSS(1−μα),Sω,gγSS(να)Sω,gγSS(να)+Sω,gγSS(1−να),[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{\gamma }^{SS}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{S_{\omega ,{g_{\gamma }^{SS}}}}({\mu _{\alpha }})}{{S_{\omega ,{g_{\gamma }^{SS}}}}({\mu _{\alpha }})+{S_{\omega ,{g_{\gamma }^{SS}}}}(1-{\mu _{\alpha }})},\frac{{S_{\omega ,{g_{\gamma }^{SS}}}}({\nu _{\alpha }})}{{S_{\omega ,{g_{\gamma }^{SS}}}}({\nu _{\alpha }})+{S_{\omega ,{g_{\gamma }^{SS}}}}(1-{\nu _{\alpha }})}\right),\end{array}\]
    where Sω,gγSS(x)=1−(∑j=1nωj(1−xj)γ)1γ[TeX:] $ {S_{\omega ,{g_{\gamma }^{SS}}}}(x)=1-{({\textstyle\sum _{j=1}^{n}}{\omega _{j}}{(1-{x_{j}})^{\gamma }})^{\frac{1}{\gamma }}}$, γ≠0[TeX:] $ \gamma \ne 0$. In particular, it is similar to the proofs of SIFWMTω,g∞SS(α)[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\infty }^{SS}}}}}}(\alpha )$, SIFWMTω,g−∞SS(α)[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{-\infty }^{SS}}}}}}(\alpha )$ and SIFWMTω,g0SS(α)[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{0}^{SS}}}}}}(\alpha )$ that
    (21)
    SIFWMSω,g∞SS(α)=min(μα)min(μα)+min(1−μα),min(να)min(να)+min(1−να),[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{\infty }^{SS}}}}}}(\alpha )=\left(\frac{\min ({\mu _{\alpha }})}{\min ({\mu _{\alpha }})+\min (1-{\mu _{\alpha }})},\frac{\min ({\nu _{\alpha }})}{\min ({\nu _{\alpha }})+\min (1-{\nu _{\alpha }})}\right),\]
    (22)
    SIFWMSω,g−∞SS(α)=max(μα)max(μα)+max(1−μα),max(να)max(να)+max(1−να),[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{-\infty }^{SS}}}}}}(\alpha )=\left(\frac{\max ({\mu _{\alpha }})}{\max ({\mu _{\alpha }})+\max (1-{\mu _{\alpha }})},\frac{\max ({\nu _{\alpha }})}{\max ({\nu _{\alpha }})+\max (1-{\nu _{\alpha }})}\right),\]
    (23)
    SIFWMSω,g0SS(α)=∏j=1nμαjωj∏j=1n(1−μαj)ωj+∏j=1nμαjωj,∏j=1nναjωj∏j=1n(1−ναj)ωj+∏j=1nναjωj.[TeX:] \[\begin{array}{r@{\hskip4.0pt}c}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{0}^{SS}}}}}}(\alpha )=& \displaystyle \left(\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\mu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}},\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\nu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}\right).\end{array}\]

  • (3) If g(t)=gγD(1−t)[TeX:] $ g(t)={g_{\gamma }^{D}}(1-t)$, then the SIFWM Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$ operator is reduced to the following form:

    (24)
    SIFWMSω,gγD(α)=Sω,gγD(μα)Sω,gγD(μα)+Sω,gγD(1−μα),Sω,gγD(να)Sω,gγD(να)+Sω,gγD(1−να),[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{\gamma }^{D}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{S_{\omega ,{g_{\gamma }^{D}}}}({\mu _{\alpha }})}{{S_{\omega ,{g_{\gamma }^{D}}}}({\mu _{\alpha }})+{S_{\omega ,{g_{\gamma }^{D}}}}(1-{\mu _{\alpha }})},\frac{{S_{\omega ,{g_{\gamma }^{D}}}}({\nu _{\alpha }})}{{S_{\omega ,{g_{\gamma }^{D}}}}({\nu _{\alpha }})+{S_{\omega ,{g_{\gamma }^{D}}}}(1-{\nu _{\alpha }})}\right),\end{array}\]
    where Sω,gγD(x)=(∑j=1nωj(xj1−xj)γ)1γ(∑j=1nωj(xj1−xj)γ)1γ+1[TeX:] $ {S_{\omega ,{g_{\gamma }^{D}}}}(x)=\frac{{({\textstyle\sum _{j=1}^{n}}{\omega _{j}}{(\frac{{x_{j}}}{1-{x_{j}}})^{\gamma }})^{\frac{1}{\gamma }}}}{{({\textstyle\sum _{j=1}^{n}}{\omega _{j}}{(\frac{{x_{j}}}{1-{x_{j}}})^{\gamma }})^{\frac{1}{\gamma }}}+1}$, γ>0[TeX:] $ \gamma >0$.

    Particularly, if γ=1[TeX:] $ \gamma =1$, then the SIFWM Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$ operator is reduced to the following form:

    (25)
    SIFWMSω,g1D(α)=Sω,g1D(μα)Sω,g1D(μα)+Sω,g1D(1−μα),Sω,g1D(να)Sω,g1D(να)+Sω,g1D(1−να),[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{1}^{D}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{S_{\omega ,{g_{1}^{D}}}}({\mu _{\alpha }})}{{S_{\omega ,{g_{1}^{D}}}}({\mu _{\alpha }})+{S_{\omega ,{g_{1}^{D}}}}(1-{\mu _{\alpha }})},\frac{{S_{\omega ,{g_{1}^{D}}}}({\nu _{\alpha }})}{{S_{\omega ,{g_{1}^{D}}}}({\nu _{\alpha }})+{S_{\omega ,{g_{1}^{D}}}}(1-{\nu _{\alpha }})}\right),\end{array}\]
    where Sω,g1D(x)=∑j=1nωjxj1−xj∑j=1nωj11−xj[TeX:] $ {S_{\omega ,{g_{1}^{D}}}}(x)=\frac{{\textstyle\sum _{j=1}^{n}}{\omega _{j}}\frac{{x_{j}}}{1-{x_{j}}}}{{\textstyle\sum _{j=1}^{n}}{\omega _{j}}\frac{1}{1-{x_{j}}}}$; it is similar to the proofs of SIFWMTω,g∞D(α)[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\infty }^{D}}}}}}(\alpha )$ and SIFWMTω,g0D(α)[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{0}^{D}}}}}}(\alpha )$ that
    (26)
    SIFWMSω,g∞D(α)=max(μα)max(μα)+max(1−μα),max(να)max(να)+max(1−να),[TeX:] \[ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{\infty }^{D}}}}}}(\alpha )=\left(\frac{\max ({\mu _{\alpha }})}{\max ({\mu _{\alpha }})+\max (1-{\mu _{\alpha }})},\frac{\max ({\nu _{\alpha }})}{\max ({\nu _{\alpha }})+\max (1-{\nu _{\alpha }})}\right),\]
    (27)
    SIFWMSω,g0D(α)=∏j=1nμαjωj∏j=1n(1−μαj)ωj+∏j=1nμαjωj,∏j=1nναjωj∏j=1n(1−ναj)ωj+∏j=1nναjωj.[TeX:] \[\begin{array}{l}\displaystyle {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{0}^{D}}}}}}(\alpha )\\ {} \displaystyle \hspace{1em}=\left(\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\mu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\mu _{{\alpha _{j}}}^{{\omega _{j}}}}},\frac{{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}{{\textstyle\textstyle\prod _{j=1}^{n}}{(1-{\nu _{{\alpha _{j}}}})^{{\omega _{j}}}}+{\textstyle\textstyle\prod _{j=1}^{n}}{\nu _{{\alpha _{j}}}^{{\omega _{j}}}}}\right).\end{array}\]

All in all, we have the following conclusion:

Corollary 3.7.

  • (1) SIFWMTω,g1H=SIFWMTω,g∞H=SIFWMTω,g0SS=SIFWMTω,g0D=SIFWMSω,g∞H=SIFWMSω,g1H=SIFWMSω,g0SS=SIFWMSω,g0D[TeX:] $ {\mathrm{SIFWM}^{{T_{\omega ,{g_{1}^{H}}}}}}={\mathrm{SIFWM}^{{T_{\omega ,{g_{\infty }^{H}}}}}}={\mathrm{SIFWM}^{{T_{\omega ,{g_{0}^{SS}}}}}}={\mathrm{SIFWM}^{{T_{\omega ,{g_{0}^{D}}}}}}={\mathrm{SIFWM}^{{S_{\omega ,{g_{\infty }^{H}}}}}}={\mathrm{SIFWM}^{{S_{\omega ,{g_{1}^{H}}}}}}={\mathrm{SIFWM}^{{S_{\omega ,{g_{0}^{SS}}}}}}={\mathrm{SIFWM}^{{S_{\omega ,{g_{0}^{D}}}}}}$;

  • (2) SIFWMSω,g∞SS=SIFWMTω,g−∞SS=SIFWMTω,g∞D[TeX:] $ {\mathrm{SIFWM}^{{S_{\omega ,{g_{\infty }^{SS}}}}}}={\mathrm{SIFWM}^{{T_{\omega ,{g_{-\infty }^{SS}}}}}}={\mathrm{SIFWM}^{{T_{\omega ,{g_{\infty }^{D}}}}}}$;

  • (3) SIFWMTω,g∞SS=SIFWMSω,g−∞SS=SIFWMSω,g∞D[TeX:] $ {\mathrm{SIFWM}^{{T_{\omega ,{g_{\infty }^{SS}}}}}}={\mathrm{SIFWM}^{{S_{\omega ,{g_{-\infty }^{SS}}}}}}={\mathrm{SIFWM}^{{S_{\omega ,{g_{\infty }^{D}}}}}}$;

  • (4) SIFWMTω,g0H=SIFWMTω,g−1SS[TeX:] $ {\mathrm{SIFWM}^{{T_{\omega ,{g_{0}^{H}}}}}}={\mathrm{SIFWM}^{{T_{\omega ,{g_{-1}^{SS}}}}}}$.

The above corollary indicates that the operators SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ and SIFWMSω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{\gamma }^{SS}}}}}}$ can well reflect the variations of the other operators. Furthermore, since SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ and SIFWMSω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{S_{\omega ,{g_{\gamma }^{SS}}}}}}$ are dual, we always use the operator SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ to neutrally aggregate the IFNs and the Eqs. (10), (11) and (12) can be considered as the cases when the decision maker is optimistic, pessimistic or impartial. With respect to the existing symmetrical intuitionistic fuzzy aggregation operators (Beliakov et al., 2011; Liao and Xu, 2015; Xia and Xu, 2012), the proposed aggregation operators based on weighted Archimedean t-norm and t-conorm possess the following advantages:

  • • these existing operators (Beliakov et al., 2011; Liao and Xu, 2015; Xia and Xu, 2012) can only treat membership and non-membership information fairly, and provide a single choice for the decision maker; the proposed ones can not only treat membership and non-membership information fairly but also provide more choices for the decision maker;

  • • the existing operator (Liao and Xu, 2015) can not reduce to the corresponding fuzzy one; the proposed ones can be considered as generalizations of the existing aggregation operators in fuzzy cases;

  • • the existing operator (Xia and Xu, 2012) is not suitable for dealing with IFNs (1,0)[TeX:] $ (1,0)$ or (0,1)[TeX:] $ (0,1)$; the proposed operators can solve the case.

4The Relationships Among the Proposed Aggregation Operators and the Existing One

The following lemma is obvious but useful to investigate the relationships among the proposed aggregation operators and the existing ones:

Lemma 4.1.

Let x,y,ωi,xi∈[0,1],λ>0[TeX:] $ x,y,{\omega _{i}},{x_{i}}\in [0,1],\lambda >0$ such that ∑i=1nωi=1[TeX:] $ {\textstyle\sum _{i=1}^{n}}{\omega _{i}}=1$, it holds that

  • (1) Ma and Xu  ( 2016) if y⩽x[TeX:] $ y\leqslant x$, then y⩽y1−x+y⩽x[TeX:] $ y\leqslant \frac{y}{1-x+y}\leqslant x$ and y⩽x1+x−y⩽x[TeX:] $ y\leqslant \frac{x}{1+x-y}\leqslant x$;

  • (2) if g is convex (concave), then g(∑i=1nωixi)⩽(⩾)∑i=1nωig(xi)[TeX:] $ g({\textstyle\sum _{i=1}^{n}}{\omega _{i}}{x_{i}})\leqslant (\geqslant ){\textstyle\sum _{i=1}^{n}}{\omega _{i}}g({x_{i}})$, the equality holds if and only if x1=x2=⋯=xn[TeX:] $ {x_{1}}={x_{2}}=\cdots ={x_{n}}$ or g is linear.

The relationships among IFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$, IFWA Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ and SIFWM Tω,g[TeX:] $ {^{{T_{\omega ,g}}}}$ can be shown as follows:

Proposition 4.2.

Let αj[TeX:] $ {\alpha _{j}}$ (j=1,2,…,n)[TeX:] $ (j=1,2,\dots ,n)$ be a collection of IFNs and Tω,g[TeX:] $ {T_{\omega ,g}}$ be a weighted Archimedean t-norm Tω,g[TeX:] $ {T_{\omega ,g}}$ with an additive generator g.

  • (1) If g is concave, then IFWM Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$ ⩽ SIFWM Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$ ⩽ IFWA Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$;

  • (2) If g is convex, then IFWA Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$ ⩽ SIFWM Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$ ⩽ IFWM Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$;

  • (3) If g has at least one inflection point, then the inequality varies with concavity-convexity of g.

Proof.

We only prove (1), and (2), (3) can be proven in a similar way.

Since g is concave, i.e. g−1[TeX:] $ {g^{-1}}$ is convex, it follows from Lemma 4.1(2) that g−1(∑i=1nωig(μαi))+g−1(∑i=1nωig(1−μαi))⩽∑i=1nωiμαi+∑i=1nωi(1−μαi)=∑i=1nωi=1[TeX:] $ {g^{-1}}({\textstyle\sum _{i=1}^{n}}{\omega _{i}}g({\mu _{{\alpha _{i}}}}))+{g^{-1}}({\textstyle\sum _{i=1}^{n}}{\omega _{i}}g(1-{\mu _{{\alpha _{i}}}}))\leqslant {\textstyle\sum _{i=1}^{n}}{\omega _{i}}{\mu _{{\alpha _{i}}}}+{\textstyle\sum _{i=1}^{n}}{\omega _{i}}(1-{\mu _{{\alpha _{i}}}})={\textstyle\sum _{i=1}^{n}}{\omega _{i}}=1$, and hence g−1(∑i=1nωig(μαi))⩽1−g−1(∑i=1nωig(1−μαi))[TeX:] $ {g^{-1}}({\textstyle\sum _{i=1}^{n}}{\omega _{i}}g({\mu _{{\alpha _{i}}}}))\leqslant 1-{g^{-1}}({\textstyle\sum _{i=1}^{n}}{\omega _{i}}g(1-{\mu _{{\alpha _{i}}}}))$, i.e. Tω,g(μα)⩽1−Tω,g(1−μα)[TeX:] $ {T_{\omega ,g}}({\mu _{\alpha }})\leqslant 1-{T_{\omega ,g}}(1-{\mu _{\alpha }})$. Using Lemma 4.1(1), we get Tω,g(μα)⩽Tω,g(μα)Tω,g(μα)+Tω,g(1−μα)⩽1−Tω,g(1−μα)[TeX:] $ {T_{\omega ,g}}({\mu _{\alpha }})\leqslant \frac{{T_{\omega ,g}}({\mu _{\alpha }})}{{T_{\omega ,g}}({\mu _{\alpha }})+{T_{\omega ,g}}(1-{\mu _{\alpha }})}\leqslant 1-{T_{\omega ,g}}(1-{\mu _{\alpha }})$. In a similar way, it yields that Tω,g(να)⩽Tω,g(να)Tω,g(να)+Tω,g(1−να)⩽1−Tω,g(1−να)[TeX:] $ {T_{\omega ,g}}({\nu _{\alpha }})\leqslant \frac{{T_{\omega ,g}}({\nu _{\alpha }})}{{T_{\omega ,g}}({\nu _{\alpha }})+{T_{\omega ,g}}(1-{\nu _{\alpha }})}\leqslant 1-{T_{\omega ,g}}(1-{\nu _{\alpha }})$. Thus, IFWM Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$ ⩽ SIFWM Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$ ⩽ IFWA Tω,g(α)[TeX:] $ {^{{T_{\omega ,g}}}}(\alpha )$. □

The relationships between IFWM Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$, IFWA Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$ and SIFWM Sω,g[TeX:] $ {^{{S_{\omega ,g}}}}$ can be shown as follows:

Proposition 4.3.

Let αj[TeX:] $ {\alpha _{j}}$ (j=1,2,…,n)[TeX:] $ (j=1,2,\dots ,n)$ be a collection of IFNs and Sω,g[TeX:] $ {S_{\omega ,g}}$ be the dual of a weighted Archimedean t-norm Tω,g[TeX:] $ {T_{\omega ,g}}$ with an additive generator g w.r.t. standard negation n(x)=1−x[TeX:] $ n(x)=1-x$.

  • (1) If g is concave, then IFWM Sω,g(α)[TeX:] $ {^{{S_{\omega ,g}}}}(\alpha )$ ⩽ SIFWM Sω,g(α)[TeX:] $ {^{{S_{\omega ,g}}}}(\alpha )$ ⩽ IFWA Sω,g(α)[TeX:] $ {^{{S_{\omega ,g}}}}(\alpha )$;

  • (2) If g is convex, then IFWA Sω,g(α)[TeX:] $ {^{{S_{\omega ,g}}}}(\alpha )$ ⩽ SIFWM Sω,g(α)[TeX:] $ {^{{S_{\omega ,g}}}}(\alpha )$ ⩽ IFWM Sω,g(α)[TeX:] $ {^{{S_{\omega ,g}}}}(\alpha )$;

  • (3) If g has at least one inflection point, then the inequality varies with concavity-convexity of g.

Proof.

We only prove (1), and (2), (3) can be proven in a similar way.

Since g is concave, i.e. g−1[TeX:] $ {g^{-1}}$ is convex, it follows from Lemma 4.1(2) that g−1(∑i=1nωig(μαi))+g−1(∑i=1nωig(1−μαi))⩽∑i=1nωiμαi+∑i=1nωi(1−μαi)=∑i=1nωi=1[TeX:] $ {g^{-1}}({\textstyle\sum _{i=1}^{n}}{\omega _{i}}g({\mu _{{\alpha _{i}}}}))+{g^{-1}}({\textstyle\sum _{i=1}^{n}}{\omega _{i}}g(1-{\mu _{{\alpha _{i}}}}))\leqslant {\textstyle\sum _{i=1}^{n}}{\omega _{i}}{\mu _{{\alpha _{i}}}}+{\textstyle\sum _{i=1}^{n}}{\omega _{i}}(1-{\mu _{{\alpha _{i}}}})={\textstyle\sum _{i=1}^{n}}{\omega _{i}}=1$, and hence g−1(∑i=1nωig(μαi))⩽1−g−1(∑i=1nωig(1−μαi))[TeX:] $ {g^{-1}}({\textstyle\sum _{i=1}^{n}}{\omega _{i}}g({\mu _{{\alpha _{i}}}}))\leqslant 1-{g^{-1}}({\textstyle\sum _{i=1}^{n}}{\omega _{i}}g(1-{\mu _{{\alpha _{i}}}}))$, i.e. Tω,g(μα)⩽1−Tω,g(1−μα)[TeX:] $ {T_{\omega ,g}}({\mu _{\alpha }})\leqslant 1-{T_{\omega ,g}}(1-{\mu _{\alpha }})$. By Lemma 4.1(1), we get Tω,g(μα)⩽1−Tω,g(1−μα)2−Tω,g(μα)−Tω,g(1−μα)=Sω,g(μα)Sω,g(μα)+Sω,g(1−μα)⩽1−Tω,g(1−μα)[TeX:] $ {T_{\omega ,g}}({\mu _{\alpha }})\leqslant \frac{1-{T_{\omega ,g}}(1-{\mu _{\alpha }})}{2-{T_{\omega ,g}}({\mu _{\alpha }})-{T_{\omega ,g}}(1-{\mu _{\alpha }})}=\frac{{S_{\omega ,g}}({\mu _{\alpha }})}{{S_{\omega ,g}}({\mu _{\alpha }})+{S_{\omega ,g}}(1-{\mu _{\alpha }})}\leqslant 1-{T_{\omega ,g}}(1-{\mu _{\alpha }})$. In a similar way, we have Tω,g(να)⩽Sω,g(να)Sω,g(να)+Sω,g(1−να)⩽1−Tω,g(1−να)[TeX:] $ {T_{\omega ,g}}({\nu _{\alpha }})\leqslant \frac{{S_{\omega ,g}}({\nu _{\alpha }})}{{S_{\omega ,g}}({\nu _{\alpha }})+{S_{\omega ,g}}(1-{\nu _{\alpha }})}\leqslant 1-{T_{\omega ,g}}(1-{\nu _{\alpha }})$. Thus IFWMSω,g(α)⩽SIFWMSω,g(α)⩽IFWASω,g(α)[TeX:] $ IFW{M^{{S_{\omega ,g}}}}(\alpha )\leqslant SIFW{M^{{S_{\omega ,g}}}}(\alpha )\leqslant IFW{A^{{S_{\omega ,g}}}}(\alpha )$. □

5An Approach to Intuitionistic Fuzzy Multi-Criteria Decision Making

For a multi-criteria decision making under intuitionistic fuzzy environment, let x={x1,x2,…,xm}[TeX:] $ x=\{{x_{1}},{x_{2}},\dots ,{x_{m}}\}$ be a set of alternatives to be selected, and C={C1,C2,…,Cn}[TeX:] $ C=\{{C_{1}},{C_{2}},\dots ,{C_{n}}\}$ be a set of criteria to be evaluated. To evaluate the performance of the alternative xi[TeX:] $ {x_{i}}$ under the criterion Cj[TeX:] $ {C_{j}}$, the decision maker is required to provide not only the information that the alternative xi[TeX:] $ {x_{i}}$ satisfies the criterion Cj[TeX:] $ {C_{j}}$, but also the information that the alternative xi[TeX:] $ {x_{i}}$ does not satisfy the criterion Cj[TeX:] $ {C_{j}}$. This two part information can be expressed by μij[TeX:] $ {\mu _{ij}}$ and νij[TeX:] $ {\nu _{ij}}$ which denote the degrees that the alternative xi[TeX:] $ {x_{i}}$ satisfies the criterion Cj[TeX:] $ {C_{j}}$ and does not satisfy the criterion Cj[TeX:] $ {C_{j}}$, then the performance of the alternative xi[TeX:] $ {x_{i}}$ under the criteria Cj[TeX:] $ {C_{j}}$ can be expressed by an IFN αij[TeX:] $ {\alpha _{ij}}$ with the condition that 0⩽μij,νij⩽1[TeX:] $ 0\leqslant {\mu _{ij}},{\nu _{ij}}\leqslant 1$ and 0⩽μij+νij⩽1[TeX:] $ 0\leqslant {\mu _{ij}}+{\nu _{ij}}\leqslant 1$. When all the performances of the alternatives are provided, the intuitionistic fuzzy decision matrix D=(αij)m×n=((μij,νij))m×n[TeX:] $ D={({\alpha _{ij}})_{m\times n}}={(({\mu _{ij}},{\nu _{ij}}))_{m\times n}}$ can be constructed. To obtain the ranking of the alternatives, the following steps are given:

  • (1) Transform the intuitionistic fuzzy decision matrix D=((μij,νij))m×n[TeX:] $ D={(({\mu _{ij}},{\nu _{ij}}))_{m\times n}}$ into the normalized one B=((βij))m×n[TeX:] $ B={(({\beta _{ij}}))_{m\times n}}$, where

    βij=αij,for benefit attributexi;αijc,for cost attributexi.[TeX:] \[ {\beta _{ij}}=\left\{\begin{array}{l@{\hskip4.0pt}l}{\alpha _{ij}},\hspace{1em}& \text{for benefit attribute }{x_{i}};\\ {} {\alpha _{ij}^{c}},\hspace{1em}& \text{for cost attribute }{x_{i}}.\end{array}\right.\]

  • (2) Aggregate the IFNs βij[TeX:] $ {\beta _{ij}}$ (j=1,2,…,n)[TeX:] $ (j=1,2,\dots ,n)$ of the alternative xi[TeX:] $ {x_{i}}$ (i=1,2,…,m)[TeX:] $ (i=1,2,\dots ,m)$, denoted as βi[TeX:] $ {\beta _{i}}$ (i=1,2,…,m)[TeX:] $ (i=1,2,\dots ,m)$, by the proposed aggregation operators SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ (9).

  • (3) Calculate the score s(βi)[TeX:] $ s({\beta _{i}})$ of βi[TeX:] $ {\beta _{i}}$ (i=1,2,…,m)[TeX:] $ (i=1,2,\dots ,m)$ by Definition 2.6, and obtain the priority of the alternatives according to the ranking of βi[TeX:] $ {\beta _{i}}$ (i=1,2,…,m)[TeX:] $ (i=1,2,\dots ,m)$, the bigger the IFN βi[TeX:] $ {\beta _{i}}$ is, the better the alternative xi[TeX:] $ {x_{i}}$ is.

To illustrate the proposed methods, an example adapted from Chen (2011), Xia et al. (2012) is given as follows:

Example 5.1.

The purchasing manager in a small enterprise considers various criteria involving C1[TeX:] $ {C_{1}}$: financial factors (e.g. economic performance, financial stability), C2[TeX:] $ {C_{2}}$: performance (e.g. delivery, quality, price), C3[TeX:] $ {C_{3}}$: technology (e.g. manufacturing capability, design capability, ability to cope with technology changes), and C4[TeX:] $ {C_{4}}$: organizational culture and strategy (e.g. feeling of trust, internal and external integration of suppliers, compatibility across levels and functions of the buyer and supplier). The set of evaluative criteria is denoted by C={C1,C2,C3,C4}[TeX:] $ \mathcal{C}=\{{C_{1}},{C_{2}},{C_{3}},{C_{4}}\}$, whose weight vector is ω=(0.34,0.23,0.22,0.21)⊤[TeX:] $ \omega ={(0.34,0.23,0.22,0.21)^{\top }}$. There are six suppliers available, and the set of all alternatives is denoted by X={x1,x2,…,x6}[TeX:] $ X=\{{x_{1}},{x_{2}},\dots ,{x_{6}}\}$. The characteristics of the suppliers xi[TeX:] $ {x_{i}}$ (i=1,2,…,6)[TeX:] $ (i=1,2,\dots ,6)$ in terms of the criteria in C are expressed by the following intuitionistic fuzzy decision matrix (see Table 1).

Table 1

Intuitionistic fuzzy decision matrix D.

C1[TeX:] $ {C_{1}}$ C2[TeX:] $ {C_{2}}$ C3[TeX:] $ {C_{3}}$ C4[TeX:] $ {C_{4}}$
x1[TeX:] $ {x_{1}}$(0.60, 0.18)(0.24, 0.44)(0.10, 0.54)(0.45, 0.23)
x2[TeX:] $ {x_{2}}$(0.41, 0.25)(0.49, 0.09)(0.10, 0.39)(0.52, 0.45)
x3[TeX:] $ {x_{3}}$(0.62, 0.18)(0.67, 0.28)(0.36, 0.42)(0.12, 0.67)
x4[TeX:] $ {x_{4}}$(0.21, 0.58)(0.76, 0.22)(0.48, 0.34)(0.15, 0.53)
x5[TeX:] $ {x_{5}}$(0.38, 0.19)(0.65, 0.32)(0.06, 0.29)(0.24, 0.39)
x6[TeX:] $ {x_{6}}$(0.56, 0.12)(0.50, 0.41)(0.21, 0.07)(0.06, 0.28)

As it has been pointed in Xia et al. (2012) that all the criteria Cj[TeX:] $ {C_{j}}$ (j=1,2,3,4)[TeX:] $ (j=1,2,3,4)$ are the benefit criteria, the IFNs of the alternatives xi[TeX:] $ {x_{i}}$ (i=1,2,…,6)[TeX:] $ (i=1,2,\dots ,6)$ do not need normalization. Thus to obtain the alternative(s), the following steps are given:

  • (1) Aggregate the IFNs βij[TeX:] $ {\beta _{ij}}$ (j=1,2,3,4)[TeX:] $ (j=1,2,3,4)$ of the alternative xi[TeX:] $ {x_{i}}$ (i=1,2,3,4,5,6)[TeX:] $ (i=1,2,3,4,5,6)$, denoted as βi[TeX:] $ {\beta _{i}}$ (i=1,2,3,4,5,6)[TeX:] $ (i=1,2,3,4,5,6)$, by the SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operator (9).

  • (2) Calculate the score s(βi)[TeX:] $ s({\beta _{i}})$ of βi[TeX:] $ {\beta _{i}}$ (i=1,2,3,4,5,6)[TeX:] $ (i=1,2,3,4,5,6)$ by item (2.2) which is shown in Figs. 1, 2 and 3.

Fig. 1.

Variation of the memberships of the aggregated results by SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operators.

Variation of the memberships of the aggregated results by SIFWMTω,gγSS$ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operators.
Fig. 2.

Variation of the nonmemberships of the aggregated results by SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operators.

Variation of the nonmemberships of the aggregated results by SIFWMTω,gγSS$ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operators.
Fig. 3.

Variation of the scores of the aggregated results by SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operators.

Variation of the scores of the aggregated results by SIFWMTω,gγSS$ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operators.

Obviously, the aggregated results and the ranking orders of the alternatives vary with the parameter γ, that is, they can be considered as the function with γ as its independent variable. Thus we can illustrate them by their functional images as follows:

  • Fig. 1 gives the variation of the memberships of the aggregated results by SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operators, denoted as μiTω,gγH[TeX:] $ {\mu _{i}^{{T_{\omega ,{g_{\gamma }^{H}}}}}}$ ( i=1,2,…,6[TeX:] $ i=1,2,\dots ,6$), respectively, with the parameter γ from −20 to 20. Particularly, when γ=0[TeX:] $ \gamma =0$, it is the result obtained by the operator in Xia et al. (2012) ; when γ=1[TeX:] $ \gamma =1$, it is the result obtained by the operator in Beliakov et al. (2011).

  • Fig. 2 indicates the variation of the nonmemberships of the aggregated results by SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operators where the values of γ increase from −20[TeX:] $ -20$ to 20. Similarly, when γ=0[TeX:] $ \gamma =0$, it is the result obtained by the operator in Xia et al. (2012) ; when γ=1[TeX:] $ \gamma =1$, it is the result obtained by the operator in Beliakov et al. (2011).

  • Fig. 3 provides the variation of the scores of the alternatives obtained by the SIFWMTω,gγSS[TeX:] $ {\text{SIFWM}\hspace{2.5pt}^{{T_{\omega ,{g_{\gamma }^{SS}}}}}}$ operator with γ from −30 to 30. When γ<−14.7[TeX:] $ \gamma <-14.7$, that is, pessimistically, the optimal alternative is x4[TeX:] $ {x_{4}}$; when −14.7<γ<−1.2[TeX:] $ -14.7<\gamma <-1.2$, relatively pessimistically, optimal one is x2[TeX:] $ {x_{2}}$; when −1.2<γ<9.2[TeX:] $ -1.2<\gamma <9.2$, impartially the optimal one is x6[TeX:] $ {x_{6}}$; when 9.2<γ[TeX:] $ 9.2<\gamma $, optimistically, the optimal one is x5[TeX:] $ {x_{5}}$. It is obvious that the alternative x5[TeX:] $ {x_{5}}$ varies from the worst one to the optimal one with the parameter γ which reflects the attitude of the decision maker.

In order to compare the ranking orders with those that are provided by SIFWG operator (Liao and Xu, 2015), we take γ=0[TeX:] $ \gamma =0$, and γ=1[TeX:] $ \gamma =1$, that is, the results obtained by the operators in Xia and Xu (2012), Beliakov et al. (2011), which are listed in Table 2.

Table 2

Ranking orders determined by different aggregation operators.

OperatorRanking order
The proposed operator with γ=−∞[TeX:] $ \gamma =-\infty $ x4≻x2≻x6≻x1≻x3≻x5[TeX:] $ {x_{4}}\succ {x_{2}}\succ {x_{6}}\succ {x_{1}}\succ {x_{3}}\succ {x_{5}}$
The proposed operator with γ=−1[TeX:] $ \gamma =-1$ x6≻x2≻x3≻x1≻x5≻x4[TeX:] $ {x_{6}}\succ {x_{2}}\succ {x_{3}}\succ {x_{1}}\succ {x_{5}}\succ {x_{4}}$
The proposed operator with γ=0[TeX:] $ \gamma =0$ (Xia and Xu, 2012) x6≻x3≻x2≻x1≻x5≻x4[TeX:] $ {x_{6}}\succ {x_{3}}\succ {x_{2}}\succ {x_{1}}\succ {x_{5}}\succ {x_{4}}$
The proposed operator with γ=1[TeX:] $ \gamma =1$ (Beliakov et al., 2011) x6≻x3≻x2≻x5≻x1≻x4[TeX:] $ {x_{6}}\succ {x_{3}}\succ {x_{2}}\succ {x_{5}}\succ {x_{1}}\succ {x_{4}}$
The proposed operator with γ=∞[TeX:] $ \gamma =\infty $ x5≻x6≻x2≻x4≻x1≻x3[TeX:] $ {x_{5}}\succ {x_{6}}\succ {x_{2}}\succ {x_{4}}\succ {x_{1}}\succ {x_{3}}$
The operator provided by Liao (Liao and Xu, 2015) x6≻x2≻x3≻x1≻x5≻x4[TeX:] $ {x_{6}}\succ {x_{2}}\succ {x_{3}}\succ {x_{1}}\succ {x_{5}}\succ {x_{4}}$

6Conclusions

Various aggregation operators have been constructed to adapt to different situations. In this paper, we proposed the SIFWM operators w.r.t. weighted Archimedean t-norms and t-conorms to neutrally deal with membership and non-membership of intuitionistic fuzzy information. Comparing the existing symmetrical operators with the proposed ones, we found that

  • (1) the existing symmetrical operators in Beliakov et al. (2011), Xia and Xu (2012) are special cases of the proposed ones with constant parameters which only reflect the impartial attitude of the decision maker;

  • (2) the proposed symmetrical operators can not only reflect the impartial attitude of the decision maker but also the optimistic or pessimistic attitude by a parameter, which provides more choices for the decision maker in the procedure of decision making.

In the future, we will utilize the symmetrical aggregation operators in other fuzzy environments such as linguistic, bipolar, Pythagorean and intuitionistic multiplicative fuzzy environment (Alghamdi et al., 2018; Alonso et al., 2013; Ma and Xu, 2016, 2018) to investigate the consensus in group decision making problems (Del Moral et al., 2018; Dong et al., 2018; Urena et al., 2019).

Acknowledgements

The authors would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper.

References

1 

Atanassov, K. ((1986) ). Intuitionistic fuzzy set. Fuzzy Sets and Systems, 20: , 87–96.

2 

Alghamdi, M.A., Alshehri, N.O., Akram, M. ((2018) ). Multi-criteria decision-making methods in bipolar fuzzy environment. International Journal of Fuzzy Systems, 20: , 2057–2064.

3 

Alonso, S., Perez, I.J., Cabrerizo, F.J., Herrera-Viedma, E. ((2013) ). A linguistic consensus model for Web 2.0 communities. Applied Soft Computing, 13: , 149–157.

4 

Beliakov, G., James, S. ((2013) ). On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts. Fuzzy Sets and Systems, 211: , 84–98.

5 

Beliakov, G., Pradera, A., Calvo, T. ((2007) ). Aggregation Functions: A Guide for Practitioners. Springer, Heidelberg, Berlin, New York.

6 

Beliakov, G., Bustince, H., Goswami, D.P., Mukherjee, U.K., Pal, N.R. ((2011) ). On averaging operators for Atanassov’s intuitionistic fuzzy sets. Information Sciences, 181: , 1116–1124.

7 

Bonferroni, C. ((1950) ). Sulle medie multiple di potenze. Bolletino Matematica Italiana, 5: , 267–270.

8 

Calvo, T., Mesiar, R. ((2003) ). Weighted triangular norms-based aggregation operators. Fuzzy Sets and Systems, 137: , 3–10.

9 

Chen, T.Y. ((2011) ). Bivariate models of optimism and pessimism in multi-criteria decision-making based on intuitionistic fuzzy sets. Information Sciences, 181: , 2139–2165.

10 

Chen, S.M., Tan, J.M. ((1994) ). Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 67: , 163–172.

11 

Del Moral, M.J., Chiclana, F., Tapia, J.M., Herrera-Viedma, E. ((2018) ). A comparative study on consensus measures in group decision making. International Journal of Intelligent Systems, 33: , 1624–1638.

12 

Deng, J.L. ((1989) ). Introduction to grey system theory. Journal of Grey Systems, 1: , 1–24.

13 

Deschrijver, G., Kerre, E.E. ((2003) ). On the relationship between some extensions of fuzzy set theory. Fuzzy Sets and Systems, 133: , 227–235.

14 

Deschrijver, G., Kerre, E.E. ((2007) ). On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision. Information Sciences, 177: , 1860–1866.

15 

Deschrijver, G., Kerre, E.E. ((2008) ). Aggregation operators in interval-valued fuzzy and Atanassov’s intuitionistic fuzzy set sheory. In: Bustince, H., Herrera, F., Montero, J. (Eds.), Fuzzy Sets and Their Extensions: Representation, Aggregation and Models. Springer, Heidelberg, New York, pp. 183–203.

16 

Dong, Y., Zhao, S., Zhang, H., Chiclana, F., Herrera-Viedma, E. ((2018) ). A self-management mechanism for non-cooperative behaviors in large-scale group consensus reaching processes. IEEE Transactions on Fuzzy Systems, 26: , 3276–3288.

17 

Dubois, D., Gottwald, S., Hajek, P., Kacprzyk, J., Prade, H. ((2005) ). Terminological difficulties in fuzzy set theory – the case of intuitionistic fuzzy sets. Fuzzy Sets and Systems, 156: , 485–491.

18 

Gau, W.L., Buehrer, D.J. ((1993) ). Vague sets. IEEE Transactions on Systems Man and Cybernetics, 23: , 610–614.

19 

Hardy, G.H., Littlewood, Pólya G, J.E. ((1934) ). Inequalities. Cambridge University Press, Cambridge.

20 

Hong, D.H., Choi, C.H. ((2000) ). Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 114: , 103–113.

21 

Klement, E.P., Mesiar, R., Pap, E. ((2000) ). Triangular Norms. Kluwer Academic Publishers, Boston, London, Dordrecht.

22 

Li, D.F. ((2010) ). Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets. Expert Systems with Applications, 37: , 8673–8678.

23 

Liao, H., Xu, Z.S. ((2014) ). Intuitionistic fuzzy hybrid weighted aggregation operators. International Journal of Intelligent Systems, 29: , 971–993.

24 

Liao, H., Xu, Z.S. ((2015) ). Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process. Applied Soft Computing, 35: , 812–826.

25 

Ma, Z.M., Xu, Z.S. ((2016) ). Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision-making problems. International Journal of Intelligent Systems, 31: , 1198–1219.

26 

Ma, Z.M., Xu, Z.S. ((2018) ). Hyperbolic scales involving appetites-based intuitionistic multiplicative preference relations for group decision making. Information Sciences, 451–452: , 310–325.

27 

Sambuc, R. (1975). Fonctions ϕ-floues. Application l’aide au diagnostic en pathologie thyroidienne. Ph.D. Thesis, Univ. Marseille, France.

28 

Tan, C., Jiang, Z., Chen, X. ((2013) ). Some issues on quasi-arithmetic intuitionistic fuzzy OWA operators. Applied Mathematics & Information Sciences, 7: , 955–961.

29 

Urena, R., Kou, G., Dong, Y., Chiclana, F., Herrera-Viedma, E. ((2019) ). A review on trust propagation and opinion dynamics in social networks and group decision making frameworks. Information Sciences, 478: , 461–475.

30 

Wang, W., Liu, X. ((2011) ). Intuitionistic fuzzy geometric aggregation operators based on Einstein operations. International Journal of Intelligent Systems, 26: , 1049–1075.

31 

Wang, W., Liu, X. ((2012) ). Intuitionistic fuzzy information aggregation using Einstein operations. IEEE Transactions on Fuzzy Systems, 20: , 923–938.

32 

Xu, Z.S. ((2007) ). Intuitionistic fuzzy aggregation operators. IEEE Transactions on Fuzzy Systems, 15: , 1179–1187.

33 

Xu, Z.S., Yager, R.R. ((2006) ). Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of Intelligent Systems, 35: , 417–433.

34 

Xu, Z.S., Yager, R.R. ((2011) ). Intuitionistic fuzzy Bonferroni means. IEEE Transactions on Systems, Man, and Cybernetics–Part B: Cybernetics, 41: , 568–578.

35 

Xu, C.Y., Ma, Z.M. ((2019) ). Symmetric intuitionistic multiplicative aggregation operator for group decision making in intuitionistic multiplicative environments. Journal of Intelligent & Fuzzy Systems, 36: , 5909–5918.

36 

Xia, M., Xu, Z.S. ((2012) ). Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment. Information Fusion, 13: , 31–47.

37 

Xia, M., Xu, Z.S., Zhu, B. ((2012) ). Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm. Knowledge-Based Systems, 31: , 78–88.

38 

Yager, R.R. ((1988) ). On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transactions on Systems Man and Cybernetics, 18: , 183–190.

39 

Yager, R.R. ((2004) a). Weighted triangular norms using generating functions. International Journal of Intelligent Systems, 19: , 217–231.

40 

Yager, R.R. ((2004) b). Generalized OWA aggregation operators. Fuzzy Optimization and Decision Making, 3: , 93–107.

41 

Yager, R.R. ((2009) ). On generalized Bonferroni mean operators for multi-criteria aggregation. International Journal of Approximate Reasoning, 50: , 1279–1286.

42 

Yager, R.R., Filev, D.P. ((1999) ). Induced ordered weighted averaging operators. IEEE Transactions on Systems Man and Cybernetics, 29: , 141–150.

43 

Yang, W., Chen, Z.P. ((2012) ). The quasi-arithmetic intuitionistic fuzzy OWA operators. Knowledge-Based Systems, 27: , 219–233.

44 

Yang, W., Jhang, S.T., Fu, Z.W., Ma, Z.M. ((2019) ). Aggregating intuitionistic fuzzy preference relations with symmetrical intuitionistic fuzzy Bonferroni mean operators in group decision making. Information Sciences.

45 

Zadeh, L.A. ((1965) ). Fuzzy sets. Information Control, 8: , 338–353.

46 

Zadeh, L.A. ((1975) ). The concept of a linguistic variable and its application to approximate reasoning. Information Sciences, 8: , 199–249.

47 

Zhao, H., Xu, Z.S., Ni, M.F., Liu, S.S. ((2010) ). Generalized aggregation operators for intuitionistic fuzzy sets. International Journal of Intelligent Systems, 25: , 1–30.

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