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Complexity
Volume 2018, Article ID 3941847, 19 pages
https://doi.org/10.1155/2018/3941847
Research Article

An Approach to Interval-Valued Hesitant Fuzzy Multiattribute Group Decision Making Based on the Generalized Shapley-Choquet Integral

1School of Economics and Trade, Hunan University, Hunan, Changsha 410079, China
2Business School, Central South University, Hunan, Changsha 410083, China
3School of Management and Economics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Correspondence should be addressed to Fanyong Meng; moc.361@eijtgnoynafgnem

Received 21 December 2017; Accepted 2 May 2018; Published 10 June 2018

Academic Editor: Danilo Comminiello

Copyright © 2018 Lifei Zhang and Fanyong Meng. 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

The purpose of this paper is to develop an approach to multiattribute group decision making under interval-valued hesitant fuzzy environment. To do this, this paper defines some new operations on interval-valued hesitant fuzzy elements, which eliminate the disadvantages of the existing operations. Considering the fact that elements in a set may be interdependent, two generalized interval-valued hesitant fuzzy operators based on the generalized Shapley function and the Choquet integral are defined. Then, some models for calculating the optimal fuzzy measures on the expert set and the ordered position set are established. Because fuzzy measures are defined on the power set, it makes the problem exponentially complex. To simplify the complexity of solving a fuzzy measure, models for the optimal 2-additive measures are constructed. Finally, an investment problem is offered to show the practicality and efficiency of the new method.

1. Introduction

The socioeconomic environment becomes more and more complex; it is impractical to require an expert to give his/her exact attribute values of every alternative. Based on fuzzy set theory [1], decision making under fuzzy environment is rapidly developed [26]. Since Zadeh [1] first introduced fuzzy sets, many extending forms are developed such as interval-valued fuzzy sets [7], type-2 fuzzy sets [8], interval type-2 fuzzy sets [9], and fuzzy multiset [10]. With the development of fuzzy set theory, the corresponding fuzzy decision-making theory is developed such as interval-valued fuzzy decision making [11, 12], type-2 fuzzy decision making [13, 14], interval type-2 fuzzy decision making [15, 16], and fuzzy multiset decision making [17].

Although there are several families of fuzzy sets, all of the above-mentioned fuzzy sets only consider the membership information. As Atanassov [18] noted, in some situations, it is insufficient to only know the membership degree for a certain fuzzy concept. Thus, Atanassov [18] introduced the concept of intuitionistic fuzzy sets (IFSs), which are characterized by a membership degree, a nonmembership degree, and a hesitancy degree. Since then, many intuitionistic fuzzy decision-making methods are proposed [1921]. To further extend the application of IFSs, Atanassov and Gargov [22] introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs), which are characterized by an interval membership function and an interval nonmembership function rather than real numbers. Such a generalization is further facilitated effectively to represent inherent imprecision and uncertainty in the human decision-making analysis. Many theories and methods on IVIFSs have been put forward and used to solve decision-making problems [2327].

Recently, Torra and Narukawa [28] noted when an expert makes a decision, there may be several possible values for one thing. To deal with this situation, Torra [29] introduced the concept of hesitant fuzzy sets (HFSs) that permit the membership to have a set of possible values. Later, Xia and Xu [30] defined some operational laws on HFSs and presented some aggregation operators for hesitant fuzzy elements. Furthermore, Xia et al. [31] defined a series of hesitant fuzzy aggregation operators with the aid of quasi-arithmetic means and developed an approach to hesitant fuzzy multiple attribute decision making. Motivated by the ideal of prioritized aggregation operators, Wei [32] developed the hesitant fuzzy prioritized weighted average (HFPWA) operator and the hesitant fuzzy prioritized weighted geometric (HFPWG) operator, whilst Zhu et al. [33] introduced the weighted hesitant fuzzy geometric Bonferroni mean (WHFGBM) operator. More researches can be seen in the literature [3437]. Just as interval type-2 fuzzy sets and IVIFSs, in some situations, it is still difficult to require an expert to give the exact possible values for one thing. Very recently, Chen et al. [38] introduced the concept of interval-valued hesitant fuzzy sets (IVHFSs) and defined some aggregation operators. Farhadinia [39] investigated the relationship between the entropy, the similarity measure, and the distance measure for HFSs and IVHFSs. Wei and Zhao [40] presented several induced hesitant interval-valued fuzzy Einstein aggregation operators and applied them to multiattribute decision making. Meanwhile, Wei et al. [41] defined two hesitant interval-valued fuzzy Choquet operators and studied their application in interval-valued hesitant multiattribute decision making. Meng and Chen [42] introduced two induced generalized interval-valued hesitant fuzzy hybrid Shapley operators that globally consider the interactions between the weights of elements in a set. It is noteworthy that all these aggregation operators are based on the operational laws presented by Chen et al. [38]. These operations cannot preserve the order relationship under multiplication by a scalar. It means that monotonicity is not always true. Thus, when these operators are used in decision making, it cannot guarantee to obtain the best choice. Furthermore, Meng et al. [43] researched the correlation coefficients of IVHFSs that need not consider the lengths of interval-valued hesitant fuzzy elements (IVHFEs). However, the correlation coefficients only consider the weights of attributes and disregard that of orders.

To address the above-mentioned issues for decision making with IVHFSs, this paper continues to study group decision making under interval-valued hesitant fuzzy environment. First, some new operations that eliminate the existing issues are defined. To deal with the situation where the elements in a set are correlative, two generalized interval-valued hesitant fuzzy dependent operators are defined, which can be seen as an extension of some hesitant fuzzy operators. Then, a distance measure on IVHFSs is defined, which does not consider the length of IVHFEs and the arrangement of their possible interval membership degrees. Based on the Shapley function and the defined distance measure, models for the optimal fuzzy measures and the optimal 2-additive measures are constructed, respectively. Finally, approach to interval-valued hesitant fuzzy multiattribute group decision making is developed. Comparing the existing methods, the new approach includes the following four features: (i) it uses the new defined operations that avoid the nonmonotonic problem; (ii) it applies the aggregation operator based on fuzzy measures that can address the interactive situations; (iii) when the weighting vector is partly known, models for the optimal fuzzy measure and the optimal 2-additive measure are built; (iv) because the experts’ knowledge, skills, and experiences are different, the new method gives the experts’ weights with respect to each attribute.

The paper is organized as follows: In Section 2, some basic concepts related to IVHFSs are reviewed, and some new operations on IVHFSs are defined. In Section 3, some generalized interval-valued hesitant fuzzy Choquet operators are defined, and some special cases are examined. Meanwhile, to simplify the complexity of solving a fuzzy measure, a generalized interval-valued hesitant fuzzy operator based on 2-additive measures is introduced. In Section 4, a new distance measure is defined, and then models for the optimal fuzzy measure and the optimal 2-additive measure on the associated set are built, respectively. After that, an approach to multiattribute group decision making under interval-valued hesitant fuzzy environment is developed. In Section 5, an illustrative example is provided to show the concrete application of the proposed procedure. Conclusions are made in the last section.

2. Some Basic Concepts

To address the situation where the membership degree of an element has several possible interval values, Chen et al. [38] presented the concept of interval-valued hesitant fuzzy sets (IVHFSs), which is an extension of hesitant fuzzy sets (HFSs) [29].

Definition 1 (see [38]). Let be a finite set, and IVHFS in is in terms of a function that when applied to returns a subset of , denoted bywhere is a finite set of all possible interval-valued membership degrees of the element to the set with being the set of all closed subintervals in . For convenience, Chen et al. [38] called an interval-valued hesitant fuzzy element (IVHFE) and is the set of all IVHFEs.

If all possible interval-valued membership degrees of each element degenerate to real numbers, it derives an HFS [29].

Similar to the operational laws on HFEs [30], Chen et al. [38] defined the following operations on IVHFEs. Let , , and be any three IVHFEs in , then(i);(ii);(iii).;(iv).

Let and be any two intervals; their order relationship is given using the possible degree formula as follows [42]: where and .

If , then ; if , then ; if , then .

Based on this possible degree formula on intervals, Chen et al. [38] introduced the following order relationship on IVHFEs.

Definition 2 (see [38]). For an IVHFE , is called the score function of with being the number of interval-valued membership degrees in , and is an interval value in . For any two IVHFEs and , if , then ; if , then .

However, the operations given by Chen et al. [38] have some undesirable properties. For example, and are not always true. See Example 3.

Example 3. Let , , and ; it derives It means .
Furthermore, take and ; it getsIt means .

In addition, as Beliakov et al. [19] noted for intuitionistic fuzzy sets, the operations given by Chen et al. [38] cannot preserve the order relationship under multiplication by a scalar: does not necessarily imply , where is a scalar. See Example 4.

Example 4. Take , , and . Because , . However, , , and , so . Thus, , does not imply .

To avoid these disadvantages, we adopt the following operations on IVHFEs. Let , , and be any three IVHFEs in ,(I);(II);(III);(IV) with being an IVHFE, namely, for all and .

It is easy to verify that the new defined operations can eliminate the issues listed above. Without special explanation, this paper adopts the operations on IVHFEs defined by (I)–(IV).

In some cases, the possible degree formula (2) fails to distinguish two distinct IVHFEs. For example, let and , then their scores are respective of and . From (2), it gets and . However, they are obviously different. To increase the identification of IVHFEs, we here adopt the following ranking method.

Let and be any two intervals; if or and , then ; otherwise, .

3. Several Generalized Interval-Valued Hesitant Fuzzy Dependent Aggregation Operators

Let us consider the following example: “We are to evaluate three companies according to three attributes: benefits, environment benefits, social , we want to give more importance to environment benefits than to economic benefits or social benefits, but on the other hand we want to give some advantage to companies that are good in environment benefits and in any of economic benefits and social benefits”. In this situation, the aggregation operator based on additive measures seems to be insufficient.

To address the situation where the elements in a set are correlative, many aggregation operators based on the Choquet integral [44] are defined [4552]. Using the Shapley function [53], Zhang et al. [54] defined the intuitionistic fuzzy Shapley weighted operator, Meng et al. [55] introduced some uncertain generalized Shapley aggregation operators, and Meng et al. [56] defined two linguistic hesitant fuzzy hybrid Shapley aggregation operators. More researches about decision making based on the Shapley function can be seen in the literature [5760].

To obtain the comprehensive attribute values and reflect the interactions between attributes as well as the ordered positions, this section introduces several interval-valued hesitant fuzzy operators based on the Choquet integral and the generalized Shapley function. First, let us review the following definitions.

Definition 5 (see [61]). A fuzzy measure on finite set is a set function satisfying (i), ,(ii)If and , then , where is the power set of .

From the definition of fuzzy measures, we know that the fuzzy measure does not only give the importance of every element but also consider the importance of all their combinations. Corresponding to fuzzy measures, fuzzy integrals are important tools to aggregate information with interactive characteristics. The Choquet integral is one of the most important fuzzy integrals, which can be seen as an extension the ordered weighted averaging (OWA) operator. Grabisch [62] gave the following expression of the Choquet integral on discrete sets.

Definition 6 (see [62]). Let be a positive real-valued function on and be a fuzzy measure on . The discrete Choquet integral of for is defined as where indicates a permutation on such that , and with .

Remark 7. From Definition 6, one can see that the fuzzy measure only relates to the positions. It does not consider which element in the position.

From Definition 6, we know that the Choquet integral only considers the correlations between the ordered subsets and (). If there are interdependences, it seems to be insufficient. To globally reflect the interactions between the ordered subsets, the generalized Shapley function [63] seems to be a good choice, denoted as where is a fuzzy measure on , and , , and denote the cardinalities of the coalitions , , and , respectively.

Form (6), we know that it is an expect value of the overall marginal contributions between the coalition and any coalition in . When , it degenerates to the famous Shapley function [53]:From (7), we know that when the elements in are uncorrelated, then their Shapley values equal to their own importance, namely, for all .

Definition 8. Let be a positive real-valued function on , and be a fuzzy measure on . The discrete generalized Shapley-Choquet integral of for is defined aswhere indicates a permutation on such that , is the generalized Shapley on , and with .

From Definition 8, one can see that the generalized Shapley-Choquet integral overall considers the interactions between any two adjacent coalitions. Now, let us introduce the generalized interval-valued hesitant fuzzy Shapley-Choquet weighted averaging (G-IVHFSCWA) operator as follows.

Definition 9. Let () be a collection of IVHFEs in and be a fuzzy measure on the ordered set . The generalized interval-valued hesitant fuzzy Shapley-Choquet weighted averaging (G-IVHFSCWA) operator is defined aswhere , indicates a permutation on such that and is the generalized Shapley value of with .

Remark 10. If , then the G-IVHFSCWA operator degenerates to the interval-valued hesitant fuzzy Shapley-Choquet weighted averaging (IVHFSCWA) operator

Remark 11. If , then the G-IVHFSCWA operator degenerates to the interval-valued hesitant fuzzy Shapley-Choquet quadratic weighted averaging (IVHFSCQWA) operatorFrom Definition 9, we know that the G-IVHFSCWA operator only gives the importance of the ordered positions. To further consider the importance of elements and reflect their correlations, we introduce the interval-valued hesitant fuzzy Shapley-Choquet hybrid operator that considers the importance of the attributes (or experts) and their ordered positions as well as reflects their interactions.

Definition 12. Let () be a collection of IVHFEs in , be a fuzzy measure on , and be a fuzzy measure on the ordered set . The generalized interval-valued hesitant fuzzy Shapley-Choquet hybrid weighted averaging (G-IVHFSCHWA) operator is defined aswhere , indicates a permutation on such that , is the Shapley value of , and is the generalized Shapley value of with .

Remark 13. If for each , then the G-IVHFSCHWA operator degenerates to the G-IVHFSCWA operator.

Remark 14. If , then the G-IVHFSCHWA operator degenerates to the interval-valued hesitant fuzzy Shapley-Choquet hybrid weighted averaging (IVHFSCHWA) operator

Remark 15. If , then the G-IVHFSCHWA operator degenerates to the interval-valued hesitant fuzzy Shapley-Choquet quadratic hybrid weighted averaging (IVHFSCQHWA) operatorAlthough the fuzzy measure can address the situation where the elements in a set are correlative, they define the power set. It makes the problem exponentially complex. Thus, it is not easy to solve a fuzzy measure when the set is large. To reflect the interactions between elements and simplify the complexity of solving a fuzzy measure, we introduce a special case of the G-IVHFSCHWA operator using 2-additive measures.
Let be a pseudo-Boolean function. Grabisch [64] noted that any fuzzy measure can be seen as a particular case of pseudo-Boolean function and put under a multilinear polynomial in variables:where , , and if and only if .
The set of coefficients () corresponds to the Möbius transform, denoted by . Because the transform is inversible, can be recovered from by .

Definition 16 (see [64]). A fuzzy measure on is said to be k-additive if its corresponding pseudo-Boolean function is a multilinear polynomial of degree , i.e., for all such that , and there exists at least one subset with elements such that .
Particularly, when , it gets a 2-additive measure. For a 2-additive measure , one can easily get [64], for any , with ,where and .
For a 2-additive measure, we only need coefficients to determine it for a set with elements.

Theorem 17 (see [64]). Let be a fuzzy measure on , then is a 2-additive measure if and only if there exist coefficients and for all that satisfy the following conditions:(i),(ii),(iii) s.t. and .

Theorem 18 (see [46]). Let be a 2-additive measure on , then the generalized Shapley function φ with respect to can be expressed asfor any such that and for any , In Definition 12, if and are both a 2-additive measure, then it derives the generalized interval-valued hesitant fuzzy 2-additive Shapley-Choquet hybrid weighted averaging (G-IVHF2SCHWA) operator.

4. An Approach to Multiattribute Group Decision Making

Because of various reasons, the weighting information may be incompletely known. To solve this situation, this section first establishes models for the optimal fuzzy measure and the optimal 2-additive measure on the associated sets. Then, an approach to multiattribute group decision making under interval-valued hesitant fuzzy environment with incomplete weighted information and interactive characteristics is developed.

Let be the set of alternatives, let be the set of attributes, and let be the set of experts. Assume that is the IVHFE of the alternative for the attribute given by the expert (). By , we denote the interval-valued hesitant fuzzy decision matrix given by the expert (). Let and be respective of the ordered sets for the attribute set and the expert set .

4.1. Models for the Optimal Fuzzy Measure

Before building models for the optimal fuzzy measure, let us first introduce a new distance measure. Let and be any two IVHFEs, Chen et al. [38] defined the following distance measures for IVHFEs, denoted aswhere is a permutation on the possible interval value in and with and being the th largest values in and , respectively; let with and being the numbers of possible interval-valued membership degrees in and . For , the authors adopted the method that extends the shorter one until both of them have the same length by adding the biggest interval several times.

Different from this distance measure, we define another one that need not consider the length of IVHFEs.

Definition 19. Let and be any two IVHFEs, then the generalized distance measure between and is defined as where and and denote the number of the possible interval value in and , respectively.

For example, let and . From (19), it derives . By (20), it gets . Furthermore, by (21) it gives for and for .

4.1.1. Models for the Optimal Fuzzy Measure on the Expert Set E

For each interval-valued hesitant fuzzy decision matrix (), we calculate the score matrix with . Because the experts’ knowledge, skills, and experiences are different, it is unreasonable to give the same weight of an expert for different attributes.

Let . With respect to the attribute , , if the weighting information on the expert set is partly known, the following model is established: where and are the coefficient matrices, and are the constant vectors, and are the known constraints, is the fuzzy measure on the expert set with respect to the attribute , is the Shapley value of the expert , and is the known weighting information.

If is a 2-additive measure, by (18) it gets the following model:where and are the coefficient matrices, and are the constant vectors, , and are the equivalent expressions of the known constraints given in model (22) with respect to the 2-additive measure .

The optimal fuzzy measure obtained from this model has the following desirable characteristics: the closer an expert’s evaluation values are to the other experts’, the larger the fuzzy measure will be. This can decrease the influence of the unduly high or low evaluation value induced by the experts’ limited knowledge or expertise.

4.1.2. Models for the Optimal Fuzzy Measure on the Ordered Set

To construct the model for the optimal fuzzy measure on the ordered set , the following procedure is needed.

Step 1. Calculate the interval-valued hesitant fuzzy Shapley weighted decision matrices , , where

Step 2. Calculate the score matrices , , where

Step 3. Calculate the mid-width matrices , , where

Step 4. For each pair , we rearrange each , , such that .

Because there is no inferior for the ordered positions with respect to the different attributes, if the weighting information on the ordered set is not exactly known, the following model for the optimal fuzzy measure is built:where and are the coefficient matrices, and are the constant vectors, and are the known constraints, is the fuzzy measure on the ordered set , is the Shapley value of the th position, and is the known weighting information.

If is a 2-additive measure, by (18) it gets the following model:where and are the coefficient matrices, and are the constant vectors, , and are the equivalent expressions of the known constraints given in model (27) with respect to 2-additive measure .

4.1.3. Models for the Optimal Fuzzy Measure on the Attribute Set C

Next, let us consider the optimal fuzzy measure on the attribute set . Assume that is the comprehensive interval-valued hesitant fuzzy decision matrix. Let and for each .

By (21), we calculate the distance between and as well as the distance between and for each pair . Because all alternatives are noninferior, if the weighting information on the attribute set is not exactly known, the following models for the optimal fuzzy measure are constructed: where and are defined in Definition 19, and are the coefficient matrices, and are the constant vectors, and are the known constraints, is the fuzzy measure on the attribute set , is the Shapley value of the attribute , and is the known weighting information.

Because models (29) and (30) have the same constraints and all alternatives are noninferior, they can be combined to formulate the following linear programming: If is a 2-additive measure, then it derives the following model:where and are the coefficient matrices, and are the constant vectors, and and are the equivalent expressions of the known constraints given in model (30) with respect to 2-additive measure .

4.1.4. Models for the Optimal Fuzzy Measure on the Ordered Set N

Letfor each pair .

For each , we rearrange such that . Similar to model for the optimal fuzzy measure on the attribute set , if the weighting vector on the ordered set is incompletely known, the following model is established: where and are the coefficient matrices, and are the constant vectors, and are the known constraints, is the fuzzy measure on the ordered set , is the Shapley value of the th position, and and is the known weighting information.

If is a 2-additive measure, then it derives the following model: