Hesitant fuzzy soft set (HFSS) is considered as a powerful tool to capture uncertain information in group decision-making process. In this communication, we propose generalized correlation coefficient for hesitant fuzzy sets (HFS) and further extend for HFSS. Using generalized correlation coefficient of HFSS, we define generalized correlation efficiency which shows the significance of the HFSS. We also propose an algorithm to apply generalized correlation coefficient in group decision-making (GDM) problem, where information is presented in the hesitant fuzzy soft environment. Finally, with the help of an example, we show the effectiveness of the proposed algorithm which uses the concept of generalized correlation efficiency.
Correlation indicates the degree of relationship between two variables and plays the vital role in many real life problems such as decision making, predicting market behavior, medical diagnosis, pattern recognition, etc. In intuitionistic fuzzy environments, Gerstenkorn and Manko [15] defined the correlation and correlation coefficient of intuitionistic fuzzy sets. Hung and Wu [18] used the concept of “expected value” to define the correlation coefficient of fuzzy numbers, which lies in [-1,1]. Hung [19] and Mitchell [8] defined the correlation coefficient of intuitionistic fuzzy sets by considering an intuitionistic fuzzy set as an ensemble of ordinary fuzzy sets. The correlation coefficient of intuitionistic fuzzy sets proposed by Hung and Wu [20] tells us the strength of the relationship between intuitionistic fuzzy sets along with positivity and negativity of the relationship. Xu [32] proposed a detailed survey on association analysis of intuitionistic fuzzy sets. Xu et al. [34] utilized a set-theoretic approach [33] to derive the association coefficients of intuitionistic fuzzy sets taking into account all the three terms (the membership degree, nonmembership degree, and hesitation margin) describing an intuitionistic fuzzy set. Szmidt and Kacprzyk [5] discussed a concept of correlation for the data represented as intuitionistic fuzzy set adopting the concepts from statistics, proposed a formula for measuring the correlation coefficient (lying in [-1,1]) of intuitionistic fuzzy sets and showed the importance to take into account membership, non-membership and hesitancy describing intuitionistic fuzzy sets.
Torra and Narukawa [16] introduced the notion of the hesitant fuzzy set as an extension to fuzzy set. They further analysed some interesting features of HFS which are not reflected by other classical extensions of fuzzt set (IVFS, IFs, Type-2 fuzzy set, etc.). Consider the case of a company, the experts decide the degree that an alternative A satisfies an attribute x; one expert assign 0.3, other assign 0.2, some assign 0.5. In fact, no unanimity is reached among these experts. In this situation, the hesitant fuzzy set provides the satisfactory degree in the form of a hesitant fuzzy element {0.3,0.2,0.5}. Thus a HFS can incorporate the opinions of all the experts and, provides a structural description pertaining to the distinct views of the group of experts. Therefore, HFS is considered as an effective tool to handle various problems related to GDM.
Xia and Xu [9] studied the aggregation operators of hesitant fuzzy sets and applied them to decision making problems. Xu and Xia [35] discussed the distance measure for hesitant fuzzy sets. Chen et al. [10] introduced some correlation coefficients formulas for hesitant fuzzy sets (HFSs) and interval valued HFSs (IVHFSs) and demonstrated their application to cluster analysis. While defining correlation coefficients of HFSs and IVHFSs, the authors considered that both HFSs and IVHFSs have the same length and their values are arranged in ascending order. When the HFSs and IVHFSs do not have the same length, the shorter one is extended up to the length of longer one. For doing so, there are two methods namely pessimistic principle in which smallest element will be added and optimistic principle in which greatest element will be added.
Recently, Dong et al. [22] proposed a novel distance based approach to determine the difference between two hesitant linguistic term sets (HFLTSs). They have also proposed a two-stage model based on optimization to obtain the optimal adjusted individual opinions in the consensus reaching process. Li et al. [1] introduced a personalized individual semantics approach to formulate and solve linguistic GDM by means of numerical scales [23] and 2-tuple linguistic model [6] to improve the management of different meanings of words of different people. Li et al. [3] applied the notion of personalized individual semantics to capture the different understanding of words for different decision makers in hesitant linguistic framework.
Due to adaptability in fuzzy techniques for a particular type of real life problem, we can not have a single model which provide solution to all problems of that type. For example, to deal with problems of pattern recognition various similarity [12] and dissimilarity measures [13] are available in the literature. But it is observed that if one of the similarity/dissimilarity measure solve one problem then it may not solve the other problem of same type due to some counter intuitive situations. Thus, in soft computing techniques for a class of problems (pattern recognition, decision making, computer vision, etc.), a new model is always desirable. A correlation measure is also a similarity measure. Xu and Xia [36] provided correlation measure between two hesitant fuzzy sets and explain its application in pattern recognition problems. Das et al. [14] extended one of the correlation coefficients [36] for HFSs and IVHFSs and derived the correlation efficiency to show the significance of the HFSS and interval-valued hesitant fuzzy soft set (IVHFSS).Due to the basic characteristics of soft computing, the correlation measure [36] and correlation measure for hesitant fuzzy soft set [14] may not be sensible to recognise similarity patterns and strict ranking respectively in certain problems.
To resolve such problems, we got motivation to propose one parametric generalization of correlation coefficients [36] and [14]. The parameter in these models may be considered as the sensitivity parameter to recognize the changes due to adaptability. The main contribution of this work is as follows:
One parametric generalization of correlation coefficient of HFSs [36] and its application in pattern recognition.
One parametric generalization of correlation coefficient of HFSSs [14] and its application in group decision-making.
Demonstration of usefulness of sensitivity parameter while applying the proposed generalized correlation coefficients in pattern recognition and GDM.
The remainder of the paper is organized as follows. Section 2 presents some basic concepts related to HFSs, HFSSs and correlation measures of HFSs. In Section 3 we introduce generalized correlation coefficient of HFS. Application of generalized correlation coefficient of HFSs in pattern recognition is shown in Section 4 through an illustrative example. Section 5 firstly defines correlation efficiency and generalized correlation efficiency for HFSSs. Then we discuss a decision-making algorithm based on HFSSs, generalized correlation measures, generalized correlation efficiency, and hesitant fuzzy-ordered weighted averaging (HFOWA) operator in Section 6. Section 7 presents the illustration of the effectiveness of the proposed algorithm with the help of a numerical example. In Section 8 we provide discussion and implications of the proposal. Finally, Section 9 summarizes this study and presents scope for future work.
Preliminaries
In this section some concepts related to HFS, HFSS, hesitant fuzzy soft matrix (HFSM) and information energy, correlation measure and correlation coefficients of HFS are given.
HFS, HFSS, and HFSM
Definition 2.1. [9] Let X be a fixed set, then a HFS N on X is given as a function fN (x) that returns a subset of [0,1] when applied to X. Mathematically, it can be represented as
where fN (x) is a set of some values in [0,1] denoting the possible membership degrees of the element x ∈ X to the set N. For convienence, Xia and Xu [9] called fN (x) as an HFE and N the set of all HFEs.
For given three HFEs, f, f1, f2, Torra and Narukawa [16] and Torra [17] defined some operations which are given below:
fc (x) = ⋃ δ∈f(x) {1 - δ},
(f1 ∪ f2) (x) = ⋃ δ1∈f1(x),δ2∈f2(x) max {δ1, δ2},
(f1 ∩ f2) (x) = ⋃ δ1∈f1(x),δ2∈f2(x) min {δ1, δ2}.
Also some operations are defined by Xia and Xu [9] on the HFEs f, f1, f2 which are given below:
Definition 2.2. [9] The score function s (f), for a HFE f is given by
where # f denotes the number of elements in f. Let f1 and f2 be two HFEs then f1 > f2 if s (f1) > s (f2); f1 = f2 if s (f1) = s (f2). Since s(f) finds the average value of every elements in f. So it gives the average of all decision maker. Higher the average value, higher will be the score which results in a better HFE. But this method of comparison fails to distinguish between two HFEs in some special cases.
Xia and Xu [9] defined some aggregation operators for HFEs by making use of HFEs for the purpose of decision making. Two of these aggregation operators are defined in Definition 2.3 and 2.4.
Definition 2.3. [9] A mapping Fn → F is said to be HFWA operator if
HWFA operator is reduced to hestant fuzzy averaging (HFA) operator, when w = .
Definition 2.4 [9] A mapping HFOWA: Fn → F is said to be HFOWA operator if
where fσ(j) is the jth largest HFE among collection of HFEs fj, j = 1, 2, … n and σ (1), σ (2), …, σ (n) is permutation of 1, 2, …, n such that fσ(j-1) ≥ fσ(j).
Definition 2.5. [11] Let X be a fixed set and HF(X) be the set of all HFSs of X. Let A be a set of parameters such that ⊆A. A HFSS over X is given by , where is a mapping given by . Let e be any parameter then is taken as HFS, which is given by
where fN (x) is a set of some values in [0,1].
Definition 2.6. [11] HFSS is given by HFSM. Let X = {x1, x2, …, xm} and A = {e1, e2, …, en} then HFSM is given by G = (gij) m×n where , i = 1, 2, …, m, j = 1, 2, …, n. Also k > 0 is the number of values which may be different for different HFEs.
Information energy, correlation measure and correlation coefficient of HFS
Let X = {x1, x2, …, xm} be universe of discourse. Let HF(X) be set of all HFSs of X. Let A = {(xi, fA (xi)), i = 1, 2, …, m} and B={(xi, fB (xi)), i = 1, 2, …, m}. Since the membership values of an HFE are not usually found in any particular order Chen et al. arranged the membership values in decreasing order for the sake of convenience. For HFE f, let σ:(1,2,...,n)→(1,2,...,n) be a permutation that satisfy fσ(i) ≥ fσ(i+1), i = 1, 2, …, n - 1, such that fσ(i) be the ith largest value in f. Since the number of values in different HFEs are normally different, to compute the correlation coefficients between two HFSs, the number of values in the corresponding HFEs are considered to be equal. Let l1 = max {l (fA (xi, ek))}, {l (fB (xi, ek))}, ∀xi ∈ X, where l (fA (xi)) and l (fB (xi)), respectively, represent the number of values in HFEs fA (xi) and fB (xi). When l (fA (xi)) ≠ l (fB (xi)), the HFE which has less number of values, some values are added to make it same. In this work, if l (fA (xi)) < l (fB (xi)), then fA (xi) is extended by adding its maximum value until it has the same length as fB (xi).
Definition 2.7. [10] Let A = {(xi, fA (xi)) |xiɛX, i = 1, 2, 3, …, m} be the HFS, then the information energy of the set A is given by
Definition 2.8. [10] Let A and B be two HFSs, then the correlation between A and B is given by
Let ρHFS (A, B) be a correlation coefficient of two HFSs A and B then
Any measure satisfying these properties is correlation coefficient.
Xu and Xia [36] defined five correlation coefficients for the hesitant fuzzy sets and obtained some relationship among these correlation coefficients. In the next section, we propose one parametric generalization of one of the correlation coefficients of Xu and Xia [36].
Generalized correlation coefficients for the hesitant fuzzy sets
Xu and Xia [36] defined five correlation coefficients for HFSs. We propose one parametric generalization of one of these five correlation coefficients for the hesitant fuzzy sets as follows:
In above defined correlation coefficients if we put α = 2, we obtain correlation coefficients defined by Xu and Xia [36].
Next, we show that above defined measure satisfies the properties CC1-CC3 of correlation coefficient.
Theorem 3.1. satisfies properties CC1-CC3.
Proof 1.
We have
Therefore, = .
≥ 0 is obvious. We only prove . Now
Now, using the Cauchy-Schwarz inequality
where are non-negative real values. So we get
Therefore,
Thus,.
We have
Now for A = B,we have
Therefore, if A=B.
Thus, is valid correlation coefficient.
Application in pattern recognition
Problem Formulation: Let E1, E2, …, En be some unknown patterns characterized by HFS in the universal set G = {z1, z2, …, zk}as follows: Ei = {〈zj, fN (zj) 〉|zj ∈ G}, j = 1, 2, …, k. Let F = {〈zj, fNF (zj) 〉|zj ∈ G}, j = 1, 2, …, k be a known pattern. The problem is to classify patterns Ei into the known patterns F.
We can solve this problem using correlation measure approach.
Correlation measure approach: Let = association of pattern F with Ei*. Then F is assigned to Ei*. .
Following example illustrates the above mentioned application of correlation coefficient in pattern recognition.
Example 1. Let
be unknown patterns and
be a known pattern.
For α = 2, Equation (2) reduces to correlation coefficient [36]. and . For α = 1.7, and . In the above example, we observe for α = 2, is unable to classify Ei to F. But for α = 1.7, ρHFS (Ei, F) (i = 1, 2) classify E2 to F. Hence, α is a sensitivity parameter which help to solve problem of this type.
In the next section, we introduce generalized correlation measure and correlation coefficient for HFSS corresponding to generalized correlation measure for HFS defined in Equation (1).
Generalized correlation measure and correlation coefficient for the HFSSs
Generalized correlation coefficient (Equation (2)) defined in Section 3 can be extended to HFSSs.
In this section, we introduce the concept of generalized correlation coefficient and generalized correlation efficiency based on HFSSs. We have generalized the correlation coefficients defined by Xu and Xia [36] in the framework of HFSs and further extend the same for HFSSs. Let and be two HFSSs defined on the set of elements {x1, x2, …, xm} and parameters {e1, e2, …, en}. fAσ(j) (xi, ek), j = 1, 2, … li be the ordered (decreasing) set of membership values in the HFE f (xi, ek), i=1,2,...,m, k=1,2,...,n. Here li = where and , respectively represent the number of values in HFEs and .
First, we present extension of generalized correlation coefficient (Equation (2)) to HFSS.
Definition 5.1. Let be two hesitant fuzzy soft sets, then the generalized correlation between two HFSSs is given by
Definition 5.2. Let be hesitant fuzzy soft set, then the generalized information energy for HFSS is defined as
Definition 5.3. Let and be two hesitant fuzzy soft sets, then the generalized correlation coefficient between two HFSSs and is defined as
Theorem 5.1. For any two HFSSs and , the generalized correlation coefficient satisfies the following properties:
= ,
0 ≤ ≤ 1,
= 1 if = .
Proof.
This proof is straightforward.
It is obvious that ≥ 0, we prove only for ≤ 1. Now
Now
Now, using the Cauchy-Schwarz inequality
where , are non-negative real values. So we get
Therefore,
Hence,
Thus
So, .
The proof is straightforward.
We propose the correlation coefficient for more realistic situations and an algorithm for GDM.
Decision making based on Generalized correlation of HFSSs
On the basis of generalized correlation coefficient formula that we propose for HFSS, we provide an algorithm for the hesitant fuzzy soft environment. Generalized correlation efficiency and generalized normalized efficiency are also defined in this section which is required for the algorithm.
Let be the set of alternatives and A = {A1, A2, …, An} be the set of attributes/criteria, and w = (w1, w2, …, wn) be the weight vector of the attributes Aj (j = 1, 2, …, n), where wj > 0 and. Let C = {C1, C2, …, Ct} be the set of decision makers. By using, HFSS = , decision makers Ck, k = 1, 2, …, t give their opinions. Let w = {w1, w2, …, wt} be the generalized normalized correlation efficiency of HFSS , k = 1,2,...,t, where Wk > 0 and . For alternatives Ai (i = 1, 2, …, m) the attribute values of a decision maker Ck are represented by the hesitant fuzzy soft matrix = , where = , Ai ∈ A, i = 1, 2, …, m, j = 1, 2, …, n.
Generalized correlation efficiency and generalized normalized efficiency:
Definition 6.1. For each HFSS , the generalized correlation efficiency is given by
Definition 6.2. For each HFSS , the generalized normalized efficiency is given by
Algorithm for GDM in HFSS environment
Das et al. [14] provided an algorithm for group decision making in hesitant fuzzy soft environment. To illustrate the efficiency of our proposed generalized correlation coefficient, we do not wish to provide a new GDM algorithm but utilize the Das et al. [14] algorithm using one parametric measures.
Algorithm
Step 1 A group of decision makers C1, C2, …, Ck, k = 1, 2, …, t give their opinions in terms of HFSS .
Step 2 For each pair of hesitant fuzzy soft set, generalized correlation coefficient is calculated by using Definition 5.3.
Step 3 For each pair of hesitant fuzzy soft set, generalized correlation efficiency and generalized normalized correlation efficiency are calculated by using Definition 6.1 and Definition 6.2.
Step 4 By using generalized normalized correlation efficiency, given by decision makers Ck, (k = 1, 2, …, t) are aggregated into a collective decision matrix given by k = 1, 2, …, t, which is used as the weight vector w =
{w1, w2, …, wt} of decision makers. (rij) m×n is calculated by making use of HFOWA operator which is given in Definition 2.4.
Step 5 HFEs fi (i = 1, 2, …, m), in collective decision matrix, for alternatives are obtained by making use of hesitant fuzzy weighted averaging(HFWA) operator, given in Definition 2.3.
Step 6 The score values s (fi) of fi, i = 1, 2, …, m are calculated by using Definition 2.2.
Step 7 We get the priorities of the alternatives Ai, i = 1, 2, 3, …, m by ranking s (fi) and select the best one.
For α = 2, the above algorithm become identical to Das et al. algorithm.
Numerical Illustration
In this section, we implement the algorithm discussed in subsection 6.2.
Example 2. Suppose , be the set of four universities and A = {A1, A2, A3, A4}, be the set of parameters (attributes) on the basis of which the universities are to be evaluated.
Let A1: Research output, A2: Placements, A3: Infrastructure and A4: Faculty-student ratio; be the parameters for evaluation of universities. The criteria for ranking universities forms a panel of four experts Ck, (k = 1, 2, 3, 4) to provide the potential information about four alternatives, with respect to attributes A = {A1, A2, A3, A4} by HFSMs = listed in Table 2. Generalized correlation measure for every pair of HFSSs i.e. , k = 1, 2, …, t, l = 1, 2, …, t, are given in Table 3. For α = 2, (it is informational energy correlations). In Table 4, we calculate correlation coefficients for each pair of HFSSs . For α = 2, the algorithm proposed in Das et al. [14] becomes a special case of algorithm in subsection 6.2.
Generalized correlation efficiency and generalized normalized correlation efficiency are obtained in Table 4. In Table 5, we by making use of HFOWA operator and weight vector w = {w1, w2, w3, w4} = {0.2471,0.24104, 0.25558,0.2563}, we compute the collective decision matrix. These weights are computed using generalized normalized correlation efficiency. Now, by using HFWA operator with the attribute weight w = {w1, w2, w3, w4} = {0.4,0.3,0.2,0.1}, each of the alternatives given in Table 5, are aggregated for obtaining aggregated HFEs for each of the alternatives , which are = (0.6,0.5,0.3,0.2), =(0.5,0.5,0.4,0.2), = (0.6,0.6,0.4,0.2) and = (0.6,0.5,0.3,0.2) as shown in Table 6. Then we calculate the score values (i = 1,2,3,4) of those HFEs, which are =0.4, = 0.4, = 0.45, = 0.4 and shown in Table 6. Since , so .
HFSMs for ,A), i=1,2,3,4
A1
A2
A3
A4
A1
A2
A3
A4
(0.2,0.5,0.3)
(0.3,0.1,0.5,0.7)
(0.7,0.2,0.6)
(0.4,0.2)
(0.4,0.3,0.1,0.2)
(0.5,0.6,0.2)
(0.4,0.2,0.7,0.2)
(0.4,0.3,0.1)
(0.5,0.3,0.7,0.9)
(0.4,0.1,0.6)
(0.4,0.1,0.4)
(0.6,0.3,0.7,0.1)
(0.4,0.5)
(0.3,0.5,0.6,0.1)
(0.4,0.6,0.1)
(0.3,0.6)
(0.4,0.2,0.1,0.6)
(0.8,0.3,0.5)
(0.5,0.6,0.4,0.7)
(0.5,0.6,0.2)
(0.8,0.4,0.8)
(0.4,0.2,0.3)
(0.7,0.2,0.4)
(0.3,0.5,0.2,0.7)
(0.5,0.4)
(0.1,0.6,0.3,0.9
(0.5,0.7)
(0.4,0.7,0.3)
(0.2,0.5,0.1,0.6)
(0.5,0.7,0.1,0.1)
(0.4,0.2,0.7)
(0.3,0.1,0.5)
A1
A2
A3
A4
A1
A2
A3
A4
(0.6,0.5)
(0.4,0.1,0.7)
(0.5,0.4,0.2,0.1)
(0.8,0.6)
(0.5,0.4,0.2,0.5)
(0.3,0.6,0.1)
(0.4,0.5,0.3)
(0.8,0.9)
(0.6,0.4,0.3,0.2)
(0.3,0.1)
(0.5,0.4,0.1)
(0.7,0.6,0.5)
(0.5,0.3)
(0.3,0.5,0.1,0.3)
(0.4,0.5,0.2,0.1)
(0.8,0.7,0.5)
(0.6,0.2,0.3)
(0.3,0.1,0.6,0.5)
(0.5,0.3)
(0.7,0.6,0.4,0.3)
(0.5,0.3,0.1)
(0.4,0.5)
(0.5,0.3)
(0.7,0.7,0.5,0.4)
(0.6,0.3,0.1)
(0.1,0.2)
(0.5,0.4,0.2,0.1)
(0.8,0.6,0.5)
(0.5,0.2,0.2,0.6)
(0.7,0.6,0.1)
(0.4,0.5,0.2)
(0.7,0.6)
Now, we implement the algorithm 6.2 for various values of α, and obtain the score values. In Table 7, score values are given for various values of α (α ≠ 2). Consequently, we have ranking . Here, we find that best alternative remains the same due to the algorithm 6.2 as well as due to algorithm 4.2 of Das et al.. But our algorithm has resolved the problem of preference ordering of alternatives to some extent in the situation when two or more alternatives have equal preference. Therefore, the algorithm 6.2 is more efficient than the algorithm 4.2 of Das et al. [14] and is suitable to solve the problem of GDM when ranking of alternatives is desired.
Now, we provide a real life example for possible implication of GDM algorithm provided in this work.
Correlation measures of HFSS pairs
3.8075
2.9358
3.345
3.7208
2.9358
3.0708
2.9592
3.1542
3.345
2.9592
3.355
3.6042
3.7208
3.1542
3.6042
3.9483
Correlation coefficients of HFSS pairs and Correlation efficiency
Corr. eff.
Normalized eff.
1
0.8585
0.9359
0.9596
0.918
0.2471
0.8585
1
0.9219
0.9059
0.8954
0.24104
0.9359
0.9219
1
0.9903
0.9494
0.25558
0.9596
0.9059
0.9903
1
0.9519
0.2563
Collective decision matrix
A1
A2
A3
A4
(0.5,0.5,0.3,0.2)
(0.7,0.5,0.3,0.1)
(0.6,0.5,0.3,0.1)
(0.6,0.6,0.6,0.2)
(0.6,0.5,0.4,0.2)
(0.4,0.4,0.3,0.1)
(0.5,0.4,0.3,0.1)
(0.7,0.7,0.5,0.2)
(0.6,0.5,0.3,0.2)
(0.6,0.5,0.4,0.1)
(0.6,0.6,0.4,0.2)
(0.7,0.6,0.4,0.2)
(0.5,0.4,0.3,0.1)
(0.6,0.5,0.2,0.1)
(0.6,0.5,0.4,0.2)
(0.7,0.7,0.5,0.2)
Example 3. “KAYAKALP” is an initiative by Ministry of Health And Welfare, Government of India to promote cleanliness in public health facilities. In this scheme some awards have been instituted for best two district hospitals in each state, best two community health centres in district and one primary health centre in every district. The awards are given on the basis of scores given by Cleanliness and Infection Control Committee based on certain set of criterion. The criterion considered for evaluation as per government directives are as follows: (1) Hospital/Facility upkeep (2) Sanitation and hygiene (3) Waste Management (4)Infection Control (5) Support Services (6) Hygiene Promotion.
The assessment of these criterion is done on the basis of certain sub-criterion. The details are available at (https://www.nhp.gov.in/sites/default/award_to_public_health_facilities_kayakalp.pdf). It has been observed that the evaluation committee assign scores to criterion/sub-criterion in crisp numbers. This kind of evaluation does not provide a realistic score to a health facility centre. In this type of study the application of the generalized correlation coefficient of HFSSs may provide a realistic scores to the health centres under investigation. Consequently, awards may be given in more fair manner.
Aggregated alternatives and score values
Alternatives
Aggregated HFE
Score value
(0.6,0.5,0.3,0.2)
0.4
(0.5,0.5,0.4,0.2)
0.4
(0.6,0.6,0.4,0.2)
0.45
(0.6,0.5,0.3,0.2)
0.4
Score values of alternatives for different values of α
Alternatives
Score values at various values of α
α = 1.2
α = 1.3
α = 1.4
α = 1.5
α = 1.6
α = 1.7
α = 1.8
α = 1.9
α = 2.1
α = 2.5
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.425
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
Discussion and implications
In the illustrative example, the problem of ranking of alternatives is not completely solved as the preference of two alternatives remain same i.e . Thus, either the preference order of are very-very close or we require more flexible measures to resolve this tie between . Also, researchers might use generalized correlation coefficient for decision-making in the uncertain environment where GDM is crucial due to lack of information, an expertise of the experts, risk amendment, etc. Some of the recent studies [2, 28–30] using the concept of hesitant fuzzy preference relations (HFPRs), hesitant fuzzy linguistic preference relations (HFLPRs), hesitant fuzzy distributions (HFDs) and hesitant fuzzy linguistic distributions (HFLDs) have provided effective models for group decision-making problems. In these studies, the preference given to one alternative over the other alternative is subjective and degree of preference of an alternative xi over an alternative xj is given according to some membership function uij but this is a matter of concerned that how to set a membership function in each particular case. The nature of membership function is extremely individual. This is possibly happening because of absence of the parametrization tool in the theory. To overcome this problem, hesitant fuzzy soft preference relations (HFSPRs), hesitant fuzzy soft linguistic preference relations (HFSLPRs) can be defined the models proposed in [2, 28–30] can be investigated in context of HFSPRs and HFSLPRs. Further, in many decision-making problem opinions of decision makers need to be revised over a period of time due to interaction among decision makers for more details refer [21, 24–27]. In such situations one decision maker give some weightage to other decision makers and finally a consensus is reached at some point of time. Therefore, another future application of existing study may be its extension to opinion dynamics. This type of study seems to be effective tool in social network analysis.
In the present study, during calculation of correlation measures the length of HFE have been made equal in accordance with optimistic principle. To do away with drawback of optimistic/pestimistic principles, some recent studies [30, 31] provided alternative method such as additive consistency and multiplicative consistency for estimation of missing values in HFE. In future, the existing study may be further investigated using multiplicative consistency and additive consistency.
Conclusion
In this paper, we have introduced generalized correlation coefficients of HFSs and HFSSs. We have obtained generalized correlation efficiency for individual HFSS which shows the importance of an HFSS in group decision-making. This paper has also proposed a decision-making algorithm which presents the application of generalized correlation coefficient and generalized correlation efficiency (used to assign importance to a decision maker) in the hesitant fuzzy environment for a more effective solution of a group decision-making problem than the existing methods. The superiority of proposed generalized correlation coefficient of HFSs in pattern recognition and the discussed algorithm in group decision-making has been established. At this stage, the anticipated implications includes: 1) development of more flexible generalized correlation coefficients to obtain strict ranking among the alternatives; 2) investigation of group decision making problems in hesitant fuzzy soft environment using multiplicative consistency based approach and additive consistency based approach; 3) to perform a real life study in hesitant fuzzy soft environment as indicated in example 3; 4) investigation of group decision-making problems using HFSPRs and HFSLPRs; 5) Social network analysis using hesitant fuzzy soft theory and its extension to opinion dynamics.
Conflict of interest
We declare that authors have no conflict of interest.
Footnotes
Acknowledgement
The authors would like to thank three anonymous refrees for their constructive suggestions.
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