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

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.


Introduction
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. 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), 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,  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 , 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  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), , , 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.

Preliminaries
To make the presentation self-contained, in what follows, we review some basic concepts.
Definition 2.4 (See Klement et al., 2000). A t-conorm S is called an Archimedean tconorm if it is continuous and S(x, x) > x for all x ∈ (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).
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)), and the same applies to its dual t-conorm, 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, ∞] such that g(1) = 0 in the following parts. For nilpotent operations the inverse changes to the pseudo-inverse.
Furthermore, let 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 A ω :

Intuitionistic Fuzzy Sets
Definition 2.5 (See Atanassov, 1986). Let X be a given universe. An intuitionistic fuzzy set (IFS) A in X is defined as follows: indicate the amounts of guaranteed membership and nonmembership of x in A, respectively, and satisfy μ A (x) + ν A (x) 1.
We recall for an intuitionistic fuzzy set the membership grade of x in A which is represented as a pair (μ A (x), ν A (x)) is called a intuitionistic fuzzy number (IFN) (Xu, 2007) and the set of all IFNs is denoted as IFN . Here, the expression π A ( , 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 , which is defined for α = (μ α , ν α ) and β = (μ β , ν β ) as β α if and only if μ β μ α and ν α ν β . 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: The larger the score s(α) 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) = μ α + ν α , where h(α) ∈ [0, 1]. When the scores are the same, the larger the accuracy h(α) is, the greater the IFN α is. However, it is obvious that h(α) + π α = 1.
Definition 2.9 (See . Let α j (j = 1, 2, . . . , n) be IFNs and T ω,g be a weighted Archimedean t-norm with an additive generator g and IFWM T ω,g : then IFWM T ω,g is called an intuitionistic fuzzy weighted mean (IFWM T ω,g ) operator of dimension n w.r.t. T ω,g .
respectively, then the following families of IFWM T ω,g operators are obtained: Note that IFWM T ω,g H γ has been investigated in . In particular, if γ = 1, then Beliakov et al. (2011) and IFWM T ω,g D 1 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 (See Beliakov and James, 2013). Given an aggregation function agg : [0, 1] n → [0, 1], the natural extension of the aggregation function agg is given by Agg :

Symmetric 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), , , 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.

Symmetric Intuitionistic Fuzzy Weighted Mean Operators w.r.t. Weighted Archimedean t-Norms
Lemma 3.1. Let α j (j = 1, 2, . . . , n) be a collection of IFNs and T ω,g be a weighted Archimedean t-norm with an additive generator g. Then is an IFN.
Especially, if ν α j = 1 − μ α j for all j = 1, 2, . . . , n, that is, all α j are reduced to μ α j , respectively, then Eq. (17) is reduced to the following form: 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) = y 1−x+y (Calvo and Mesiar, 2003) to aggregate fuzzy information.
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) = g H γ (1 − t), then the SIFWM S ω,g operator is reduced to the following form: (1 − x j ) ω j is the symmetric form of intuitionistic fuzzy weighted averaging (IFWA) operator defined by Xu (2007) is the symmetric form of IFWG operator based on Einstein t-norm and t-conorm defined by Wang and Liu (2012) 2. If g(t) = g SS γ (1 − t), then the SIFWM S ω,g operator is reduced to the following form: where (3) If g(t) = g D γ (1 − t), then the SIFWM S ω,g operator is reduced to the following form: where Particularly, if γ = 1, then the SIFWM S ω,g operator is reduced to the following form: where S ω,g D All in all, we have the following conclusion: Corollary 3.7.
(1) SIFWM (2) SIFWM S ω,g SS (3) SIFWM (4) SIFWM The above corollary indicates that the operators SIFWM  (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;, the proposed aggregation operators based on weighted Archimedean tnorm and t-conorm possess the following advantages: • these existing operators (Beliakov et al., 2011;Liao and Xu, 2015; 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  is not suitable for dealing with IFNs (1, 0) or (0, 1); the proposed operators can solve the case.

The 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, ω i , x i ∈ [0, 1], λ > 0 such that n i=1 ω i = 1, it holds that (1) Ma and Xu (2016) if y x, then y y 1−x+y x and y x 1+x−y x; (2) if g is convex (concave), then g( n i=1 ω i x i ) ( ) n i=1 ω i g(x i ), the equality holds if and only if x 1 = x 2 = · · · = x n or g is linear.
The relationships among IFWM T ω,g , IFWA T ω,g and SIFWM T ω,g can be shown as follows: Proposition 4.2. Let α j (j = 1, 2, . . . , n) be a collection of IFNs and T ω,g be a weighted Archimedean t-norm T ω,g with an additive generator g.

An Approach to Intuitionistic Fuzzy Multi-Criteria Decision Making
For a multi-criteria decision making under intuitionistic fuzzy environment, let x = {x 1 , x 2 , . . . , x m } be a set of alternatives to be selected, and C = {C 1 , C 2 , . . . , C n } be a set of criteria to be evaluated. To evaluate the performance of the alternative x i under the criterion C j , the decision maker is required to provide not only the information that the alternative x i satisfies the criterion C j , but also the information that the alternative x i does not satisfy the criterion C j . This two part information can be expressed by μ ij and ν ij which denote the degrees that the alternative x i satisfies the criterion C j and does not satisfy the criterion C j , then the performance of the alternative x i under the criteria C j can be expressed by an IFN α ij with the condition that 0 μ ij , ν ij 1 and 0 μ ij + ν ij 1. When all the performances of the alternatives are provided, the intuitionistic fuzzy decision matrix D = (α ij ) m×n = ((μ ij , ν ij )) m×n can be constructed. To obtain the ranking of the alternatives, the following steps are given: (1) Transform the intuitionistic fuzzy decision matrix D = ((μ ij , ν ij )) m×n into the normalized one B = ((β ij )) m×n , where β ij = α ij , for benefit attribute x i ; α c ij , for cost attribute x i .
To illustrate the proposed methods, an example adapted from Chen (2011),  is given as follows: Example 5.1. The purchasing manager in a small enterprise considers various criteria involving C 1 : financial factors (e.g. economic performance, financial stability), C 2 : performance (e.g. delivery, quality, price), C 3 : technology (e.g. manufacturing capability, design capability, ability to cope with technology changes), and 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 = {C 1 , C 2 , C 3 , C 4 }, whose weight vector is ω = (0.34, 0.23, 0.22, 0.21) . There are six suppliers available, and the set of all alternatives is denoted by X = {x 1 , x 2 , . . . , x 6 }. The characteristics of the suppliers x i (i = 1, 2, . . . , 6) in terms of the criteria in C are expressed by the following intuitionistic fuzzy decision matrix (see Table 1).
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:  from −20 to 20. Particularly, when γ = 0, it is the result obtained by the operator in ; when γ = 1, it is the result obtained by the operator in Beliakov et al. (2011).  operator with γ from −30 to 30. When γ < −14.7, that is, pessimistically, the optimal alternative is x 4 ; when −14.7 < γ < −1.2, relatively pessimistically, optimal one is x 2 ; when −1.2 < γ < 9.2, impartially the optimal one is x 6 ; when 9.2 < γ , optimistically, the optimal one is x 5 . It is obvious that the alternative 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, and γ = 1, that is, the results obtained by the operators in , Beliakov et al. (2011), which are listed in Table 2.

Conclusions
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 Table 2 Ranking orders determined by different aggregation operators.

Operator
Ranking order The proposed operator with γ = −∞ x 4 x 2 x 6 x 1 x 3 x 5 The proposed operator with γ = −1 x 6 x 2 x 3 x 1 x 5 x 4 The proposed operator with γ = 0  x 6 x 3 x 2 x 1 x 5 x 4 The proposed operator with γ = 1 (Beliakov et al., 2011) x 6 x 3 x 2 x 5 x 1 x 4 The proposed operator with γ = ∞ x 5 x 6 x 2 x 4 x 1 x 3 The operator provided by Liao (Liao and Xu, 2015) x 6 x 2 x 3 x 1 x 5 x 4 (1) the existing symmetrical operators in Beliakov et al. (2011),  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;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).

Z.M.
Ma is an associate professor at School of Mathematics and Statistics, Linyi University. He has received his MS degree in systems theory and PhD in computational mathematics from the School of Mathematics and Statistics, Wuhan University. His current research is focused on decision-making theory and intelligent decision support systems.
W. Yang is a lecturer at School of Mathematics and Statistics, Linyi University. She has received her MS degree in applied mathematics from the School of Scineces, University of Science and Technology, Beijing. Her current research is focused on decision-making theory.