Generalized Triangular Intuitionistic Fuzzy Geometric Averaging Operator for Decision Making in Engineering

Intuitionistic fuzzy set, which can be represented using the triangular intuitionistic fuzzy number (TIFN), is a more generalized platform for expressing imprecise, incomplete and inconsistent information when solving multi-criteria decision-making problems, as well as for reflecting the evaluation information exactly in different dimensions. In this paper, the TIFN has been applied for solving some multi-criteria decision-making problems by developing a new triangular intuitionistic fuzzy geometric aggregation operator, that is the generalized triangular intuitionistic fuzzy ordered weighted geometric averaging (GTIFOWGA) operator, and defining some triangular intuitionistic fuzzy geometric aggregation operators including the triangular intuitionistic fuzzy weighted geometric averaging (TIFWGA) operator, the ordered weighted geometric averaging (TIFOWGA) operator and the hybrid geometric averaging (TIFHWGA) operator. Based on these operators, a new approach for solving multicriteria decision-making problems when the weight information is fixed has been proposed. Finally, the proposed method has been compared with some similar existing computational approaches by virtue of a numerical example to verify its feasibility and rationality.


Introduction
In solving multi-criteria decision-making (MCDM) problems, it is often required that several criteria are considered simultaneously before selecting or ranking alternatives.Since the information required for solving the MCDM problems is often incomplete, inconsistent and indeterminate, the manner in which it is expressed, therefore, has remained a major task and of great interest among researchers over the past several years.In handling these issues, Zadeh [1], who introduced the concept of fuzzy set theory, has outlined how the fuzzy set (FS) concept could be used for expressing such decision-making problems.However, the FS theory, which is characterized by only one function, "the membership function ( )", in most cases cannot be used fully to express some kind of complex fuzzy information."For example, during voting, if there are ten persons voting for an issue, and three of them give the "agree'', four of them give the "disagree", and the others abstain.Obviously, FS cannot fully express the polling information" [2].To solve this kind of problem, Atanassov [3] extended the fuzzy set theory by adding a new function "the nonmembership function ( )", in order to form the intuitionistic fuzzy set (IFS) theory.
The membership and non-membership functions of the IFS theory are represented by an intuitionistic fuzzy number (IFN), are more or less independent and are constrained with the conditions that the sum of the membership and non-membership must not exceed one [4].These constraints, however, have been challenged recently by Despi [5] who defined a new IFS in which the sum of the membership and non-membership functions are more than one and their differences are either positive or negative.This new IFS has been justified and supported by Li [6] and Marasini et al. [7].The computation of membership and non-membership function in this study will be based on the new IFS.
The application of TIFN in MCDM is based on its ability to express decision information in several dimensions and to reflect the assessment information in a more holistic manner [24].Several research efforts have been made in the advancement of TIFN over the past few years.Among them, we can mention the characterization of membership and non-membership degrees in intuitionistic fuzzy sets (IFS) using the triangular fuzzy numbers by Shu et al. [26].Chen and Li [27] developed a new distance measurement between two TIFNs for determining attribute weights, as well as weighted arithmetic averaging (TIFN-WAA) operators on TIFNs.Zhang and Nan [28] developed a methodology for ranking TIFNs by considering the concept of a TIFN as a special case of the IFN.Wan et al. [29], using the TIFN, extended the classical VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method for solving multi-attributes group decision-making (MAGDM) problems, while Li et al. [30] investigated the arithmetic operations and cut sets over TIFNs, and defined the values and ambiguities of the membership degree and non-membership degree for the TIFNs, as well as the value index and ambiguity index.
Other contributions to the study of TIFN are in the area of information fusion operators (aggregation), where Chen and Li [27] introduced the weighted arithmetic averaging operator on TIFNs (TIFN-WAA).Wan et al. [29] presented the triangular intuitionistic fuzzy weighted average (TIF-WA) operator for the selection of personnel.The triangular intuitionistic fuzzy weighted average (TIFWA) operator, the ordered weighted average (TIFOWA) operator, the hybrid weighted average (TIFHWA) operator, the triangular intuitionistic fuzzy generalized ordered weighted average (TIFGOWA) operator, and the generalized hybrid weighted average (TIFGHWA) operator were developed by Wan et al. [31].
Upon investigation of the different aggregation operators of TIFN, it has been revealed that the ranking of TIFNs are a bit complicated and cannot be easily compared with other TIFNs [32] as well as account for attitudinal character or risk attitude of the DMs.In order to further advance the study of aggregation operators of TIFN, simplify its comparison and application in MCDMs, and to express the risk attitude of the DMs in the decision-making process, this paper attempts to do the following: (1) Define some triangular intuitionistic fuzzy aggregation operators, that is, the triangular intuitionistic fuzzy weighted geometric averaging (TIFWGA) operator, ordered weighted geometric averaging (TIFOWGA) operator and the hybrid weighted geometric averaging (TIFHWGA) operator; (2) Develop a new generalized triangular intuitionistic fuzzy aggregation operator, that is, the generalized triangular intuitionistic fuzzy ordered weighted geometric averaging (GTIFOWGA) operator.This is mainly to allow for more attitudinal information to be expressed or used in accordance with the different DMs interests or preference; (3) Propose a simple and straightforward approach for solving MCDM problems when the performance ratings are expressed in triangular intuitionistic fuzzy numbers (TIFNs).
The rest of this paper is organized as follows: in Section 2, the concepts of intuitionistic fuzzy set theory and triangular intuitionistic fuzzy sets are presented.In Section 3, some triangular intuitionistic fuzzy weighted geometric operators are defined, and the GTIFOWGA operator is developed.In Section 4, the algorithm of the proposed method is presented and applied to solving MCDM problem.Finally, in Section 5, some conclusions are presented.

Preliminaries
In this section, the fundamental definitions and concepts of TIFN and IFS as described by Liang et al. [32] and Despic and Simonovic [33] are presented.

The Triangular Intuitionistic Fuzzy Number (TIFN)
The TIFN is basically the use of the traditional triangular fuzzy number to express the membership ( ) and non-membership degree ( ) such that the intuitionistic fuzzy number is based on the triangular fuzzy number, which is termed the triangular intuitionistic fuzzy number (TIFN).In the following, the basic concepts relating to the TIFN are introduced: Definition 2. Let be a TIFN, where the membership and non-membership function for are defined as follows [28,31,32]: Membership function: For non-membership function, it is given as: where The TIFN is therefore denoted as ́= 〈([ , , ]; ), ([ , , ]; )〉, when = 1, and = 0, and ́ will change into the traditional triangular fuzzy number (TFN).Generally, the TIFN ́ is defined as ́= ([ , , ]; , ) for convenience.In the following, the operational rules for any two TIFNs are presented.Peer-reviewed version available at Information 2017, 8, 78; doi:10.3390/info8030078 1.
The operational results for the rules given in the Definition 4 for the two TIFNs are given in the operations: 1.

Some Weighted Geometric Operators and the Generalized Ordered Weighted Geometric Operators of TIFNs
In this section, motivated by existing achievements [32,36,37], we develop some triangular intuitionistic fuzzy geometric averaging operators and discuss some of their useful properties and then introduce new generalized geometric operators for TIFNs.

Some Weighted Geometric Aggregation Operators on TIFNs
) .

The Generalized Ordered Geometric Operator of TIFNs
The TIFOWGA operator is extended to develop a new generalized aggregation operator for TIFN.The new GTIFOWGA has been inspired by the work of Tan [37] and Qi et al. [38].

Suppose
For = + 1, we then have: It confirms that the result is true for = + 1, and thus it holds for all of .Hence, The theorem is true for any number of TIFN, which completes the proof.□ Proof: Since = , then we have It follows that,
Step 4: Rank the alternatives by virtue of Definition 6.

Numerical Example
Suppose the product development team of a design company "X" has generated four new design alternatives (A1, A2, A3, and A4) for a new crane machine during the conceptual design phase.A group of experts (E1, E2, E3, and E4), within the company have given their aggregated assessment (see Table 2) of the design alternatives with respect to the criteria: Expected mechanical safety C1, Amount of wear C2, Operating and maintenance cost C3, and Mass and size C4, whose weight vectors are given as = {0.15,0.25, 0.32, 0.28}.We select the best alternative design using the proposed algorithm.Since the experts' assessments have already been aggregated, we jump to step 2 in the algorithm to derive the overall preference values.Using the GTIFOWGA operator when the criteria weighting vector are given as = {0.15,0.25, 0.32, 0.28}, the comprehensive evaluation for the four design alternatives are shown in Table 3.By applying Definitions 6, we can obtain the ranking of all the design alternatives as shown in Table 4.In addition, from Tables 3 and 4, we can see that, when the value of λ changes, the rankings of the design alternatives also change.Furthermore, if we decide to use any of the operators (TIFOWGA, TIFWGA or the TIFHWGA) in step 2, the ranking of the design alternatives will, therefore, be in the order > > > , where the best alternative is .

Table 2 .
Aggregation of all the experts' assessments (group intuitionistic fuzzy decision matrix).

Table 3 .
The overall preference values (comprehensive evaluations for four alternatives).

Table 4 .
The rankings of all design alternatives.