#### Abstract

In many countries, green products play a critical role in energy recycling and environment protection. The selection of green products can be regarded as a multiple attribute decision making (MADM) problem. Due to the complexity and uncertainty of the problem, decision makers may give their personal preference values to different attributes of alternatives by intuitionistic unbalanced linguistic term sets. The main purpose of this paper is to put forward a new generalized multiple attribute group decision making (GMAGDM) approach based on the intuitionistic unbalanced linguistic dependent weighted generalized Heronian mean (IULDWGHM) operator and the intuitionistic unbalanced linguistic dependent weighted generalized geometric Heronian mean (IULDWGGHM) operator. The proposed method can not only relieve the influence of unfair assessments, but also consider the interaction effects of attributes. Furthermore, the appropriate parameter values and operators can be selected to meet the different risk preference of decision makers and actual requirements. Finally, a green products selection case is given to illustrate the effectiveness and universality of the developed approach.

#### 1. Introduction

Zadeh [1–3] introduced the concept of linguistic variable in 1975. It can deal with qualitative situation in form of words and sentences. For example, the performance of a car is a linguistic term rather than numeric, i.e., very good, good, medium, bad, very bad, quite bad,...and so on. Generally, the linguistic variable is the element of the linguistic term set with uncertain granularity.

Soon afterwords, series linguistic models have been presented to manage decision making problems in uncertain circumstance. Herrera and Martínez [4] initiated the 2-tuple fuzzy linguistic representation model with a linguistic term and a numeric value assessed in -0.5, 0.5, so that the loss of information in the fusion process is avoided. Xu [5] proposed the virtual linguistic model and defined some operational laws. It overcomes the problem that operational results of linguistic variables exceed bounds of the original linguistic term set. Wang and Li [6] put forward the concept of the intuitionistic linguistic sets which combines the intuitionistic fuzzy set and the the linguistic set to express the fuzzy information. Rodríguez et.al. [7] introduced the hesitant fuzzy linguistic model to handle conditions where experts may hesitate among several consecutive qualitative linguistic terms. Ji, Zhang and Wang [8] considered the outranking method with multi-heditant fuzzy linguistic term set by introducing the projection. Wang and Peng [9] put forword hesitant linguistic intuitionistic fuzzy sets (HLIFSs) based on hesitant fuzzy sets (HFSs) and linguistic intuitionistic fuzzy numbers (LIFNs) which can depict complex and uncertain decision-making information and reflect the hesitancy of decision-makers. A computational model based on type-2 fuzzy sets was proposed by Mendel and Türkşen [10] which maintained the uncertainty and reduced the computational efforts when aggregating them. Recently, Herrera, et al. [11] presented a new fuzzy linguistic methodology called unbalanced linguistic term sets, in which linguistic labels are non-uniformly and asymmetrically distributed.

After the concept of unbalanced linguistic term sets was presented, numerous studies have been developed on both theoretical basis and practical applications. Bartczuk, et al. [12] introduced a new methodology to handle unbalanced linguistic information with a linguistic label and a value of the correction factor. Dong, Li and Herrera [13] proposed a novel numerical scale model to address the hesitant fuzzy unbalanced linguistic term sets. Dong, et al. [14] put forward the unbalanced linguistic assessments with interval symbolic proportions under multi-granular linguistic contexts. Wang, Liang and Qian [15] built a normalized numerical scaling approach to determine semantics of multigranular linguistic terms, which can lower the complexity of computation and the subjectivity in transformation process. Jiang, Liu et al. [16] gave an aggregation method for unbalanced fuzzy linguistic information by using the linguistic proportional 2-tuple power average operator, while Mata, et al. [17] proposed the type-1 OWA operator to aggregate linguistic values of unbalanced linguistic terms. Dong, et al [18, 19] made studies on the preference relations with unbalanced linguistic information to obtain a required consistency level. By using the 2-tuple model, Wang et al.[20] designed a new onling recommendation model based on unbalanced variables and integrated cloud. More related works can be seen in Refs. [21–27]

It is noteworthy that the unbalanced linguistic term sets can represent the saltation and nonlinear performance of human thought. Thus, it has real meaning to study the applications of unbalanced linguistic information. Among these applications, the aggregation of this linguistic variables is of great important. A lot of aggregatopn operators have been introduced in [28–34], Herrera, et al. [11] put forword the arithmetic mean of linguistic 2-tuples to for unbalanced linguistic variables. Isern, et al. [35] utilized the concepts of ordered weighted averaging operators to aggregate unbalanced linguistic variables. Han et al. [36] gave an aggregation method for unbalanced linguistic information by untilizing the generalized Heronian mean operators. Han et al. [37] processed the unbalanced linguistic information with a generalized dependent OWA operator which can relieve the influence of unfair linguistic variables by assigning low weights to the biased ones, and make the decision results more reasonable.

Noting that discussions on MADM problems are always in the situation that the attribute sets faced by decision makers are the same. However, the decision makers may consider attribute sets that are not the same due to the different knowledge background. That is to say, they may make mistakes when they give preference values out of their expertised fields. For example, the government tried to invest a new green product, four suppliers are selected for further consideration. The assessments are provided by four departments. The environment department may consider the recyclability level , the contamination degree ; the comprehensive department could focus on the public satisfaction and company scale ; the technology sector may care about the speed of reusability , the quality of maintenance and the level of technical advice ; the financial department have to think about reasonableness of the charge . Herein, four departments take into account alternatives via their own attribute information, respectively. Thus it is necessary to consider the generalized MADM (GMADM) in which the decision making attributes are changeable for different decision makers.

Comparing to real number and the fuzzy set, intuitionistic unbalanced linguistic numbers (IULNs),which act as elements of intuitionistic unbalanced linguistic term set, could describe the uncertain and incomplete assessments more effectively. For instance, when the recycling characteristics of a green product is evaluated, the decision maker may state that the product is “Quite Good” with probability of truth, falsity, uncertainty of 40%, 30%, 30%. The assessment can be expressed as using the IULN. Thus, decision making information is vital to be described by IULNs.

Up to now, the applications of unbalanced linguistic variables in intuitionistic fuzzy situation has not been studied. Besides, the decision makers may assign the high preference values to their preferred alternatives as well as the low evaluation values to their disgusting one. In the meantime, the affecting factors of the green products selection have some relevance, such as the recycling degree and the environment pollution. Based on above analysis, it is very important and necessary to extend the dependent operator and the HM operator to cope with the generalized MAGDM in intuitionistic unbalanced linguistic environment. Thus, the aim of this paper is to solve geen product selection GMAGDM problems in which the evaluation values are correlative intuitionistic unbalanced linguistic information. We will introduce the intuitionistic unbalanced linguistic dependent weighted generalized Heronian mean (IULDWGHM) operator and the intuitionistic unbalanced linguistic dependent weighted generalized geometric Heronian mean(IULDWGGHM) operator by combining the dependent operator and the Heronian mean operator under intuitionistic unbalanced linguistic situations. The most crucial advantages of these operators are that they could take into account correlation of input variables, relieve the influence of unfair assessment values and deal with intuitionistic unbalanced linguistic information. For the situation in which the attribute sets considered by DMs are not identical on account of their different knowledge background, the generalized MAGDM with intuitionistic unbalanced linguistic information is proposed. The constributions of this paper are as follows:

(i) The selection of green products is a generalized multiple attribute group decision making (GMAGDM) problem with intuitionistic unbalanced linguistic numbers due to that the attribute sets provided by decision makers are not identical.

(ii) The unbalanced linguistic representation model and the concept of distance between any two intuitionistic unbalanced linguistic numbers are very convenient to translate the qualitative assessments to quantitative ones.

(iii) The intuitionistic unbalanced linguistic dependent weighted generalized Heronian mean (IULDWGHM) operator and the intuitionistic unbalanced linguistic dependent weighted generalized geometric Heronian mean(IULDWGGHM) operator are proposed to deal with the case of green products selection. The above operators can not only relieve the influence of unfair evaluations, but also reflect the relationship of both the different criteria values and the criteria value itself. In addition, it has flexible parameter values, we could select the appropriate parameter values to meet the different actual requirements.

The rest of the paper is arranged as follows: Section 2 introduces some basic concepts and notions. Section 3 proposes a GMAGMD approach for selecting the optimal green product based on intuitionistic unbalanced linguistic dependent weighted generalized Heronian mean operartor and intuitionistic unbalanced linguistic dependent weighted generalized geometric Heronian mean operartor, investigates the properities and some particular cases. Section 4 describes the GMAGDM problem with intuitionistic unbalanced linguistic information, a detailed procedure is proposed for managing the GMAGDM in the following. Subsequently, an example of a green product selecton is given to illustrate the effectiveness and universality of the developed approach in Section 5. Section 6 concludes the comparison analyses with other methods. Finally, the paper is summarized in Section 7.

#### 2. Preliminaries

In this section, we briefly review the concepts of the intuitionistic linguistic term set, the unbalanced linguistic term set, the dependent ordered weighted average (DOWA) operator and the Heronian mean (HM) operator.

##### 2.1. The Intuitionistic Linguistic Set

*Definition 1 (see [6]). *Let be the universe of discourse. An intuitionistic linguistic term set A on X can be defined aswhere belongs to the continuous linguistic set , the function and stand for the membership degree and non-membership degree of to .

For the sake of convenience, Wang et.al. [6] named an intuitionistic linguistic number (ILN). Some operational laws were given as follows:

Let , be any two ILNs, be any positive scalar, then

Obviously, the results of above operations are still ILNs.

Liu [38] proposed the concept of the score function and the accuracy function of ILNs. Furthermore, the method to compare any two ILNs is proposed as follows.

*Definition 2 (see [38]). *Let be an ILN, then the score function and the accuracy function of can be given aswhere is the granularity of the linguistic term set.

*Definition 3 (see [38]). *Suppose that are any two ILNs, then(1)If , then ;(2)If and , then .

*Definition 4 (see [38]). *Assume that , are any two ILNs, then the Hamming distance between and is defined as

##### 2.2. The Unbalanced Linguistic Representation Model

Herrera et.al [11] introduced the concept of the unbalanced linguistic term set to reflect the jumpy of human thinking, i.e., the linguistic assessment variables are non-uniformly and non-symmetrically distributed.

*Definition 5 (see [11]). *An unbalanced linguistic term set S can be expressed as where is the set of all left labels of the central label, contains the central label merely, is the set of all right labels of the central one.

*Example 6. *An unbalanced linguistic term set with 9 granularity S=N(none), NG(not good), M(middle), AG(almost good), G(good), QG(quite good), VG(very good), AT(almost total), T(total), which is used to evaluate the comprehensive quality of cars. We obtain , , . It can be seen that there are fewer labels in . The distribution of the semantics can be shown in Figure 1.

The model to handle unbalanced linguistic information is based on linguistic hierarchies and the 2-tuple model. The semantic representation of unbalanced linguistic terms are derived via linguistic hierarchies and the computational model based on 2-tuple is defined to accomplish process of computing with words.

A linguistic hierarchy [39, 40] is a set of levels where each level is a linguistic term set with a different granularity from the remaining levels of the hierarchy. Each level belonging to a linguistic hierarchy is denoted as with being a number that indicates the level of the hierarchy and the granularity of the linguistic term set of the level. For example, a linguistic hierarchy of level 4 is represented by , its graphics is shown in Figure 2 and Table 1.

In a linguistic hierarchy, the transformation functions (TF) [41] between variables from different levels is defined as follows. For example: .

Any unbalanced linguistic 2-tuple can be transformed into the term in linguistic hierarchies by the following unbalanced linguistic transformation function LH and vice versa. The detailed transformation process is given as follows.Representation of unbalanced linguistic terms in linguistic hierarchies. To accomplish the process of computing with words, the first step is to transform the unbalanced linguistic information in S into the term in linguistic hierarchies. The 2-tuple in linguistic hierarchies associating with respective unbalanced linguistic 2-tuple can be obtained by unbalanced linguistic transformation function LH, i.e. such that .Computational process. The process of computing with words can be accomplished by the computation model in linguistic hierarchies. To obtain the expression in the same linguistic domains, we first translate into linguistic 2-tuple in the maximum level of linguistic hierarchies, noted as by the transformation function TF Eq.(10). Then the linguistic 2-tuple computation model is used with the result as .Retransformation phase in unbalanced linguistic term set. By the retransformation phase, the result is translated into the unbalanced linguistic term via the transformation function :such that , can be determined by cases as follows:

*Case 1. *If is represented by merely one label in LH, then , that is ;

*Case 2. *If is represented by two labels in LH, then or

*Case 3. *If there exists no such that , we convert into another level, that is

, then it is returned to Case 1 or Case 2.

*Example 7. *Continuing Example 6, we have

##### 2.3. The Dependent Ordered Weighted Average (DOWA) Operator and the Heronian Mean (HM) Operator

The dependent ordered weighted average (DOWA) operator was developed by Xu [28], it can be defined as follows

*Definition 8 (see [28]). *Assuming to that is a collection of arguments, is the average value, i.e. , is a permutation of , such that for all , ifThen DOWA is called the dependent OWA operator.

The Heronian mean operator has the capacity of capturing the interactions between the input arguments. It can be defined as follows:

*Definition 9 (see [32]). *Let be a collection of non-negative numbers, . Then a HM operator of dimension n is a mapping which satisfies

A series of HM operators are provided, such as the generalized HM (GHM) operator and the generalized geometric HM (GGHM) operator.

*Definition 10 (see [32]). *Let , and be a collection of non-negative numbers, , then a GHM operator of dimension n is a mapping which satisfies

*Definition 11 (see [33]). *Let , , and be a collection of non-negative numbers, , then a GGHM operator of dimension n is a mapping which satisfies

#### 3. The IULDWGHM Operator and the IULDWGGHM Operator

##### 3.1. The IULDWGHM Operator

Inspired by Xu [28], we will define the intuitionistic unbalanced linguistic term set (IULTS), the intuitionistic unbalanced linguistic number(IULN), then the IULDWGHM operator by combining the dependent operator and the weighted generalized Heronian mean operator will be proposed in the intuitionistic unbalanced linguistic environment.

*Definition 12. *Let be the universe of discourse. An intuitionistic unbalanced linguistic set on X can be defined aswhere , belongs to the unbalanced linguistic set, the function and stand for the membership degree and the non-membership degree of to .

Similarly, is called an intuitionistic unbalanced linguistic number (IULN).

*Definition 13. *Given that is a set of intuitionistic unbalanced linguistic numbers, the mean value of intuitionistic unbalanced linguistic numbers can be defined aswhere , , .

*Definition 14. *Assuming that is an intuitionistic unbalanced linguistic number (IULN), then the score value and the accurate value of can be defined aswhere is the granularity of the lever in the linguistic hierarchies.

*Example 15. *Let , , be three intuitionistic unbalanced linguistic numbers which the unbalanced linguistic term set with 9 granularity is S=N(none), NG(not good), M(middle), AG(almost good), G(good), QG(quite good), VG(very good), AT(almost total), T(total), the score values and the accurate values of , , are as follows:

Then we can compare any two IULNs on the basis of score function and accuracy function.

*Definition 16. *Supposing that and are any two intuitionistic unbalanced linguistic numbers(1)If , then ;(2)If and , then .

*Definition 17. *Let and be any two intuitionistic unbalanced linguistic numbers, the Hamming distance between and iswhere is the granularity of lever in the linguistic hierarchy.

Similarly, the Euclidean distance can be shown as

*Definition 18. *Let be a set of intuitionistic unbalanced linguistic numbers, is the arithmetic mean, the similarity between and is

In real life situation, different experts can provide their preference values in the form of the intuitionistic unbalanced linguistic numbers . Some experts may assign unduly high preference values to their enjoyed objects while low values to their detested one. The above “false” opinions should be assigned very low weights. In other words, the distance between a preference value and the mean one is lager, the weights should be smaller. Based on Eq. (27), the dependent weight of intuitionistic unbalanced linguistic numbers is

Based on the above dependent weight, we can provide the definition of the intuitionistic unbalanced linguistic dependent weighted generalized Heronian mean (IULDWGHM) operator.

*Definition 19. *Let be a set of IULNs and , IUL be the set of all intuitionistic unbalanced linguistic numbers, an IULDWGHM operator is a mapping where ,

The properties of the IULDWGHM operator can be shown as follows. For convenience, we denote that in the following.

Lemma 20. *Let be a set of IULNs and , then we have*

*Proof. *we prove Eq. (29) by means of mathematical induction.

For , according to Eqs. (2) – (4), we haveTherefore, Eq. (29) holds for n=2.

If Eq. (29) holds for ,

Then when , we can obtainwhich means that Eq. (29) also holds for n=k+1. We complete the proof.

Theorem 21. *Let be a set of IULNs and , the aggregated result of Eq. (28) is still an IULN and *

Where

*Proof. *by Eq. (4) and (5), we can haveIn the following, we need to prove following Eq. (35) by mathematical induction on n.For n=2, If Eq.(35) for n=k, i.e.,Then, for n=k+1, we obtainBy Eq. (29), (37), we can transform (38) asFurthermore, we haveThus, where Next, we prove that the sum of the membership degree and the non-membership degree belongs to .

SinceThusThis completes the proof of Theorem 21.

Theorem 22 (idempotency). *If for all , then *

*Proof. *Since for all , , then we have

Theorem 23 (monotonicity). *Let and be any two collections of IULNs, if , , for all , then *

*Proof. *Since , , for all , then we haveThus i.e.

Theorem 24 (boundedness). *The IULDWGHM operator lies between the maximum operator and the minimum operator, i.e.*

*Proof. *Since , then we haveThus