Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 9364987, 11 pages

https://doi.org/10.1155/2018/9364987

## Interval-Valued Intuitionistic Fuzzy Einstein Geometric Choquet Integral Operator and Its Application to Multiattribute Group Decision-Making

Logistics and E-Commerce College, Zhejiang Wanli University, Ningbo 315100, China

Correspondence should be addressed to Qifeng Wang; moc.361@fqwyhl

Received 20 July 2017; Revised 4 November 2017; Accepted 7 November 2017; Published 14 January 2018

Academic Editor: Peide Liu

Copyright © 2018 Qifeng Wang and Haining Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With respect to the multiattribute decision-making (MADM) problem in which the attributes have interdependent or interactive phenomena under the interval-valued intuitionistic fuzzy environment, we propose a group decision-making approach based on the interval-valued intuitionistic fuzzy Einstein geometric Choquet integral operator (IVIFEGC). Firstly, the Einstein operational laws and some basic principle on interval-valued intuitionistic fuzzy sets are introduced. Then, the IVIFEGC is developed and some desirable properties of the operator are studied. Further, an approach to multiattribute group decision-making with interval-valued intuitionistic fuzzy information is developed, where the attributes have interdependent phenomena. Finally, an illustrative example is used to illustrate the developed approach.

#### 1. Introduction

The intuitionistic fuzzy set (IFS) [1] is the generalization of fuzzy set theory proposed by Zadeh [2]. The IFS includes the membership degree, nonmembership degree, and hesitancy degree which is more flexible in dealing with the imprecise and vague information. Presently, the IFS has been widely used in many fields, such as machine learning, decision-making, and pattern recognition. Meanwhile, many experts study the IFS theory and have many achievements. Yu et al. [3–5] make some reviews on the development of the IFS. In many real decision-making problems, the decision-makers often have difficulties in determining the membership degree and nonmembership degree with the crisp numbers. Atanassov and Gargov [6] generalized the concept of the IFS to propose the interval-valued intuitionistic fuzzy set (IVIFS). The membership degree, nonmembership degree, and hesitancy degree of the IVIFS take the form of the interval values, which have more advantages in expressing the decision-makers’ preference information.

For multiattribute decision-making problems with IVIF information, an important topic is the aggregation of IVIF information. Based on the algebraic operational laws on the IVIFS, Yu [7] defined the generalized interval-valued intuitionistic fuzzy weighted averaging operator and the generalized interval-valued intuitionistic fuzzy weighted geometric operator. Zhao et al. [8] proposed the generalized interval-valued intuitionistic fuzzy ordered weighted averaging operator and the generalized interval-valued intuitionistic fuzzy hybrid averaging operator. Lin and Zhang [9] proposed some interval-valued intuitionistic fuzzy continuous operators. Wei [10] developed the induced interval-valued intuitionistic fuzzy weighted geometric operator and the induced interval-valued intuitionistic fuzzy ordered weighted geometric operators. Yu et al. [11] developed the interval-valued intuitionistic fuzzy prioritized weighted averaging operator and the interval-valued intuitionistic fuzzy prioritized weighted geometric operator. Wu and Su [12] proposed the interval-valued prioritized hybrid weighted operator. Zhou and He [13] proposed the interval-valued intuitionistic fuzzy precise weighted operator. Zhao and Xu [14] presented some new synthesized interval-valued intuitionistic fuzzy aggregation operators. Xu and Gou [15] made an overview of interval-valued intuitionistic fuzzy aggregation operator. The operators above are based on the algebraic operational laws (i.e., the algebraic product and the algebraic sum) of the IVIFS. However, the algebraic product and the algebraic sum are not the only operational laws. The Einstein operational laws are good alternatives for information aggregation [16]. Wang and Liu [17] extended the Einstein operational laws to accommodate the environment where the aggregated information is interval-valued intuitionistic fuzzy number and proposed the interval-valued intuitionistic fuzzy Einstein weighted averaging operator, the interval-valued intuitionistic fuzzy Einstein ordered weighted averaging operator, and the interval-valued intuitionistic fuzzy Einstein hybrid weighted operator. Wang and Liu [18] proposed the interval-valued intuitionistic fuzzy Einstein weighted geometric operator, the interval-valued intuitionistic fuzzy Einstein ordered weighted geometric operator, and the interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric operator. Yang and Yuan [19] presented the induced interval-valued intuitionistic fuzzy Einstein ordered weighted geometric operator. Cai and Han [20] developed the induced interval-valued Einstein ordered weighted averaging operator. The operators proposed above are based on the premise that all of the attributes are independent. However, in many real decision-making problems, the attributes are correlative or interdependent. For example, Grabisch [21] and Torra [22] gave the following classical example: ‘‘We are to evaluate a set of students in relation to three subjects: , we want to give more importance to science-related subjects than to literature, but on the other hand we want to give some advantage to students that are good both in literature and in any of the science-related subjects.” In order to solve the problem where the attributes are correlative or interactive, Choquet [23] developed the Choquet integral. Based on the Choquet integral and interval-valued intuitionistic fuzzy set, Xu [24] proposed the interval-valued intuitionistic fuzzy correlative averaging operator and the interval-valued intuitionistic fuzzy correlative geometric operator. Tan [25, 26] proposed the interval-valued intuitionistic fuzzy geometric Choquet integral operator and the interval-valued intuitionistic fuzzy Choquet integral operator. Xu and Xia [27] presented the induced generalized interval-valued intuitionistic fuzzy correlative averaging operator. Meng et al. [28] developed the generalized Banzhaf interval-valued intuitionistic fuzzy geometric Choquet integral operator. Cheng and Tang [29] proposed the interval-valued intuitionistic fuzzy generalized Shapley geometric Choquet integral operator. Gu et al. [30] proposed the interval-valued intuitionistic fuzzy Einstein correlative averaging operator. Liu et al. [31–41] proposed some Heronian mean aggregation operators and Bonferroni mean operators to solve decision-making problems. Liu et al. [42] developed the IVIFOWCS operator which combines the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized ordered weighted averaging operator and applied it in the investment decision-making. Krishankumar et al. [43] developed a scientific decision framework with interval-valued intuitionistic fuzzy operator and applied it in supplier selection.

In the above studies, most interval-valued intuitionistic fuzzy Einstein operators are still used to solve the problems where the attributes are independent. The interval-valued intuitionistic fuzzy Einstein operators are seldom applied to deal with the problems where the attributes are correlative. In this paper, we developed the interval-valued intuitionistic fuzzy Einstein geometric Choquet integral operator and applied it in multiattribute group decision-making problems with interval-valued intuitionistic fuzzy information, where the attributes have interdependent phenomena. To do that, this paper is organized as follows. In Section 2, the basic knowledge of the IFS and IVIFS, Einstein operational laws of the IVIFS, and fuzzy measure are introduced. Section 3 proposes the interval-valued intuitionistic fuzzy Einstein geometric Choquet integral operator and investigates the fundamental properties of the operator. Section 4 develops a multiattribute group decision-making method based on the operator. An illustrative example is used to verify the effectiveness of the method in Section 5. The comparison with other methods is discussed in Section 6. Section 7 concludes the paper.

#### 2. Preliminaries

In this section, some basic knowledge is introduced including the definitions of the intuitionistic fuzzy set (IFS), the interval-valued intuitionistic fuzzy set (IVIFS), the Einstein operational laws of IVIFS, and fuzzy measure.

*Definition 1 (see [1]). *Let be a universal set; an IFS in is given bywhere represents the membership degree, represents the nonmembership degree, and is called the hesitancy degree.

Atanassov and Gargov [6] extended the IFS to propose the IVIFS; the definition of the IVIFS is given as follows.

*Definition 2 (see [6]). *Let be a universal set; an IVIFS in is given bywhere the intervals are, respectively, called the degree of membership and degree of nonmembership. , is called the degree of hesitancy.

Presently, most proposed operators are based on the algebraic product and the algebraic sum of the IVIFS, while the algebraic product and sum are not the unique operational laws chosen to model the intersection and union on the IVIFS. Einstein product and sum are also used in interval-valued intuitionistic fuzzy information aggregation [17].

Let and be two IVIFNs; some operations of Einstein product and sum on the IVIFS are given as follows:

To compare two IVIFNs, Xu [24] defined the score function and accuracy function ; then(1)if , then ;(2)if , then①if , then ;②if , then ;③if , then .

Wang and Liu [18] proposed the interval-valued intuitionistic fuzzy Einstein weighted geometric operator based on the Einstein operational laws of the IVIFS.

*Definition 3 (see [18]). *Let , be the IVIFS. An operator of dimension is a mapping : which has a weighting vector with and , according to the following formula:

In real decision-making problems, the attributes of the alternatives are often interdependent or interactive. In order to solve these problems, Sugeno [44] introduced the concept of the fuzzy measure to model interaction phenomenon among combinations.

*Definition 4 (see [44]). *Let be a universe of discourse and let be the power set of . A fuzzy measure on is a set function , satisfying the following conditions:(1).(2).Since the fuzzy measure is defined on the power set, it is not easy to get the fuzzy measure of each combination in a set when it is large. To increase the practicability of the fuzzy measure, Sugeno [44] introduced a special kind of fuzzy measure named the -fuzzy measure , which is expressed by the following form:where and , with .

It is apparent that when , then is an additive measure, which means that there is no interaction between subsets and . If , then , which means that is a superadditive measure and there exists complementary interaction between subsets and . If , then , which implies that is a subadditive measure and there exists redundancy interaction between subsets and .

For finite set , the -fuzzy measure can be equivalently expressed byFrom , we know that is determined by . When each is given, the value of can be derived from (6). For the set with elements, we only need values to get the fuzzy measure of each subset in . Furthermore, if , then .

*Definition 5 (see [20]). *Let be a universe of discourse. Let be a positive real-valued function on , and let be the fuzzy measure on . The discrete Choquet integral of with respect to is defined bywhere is a permutation of , satisfying . And ; .

#### 3. Interval-Valued Intuitionistic Fuzzy Einstein Geometric Choquet Integral Operator

In many decision-making problems, the criteria of alternatives often are correlative or interdependent. To solve these problems, the interval-valued intuitionistic fuzzy Einstein geometric Choquet integral operator is proposed and the properties of the operator are investigated as follows.

*Definition 6. *Let , be an IVIFS on ; let be the fuzzy measure on . An interval-valued intuitionistic fuzzy Einstein geometric Choquet integral operator of dimension is a mapping : such thatwhere is a permutation of , satisfying and and .

The IVIFEGC operator can also be represented by the following equation:where .

Next the properties of the operator are further investigated.

Theorem 7 (idempotency). *Let , be an IVIFS on , and let be a fuzzy measure on . If all IVIFNs are equal (i.e., ), then*

*Proof. *According to Theorem 7, if all IVIFNs are equal, thenSincethen

Theorem 8 (monotonicity). *Let , be two IVIFSs on , let be a fuzzy measure on , and let be a permutation on X such that , ; if , then*

*Proof. *Since , then .

For any , and ; thenthat is,Also,For any , ; thenthat is,Also,According to the score function and accuracy function, we have

Theorem 9 (boundedness). *Let , be an IVIFS on , and let be a fuzzy measure on ; if then*

*Proof. *For all , and are IVIFNs. Since , then .

Since thenSinceand according to Theorem 7, we have; that is,

Theorem 10 (commutativity). *Let , be an IVIFS on , and let be a fuzzy measure on ; is any permutation of ; then*

*Proof. *According to the definition of the IVIFEGC operator, Theorem 10 can be proven.

#### 4. The Multiattribute Group Decision-Making Approach Based on Interval-Valued Intuitionistic Fuzzy Einstein Geometric Choquet Integral Operator

Considering the multiattribute group decision-making problem under interval-valued intuitionistic fuzzy environment, let be the set of experts, let be a set of alternatives, and let be a set of attributes. The decision-making matrix is given by expert , where is the evaluation value given by expert on the alternatives with respect to the attribute .

*Step 1. *Determine the fuzzy measure of the individual expert and subset of the experts.

By employing the experience of the experts, the fuzzy measure of the individual expert can be confirmed (i.e., ). According to (6), the parameter and the fuzzy measure of the subset of the experts can be determined.

*Step 2. *Utilize the IVIFEGC operator to aggregate the decision-making matrix , , into the comprehensive decision-making matrix .

*Step 3. *Determine the fuzzy measure of the criterion and the subset of criteria.

The experts confirm the fuzzy measure of the criterion empirically (i.e., ). By utilizing (6), the parameter and the fuzzy measure of the subset of the criteria can be determined.

*Step 4. *Utilize the IVIFEGC operator to obtain the overall preference of the alternative .

*Step 5. *Rank the alternatives according to the score of the alternatives, the greater the score of , the better the alternative .

#### 5. An Application Example

With the increasingly fierce competition in the market, many traditional manufacturers usually outsource their noncore businesses and focus on core business, such as production design, marketing, and after-sale service. AUX is a famous air conditioner manufacturer in China; it wants to outsource the logistics to reduce the logistics cost and improve the satisfaction of the customers. After the first round screening, four alternatives are selected, which are .The evaluation criteria include service quality (), response ability (), flexibility (), technology (), and cost (). Obviously, these evaluation criteria are mutually interactive. For example, there exists tradeoff between service quality and cost. Response ability, flexibility, and technology still have some effects on the service quality. The manufacturer invites five experts to evaluate and select the appropriate logistics service provider. The alternatives are to be evaluated using the interval-valued intuitionistic fuzzy number by experts, as listed in Table 1.