#### Abstract

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.

#### 1. Introduction

As extension of Zadeh’s fuzzy set [1] whose basic component is only a membership function, Atanassov [2–4] 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 [10–14]. 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.

#### 2. Preliminaries

In this section, we introduce some basic concepts related to intuitionistic fuzzy sets. Atanassov [2–4] 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 and a nonmembership function , with the condition where the numbers and 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 .

For convenience, Xu and Yager [15] called an intuitionistic fuzzy value (IFV), where , , and . For convenience, let be the set of all IFVs.

Let be an IFV; Chen and Tan [41] introduced a score function , which can be represented as follows: where .

For an IFV , it is clear that if the deviation between and gets greater, which means the value gets bigger and the value 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 . 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 .

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 :(i) if , then ;(ii) if , the(1) if , then ;(2) if , then .

To aggregate intuitionistic preference information, Xu [16] defined the following operations.

*Definition 1 (see [16]). *Let and two IFVs; then(1);(2);(3), ;(4), .

*Definition 2 (see [16]). *Let be a collection of IFVs and an intuitionistic fuzzy weighed averaging operator of dimension is a mapping IFWA: , if
where is the weighing vector of such that , and .

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 and , 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.

*Definition 4 (see [16]). *Let be a collection of IFVs and an intuitionistic fuzzy ordered weighed averaging operator of dimension is a mapping IFOWA: , if
where is a permutation of such that for all ; is the weighing vector of such that , , and .

#### 3. 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.

*Definition 5 (see [44]). *Let and be two IFVs and the complement of ; then
is called the similarity measure between and , where
is the Hamming distance between and .

*Definition 6. *Let be a collection of IFVs; the intuitionistic fuzzy arithmetic mean is computed as

In real-life decision making problems, the decision making experts may have personal biases and some individuals may give unduly high or unduly low preference values to their preferred or repugnant objects. In such a case, we will assign very low weights to these false or biased opinions that is to say, the closer a preference value (argument) is to the mid one(s), the more the weight is. As a result, based on (8) and (10), we define the IFOWA weights as Obviously, and . In particular, if for , then by (12), we have .

Theorem 7. *Let be a set of IFV and the arithmetic mean of these IFV is a permutation of , such that for all . If , then .*

By (11), we have Since 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 8 (commutativity). *Let be any permutation of ; then
*

*Proof. *Let
Since is any permutation of , we have
Thus

Theorem 9 (idempotency). *Let ; then
*

*Proof. *Since for all , we have
This completes the proof of Theorem 9.

Theorem 10 (boundedness). *The IFDOWA operator lies between the max and min operators; that is,
*

*Proof. *Let
Since , we have
That is,
thus

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.

*Definition 11. *Let be a collection of the IFV and DIFHWA: . If
where , is a permutation of , such that for all and is the weight vector of with and ; then DIFHWA is called the dependent intuitionistic fuzzy hybrid weighed aggregation operator. In particular, if , then DIFHWA is reduced to the DIFOWA operator.

Since
So we can replace (27) by

From (29), we can see that the DIFHWA operator can not only consider the object weight but also relieve the influence of unfair arguments on the aggregated results by assigning low weights to those “false” and “biased” ones.

Theorem 12. *The DIFOWA operator is a special case of the DIHHWA operator.*

*Proof. *Let ; then for all , and we have
This completes the proof of Theorem 12.

#### 4. 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.

Let be a discrete set of alternatives, let be the set of attributes, let be the weighing vector of the attribute, where , , be the set of decision makers, and, be the expert weight, with and . Suppose that is the decision matrix, where takes the form of the IFV, given by the decision maker , for alternative with respect to the attribute . The methods involve the following steps.

*Step 1. *Utilize the decision information given in matrix and the IFWA operator
to derive the individual overall preference value of the alternative .

*Step 2. *Utilize (9)–(11) to calculate the degree of similarity :
where is mean of the .

*Step 3. *Utilize the DIFHWA operator:
to derive the collective overall preference values of the alternative .

*Step 4. *Rank all the alternatives and select the best one(s) in accordance with the collective overall preference values .

*Step 5. *End.

#### 5. Illustrative Example

In this section, we discuss a problem concerning a manufacturing company, searching the best global supplier for one of its most critical parts used in assembling process (adapted from Chan and Kumar [50]). The attributes which are considered here in selection of five potential global suppliers are (1) : overall cost of the product; (2) : quality of the product; (3) : service performance of supplier; (4) : supplier’s profile; and (5) : risk factor. The five alternatives are to be evaluated using IFVs by four decision makers (whose weighing vector ) under the above five attributes (whose weighing vector ), and construct, respectively, the intuitionistic fuzzy decision matrices as listed in Tables 1, 2, 3, and 4.

To get the best alternative(s), the following steps are involved.(1) Calculate the comprehensive evaluation values : (2) Calculate the degree of similarity : (3) Calculate the comprehensive evaluation value of each alternative: (4) Calculate the score function and rank . Since Then (5) Rank all the alternatives.

According to the ranking of score function , the ranking is Thus the best alternative is .

#### 6. 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.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This paper is supported by the Statistical Scientific Key Research Project of China (no. 2013363), the National Funds of Social Science of China (nos. 12ATJ001, 12&ZD211), the Key Research Center of Philosophy and Social Science of Zhejiang Province—Modern Port Service Industry and Creative Culture Research Center, Zhejiang Provincial Key Research Base for Humanities and Social Science Research (Statistics), Projects in Science and Technique of Ningbo Municipal (no. 2012B82003), Ningbo Soft Science Fund (no. 2011A1012), Ningbo Natural Science Foundation (no. 2011A610106), Zhejiang new generation mobile Internet client innovation team (no. 2012R10009-07), Projects in Association with Social Sciences Research of Zhejiang (no. 2013B069), Soft Science Key Project in Zhejiang (no. 2012C25099), Ningbo Natural Science Foundation (no. 2013A610286), the MOE Project of Key Research Institute of Humanities and Social Sciences in Universities (no. 13JJD910002), and the key research project of Zhejiang Wanli University.