A Method Based on Intuitionistic Fuzzy Dependent Aggregation Operators for Supplier Selection

. Recently, resolving the decision making problem of evaluation and ranking the potential suppliers have become as a key strategic factor for business firms. In this paper, two new intuitionistic fuzzy aggregation operators are developed: dependent intuitionistic fuzzy ordered weighed averaging (DIFOWA) operator and dependent intuitionistic fuzzy hybrid weighed aggregation (DIFHWA) operator. Some of their main properties are studied. A method based on the DIFHWA operator for intuitionistic fuzzy multiple attribute decision making is presented. Finally, an illustrative example concerning supplier selection is given.


Introduction
As extension of Zadeh's fuzzy set [1] whose basic component is only a membership function, Atanassov [2][3][4] introduced the concept of intuitionistic fuzzy set (IFS).Bustince and Burillo [5] showed that IFS are vague sets [6].IFS has been proven to be highly useful to deal with uncertainty and vagueness, and a lot of work has been done to develop and enrich the IFS theory [7,8].In many complex decision making problems, the decision information provided by the decision maker is often imprecise or uncertain [9] due to time pressure, lack of data, or the decision maker's limited attention and information processing capabilities.Thus, IFS is a very suitable tool to be used to describe imprecise or uncertain decision information.Recently, some approaches were investigated to multiple attribute decision making (MADM) problems based on intuitionistic fuzzy sets [10][11][12][13][14].One of the important things of the MADM problems is to aggregate the information provided by the experts.Aggregating intuitionistic fuzzy information has received more and more attention in recent years.Xu and Yager [15] developed some geometric aggregation operators based on intuitionistic fuzzy sets, such as intuitionistic fuzzy weighed geometric (IFWG) operator, intuitionistic fuzzy ordered weighed geometric (IFOWG) operator, intuitionistic fuzzy hybrid geometric (IFHG) operator and applied them to multiple attribute decision making.Xu [16] also developed some intuitionistic fuzzy aggregation operators, such as the intuitionistic fuzzy weighed averaging (IFWA) operator, intuitionistic fuzzy ordered weighed averaging (IFOWA) operator, and intuitionistic fuzzy hybrid averaging (IFHA) operator, Xu and Yager [17] developed an operator called dynamic intuitionistic fuzzy weighed averaging (DIFWA) operator and procedure to deal with the situations where all the attribute values are collected at different periods.Wei [18] proposed the dynamic intuitionistic fuzzy weighed geometric (DIFWG) operator and induced intuitionistic fuzzy ordered weighed geometric (I-IFOWG) operator [19].Zhao et al. [20] proposed the generalized intuitionistic fuzzy weighed averaging (GIFWA) operator, generalized intuitionistic fuzzy ordered weighed averaging (GIFOWA) operator, and generalized intuitionistic fuzzy hybrid averaging (GIFHA) operator.Based on the correlation properties of the Choquet integral, Xu [21] and Tan and Chen [22] proposed the intuitionistic fuzzy Choquet integral operator, respectively.Xia and Xu [23] developed a series of intuitionistic fuzzy point aggregation operators based on the idea of generalized aggregation.Xu and Yager [24] developed an intuitionistic fuzzy Bonferroni mean (IFBM) and applied the weighed IFBM to MADM.Xu [25] developed a series of intuitionistic fuzzy aggregation operators, whose weighing vectors depend upon the input arguments and allow values being aggregated to support and reinforce each other.Xu and Wang [26] developed the intuitionistic fuzzy induced generalized aggregation operators.On the basis of the idea of the ordered weighed averaging distance (OWAD) operator [27,28], Zeng and Su [29] developed an intuitionistic fuzzy ordered weighed distance (IFOWD) operator.Zeng [30] developed the intuitionistic fuzzy hybrid weighed distance measure, and presented a consensus reaching process for group decision making with intuitionistic fuzzy preference information.Yu [31] developed the intuitionistic fuzzy prioritized weighed average (IFPWA) and the intuitionistic fuzzy prioritized weighed geometric (IFPWG) operators.Yu [32] developed some new aggregation operators for intuitionisticfuzzy information are proposed, including the intuitionistic fuzzy geometric Heronian mean (IFGHM) operator and the intuitionistic fuzzy geometric weighed Heronian mean (IFGWHM) operator.Wei and Merigó [33] developed some probability intuitionistic fuzzy aggregation operators.All the above operators are based on the algebraic operational laws of IFSs for carrying the combination process and are not consistent with the limiting case of ordinary fuzzy sets [34].Recently, Wang and Liu [35,36] developed some intuitionistic fuzzy aggregation operators based on Einstein operations.
However, most of the existing aggregation operators do not take into account the relationship between the values being fused.Xu [37] proposed some dependent OWA operators, in which the associated weights depend on the aggregated arguments.The prominent characteristic of this dependent OWA operator is that it can relieve the influence of unfair arguments on the aggregated results.Furthermore, Xu [38] developed some dependent uncertain ordered weighed aggregation operators, including dependent uncertain ordered weighed averaging (DUOWA) operators and dependent uncertain ordered weighed geometric (DUOWG) operators, in which the associated weights only depend on the aggregated interval arguments.Wei and Zhao [39] developed a dependent uncertain linguistic ordered weighed geometric (DULOWG) operator to aggregate uncertain linguistic variable.Liu [40] developed the intuitionistic linguistic generalized dependent ordered weighed average (ILGDOWA) operator and the intuitionistic linguistic generalized dependent hybrid weighed aggregation (ILGDHWA) operator.
Nowadays, the problem of supplier selection has emerged as an active research field where numerous research papers have been published around this area within the last few years.Supplier selection plays a key role in supply chain management (SCM) and deals with evaluation, ranking, and selection of the best option from a pool of potential suppliers especially in the presence of conflicting attribute.In the literature, supplier selection has been treated as a multiple attribute decision making (MADM) and a wide range of mathematical methods have been undertaken to provide the problems with sufficient and more accurate solutions.In this paper, motivated by the idea of dependent aggregation operator proposed by Xu [37,38], we develop some new intuitionistic fuzzy aggregation operators, including dependent intuitionistic fuzzy ordered weighed averaging (DIFOWA) operator and dependent intuitionistic fuzzy hybrid weighed aggregation (DIFHWA) operator.Furthermore, we study some of their main desirable properties.We also apply the developed operators to multiple attribute decision making (MADM) problems concerning the supplier selection with intuitionistic fuzzy information.

Preliminaries
In this section, we introduce some basic concepts related to intuitionistic fuzzy sets.Atanassov [2][3][4] introduced a generalized fuzzy set called intuitionistic fuzzy set, shown as follows.
An IFS in  is given by which is characterized by a membership function   :  → [0, 1] and a nonmembership function where the numbers   () and V  () represent, respectively, the degree of membership and the degree of nonmembership of the element  to the set .
For each IFS  in , if then   () is called the indeterminacy degree or hesitation degree of  to .
Let  = (  , V  ) be an IFV; Chen and Tan [41] introduced a score function , which can be represented as follows: where For an IFV  = (  , V  ), it is clear that if the deviation between   and V  gets greater, which means the value   gets bigger and the value V  gets smaller, then the IFV  gets greater.
Later, Hong and Choi [42] noted that the score function alone cannot differentiate many IFVs even though they are obviously different.To make the comparison method more discriminatory, an accuracy function  to evaluate the degree of accuracy of the intuitionistic fuzzy value can be represented as follows: where () ∈ [0, 1].The larger the value of (), the higher the degree of accuracy of the degree of membership of the IFV .
The score function  and the accuracy function  are, respectively, defined as the difference and the sum of the membership function   () and the nonmembership function V  ().
To rank IFVs, Xu and Yager [15] and Xu [16] developed a method for the comparison between two IFVs, which is based on the score function  and the accuracy function : To aggregate intuitionistic preference information, Xu [16] defined the following operations.
The OWA operator introduced by Yager [43] is an aggregation operator that provides a parameterized family of aggregation operators between the maximum and the minimum.Since its introduction, lots of extensions of the OWA operator have been studied, such as the weighed OWA (WOWA) [44], the hybrid averaging (HA) operator [45], the ordered weighed averaging weighed averaging (OWAWA) operator [46], and the immediate weighed OWA distance (IWOWAD) operator [47].It can be defined as follows.
Definition 3 (see [43]).An OWA operator of dimension  is a mapping OWA:   →  that has an associated weighing  with   ∈ [0, 1] and ∑  =1   = 1, such that where   is the th largest of the   .
The OWA operator has been used in a wide range of applications, such as engineering, neural networks, data mining, decision making, image process, and expert systems.Consider that the OWA operator aggregates only the exact inputs having been reordered; Xu [16] extended the OWA operator to accommodate the situations where the input arguments are intuitionistic fuzzy numbers and developed the intuitionistic fuzzy ordered weighed averaging (IFOWA) operator.

Some Dependent Intuitionistic Fuzzy Aggregation Operators
As an interesting and important research topic in IFS theory, similarity measure between intuitionistic fuzzy set (IFS) has been receiving more and more attention in recent years.
Recently, motivated by the idea of the TOPSIS of Hwang and Yoon [48], Xu and Yager [49] introduced an intuitionistic fuzzy similarity measure combining the distance measure as follows.
By (11), we have then we can replace (13) by We call (15) a dependent intuitionistic fuzzy ordered weighed averaging (DIFOWA) operator, which is a generalization of the dependent ordered weighed averaging (DOWA) operator [37].Consider that the aggregated value of the DIFOWA operator is independent of the ordering; thus it is also a neat operator.From (15) we know that all the associated weights of the DIFOWA operator only depend on the aggregated IFVs and can relieve the influence of unfair arguments on the aggregated results by assigning low weights to those "false" and "biased" ones and thus make the aggregated results more reasonable in the practical applications.
Similar to the DOWA operator, the DIFOWA operator has the following properties.
Theorem 10 (boundedness).The IFDOWA operator lies between the max and min operators; that is, Proof.Let Since  ≤   ≤ , we have That is, The IFWA operator only considers the weight of the aggregated IFVs, and in IFDOWA operator, we assumed that all of the IFVs being aggregated were of equal importance.However, in many cases, the importance degrees should not be treated as equally important and thus need to be assigned different weights.Here, we will consider the effect on the dependent operations of having differing importance of the objects.So, in what follows, we will develop a new aggregation operator to process this case.
This completes the proof of Theorem 12.

An Approach to Multiple Attribute Decision Making Based on the DIFHWA Operator
For the multiple attribute decision making problems, in which both the attribute weights and the expert weights take the form of real numbers, and the attribute preference values take the form of IFVs, we will develop an approach based on the IFWA and DIFHWA operators to multiple attribute group decision making based on intuitionistic fuzzy information processing.
Step 1. Utilize the decision information given in matrix  ()  and the IFWA operator to derive the individual overall preference value  ()  of the alternative   .
According to the ranking of score function (  ), the ranking is Thus the best alternative is  4 .

Conclusion
In this paper, we have investigated the multiple attribute decision making (MADM) problems in which both the attribute weights and the expert weights take the form of real numbers and attribute values take the form of intuitionistic fuzzy information.Motivated by the ideal of dependent aggregation operator, we develop two dependent intuitionistic fuzzy aggregation operators: the dependent intuitionistic fuzzy ordered weighed averaging (DIFOWA) operator and the dependent intuitionistic fuzzy hybrid weighed aggregation (DIFHWA) operator, in which the associated weights only depend on the aggregated intuitionistic fuzzy numbers.Furthermore, some desirable properties of the DIFOWA operator, such as commutativity and idempotency, are studied.Based on the DIFHWA operator, an approach to multiple attribute group decision making with intuitionistic fuzzy information is proposed.Because the associated weights only depend on the aggregated input arguments, the method can relieve the influence of unfair input arguments on the aggregated results by assigning low weights to those "false" and "biased" ones.Finally, an illustrative example concerning the supplier selection is given to verify the developed approach.
In the future, we will continue working in the extension and application of the developed operators to other domains.