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Correlation Coefficient Measures of Interval Bipolar Neutrosophic Sets for Solving Multi-Attribute Decision Making Problems.

1 Introduction

Correlation coefficient is an important decision making apparatus in statistics to evaluate the relation between two sets. In neutrosophic environment [1], Hanafy et al. [2] derived a formula for correlation coefficient between two neutrosophic sets (NSs). Hanafy et al. [3] obtained the correlation coefficient of NSs by using centroid strategy which lies in [-1, 1]. The correlation coefficient obtained from [3] provides the information about the degree of the relationship between two NSs and also informs us whether the NSs are positive or negatively related. In 2013, Ye [4] defined correlation, correlation coefficient, weighted correlation coefficient in single valued neutrosophic set (SVNS) [5] environment and established a multi-criteria decision making (MCDM) based on the proposed weighted correlation coefficient measure. Broumi and Smarandache [6] introduced the concept of correlation coefficient and weighted correlation coefficient between two interval neutrosophic sets (INSs) [7] and established some of their basic properties. Hanafy et al. [8] studied the notion of correlation and correlation coefficient of neutrosophic data under probability spaces. Ye [9] suggested an improved correlation coefficient between two SVNSs in order to overcome the drawbacks of the correlation coefficient discussed in [4] and investigated its properties. In the same study, Ye [9] extended the concept of correlation coefficient measure of SVNS to correlation coefficient measure of INS environment. Furthermore, Ye [9] developed strategies for solving multi-attribute decision making (MADM) problems with single valued neutrosophic and interval neutrosophic environments based on the proposed correlation coefficient measures. Broumi and Deli [10] defined correlation measure of two neutrosophic refined (multi) sets [11] by extending the correlation measure of two intuitionistic fuzzy multi-sets proposed by Rajarajeswari and Uma [12] and proved some of its basis properties. Zhang et al. [13] defined an improved weighted correlation coefficient on the basis of integrated weight for INSs and a decision making strategy is developed. Karaaslan [14] proposed a strategy to compute correlation coefficient between possibility neutrosophic soft sets and presented several properties related to the proposed strategy. Karaaslan [15] defined a new mathematical structure called single-valued neutrosophic refined soft sets (SNRSSs) and presented its set theoretical operations such as union, intersection and complement and proved some of their basic properties. In the same study [15], two formulas to determine correlation coefficient between two SNRSSs are proposed and the developed strategy is used to solve a clustering analysis problem. Sahin and Liu [16] defined single valued neutrosophic hesitant fuzzy sets (SVNHFSs) and established some basic properties and finally proposed a decision making strategy. Liu and Luo [17] defined correlation coefficient and weighted correlation coefficient for interval-valued neutrosophic hesitant fuzzy sets (INHFSs) due to Liu and Shi [18] and studied their properties. Then, Liu and Luo [17] developed a MADM strategy within the framework of INHFSs based on weighted correlation coefficient. Ye [19] suggested a dynamic single valued neutrosophic multiset (DSVNM) based on dynamic information obtained from different time intervals in several practical situations in order to express dynamical data and operational relations of DSVNMs. In the same study [19], correlation coefficient and weighted correlation coefficient measures between DSVNMs are proposed and a MADM strategy is developed on the basis of the proposed weighted correlation coefficient under DSVNM setting. Recently, Ye [20] proposed two correlation coefficient between normal neutrosophic sets (NNSs) based on the score functions of normal neutrosophic numbers and investigated their essential properties. In the same study, Ye [20] formulated a MADM strategy by employing correlation coefficient of NNSs in normal neutrosophic environment. Pramanik et al. [21] defined correlation coefficient and weighted correlation coefficient between two rough neutrosophic sets and proved their basic properties. In the same study, Pramanik et al. [21] developed a multi-criteria decision making strategy based on the proposed correlation coefficient measure and solved an illustrative example in medical diagnosis.

In 2015, Deli et al. [22] introduced a novel concept called bipolar neutrosophic sets (BNSs) by generalizing the concepts of bipolar fuzzy sets [23, 24] and bipolar intuitionistic fuzzy sets [25]. In the same study, Deli et al. [22] defined score, accuracy and certainty functions to compare BNSs and formulated a MCDM approach based on the score, accuracy and certainty functions and bipolar neutrosophic weighted average operator ([A.sub.w]) and bipolar neutrosophic weighted geometric operator ([G.sub.w]). In bipolar neutrosophic environment, Dey et al. [26] developed a MADM approach based on technique for order of preference by similarity to ideal solution (TOPSIS) strategy. Deli and Subas [27] and Sahin et al. [28] developed MCDM strategies based on correlation coefficient and Jaccard similarity measures, respectively in BNS environment. Ulugay et al. [29] defined Dice, weighted Dice similarity measures, hybrid and weighted hybrid similarity measures for MCDM problems with bipolar neutrosophic information. Pramanik et al. [30] defined projection, bidirectional projection and hybrid projection measures between BNSs and proved their basic properties and then, three new MADM models are developed based on proposed measures.

Mahmood et al. [31] and Deli et al. [32] incorporated the notion of interval bipolar neutrosophic sets (IBNSs) and defined some operations and operators for IBNSs. Recently, Pramanik et al. [33] defined new cross entropy and weighted cross entropy measures in BNS and IBNS environment and discussed some of their essential properties. In the same study, Pramanik et al. [33] developed two novel MADM strategies on the basis of the proposed weighted cross entropy measures.

Research gap:

MADM strategy based on correlation coefficient under IBNSs environment.

This paper answers the following research questions:

i. Is it possible to introduce a novel correlation coefficient measure for IBNSs?

ii. Is it possible to introduce a novel weighted correlation coefficient measure for IBNSs?

iii. Is it feasible to formulate a novel MADM strategy based on the proposed correlation coefficient measure in IBNS environment?

iv. Is it feasible to formulate a novel MADM strategy based on the proposed weighted correlation coefficient measure in IBNS environment?

Motivation:

The aforementioned analysis presents the motivation behind developing correlation coefficient-based strategy for handling MADM problems with IBNS information.

The objectives of the paper are as follows:

1. To define a new correlation coefficient measure and a new weighted correlation coefficient measure in IBNS environment and prove their basic properties.

2. To develop a new MADM strategy based on weighted correlation coefficient measure in IBNS environment.

In order to fill the research gap, we propose correlation coefficient-based MADM strategy in IBNS environment.

Rest of the article is organized as follows. Section 2 provides the preliminaries of bipolar fuzzy sets, bipolar intuitionistic fuzzy sets, BNSs and IBNSs. Section 3 defines the correlation coefficient and weighted correlation coefficient measures in IBNS environment and establishes their basic properties. In section 4, a new MADM strategy based on the proposed weighted correlation coefficient measure is developed. In section 5, we solve a numerical example and comparison analysis is given. Finally, in the last section, conclusions are presented.

2 Preliminaries

2.1 Bipolar fuzzy sets

A bipolar fuzzy set [23, 24] B in X is characterized by a positive membership functional [[alpha].sup.+.sub.B](x) and a negative membership function [[alpha].sup.-.sub.B](x). A bipolar fuzzy set B is expressed in the following way.

B - {x, <[[alpha].sup.+.sub.B](x), [[alpha].sup.-.sub.B](x)> | x [member of] X}

where [[alpha].sup.+.sub.B](x): X [right arrow] [0, 1] and [[alpha].sup.- .sub.B](x): X [right arrow] [-1, 0] for each point x [member of] X.

2.2 Bipolar intuitionistic fuzzy sets

Consider X be a non-empty set, then a BIFS [25] E is expressed in the following way.

E = {x, <[[alpha].sup.+.sub.E](x), [[alpha].sup.-.sub.E](x), [[beta].sup.+.sub.E](x), [[beta].sup.-.sub.E](x)> | x [member of] X}

where [[alpha].sup.+.sub.E](x), [[beta].sup.+.sub.E](x): X [right arrow] [0, 1] and [[alpha].sup.-.sub.E](x), [[beta].sup.-.sub.E](x): X [right arrow] [-1, 0] for each point x [member of] X such that 0 [less than or equal to] [[alpha].sup.+.sub.E](x) + [[beta].sup.+.sub.E](x) [less than or equal to] 1 and -i [less than or equal to] [[alpha].sup.-.sub.E](x) + [[beta].sup.-.sub.E](x) [less than or equal to] 0.

2.3 Bipolar neutrosophic sets

A BNS [22]M in X is presented as follows:

[mathematical expression not reproducible]

where [mathematical expression not reproducible]. The positive membership degrees [[alpha].sup.+.sub.M](x), [[beta].sup.+.sub.M](x), [[gamma].sup.+.sub.M](x) denote the truth membership, indeterminate membership, and false membership functions of an object x [member of] X corresponding to a BNS M and the negative membership degrees [[alpha].sup.-.sub.M](x), [[beta].sup.-.sub.M](x), [[gamma].sup.- .sub.M](x) denote the truth membership, indeterminate membership, and false membership of an object x [member of] X to several implicit counter property associated with a BNS M.

Definition 2.3.1

Let, [mathematical expression not reproducible] be any two BNSs. Then, a BNS [M.sub.1] is contained in another BNS [M.sub.2], represented by [M.sub.1] [subset or equal to] [M.sub.2] if and only if [mathematical expression not reproducible].

Definition 2.3.2

Let, [mathematical expression not reproducible] for all x [member of] X.

Definition 2.3.3

The complement of a BNS [33] M is [mathematical expression not reproducible]

where

[mathematical expression not reproducible].

Definition 2.3.4

The union [30]of two BNSs [M.sub.1] and [M.sub.2] represented by [M.sub.1], [union] [M.sub.2] is defined as follows:

[mathematical expression not reproducible].

Definition 2.3.5

The intersection [30] of two BNSs [M.sub.1] and [M.sub.2] denoted by [M.sub.1] [intersection] [M.sub.2] is defined as follows:

[mathematical expression not reproducible].

2.4 Interval bipolar neutrosophic sets

Consider X be the space of objects, then an IBNS [31, 32] L in X is is represented as follows:

[mathematical expression not reproducible]

where L is characterized by positive and negative truth-membership [[alpha].sup.+.sub.L](x), [[alpha].sup.-.sub.L](x); inderterminacy-membership [[beta].sup.+.sub.L](x), [[beta].sup.-.sub.L](x); falsity-membership [[gamma].sup.+.sub.L](x), [[gamma].sup.-.sub.L](x) functions respectively. Here, [mathematical expression not reproducible] for all x [member of] X with the conditions 0 [less than or equal to] sup [[alpha].sup.+.sub.L](x) + sup [[beta].sup.+.sub.L](x) + sup [[gamma].sup.+.sub.L](x) [less than or equal to] 3, and -3 [less than or equal to] sup [[alpha].sup.-.sub.L](x) + sup [[beta].sup.-.sub.L](x) + sup [[gamma].sup.-.sub.L](x) [less than or equal to] 0.

Definition 2.4.1: [mathematical expression not reproducible].

Definition 2.4.2: [mathematical expression not reproducible].

Definition 2.4.3: [mathematical expression not reproducible].

3 Correlation coefficient measures under IBNSs setting

Definition 3.1: Let [L.sub.1] and [L.sub.2] be two IBNSs in X = {[x.sub.1], [x.sub.2], ..., [x.sub.n]}, then the correlation between [L.sub.1] and [L.sub.2] is defined as follows: R ([L.sub.1], [L.sub.2) =

[mathematical expression not reproducible]

Definition 3.2: Consider [L.sub.1] and [L.sub.2] be two IBNSs in X = {[x.sub.1], [x.sub.2], ..., [x.sub.n]}, then the correlation coefficient between [L.sub.1] and [L.sub.2] is defined as follows:

[mathematical expression not reproducible] (1)

where

[mathematical expression not reproducible]

Theorem 1. The correlation coefficient measure Cor ([L.sub.1], [L.sub.2]) between two IBNSs [L.sub.1], [L.sub.2] satisfies the following properties:

[mathematical expression not reproducible].

Proof:

(1) [mathematical expression not reproducible].

(2) Since, R ([L.sub.1], [L.sub.2]) [greater than or equal to] 0, R ([L.sub.1], [L.sub.1]) [greater than or equal to] 0, R ([L.sub.2], [L.sub.2]) [greater than or equal to] 0 and using Cauchy-Schwarz inequality we can easily prove that Cor ([L.sub.1], [L.sub.2]) [less than or equal to] 1, therefore, 0 [less than or equal to] Cor ([L.sub.1], [L.sub.2]) [less than or equal to] 1.

(3) [mathematical expression not reproducible].

Definition 3.3: Let [w.sub.i], = ([w.sub.1], [w.sub.2], ..., [w.sub.n]) [member of] [0, 1] be the weight vector of the elements [x.sub.j] (j = 1, 2, ..., n), the weighted correlation coefficient between two IBNSs [L.sub.1], [L.sub.2] can be defined by the following formula

[mathematical expression not reproducible] (2)

where

[mathematical expression not reproducible]

If w = [(1/n, 1/n, ..., 1/n).sup.T], the Eq. (2) is reduced to Eq. (1).

Theorem 2. The weighted correlation coefficient measure [Cor.sub.w] ([L.sub.1], [L.sub.2]) between two IBNSs [L.sub.1], [L.sub.2] also satisfies the following properties:

(C1) [Cor.sub.w] ([L.sub.1], [L.sub.2]) = [Cor.sub.w] ([L.sub.2], [L.sub.1]);

(C2) 0 [less than or equal to] [Cor.sub.w] ([L.sub.1], [L.sub.2]) [less than or equal to] 1;

(C3) [Cor.sub.w] ([L.sub.1], [L.sub.2]) = 1, if [L.sub.1] = [L.sub.2].

Proof:

(1) [mathematical expression not reproducible].

(2) Since, [R.sub.ww] ([L.sub.1], [L.sub.2]) [greater than or equal to] 0, [R.sub.w] ([L.sub.1], [L.sub.1]) [greater than or equal to] 0, [R.sub.w] ([L.sub.2], [L.sub.2]) [greater than or equal to] 0 and using Cauchy-Schwarz inequality we can easily prove that [Cor.sub.w] ([L.sub.1], [L.sub.2]) [less than or equal to] 1, so, 0 [less than or equal to] [Cor.sub.w] ([L.sub.1], [L.sub.2]) [less than or equal to] 1.

(3) [mathematical expression not reproducible].

Example 1. Suppose that [L.sub.1] = < [0.3, 0.7], [0.3, 0.8], [0.5, 0.9], [-0.9, -0.3], [-0.6, -0.2], [-0.8, -0.4] > and [L.sub.2] = < [0.1, 0.6], [0.2, 0.7], [0.3, 0.5], [-0.8, -0.2], [-0.8, -0.3], [0.7, -0.4] > be two IBNSs, then correlation coefficient between [L.sub.1] and [L.sub.2] is obtain using Eq. (1) as follows:

Cor ([L.sub.1], [L.sub.2]) = 0.4870391.

Example 2. If w = 0.4, then the weighted correlation coefficient between [L.sub.1] = < [0.3, 0.7], [0.3, 0.8], [0.5, 0.9], [-0.9, -0.3], [-0.6, -0.2], [-0.8, -0.4] > and [L.sub.2] = < [0.1, 0.6], [0.2, 0.7], [0.3, 0.5], [-0.8, -0.2], [-0.8, -0.3], [-0.7, -0.4] > is calculated by using Eq. (2) as follows.

[Cor.sub.w] ([L.sub.1], [L.sub.2]) = 0.5689123.

4. MADM strategy based on weighted correlation coefficient measure in IBNS environment

In this section, we have developed a novel MADM strategy based on weighted correlation coefficient measure in interval bipolar neutrosophic environment. Let, F = {[F.sub.1], [F.sub.2], ..., [F.sub.m]}, (m [greater than or equal to] 2) be a discrete set of m feasible alternatives, G = {[G.sub.1], [G.sub.2], ..., [G.sub.n[}, (n [greater than or equal to] 2) be a set of n predefined attributes and [w.sub.j] be the weight vector of the attributes such that 0 [less than or equal to] [w.sub.j] [less than or equal to] 1 and [n.summation over (j=1)][w.sub.j] = 1. The steps for solving MADM problems in IBNS environment are presented as follows.

Step 1. The evaluation of the performance value of alternative [F.sub.i] (i = 1, 2, ..., m) with regard to the predefined attribute [G.sub.j], (j = 1, 2, ..., n) provided by the decision maker or expert can be presented in terms of interval bipolar neutrosophic values [mathematical expression not reproducible]. The interval bipolar neutrosophic decision matrix [[[[??].sub.ij]].sub.mxn] is presented as given below.

[mathematical expression not reproducible]

Step 2. The interval bipolar neutrosophic positive ideal solution (IBN-PIS) can be defined as follows: [mathematical expression not reproducible] j = 1, 2, ..., n, where [J.sup.+], [J.sup.-] denote the benefit and cost type attributes, respectively.

Step 3. The weighted correlation coefficient of IBNS between alternative [F.sub.i] (i = 1, 2, ..., m) and the ideal alternative [Q.sup.*] can be derived as follows:

[mathematical expression not reproducible]

where,

[mathematical expression not reproducible].

Step 4: The biggest value of [Cor.sub.w] ([F.sub.i], [Q.sup.*]), i = 1, 2, ..., m implies [F.sub.i], (i = 1, 2, ..., m) is the better alternative.

In Fig 1. we represent the steps for solving MADM problems based on weighted correlation coefficient measure in IBNS environment.

5. Numerical example

In this section, an illustrative numerical problem is solved to illustrate the proposed strategy. We consider an MADM studied in [31, 33] where there are four possible alternatives to invest money namely, a food company ([F.sub.1]), a car company ([F.sub.2]), a arm company ([F.sub.3]), and a computer company ([F.sub.4]). The investment company must take a decision based on the three predefined attributes namely growth analysis ([G.sub.1]), risk analysis ([G.sub.2]), and environment analysis ([G.sub.3]) where [G.sub.1], [G.sub.2] are the benefit type and [G.sub.3] is the cost type attribute [34] and the weight vector of [G.sub.1], [G.sub.2], and [G.sub.3] is given by w = ([w.sub.1], [w.sub.2], [w.sub.3]) = (0.35, 0.25, 0.4) [31].

The proposed strategy consisting of the following steps:

Step 1. The evaluation of performance value of the alternatives with respect to the attributes provided by the decision maker can be expressed by interval bipolar neutrosophic values and the decision matrix is presented as follows:

Interval bipolar neutrosophic decision matrix

[mathematical expression not reproducible]

Step 2. Determine the IBN-PIS ([Q.sup.*]) from interval bipolar neutrosophic decision matrix as follows:

[mathematical expression not reproducible].

Step 3. The weighted correlation coefficient [Cor.sub.w] ([F.sub.i], [Q.sup.*]) between alternative [F.sub.i] (i = 1, 2, ..., m) and IBN-PIS [Q.sup.*] is obtained as given below.

[mathematical expression not reproducible].

We observe that [Cor.sub.w] ([F.sub.4], [Q.sup.*]) > [Cor.sub.w] ([F.sub.2], [Q.sup.*]) > [Cor.sub.w] ([F.sub.3], [Q.sup.*]) > [Cor.sub.w] ([F.sub.1], [Q.sup.*]).

Step 4. According to the weighted correlation coefficient values, the ranking order of the companies is presented as:

[F.sub.4] > [F.sub.2] > [F.sub.3] > [F.sub.1].

Hence, the most desirable investment company is [F.sub.4].

In Fig 2. we represent the graphical representation of alternatives versus weighted correlation coefficient values.

Next, we compare the obtained results with the results of Mahmood et al. [31] and Pramanik et al. [33] in Table 1 where the weight vector of the attributes is w = (0.35, 0.25, 0.4) [31]. We see that ranking orders of alternatives derived by the proposed strategy and the strategies discussed by Mahmood et al. [31] and Pramanik et al. [33] are different. We also observe that [F.sub.4] is the best option obtained by the proposed strategy as well as the strategy discussed by Mahmood et al. [31]. However, Pramanik et al. [33] found that [F.sub.2] is the most desirable alternative based on weighted cross entropy measure.

6 Conclusion

In the study, we have defined correlation coefficient and weighted correlation coefficient measures in interval bipolar neutrosophic environments and prove their basic properties. Using the proposed weighted correlation coefficient measure, we have developed a novel MADM strategy in interval bipolar neutrosophic environment. We have solved an investment problem with interval bipolar neutrosophic information. Comparison analysis with other existing strategies is presented to demonstrate the feasibility and applicability of the proposed strategy. We hope that the proposed correlation coefficient measures can be employed to tackle realistic multi attribute decision making problems such as clustering analysis [15], medical diagnosis [21], weaver selection [35-37], fault diagnosis [38], brick selection [39- 40], data mining [41], logistic centre location selection [42- 43], school selection [44], teacher selection [45-47], image processing, information fusion, etc. in interval bipolar neutrosophic environment. Using aggregation operators, the proposed strategy can be extended to multi attribute group decision making problem in interval bipolar neutrosophic set environment.

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Received : February 9, 2018. Accepted : March 26, 2018.

Surapati Pramanik (1), Partha Pratim Dey (2) and Florentin Smarandache (3)

(1) Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-- Narayanpur, District--North 24 Parganas, Pin Code-743126, West Bengal, India. E-mail: sura_pati@yahoo.co.in

(2) Department of Mathematics, Patipukur Pallisree Vidyapith, Patipukur, Kolkata-700048, West Bengal, India. E-mail: parsur.fuzz@gmail.com

(3) Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA. E-mail: smarand@unm.edu

* Corresponding author's E-mail: sura_pati@yahoo.co.in

Caption: Figure. 1 Decision making procedure of proposed MADM strategy
Table 1. The results derived from different strategies

        strategy           Ranking results     Best
                                              choice

The proposed               [F.sub.4] >       [F.sub.4]
weighted correlation       [F.sub.2] >
coefficient strategy       [F.sub.3] >
                           [F.sub.1]

Mahmood et al.'s           [F.sub.4] >       [F.sub.4]
strategy [31]              [F.sub.1] >
                           [F.sub.3] >
                           [F.sub.2]

Weighted cross             [F.sub.1] <       [F.sub.2]
entropy measure [33]       [F.sub.3] <
                           [F.sub.4] <
                           [F.sub.2]

Fig 2. Graphical representation of alternatives versus
weighted correlation coefficient values.

Alternatives    Weighted correlation
                coefficient values

F1                   0.33195
F2                   0.35261
F3                   0.34567
F4                   0.35654

Note: Table made from bar graph.
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Author:Pramanik, Surapati; Dey, Partha Pratim; Smarandache, Florentin
Publication:Neutrosophic Sets and Systems
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Date:Mar 1, 2018
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