Interval-valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets

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University of New Mexico

Interval-valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets and some uncertainty based measures on them Rajashi Chatterjee1 , Pinaki Majumdar 2 , Syamal Kumar Samanta 3 1 Department 2 Department

of Mathematics, Visva-Bharati, Santiniketan, 731235, India. E-mail: rajashi.chatterjee@gmail.com

of Mathematics, M. U. C. Women’s College, Burdwan, 713104, India. E-mail: pmajumdar@gmail.com

3 Department

of Mathematics, Visva-Bharati, Santiniketan, 731235, India. E-mail: syamal123@yahoo.co.in ued neutrosophic soft sets. Some basic set-theoretic operations have been defined on them. Some distance, similarity, entropy and inclusion measures for possibility quadripartitioned single valued neutrosophic sets have been proposed. An application in a decision making problem has been shown.

Abstract: The theory of quadripartitioned single valued neutrosophic sets was proposed very recently as an extension to the existing theory of single valued neutrosophic sets. In this paper the notion of possibility fuzzy soft sets has been generalized into a new concept viz. interval-valued possibility quadripartitioned single val-

Keywords: Neutrosophic set, entropy measure, inclusion measure, distance measure, similarity measure.

1

Introduction

The theory of soft sets (introduced by D. Molodstov, in 1999) ([10],[15]) provided a unique approach of dealing with uncertainty with the implementation of an adequate parameterization technique. In a very basic sense, given a crisp universe, a soft set is a parameterized representation or parameter-wise classification of the subsets of that universe of discourse with respect to a given set of parameters. It was further shown that fuzzy sets could be represented as a particular class of soft sets when the set of parameters was considered to be [0, 1]. Since soft sets could be implemented without the rigorous process of defining a suitable membership function, the theory of soft sets, which seemed much easier to deal with, underwent rapid developments in fields pertaining to analysis as well as applications (as can be seen from the works of [1],[6],[7],[12],[14],[16],[17] etc.) On the otherhand, hybridized structures, often designed and obtained as a result of combining two or more existing structures, have most of the inherent properties of the combined structures and thus provide for a stronger tool in handling application oriented problems. Likewise, the potential of the theory of soft sets was enhanced to a greater extent with the introduction of hybridized structures like those of the fuzzy soft sets [8], intuitionistic fuzzy soft sets [9], generalized fuzzy soft sets [13], neutrosophic soft sets [11], possibility fuzzy soft sets [2], possibility intuitionistic fuzzy soft sets [3] etc. to name a few. While in case of generalized fuzzy soft sets, corresponding to each parameter a degree of possibility is assigned to the corresponding fuzzy subset of the universe; possibility fuzzy sets, a further modification of the generalized fuzzy soft sets, characterize each element of the universe with a possible degree of belongingness along with a degree of membership. Based on Belnap’s four-valued logic [4] and Smarandache’s n-valued refined

neutrosophic set [18], the theory of quadripartitioned single valued neutrosophic sets [5] was proposed as a generalization of the existing theory of single valued neutrosophic sets [19]. In this paper the concept of interval valued possibility quadripartitioned single valued neutrosophic soft sets (IPQSVNSS, in short) has been proposed. In the existing literature studies pertaining to a possibility degree has been dealt with so far. Interval valued possibility assigns a closed sub-interval of [0, 1] as the degree of chance or possibility instead of a number in [0, 1] and thus it is a generalization of the existing concept of a possibility degree. The proposed structure can be viewed as a generalization of the existing theories of possibility fuzzy soft sets and possibility intuitionistic fuzzy soft sets. The organization of the rest of the paper is as follows: a couple of preliminary results have been stated in Section 2, some basic set-theoretic operations on IPQSVNSS have been defined in Section 3, some uncertainty based measures viz. entropy, inclusion measure, distance measure and similarity measure, have been defined in Section 4 and their p roperties, applications and inter-relations have been studied. Section 5 concludes the paper.

2

Preliminaries

In this section some preliminary results have been outlined which would be useful for the smooth reading of the work that follows.

2.1

An outline on soft sets and possibility intuitionistic fuzzy soft sets

Definition 1 [15]. Let X be an initial universe and E be a set of parameters. Let P(X) denotes the power set of X and A ⊂ E. A pair (F, A) is called a soft set iff F is a mapping of A into

R. Chatterjee, P. Majumdar and S. K. Samanta, Interval-valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets and some uncertainty based measures on them

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P(X). The following results are due to [3].

WhenP X is discrete, A is represented as, n A = i=1 hTA (xi ), CA (xi ), UA (xi ), FA (xi )i /xi , xi X. However, when the universe of discourse is continuous, A is Definition 2 [3]. Let U = {x1 , x2 , ..., xn } be the univer- represented as, sal sets of elements and let E = {e1 , e1 , ..., em } be the universal A = hTA (x), CA (x), UA (x), FA (x)i /x, x X set of parameters. The pair (U, E) will be called a soft universe. U U Let F : E → (I × I) × I U where (I × I) is the collection of Definition 7 [5]. A QSVNS is said to be an absolute QSVNS, U all intuitionistic fuzzy subsets of U and I is the collection of denoted by A, iff its membership values are respectively defined all fuzzy subsets of U . Let p be a mapping such that p : E → I U as TA (x) = 1, CA (x) = 1, UA (x) = 0 and FA (x) = 0, ∀x X. U and let Fp : E → (I × I) × I U be a function defined as Definition 8 [5]. A QSVNS is said to be a null QSVNS, follows: Fp (e) = (F (e)(x), p(e)(x)), where F (e)(x) = denoted by Θ, iff its membership values are respectively defined as TΘ (x) = 0, CΘ (x) = 0, UΘ (x) = 1 and FΘ (x) = 1, ∀x X (µe (x) , νe (x)) ∀x U . Then Fp is called a possibility intuitionistic fuzzy soft set (PIFSS in short) over the soft universe (U, E). For each parameter ei , Definition 9 [5]. Let A and B be two QSVNS over X. Then the following operations can be defined: Fp (ei ) cann be represented as: o Containment: A ⊆ B iff TA (x) ≤ TB (x), CA (x) ≤ CA (x), xn x1 Fp (ei ) = F (ei )(x1 ) , p(ei ) (x1 ) , ..., F (ei )(xn ) , p(ei ) (xn ) UA (x) ≥ UA (x) and PFnA (x) ≥ FA (x), ∀x X. Complement:Ac = i=1 hFA (xi ), UA (xi ), CA (xi ), TA (xi )i /xi, xi X Definition 3 [3]. Let Fp and Gq be two PIFSS over (U, E). Then i.e. TAc (xi ) = FA (xi ), CAc (xi ) = UA (xi ) , UAc (xi ) = CA (xi ) the following operations were defined over PIFSS as follows: and FAc (xi ) = TA (xi ), xi X Pn Containment: Fp is said to be a possibility intuitionistic fuzzy Union: A ∪ B = < i=1 soft subset (PIFS subset) of Gq and one writes Fp ⊆ Gq if (TA (xi ) ∨ TB (xi )) , (CA (xi ) ∨ CB (xi )) , (UA (xi ) ∧ UB (xi )) , (i) p(e) is a fuzzy subset of q(e), for all e E, (FA (x) ∧ FB (x)) > /xi, xi X Pn (ii)F (e) is an intuitionistic fuzzy subset of G(e), for all e E. Intersection: A ∩ B = < i=1 Equality: Fp and Gq are said to be equal and one writes Fp = Gq (TA (xi ) ∧ TB (xi )) , (CA (xi ) ∧ CB (xi )) , (UA (xi ) ∨ UB (xi )) , if Fp is a PIFS subset of Gq and Gq is a PIFS subset of Fp (FA (xi ) ∨ FB (xi )) > /xi, xi X Ëœ Gq = Hr , Hr : E → (I × I)U × I U is deUnion: Fp ∪ fined by Hr (e) = (H (e) (x) , r (e) (x)), ∀e E such that Proposition 1[5]. Quadripartitioned single valued neutrosophic H (e) = ∪Atan (F (e) , G (e)) and r (e) = s (p (e) , q (e)), sets satisfy the following properties under the aforementioned where ∪Atan is Atanassov union and s is a triangular conorm. set-theoretic operations: U U Ëœ Gq = Hr , Hr : E → (I × I) × I is Intersection: Fp ∩ defined by Hr (e) = (H (e) (x) , r (e) (x)), ∀e E such that 1.(i) A ∪ B = B ∪ A H (e) = ∩Atan (F (e) , G (e)) and r (e) = t (p (e) , q (e)), (ii) A ∩ B = B ∩ A where ∩Atan is Atanassov intersection and t is a triangular norm. 2.(i) A ∪ (B ∪ C) = (A ∪ B) ∪ C (ii) A ∩ (B ∩ C) = (A ∩ B) ∩ C Definition 4 [3]. A PIFSS is said to be a possibility abso3.(i) A ∪ (A ∩ B) = A lute intuitionistic fuzzy soft set, denoted by A1 , if A1 : E → (ii) A ∩ (A ∪ B) = A U (I × I) × I U is such that A1 (e) = (F (e) (x) , P (e) (x)), c 4.(i) (Ac ) = A ∀e E where F (e) = (1, 0) and P (e) = 1, ∀e E. (ii) Ac = Θ (iii) Θc = A Definition 5 [3]. A PIFSS is said to be a possibility null intuitionc (iv) De-Morgan’s laws hold viz. (A ∪ B) = Ac ∩ B c ; U istic fuzzy soft set, denoted by φ0 , if φ0 : E → (I × I) × I U c (A ∩ B) = Ac ∪ B is such that φ0 = (F (e) (x) , p (e) (x)), ∀e E where 5.(i) A ∪ A = A F (e) = (0, 1) and p (e) = 0, ∀e E. (ii) A ∩ A = A (iii) A ∪ Θ = A 2.2 An outline on quadripartitioned single valued (iv) A ∩ Θ = Θ

neutrosophic sets Definition 6 [5]. Let X be a non-empty set. A quadripartitioned neutrosophic set (QSVNS) A, over X characterizes each element x in X by a truth-membership function TA , a contradictionmembership function CA , an ignorance-membership function UA and a falsity membership function FA such that for each x X, TA , CA , UA , FA [0, 1] R. Chatterjee, P. Majumdar and S. K. Samanta, Interval-valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets and some uncertainty based measures on them

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Interval-valued possibility quadripartitioned single valued neutrosophic soft sets and some of their properties

Definition 12. The null IPQSVNSS over (X, E) is denoted by θËœÂŻ0 such that for each e E and ∀x X, θËœe (x) = h0, 0, 1, 1i and ÂŻ0e (x) = [0, 0]

3.1

Operations over IPQSVNSS

Definition 10. Let X be an initial crisp universe and E be a set of parameters. Let I = [0, 1] , QSV N S(X) represents the collec- Definition 13. Let FĎ and GÂľ be two IPQSVNSS over the tion of all quadripartitioned single valued neutrosophic sets over common soft universe (X, E). Some elementary set-theoretic X , Int([0, 1]) denotes the set of all closed subintervals of [0, 1] operations on IPQSVNSS are defined as, Ëœ GÂľ = HΡ such that for each e E and ∀x X, and (Int([0, 1]))X denotes the collection of interval valued fuzzy (i) Union: FĎ âˆŞ H (x) = hteF (x) ∨ teG (x) , ceF (x) ∨ ceG (x) , ueF (x) ∧ e subsets over X . An interval-valued possibility quadripartitioned e e e single valued neutrosophic soft set (IPQSVNSS, in short) is a uG (x) , fF (x) ∧−fG (x)i−and + + mapping of the form FĎ : E → QSV N S(X) Ă— (Int([0, 1]))X Ρe (x) = [sup (Ď e (x) , Âľe (x)) , sup (Ď e (x) , Âľe (x))]. Ëœ GÂľ = HΡ such that for each e E and and is defined as FĎ (e) = (Fe , Ď e ) , e E, where, for each x X, (ii) Intersection: FĎ e∊ e e e e ∀x X, H (x) = ht e F (x) ∧ tG (x) , cF (x) ∧ cG (x) , uF (x) ∨ Fe (x) is the quadruple which represents the truth membership, e e e the contradiction-membership, the ignorance-membership and uG (x) , fF (x) ∨−fG (x)i−and + + the falsity membership of each element x of the universe of dis- Ρe (x) = [inf (Ď e (x) , Âľe c(x)) , infc (Ď e (x) , Âľe (x))]. (FĎ ) = FĎ such that for each e E course X viz. Fe (x) = hteF (x) , ceF (x) , ueF (x) , fFe (x)i (iii) Complement: c (x) = hfFe (x), ueF (x), ceF (x), teF (x)i and and ∀x X, F + − e If ,∀x X and Ď e (x) = [Ď e (x) , Ď e (x)] Int([0, 1]). − c + X = {x1 , x2 , ..., xn } and E = {e1 , e2 , ..., em }, an interval- Ď e (x) = [1 − Ď e (x) , 1 − Ď e (x)] Ëœ Âľ if for each e E and ∀x X, teF (x) ≤ Containment: FĎ âŠ†G valued possibility quadripartitioned single valued neutrosophic (iv) e e e e (x) cG (x) , ueF (x) ≼ ueG (x) , fFe (x) ≼ fG soft set over as, tG (x)−, cF (x) ≤ the soft universe (X, E) is represented + + − (x). (x) ≤ Âľ (x) , Ď (x) ≤ Âľ and Ď e e e e 1 2 FĎ (ei ) = { Fe x(x , Ď ei (x1 ) , Fe x(x , Ď ei (x2 ) , ..., 1) 2) i i xn Example 2. Consider the IPQSNSS FĎ and GÂľ over the Fei (xn ) , Ď ei (xn ) } viz. same soft universe (X, E) defined in example 1. Then, FĎ c is x1 FĎ (ei ) = { tei (x ),cei (x ),u , [Ď âˆ’ (x ) , Ď + ei (x1 )] , obtained as, h F 1 F 1 eFi (x1 ),fFei (x1 )i ei 1 x1 , [0.4, 0.5] , FĎ c (e1 ) = { h0.5,0.4,0.1,0.3i xn ..., tei (x ),cei (x ),u , [Ď âˆ’ (x ) , Ď + ei (xn )] }, ei E, x2 x3 h F n F n eFi (xn ),fFei (xn )i ei n , [0.7, 0.75] , , [0.3, 0.4] } h0.01,0.1,0.2,0.6i h0.6,0.4,0.3,0.7i i = 1, 2, ..., m. x1 FĎ c (e2 ) = { h0.2,0.5,0.3,0.7i , [0.8, 0.9] , x3 x2 , [0.4, 0.55] , h0.2,0.3,0.5,0.5i , [0.6, 0.7] } h0.7,0.6,0.2,0.1i Example 1. Let X = {x1 , x2 , x3 } and E = {e1 , e2 }. Ëœ GÂľ is obtained as, Define an IPQSVNSS over the soft universe (X, E), HΡ = FĎ âˆŞ x1 X HΡ (e1 ) = { h0.8,0.6,0.3,0.4i , [0.8, 0.85] , FĎ : E → QSV as, N S(X) Ă— (Int([0, 1])) x1 x2 x3 FĎ (e1 ) = { h0.3,0.1,0.4,0.5i , [0.5, 0.6] , , [0.4, 0.5] , , [0.6, 0.7] } h0.6,0.2,0.1,0.01i h0.7,0.5,0.3,0.4i x2 x3 x1 h0.6,0.2,0.1,0.01i , [0.25, 0.3] , h0.7,0.3,0.4,0.6i , [0.6, 0.7] } HΡ (e2 ) = { h0.7,0.6,0.3,0.2i , [0.6, 0.75] , x1 x x3 FĎ (e2 ) = { h0.7,0.3,0.5,0.2i , [0.1, 0.2] , 2 , [0.8, 0.9] , , [0.35, 0.5] } h0.4,0.2,0.2,0.7i h0.9,0.7,0.1,0.2i x2 x3 Ëœ GÂľ is defined as, Also, the intersection Kδ = FĎ âˆŠ h0.1,0.2,0.6,0.7i , [0.45, 0.6] , h0.5,0.5,0.3,0.2i , [0.3, 0.4] } x1 Kδ (e1 ) = { h0.3,0.1,0.4,0.5i , [0.5, 0.6] , Another IPQSVNSS GÂľ can be defined over (X, E) as x2 x3 , [0.25, 0.3] , , [0.4, 0.6] } x1 h0.2,0.1,0.1,0.6i h0.5,0.3,0.4,0.6i GÂľ (e1 ) = { h0.8,0.6,0.3,0.4i , [0.8, 0.85] , x1 Kδ (e2 ) = { h0.2,0.3,0.5,0.7i , [0.1, 0.2] , x2 x3 h0.2,0.1,0.1,0.6i , [0.4, 0.5] , h0.5,0.5,0.3,0.4i , [0.4, 0.6] } x2 x3 , [0.45, 0.6] , , [0.3, 0.4] } x1 h0.1,0.2,0.6,0.7i h0.5,0.5,0.3,0.6i GÂľ (e2 ) = { h0.2,0.6,0.3,0.7i , [0.6, 0.75] , x2 x3 Proposition 2. For any FĎ , GÂľ , HΡ IP QSV N SS(X, E), h0.4,0.2,0.2,0.7i , [0.8, 0.9] , h0.9,0.7,0.1,0.6i , [0.35, 0.5] } the following results hold: Ëœ GÂľ = GÂľ âˆŞ Ëœ FĎ Definition 11. The absolute IPQSVNSS over (X, E) is denoted 1. (i) FĎ âˆŞ Ëœ GÂľ = GÂľ ∊ Ëœ FĎ by AËœÂŻ1 such that for each e E and ∀x X, AËœe (x) = h1, 1, 0, 0i (ii) FĎ âˆŠ Ëœ (GÂľ âˆŞ Ëœ HΡ ) = (FĎ âˆŞ Ëœ GÂľ ) âˆŞ Ëœ HΡ and ÂŻ1e (x) = [1, 1] 2. (i) FĎ âˆŞ Ëœ (GÂľ ∊ Ëœ HΡ ) = (FĎ âˆŠ Ëœ GÂľ ) ∊ Ëœ HΡ (ii) FĎ âˆŠ R. Chatterjee, P. Majumdar and S. K. Samanta, Interval-valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets and some uncertainty based measures on them

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Ëœ θËœÂŻ0 = FĎ 3. (i) FĎ âˆŞ Ëœ θËœÂŻ0 = θËœÂŻ0 (ii) FĎ âˆŠ Ëœ AËœÂŻ1 = AËœÂŻ1 (iii) FĎ âˆŞ Ëœ (iv) FĎ âˆŠAËœ ÂŻ1 = FĎ c 4. (i) FĎ c = FĎ (ii) AËœc1ÂŻ = θËœÂŻ0 c (iii) θËœÂŻ0 = AËœÂŻ1 Ëœ GÂľ )c = (FĎ )c ∊ Ëœ (GÂľ )c 5. (i) (FĎ âˆŞ c c Ëœ GÂľ ) = (FĎ ) âˆŞ Ëœ (GÂľ )c (ii) (FĎ âˆŠ Proofs are straight-forward.

4 4.1

Some uncertainty-based measures on IPQSVNSS Entropy measure

Definition 14. Let IP QSV N SS(X, E) denotes the set of all IPQSVNSS over the soft universe (X, E). A mapping Îľ : IP QSV N SS(X, E) → [0, 1] is said to be a measure of entropy if it satisfies the following properties: (e1) Îľ FĎ c = Îľ (FĎ ) e Ëœ Âľ with fFe (x) ≼ fG (x) ≼ (e2)Îľ (FĎ ) ≤ Îľ (GÂľ ) whenever FĎ âŠ†G e e e e e e tG (x) ≼ tF (x), uF (x) ≼ uG (x) ≼ cG (x) ≼ cF (x) and + Ď âˆ’ e (x) + Ď e (x) ≤ 1. (e3) Îľ (FĎ ) = 1 iff teF (x) = fFe (x), ceF (x) = ueF (x) and + Ď âˆ’ e (x) + Ď e (x) = 1, ∀x X and ∀e E.

Neutrosophic Sets and Systems, Vol. 14, 2016 − |ceG (x)−ueG (x)| ≤ |ceF (x)−ueF (x)| and |1−{Âľ+ e (x)+Âľe (x)}| ≤ + − |1 − {Ď e (x) + Ď e (x)}|, ∀x X, ∀e E. Then, e − |teG (x) − fG (x)|.|ceG (x) − ueG (x)|.|1 − {Âľ+ e (x) + Âľe (x)}| e e e e + ≤ |tF (x) − fF (x)|.|cP − {Ď e (x) + Ď âˆ’ e (x)}| F (x) − PuF (x)|.|1 1 e ⇒ 1 − ||X||.||E|| e E x X |tF (x) − fFe (x)|.|ceF (x) − − ueF (x)|.|1 − {Ď + e (x) + PĎ e (x)}| P 1 e e e ≤ 1 − ||X||.||E|| e E x X |tG (x) − fG (x)|.|cG (x) − e + − uG (x)|.|1 − {Âľe (x) + Âľe (x)}| ⇒ Îľ (FĎ ) ≤ Îľ (GÂľ )

(iii) Îľ (FĎ ) = 1 P P 1 e e e ⇔ 1 − ||X||.||E|| e E x X |tF (x) − fF (x)|.|cF (x) − + − ueF (x)|.|1 − {Ď =1 Pe (x)P+ Ď e (x)}| 1 e ⇔ ||X||.||E|| |t (x) − fFe (x)|.|ceF (x) − ueF (x)|.|1 − e E x X F + − {Ď e (x) + Ď e (x)}| = 0 ⇔ |teF (x) − fFe (x)| = 0, |ceF (x) − ueF (x)| = 0, − |1 − {Ď + e (x) + Ď e (x)}| = 0, for each x X and each e E. e e − ⇔ tF (x) = fF (x), ceG (x) = ueG (x), Ď + e (x) + Ď e (x) = 1, for each x X and each e E. Remark from Theorem 1, it follows that, 1. In particular, Ëœ Ëœ Îľ AÂŻ1 = 0 and Îľ θ¯0 = 0. Proof is straight-forward. 4.1.1

An application of entropy measure in decision making problem

The entropy measure not only provides an all over information about the amount of uncertainty ingrained in a particular structure, it can also be implemented as an efficient tool in decision making processes. Often while dealing with a selection process subject to a predefined set of requisitions, the procedure involves allocation of weights in order to signify the order of preference of the criteria under consideration. In what follows next, the Proof: entropy measure corresponding to an IPQSVNSS has been uti P P 1 e lized in defining weights corresponding to each of the elements = 1 − ||X||.||E|| (i) Îľ FĎ c |f (x) − e E x X F e e e − + of the parameter set over which the IPQSVNSS has been defined. tF (x)|.|uF (x) − cF (x)|.|1 P − P{(1 − eĎ e (x)) + (1e − Ď e (x))}| 1 e = 1 − ||X||.||E|| e E x X |tF (x) − fF (x)|.|cF (x) − The algorithm is defined as follows: − ueF (x)|.|1 − {Ď + e (x) + Ď e (x)}| = Îľ (FĎ ).

Theorem 1. The mapping e : IP QSV P N SS(X, [0, 1] P E) → 1 e (x) − |t defined as, Îľ (FĎ ) = 1 − ||X||.||E|| e E x X F − (x)}| is an entropy (x) + Ď fFe (x)|.|ceF (x) − ueF (x)|.|1 − {Ď + e e measure for IPQSVNSS.

e Ëœ Âľ and fG (ii) Suppose that FĎ âŠ†G (x) ≼ teG (x), e e − + uG (x) ≼ cG (x) , Ď e (x) + Ď e (x) ≤ 1. Automatically, + e e e e Âľâˆ’ e (x) + Âľe (x) ≤ 1. Thus, fF (x) ≼ fG (x), tG (x) ≼ tF (x), e e e e − − uF (x) ≼ uG (x), cG (x) ≼ cF (x), Âľe (x) ≼ Ď e (x) , + e e e e Âľ+ e (x) ≼ Ď e (x), and fG (x) ≼ tG (x), uG (x) ≼ cG (x) , − + Ď e (x) + Ď e (x) ≤ 1. e ⇒ fFe (x) ≼ fG (x) ≼ teG (x) ≼ teF (x), ueF (x) ≼ ueG (x) ≼ e e − + + cG (x) ≼ cF (x) , Âľâˆ’ e (x) ≼ Ď e (x) , Âľe (x) ≼ Ď e (x) and − + − + Ď e (x) + Ď e (x) ≤ 1, Âľe (x) + Âľe (x) ≤ 1. e From the above relations it follows that teG (x) − fG (x) ≼ e e e e e e tF (x) − fF (x) but tG (x) − fG (x) ≤ 0, tF (x) − fF (x) ≤ 0 e ⇒ |teG (x) − fG (x)| ≤ |teF (x) − fFe (x)|. Similarly,

Step 1: Represent the data in hand in the form of an IPQSVNSS, say FĎ . Step 2: Calculate the entropy measure Îľ (FĎ ), as defined in Theorem A. Step 3: For each Îą E, assign weights ωF (Îą), given by the formula, P Îľ(FĎ ) 1 Îą ωF (Îą) = ÎşF (Îą) , where ÎşF (Îą) = 1 − ||X||.||E|| x X |tF (x) − Îą + − fFÎą (x)|.|cÎą F (x) − uF (x)|.|1 − {Ď Îą (x) + Ď Îą (x)}|. Step 4: Corresponding to each option x X, calculate the net score, defined P as, Îą Îą Îą score(xi ) = e ωF (Îą).[tF (xi ) + cF (xi ) + {1 − uF (xi )} + {1 − fFÎą (xi )}].{

− Ď + Îą (xi )+Ď Îą (xi ) }. 2

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Step 5: Arrange score(xi ) in the decreasing order of values. 0.99, ωF (l) = 0.984 Step 6: Select maxi {score(xi )}. If maxi {score(xi )} = (4) score(x1 ) = 7.193, score(x2 ) = 9.097, score(x3 ) = 8.554 score(xm ), xm X , then xm is the selected option. (5) score(x2 ) > score(x3 ) > score(x1 ) (6) x2 is the chosen model. Theorem 2. Corresponding to each parameter Îą E, Îľ(FĎ ) ωF (Îą) = ÎşF (Îą) is such that 0 ≤ ωF (Îą) ≤ 1. 4.2 Inclusion measure Definition 15. A mapping I : IP QSV N SS(X, E) Ă— IP QSV N SS(X, E) → [0, 1] is said to be an inclusion measure From the definition of ÎşF (Îą) and Îľ (FĎ ), it is clear that for IPQSVNSS over the soft universe (X, E) if it satisfies the following ωF (Îą) ≼ 0. properties: Îą Îą Îą + Consider |tÎą (x) − f (x)|.|c (x) − u (x)|.|1 − {Ď (x) + (I1) I AËœÂŻ1 , θËœ0 = 0 Îą F F F F − Ď that, Îą (x)}|. Ëœ Âľ P P It follows (I2) I (FĎ , GÂľ ) = 1 ⇔ FĎ âŠ†G Îą Îą + |t (x) − fFÎą (x)|.|cÎą F F (x) − uF (x)|.|1 − {Ď Îą (x) + (I3) if F ⊆G Îą E x X P Ëœ Ëœ ≤ I (GÂľ , FĎ ) and Ď Âľ ⊆HΡ then I (HΡ , FĎ ) Îą Îą Îą Îą Ď âˆ’ Îą (x)}| ≼ x X |tF (x) − fF (x)|.|cF (x) − uF (x)|.|1 − I (H , F ) ≤ I (H , G ) Ρ Ď Îˇ Âľ + − {Ď Îą (x) + Ď Îą (x)}|, whenever 1. P P||X|| ≼ 1 Îą |t (x) − fFÎą (x)|.|cÎą ⇒ 1 − ||X||.||E|| F (x) − Theorem 3. The mapping I : IP QSV N SS(X, E) → [0, 1] Îą E x X F P 1 − + Îą uF (x)|.|1 − {Ď Îą (x) + Ď Îą (x)}| ≤ 1 − ||X||.||E|| x X |tÎą F (x) − defined as, P P − + Îą 1 e (x)}| (x) + Ď (x)|.|1 − {Ď (x) − u fFÎą (x)|.|cÎą I (FĎ , GÂľ ) = 1 − 6||X||.||E|| Îą Îą F F e E x X [|tF (x) − e e e e e ⇒ Îľ (FĎ ) ≤ ÎşF (Îą) min{tF (x), tG (x)}| + |cF (x) − min{cF (x), cG (x)}| + Îľ(FĎ ) e ⇒ ωF (Îą) = ÎşF (Îą) ≤ 1, for each Îą E. (x)} − |max{ueF (x), ueG (x)} − ueF (x)| + |max{fFe (x), fG e − − − fF (x)| + |Ď e (x) − min{Ď e (x), Âľe (x)}| + |Ď + e (x) − + Example 3. Suppose a person wishes to buy a phone and min{Ď + e (x), Âľe (x)}|], is an inclusion measure for IPQSVNSS. the judging parameters he has set are a: appearance, c: cost, b: battery performance, s: storage and l: longevity. Further suppose Proof: that he has to choose between 3 available models, say x1 , x2 , x3 of the desired product. After a survey has been conducted by (i) Clearly, according to the definition of the proposed the buyer both by word of mouth from the current users and measure, I AËœÂŻ , θËœ = 0 1 0 the salespersons, the resultant information is represented in the form of an IPQSVNSS, say FĎ as follows, where it is assumed (ii) From the definition of the proposed measure, it folthat corresponding to an available option, a higher degree of lows that, belongingness signifies a higher degree of agreement with the I (FĎ , GÂľ ) = 1, P P e concerned parameter: ⇔ − min{teF (x), teG (x)}| + e E x X [|tF (x) e e e (x)}| + |max{ueF (x), ueG (x)} − (x), c (x) − min{c |c G F F x1 e e e FĎ (a) = { h0.4,0.3,0.1,0.5i , [0.5, 0.6] , u (x)| + |max{f (x), f (x)} − fFe (x)| + |Ď âˆ’ e (x) − F F G − − + + + x3 x2 min{Ď (x), Âľ (x)}| + |Ď (x) − min{Ď (x), Âľ (x)}|] = e e e e e h0.8,0.1,0.0,0.01i , [0.6, 0.7] , h0.6,0.3,0.2,0.5i , [0.45, 0.5] } 0, ∀x X, ∀e E. x1 FĎ (c) = { h0.8,0.1,0.1,0.2i , [0.7, 0.75] , ⇔ |teF (x) − min{teF (x), teG (x)}| = 0, |ceF (x) − e e e e min{c (x), c (x)}| = 0, |max{u (x), u (x)} − ueF (x)| = 0, x2 x3 F G F G e e e h0.5,0.01,0.1,0.6i , [0.4, 0.55] , h0.7,0.2,0.1,0.1i , [0.6, 0.65] } |max{f (x), f (x)} − f (x)| = 0, |Ď âˆ’ e (x) − F G F − − + + + x1 min{Ď (x), Âľ (x)}| = 0 and |Ď (x)−min{Ď (x), Âľ FĎ (b) = { h0.65,0.3,0.1,0.2i , [0.6, 0.65] , e e e e e (x)}| = 0, ∀x X, ∀e E. x2 x3 h0.8,0.2,0.1,0.0i , [0.75, 0.8] , h0.4,0.5,0.3,0.6i , [0.7, 0.8] } Now, |teF (x) − min{teF (x), teG (x)}| = 0⇔ teF (x) ≤ teG (x). x1 Similarly, it can be shown that, ceF (x) ≤ ceG (x), ueF (x) ≼ FĎ (s) = { h0.5,0.4,0.3,0.6i , [0.7, 0.8] , e − + ueG (x), fFe (x) ≼ fG (x), Ď âˆ’ e (x) ≤ Âľe (x) and Ď e (x) ≤ x2 x3 + Ëœ Âľ. Âľe (x), ∀x X, ∀e E which proves FĎ âŠ†G h0.85,0.1,0.0,0.01i , [0.8, 0.85] , h0.8,0.2,0.1,0.02i , [0.85, 0.9] } x1 , [0.45, 0.55] , FĎ (l) = { h0.6,0.3,0.2,0.5i Ëœ Âľ ⊆H Ëœ Ρ . Thus we have, teF (x) ≤ teG (x) ≤ (iii) Suppose, FĎ âŠ†G x2 x3 e e e tH (x), cF (x) ≤ cG (x) ≤ ceH (x), ueF (x) ≼ ueG (x) ≼ ueH (x), h0.75,0.3,0.3,0.2i , [0.67, 0.75] , h0.75,0.3,0.2,0.2i , [0.7, 0.75] } e e − − fFe (x) ≼ fG (x) ≼ fH (x), Ď âˆ’ e (x) ≤ Âľe (x) ≤ Ρe (x) and + + + Following steps 2-6, we have the following results: Ď e (x) ≤ Âľe (x) ≤ Ρe (x) for all x X and e E. ⇒ I (HΡ , FĎ ) ≤ I (GÂľ , FĎ ). (2) Îľ (FĎ ) = 0.982 In an exactly analogous manner, it can be shown that, (3) ωF (a) = 0.984, ωF (c) = 0.983, ωF (b) = 0.988, ωF (s) = I (HΡ , FĎ ) ≤ I (HΡ , GÂľ ). This completes the proof. Proof:

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Example 4. Consider IPQSVNSS FĎ , GÂľ in Example 1, 4.4 Similarity measure then I (FĎ , GÂľ ) = 0.493. Definition 19. A mapping s : IP QSV N SS(X, E) Ă— IP QSV N SS(X, E) → R+ is said to be a quasi4.3 Distance measure similarity measure between IPQSVNSS if for any Definition 16. A mapping d : IP QSV N SS(X, E) Ă— FĎ , GÂľ , HΡ IP QSV N SS(X, E) it satisfies the following IP QSV N SS(X, E) → R+ is said to be a distance measure be- properties: tween IPQSVNSS if for any FĎ , GÂľ , HΡ IP QSV N SS(X, E) (s1) s (FĎ , GÂľ ) = s (GÂľ , FĎ ) (s2) 0 ≤ s (FĎ , GÂľ ) ≤ 1 and s (FĎ , GÂľ ) = 1 ⇔ FĎ = GÂľ it satisfies the following properties: In addition, if it satisfies (d1) d (FĎ , GÂľ ) = d (GÂľ , FĎ ) Ëœ Âľ ⊆H Ëœ Ρ then s (FĎ , HΡ ) ≤ s (FĎ , GÂľ ) ∧ s (GÂľ , HΡ ) (s3) if FĎ âŠ†G (d2) d (FĎ , GÂľ ) ≼ 0 and d (FĎ , GÂľ ) = 0 ⇔ FĎ = GÂľ then it is known as a similarity measure between IPQSVNSS. (d3) d (FĎ , HΡ ) ≤ d (FĎ , GÂľ ) + d (GÂľ , HΡ ) In addition to the above conditions, if the mapping d satisfies the Various similarity measures for quadripartitioned single condition valued neutrosophic sets were proposed in [5]. Undertaking a (d4) d (FĎ , GÂľ ) ≤ 1, ∀FĎ , GÂľ IP QSV N SS(X, E) similar line of approach, as in our previous work [5] we propose it is called a Normalized distance measure for IPQSVNSS. a similarity measure for IPQSVNSS as follows: Theorem 4. The mapping dh : IP QSV N SS(X, E) Ă— Definition 20. Consider FĎ , GÂľ IP QSV N SS(X, E). Define IP QSV N SS(X, P E) → P R+ defined as, F,G e e e : X → [0, 1], i = 1, 2, .., 5 such that for each dh (FĎ , GÂľ ) = e E x X (|tF (x) − tG (x)| + |cF (x) − functions Ď„i,e e e e e e − x X, e E cG (x)| + |uF (x) − uG (x)| + |fF (x) − fG (x)| + |Ď e (x) − F,G e e + + Âľâˆ’ e (x)|+|Ď e (x)âˆ’Âľe (x)|) is a distance measure for IPQSVNSS. Ď„1,e (x) = |tG (x) − tF (x)| F,G e e It is known as the Hamming Distance. Ď„2,e (x) = |fF (x) − fG (x)| F,G e e Ď„3,e (x) = |cG (x) − cF (x)| Proofs are straight-forward. F,G Ď„4,e (x) = |ueF (x) − ueG (x)| F,G − − Definition 17. The corresponding Normalized Hamming Ď„5,e (x) = |Ď e (x) − Âľe (x)| F,G + distance for IPQSVNSS is defined as dN = Ď„6,e (x) = |Ď + h (FĎ , GÂľ ) e (x) − Âľe (x)| 1 denotes the cardinality d (F , G ), where ||.|| Finally, define a mapping s : IP QSV N SS(X, E) Ă— Ď Âľ 6||X||.||E|| h IP QSV N SS(X, E) → R+ as, s (FĎ , GÂľ ) = 1 − of a set. P P P6 F,G 1 e E x X i=1 Ď„i,e (x) 6||X||.||E|| Theorem 5. The mapping dE : IP QSV N SS(X, E) Ă— IP QSV N SS(X,P E) →P R+ defined as, Theorem 6. The mapping s (FĎ , GÂľ) defined above is a e e 2 e dE (FĎ , GÂľ ) = e E x X {(tF (x) − tG (x)) + (cF (x) − similarity measure. e ceG (x))2 + (ueF (x) − ueG (x))2 + (fFe (x) − fG (x))2 + (Ď âˆ’ e (x) − − 2 + + 2 12 Âľe (x)) + (Ď e (x) − Âľe (x)) } is a distance measure for Proof: IPQSVNSS. It is known as the Euclidean Distance. (i) It is easy to prove that s(FĎ , GÂľ ) = s(GÂľ , FĎ ). Proofs are straight-forward. (ii) We have, teF (x), ceF (x), ueF (x), fFe (x) [0, 1] and F,G Definition 18. The corresponding Normalized Hamming Ď e (x), Âľe (x) Int([0, 1]) for each x X, e E. Thus, Ď„1,e (x) N e e distance for IPQSVNSS is defined as dE (FĎ , GÂľ ) = attains its maximum value if either one of tF (x) or tG (x) is equal 1 to 1 while the other is 0 and in that case the maximum value is 1. 6||X||.||E|| dE (FĎ , GÂľ ). Similarly, it attains a minimum value 0 if teF (x) = teG (x). So, it F,G Ëœ Ëœ Proposition 3. FĎ âŠ†GÂľ ⊆HΡ iff follows that 0 ≤ Ď„1,e (x) ≤ 1, for each x X. Similarly it can be (i) dh (FĎ , HΡ ) = dh (FĎ , GÂľ ) + dh (GÂľ , HΡ ) F,G shown that Ď„ (x), i = 2, ..., 6 lies within [0, 1] for each x X. i,e N N (ii) dN h (FĎ , HΡ ) = dh (FĎ , GÂľ ) + dh (GÂľ , HΡ ) So, P 6 F,G 0 ≤ i=1 Ď„i,e (x) ≤ 6 Proofs are straight-forward. P P Pn F,G (x) ≤ 6||X||.||E|| ⇒ 0 ≤ e E x X i=1 Ď„i,e which implies 0 ≤ s(F , G ) ≤ Ď Âľ Example 5. Consider the IPQSVNSS given in Example 1. Pn 1. Now s(F , G ) = 1 iff Ď Âľ i=1 Ď„i,e (x) = 0 for each x X, e E The various distance measures between the sets are obtained e e e ⇔ t (x) = t (x), c (x) = ceG (x), ueF (x) = ueG (x), N F G F as, d (F , G ) = 5.29, d (F , G ) = 0.882,d (F , G ) = h

4.387,

Ď N dE

Âľ

h

Ď

Âľ

E

Ď

Âľ

(FĎ , GÂľ ) = 0.731

R. Chatterjee, P. Majumdar and S. K. Samanta, Interval-valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets and some uncertainty based measures on them

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Neutrosophic Sets and Systems, Vol. 14, 2016 e − + + fFe (x) = fG (x) and Ď âˆ’ e (x) = Âľe (x), Ď e (x) = Âľe (x) , for all x X, e E i.e.. iff FĎ , GÂľ .

Theorem 7. sω (FĎ , GÂľ ) is a similarity measure. Proof is similar to that of Theorem 6.

Ëœ Âľ ⊆H Ëœ Ρ . then, we have, teF (x) ≤ teG (x) ≤ (iii) Suppose FĎ âŠ†G e e e tH (x), cF (x) ≤ cG (x) ≤ ceH (x), ueF (x) ≼ ueG (x) ≼ ueH (x), e e − − fFe (x) ≼ fG (x) ≼ fH (x), Ď âˆ’ e (x) ≤ Âľe (x) ≤ Ρe (x) and + + + Ď e (x) ≤ Âľe (x) ≤ Ρe (x) for all x X and e E. ConF,G F,G (x). Since teF (x) ≤ teG (x) holds, (x) and Ď„2,e sider Ď„1,e e it follows that, |tG (x) − teF (x)| ≤ |teH (x) − teF (x)|⇒ F,G F,H Ď„1,e (x) ≤ Ď„1,e (x). Similarly it can be shown that F,G F,H Ď„i,e (x) ≤ Ď„i,e (x), for i = 3, 5, 6 and all x X. Next, F,G (x). consider Ď„2,e e e e Since, fF (x) ≼ fG (x) ≼ fH (x), it follows that e e e e e fF (x) − fG (x) ≤ fF (x) − fH (x) where fFe (x) − fG (x) ≼ 0, e e e e e e fF (x)−fH (x) ≼ 0. Thus, |fF (x)−fG (x)| ≤ |fF (x)−fH (x)|⇒ F,H F,G Ď„3,e (x) ≤ Ď„3,e (x). F,G F,H Also, it can be shown that Ď„4,e (x) ≤ Ď„4,e (x) respectively for each x X. P P Pn F,G Thus, we have, ≤ e E x X i=1 Ď„i,e (x) P P Pn F,H Ď„ (x) e E x X i=1 i,e P P Pn F,H 1 ⇒ 1 − 6||X||.||E|| ≤ e E x X i=1 Ď„i,e (x) P P P n F,G 1 1 − 6||X||.||E|| e E x X i=1 Ď„i,e (x) ⇒ s (FĎ , HΡ ) ≤ s (FĎ , GÂľ ) In an analogous manner, it can be shown that s (FĎ , HΡ ) ≤ s (GÂľ , HΡ ). Thus, we have, s (FĎ , HΡ ) ≤ s (FĎ , GÂľ ) ∧ s (GÂľ , HΡ )

Remark 3. sω (FĎ , GÂľ ) is the weighted similarity measure between any two IPQSVNSS FĎ and GÂľ . 4.4.1

Allocation of entropy-based weights in calculating weighted similarity

It was shown in Section 4.1.1 how entropy measure could be implemented to allocate specific weights to the elements of the parameter set. In this section, it is shown how the entropy-based weights can be implemented in calculating weighted similarity. Consider an IPQSVNSS FĎ defined over the soft universe (X, E). Let ωF (e) [0, 1] be the weight allocated to an element e E, w.r.t. the IPQSVNSS FĎ . Define ωF (Îą) as before, viz. P Îľ(FĎ ) 1 Îą , where ÎşF (Îą) = 1 − ||X||.||E|| ωF (Îą) = ÎşF (Îą) x X |tF (x) − − + Îą Îą Îą fF (x)|.|cF (x) − uF (x)|.|1 − {Ď Îą (x) + Ď Îą (x)}| Consider any two IPQSVNSS FĎ , GÂľ IP QSV N SS(X). Following Definition C, the weighted similarity measure between these two sets can be defined P P6 P as Ď„ F,G (x)} x X e E ω(Îą){ P i=1 i sω (FĎ , GÂľ ) = 1 − , where 6||X||.||E|| ω(Îą) e E

Îľ(G )

¾ G (ι) ω(ι) = ωF (ι)+ω , and ωG (ι) = κG (ι) is the weight 2 allocated to the parameter ι E w.r.t. the IPQSVNSS G¾ . From previous results clearly, ωF (ι), ωG (ι) [0, 1] ⇒ ω(ι) [0, 1].

Remark 2. s(AËœÂŻ1 , θËœÂŻ0 ) = 0.

(x)

Example 6. Consider FĎ , GÂľ IP QSV N SS(X) as defined in Example 1. Then s (FĎ , GÂľ ) = 0.738. Also, ωF (e1 ) = 0.983, ωG (e1 ) = 0.987, ωF (e2 ) = 0.993, ωG (e2 ) = 0.988, which gives, ω(e1 ) = 0.985, ω(e2 ) = 0.991 which finally yields = sω (FĎ , GÂľ ) = 0.869.

(x)

=

Proof : For each x X and e E, Ëœ ,θËœ A Ď„1 ÂŻ1 ÂŻ0 (x) = |teθËœ (x) − teAËœÂŻ (x)| ÂŻ 1 0 |fAeËœÂŻ (x) − fθËœe (x)| = 1 1

Ëœ ,θËœ A

1

Ëœ ,θËœ A

1, Ď„2

=

1, Ď„4

ÂŻ 0

Ď„3 ÂŻ1 ÂŻ0 (x) = |ceθËœ (x) − ceAËœÂŻ (x)| ÂŻ 1 0 |ueAËœÂŻ (x) − ueθËœ (x)| = 1 ÂŻ 0

ËœÂŻ1 ,θËœÂŻ0 A

=

ËœÂŻ1 ,θËœÂŻ0 A

5

Relation between the various uncertainty based measures

Ëœ ,θËœ A

− Ď„5 ÂŻ1 ÂŻ0 (x) = |Ď âˆ’ = 1, Ď„6 ÂŻ1 ÂŻ0 (x) = Theorem 8. s1 (F , G ) = 1 − dN (F , G ) is a similarity e (x) − Âľe (x)| Ď Âľ Ď Âľ d h + + |Ď e (x) − Âľe (x)| = 1 measure. P P P6 Ëœ Ëœ A ,θ which yields e E x X i=1 Ď„i ÂŻ1 ÂŻ0 (x) = 6||X||.||E|| P P P6 ËœÂŻ1 ,θËœÂŻ0 A 1 ⇒ s(AËœÂŻ1 , θËœÂŻ0 ) = 1 − 6||X||.||E|| (x) = Proof: e E x X i=1 Ď„i 0. N 1 1 (i) dN h (FĎ , GÂľ ) = dh (GÂľ , FĎ ) ⇒ sd (FĎ , GÂľ ) = sd (GÂľ , FĎ ) N 1 Definition 21. Suppose FĎ , GÂľ IP QSV N SS(X, E). (ii) 0 ≤ dh (FĎ , GÂľ ) ≤ 1 ⇒ 0 ≤ sd (FĎ , GÂľ ) ≤ 1 1 N F,G Consider functions Ď„i,e : X → [0, 1], i = Also, sd (FĎ , GÂľ ) = 1 ⇔ dh (FĎ N, GÂľ ) = 0 ⇔ FĎ N= GÂľ . Ëœ Âľ ⊆H Ëœ Ρ , dh (FĎ , HΡ ) = dh (FĎ , GÂľ ) + Whenever FĎ âŠ†G 1, 2, .., 5 as in Definition 1. Define a mapping sω : (iii) N d (G , H ). Thus, + Âľ Ρ IP QSV N SS(X, E) Ă— IP QSV N SS(X, E) → R as, 1h P P P6 F,G sd (FĎ , GÂľ ) − s1d (FĎ , HΡ ) = 1 − dN ω(e)Ď„i,e (x) h (FĎ , GÂľ ) − 1 + e E x X i=1 P sω (FĎ , GÂľ ) = 1 − , where ω(e) is dN (F , H ) = dN (F , H ) − dN (F , G ) = dN (G , H ) ≼ 6||X||.||E|| e E ω(e) Ď Îˇ Ď Îˇ Ď Âľ Âľ Ρ h h h h the weight allocated to the parameter e E and ω(e) [0, 1], for 0, from property of distance measure. each e E. ⇒ s1d (FĎ , HΡ ) ≤ s1d (FĎ , GÂľ ). Similarly, it can be shown that, s1d (FĎ , HΡ ) ≤ s1d (GÂľ , HΡ ).

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Hence, s1d (FĎ , HΡ ) ≤ s1d (FĎ , GÂľ ) ∧ s1d (GÂľ , HΡ ).

|Îľ(FĎ ) − Îľ(GÂľ )| is a distance measure.

Remark 4. For any similarity measures (FĎ , GÂľ ) , 1−s (FĎ , GÂľ ) Proof: may not be a distance measure. (i) |Îľ(FĎ ) − Îľ(GÂľ )| = |Îľ(GÂľ ) − Îľ(FĎ )| 1 Theorem 9.s2d (FĎ , GÂľ ) = 1+dh (F is a similarity measure. (ii) |Îľ(FĎ ) − Îľ(GÂľ )| ≼ 0 and in particular, |Îľ(FĎ ) − Îľ(GÂľ )| = Ď ,GÂľ ) 0 ⇔ Îľ(FĎ ) = Îľ(GÂľ ) ⇔ Îľ(FĎ ) ≤ Îľ(GÂľ ) and Proof: Îľ(FĎ ) ≼ Îľ(GÂľ ) ⇔ FĎ = GÂľ (iii) Triangle inequality follows from the fact that, (i) dh (FĎ , GÂľ ) = dh (GÂľ , FĎ ) ⇒ s2d (FĎ , GÂľ ) = s2d (GÂľ , FĎ ) |Îľ(FĎ ) − Îľ(HΡ )| ≤ |Îľ(FĎ ) − Îľ(GÂľ )| + |Îľ(GÂľ ) − Îľ(HΡ )| (ii) dh (FĎ , GÂľ ) ≼ 0 ⇒ 0 ≤ s2d (FĎ , GÂľ ) ≤ 1. Also, for any FĎ , GÂľ , HΡ IP QSV N SS(X, E). s2d (FĎ , GÂľ ) = 1 ⇔ dh (FĎ , GÂľ ) = 0 ⇔ FĎ = GÂľ . (iii) dh (FĎ , HΡ ) = dh (FĎ , GÂľ ) + dh (GÂľ , HΡ ) whenever Ëœ Âľ ⊆H Ëœ Ρ. FĎ âŠ†G 6 Conclusions and Discussions ⇒ dh (FĎ , HΡ ) ≼ dh (FĎ , GÂľ ) and dh (FĎ , HΡ ) ≼ dh (GÂľ , HΡ ). 1 1 ≤ 1+dh (F ⇒ s2d (GÂľ , FĎ ) ≤ s2d (FĎ , GÂľ ). In this paper, the concept of interval possibility quadripartitioned ⇒ 1+dh (F Ď ,HΡ ) Ď ,GÂľ ) single valued neutrosophic sets has been proposed. In the present Similarly, it can be shown that, s2d (GÂľ , FĎ ) ≤ s2d (GÂľ , HΡ ). set-theoretic structure an interval valued gradation of possibil1 ity viz. the chance of occurrence of an element with respect to Corollary 1. s3d (FĎ , GÂľ ) = 1+dN (F is a similarity Ď ,GÂľ ) h a certain criteria is assigned and depending on that possibility of measure. occurrence the degree of belongingness, non-belongingness, conProofs follow in the exactly same way as the previous the- tradiction and ignorance are assigned thereafter. Thus, this structure comes as a generalization of the existing structures involvorem. ing the theory of possibility namely, possibility fuzzy soft sets Remark 5. For any similarity measure s (FĎ , GÂľ ) , s(FĎ 1,GÂľ ) − 1 and possibility intuitionistic fuzzy soft sets. In the present work, the relationship between the various uncertainty based measures may not be a distance measure. have been established. Applications have been shown where the Theorem 10 Consider the similarity measure s (FĎ , GÂľ ). entropy measure has been utilized to assign weights to the elements of the parameter set which were later implemented in a Ëœ GÂľ )is an inclusion measure. s (FĎ , FĎ âˆŠ decision making problem and also in calculating a weighted similarity measure. The proposed theory is expected to have wide Proof: applications in processes where parameter-based selection is inËœ Ëœ Ëœ (i) Choose FĎ = A1ÂŻ and GÂľ = θ0ÂŻ . Then, s (FĎ , FĎ âˆŠGÂľ ) = volved. s(AËœÂŻ1 , θËœÂŻ0 ) = 0, from previous result. Ëœ Âľ. Ëœ GÂľ ) = 1 ⇔ FĎ = FĎ âˆŠ Ëœ GÂľ ⇔ FĎ âŠ†G (ii) s (FĎ , FĎ âˆŠ Ëœ Ëœ (iii) Let FĎ âŠ†GÂľ ⊆HΡ . Then, s (FĎ , HΡ ) ≤ s (FĎ , GÂľ ) and 7 Acknowledgements s (FĎ , HΡ ) ≤ s (GÂľ , HΡ ) hold. Consider s (FĎ , HΡ ) ≤ s (FĎ , GÂľ ). From commutative property of similarity measure, The research of the first author is supported by University JRF Ëœ FĎ ) ≤ (Junior Research Fellowship). it follows that, s (HΡ , FĎ ) ≤ s (GÂľ , FĎ ) ⇒ s (HΡ , HΡ âˆŠ The research of the third author is partially supported by the SpeËœ Ëœ Ëœ s (GÂľ , GÂľ ∊FĎ ). Similarly, s (HΡ , HΡ âˆŠFĎ ) ≤ s (FĎ , FĎ âˆŠGÂľ ). cial Assistance Programme (SAP) of UGC, New Delhi, India [Grant no. F 510/3/DRS-III/(SAP-I)]. Ëœ Theorem 11.1 − d (F , F ∊G ) is an inclusion measure. h

Ď

Ď

Âľ

Proof follows from the results of Theorem 8 and Theorem 10. Theorem 12. 1+dh (FĎ 1,FĎ âˆŠG Ëœ Âľ ) and clusion measures.

1 Ëœ 1+dN h (FĎ ,FĎ âˆŠGÂľ )

are in-

Proofs follow from Theorem 9,Corollary 1 and Theorem 10. Theorem 13. Let e : IP QSV N SS(X, E) → [0, 1] be a Ëœ Âľ . Then measure of entropy such that Îľ(FĎ ) ≤ Îľ(GÂľ ) ⇒ FĎ âŠ†G

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R. Chatterjee, P. Majumdar and S. K. Samanta, Interval-valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets and some uncertainty based measures on them

Neutrosophic Sets and Systems, Vol. 14, 2016

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Received: November 15, 2016. Accepted: November 22, 2016

R. Chatterjee, P. Majumdar and S. K. Samanta, Interval-valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets and some uncertainty based measures on them

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