Intuitionistic Linguistic Multiple Attribute Decision-making Based on Heronian Mean Method and Its Application to Evaluation of Scientific Research Capacity

The main focus of this paper is to investigate intuitionistic linguistic information fusion based on Heronian mean. Two new intuitionistic linguistic aggregation operators called intuitionistic linguistic generalized Heronian mean (ILGHM) and intuitionistic linguistic generalized weighted Heronian mean (ILGWHM) operators, are introduced. The ILGHM and ILGWHM operators are characterized by the ability to deal with the intuitionistic linguistic multiple attribute decision making problems in which the attributes are interactive. Some desired properties and special cases with respect to the different parameter values in the developed operators are studied. Furthermore, a method based on the proposed operators is developed to deal with multiple attribute group decision making (MAGDM) method problems. Finally, an illustrative example concerning evaluation of scientific research capacity is provided to illustrate the decision-making process and to discuss the influences of different parameters on the decision-making results.


INTRODUCTION
Because the objects are often fuzzy and uncertain, the available information involved in multiple attribute group decision-making (MAGDM) problems are not always expressed as real numbers, and sometimes it is better suited to use another approach to deal with this information such as fuzzy set (Zadeh, 1965), linguistic information (Atanassov, 1986) and intuitionistic fuzzy set (IFS) (Herrera and Herrera-Viedma, 2000).Among all the tools, the intuitionistic fuzzy set (IFS) proposed by Atanassov (Herrera and Herrera-Viedma, 2000), is a useful tool to describe and deal with vagueness.A prominent characteristic of IFS is that it assigns to each element a membership degree and a non-membership degree, and thus, it is more powerful to deal with uncertainty and vagueness in real applications than fuzzy set which is only assigns to each element a membership degree (Hsieh and Chan, 2016).The IFS has received more and more attention since its appearance (Xu, 2007;Yu, 2015;Boran and Akay, 2014;Wan, Wang and Dong, 2016;Zeng and Chen, 2015;Zeng and Xiao, 2016;Zeng, Su and Zhang, 2016).
However, in real decision-making problems, it is difficult for decision makers to provide exact numbers for the membership and non-membership degrees of an intuitionistic fuzzy set while it is easy to provide linguistic assessment values.On the basis of the intuitionistic fuzzy set and the linguistic assessment set, Wang and Li (2010) proposed the concept of intuitionistic linguistic set (ILS), whose basic elements are intuitionistic linguistic numbers (ILNs).As a generalization of intuitionistic fuzzy numbers and linguistic variables, the ILN is able to handle the vague characters of things more accurately than linguistic term sets and intuitionistic fuzzy numbers.Wang and Li (2010) also proposed the score function, accuracy function and some operational laws of the intuitionistic linguistic set.The intuitionistic linguistic information fusion and aggregation method has received more and more attention.For example, based on the ordered weighted average (OWA) operator (Yager, 1988), Liu (2013) proposed the intuitionistic linguistic generalized dependent aggregation operator and applied it to group decision making.Liu and Wang (2014) proposed some intuitionistic linguistic power aggregation operators.Su et al. (2014) presented the intuitionistic linguistic OWA dis-tance (ILOWAD) operator.Ju et al. (2016) extended Maclaurin symmetric mean aggregation operators to intuitionistic linguistic environment.
The generalized OWA (GOWA) operator introduced by Yager ( 2004) is a very common aggregation method, which uses generalized means in the aggregation process.It generalizes a wide range of aggregation operators such as the generalized mean, the OWA and the ordered weighted geometric (OWG) operator.The GOWA operator has been studied by various authors (Beliakov, Pradera and Calvo, 2007;Merigo and Yager, 2013;Peng, Gao and Gao, 2013;Zeng, Chen and Li, 2016).Another interesting aggregation operator is the Heronian mean (HM), which is developed to deal with the exact numerical values (Beliakov, Pradera and Calvo, 2007).The desirable characteristic of the HM is that it can capture the interrelationship of the input arguments, which makes it very useful in decisionmaking.
Motivated by the idea of GOWA operator and Heronian mean, in this paper, we propose two new intuitionistic linguistic aggregation operators: intuitionistic linguistic generalized Heronian mean (ILGHM) and intuitionistic linguistic generalized weighted Heronian mean (ILGWHM) operator.Furthermore, some desirable properties of the ILGHM and ILGWHM operators are studied.At the same time, some special cases of the generalized parameters in these operators are analyzed.To do this, the remainder of this paper is organized as follows.In Section 2, we briefly review some basic concepts.Section 3 presents the ILGHM and ILGWHM operators and analyzes a wide range of particular cases.In Section 4, we develop a method for multiple attribute decision making based on the ILGWHM operator.Section 5 presents an illustrative example about scientific research capacity evaluation and Section 6 summarizes the main conclusions found in the paper.

PRELIMINARIES
This section briefly reviews the intuitionistic linguistic set, the GOWA operator and the HM operator.

The Intuitionistic Linguistic Set
The linguistic method is an approximate technique, which represents qualitative aspects as linguistic values by means of linguistic variables (Atanassov, 1986).For convenience, let  = {  | = 0,1, … , } be a finite and totally ordered discrete term set, where   represents a possible value for a linguistic variable,  + 1 is the cardinality of .For example, a set of seven terms  could be given as follows: In these cases, it is usually required that there exist the following (Xu, 2015): (1) The set is ordered: (2) There is the negation operator: nes(  ) =   , such that  =  − .
In order to preserve all the given information, Xu (2015) extended the discrete term set  to a continuous term set  ̅ = {  | ∈ [0, ]}, where, if   ∈ , then we call   the original term, otherwise, we call   the virtual term.

Contribution of this paper to the literature
• This study explores the intuitionistic linguistic Heronian mean method, which can fully capture the interrelationship of the input arguments.
• This study develops an approach to group decision-making based on the ILGWHM operator, which is able to deal with the interrelationships between the attributes with intuitionistic linguistic information.
• The developed evaluation method for scientific research capacity provides a lot of different scenarios for decision makers to select the best one(s) by picking the particular values according to their interests.
For each ILS  in , if then   () is called the indeterminacy degree or hesitation degree of  to linguistic index  () .

The GOWA Operator
The generalized OWA (GOWA) operator is an extension of the OWA operator with generalized means (Yager, 2004).It provides a wide range of aggregation operators including the OWA operator with its particular cases.The GOWA operator can be defined as follows.
Definition 4. A GOWA operator of dimension  is a mapping GOWA:   →  with an associated weighting vector  of dimension  with   ∈ [0,1] and ∑   = 1  =1 , such that: The GOWA operator is commutative, monotonic, bounded and idempotent.If we look to different values of the parameter , we can obtain other special cases such as the usual OWA operator and the OWGA operator.

The Heronian Mean (HM)
The Heronian mean (HM) operator (Beliakov, Pradera and Calvo, 2007) is an important aggregation operator which can capture effectively the relevance between the aggregated arguments.It can be defined as follows.
The desirable characteristic of the HM is that it can capture the interrelationship of the input arguments, which makes it very effective in decision making.However, the existed HM is only suitable to aggregate input data taken by the forms of crisp numbers rather than any other types of arguments, which restricts the its potential applications to more extensive areas.In next section, we should the HM to process intuitionistic linguistic environment.

INTUITIONSITIC LINGUISTIC GENERALIZED HERONIAN MEAN
Based on the Heronian mean and the GOWA operator, in this section we develop the intuitionistic linguistic generalized Heronian mean (ILGHM) operator.The main advantage of the ILGHM operator is that that it can capture the interrelationship of the input arguments, which makes it very suitable in decision making with intuitionistic linguistic information.It can be defined as follows.Definition 6.Let  �  = �   , (  ,   )� ( = 1,2, . . ., ) be a collection of ILNs, ,  ≥ 0 and ,  do not take the value 0 simultaneously, if then ILGHM is called the intuitionistic linguistic generalized Heronian mean (ILGHM).
Based on the operational laws of the ILNs described earlier, we can derive the result shown as the Theorem 1.
Theorem 1.Let  �  = �   , (  ,   )� ( = 1,2, . . ., ) be a collection of ILNs, ,  ≥ 0 and ,  do not take the value 0 simultaneously.Then the result aggregated value by the ILGHM is still an ILNs, and even and Then we have which completes the proof of Theorem 1.
By assigning different values to the parameters  and , we can get some special cases of the ILGHW, such as: (1) If  =  = 1 2 ⁄ , then the  , reduces to which we call the intuitionistic linguistic Heronian mean.
(2) if  =  = 1, then the  , reduces to which we call the intuitionistic linguistic generalized interrelated square mean.
Similar to Theorem 1, the theorem 2 can be derived easily.
Theorem 3. The ILGHM operator is a special case of the ILWGHM operator.
which completes the proof of the theorem.

AN APPROACH TO GROUP DECISION-MAKING BASED ON THE ILGWHM OPERATOR
In many actual decision problems, there exist the interrelationships between the attributes (Chang & Wang, 2016).At the same time, because of fuzziness of the attributes, they can be easily expressed by the intuitionistic linguistic variables.Thus, it is necessary to propose a decision-making approach based on the ILGWHM operator to deal with the interrelationships between the attributes with intuitionistic linguistic information.

EXAMPLE
Next, we give an example concerning evaluation of scientific research capacity to illustrate the proposed method.The scientific research capacity evaluation is crucial to the development and planning of discipline, thus raising the ability of scientific research is a central task of discipline construction.Currently, Zhejiang Gongshang University will increase the intensity of investment in scientific research, and tries to select a subject from all disciplines for pilot study.Four alternatives (  ,  = 1,2,3,4) will be evaluated to confirm which is fitted best.Three experts (  ,  = 1,2,3) from administration establishes the panel of decision makers that will take the whole responsibility for this evaluation (the weight vector of experts is  = (0.4,0.28,0.32) ).They evaluate the subjects (  ,  = 1,2, . . ., ) according to following four aspects, size of the research team  1 , scientific influence of research output  2 , research platform construction level  3 and achievement transformation  4 , respectively, the intuitionistic linguistic decision matrices   = � �   � 4×4 ( = 1,2,3) as shown in Tables 1-3.
If we use the different values  and  in above methods to rank the alternatives, the ordering of the alternatives may be different.The ranking results are listed in Table 5.
As we can see, depending on different  and  in this example, the ranking of the alternatives may be different.Comparing with evaluation method proposed by Yu (2013), the main advantage of the method in this paper is that it can deal with the interactions between the attributes.Moreover, it provides a lot of different scenarios for decision makers to select the best one(s) by picking the particular values  and  according to their interests.

CONCLUSION
In this paper, we have extended the traditional Heronian mean operator to process the intuitionistic linguistic information.We have proposed the intuitionistic linguistic generalized Heronian mean (ILGHM) and the intuitionistic linguistic generalized weighted Heronian mean (ILGWHM).The prominent char-acteristic of the ILGHM and ILGWHM operators is that they can not only accommodate the intuitionistic linguistic environment, but also consider the inter-dependent phenomena among the criteria.Based on the developed operators, we have also presented an application of the new approach to a group decision-making problem about the evaluation of scientific research capacity.The example shows that the proposed method is very flexible because it can provide the decision makers more choices to select the particular cases by assigning different parameter values for the operator.
In future research we expect to develop further extensions by adding new characteristics in the problem such as the use of order-inducing variables.We will also consider other decision-making applications such as human resource management and product management.