Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making

Highlights

• The weighted possibility means of TrIFNs are firstly introduced.

• A new ranking method for TrIFNs is thereby presented.

• Minkowski distance for TrIFNs is defined.

• Four kinds of power geometric operators of TrIFNs are developed.

• The collective overall values of alternatives are obtained by goal programming model.

Abstract

As a special intuitionistic fuzzy set on a real number set, trapezoidal intuitionistic fuzzy numbers (TrIFNs) have the better capability to model ill-known quantities. The purpose of this paper is to develop some power geometric operators of TrIFNs and apply to multi-attribute group decision making (MAGDM) with TrIFNs. First, the lower and upper weighted possibility means of TrIFNs are introduced as well as weighted possibility means. Hereby, a new lexicographic method is developed to rank TrIFNs. The Minkowski distance between TrIFNs is defined. Then, four kinds of power geometric operators of TrIFNs are investigated including the power geometric operator of TrIFNs, power weighted geometric operator of TrIFNs, power ordered weighted geometric operator of TrIFNs and power hybrid geometric operator of TrIFNs. Their desirable properties are discussed. Four methods for MAGDM with TrIFNs are respectively proposed for the four cases whether the weight vectors of attributes and DMs are known or unknown. In these methods, the individual overall attribute values of alternatives are generated by using the power geometric or power weighted geometric operator of TrIFNs. The collective overall attribute values of alternatives are determined through constructing the multi-objective optimization model, which is transformed into the goal programming model to solve. Thus, the ranking order of alternatives is obtained according to the collective overall attribute values of alternatives. Finally, the green supplier selection problem is illustrated to demonstrate the application and validation of the proposed method.

Introduction

A human being who expresses the degree of membership of a given element in a fuzzy set (FS) [1] very often does not express corresponding degree of non-membership as the complement to 1. Thus, Atanassov [2] proposed the intuitionistic fuzzy set (IFS) using two characteristic functions expressing the degree of membership and the degree of non-membership of elements of the universal set to the IFS. It can effectively handle the presence of vagueness and hesitancy originating from imprecise knowledge or information. Many researchers have paid great attention to discussion on possible application of the IFS to the fields of multi-attribute decision making (MADM) and multi-attribute group decision making (MAGDM) [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. They achieved many research results, such as aggregation operators [3], [4], [5], [6], [7], entropy measures [8], extension of classical decision making methods [9], [10], [11], [12], [13] and new decision making methods [14], [15], [16], [17].

As a generalization of fuzzy numbers [18], the intuitionistic fuzzy number (IFN) is a special IFS defined on the real number set, which seems to suitably describe an ill-known quantity [19]. Currently, there are three kinds of typical IFNs: triangular IFN (TIFN) [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], trapezoidal IFN (TrIFN) [29], [30], [31], [32], [33], [34], [35], [36], [37], [38] and interval-valued trapezoidal IFN (IVTrIFN) [39], [40], [41], which have attracted much research interest recently. Shu et al. [20] defined the concept of a TIFN in a similar way to that of the fuzzy number [18] and developed an algorithm for intuitionistic fuzzy fault tree analysis. Li [19] pointed out and corrected some errors in the definition of the four arithmetic operations over the TIFNs in [20]. Li [21] discussed the concept of the TIFN and ranking method on the basis of the concept of a ratio of the value index to the ambiguity index as well as applications to MADM problems in depth. Li et al. [22] defined the values and ambiguities of the membership degree and the non-membership degree for TIFNs as well as the value-index and ambiguity-index. Hereby a value and ambiguity based method is developed to rank TIFNs and applied to solve MADM problems in which the ratings of alternatives on attributes are expressed using TIFNs. Nan et al. [23] defined the ranking order relations of TIFNs, which are applied to matrix games with payoffs of TIFNs. Wang et al. [24] proposed new arithmetic operations and logic operators for TIFNs and applied them to fault analysis of a printed circuit board assembly system. Wan et al. [25] proposed the extended VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method for solving MAGDM with TIFNs. Wan et al. [26] investigated the possibility mean, variance and covariance of TIFNs. Wan [27] defined the possibility variance coefficient of TIFNs and applied to MADM with TIFNs. Wan and Dong [28] proposed the possibility method for MAGDM with TIFNs and incomplete weight information.

As the extensions of the TIFNs, Wang [29] defined the TrIFN and IVTrIFN. Wang and Zhang [30] investigated the weighted arithmetic averaging operator and weighted geometric averaging operator on TrIFNs and their applications to MADM problems. Wei [31] investigated some arithmetic aggregation operators with TrIFNs and their applications to MAGDM problems. Du and Liu [32] extended fuzzy VIKOR method with TrIFNs. Wu and Cao [33] developed some families of geometric aggregation operators with TrIFNs and applied to MAGDM problems. Wan and Dong [34] defined the expectation and expectant score, ordered weighted aggregation operator and hybrid aggregation operator for TrIFNs and employed to MAGDM. Ye [35] developed the expected value method for intuitionistic trapezoidal fuzzy multicriteria decision-making problems. Ye [36] proposed the MAGDM method using vector similarity measures for TrIFNs. Zhang et al. [37] proposed a grey relational projection method for MAGDM based on TrIFNs. Wan [38] investigated four kinds of power average operators of TrIFNs, involving the power average operator (TrIFPA) of TrIFNs, weighted power average (TrIFWPA) operator of TrIFNs, power order weighted average (TrIFPOWA) operator of TrIFNs, and power hybrid average (TrIFPHA) operator of TrIFNs. Wan [39] firstly defined some operational laws of IVTrIFNs and developed the IVTrIFN weighted arithmetical average operator and weighted geometrical average operator. An approach to ranking IVTrIFNs is presented based on the score function and accurate function. The MAGDM method using IVTrIFNs is then proposed. Wan [40] defined the Hamming and Euclidean distances for IVTrIFNs and proposed the fractional programming method for the MADM problems using IVTrIFNs. Wu and Liu [41] defined some interval-valued trapezoidal intuitionistic fuzzy geometric aggregation operators and applied to MAGDM with IVTrIFNs.

The above researches about IFNs mainly focus on the operation laws [19], [20], [24], [37], [38], aggregation operators [24], [30], [33], [34], [38], [41], ranking methods [22], [23], [27], [28], [29], [39], [41], extension of classical decision making methods [25], [32], [37] and new decision making methods [27], [28], [35], [40]. It is worthwhile to mention that the domains of the IFSs are discrete sets, which are also the same as fuzzy sets. TIFNs, TrIFNs and IVTrIFNs extend the domain of IFSs from the discrete set to the continuous set. They are the extensions of fuzzy numbers [30]. Compared with the IFSs, TrIFNs are defined by using trapezoidal fuzzy numbers expressing their membership and non-membership functions. Hence, TrIFNs may better reflect the information of decision problems than IFSs. With the increasing complexity of modern society, continued expansion of the scale and the diversification of business, many large and important management decision optimization problems require many experts to participate in making decisions together. Therefore, the MAGDM problems with TrIFNs are of great importance for scientific researches and real applications.

It should be pointed out that the aggregation operators of IFNs [25], [30], [31], [33], [34], [40], [41] did not consider the information about the relationship between the values being fused. In fact, the relationship between the variables being fused is very important to the decision making results for some real-life decision problems [5], [42], [43], [44], [45], [46], [47]. In this respect, the power-average (PA) operator and power OWA (POWA) operator developed by Yager [42] sufficiently consider the relationship between the variables being fused since their weighting vectors depend upon the input arguments and allow values being aggregated to support. Based on this, Xu and Yager [43] developed the power-geometric (PG) operator, weighted PG (WPG) operator, and power-ordered-weighted-geometric (POWG) operator and extended the PG and POWG operators to uncertain environments. Zhou and Chen [44] presented the generalized power average (GPA) operator and generalized power ordered weighted average (GPOWA) operator. They also proposed the linguistic generalized power average (LGPA) operator, the weighted linguistic generalized power average (WLGPA) operator and the linguistic generalized power ordered weighted average (LGPOWA) operator. Xu and Cai [45] proposed the uncertain power average operators for aggregating interval fuzzy preference relations. Xu and Wang [46] developed 2-tuple linguistic power average (2TLPA) operator, 2-tuple linguistic weighted PA operator (2TLWPA) and 2TLPOWA operator. Zhou et al. [47] presented the uncertain generalized power average (UGPA) operator and its weighted form, and the uncertain generalized power ordered weighted average (UGPOWA) operator. They also extended the GPA operator and the GPOWA operator to generalized intuitionistic fuzzy (IF) environment, and defined the generalized IF power averaging (GIFPA) operator and the generalized IF power ordered weighted averaging (GIFPOWA) operator. Xu [5] developed the IF power aggregation operators and the interval-valued IF power aggregation operators.

However, the above power average and power geometric operators [12], [42], [43], [44], [45], [46], [47] cannot directly be applied to the case where the aggregation arguments are IFNs, such as TIFNs, TrIFNs and IVTrIFNs. Especially, the PG, WPG and POWG operators developed in [43] can only deal with real numbers and intervals, but are invalid if the aggregation arguments are IFNs. Therefore, the focus of this paper is to develop some power geometric operators of TrIFNs. Firstly, the weighted possibility means of TrIFNs are introduced and thereby a new method is presented to rank the TrIFNs. The Minkowski distance between TrIFNs is defined. Then, four kinds of power geometric operators of TrIFNs are developed, including the power geometric (TrIFPG) operator of TrIFNs, power weighted geometric (TrIFPWG) operator of TrIFNs, power ordered weighted geometric (TrIFPOWG) operator of TrIFNs and power hybrid geometric (TrIFPHG) operator of TrIFNs. Their desirable properties are discussed in detail. These power geometric operators can take all the decision arguments and their relationships into consideration. Finally, four methods for MAGDM with TrIFNs are proposed for the four cases whether the weight vectors of attributes and DMs are unknown or known. Compared with the pertinent literature [5], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], this paper has the following difference and features:

• Wan [36] investigated power average operators of TrIFNs, while this paper develops power geometric operators of TrIFNs. The fusion principles of power average operators and power geometric operators of TrIFNs are significantly different. The latter are the useful complements and extensions of the former. The decision methods proposed by Wan [36] and this paper are also remarkably different. The former utilized TrIFPHA operator to obtain the collective overall values of alternatives, whereas the latter constructs the goal programming model to derive objectively the collective ones. Furthermore, the former is only suitable to the situation whether the weight vector of DMs is known, while the latter can deal with the situation where the weight vector of DMs is known or unknown.

• The four methods proposed in this paper are based on power geometric operators of TrIFNs which can sufficiently consider the relationship among input TrIFNs arguments, effectively relieve the influence of unfair data on the fusion results, and thus make the decision results more reasonable. Whereas the methods proposed in [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [39], [40], [41] are based on the weighted arithmetic (or geometric) averaging operators of TrIFNs which consider the input arguments as independent, i.e., the inter-relationship among the individual arguments has not been captured by those operators.

• Wei [31] and Wu and Cao [33] simply calculated the distances between the TrIFNs and positive ideal solution to rank the TrIFNs. Such a ranking method of TrIFNs is a single-index approach, which is not always feasible and effective. Using the weighted possibility means, this paper proposes the lexicographic ranking method of TrIFNs, which is a two-index approach and more reasonable than the methods of [31], [33]. The four methods proposed in this paper have wider range of real application, better flexibility and agility than the methods [5], [33].

The main contributions of this work are outlined as follows:

• Define the weighted possibility means of the membership and non-membership functions for TrIFNs and thereby present a new lexicographic method to rank TrIFNs.

• Develop four kinds of power geometric operators of TrIFNs (i.e., TrIFPG, TrIFPWG, TrIFPOWG, and TrIFPHG operators) and discuss their desirable properties.

• Propose four methods for MAGDM with TrIFNs respectively aimed at the four cases whether the weight vectors of attributes and DMs are unknown or known.

The remainder of this paper unfolds as follows: Section 2 introduces the operation laws, weighted possibility means, ranking method and distance for TrIFNs. Four kinds of power geometric operators of TrIFNs are developed in Section 3. Section 4 presents the problems of MAGDM with TrIFNs and proposes the corresponding decision methods. In Section 5, a green supplier selection example and the comparison analyzes are given. Section 6 ends the paper with some concluding remarks.