Jaccard Vector Similarity Measure of Bipolar Neutrosophic Set Based on Multi-Criteria Decision Makin

ICNASE’16

International Conference on Natural Science and Engineering (ICNASE’16) March 19-20, 2016, Kilis

Jaccard Vector Similarity Measure of Bipolar Neutrosophic Set Based on Multi-Criteria Decision Making Mehmet Şahin Gaziantep University, Department of Mathematics, Gaziantep 27310-Turkey Irfan Deli Kilis 7 AralĹk University, Muallim RĹfat Faculty of Education, Kilis 79000, Turkey Vakkas Uluçay Gaziantep University, Department of Mathematics, Gaziantep 27310-Turkey

ABSTRACT The main aim of this study is to present a novel method based on multi-criteria decision making for bipolar neutrosophic sets. Therefore, Jaccard vector similarity and weighted Jaccard vector similarity measure is defined to develop the bipolar neutrosophic decision making method. In addition, the method is applied to a numerical example in order to confirm the practicality and accuracy of the proposed method.

Keywords: Neutrosophic set, bipolar neutrosophic set, Jaccard vector similarity measure, multicriteria decision making.

1. Introduction As generalization of fuzzy set [20] and intuitionistic fuzzy set [1], Smarandache [10,11] initiated the notation of neutrosophic set which has a truth-membership, a indeterminacy membership and a falsemembership function in ି [0,1áˆżŕŹž . After Smarandache, many extensions and examples of neutrosophic sets have been introduced by many researcher such as; single valued neutrosophic sets [13], interval neutrosophic sets [14], single valued neutrosophic multi-sets [5,16], N-valued interval neutrosophic sets [2], neutrosophic soft sets [9], interval neutrosophic soft sets [7], possibility neutrosophic soft sets [8], rough neutrosophic sets [12], and so on. As a significant content in fuzzy sets, the similarity measure between the these sets have received more attention to calculate the degree of similarity measure between proposed the sets in [2,3,4,12,15,17,18,19].

Recently, different a generalization of neutrosophic sets is proposed by Deli et a.[6] is called bipolar neutrosophic sets. The bipolar neutrosophic set can be effectively used to evaluate information during decision making process. Therefore, in this study we present a novel method by extending the Jaccard vector similarity measures of neutrosophic sets to bipolar neutrosophic sets.

2. Preliminaries In the subsection, we give some concepts related to neutrosophic sets and bipolar neutrosophic sets.

Definition 2.1 [10-11] Let be a universe of discourse. Then a neutrosophic set N is defined as: Ü° ŕľŒ áˆźâ€ŤÂšŰƒâ€ŹÇĄ ŕ­’ áˆşÂšáˆťÇĄ ŕ­’ áˆşÂšáˆťÇĄ ŕ­’ áˆşÂšáˆťâ€ŤŰ„â€ŹÇŁ š ‍ ×?â€Źáˆ˝ÇĄ

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which is characterized by a truth-membership function ŕ­’ ÇŁ Őœ áˆżÍ˛ŕŹż ÇĄ ͳା ሞ, an indeterminacymembership function ŕ­’ ÇŁ Őœ áˆżÍ˛ŕŹż ÇĄ ͳା ሞand a falsity-membership function ŕ­’ ÇŁ Őœ áˆżÍ˛ŕŹż ÇĄ ͳା ሞ.

There is not restriction on the sum of ŕ­’ áˆşÂšáˆť, ŕ­’ áˆşÂšáˆť and ŕ­’ áˆşÂšáˆť, so Ͳି ŕľ‘ •—’ ŕ­’ áˆşÂšáˆť ŕľ‘ •—’ ŕ­’ áˆşÂšáˆť ŕľ‘ •—’ ŕ­’ áˆşÂšáˆť ŕľ‘ ͵ା .

Definition 2.2 [15] Let be a universe of discourse. Then a single valued neutrosophic set(SVNset) is defined as: Ü° ŕľŒ áˆźâ€ŤÂšŰƒâ€ŹÇĄ ŕ­’ áˆşÂšáˆťÇĄ ŕ­’ áˆşÂšáˆťÇĄ ŕ­’ áˆşÂšáˆťâ€ŤŰ„â€ŹÇŁ š ‍ ×?â€Źáˆ˝ÇĄ which is characterized by a truth-membership function ŕ­’ ÇŁ Őœ áˆžÍ˛ÇĄÍłáˆż, an indeterminacy-membership function ŕ­’ ÇŁ Őœ áˆžÍ˛ÇĄÍłáˆżand a falsity-membership function ŕ­’ ÇŁ Őœ áˆžÍ˛ÇĄÍłáˆżÇ¤ There is not restriction on the sum of ŕ­’ áˆşÂšáˆť, ŕ­’ áˆşÂšáˆť and ŕ­’ áˆşÂšáˆť, so Ͳ ŕľ‘ •—’ ŕ­’ áˆşÂšáˆť ŕľ‘ •—’ ŕ­’ áˆşÂšáˆť ŕľ‘ •—’ ŕ­’ áˆşÂšáˆť ŕľ‘ Íľ.

Definition 2.3 [6] Let be a universe of discourse. A bipolar neutrosophic set ୆୒ୗ in X is defined as an object of the form ABNS

^

x , T ( x ), I ( x ), F ( x ), T

x ,I x ,F

(x) : x Â? X

`,

where T , I , F : X o > 1, 0 @ and T , I , F : X o > 1, 0 @ .

The positive membership degree T ( x ), I ( x ), F ( x ) denotes the truth membership, indeterminate membership and false membership of an element x Â? X corresponding to a bipolar neutrosophic set

୆୒ୗ and the negative membership degree T ( x ), I ( x ), F ( x )

denotes the truth membership,

indeterminate membership and false membership of an element x � X to some implicit counter-property corresponding to a bipolar neutrosophic set ୆୒ୗ .

Set- theoretic operations, for two bipolar neutrosophic set ABNS

^

x , T1 ( x ), I 1 ( x ), F1 ( x ), T1

^

x , T 2 ( x ), I 2 ( x ), F 2 ( x ), T 2

x , I 1 x , F1

(x) : x Â? X

`

and B BNS

1. The subset;

F R if W

F R WC

T

x , I 2 x , F2

( x) : x Â? X

`

are given as;

and only if

1

( x ) d T2 ( x ) I

1

(x) d I2 (x) , F

1

( x ) t F2 ( x ) ,

and T

1

( x ) t T2 ( x ) , I

for all x Â? X .

2.

F R WL

FR W

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,

1

(x) t I

2

( x ) , F1 ( x ) d F 2 ( x )

International Conference on Natural Science and Engineering (ICNASE’16)

T

1

(x)

T2 ( x ) , I

(x)

T2 ( x ) , I

1

(x)

I2 (x) , F

(x)

I

1

F2 ( x ) ,

(x)

and T

1

1

2

( x ) , F1 ( x )

F2 ( x )

for all x Â? X .

3. The complement of ୆୒ୗ is denoted by ୭୆୒ୗ and is defined by T

A

(x)

{1 } T A ( x ) , I

(x)

{1 } T A ( x ) , I c ( x ) A

c

A

c

(x)

{1 } I A ( x ) , F c ( x ) A

{1 } F A ( x )

and T

A

c

{1 } I A ( x ) , F

A

c

(x)

{1 } F A ( x ) ,

for all x Â? X .

4. The intersection

( A B N S ˆ B B N S )( x )

­ ½ I1 ( x ) I 2 ( x )

, m a x (( F1 ( x ), F 2 ( x )), m a x (T 1 ( x ), T 2 ( x )), ° ° x , m in ( T1 ( x ), T 2 ( x )), ° ° 2 Ž ž

° I1 ( x ) I 2 ( x ) ° , m in (( F1 ( x ), F 2 ( x )) :x� X °¯ °¿ 2

5. The union

( ABNS ‰ B BNS )( x )

­ ½ I1 ( x ) I 2 ( x )

, m in ( ( F1 ( x ) , F 2 ( x ) ) , ° x , m a x ( T1 ( x ) , T 2 ( x ) ) , ° ° ° 2 Ž ž

I1 ( x ) I 2 ( x )

° ° m in ( T1 ( x ) , T 2 ( x ) ) , , m a x ( ( F1 ( x ) , F 2 ( x ) ) :x� X ° ° ¯ ¿ 2

.

Definition 2.3 [19] Let ‍ ÜŁâ€ŹŕľŒ â€ŤÜśŰƒâ€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťÇĄ â€ŤÜŤâ€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťÇĄ â€ŤÜ¨â€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťâ€Ť ۄ‏and ‍ Ü¤â€ŹŕľŒ â€ŤÜśŰƒâ€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆťÇĄ â€ŤÜŤâ€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆťÇĄ â€ŤÜ¨â€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆťâ€Ť ۄ‏be two SVNSs in a universe of discourse Üş ŕľŒ áˆşâ€ŤÝ”â€ŹŕŹľ ÇĄ â€ŤÝ”â€ŹŕŹś ÇĄ ÇĽ ÇĄ ‍ݔ‏௥ áˆťÇ¤ Then the Jaccard similarity measure between SVNsets A and B in the vector space is defined as follows: ௥

â€ŤÜŹâ€Źáˆşâ€ŤÜŁâ€ŹÇĄ â€ŤÜ¤â€Źáˆť

‍ۇ‏ ‍ۊ‏ Íł ܶ஺ áˆşâ€ŤÝ”â€ŹŕŻœ áˆťÜśŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆť ŕľ… â€ŤÜŤâ€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťâ€ŤÜŤâ€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆť ŕľ… â€ŤÜ¨â€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťâ€ŤÜ¨â€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆť ŕˇ?ŕľˆ â€ŤŰˆâ€Ź ଶ ଶ ଶ ଶ ଶ ଶ ‍ۋ‏ ÝŠ ൫ܶ஺ áˆşâ€ŤÝ”â€ŹŕŻœ áˆťŕľŻ ŕľ… ŕľŤâ€ŤÜŤâ€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťŕľŻ ŕľ… ŕľŤâ€ŤÜ¨â€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťŕľŻ ŕľ… ൫ܶ஻ áˆşâ€ŤÝ”â€ŹŕŻœ áˆťŕľŻ ŕľ… ŕľŤâ€ŤÜŤâ€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆťŕľŻ ŕľ… ŕľŤâ€ŤÜ¨â€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆťŕľŻ ŕŻœŕ­€ŕŹľ ྼ ྊ ŕľ†ŕľŤÜśŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťÜśŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆť ŕľ… â€ŤÜŤâ€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťâ€ŤÜŤâ€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆť ŕľ… â€ŤÜ¨â€ŹŕŽş áˆşâ€ŤÝ”â€ŹŕŻœ áˆťâ€ŤÜ¨â€ŹŕŽť áˆşâ€ŤÝ”â€ŹŕŻœ áˆťŕľŻ ‍ۉ‏ â€ŤŰŒâ€Ź

Then, this similarity measure satisfies the following properties: 1. Ͳ ŕľ‘ â€ŤÜŹâ€Źáˆşâ€ŤÜŁâ€ŹÇĄ â€ŤÜ¤â€Źáˆť ŕľ‘ ͳǥ 2. â€ŤÜŹâ€Źáˆşâ€ŤÜŁâ€ŹÇĄ â€ŤÜ¤â€Źáˆť ŕľŒ â€ŤÜŹâ€Źáˆşâ€ŤÜ¤â€ŹÇĄ â€ŤÜŁâ€Źáˆť, 3. â€ŤÜŹâ€Źáˆşâ€ŤÜŁâ€ŹÇĄ â€ŤÜ¤â€Źáˆť ŕľŒ Íł for ‍ ÜŁâ€ŹŕľŒ ‍ ܤ‏i.e. ŕ­… áˆşÂšáˆť ŕľŒ ŕ­† áˆşÂšáˆťÇĄ ŕ­… áˆşÂšáˆť ŕľŒ ŕ­† áˆşÂšáˆťÇĄ ŕ­… áˆşÂšáˆť ŕľŒ ŕ­† áˆşÂšáˆťÇĄ Ý… ŕľŒ áˆşÍłÇĄÍ´ ÇĽ ÇĄ ÝŠáˆť ‍ܺ ×?‏Ǥ

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3. Jaccard vector similarity measure of Bipolar Neutrosophic Set In this section, we present a Jaccard vector similarity and weighted Jaccard vector similarity measure for bipolar neutrosophic sets by extending the approach of SVN-set [19] to bipolar neutrosophic set.

Definition 3.1 Let ‫ ܣ‬ൌ ‫ܶۃ‬஺ା ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஺ା ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஺ା ሺ‫ݔ‬௜ ሻǡ ܶ஺ି ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஺ି ሺ‫ݔ‬௜ ሻ‫ۄ‬ and ‫ ܤ‬ൌ ‫ܶۃ‬஻ା ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஻ା ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஻ା ሺ‫ݔ‬௜ ሻǡ ܶ஻ି ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஻ି ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ‫ ۄ‬be two BNSsǤ Then, Jaccard similarity measure between BNS A and B, denoted ‫ܬ‬ሺ‫ܣ‬ǡ ‫ܤ‬ሻǡ is defined as follows:

‫ܬ‬ሺ‫ܣ‬ǡ ‫ܤ‬ሻ

‫ۇ‬ ‫ۊ‬ ‫ۈ‬ ‫ۋ‬ ା ሺ‫ ݔ‬ሻܶ ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܫ‬ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܨ‬ା ሺ‫ ݔ‬ሻ ‫ۈ‬ ‫ۋ‬ ܶ ൅ ‫ܫ‬ ൅ ‫ܨ‬ െ ௜ ஻ ௜ ௜ ஻ ௜ ௜ ஻ ௜ ஺ ஺ ஺ ௡ ‫ۈ‬ ‫ۋ‬ ି ି ି ି ି ି ͳ ͳ ሺܶ஺ ሺ‫ݔ‬௜ ሻܶ஻ ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஺ ሺ‫ݔ‬௜ ሻ‫ܫ‬஻ ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஺ ሺ‫ݔ‬௜ ሻ‫ܨ‬஻ ሺ‫ݔ‬௜ ሻሻ ‫ۋ‬ ෍ൈ ‫ۈ‬ ଶ ଶ ଶ ଶ ଶ ଶ ݊ ʹ ‫ ۈ‬൫ܶ ା ሺ‫ ݔ‬ሻ൯ ൅ ൫‫ ܫ‬ା ሺ‫ ݔ‬ሻ൯ ൅ ൫‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ ൅ ൫ܶ ା ሺ‫ ݔ‬ሻ൯ ൅ ൫‫ ܫ‬ା ሺ‫ ݔ‬ሻ൯ ൅ ൫‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ ൅ ‫ۋ‬ ௜ ௜ ௜ ௜ ௜ ஺ ஺ ஺ ஻ ஻ ௜ ஻ ௜ୀଵ ‫ۋې‬ ‫ۍۈ‬ ‫ێ‬ ି ሺ‫ ݔ‬ሻ൯ଶ ି ሺ‫ ݔ‬ሻ൯ଶ ି ሺ‫ ݔ‬ሻ൯ଶ ି ሺ‫ ݔ‬ሻ൯ଶ ି ሺ‫ ݔ‬ሻ൯ଶ ି ሺ‫ ݔ‬ሻ൯ଶ ‫ۑ‬ ‫ ێۈ‬൫ܶ஺ ௜ ൅ ൫‫ܫ‬஺ ௜ ൅ ൫‫ܨ‬஺ ௜ ൅ ൫ܶ஻ ௜ ൅ ൫‫ܫ‬஻ ௜ ൅ ൫‫ܨ‬஻ ௜ ‫ۋۑ‬ ‫ێۈ‬ ା ሺ‫ ݔ‬ሻܶ ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܫ‬ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ ‫ۋۑ‬ െ൫ܶ஺ ௜ ஻ ௜ ൅ ‫ܫ‬஺ ௜ ஻ ௜ ൅ ‫ܨ‬஺ ௜ ஻ ௜ ‫ێ‬ ‫ۑ‬ െ൫ܶ஺ି ሺ‫ݔ‬௜ ሻܶ஻ି ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻ‫ܫ‬஻ି ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஺ି ሺ‫ݔ‬௜ ሻ‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ൯ ‫ۏۉ‬ ‫یے‬

Definition 3.2 Let ‫ ܣ‬ൌ ‫ܶۃ‬஺ା ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஺ା ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஺ା ሺ‫ݔ‬௜ ሻǡ ܶ஺ି ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஺ି ሺ‫ݔ‬௜ ሻ‫ۄ‬ and ‫ ܤ‬ൌ ‫ܶۃ‬஻ା ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஻ା ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஻ା ሺ‫ݔ‬௜ ሻǡ ܶ஻ି ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஻ି ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ‫ ۄ‬be two BNSs and ‫ݓ‬௜ ‫ א‬ሾͲǡͳሿ be the weight of each element ‫ݔ‬௜ for ǡ ݅ ൌ ሺͳǡʹ ǥ ǡ ݊ሻ such that σ௡௜ୀଵ ‫ݓ‬௜ ൌ ͳǤ Then, weighted Jaccard similarity measure between BNS A and B, denoted ‫ܬ‬௪ ሺ‫ܣ‬ǡ ‫ܤ‬ሻǡ is defined as follows:

‫ܬ‬௪ ሺ‫ܣ‬ǡ ‫ܤ‬ሻ

‫ۇ‬ ‫ۊ‬ ‫ۈ‬ ‫ۋ‬ ା ሺ‫ ݔ‬ሻܶ ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܫ‬ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܨ‬ା ሺ‫ ݔ‬ሻ ‫ۈ‬ ‫ۋ‬ ܶ ൅ ‫ܫ‬ ൅ ‫ܨ‬ െ ௜ ஻ ௜ ௜ ஻ ௜ ஺ ஺ ௡ ‫ݓ‬௜ ൤ ஺ ି ௜ ஻ ି ௜ ൨ ‫ۋ‬ ି ି ି ି ͳ‫ۈ‬ ሺܶ஺ ሺ‫ݔ‬௜ ሻܶ஻ ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஺ ሺ‫ݔ‬௜ ሻ‫ܫ‬஻ ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஺ ሺ‫ݔ‬௜ ሻ‫ܨ‬஻ ሺ‫ݔ‬௜ ሻሻ ‫ۋ‬ ෍ൈ ‫ۈ‬ ଶ ଶ ଶ ଶ ଶ ଶ ʹ ‫ ۈ‬൫ܶ ା ሺ‫ ݔ‬ሻ൯ ൅ ൫‫ ܫ‬ା ሺ‫ ݔ‬ሻ൯ ൅ ൫‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ ൅ ൫ܶ ା ሺ‫ ݔ‬ሻ൯ ൅ ൫‫ ܫ‬ା ሺ‫ ݔ‬ሻ൯ ൅ ൫‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ ൅ ‫ۋ‬ ௜ ௜ ௜ ௜ ௜ ஺ ஺ ஺ ஻ ஻ ௜ ஻ ௜ୀଵ ‫ې‬ ‫ۍۈ‬ ଶ ଶ ଶ ଶ ଶ ଶ ‫ۋۑ‬ ‫ێ‬ ି ି ି ି ି ି ‫ ێۈ‬൫ܶ஺ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܫ‬஺ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܨ‬஺ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫ܶ஻ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܫ‬஻ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܨ‬஻ ሺ‫ݔ‬௜ ሻ൯ ‫ۋۑ‬ ‫ێۈ‬ ‫ۋۑ‬ െ൫ܶ஺ା ሺ‫ݔ‬௜ ሻܶ஻ା ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஺ା ሺ‫ݔ‬௜ ሻ‫ܫ‬஻ା ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஺ା ሺ‫ݔ‬௜ ሻ‫ܨ‬஻ା ሺ‫ݔ‬௜ ሻ൯ ‫ێ‬ ‫ۑ‬ െ൫ܶ஺ି ሺ‫ݔ‬௜ ሻܶ஻ି ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻ‫ܫ‬஻ି ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஺ି ሺ‫ݔ‬௜ ሻ‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ൯ ‫ۏۉ‬ ‫یے‬

Proposition 3.3 Let ‫ܬ‬௪ ሺ‫ܣ‬ǡ ‫ܤ‬ሻ be a Jaccard similarity measure between bipolar neutrosophic sets A and B. Then, we have 1. Ͳ ൑ ‫ܬ‬௪ ሺ‫ܣ‬ǡ ‫ܤ‬ሻ ൑ ͳǡ 2. ‫ܬ‬௪ ሺ‫ܣ‬ǡ ‫ܤ‬ሻ ൌ ‫ܬ‬௪ ሺ‫ܤ‬ǡ ‫ܣ‬ሻ, 3. ‫ܬ‬௪ ሺ‫ܣ‬ǡ ‫ܤ‬ሻ ൌ ͳ for ‫ ܣ‬ൌ ‫ ܤ‬i.e. ܶ஺ା ሺ‫ݔ‬௜ ሻ ൌ ܶ஻ା ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஺ା ሺ‫ݔ‬௜ ሻ ൌ ‫ܫ‬஻ା ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஺ା ሺ‫ݔ‬௜ ሻ ൌ ‫ܨ‬஻ା ሺ‫ݔ‬௜ ሻǡ ܶ஺ି ሺ‫ݔ‬௜ ሻ ൌ ܶ஻ି ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻ ൌ ‫ܫ‬஻ି ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஺ି ሺ‫ݔ‬௜ ሻ ൌ ‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ ݅ ൌ ሺͳǡʹ ǥ ǡ ݊ሻ ‫ܺ א‬Ǥ

Proof:

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International Conference on Natural Science and Engineering (ICNASE’16)

1. It is clear from Definition 3.2.

2.

‫ܬ‬௪ ሺ‫ܣ‬ǡ ‫ܤ‬ሻ

‫ۇ‬ ‫ۊ‬ ‫ۈ‬ ‫ۋ‬ ା ሺ‫ ݔ‬ሻܶ ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܫ‬ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܨ‬ା ሺ‫ ݔ‬ሻ ‫ۈ‬ ‫ۋ‬ ܶ ൅ ‫ܫ‬ ൅ ‫ܨ‬ െ ௜ ௜ ௜ ௜ ௜ ௜ ஻ ஻ ஻ ஺ ஺ ஺ ௡ ‫ݓ‬௜ ൤ ି ൨ ‫ۈ‬ ‫ۋ‬ ି ି ି ି ି ͳ ሺܶ஺ ሺ‫ݔ‬௜ ሻܶ஻ ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஺ ሺ‫ݔ‬௜ ሻ‫ܫ‬஻ ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஺ ሺ‫ݔ‬௜ ሻ‫ܨ‬஻ ሺ‫ݔ‬௜ ሻሻ ‫ۋ‬ ෍ൈ ‫ۈ‬ ʹ ‫ ۈ‬൫ܶ ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫‫ ܫ‬ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫ܶ ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫‫ ܫ‬ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ‫ۋ‬ ௜ ௜ ௜ ௜ ௜ ஺ ஺ ஺ ஻ ஻ ௜ ஻ ௜ୀଵ ‫ۍ‬ ‫ې‬ ‫ۈ‬ ଶ ଶ ଶ ଶ ଶ ଶ ‫ۋ‬ ‫ ێێۈ‬൫ܶ஺ି ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܨ‬஺ି ሺ‫ݔ‬௜ ሻ൯ ൅ ൫ܶ஻ି ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܫ‬஻ି ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ൯ ‫ۋۑۑ‬ ‫ێۈ‬ ‫ۋۑ‬ െ൫ܶ஺ା ሺ‫ݔ‬௜ ሻܶ஻ା ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஺ା ሺ‫ݔ‬௜ ሻ‫ܫ‬஻ା ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஺ା ሺ‫ݔ‬௜ ሻ‫ܨ‬஻ା ሺ‫ݔ‬௜ ሻ൯ ‫ێ‬ ‫ۑ‬ െ൫ܶ஺ି ሺ‫ݔ‬௜ ሻܶ஻ି ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻ‫ܫ‬஻ି ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஺ି ሺ‫ݔ‬௜ ሻ‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ൯ ‫ۏۉ‬ ‫یے‬

‫ۇ‬ ‫ۊ‬ ‫ۈ‬ ‫ۋ‬ ା ሺ‫ ݔ‬ሻܶ ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܫ‬ା ሺ‫ ݔ‬ሻ ା ሺ‫ ݔ‬ሻ‫ ܨ‬ା ሺ‫ ݔ‬ሻ ‫ۈ‬ ‫ۋ‬ ܶ ൅ ‫ܫ‬ ൅ ‫ܨ‬ െ ௜ ௜ ஺ ௜ ஻ ௜ ஺ ஻ ௡ ൨ ‫ݓ‬௜ ൤ ஻ ି ௜ ஺ ି ௜ ‫ۋ‬ ି ି ି ି ͳ‫ۈ‬ ሺܶ஻ ሺ‫ݔ‬௜ ሻܶ஺ ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஻ ሺ‫ݔ‬௜ ሻ‫ܫ‬஺ ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஻ ሺ‫ݔ‬௜ ሻ‫ܨ‬஺ ሺ‫ݔ‬௜ ሻሻ ‫ۋ‬ ൌ ෍ൈ ‫ۈ‬ ʹ ‫ ۈ‬൫ܶ ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫‫ ܫ‬ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫ܶ ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫‫ ܫ‬ା ሺ‫ ݔ‬ሻ൯ଶ ൅ ൫‫ ܨ‬ା ሺ‫ ݔ‬ሻ൯ଶ െ ‫ۋ‬ ௜ ௜ ௜ ௜ ௜ ஻ ஻ ௜ ஻ ஺ ஺ ஺ ௜ୀଵ ‫ې‬ ‫ۍۈ‬ ଶ ଶ ଶ ଶ ଶ ଶ ‫ۋۑ‬ ‫ێ‬ ି ି ି ି ି ି ‫ ێۈ‬൫ܶ஻ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܫ‬஻ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܨ‬஻ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫ܶ஺ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܫ‬஺ ሺ‫ݔ‬௜ ሻ൯ ൅ ൫‫ܨ‬஺ ሺ‫ݔ‬௜ ሻ൯ ‫ۋۑ‬ ‫ێۈ‬ ‫ۋۑ‬ െ൫ܶ஻ା ሺ‫ݔ‬௜ ሻܶ஺ା ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஻ା ሺ‫ݔ‬௜ ሻ‫ܫ‬஺ା ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஻ା ሺ‫ݔ‬௜ ሻ‫ܨ‬஺ା ሺ‫ݔ‬௜ ሻ൯ ‫ێ‬ ‫ۑ‬ െ൫ܶ஻ି ሺ‫ݔ‬௜ ሻܶ஺ି ሺ‫ݔ‬௜ ሻ ൅ ‫ܫ‬஻ି ሺ‫ݔ‬௜ ሻ‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻ ൅ ‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ‫ܨ‬஺ି ሺ‫ݔ‬௜ ሻ൯ ‫ۏۉ‬ ‫یے‬ ൌ ‫ܬ‬௪ ሺ‫ܤ‬ǡ ‫ܣ‬ሻ.

3. Since ܶ஺ା ሺ‫ݔ‬௜ ሻ ൌ ܶ஻ା ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஺ା ሺ‫ݔ‬௜ ሻ ൌ ‫ܫ‬஻ା ሺ‫ݔ‬௜ ሻǡ ‫ܨ‬஺ା ሺ‫ݔ‬௜ ሻ ൌ ‫ܨ‬஻ା ሺ‫ݔ‬௜ ሻǡ ܶ஺ି ሺ‫ݔ‬௜ ሻ ൌ ܶ஻ି ሺ‫ݔ‬௜ ሻǡ ‫ܫ‬஺ି ሺ‫ݔ‬௜ ሻ ൌ ି ሺ‫ ݔ‬ሻǡ ି ሺ‫ ݔ‬ሻ ‫ܫ‬஻ ௜ ‫ܨ‬஺ ௜ ൌ ‫ܨ‬஻ି ሺ‫ݔ‬௜ ሻ ݅ ൌ ሺͳǡʹ ǥ ǡ ݊ሻ ‫ܺ א‬ǡ we have ‫ܬ‬௪ ሺ‫ܣ‬ǡ ‫ܤ‬ሻ ൌ ͳǤ The proof is completed.

4. BN- Multi-criteria Decision Making Method In this section, we developed BN- Multi-criteria Decision Making Method based on weighted Jaccard vector similarity for bipolar neutrosophic sets by extending the some definitions of SVN-set [19,21] to bipolar neutrosophic set.

Definition 4.1 Let ܷ ൌ ሺ‫ݑ‬ଵ ǡ ‫ݑ‬ଶ ǡ ǥ ǡ ‫ݑ‬௠ ሻ be a set of alternatives, ‫ ܣ‬ൌ ሺܽଵ ǡ ܽଶ ǡ ǥ ǡ ܽ௡ ሻ be the set of criteria, ‫ ݓ‬ൌ ሺ‫ݓ‬ଵ ǡ ‫ݓ‬ଶ ǡ ǥ ǡ ‫ݓ‬௡ ሻ் be the weight vector of the ܽ௝ ሺ݆ ൌ ͳǡʹ ǥ ǡ ݊ሻ such that ‫ݓ‬௝ ൒ Ͳ and ା ା ି ି ି‫ۄ‬ σ௡௜ୀଵ ‫ݓ‬௜ ൌ ͳ and ሾܾ௜௝ ሿ௠௫௡ ൌ ‫ܶۃ‬௜௝ା ǡ ‫ܫ‬௜௝ ǡ ‫ܨ‬௜௝ ǡ ܶ௜௝ ǡ ‫ܫ‬௜௝ ǡ ‫ܨ‬௜௝ be the decision matrix in which the rating values of the alternatives. Then

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International Conference on Natural Science and Engineering (ICNASE’16) a1

ª b ij º ¬ ¼ mun

a2

u 1 § b1 1 ¨ u 2 b21 ¨ ¨ ¨ u m ¨ bm1 ¨ ©

an b1 n · ¸ b2 n ¸ ¸ ¸ bm n ¸ ¸ ¹

b1 2 b22

bm 2

is called a BN-multi-attribute decision making matrix of the decision maker.

Also; ܾ௝‫ כ‬is positive ideal bipolar neutrosophic solution of decision matrix ª¬ b i j º¼ ܾ௝‫כ‬

ൌ

as form:

mun

ା ି ‫ ۃ‬௜ ሼ ܶ௜௝ା ሽǡ ௜ ሼ ‫ܫ‬௜௝ ሽǡ ௜ ሼ ‫ܨ‬௜௝ା ሽǡ ௜ ሼ ܶ௜௝ି ሽǡ ݉ܽ‫ݔ‬௜ ሼ ‫ܫ‬௜௝ ሽǡ ௜ ሼ ܶ௜௝ି ሽ‫ۄ‬

and ܾത௝‫ כ‬is negative ideal bipolar neutrosophic solution of decision matrix

ª b ij º ¬ ¼ mun

as form:

ା ି ܾത௝‫ כ‬ൌ ‫ ۃ‬௜ ሼ ܶ௜௝ା ሽǡ ௜ ሼ ‫ܫ‬௜௝ ሽǡ ௜ ሼ ‫ܨ‬௜௝ା ሽǡ ௜ ሼ ܶ௜௝ି ሽǡ ݉݅݊௜ ሼ ‫ܫ‬௜௝ ሽǡ ௜ ሼ ܶ௜௝ି ሽ‫ۄ‬.

Algorithm: Step1. Give the decision-making matrix

ª b ij º ¬ ¼ mun

; for decision;

Step2. Compute the positive ideal (or negative ideal) bipolar neutrosophic solution ܾ௝‫ כ‬ൌ ሼ ܾଵ‫ כ‬ǡ ܾଶ‫ כ‬ǡ ǥ ܾ௡‫ כ‬ሽ fo r ª¬ b i j º¼

mun

;

Step3. Calculate the weighted Jaccard vector similarity measure ܵ௜ between positive ideal (or ା ା ି ି ି‫ۄ‬ negative ideal) bipolar neutrosophic solution ܾ௝‫ כ‬and ܾ௜ ൌ ‫ܶۃ‬௜௝ା ǡ ‫ܫ‬௜௝ ǡ ‫ܨ‬௜௝ ǡ ܶ௜௝ ǡ ‫ܫ‬௜௝ ǡ ‫ܨ‬௜௝ and ሺ݅ ൌ ͳǡʹ ǥ ǡ ݉ሻ ሺ݆ ൌ ͳǡʹ ǥ ǡ ݊ሻ as;

ܵ௜ = ‫ܬ‬௪ ൫ܾ௝‫ כ‬ǡ ܾ௜ ൯

‫ۇ‬ ‫ۊ‬ ‫ۈ‬ ‫ۋ‬ శ శ శ శ శ శ ‫כ‬ ‫כ‬ ‫כ‬ ‫ۈ‬ ‫ۋ‬ ቁ ାቀூೕ‫כ‬ቁ ቀூ೔ೕ ቁ ାቀிೕ‫ כ‬ቁ ቀி೔ೕ ቁ ି ቀ்ೕ‫כ‬ቁ ቀ்೔ೕ ௪೔ ቎ ష ష ష ష ష ష ቏ ‫ۈ‬ ‫ۋ‬ ‫כ‬ ‫כ‬ ‫כ‬ ቀ்ೕ‫כ‬ቁ ቀ்೔ೕ ቁ ିቀூೕ‫ כ‬ቁ ቀூ೔ೕ ቁ ିቀிೕ‫ כ‬ቁ ቀி೔ೕ ቁ ‫ۋ‬ σ௡௝ୀଵൈ ‫ۈ‬ మ మ మ మ మ మ ‫ۍۈ‬൬ቀ்ೕ‫כ‬ቁశ൰ ା൬ቀூೕ‫כ‬ቁశ൰ ା൬ቀிೕ‫ כ‬ቁశ൰ ା൬ቀ்೔ೕ‫ כ‬ቁశ൰ ା൬ቀூ೔ೕ‫ כ‬ቁశ൰ ା൬ቀி೔ೕ‫כ‬ቁశ൰ ା‫ۋې‬ ‫ۑ‬ ‫ێۈ‬ ష మ ష మ ష మ ష మ ష మ ష మ ‫ۋ‬ ‫ ێێۈ‬ቀቀ்ೕ‫כ‬ቁ ቁ ାቀቀூೕ‫כ‬ቁ ቁ ାቀቀிೕ‫כ‬ቁ ቁ ାቀቀ்೔ೕ‫ כ‬ቁ ቁ ାቀቀூ೔ೕ‫כ‬ቁ ቁ ାቀቀி೔ೕ‫ כ‬ቁ ቁ ‫ۋۑۑ‬ శ శ శ శ శ శ ‫ێۈ‬ ‫ۋۑ‬ ି൬ቀ் ‫ כ‬ቁ ቀ் ‫ כ‬ቁ ାቀூ‫ כ‬ቁ ቀூ ‫ כ‬ቁ ାቀி‫כ‬ቁ ቀி‫ כ‬ቁ ൰ ‫ێ‬ ‫ێ‬ ‫ۏۉ‬

ೕ

೔ೕ

ష

ೕ

ష

೔ೕ

ష

ೕ

ష

೔ೕ

ష

ష

‫ כ‬ቁ ାቀூ‫ כ‬ቁ ቀூ ‫ כ‬ቁ ାቀி‫ כ‬ቁ ቀி‫ כ‬ቁ ቁ ିቀቀ்ೕ‫ כ‬ቁ ቀ்೔ೕ ೕ ೔ೕ ೕ ೔ೕ

‫ۑ‬ ‫ۑ‬ ‫یے‬

Step 4. Determine the nonincreasing order of ܵ௜ = ‫ܬ‬௪ ൫ܾ௝‫ כ‬ǡ ܾ௜ ൯ ሺ݅ ൌ ͳǡʹ ǥ ǡ ݉ሻ ሺ݆ ൌ ͳǡʹ ǥ ǡ ݊ሻ and select the best alternative.

Now, we give a numerical example as follows; Example 4.2 Let us consider decision making problem adapted from Xu and Cia [21]. We consider Gaziantep hospital who intends to buy bed. Four types of beds (alternatives) ‫ݑ‬௜ ሺ݅ ൌ ͳǡʹǡ͵ǡͶሻ are available.

The customer takes into account four attributes to evaluate the alternatives; ܽଵ ൌ air bed; ܽଶ ൌ moving bed; ܽଷ ൌ two motorized bed and use the bipolar neutrosophic values to evaluate the four possible alternatives ‫ݑ‬୧ ሺ ൌ ͳǡ ʹǡ ͵ǡ Ͷሻ under the above four attributes. Also, the weight vector of the attributes ܽ௝ ሺ݆ ൌ ͳǡʹǡ͵ሻ is ɘ ൌ ሺͲǤ͸ǡͲǤ͵ǡͲǤͳሻ୘ . Then, Algorithm Step1. Constructed the decision matrix provided by the Gaziantep hospital as;

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International Conference on Natural Science and Engineering (ICNASE’16)

Table 1: Decision matrix given by Hospital ܽ

5

6

7

5

ˆr wÆr yÆr tÆF r yÆF r uÆF r x˜ ˆr vÆr vÆr wÆF r yÆF r zÆF r v˜ ˆr yÆr yÆr wÆF r zÆF r yÆF r x˜

6

ˆr {Ær yÆr wÆF r yÆF r yÆF r s˜ ˆr yÆr xÆr zÆF r yÆF r wÆF r s˜ ˆr {Ær vÆr xÆF r sÆF r yÆF r w˜

7

ˆr uÆr vÆr tÆF r xÆF r uÆF r y˜ ˆr tÆr tÆr tÆF r vÆF r yÆF r v˜ ˆr {Ær wÆr wÆF r xÆF r wÆF r t˜

8

ˆr {Ær yÆr tÆF r zÆF r xÆF r s˜ ˆr uÆr wÆr tÆF r wÆF r wÆF r t˜ ˆr wÆr vÆr wÆF r sÆF r yÆF r t˜

Step2. Computed the positive ideal bipolar neutrosophic solution as; > L <ˆr {Ær vÆr tÆF r zÆF r uÆF r s˜Æˆr yÆr tÆr tÆF r yÆF r wÆF r s˜Æˆr {Ær vÆr wÆF r zÆF r wÆF r t˜= Step3. Calculated the weighted Jaccard vector similarity measures 5= ,Œ k> Æ> oas; 55 L r r us t x 56 L r r wz r { 57 L r r wr u u 58 L r u rt t w Step4. Rank all the software systems of ( L sÆtÆuÆv) according to the weighted Jaccard vector similarity measure as; 6 8 7 8 5 8 8 and thus 6is the most desirable alternative.

5. Conclusions In this paper, we developed a multi-criteria decision making for bipolar neutrosophic sets based on Jaccard vector similarity measures and applied to a numerical example in order to confirm the practicality and accuracy of the proposed method. In the future, the method can be extend with different similarity and distance measures in fuzzy set, intuitionistic fuzzy set and neutrosophic set.

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