Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process

The objective of this work is to present novel correlation coefficients under the intuitionistic multiplicative preference relation (IMPR), for measuring the relationship between the two intuitionistic multiplicative sets, instead of intuitionistic fuzzy preference relation (IFPR). As IFPR deals under the conditions that the attribute values grades are symmetrical and uniformly distributed. But in our day-to-day life, these conditions do not fulfill the decision maker requirement and hence IFPR theory is not applicable in that domain. Thus, for handling this, an intuitionistic multiplicative set theory has been utilized where grades are distributed asymmetrical around 1. Further, under this environment, a decision making method based on the proposed novel correlation coefficients has been presented. Pairs of membership and non-membership degree are considered to be a vector representation during formulation. Three numerical examples have been taken to demonstrate the efficiency of the proposed approach.


1.
Introduction. In many decision-making problems, it is difficult for a decision maker to give his assessments towards the object in crisp values due to ambiguity and incomplete information. Instead, it has become popular that these assessments are presented by a fuzzy set or extensions of the fuzzy set. Fuzzy set (FS) [37], proposed by Zadeh, is a powerful tool to deal with vagueness and has received much attention. After it, some extensions such as intuitionistic fuzzy sets (IFSs) [2], interval-valued intuitionistic fuzzy sets (IVIFSs) [1] etc., have been proposed by the researchers. As IFS has been defined on a intuitionistic fuzzy preference relation (IFPR) R = (α ij ) n×n with the condition that α ij + α ji = 1 and 0 ≤ α ij ≤ 1, where α ij indicates the alternative x i is preferred to x j . Here, each alternative α ij , in an intuitionistic fuzzy preference relation R, is expressed on a scale of 0.1 -0.9 which assumes that the grades lie between "extremely not preferred" and "extremely preferred" are distributed uniformly and symmetrically. Using these, during the last decades, various researchers have analyzed this theory in the process of decision making using aggregation operators [14,31,30,5,3,16], similarity measures [4,34,23,21,22,18], distance measures [25], score and accuracy functions 1502 HARISH GARG [33,8,12,13,7], correlation coefficients [17,32,26,9,34,29] etc. But due to the large complexity of the system day-to-day, it is difficult for the system analyst to represent the data in a symmetrical form and hence, IFPR theory in sometimes may have unable to get the correct option to the decision-maker to opt for the system representation.
To resolve this issue, Saaty [24] presented a different scale named as 1-9 scale, given in Table 1, where the decision maker gives their preferences on the scale of 1/9− 9 instead of 0−1 to represent the data. However, it can only represent the degree of acceptance of the alternative but unable to represent the degree of rejection. To cope this situation, the intuitionistic multiplicative preference relation (IMPR) [27] has been proposed in which the representation of each alternative is considered as a pair of acceptance and rejection degrees and hence, their corresponding preference relation is called IMPR, denoted by S = (α ij ) m×n with the condition that α ij α ji = 1 and 1 9 ≤ α ij ≤ 9 where α ij represents the asymmetrical distribution degree of the alternative x i w.r.t. x j . Under this preference environment, Xia et al. [27] introduced an aggregation operator for different intuitionistic multiplicative sets (IMSs) in the decision making process while Yu et al. [35] extended it to an interval-valued IMS environment. Xia and Xu [28] introduced the aggregation operator based on the Choquet integral and applied them to the group decision-making problems. Yu and Xu [36] introduced an aggregation operator under the triangular intuitionistic multiplicative numbers. Garg [6,11] presented some generalized weighted, ordered weighted and hybrid averaging and geometric aggregation operators under the intuitionistic fuzzy multiplicative preference relation. Jiang et al. [20] proposed a method for ranking the IMSs based on distance measures. Garg [10] presented the distance and similarity measures for intuitionistic multiplicative preference relation and applied them to solve the decision-making problems. Further, Garg [15] presented an improved score based ranking method for intuitionistic multiplicative sets under the crisp, and interval environment. From these, it has been concluded that IMPR describes the characteristics of the alternative in a better way than IFPR. Intermediate value used to between 1/9 and 9 between 0 and 1 present compromise Thus, keeping inspiration from the fact that IMPR has powerful ability to model the imprecise and ambiguous information in real-world applications, the main purposes of this paper is to extend their operations from an aggregation process to the correlation coefficient. To the best of author knowledge, no work has been carried out so far on the correlation coefficient between the two IMSs. Thus, the present manuscript studies the correlation coefficient between these sets. For this, some operational laws on IMSs and hence new informational energies, as well as the covariance between the two IMSs, have been defined. Based on these, correlation coefficients have been proposed by considering the set of pairs of the degree of membership and non-membership toward an element of the universe. Also, in order to deal with the situations where the elements in a set are correlative, weighted correlation coefficients have been defined. Using the proposed correlation coefficients between the IMSs, the ranking orders of all the alternatives are obtained. At length, we provide multi-criteria decision-making (MCDM) problem from the fields of decision making to validate the effectiveness and applicability of the proposed decision method.
The rest of the article is organized as follows: In section 2, we briefly introduce some basic concepts of IFS and IMS. In section 3, we establish the correlation and weighted correlation coefficients between the pairs of IMSs along with their desirable properties. In Section 4, we develop an approach based on the proposed correlation coefficients to decision-making process under the intuitionistic multiplicative environment. In Section 5, three illustrative examples have been presented to validate the proposed approach and comparative analysis is shown to explore its effectiveness. Section 6 gives the concluding remarks.
2. Basic concepts about IFSs and IMSs. In this section, some basic concepts about the IFSs and IMSs are defined.
Definition 2.1. Intuitionistic fuzzy sets (IFSs): Let X be a non-empty reference set, an IFS [2] A in X is defined as where µ A , ν A : X → [0, 1] represent the degree of membership and non-membership of x to A such that for any x ∈ X, µ A (x) + ν A (x) ≤ 1, and in turn, the intuitionistic index of x to A is defined as is called an intuitionistic fuzzy number (shorted by IFN), and it is often simplified as α = µ, ν where µ ∈ [0, 1], ν ∈ [0, 1], µ + ν ≤ 1. The score value corresponding to IFN α is defined as sc(α) = µ − ν [30].
. . , n) be collection of an IMS, then (i) the extended distance function is given as (ii) the extended accuracy function is given as . . , n) be collection of an IMS, then (i) the extended weighted distance function is given as (ii) the extended weighted accuracy function is given as where ω = (ω 1 , ω 2 , . . . , ω n ) T be the weight vector of x i (i = 1, 2, . . . , n), such that 3. Correlation coefficient of IMSs. In this section, we have proposed some correlation coefficients under the IMS environment which can be applied to numerous engineering and scientific fields.
for every x i ∈ X, the informational intuitionistic energies of IMSs A and B are defined as and The correlation of the IMSs A and B is defined as It is obvious that the Eq. (10) satisfy the following properties: Therefore, the correlation coefficient between two IMSs A and B is defined as follows: any two IMSs defined on X, then the correlation coefficient between them, denoted by K 1 (A, B) is defined as and it satisfy the following properties: Theorem 3.1. For any two IMSs A and B in X, the correlation coefficient K 1 (A, B) satisfy the property:

Proof. For any two IMSs
Similarly, for non-membership σ A (x i ), we have Hence, the informational energy corresponding to element x i ∈ X for set A is given by Therefore, by using Eq. (8) for all x i ∈ X, we get the informational energy between the IMS A as Similarly, we can obtain the informational energy between the IMS B as 1 20 Further, for any x i ∈ X, it can be easily deduced that and thus for all x i (i = 1, 2, . . . , n), we follows that Similarly, for non-membership degrees, we can obtain the following results By adding Eqs. (13) and (14), and using Eq. (10), we get Therefore, by Eq. (11), we get Now, by using the well-known Cauchy-Schwarz inequality: with equality if and only if the two vectors a = (a 1 , a 2 , . . . , a n ) and b = (b 1 , b 2 , . . . , b n ) are linearly dependent.
In terms of membership and non-membership functions for two IMSs A and B, Eq. (16) yields to Thus, from Eq. (15) we get which completes the proof.
Then by using Eq. (8), the informational energy of A is and the informational energy of B is By using Eq. (10), the correlation between them is Hence, by Eq. (11), we get the correlation coefficient between the IMSs A and B as Definition 3.2. Let A and B be two IMSs, then the correlation coefficient is defined as Theorem 3.2. The correlation coefficient, K 2 (A, B), satisfy the following properties: Proof. Since A and B be any two IMSs, then we have As similar to Eq. (15), we can obtain Now, the inequality 1 9 ≤ K 2 (A, B) ≤ 9 can be proved directly by using the well-known Cauchy-Schwarz inequality: with equality if and only if the two vectors a = (a 1 , a 2 , . . . , a n ) and b = (b 1 , b 2 , . . . , b n ) are linearly dependent.
From the Definition 3.1 and Definition 3.2, we observe that the correlation coefficient defined by Eq. (11) uses the geometric mean of the informational energies of the IMSs A and B, and the correlation coefficient defined by Eq. (17) applies the maximum between them. For the optimistic decision makers, they tend to use the correlation coefficient defined by Eq. (11). Contrary to the optimism decision makers, the pessimistic decision makers tend to apply the correlation coefficient defined by Eq. (17).
However, in many practical situations, the different set may have taken different weights and thus, weight ω i of the element x i ∈ X(i = 1, 2, . . . , n) should be taken into account. In the following, we develop some weighted correlation coefficients between the two IMSs. For it, let ω = (ω 1 , ω 2 , . . . , ω n ) T be the weight vector of x i (i = 1, 2, . . . , n) with ω i > 0 and n i=1 ω i = 1, then we have extended the above formulated correlation coefficients K 1 and K 2 to weighted correlation coefficients K 3 and K 4 , respectively, as follows: It can be easily verified that, if ω = ( 1 n , 1 n , . . . , 1 n ) T then (20) and (21) reduces to (11) and (17) respectively. Further, it is easy to check that the weighted correlation coefficients, K 3 between two IMSs A and B also satisfy the property  with ω i > 0 and n i=1 ω i = 1, then the weighted correlation coefficient between IMSs A and B defined by Eq. (20), satisfy the following properties: Proof. Similar, to the proof of Theorem 3.1 we can proof the above properties. 4. Proposed decision making approach based on correlation coefficients under intuitionistic multiplicative environment. In this section, we shall utilize the proposed correlation coefficients to multi-criteria decision making under the intuitionistic multiplicative environment. For it, the following assumptions or notations are used to present the MCDM problems for evaluating of these with an intuitionistic multiplicative environment. Let A = {A 1 , A 2 , . . . , A m } be the set of m different alternatives which have to be evaluated under the set of different criteria. Assume that these alternatives are evaluated by the set of the q decision makers D = (D (1) , D (2) , . . . , D (q) ) who will receive the full responsibility for the whole process. The weight vector information corresponding to each expert is denoted by w = (w 1 , w 2 , . . . , w q ) T with w k > 0 and q k=1 w k = 1. Each expert give their preferences related to each alternative A i (i = 1, 2, . . . , m) under the intuitionistic multiplicative environment, and these values can be considered as IMNs represents the priority values of alternative A i given by decision maker D (k) (k = 1, 2, . . . , q) such that 1 9 ij ≤ 1. Then, the proposed method has been summarized into the various steps which are described as follows. (Step 1:) Construct the intuitionistic multiplicative decision matrix D (k) = (α (k) ij ) m×m , k = 1, 2, . . . , q.   Here, in the first decision matrix, D (1) , for example, the first preference is 1, 1 implies that when the first candidate X 1 compares with himself then the preference is (1, 1). On the other hand, the IMN 1/3, 1/4 indicates that the first expert argued that the degree of the first candidate is a priority to the second candidate is 1/3 while at the same time, he thinks the degree of the first candidate is not a priority to the second candidate is 1/4. Similarly, the other observations have their meaning. (Step 2:) Aggregate these different preference values of each expert D (k) , k = 1, 2, 3, 4 towards the alternatives X i (i = 1, 2, 3, 4) into the collected decision matrix R = (α ij ) by using Eq. (3) Step 3:) In order to access the best alternative in the decision set, a concept of an ideal point has been used. Although the ideal alternative does not exist in real world, it does provide a useful theoretical construct against which to evaluate alternatives. Hence, we define the fixed ideal alternative in terms of pessimistic view i.e., X * = 1/9, 9 as the IMS. Now, utilizing the proposed correlation coefficients (11) and (17) to compute the measurement values of each candidate from its ideal alternative and hence get K 4 (X 3 , X * ) = 0.1965 ; K 4 (X 4 , X * ) = 0.1297 (Step 4:) Thus, based on these measurement values, the ranking order of the alternatives X i (i = 1, 2, 3, 4) is X 4 X 1 X 3 X 2 corresponding to pessimistic view and X 2 X 3 X 1 X 4 corresponding to optimistic view. Therefore, based on the decision-maker choice regarding pessimistic or optimistic, they can select the best candidate either X 4 or X 2 for the required post of the Professor.

5.2.
Example 2: Pattern recognition. Consider three unknown patterns C 1 , C 2 and C 3 which are represented by the following IMSs in a given universe X = {x 1 , x 2 , x 3 } as Consider a known pattern P which will be recognized as an IMS in X, where P = {(x 1 , 1, 1/5 ), (x 2 , 3, 1/7 ), (x 3 , 7, 1/9 )} The target of this problem is to classify the pattern P in one of the classes C 1 , C 2 and C 3 . For it, proposed correlation coefficient index, K 1 and K 2 defined in (11) and (17) respectively, have been computed from P to C k (k = 1, 2, 3) and their measurement values are given as follows.
From these computation results, it has been observed that all the results are same and coincides with the proposed one which validates the proposed measure to rank the IMNs.
On the other hand, if we assign weights 0.15, 0.25, 0.20, 0.15 and 0.25 corresponding to s j (j = 1, 2, . . . , 5) respectively, then by applying the correlation coefficients (20) and (21), we get the following values: Thus, ranking order of the diagnoses is Q 2 Q 3 Q 1 Q 5 Q 4 and Q 2 Q 1 Q 3 Q 5 Q 4 corresponding to K 3 and K 4 . Hence, based on the maximum recognition principle we compute that patient P diagnosis with Q 2 . Also, we have seen that the ranking order is different for the different indices. Therefore, based on the inherent properties of the proposed correlation coefficients, the decision maker may choose their goals according to their desired, while the best one remains the same by all the four kinds of proposed measures.

5.3.1.
Comparison of results with those obtained through IMS. In order to compare these results with the existing approaches [20,27] under IMS environment defined in Eqs. (2)- (7), an analysis has been conducted which are summarized as follows.

5.3.2.
Comparison of results with those obtained through IFS. In order to compare the proposed approach with the results obtained through IFS environment. For this, firstly Definition 2.6 has been used for converting the preferences of the pattern Q k (k = 1, 2, . . . , 5) from IMS to IFS and then existing measures [30,25,33,34,4,18,7] have been utilized for finding the classifying diagnoses Q k 's. The results corresponding to its have been presented in Table 3. From these results, it has been clearly seen that some of the existing approaches are unable to find the diagnoses of the patient P . For instance, cosine similarity measures as proposed by [34] is unable to distinguish between the diagnoses Q 2 and Q 3 . Furthermore, all these analyses have investigated under the IFPR scale which assumes that the grades are lies between 0.1 -0.9 and are distributed uniformly and symmetrical. Due to this reason, the different approaches have produced different diagnoses as the best. However, the results computed under the IMPR scale produces the uniform result and found that patient P suffers from the Q 2 diagnosis, as explained in section 5.3.1.  [33] 0.3375 0.7410 0.1582 0.2066 0.5198 Q 2 Q 5 Q 1 Q 4 Q 3 Correlation coefficient [34] 0.7208 0.7047 0.7207 0.6103 0.6957 Q 1 Q 3 Q 2 Q 5 Q 4 Similarity measure (S C ) [4] 0.6856 0.6767 0.7249 0.5701 0.6490 Q 3 Q 1 Q 2 Q 5 Q 4 Similarity measure (S H ) [18] 0.6742 0.6713 0.7249 0.5701 0.6490 Q 3 Q 1 Q 2 Q 5 Q 4 Cosine Similarity measure [34] 0.4982 0.5091 0.5091 0.4328 0.5076 Q 2 = Q 3 Q 5 Q 1 Q 4 Improved score function [7] 0.6894 0.7343 0.5248 0.4676 0.5846 Q 2 Q 1 Q 5 Q 3 Q 4 6. Conclusion. IMPR is one of the successful extension of the IFPR in which a different scale named as the 1-9 scale has been used instead of 0 -1 to represent the data. As the IFPR has been valid only for those grades which are uniformly distributed and symmetrical while in IMPR, the grades are distributed asymmetrically around 1. By keeping the advantages of IMPR, in this paper, we have presented novel correlation and weighted correlation coefficients for measuring the relationships between two or more values in IMSs. Some of its desirable properties have also been studied. Further, based on these correlation coefficients, a decision-making approach has been presented under the intuitionistic multiplicative environment. The applicability of the proposed approach has been demonstrated through some numerical examples and results are compared with the existing approaches results under the IMPR, as well as the IFPR, environments. From these, we conclude that the proposed correlation coefficients can suitably be handled the real-life decision-making problem with their targets. In our further research, we will focus on adopting this approach to some more complicated applications from the fields of cluster analysis, uncertain programming, and mathematical programming.