GROUP DECISION MAKING APPROACH BASED ON POSSIBILITY DEGREE MEASURE UNDER LINGUISTIC INTERVAL-VALUED INTUITIONISTIC FUZZY SET ENVIRONMENT

. In the present article, we extended the idea of the linguistic intuitionistic fuzzy set to linguistic interval-valued intuitionistic fuzzy (LIVIF) set to represent the data by the interval-valued linguistic terms of membership and non-membership degrees. Some of the desirable properties of the proposed set are studied. Also, we propose a new ranking method named as possibility degree measures to compare the two or more diﬀerent LIVIF numbers. During the aggregation process, some LIVIF weighted and ordered weighted aggregation operators are proposed to aggregate the collections of the LIVIF numbers. Finally, based on these proposed operators and possibility degree measure, a new group decision making approach is presented to rank the diﬀerent alternatives. A real-life case has been studied to manifest the practicability and feasibility of the proposed group decision making method.


1.
Introduction. In real life, multi attributes group decision making (MAGDM) problems are an important part of decision theory in which we choose the best one from the set of finite alternatives to the different attributes. For solving the decisionmaking (DM) problems, decision-makers provide their judgments or ratings towards the object. But it is not possible that their judgments may be in the form of crisp data due to the fuzziness or vagueness of the data [15,25]. Instead, it has become popular that these assessments are presented by a fuzzy set or extensions of the fuzzy set. Fuzzy set (FS) [49], proposed by Zadeh, is an effective tool to deal with vagueness and has received much attention. After their successful study, researchers are engaged in their extensions and out of that, intuitionistic fuzzy set (IFS) and interval-valued intuitionistic fuzzy set (IVIFS) theories as proposed by Atanassov [4], Atanassov and Gargov [3] respectively, which takes the degree of membership as well as non-membership simultaneously are widely used in the decision making fields. During the last decades, the researchers were paying more attention to these theories and successfully applied it to the various situations in the decision-making process. The two important aspects of solving the MAGDM problem are (i) to design an 446 HARISH GARG AND KAMAL KUMAR appropriate function which aggregates the different preference of the decision makers into the collective ones, (ii) to design appropriate measures to rank the alternatives. For the former part, an aggregation operator is an important part of the decision making which usually takes the form of mathematical function to aggregate all the individual input data onto a single one. Over the last decade, numerable attempts have been made by different researchers in processing the information values using different aggregation operators under IFS and IVIFS environments [1,2,46,13,14,38,18,17,8,36,10,41,28,27,26,32,34].
In the aspect of ranking the numbers, Xu and Da [47] defined the possibility degree measure method for ranking the interval numbers. Garg [9] presented a generalized improved score function of ranking the IVIFSs. Zhang et al. [52] presented possibility degree measure method to rank the different interval-valued intuitionistic fuzzy numbers (IVIFNs). Wan and Dong [35] proposes a new ranking method of interval-valued IFNs based on the possibility degree measures from the probability viewpoint and their corresponding decision making method. Garg and Kumar [19] presented an improved possibility degree measure method for ranking the intuitionistic fuzzy numbers. The overview of the different approaches related to possibility degree measures under these existing theories are summarized in [6,7].
Since all the above existing theories deal with the uncertain information only by quantitative aspects. But in real-life problems, there are many attribute values which are qualitative in nature and cannot be expressed by a numeric value. In such cases, it is easy to describe the preference values as a linguistic variable. For this, Zadeh [50] proposed the concept of a linguistic variable (LV) to represent the rating values towards the object. For instance, in order to measure the performance of a student in an academic year, he may use some of the linguistic terms such as "excellent", "good", "average", "poor". However, due to the complexity of decision environment and the subjective nature of human thinking, sometimes it is very difficult to express the membership and non-membership degrees by numeric values and is possible to be expressed as LVs. Linguistic approaches [24,23,40] provide us more degree of freedom to analyze the imprecise and vague information. In that direction, Xu [43] proposed some linguistic weighted geometric averaging operators while Xu [39] presented a decision-making approach based on possibility degree measures under the uncertain linguistic information. Later on, Zhang [51] introduced a new concept of the linguistic intuitionistic fuzzy set (LIFS) in which the membership degrees are expressed by the LVs. Chen et al. [5] defined averaging and geometric aggregation operators for aggregating LIFNs information and presented a MAGDM approach. Liu and Wang [30] defined some improved operational laws for LIFNs and aggregation operators based on it. Garg and Kumar [20] presented some aggregation operators for LIFSs by using set pair analysis theory. Liu and Qin [29] presented power averaging operator for aggregating the LIFNs and MAGDM method based on it. Peng et al. [31] defined Frank operations and aggregating operator based on it. Xian et al. [37] presented a new hybrid aggregation operator and decision-making approach based on it. Recently, Garg [12], Garg and Nancy [21] presented some new linguistic decision making approaches for solving the decision making problems under different environments.
The above theories have been successfully applied, but sometimes due to the complex fuzzy information, decision-makers cannot provide their opinion by LIFNs in terms of single-valued linguistic terms. Therefore to provide more degree of freedom from decision-makers to express their opinion and better dealing with the complex fuzzy information, in this article motivated by IVIFS, we have extended the idea of LIFN to the linguistic interval-valued intuitionistic fuzzy number (LIVIFN) by expressing membership and non-membership grades in form of interval-valued linguistic terms. For developing any other theory about the field of the multi-attribute DM, information aggregating techniques and ranking methods have played an important roles. For it, an improved possibility degree measure of comparison between two different LIVIFNs is defined by using the notion of the 2-dimensional random vector, and a new method is then developed to rank the numbers. Afterward, for solving the MAGDM problems, we need some aggregating techniques to collect the information of different attributes. For it, we proposed some geometric and ordered weighted geometric aggregation operators under the LIVIFS environment where preferences are expressed in terms of LIVIFNs. The various desirable properties of the proposed operators are investigated. Therefore under the LIVIFS environment, the objective of this paper is divided into the four parts: (i) to propose the concept of the LIVIFS, by combining the features of linguistic variables and IVIFS, in which the importance of each object is expressed in the form of the interval-valued linguistic membership and non-membership terms. (ii) to introduce a new possibility degree measure for ranking the different LIV-IFNs and to propose two new weighted geometric aggregation operators namely, LIVIF weighted geometric (LIVIFWG) and LIVIF ordered weighted geometric (LIVIFOWG) of the different LIVIFNs. (iii) to establish a MAGDM approach based on these proposed operator and possibility degree measure method. (iv) to illustrate the developed approach with a numerical example and validate it with some validity test criteria. The remainder of this paper is formulated as follows. In Section 2, we briefly describe the concepts of IVIFS and the linguistic variable. In Section 3, we defined the concept of the linguistic interval-valued intuitionistic fuzzy set and a new possibility degree measure method for ranking different LIVIFSs. In Section 4, we present some new weighted geometric and ordered weighted aggregation operators for aggregating the collections of the LIVIFNs. Various desirable properties of these operators are also investigated. In section 5, a group decision making approach to solve the MAGDM problems is presented based on the proposed operators and possibility degree measure method. Section 6 deals with an illustrative example to show the effectiveness and feasibility of the approach. Finally, section 7 concludes the paper.

2.
Preliminaries. In this section, we briefly review some basic concepts of the IVIFSs and linguistic set approaches.
Definition 2.1. [3,44] Let X be the nonempty finite universal set, an IVIFS A in X is defined as where, for each x, 1], and represents membership and nonmembership grades of x to A respectively, such that 0 is called an IVIF number (IVIFN). Sometimes, a decision maker may give his preferences in terms of a linguistic number than a numerical number. For it, a linguistic term set (LTS) is defined as: [24] Let S = s t | t = 0, 1, 2, . . . , h be a finite odd cardinality LTS, where s t represents a possible value for a linguistic variable and h is the positive integer. Each linguistic term s t must have the following characteristics.
(i) The set is ordered: Later on, Xu [43] extended the discrete LTS to a continuous LTS as h] . Addition and multiplication operation laws for linguistic variables based on the t-norm and t-conorm are defined as follows: where, ∀ x, s τ , s θ ∈ S [0,h] represent the linguistic membership and nonmembership degrees of x to A respectively, such that 0 ≤ τ + θ ≤ h holds. Usually, the pair (s τ , s θ ) is called a linguistic intuitionistic fuzzy number (LIFN).
3. Linguistic interval-valued intuitionistic fuzzy set. In this section, we propose the concept of the linguistic interval-valued intuitionistic fuzzy set (LIVIFS) by extending it from the linguistic intuitionistic fuzzy set. The concept of the LIVIFS is given as follows.
where [s τ , s η ] and [s θ , s υ ] are all subsets of [s 0 , s h ] and represent the linguistic membership and nonmembership degrees of x to A respectively. For any x ∈ X, s η(x) + s υ(x) ≤ s h (i.e., η + υ ≤ h) is always satisfied, and in turn, the linguistic intuitionistic index of x to A is defined as Motivated from the idea as presented by Xu and Da [46], we proposed a new possibility degree measure to compare any two LIVIFNs under LIVIFS environment as follows.
Further, in order to rank the different LIVIFNs, the inclusion-comparison probability of LIVIFNs γ k γ t ; k, t ∈ {1, 2, . . . , n} is denoted by p(γ k γ t ) and their corresponding likelihood possibility degree measure matrix is denoted by P = (p kt ) n×n where p kt = p(γ k γ t ); (k, t = 1, 2, . . . , n) given by [45] Now, the ranking value which represents the optimal degrees of the membership for the numbers γ k (k = 1, 2, . . . , n) is given as follows: Then, the ranking order of all alternatives γ k , (k = 1, 2, . . . , n) is found according to decreasing order of the values of r k 's and hence choose the best alternative.
) be four LIVIFNs defined over the continuous linguistic term set S [0,8] . In order to rank these given numbers by using our proposed possibility degree measure, we construct the possibility degree matrix P = (p kt ) 4×4 by using Eq. (4)  Based on this matrix, the optimal membership degrees of the numbers are computed by using Eq. (7) and get r 1 = 1.0833, r 2 = 0.4167, r 3 = 1.8611 and r 4 = 2.3056. From these values, it is observed that r 4 > r 3 > r 1 > r 2 and thus the ranking order of the given LIVIFNs is γ 4 γ 3 γ 1 γ 2 , where " " refers to "preferred to".

4.
Aggregating operators for LIVIFNs. In this section, we define some new geometric aggregation operators for LIVIFNs.
4.1. Some operational laws. Motivated by t-norm and t-conorm, we propose the following operations laws for LIVIFNs.
Proof. Here, we shall prove only γ 1 ⊕ γ 2 is LIVIFN, while other can be proven similar.
For two LIVIFNs Similarly, we have On the other hand, we have Hence, γ 1 ⊕ γ 2 is a LIVIFN.
Here, we prove the parts (i)-(iii), while others can be proven similarly.
(i) According to Definition 4.1, we have (ii) For a real number λ > 0, we have HARISH GARG AND KAMAL KUMAR Therefore, (iii) For positive real numbers λ 1 , λ 2 and LIVIFN γ, we get Weighted geometric operator. In this section, motivated from the idea of geometric operators as presented by Xu and Yager [48], we define LIVIF weighted geometric aggregation operator for a collection of LIVIFNs denoted by Ω.
Definition 4.4. Let γ t (t = 1, 2, . . . , n) be a collection of LIVIFNs. A LIVIFWG operator is a mapping LIVIFWG : Ω n → Ω defined as where ω = (ω 1 , ω 2 , . . . , ω n ) T be the weight vector of γ t such that ω t > 0, LIVIFNs, and ω t be the weight vector of γ t such that ω t > 0, n t=1 ω t = 1, then the aggregated value by using LIVIFWG operator is also a LIVIFN, and given by Proof. The first result holds immediately from Theorem 4.2. Now to prove Eq. (9), we use the principle of mathematical induction on n.
For n = 2 and by Definition 4.1, we have Therefore, .
Hence, Eq. (9) hold for n = 2. Secondly, assume that the Eq. (9) hold for n = k, that is Now, for n = k + 1, by Definition 4.1, we have which is true for n = k + 1. Hence, by the principle of mathematical induction, result given in Eq. (9) hold for positive integer n, which completes the proof of the theorem.
The following example illustrate the working of the proposed geometric aggregation operator.
Similarly, we have On the other hand, for non-membership part, we have Similarly, Hence, according to Definition 3.2, we obtain (P3) The monotonicity of the LIVIFWG operator can be obtained by similar proving method.

4.3.
Ordered weighted geometric operator. In this section, we define ordered weighted LIVIF geometric aggregation operator for a collection of LIVIFNs denoted by Ω.
Definition 4.7. Let γ t (t = 1, 2, . . . , n) be a collection of LIVIFNs, then a linguistic interval-valued intuitionistic fuzzy ordered weighted geometric (LIVIFOWG) operator is a mapping LIVIFOWG : Ω n → Ω defined as ) is the t th largest value of the γ t and w = (w 1 , w 2 , . . . , w n ) T be the weight vector of LIVIFOWG operator with w t > 0, n t=1 w t = 1. . . , n) by using LIVIFOWG operator is also a LIVIFN, and is given by Proof. The proof of this theorem is similar to Theorem 4.5, so we omit here. From these values, it is seen that r 2 > r 1 > r 3 and thus ordering of the given numbers is γ 2 γ 1 γ 3 . Therefore, γ σ(1) = γ 2 , γ σ(2) = γ 1 and γ σ (3) = γ 3 . Now, based on these ordering numbers and by using the proposed LIVIFOWG operator we get Then, in the following, we develop a method based on the proposed operators and the possibility degree measure to solve the decision making problems with LIVIFS information, which involves the following steps.
Step 1. Arrange the collective information of the alternatives given by the decision makers in the form of the decision matrices R (q) = γ (q) kt m×n (q = 1, 2, . . . , l) as Step 2. Normalize the collective information, if required, by converting the cost type attributes into the benefit type by using Eq. (12) to balance the physical dimensions of the rating values as kt ]) ; for benefit type attribute kt ]) ; for cost type attribute and get the normalized decision matrices R (q) = (γ (q) kt ) m×n .
Step 6. The ranking value which represents the optimal degree of the membership for alternative A k (k = 1, 2, . . . , m) is computed by using Thus, based on these membership values, the ranking order of all alternatives is found according to decreasing order of the values of r k (k = 1, 2, . . . , m)'s and hence choose the best alternative.
6. Illustrative example. The above mentioned approach is illustrated with a numerical example under the LIVIFNs environment which are stated below. Consider a problem of a pharmaceutical company which wants to select a lab technician for the micro-bio laboratory. For this, company published notification in the newspaper and consider the four attributes required for technician selection, namely, academic record (G 1 ), personal interview evaluation (G 2 ), experience (G 3 ) and technical capability (G 4 ). The relative importance of these attributes is taken in the form of weight ω = (ω 1 , ω 2 , ω 3 , ω 4 ). On the basis of the notification conditions, the four candidates A 1 , A 2 , A 3 and A 4 are shortlisted for the interview and considered as alternatives. Then, the main object of the company is to choose the best candidate among them for the task. For it, a panel of three experts D (1) ("Director"), D (2) ("Head of the Department"), D (3) ("Human resources manager") are invited to evaluate the given alternatives under each attribute according to linguistic variables defined as s 0 = "extremely poor", s 1 = "very poor", s 2 = "poor", s 3 = "slightly poor", s 4 = "fair", s 5 = "slightly good", s 6 = "good", s 7 = "very good", s 8 = "extremely good". In order to fulfill it, they have evaluated these and give their preferences in the term of LIVIFNs. 6.1. By proposed approach. The following steps of the proposed approach are executed to find the best alternative(s) for the required post.
Step 1. The rating values of each expert on the alternatives under set of the attributes are summarized in Tables 1, 2 and 3 in the form of LIVIFNs.
Step 2. Since all attributes are of the same types, so there is no need of the normalization process.
Step 3. Aggregate the preference of each alternative A k (k = 1, 2, 3, 4) by using LIVIFWG operator defined in Eq. (12). The result corresponding to it is summarized in Table 4.
Step 4. If we assign the weights vector w = (0.25, 0.55, 0.20) T of decision maker's then collective overall performance γ k corresponding to each alternative A k    Step 6. The optimal value r k of the membership degree of the alternatives A k (k = 1, 2, 3, 4) can be computed by using Eq. (16) and get r 1 = 1.7058, r 2 = 1.1829, r 3 = 1.8623 and r 4 = 0.9157. Since r 3 > r 1 > r 2 > r 4 , therefore ranking order of the alternatives is A 3 A 1 A 2 A 4 . Thus, the best alternative for the required task is A 3 .
6.2. Validity test. Wang and Triantaphyllou [36] established the following testing criterions to evaluate the validity of MAGDM methods. Test criterion 1: "An effective MAGDM method does not change the index of the best alternative by replacing a non optimal alternative with a worse alternative without shifting the corresponding importance of every decision attribute". Test criterion 2: "To an effective MAGDM method must satisfy transitive property". Test criterion 3: "If we decomposed a MAGDM problem into the smaller DM problems by deleting some of the alternatives and same MAGDM method is utilized on these problems to rank alternatives, collective ranking of alternatives must be identical to ranking of un-decomposed DM problem".
Validity of the proposed approach is tested by using these criteria as follows: 6.2.1. Validity test by criterion 1. For testing the validity of proposed MAGDM approach under the test criterion 1, we replace the non optimal alternative A 4 with the worse alternative A 4 in original decision matrices for each decision maker D (q) , q = 1, 2, 3. The rating values of A 4 is chosen as an arbitrary and summarized in Table  5. Then, by applying the proposed MAGDM approach to transform data we get the A 1 A 4 and A 3 A 2 A 4 respectively. After combining together the ranking of the alternatives of these smaller problem, we get the final ranking order as A 3 A 1 A 2 A 4 which is same as un-decomposed DM problem and shows transitive property. Hence, the proposed MAGDM approach is valid under the test criteria 2 and 3.
6.3. Further discussion. In the following we give some characteristics comparison of our proposed method and the aforementioned methods, which are listed in Table 6. The method proposed by Xu and Yager [48] adopts IFNs to aggregate the uncertain information using geometric operators only by quantitative aspects. On the other hand, the method described by the author in Xu [44] represent the Whether flexibly to Whether describe Whether describe Whether have the express a wider range information using information by characteristic of of information linguistic features interval-valued numbers generalization Xu and Yager [48] no no no no Xu [44] yes no yes no Zhang [51] no yes no yes The proposed method yes yes yes yes wider range of the information in terms of the interval-valued membership degrees. But their approach is also limited to only quantitative aspects and does not apply the linguistic information. Apart from these, the method proposed by Zhang [51] adopts LIFNs to describe the uncertainties in the data as a crisp number. However, in the present study, we proposed the LIVIFNs to describe the uncertainties in terms of linguistic interval pairs of the membership degrees which can easily express the information in a more semantics and concise way and hence can reduce the information loss.
In addition, LIVIFNs used in the new method can model the uncertain and fuzzy information more flexible by its linguistic interval-valued intuitionistic fuzzy numbers during the evaluation process, which can reflect the inherent thoughts of decision makers more accurately. Further, it has been analyzed that the operators defined by the author in [51] can be considered as a special case of the proposed operator by setting the lower and upper bound of membership degrees are equal. Thus, the proposed aggregation operators are more generalized and capture the more information during the analysis.

7.
Conclusion. LIVIFS is the generalization of LIFS in which membership and nonmembership degree represented by the interval-valued linguistic terms for better dealing with fuzzy information under the qualitative aspect. In this paper, we have presented a MAGDM approach under the LIVIFS environment. For it, firstly, we defined possibility degree measure method to compare the LIVIFNs along with some properties of it. Afterward, we proposed some weighted and ordered weighted geometric aggregation operators for the collection of the different LIVIF information. Thee desirable properties, namely, idempotency, monotonicity, and boundedness are investigated in details. Finally, a real-life case has been discussed to describe the decision step and illustrate the feasibility of the proposed approach. From the results of the validity test and numerical example, we have concluded that the proposed approach solve the real-life problem effectively. The special advantage of the proposed approach is that possibility degree ranking method reflects the uncertainty of the information and provide the more reasonable results. In the future, the result of this paper can be extended to some other uncertain and fuzzy environments [11,16,22,33].