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Computational Intelligence and Neuroscience
Volume 2018, Article ID 1404067, 25 pages
https://doi.org/10.1155/2018/1404067
Research Article

An Approach to Linguistic Multiple Attribute Decision-Making Based on Unbalanced Linguistic Generalized Heronian Mean Aggregation Operator

1School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, China
2School of Business, Anhui University, Hefei, Anhui 230601, China
3Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695, USA

Correspondence should be addressed to Huayou Chen; moc.621@cuoyauh

Received 15 December 2017; Accepted 8 April 2018; Published 12 June 2018

Academic Editor: Elio Masciari

Copyright © 2018 Bing Han et al. 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

This paper proposes an approach to linguistic multiple attribute decision-making problems with interactive unbalanced linguistic assessment information by unbalanced linguistic generalized Heronian mean aggregation operators. First, some generalized Heronian mean aggregation operators with unbalanced linguistic information are proposed, involving the unbalanced linguistic generalized arithmetic Heronian mean operator and the unbalanced linguistic generalized geometric Heronian mean operator. For the situation that the input arguments have different degrees of importance, the unbalanced linguistic generalized weighted arithmetic Heronian mean operator and the unbalanced linguistic generalized weighted geometric Heronian mean operator are developed. Then we investigate their properties and some particular cases. Finally, the effectiveness and universality of the developed approach are illustrated by a low-carbon tourist instance and comparison analysis. A sensitivity analysis is performed as well to test the robustness of proposed methods.

1. Introduction

As an important part of multicriteria decision-making, multiple attribute decision-making (MADM) [1] and multiobjective decision-making build up the multicriteria decision-making system. The MADM concentrates research on discrete finite alternatives. The essence of MADM is ranking for the given alternatives and selecting the most desirable one. In order to integrate the individual preference value into a collective one, various operators have been presented during the past few years, such as the ordered weighted average (OWA) operator [2] which pays attention to the ordered position of each input datum, the ordered weighted geometric (OWG) operator [3], the dependent uncertain OWA (DUOWA) operator [4, 5], and the generalized OWA (GOWA) operator by adding an attitude parameter [6]. Zhou and Chen [7] investigated the continuous generalized OWA operator. Merigo [8, 9] presented the induced uncertain heavy OWA operators and induced generalized OWA (IGOWA) operator by using induced variables. Liao and Xu [10, 11] investigated the hybrid aggregation operators which consider the weight of arguments and their positions simultaneously. Liu et al. [12] presented some q-Rung Orthopair Fuzzy Aggregation Operators which could describe the space of uncertain information broadly.

However, the above aggregation operators have one thing in common: all input arguments are irrelevant, which is not realistic. The Heronian mean (HM) operator can overcome the drawback and has been improved to be an aggregation operator in [13]. Subsequently, a new range of extensions have been proposed, like the generalized Heronian mean (GHM) operator [14, 15], the intuitionistic fuzzy geometric HM (IFGHM) operator [16], the uncertain linguistic Heronian mean (ULHM) aggregation operators [17, 18], partitioned Heronian means operator [19], and Heronian aggregation operators of intuitionistic fuzzy numbers [20]. The Heronian mean operator has some particular characteristics that the others do not have. Contrasting the Choquet integral or power average operator which stresses on weight changes subjectively or objectively, Heronian mean focuses on aggregated arguments themselves. For a set of criteria values , the Bonferroni mean operator only considers the correlation between and . However, the relationship between and itself can not be considered. The Heronian mean operator can solve the correlation of both the different criteria values , and the criteria value itself.

With the development of society, the decision-making information is more and more fuzzy or uncertain [21, 22]. It is more suitable and reasonable to express the preference in the form of linguistic information rather than real number. Some fuzzy linguistic approaches were firstly introduced by Zadeh [23]. Later on, a series of extended linguistic term sets have been developed, such as intuitionistic linguistic term set (ILTS) [2428], 2-tuple linguistic term set (2TLTS) [19, 2933], virtual linguistic term set (VLTS) [34], probabilistic linguistic term set (PLTS) [35], and hesitant fuzzy linguistic tern set (HFLTS) [36]. The ILTS was introduced by Wang and Li [36] which has three main parts: linguistic terms, membership function, and nonmembership function. Herrera and Martinez [37] presented 2-tuple linguistic (2TL) model which can avoid information loss validly. To preserve all the given information, Xu [34] extended the original linguistic term set to a continuous linguistic term set and introduced the concept of the virtual linguistic term. Some researchers have reported that the computational models of both the 2-tuple linguistic model and the virtual linguistic model are equivalent [38, 39]. In consideration of the possible uncertainties in linguistic expression, the probability linguistic term set (PLTS) [35] was developed through adding the probabilities without loss of any original linguistic variables. The HFLTS, combining the LTS and the HFS, was introduced by Rodríguez et al. [40]. It is a more reasonable information expression form, which can be used to describe people’s subjective cognitions.

Obviously, the above linguistic aggregation operators are based upon symmetrically and uniformly placed linguistic term set. However, it is necessary to give evaluations by using nonsymmetrically and nonuniformly distributed linguistic terms [41] in some cases. For example, when assessing a person’s ability, the linguistic term set used by experts is extremely bad, bad, medium, almost good, good, quite good, very good, extremely good, perfect. The number of the terms lying on the left of the central term “medium” (two) is less than that on the right one (six). To overcome the drawback, the unbalanced linguistic representation model has been presented in [42]. Subsequently, the unbalanced linguistic aggregation operators were introduced, for instance, the unbalanced linguistic OWG (ULOWG) operator [43], the unbalanced linguistic weighted OWA (ULWOWA) operator [41], and the unbalanced linguistic power average (ULPA) operator [44]. Furthermore, unbalanced linguistic aggregation operators in risk analysis were also investigated in [45, 46].

Through the above analysis, it is very important and necessary to develop the Heronian mean operator to cope with unbalanced linguistic information. Thus, the aim of this paper is to solve multiple attribute decision-making problems in which the evaluation information is correlative unbalanced linguistic information by combining the Heronian mean operator with unbalanced linguistic variables. We first introduce the unbalanced linguistic generalized arithmetic Heronian mean (ULGAHM) operator and the unbalanced linguistic generalized geometric Heronian mean (ULGGHM) operator. The most crucial advantage of these operators is that they could take into account correlation of input variables and deal with unbalanced linguistic information. For the situation that different attributes have different degrees of importance, the unbalanced linguistic generalized weighted arithmetic Heronian mean (ULGWAHM) operator and the unbalanced linguistic generalized weighted geometric Heronian mean (ULGWGHM) operator are presented and applied to MADM problems. The motivation of this paper is reposed on the following facts:

(i) The existing aggregation operators with unbalanced linguistic information are mainly concentrated on the OWA and OWG operator. There was less research about Heronian mean operator with unbalanced linguistic information.

(ii) The generalized Heronian mean aggregation operators can reflect the relationship of both the different criteria values , and the criteria value itself. In addition, they have flexible parameters and , and we could select the appropriate and to meet the different actual requirements.

(iii) Zou [43] just considered the weights of criteria in unbalanced linguistic environment. Meng [44] considered the weights of both experts and attributes. However, both of them ignore the relationship of input arguments. The multiple attribute decision-making [43, 44] can not deal with the situation where the assessment is in form of interrelated unbalanced linguistic information. Jiang [45] emphasized the changing of the weight of aggregation operator not the input arguments themselves. These new Heronian mean operators with unbalanced linguistic information could be used to solve above cases effectively.

The rest of the paper is arranged as follows: Section 2 introduces some basic concepts and notions. Some operational laws for unbalanced linguistic 2-tuple are defined in Section 3. In Section 4, some existing Heronian mean operators are reviewed and further we developed some new unbalanced linguistic generalized Heronian mean operators and investigated the properties and particular cases. Section 5 presents the multiple attribute decision-making problem with unbalanced linguistic information. Subsequently, an actual example is given in Section 6. Section 7 concludes the comparison analyses with other methods. Finally, the paper is summarized in Section 8.

2. Preliminaries

In this section, we briefly review the linguistic approach and the unbalanced linguistic representation model.

2.1. The Linguistic Approach

As an approximate technique, the linguistic approach [23] expresses the qualitative information in form of linguistic values of linguistic labels. Let be a linguistic term set. The label represents a possible value of linguistic labels. For instance, a linguistic term set of seven labels could be given as follows:where the central label represents the mediocre comment and the others sit on either side of the central one symmetrically and uniformly. Generally, should meet the following features:(1)A negation operator: Neg.(2)An order: if and only if .(3)A max operator: max if ; a min operator: min if .

In order to avoid information loss effectively, Herrera [37] introduced the 2-tuple fuzzy linguistic representation model which is composed of a linguistic label and a real number denoting the value of symbolic translation.

Definition 1 (see [37]). Let be a linguistic term set and let be a number value representing the aggregation result of linguistic symbolic. Then the function is defined as follows:where round () is the integer operator, is the closest index label to , and is the value of symbolic translation.

Definition 2 (see [37]). Let be a linguistic term set and let be a linguistic 2-tuple. Then the equivalent numerical value to a 2-tuple can be obtained by the following function

We can convert a linguistic term to a linguistic 2-tuple by adding a value 0 as symbolic translation:

The computational model of 2-tuple linguistic information has been developed, such as 2-tuple comparison operator, 2-tuple negation operator, and a wide range of 2-tuple aggregation operators.

2.2. The Unbalanced Linguistic Representation Model

The unbalanced linguistic representation model was introduced by Herrera [42]. The advantage of this model is that it can manage the linguistic assessment variables which are nonuniformly and nonsymmetrically distributed.

Definition 3 (see [42]). If a linguistic term set S has a maximum linguistic term, a minimum linguistic term, and a central linguistic term and other terms are nonuniformly and nonsymmetrically distributed around the central one on both left and right lateral sets, i.e., the different discrimination levels on both sides of central linguistic term, then this type of linguistic term sets is called unbalanced linguistic term sets. An unbalanced linguistic term set S can be noted as , in which contains the central linguistic term merely and contains all left linguistic terms lower than the central one. Similarly, contains all right linguistic terms higher than the central one.

Example 4. When experts try to evaluate the “comfort” of a car, the linguistic assessment set is S= N (none), M (middle), H (high), VH (very high), P (perfect), in which , , . Obviously, it has the minimum linguistic term N, the maximum linguistic term P, and the central linguistic term M, and the number of terms in the left is 1 which is lower than that in the right (3). In other words, discrimination levels on both sides of central linguistic term are different. So S is an unbalanced linguistic term set (Figure 1).

Figure 1: Unbalanced linguistic term set.

In order to transmit the unbalanced linguistic terms into linguistic 2-tuple information, the concept of a linguistic hierarchy was defined as follows.

Definition 5 (see [47, 48]). A linguistic hierarchy is a set of all levels with each level being a linguistic term set of different granularity. It can be noted as , where is a level belonging to the linguistic hierarchy, is a number that indicates the level of the hierarchy, and is the granularity of the linguistic term set of t. The set is called former modal points set of the level t. The construction of a LH must satisfy linguistic hierarchy basic rules:Rule 1: to preserve all former modal points of the membership functions of each linguistic term from one level to the following one.Rule 2: to make smooth transitions between successive levels. The aim is to add a new linguistic term set in the hierarchy such that a new linguistic term will be added between each pair of terms belonging to the term set of the previous level t.

Example 6. A linguistic hierarchy of level 3 could be given as follows:
, where n(1)=3, n(2)=5, n(3)=9; that is, the first level is a linguistic term set of granularity 3, the second level is a linguistic term set of granularity 5, and the third level is a linguistic term set of granularity 9. It can be graphically shown in Figure 2 with the granularity for each linguistic term set of a LH according to the rules in Table 1.

Table 1: Linguistic hierarchies.
Figure 2: Linguistic hierarchies of 3, 5, and 9 labels.

Definition 7 (see [49]). Let be a linguistic term of the level , then the transformation function from a linguistic level to another level is defined as follows:

Example 8. Let be a linguistic 2-tuple representation of level 2, and its linguistic 2-tuple representation of level 3 isFor each label of unbalanced linguistic term set, the semantic representation can be obtained by using linguistic hierarchies. The transformation process is illustrated by the following example.

Example 9. For an unbalanced linguistic term set S = N, M, H, VH, P, it can be transformed to 2-tuple representation according to the following steps.
Step 1. Due to , assume represents the number of linguistic terms in , .   such that , and can be represented by a label of level 1 in LH, i.e., .
Step 2. Due to , suppose is the number of linguistic labels in , . , with , and the semantic representation of can be got from labels of level 3, 4 in LH. Assume is the number of assigned labels of level i in LH; according to proposition 3 in [42], and can be obtained; that is, can be represented by three labels of level 3 and two labels of level 4; i.e., .
Step 3. To bridge representation gaps defined in [42], can be represented by the upside of the central label in level 1 and the downside of level 3, respectively; i.e., .
Step 4. The ultimate representations are as follows:

Definition 10 (see [42]). The transformation function from an unbalanced linguistic 2-tuple to its corresponding linguistic 2-tuple representation in LH is a mapping , such that ,  .Conversely, we can obtain the linguistic 2-tuple representation from the unbalanced linguistic 2-tuple: can be determined by cases as follows:

(1) If is represented by merely one label in LH, then ,  .

(2) If is represented by two labels in LH, then or

(3) If there exists no such that , we convert to another level; that is, , then return to (1) or (2).

Example 11. Continuing Example 9, we have

3. Some Operational Laws for Unbalanced Linguistic 2-Tuple

Based on 2-tuple representation model, we propose some operational laws and properties of unbalanced linguistic 2-tuple.

Definition 12. Let and be two unbalanced linguistic 2-tuples, , then one has(1);(2);(3);(4).

Theorem 13. Assume that and are two unbalanced linguistic 2-tuples, , then(1);(2);(3);(4);(5);(6);(7);(8).

Proof. (1)(2)(3)(4)(5)(6)(7)(8)

4. Some Heronian Mean Operators

4.1. The Existing Heronian Mean Operators

The Heronian mean operator has the capacity of capturing the interaction between the input arguments.

Definition 14 (see [15]). Let be a collection of nonnegative numbers; the Heronian mean operator is a mapping HM: which satisfiesA series of HM operators are provided, such as the generalized HM (GHM) operator and the generalized geometric HM (GGHM) operator.

Definition 15 (see [15]). Let be a collection of nonnegative numbers and ; the generalized Heronian mean operator is a mapping GHM: which satisfies

Definition 16 (see [17]). Let be a collection of nonnegative numbers and ; the generalized geometric Heronian mean operator is a mapping GGHM: which satisfies

4.2. The Proposed Heronian Mean Operators

The Heronian mean operator can capture the relevance of individual argument. However, it is rarely applied in unbalanced linguistic information. In this section, we shall extend the Heronian mean operator to the situation in which the input arguments are unbalanced linguistic information and shall develop some unbalanced linguistic Heronian mean operators, such as the unbalanced linguistic generalized arithmetic Heronian mean (ULGAHM) operator, the unbalanced linguistic generalized geometry Heronian mean (ULGGHM) operator, the unbalanced linguistic generalized weighted Heronian mean (ULGWHM) operator, and the unbalanced linguistic generalized weighted geometric Heronian mean (ULGWGHM) operator. Moreover, some properties of these operators are investigated; some special cases with respect to the parameter values are discussed simultaneously.

Definition 17. Let and be a collection of unbalanced linguistic 2-tuple variables, then the unbalanced linguistic generalized arithmetic Heronian mean operator of dimension n is a mapping ULGAHM: , which satisfieswhere is the set of all unbalanced linguistic 2-tuple variables and is the level of the maximum granularity in LH.

Now, we explore some properties of the ULGAHM operator.

Theorem 18. Let be a collection of unbalanced linguistic 2-tuples and , then the properties of the ULGAHM operator are given as follows:(1)Monotonicity: let and be two collections of unbalanced linguistic 2-tuples and for all , then(2)Idempotency: if for all , then(3)Boundedness: ULGAHM operator lies between maximum and minimum operator; i.e.,

Proof. (1) Since for all , according to the definition of LH and ,
we have for all ; based on Definition 12,Thus, .
(2) Since for all , we have(3) Let ; according to the property of idempotency, we have , , since for all .
Thus, .
It is easy to see that the unbalanced linguistic generalized arithmetic Heronian mean operator does not satisfy the property of commutativity.
We can get a series of special cases by assigning different values to the parameters and of the ULGAHM operator.(1)If , we getwhich is called the unbalanced linguistic generalized weighted mean (ULGWM) operator with the descending weight vector .(2)If , we haveThe ULGAHM operator reduces to the unbalanced linguistic weighted mean (ULWM) operator.(3)If , we obtainwhich is called the unbalanced linguistic weighted square mean (ULWSM) operator.(4)If , we haveObviously, the ULGAHM operator reduces to the unbalanced linguistic generalized weighted mean (ULGWM) operator with the ascending weight vector .(5)If , we obtain(6)If , we getwhich we call general unbalanced linguistic Heronian mean (ULHM) operator in this case.

We introduce the concept of the unbalanced linguistic generalized geometric Heronian mean operator as follows.

Definition 19. Let ,  , and be a collection of unbalanced linguistic 2-tuples, then the unbalanced linguistic generalized geometric Heronian mean operator of dimension n is a mapping ULGGHM:  , which satisfieswhere is the set of all the unbalanced linguistic 2-tuples and is a level of the maximum granularity in LH.

Some properties of the unbalanced linguistic generalized geometric Heronian mean operator are investigated as follows.

Theorem 20. Let be a collection of unbalanced linguistic 2-tuples and , then the properties of the ULGGHM operator are given as follows:(1)Monotonicity: let and be two collections of unbalanced 2-tuple linguistic variables and for all , then(2)Idempotency: if for all , then(3)Boundedness: let , then .

Proof. The proof of Theorem 20 can be seen in the Appendix.

Similarly, the unbalanced linguistic generalized geometric Heronian mean operator does not satisfy the property of commutativity.

Next, we analyze some particular cases in regard to parameters and .(1)If , thenwhich we call the unbalanced linguistic geometric mean (ULGM) operator with the descending weight vector. It has no relationship with p while .(2)If , thenThe ULGGHM operator reduces to the unbalanced linguistic geometric mean (ULGM) operator with the ascending weight vector. It has no relationship with q while .(3)If , then(4)If , thenwhich we call general unbalanced linguistic geometric Heronian mean (ULHM) operator in this case.

In (22) and (34), all aggregated arguments have the same importance. However, different parameters have different importance because of the different attitudes of decision-makers. Considering the importance of each argument, so we introduce the unbalanced linguistic generalized weighted arithmetic Heronian mean (ULGWAHM) operator and the unbalanced linguistic generalized weighted geometric Heronian mean (ULGWGHM) operator as follows.

Definition 21. Let , and be a collection of unbalanced linguistic 2-tuples, then the unbalanced linguistic generalized weighted arithmetic Heronian mean operator of dimension n is a mapping ULGWAHM: , which satisfieswhere is the set of all the unbalanced linguistic 2-tuples, is a level of the maximum granularity in LH, and is the weight vector of , satisfying .

Now, we discuss some properties of the unbalanced linguistic generalized weighted arithmetic Heronian mean operator.

Theorem 22. Assume that is a collection of unbalanced linguistic 2-tuples and , then the properties of the unbalanced linguistic generalized weighted arithmetic Heronian mean operator are given as follows:(1)Reducibility: let , then (2)Monotonicity: let and be two collections of unbalanced linguistic 2-tuples and for all , then(3)Idempotency: if for all , then(4)Boundedness: let , then

It is easy to see that the unbalanced linguistic generalized weighted arithmetic Heronian mean operator does not satisfy the property of commutativity.

Proof. The proof of Theorem 22 and some special cases of the unbalanced linguistic generalized weighted arithmetic Heronian mean operator in regard to parameters and can be seen in the Appendix.

Considering the importance of input arguments and unbalanced linguistic generalized geometric Heronian mean operator, we further introduce the unbalanced linguistic generalized weighted geometric Heronian mean (ULGWGHM) operator.

Definition 23. Let ,  , and be a collection of unbalanced linguistic 2-tuples, then the unbalanced linguistic generalized weighted geometric Heronian mean operator of dimension n is a mapping ULGWGHM: , which satisfieswhere is the set of all the unbalanced linguistic 2-tuples, is a level of LH which has the maximum granularity, and is the weight vector of