3
Neutrosophic Sets and Systems, 18/2017
University of New Mexico
1HXWURVRSKLF 6RIW 0DWUL[ DQG LWV DSSOLFDWLRQ WR 'HFLVLRQ 0DNLQJ Tuhin Bera1 , Nirmal Kumar Mahapatra 2 1 2 Department
Department of Mathematics, Boror S. S. High School, Bagnan, Howrah-711312,WB, India. E-mail: tuhin78bera@gmail.com
of Mathematics, Panskura Banamali College, Panskura RS-721152,WB, India. E-mail: nirmal hridoy@yahoo.co.in (corresponding author)
Abstract: The motivation of this paper is to extend the concept of Neutrosophic soft matrix (NSM) theory. Some basic definitions of classical matrix theory in the parlance of neutrosophic soft set theory have been presented with proper examples. Then, a theoretical studies of some traditional operations of NSM have been developed.
Finally, a decision making theory has been proposed by developing an appropriate solution algorithm, namely, score function algorithm and it has been illustrated by suitable examples.
Keywords: Intuitionistic fuzzy soft matrix, Neutrosophic soft set, Neutrosophic soft matrix and different operators, Application in decision making.
1
Introduction
Researchers in economics, sociology, medical science, engineering, environment science and many other several fields deal daily with the vague, imprecise and occasionally insufficient information of modeling uncertain data. Such uncertainties are usually handled with the help of the topics like probability, fuzzy sets [1], intuitionistic fuzzy sets [2], interval mathematics, rough sets etc. But, Molodtsov [3] has shown that each of the above topics suffers from inherent difficulties possibly due to inadequacy of their parametrization tool and there after, he initiated a novel concept ‘soft set theory’ for modeling vagueness and uncertainties. It is completely free from the parametrization inadequacy syndrome of different theories dealing with uncertainty. This makes the theory very convenient, efficient and easily applicable in practice. Molodtsov [3] successfully applied several directions for the applications of soft set theory, such as smoothness of functions, game theory, operation research, Riemann integration, Perron integration and probability etc. In 2010, Cagman and Enginoglu [4] introduced a new soft set based decision making method which selects a set of optimum elements from the alternatives. Maji et al. [5, 6] have done further research on soft set theory. Presence of vagueness demanded ‘fuzzy soft set’ [7] to come into picture. But satisfactory evaluation of membership values is not always possible because of the insufficiency in the available information situation. For that, Maji et al. [8, 9] have introduced the notion ‘intuitionistic fuzzy soft set’ in 2001. Matrices play an important role in the broad area of science and engineering. Classical matrix theory sometimes fails to solve the problems involving uncertainties. Hence, several authors proposed the matrix representation of soft set, fuzzy soft set, intuitionistic fuzzy soft set and applied these in certain decision making problems, for instance Cagman and Enginoglu [10], Yong and Chenli [11], Borah et al. [12], Neog and Sut [13], Broumi et al. [14], Mondal and Roy [15], Chetia and Das [16], Basu et al. [17], Rajarajeswari and Dhanalakshmi [18].
Evaluation of non-membership values is also not always possible for the same reason as in case of membership values and so, there exist an indeterministic part upon which hesitation survives. As a result, Smarandache [19, 20] has introduced the concept of Neutrosophic Set (NS) which is a generalisation of classical sets, fuzzy set, intuitionistic fuzzy set etc. Later, Maji [21] has introduced a combined concept Neutrosophic soft set (NSS). Using this concept, several mathematicians have produced their research works in different mathematical structures, for instance Deli [22, 24], Broumi and Smarandache [25]. Later, this concept has been modified by Deli and Broumi [26]. Accordingly, Bera and Mahapatra [23, 27-31] introduce some view on algebraic structure on neutrosophic soft set. The development of decision making algorithms over neutrosophic soft set theory are seen in the literatures [32-37]. The present study aims to extend the NSM theory by developing the basic definitions of classical matrix theory and by establishing some results in NSS theory context. The organisation of the paper is as following : Section 2 deals some preliminary necessary definitions which will be used in rest of this paper. In Section 3, the concept of NSM has been discussed broadly with suitable examples. Then, some traditional operators of NSM are proposed along with some properties in Section 4. In Section 5, a decision making algorithm has been developed and applied in two different situations. Firstly, it has been adopted in a class room to select the best student in an academic year and then in national security system to emphasize the security management in five mega cities. This algorithm is much more brief and simple rather than others. Moreover, a decision can be made with respect to a lot of parameters concerning the fact easily by that. That is why, this algorithm is more generous, we think. Finally, the conclusion of the present work has been stated in Section 6.
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making
Neutrosophic Sets and Systems, 18/2017
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4
Preliminaries
2.4
Definition [20]
In this section, we recall some necessary definitions related to Let X be a space of points (objects), with a generic element fuzzy set, intuitionistic fuzzy soft matrix, neutrosophic set, neu- in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function TA , an indeterminacytrosophic soft set, NSM for the sake of completeness. membership function IA and a falsity-membership function FA . TA (x), IA (x) and FA (x) are real standard or non-standard subsets of ]− 0, 1+ [. That is TA , IA , FA : X →]− 0, 1+ [. There 2.1 Definition [28] is no restriction on the sum of TA (x), IA (x), FA (x) and so, − 0 ≤ sup TA (x) + sup IA (x) + sup FA (x) ≤ 3+ . A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is continuous t norm if ∗ satisfies the following conditions: (i) ∗ is commutative and associative. (ii) ∗ is continuous. 2.5 Definition [3] (iii) a ∗ 1 = 1 ∗ a = a, ∀a ∈ [0, 1]. Let U be an initial universe set and E be a set of parameters. Let (iv) a ∗ b ≤ c ∗ d if a ≤ c, b ≤ d with a, b, c, d ∈ [0, 1]. A few examples of continuous t-norm are a ∗ b = ab, a ∗ b = P (U ) denote the power set of U . Then for A ⊆ E, a pair (F, A) is called a soft set over U , where F : A → P (U ) is a mapping. min{a, b}, a ∗ b = max{a + b − 1, 0}.
2.2
Definition [28]
2.6
A binary operation : [0, 1] × [0, 1] → [0, 1] is continuous t conorm (s - norm) if satisfies the following conditions : (i) is commutative and associative. (ii) is continuous. (iii) a 0 = 0 a = a, ∀a ∈ [0, 1]. (iv) a b ≤ c d if a ≤ c, b ≤ d with a, b, c, d ∈ [0, 1]. A few examples of continuous s-norm are a b = a + b − ab, a b = max{a, b}, a b = min{a + b, 1}.
Let U be an initial universe set and E be a set of parameters. Let N S(U ) denote the set of all NSs of U . Then for A ⊆ E, a pair (F, A) is called an NSS over U , where F : A → N S(U ) is a mapping. This concept has been modified by Deli and Broumi [26] as given below.
2.7 2.3
Definition [16]
Let U be an initial universe, E be the set of parameters and A ⊆ E. Let, (fA , E) be an intuitionistic fuzzy soft set over U . Then a subset of U × E is uniquely defined by RA = {(u, e) : e ∈ A, u ∈ fA (e)} which is called a relation form of (fA , E). The membership function and non-membership functions are written by μRA : U × E → [0, 1] and νRA : U × E → [0, 1] where μRA (u, e) ∈ [0, 1] and νRA (u, e) ∈ [0, 1] are the membership value and non-membership value, respectively of u ∈ U for each e ∈ E. If (μij , νij ) = (μRA (ui , ej ), νRA (ui , ej )), we can define a matrix [(μij , νij )]m×n = ⎛ ⎜ ⎜ ⎜ ⎝
(μ11 , ν11 ) (μ21 , ν21 ) .. .
(μ12 , ν12 ) (μ22 , ν22 ) .. .
... ... .. .
(μ1n , ν1n ) (μ2n , ν2n ) .. .
(μm1 , νm1 )
(μm2 , νm2 )
...
(μmn , νmn )
⎞ ⎟ ⎟ ⎟ ⎠
Definition [21]
Definition [26]
Let U be an initial universe set and E be a set of parameters. Let N S(U ) denote the set of all NSs of U . Then, a neutrosophic soft set N over U is a set defined by a set valued function fN representing a mapping fN : E → N S(U ) where fN is called approximate function of the neutrosophic soft set N . In other words, the neutrosophic soft set is a parameterized family of some elements of the set N S(U ) and therefore it can be written as a set of ordered pairs, N = {(e, {< x, TfN (e) (x), IfN (e) (x), FfN (e) (x) >: x ∈ U }) : e ∈ E} where TfN (e) (x), IfN (e) (x), FfN (e) (x) ∈ [0, 1] are respectively called truth-membership, indeterminacy-membership, falsitymembership function of fN (e). Since supremum of each T, I, F is 1 so the inequality 0 ≤ TfN (e) (x)+IfN (e) (x)+FfN (e) (x) ≤ 3 is obvious.
which is called an m × n IFSM of the IFSS (fA , E) over U . Therefore, we can say that a IFSS (fA , E) is uniquely charac- 2.7.1 Example terised by the matrix [(μij , νij )]m×n and both concepts are interchangeable. The set of all m × n IFS matrices over U will be Let U = {h1 , h2 , h3 } be a set of houses and E = {e1 (beautiful), e2 (good location), e3 , (green surrounding)} be a denoted by IFSMm×n .
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making
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set of parameters describing the nature of houses. Let,
2.9
fN (e1 )
=
fN (e2 )
=
1. Let N be a neutrosophic soft set over N (U ). Then a subset of N (U ) × E is uniquely defined by : RN = {(fN (x), x) : x ∈ E, fN (x) ∈ N (U )} which is called a relation form of (N, E). The characteristic function of RN is written as :
fN (e3 )
=
{< h1 , (0.5, 0.6, 0.3) >, < h2 , (0.4, 0.7, 0.6) >, < h3 , (0.6, 0.2, 0.3) >} {< h1 , (0.6, 0.3, 0.5) >, < h2 , (0.7, 0.4, 0.3) >, < h3 , (0.8, 0.1, 0.2) >} {< h1 , (0.7, 0.4, 0.3) >, < h2 , (0.6, 0.7, 0.2) >, < h3 , (0.7, 0.2, 0.5) >}
Then N = {[e1 , fN (e1 )], [e2 , fN (e2 )], [e3 , fN (e3 )]} is an NSS over (U, E). The tabular representation of the NSS N is given in Table 1. h1 h2 h3
fN (e1 ) fN (e2 ) fN (e3 ) (0.5,0.6,0.3) (0.6,0.3,0.5) (0.7,0.4,0.3) (0.4,0.7,0.6) (0.7,0.4,0.3) (0.6,0.7,0.2) (0.6,0.2,0.3) (0.8,0.1,0.2) (0.7,0.2,0.5) Table 1 : Tabular form of NSS N .
ΘRN : N (U ) × E → [0, 1] × [0, 1] × [0, 1] by ΘRN (u, x) = (TfN (x) (u), IfN (x) (u), FfN (x) (u)) where TfN (x) (u), IfN (x) (u), FfN (x) (u) are truth-membership, indeterminacy-membership and falsity-membership of u ∈ U , respectively. 2. Let U = {u1 , u2 , · · · , um }, E = {x1 , x2 , · · · , xn } and N be a neutrosophic soft set over N (U ). Then, RN fN (x1 ) fN (x2 ) ··· fN (xn ) u1 ΘRN (u1 , x1 ) ΘRN (u1 , x2 ) · · · ΘRN (u1 , xn ) ΘRN (u2 , x1 ) ΘRN (u2 , x2 ) · · · ΘRN (u2 , xn ) u2 .. .. .. .. .. . . . . . um
2.8
Definition [26]
Definition [26]
ΘRN (um , x1 )
ΘRN (um , x2 )
···
ΘRN (um , xn )
If aij = ΘRN (ui , xj ), we can define a matrix ⎛
1. The complement of a neutrosophic soft set N is denoted by N o and is defined by :
⎜ ⎜ [aij ] = ⎜ ⎝
N o = {(e, {< x, FfN (e) (x), 1 − IfN (e) (x), TfN (e) (x) >: x ∈ U }) : e ∈ E}
a11 a21 .. .
a12 a22 .. .
··· ··· .. .
a1n a2n .. .
am1
am2
···
amn
⎞ ⎟ ⎟ ⎟ ⎠
2. Let N1 and N2 be two NSSs over the common universe (U, E). such that aij = (TfN (xj ) (ui ), IfN (xj ) (ui ), FfN (xj ) (ui )) = a a a Then N1 is said to be the neutrosophic soft subset of N2 if ∀e ∈ (Tij , Iij , Fij ), which is called an m × n neutrosophic soft matrix (NS-matrix) of the neutrosophic soft set N over N (U ). E and x ∈ U According to this definition, a neutrosophic soft set N is uniquely characterised by a matrix [aij ]m×n . Therefore, we shall TfN1 (e) (x) ≤ TfN2 (e) (x), IfN1 (e) (x) ≥ IfN2 (e) (x), identify any neutrosophic soft set with it’s soft NS-matrix and FfN1 (e) (x) ≥ FfN2 (e) (x). use these two concepts as interchangeable. The set of all m × n ˜m×n . From now on We write N1 ⊆ N2 and then N2 is the neutrosophic soft superset NS-matrix over N (U ) will be denoted by N we shall delete the subscripts m × n of [a of N1 . ij ]m×n , we use [aij ] ˜ 3. Let N1 and N2 be two NSSs over the common universe (U, E). instead of [aij ]m×n , since [aij ] ∈ Nm×n means that [aij ] is an Then their union is denoted by N1 ∪ N2 = N3 and is defined by m × n NS-matrix for i = 1, 2, · · · , m and j = 1, 2, · · · , n. N3 = {(e, {< x, TfN3 (e) (x), IfN3 (e) (x), FfN3 (e) (x) >: x ∈ U }) : e ∈ E}
2.10
Definition [26]
˜m×n . Then, Let [aij ], [bij ] ∈ N 1. [aij ] is a zero NS-matrix, denoted by [˜ 0], if aij = (0, 1, 1), ∀i, j. 4. Let N1 and N2 be two NSSs over the common universe (U, E). 2. [aij ] is a universal NS-matrix, denoted by [˜1], if aij = Then their intersection is denoted by N1 ∩ N2 = N4 and is de- (1, 0, 0), ∀i, j. fined by : ˜ ij ], if 3. [aij ] is an NS-submatrix of [bij ], denoted by [aij ]⊆[b a b a b a b Tij ≤ Tij , Iij ≥ Iij , Fij ≥ Fij , ∀i, j. N4 = {(e, {< x, TfN4 (e) (x), IfN4 (e) (x), FfN4 (e) (x) >: x ∈ 4. [aij ] and [bij ] are equal NS- matrices, denoted by [aij ] = [bij ], U }) : e ∈ E} if aij = bij , ∀i, j. where TfN4 (e) (x) = TfN1 (e) (x) ∗ TfN2 (e) (x), IfN4 (e) (x) = 5. Complement of [aij ] is denoted by [aij ]o and is defined as a , Tija ), ∀i, j. IfN1 (e) (x) IfN2 (e) (x), FfN4 (e) (x) = FfN1 (e) (x) FfN2 (e) (x). another NS-matrix [cij ] such that cij = (Fija , 1 − Iij where TfN3 (e) (x) = TfN1 (e) (x) TfN2 (e) (x), IfN3 (e) (x) = IfN1 (e) (x) ∗ IfN2 (e) (x), FfN3 (e) (x) = FfN1 (e) (x) ∗ FfN2 (e) (x).
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making
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Neutrosophic Sets and Systems, 18/2017
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Neutrosophic soft matrix h1 h2 h3
In this section, we have introduced some definitions and have included some new operations related to NSM.
3.1
e1 (0.5,0.6,0.3) (0.4,0.7,0.6) (0.6,0.2,0.3)
e2 (0.6,0.3,0.5) (0.7,0.4,0.3) (0.8,0.1,0.2)
e3 (0.7,0.4,0.3) (0.6,0.7,0.2) (0.7,0.2,0.5)
Hence, the NSM corresponding to this NSS N over (U, E) is :
Definition
⎛
⎞ (0.5, 0.6, 0.3) (0.6, 0.3, 0.5) (0.7, 0.4, 0.3) = ⎝ (0.4, 0.7, 0.6) (0.7, 0.4, 0.3) (0.6, 0.7, 0.2) ⎠ (0.6, 0.2, 0.3) (0.8, 0.1, 0.2) (0.7, 0.2, 0.5)
Let U = {u1 , u2 , · · · , um } and E = {e1 , e2 , · · · , en } be the universal set of objects and the parametric set, respectively. Sup- [aij ]3×3 pose, N be a neutrosophic soft set over (U, E) given by N = {< (e, fN (e)) >: e ∈ E} where Next, let E1 = {e1 (cheap), e2 (moderate), e3 (high), e4 (very high)} be another set of parameters describing the cost of houses in U . fN (e) = {< u, (TfN (e) (u), IfN (e) (u), FfN (e) (u)) >: u ∈ U }. The relation form of an NSS M over (U, E1 ) is written as : Thus, fN (e) corresponds a relation on {e} × U i.e., fN (e) = e1 e2 e3 e4 {(e, ui ) : 1 ≤ i ≤ m} for each e ∈ E. It is obviously a symh1 (.4, .5, .5) (.5, .7, .6) (.2, .5, .8) (.5, .6, .4) metric relation. Now, consider a relation RE on U × E given h2 (.6, .4, .7) (.6, .3, .4) (.7, .6, .5) (.8, .4, .3) by RE = {(u, e) : e ∈ E, u ∈ fN (e)}. It is called a relation h3 (.7, .3, .4) (.5, .2, .5) (.8, .4, .4) (.1, .6, .6) form of the NSS N over (U, E). The characteristic function of RE is χRE : U × E → [0, 1] × [0, 1] × [0, 1] and is defined as : χRE (u, e) = (TfN (e) (u), IfN (e) (u), FfN (e) (u)). The tabular Here, the NSM corresponding to the NSS M over (U, E1 ) is [bij ]3×4 = representation of RE is given in Table 2.
u1 u2 .. .
e1 χRE (u1 , e1 ) χRE (u2 , e1 ) .. .
e2 χRE (u1 , e2 ) χRE (u2 , e2 ) .. .
··· ··· ··· .. .
en χRE (u1 , en ) χRE (u2 , en ) .. .
um
χRE (um , e1 )
χRE (um , e2 )
···
χRE (um , en )
Table 2 : Tabular form of RE If aij = χRE (ui , ej ), then we can define a matrix ⎛ a11 a12 · · · a1n ⎜ a21 a22 · · · a2n ⎜ [aij ]m×n = ⎜ . .. .. .. ⎝ .. . . . am1
am2
···
⎞ ⎟ ⎟ ⎟ ⎠
amn
where aij = (TfN (ej ) (ui ), IfN (ej ) (ui ), FfN (ej ) (ui )) = a , Fija ). (Tija , Iij Thus, we shall identify any neutrosophic soft set with it’s NSM and use these two concepts as interchangeable. Since we consider the full parametric set E, so each NSS N over (U, E) corresponds a unique NSM [aij ]m×n where cardinality of U and E are m and n, respectively. To get another NSM of the same order over (U, E), we need to define another NSS over (U, E). The set of all NSMs of order m × n is denoted by N SMm×n . Whenever U and E are fixed, we get all NSMs of unique order i.e., to obtain an NSM of distinct order, at least any of U and E will have to be changed.
⎛
⎞ (.4, .5, .5) (.5, .7, .6) (.2, .5, .8) (.5, .6, .4) ⎝ (.6, .4, .7) (.6, .3, .4) (.7, .6, .5) (.8, .4, .3) ⎠ (.7, .3, .4) (.5, .2, .5) (.8, .4, .4) (.1, .6, .6)
3.2
Definition
a Let A = [aij ] ∈ N SMm×n where aij = (Tija , Iij , Fija ). Then, 1. A is called a square NSM if m = n i.e., if the number of rows and the number of columns are equal. The NSS corresponding to this NSM has the same number of objects and parameters. 2. A square NSM A = [aij ]n×n is called upper triangular NSM if aij = (0, 1, 1), ∀i > j and is called lower triangular NSM if aij = (0, 1, 1), ∀i < j. A is called triangular NSM if it is either neutrosophic soft upper triangular or neutrosophic soft lower triangular matrix. 3. The transpose of a square NSM A = [aij ]n×n is another square NSM of same order obtained from [aij ] by interchanging it’s rows and columns. It is denoted by At . Thus At = [aij ]t = a a a a , Fija )]t = [(Tji , Iji , Fji )]. The NSS corresponding to [(Tija , Iij t A becomes a new NSS over the same universe and the same parametric set. 4. A square NSM A = [aij ]n×n is said to be a symmeta , Fija ) = ric NSM if At = A i.e., if aij = aji or (Tija , Iij a a a (Tji , Iji , Fji ), ∀i, j.
3.3 Definition 3.1.1 Example Consider the Example [2.7.1]. The relation form of the NSS N over the said (U, E) is
a Let A = [aij ] ∈ N SMm×n , where aij = (Tija , Iij , Fija ). Then, the scalar multiple of NSM A by a scalar k is defined by kA = [kaij ]m×n where 0 ≤ k ≤ 1.
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making
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Example
3.3.1 Let
3.6
A = [aij ]2×3 = (0.4, 0.5, 0.5) (0.5, 0.7, 0.6) (0.6, 0.4, 0.7) (0.7, 0.3, 0.4)
(0.5, 0.6, 0.4) (0.8, 0.4, 0.3)
be an NSM. Then the scalar multiple of this matrix by k = 0.5 is kA = [kaij ]2×3 = (0.20, 0.25, 0.25) (0.25, 0.35, 0.30) (0.25, 0.30, 0.20) (0.30, 0.20, 0.35) (0.35, 0.15, 0.20) (0.40, 0.20, 0.15)
3.4
Definition
Let A = [aij ] ∈ N SMm×n , where m = n and aij = a , Fija ). Then, the trace of NSM A is denoted by tr(A) (Tija , Iij
m and is defined as tr(A) = i=1 [Tiia − (Iiia + Fiia )]. 3.6.1
Example
Let
A = [aij ]3×3 = ⎛
(0.5, 0.6, 0.3) ⎝ (0.4, 0.7, 0.6) (0.6, 0.2, 0.3)
Proposition
(0.6, 0.3, 0.5) (0.7, 0.4, 0.3) (0.8, 0.1, 0.2)
⎞ (0.7, 0.4, 0.3) (0.6, 0.7, 0.2) ⎠ (0.7, 0.2, 0.5)
Let A = [aij ], B = [bij ] ∈ N SMm×n where aij = a , Fija ). For two scalars s, k ∈ [0, 1], (Tija , Iij be an NSM. Then tr(A) = (0.5−0.6−0.3)+(0.7−0.4−0.3)+ (i) s(kA) = (sk)A. (ii) s ≤ k ⇒ sA ≤ kA. (iii) A ⊆ B ⇒ (0.7 − 0.2 − 0.5) = −0.4 sA ⊆ sB. Proof.
3.7
(i) s(kA)
a , kFija )] = s[kaij ] = s[(kTija , kIij a a , skFija )] = sk[(Tija , Iij , Fija )] = [(skTija , skIij = sk[aij ] = (sk)A.
a a , Fija ∈ [0, 1], ∀i, j so, sTija ≤ kTija , sIij ≤ (ii) Since Tija , Iij a a a kIij , sFij ≤ kFij . a a Now, sA = [(sTija , sIij , sFija )] ≤ [(kTija , kIij , kFija )] = kA.
(iii) A ⊆ B
⇒ [aij ] ⊆ [bij ] a b ≥ Iij , Fija ≥ Fijb , ∀i, j ⇒ Tija ≤ Tijb , Iij a b sTija ≤ sTijb , sIij ≥ sIij , sFija ≥ sFijb , ∀i, j ⇒ s[aij ] ⊆ s[bij ] ⇒ sA ⊆ sB
⇒
3.5
Theorem
Proposition
a Let A = [aij ] ∈ N SMn×n , where aij = (Tija , Iij , Fija ). If k ∈ [0, 1] is a scalar, then tr(kA) = k tr(A).
n
n Proof. tr(kA) = i=1 [kTiia − (kIiia + kFiia )] = k i=1 [Tiia − (Iiia + Fiia )] = k tr(A).
3.8
Max-Min Product of NSMs
Two NSMs A and B are said to be conformable for the product A ⊗ B if the number of columns of the NSM A be equal to the number of rows of the NSM B and this product becomes also an NSM. If A = [aij ]m×n and B = [bjk ]n×p , then A ⊗ B = [cik ]m×p where a b b b , Fija ), bjk = (Tjk , Ijk , Fjk ) and cik = aij = (Tija , Iij a b a b b (maxj min(Tij , Tjk ), minj max(Iij , Ijk ), minj max(Fija , Fjk )). Clearly, B ⊗ A can not be defined here.
a Let A = [aij ]m×n be an NSM where aij = (Tija , Iij , Fija ). Then, 3.8.1 Example (i) (kA)t = k At for k ∈ [0, 1] being a scalar. ⎛ (ii) (At )t = A. (0.5, 0.6, 0.3) (iii) If A = [aij ]n×n is an upper triangular (lower triangular) Let A = [aij ]3×2 = ⎝ (0.4, 0.7, 0.6) NSM, then At is lower triangular (upper triangular) NSM. (0.6, 0.2, 0.3)
Proof.(i) Here (kA)t , k At ∈ N SMn×m . Now, (kA)t
=
a a a a [(kTija , kIij , kFija )]t = [(kTji , kIji , kFji )]
=
a a a a , Iji , Fji )] = k[(Tija , Iij , Fija )]t = k At . k[(Tji
(ii) Here At ∈ N SMn×m and so (At )t ∈ N SMm×n . Now, (At )t
= =
a a a a t ([(Tija , Iij , Fija )]t )t = [(Tji , Iji , Fji )] a a a [(Tij , Iij , Fij )] = A.
(iii) Straight forward.
and
⎞ (0.6, 0.3, 0.5) (0.7, 0.4, 0.3) ⎠ (0.8, 0.1, 0.2)
B = [bjk ]2×3 =
(0.4, 0.5, 0.5) (0.6, 0.4, 0.7)
(0.5, 0.7, 0.6) (0.7, 0.3, 0.4)
(0.5, 0.6, 0.4) (0.8, 0.4, 0.3)
be two NSMs. Then, A ⊗ B = [cik ]3×3 = ⎛
(0.6, 0.4, 0.5) ⎝ (0.6, 0.4, 0.6) (0.6, 0.4, 0.5)
(0.6, 0.3, 0.5) (0.7, 0.4, 0.4) (0.7, 0.3, 0.4)
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making
⎞ (0.6, 0.4, 0.4) (0.7, 0.4, 0.3) ⎠ (0.8, 0.4, 0.3)
Neutrosophic Sets and Systems, 18/2017
8
One calculation is provided herewith for convenience of A ⊗ B. c T21
a b a b max{min(T21 , T11 ), min(T22 , T21 )}
=
Fijc =
(Fija )w1 · (Fijb )w2 , ∀i, j and w1 , w2 > 0.
(w1 +w2 )
(vii) Harmonic mean A B = C where Tijc =
j
c I21
= =
max{min(0.4, 0.4), min(0.7, 0.6)} = 0.6 a b a b min{max(I21 , I11 ), max(I22 , I21 )}
c F21
= =
min{max(0.7, 0.5), max(0.4, 0.4)} = 0.4 a b a b min{max(F21 , F11 ), max(F22 , F21 )}
=
min{max(0.6, 0.5), max(0.3, 0.7)} = 0.6
a b 2Iij Iij a +I b Iij ij
, Fijc =
a b 2Fij Fij a +F b Fij ij
a b 2Tij Tij c a +T b , Iij Tij ij
=
, ∀i, j.
(viii) Weighted harmonic mean A w B = C where Tijc = w1 +w2 w1 +w2 w1 +w2 c c w1 w2 , Iij = w1 w2 , Fij = w1 w2 , ∀i, j and w1 , w2 > 0. a + a + a +
j
T
ij
Tb ij
I
ij
Ib ij
F
ij
Fb ij
j
4.1
Proposition
Let A = [aij ], B = [bij ] ∈ N SMm×n , where aij = a , Fija ). Then, (Tija , Iij (i) (A ∪ B)t = At ∪ B t , (A ∩ B)t = At ∩ B t . 3.9 Theorem (ii) (A B)t = At B t , (A w B)t = At w B t . t t t w t t w t Let A = [aij ]m×n , B = [bjk ]n×p be two NSMs where (iii) (A B)t = At Bt , (A w B)t = At w Bt . (iv) (A B) = A B , (A B) = A B . a , Fija ). Then, (A ⊗ B)t = B t ⊗ At aij =(Tija , Iij Thus, c21 = (0.6, 0.4, 0.6) and so on.
t t t t t Proof. Let A ⊗ B = [cik ]m×p . Then (A ⊗ B)t = [cki ]p×m , At = Proof. (i) Here A ∪ B, (A ∪ B) , A , B , A ∪ B ∈ N SMm×n . [aji ]n×m , B t = [bkj ]p×n and so the order of (B t ⊗At ) is (p×m). Now, Now, a b ∗ Iij , Fija ∗ Fijb )]t (A ∪ B)t = [(Tija Tijb , Iij
= =
(A ⊗ B)t c c c [(Tki , Iki , Fki )]p×m b a b a [(max min(Tkj , Tji ), min max(Ikj , Iji ), b a , Fji ))]p×m min max(Fkj
j
j
j
=
a a a b b b , Iji , Fji )]n×m = B t ⊗ At . [(Tkj , Ikj , Fkj )]p×n ⊗ [(Tji
Operators of NSMs
a b Let A = [(Tija , Iij , Fija )], B = [(Tijb , Iij , Fijb )] ∈ N SMm×n . Then, c a (i) Union A ∪ B = C where Tijc = Tija Tijb , Iij = Iij ∗ b c a b Iij , Fij = Fij ∗ Fij , ∀i, j. c = (ii) Intersection A ∩ B = C where Tijc = Tija ∗ Tijb , Iij a b c a b Iij Iij , Fij = Fij Fij , ∀i, j.
(iii)Arithmetic mean A B = C where Tijc = a b Iij +Iij , 2
F a +F b
a b Tij +Tij , 2
c Iij =
Fijc = ij 2 ij , ∀i, j. (iv) Weighted arithmetic mean A w B = C where Tijc = a b w1 Tij +w2 Tij c , Iij w1 +w2 w1 , w2 > 0.
=
a b w1 Iij +w2 Iij , w1 +w2
Fijc =
(v) Geometric mean A B = C where a · Ib , F c = Iij Fija · Fijb , ∀i, j. ij ij
a b w1 Fij +w2 Fij , w1 +w2
Tijc
∀i, j and
c = Tija · Tijb , Iij =
(vi) Weighted geometric mean A w B = C where Tijc = (w1 +w2 ) (Tija )w1 · (Tijb )w2 , c a )w1 · (I b )w2 , = (w1 +w2 ) (Iij Iij ij
a b a b a b [(Tji Tji , Iji ∗ Iji , Fji ∗ Fji )]
=
b b a a a b [(Tji , Iji , Fji )] , Iji , Fji )] ∪ [(Tji
=
a b [(Tija , Iij , Fija )]t ∪ [(Tijb , Iij , Fijb )]t
=
At ∪ B t .
Next A ∩ B, (A ∩ B)t , At ∩ B t ∈ N SMm×n . Now, (A ∩ B)t
4
=
=
a b [(Tija ∗ Fijb , Iij (1 − Iij ), Fija Tijb )]t
=
a b a b a b [(Tji ∗ Fji , Iji (1 − Iji ), Fji Tji )]
=
a a a b b b [(Tji , Iji , Fji )] ∩ [(Tji , Iji , Fji )]
=
a b [(Tija , Iij , Fija )]t ∩ [(Tijb , Iij , Fijb )]t
=
At ∩ B t .
Remaining others can be proved in the similar manner.
4.2
Proposition
Let A = [aij ], B = [bij ] are upper triangular (lower triangular) NSMs of same order. Then (i) A ∪ B, A ∩ B (ii) A B, A w B (iii) A B, A w B all are upper triangular (lower triangular) NSMs. Proof. Straight forward.
4.3
Theorem
Let A = [aij ], B = [bij ] be two symmetric NSMs of same order. Then, (i) A ∪ At , A ∪ B, A ∩ B, A B, A w B, A B, A w B, A B, A w B are so.
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Neutrosophic Sets and Systems, 18/2017
(ii) A ⊗ B is symmetric iff A ⊗ B = B ⊗ A. (iii) A ⊗ At , At ⊗ A both are symmetric.
=
a b , Tija )] [(Fijb , 1 − Iij , Tijb )] [(Fija , 1 − Iij
=
a b [(Tija , Iij , Fija )]o [(Tijb , Iij , Fijb )]o
= Ao B o . Proof. Here At = A and B t = B as both are symmetric NSMs. t w w Clearly A âˆŞ A , A âˆŞ B, A ∊ B, A B, A B, A B, A B, A B, A w B, A ⊗ B, B ⊗ A, A ⊗ At , At ⊗ A all are well Next, for w1 , w2 > 0, we have, deďŹ ned as both the NSMs are same order and square. Now, (A w B)o (i) These are left to the reader. a b + w2 Iij w1 Tija + w2 Tijb w1 Iij w1 Fija + w2 Fijb o (ii) (A ⊗ B)t = B t ⊗ At = B ⊗ A = A ⊗ B. = [( , , )] w1 + w2 w1 + w2 w1 + w2 (iii) (A ⊗ At )t = (At )t ⊗ At = A ⊗ At and (At ⊗ A)t = a b At ⊗ (At )t = At ⊗ A. + w2 Iij w1 Fija + w2 Fijb w1 Iij w1 Tija + w2 Tijb = [( ,1 − , )] w1 + w2 w1 + w2 w1 + w2 a b ) + w2 (1 − Iij ) w1 Fija + w2 Fijb w1 (1 − Iij 4.4 Proposition = [( , , w1 + w2 w1 + w2 a b Let A = [(Tija , Iij , Fija )], B = [(Tijb , Iij , Fijb )] ∈ N SMmĂ—n . w1 Tija + w2 Tijb Then, )] w1 + w2 (i) (A âˆŞ B)o = Ao ∊ B o , (A ∊ B)o = Ao âˆŞ B o . a b , Tija )] w [(Fijb , 1 − Iij , Tijb )] = [(Fija , 1 − Iij (ii) (A B)o = Ao B o , (A w B)o = Ao w B o . Proof. (i) Here (A âˆŞ B)o , Ao ∊ B o ∈ N SMmĂ—n . Now, (A âˆŞ B)o
=
a b [(Tija Tijb , Iij ∗ Iij , Fija ∗ Fijb )]o
=
a b [(Fija ∗ Fijb , 1 − (Iij ∗ Iij ), Tija Tijb )]
=
b a [(Fija ∗ Fijb , (1 − Iij ), Tija Tijb )] ) (1 − Iij
=
a b [(Fija , 1 − Iij , Tija )] ∊ [(Fijb , 1 − Iij , Tijb )]
=
a b [(Tija , Iij , Fija )]o ∊ [(Tijb , Iij , Fijb )]o
=
Ao ∊ B o .
Next, (A ∊ B)o , Ao âˆŞ B o ∈ N SMmĂ—n . Now, (A ∊ B)
o
= = = = = =
a b [(Tija ∗ Tijb , Iij Iij , Fija Fijb )]o a b [(Fija Fijb , 1 − (Iij Iij ), Tija ∗ Tijb )] a b [(Fija Fijb , (1 − Iij ) ∗ (1 − Iij ), Tija ∗ Tijb )] a b [(Fija , 1 − Iij , Tija )] âˆŞ [(Fijb , 1 − Iij , Tijb )] a b [(Tija , Iij , Fija )]o âˆŞ [(Tijb , Iij , Fijb )]o o o
A âˆŞB .
=
4.5
(A B)o
= = =
a b + Iij Tija + Tijb Iij Fija + Fijb o [( , , )] 2 2 2 a b + Iij Iij Tija + Tijb Fija + Fijb ,1 − , )] [( 2 2 2 a b Fija + Fijb (1 − Iij ) + (1 − Iij ) Tija + Tijb [( , , )] 2 2 2
Proposition (Commutative law)
a b Let A = [(Tija , Iij , Fija )], B = [(Tijb , Iij , Fijb )] ∈ N SMmĂ—n . Then, (i) A âˆŞ B = B âˆŞ A, A ∊ B = B ∊ A (ii) A B = B A, A w B = B w A (iii) A B = B A, A w B = B w A (iv) A B = B A, A w B = B w A.
Proof. Obvious
4.6
Proposition (Associative law)
a b Let A = [(Tija , Iij , Fija )], B = [(Tijb , Iij , Fijb )], C = c c c [(Tij , Iij , Fij )] ∈ N SMmĂ—n . Then, (i) (A âˆŞ B) âˆŞ C = A âˆŞ (B âˆŞ C) (ii) (A ∊ B) ∊ C = A ∊ (B ∊ C) (iii) (A B) C = A (B C) (iv) (A B) C = A (B C) (v) (A B) C = A (B C).
Proof. (i) Clearly (A âˆŞ B) âˆŞ C, A âˆŞ (B âˆŞ C) ∈ N SMmĂ—n . Now,
a b a b ) (1 − Iij ) = 1 − (Iij ∗ Iij ) and (1 − Note : Here, (1 − Iij a b a b Iij ) ∗ (1 − Iij ) = 1 − (Iij Iij ) hold for dual pairs of nonparameterized t-norms and s-norms e.g., a ∗ b = min{a, b} and a b = max{a, b}, a ∗ b = max{a + b − 1, 0} and a b = min{a + b, 1} etc.
(ii) Here (A B)o , Ao B o ∈ N SMmĂ—n .
a b [(Tija , Iij , Fija )]o w [(Tijb , Iij , Fijb )]o = Ao w B o .
=
(A âˆŞ B) âˆŞ C a b c ∗ Iij , Fija ∗ Fijb )] âˆŞ [(Tijc , Iij , Fijc )] [(Tija Tijb , Iij
=
a b c [((Tija Tijb ) Tijc , (Iij ∗ Iij ) ∗ Iij , (Fija ∗ Fijb ) ∗ Fijc )]
=
a b c [(Tija (Tijb Tijc ), Iij ∗ (Iij ∗ Iij ), Fija ∗ (Fijb ∗ Fijc ))]
=
A âˆŞ (B âˆŞ C)
Similarly, the other results can be veriďŹ ed.
4.7
Proposition (Distributive law)
a b Let A = [(Tija , Iij , Fija )], B = [(Tijb , Iij , Fijb )], C = c c c [(Tij , Iij , Fij )] ∈ N SMmĂ—n . Then, (i) A ∊ (B C) = (A ∊ B) (A ∊ C), (A B) ∊ C =
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Neutrosophic Sets and Systems, 18/2017
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(A C) ∪ (B C). (A ∩ C) (B ∩ C). (ii) A ∪ (B C) = (A ∪ B) (A ∪ C), (A B) ∪ C = A (B ∩ C) = (A B) ∩ (A C), (A ∩ B) C = (A C) ∩ (B C). (A ∪ C) (B ∪ C). (v) A (B ∪ C) = (A B) ∪ (A C), (A ∪ B) C = Proof. (i) Here A ∩ (B C), (A ∩ B) (A ∩ C) ∈ N SMm×n . (A C) ∪ (B C). Now, A (B ∩ C) = (A B) ∩ (A C), (A ∩ B) C = (A C) ∩ (B C). A ∩ (B C) = = =
= =
b c + Iij Fijb + Fijc Tijb + Tijc Iij , , )] 2 2 2 b c + Iij Tijb + Tijc a Iij Fijb + Fijc [(Tija ∗ , Iij , Fija )] 2 2 2 a b a c Tija ∗ Tijb + Tija ∗ Tijc Iij Iij + Iij Iij [( , , 2 2 Fija Fijb + Fija Fijc )] 2 a b a b [(Tij ∗ Tij , Iij Iij , Fija Fijb )] a c Iij , Fija Fijc )] [(Tija ∗ Tijc , Iij (A ∩ B) (A ∩ C)
Proof. We shall here prove (i), (iv) and (v) only. The others can be proved in the similar fashion. (i) Here A ∩ (B ∪ C), (A ∩ B) ∪ (A ∩ C) ∈ N SMm×n . Now,
a [(Tija , Iij , Fija )] ∩ [(
=
min{Fijb , Fijc })] = = = =
a b [(min{Tija , Tijb }, max{Iij , Iij }, max{Fija , Fijb })]
a c , Iij }, max{Fija , Fijc })] ∪[(min{Tija , Tijc }, max{Iij (A ∩ B) ∪ (A ∩ C)
Next (A ∪ B) ∩ C, (A ∩ C) ∪ (B ∩ C) ∈ N SMm×n . Now, =
(A ∪ B) ∩ C a b , Iij }, min{Fija , Fijb })] [(max{Tija , Tijb }, min{Iij c , Fijc )] ∩[(Tijc , Iij
=
a b c [(min{max{Tija , Tijb }, Tijc }, max{min{Iij , Iij }, Iij },
max{min{Fija , Fijb }, Fijc })] =
b c [(Tijb ∗ Tijc , Iij Iij , Fijb Fijc )] (A ∩ C) (B ∩ C)
a c [(max{min{Tija , Tijc }, min{Tijb , Tijc }}, min{max{Iij , Iij }, b c max{Iij , Iij }}, min{max{Fija , Fijc }, max{Fijb , Fijc }})]
=
In a similar way, the remaining can be established.
a c , Iij }, max{Fija , Fijc })] [(min{Tija , Tijc }, max{Iij
b c ∪[(min{Tijb , Tijc }, max{Iij , Iij }, max{Fijb , Fijc })]
=
Proposition (Distributive law)
a b Let A = [(Tija , Iij , Fija )], B = [(Tijb , Iij , Fijb )], C c c c [(Tij , Iij , Fij )] ∈ N SMm×n . If a ∗ b = min{a, b} and a b = max{a, b}, then (i) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), (A ∪ B) ∩ C (A ∩ C) ∪ (B ∩ C). (ii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), (A ∩ B) ∪ C (A ∪ C) ∩ (B ∪ C). (iii)A (B ∪ C) = (A B) ∪ (A C), (A ∪ B) C (A C) ∪ (B C). A (B ∩ C) = (A B) ∩ (A C), (A ∩ B) C (A C) ∩ (B C). (iv) A (B ∪ C) = (A B) ∪ (A C), (A ∪ B) C
a b [(max{min{Tija , Tijb }, min{Tija , Tijc }}, min{max{Iij , Iij }, a c max{Iij , Iij }}, min{max{Fija , Fijb }, max{Fija , Fijc }})]
(A B) ∩ C a b + Iij Tija + Tijb Iij Fija + Fijb c = [( , , )] ∩ [(Tijc , Iij , Fijc )] 2 2 2 a b + Iij Tija + Tijb Iij Fija + Fijb c = [( ∗ Tijc , Iij Fijc )] , 2 2 2 a c b c Iij + Iij Iij Tija ∗ Tijc + Tijb ∗ Tijc Iij = [( , , 2 2 Fija Fijc + Fijb Fijc )] 2 a c Iij , Fija Fijc )] = [(Tija ∗ Tijc , Iij
4.8
a b c [(min{Tija , max{Tijb , Tijc }}, max{Iij , min{Iij , Iij }},
max{Fija , min{Fijb , Fijc }})]
Next (A B) ∩ C, (A ∩ C) (B ∩ C) ∈ N SMm×n . Now,
=
A ∩ (B ∪ C) a b c , Fija )] ∩ [(max{Tijb , Tijc }, min{Iij , Iij }, [(Tija , Iij
=
= =
(A ∩ C) ∪ (B ∩ C)
(iv) Here A (B ∪ C), (A B) ∪ (A C) ∈ N SMm×n . Now, = =
= = =
=
A (B ∪ C) a b c , Fija )] [(max{Tijb , Tijc }, min{Iij , Iij }, [(Tija , Iij min{Fijb , Fijc })] a · min{I b , I c }, [( Tija · max{Tijb , Tijc }, Iij ij ij Fija · min{Fijb , Fijc })] a · Ib , [(max{ Tija · Tijb , Tija · Tijc }, min{ Iij ij a · I c }, min{ F a · F b , Iij Fija · Fijc })] ij ij ij
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making
Neutrosophic Sets and Systems, 18/2017
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=
=
a · Ib , Tija · Tijb , Iij Fija · Fijb )] ij a · Ic , ∪[( Tija · Tijc , Iij Fija · Fijc )] ij
[(
=
min{Fijb , Fijc } + Fijc
Next (A ∪ B) C, (A C) ∪ (B C) ∈ N SMm×n . Now,
=
(A C) ∪ (B C)
(v) Here A (B ∪ C), (A B) ∪ (A C) ∈ N SMm×n . Now, =
[(
a b c · min{Iij , Iij } 2 · Tija · max{Tijb , Tijc } 2 · Iij , a , a b c b c Tij + max{Tij , Tij } Iij + min{Iij , Iij }
Fija + min{Fijb , Fijc }
[(max{
2Tija Tijb
Tija
+
Tijb
,
)]
a b Iij 2Tija Tijc 2Iij }, min{ , a c a b Tij + Tij Iij + Iij
a c 2Iij Iij 2Fija Fijb 2Fija Fijc }, min{ , })] a + Ic Iij Fija + Fijb Fija + Fijc ij
2Tija Tijb
a b Iij 2Iij
2Fija Fijb
=
)] a + Ib F a + F b Tija + Tijb Iij ij ij ij a c Iij 2Fija Fijc 2Tija Tijc 2Iij ∪[( a , a c c , F a + F c )] Tij + Tij Iij + Iij ij ij
=
(A B) ∪ (A C)
[(
,
,
Next (A ∪ B) C, (A C) ∪ (B C) ∈ N SMm×n . Now, =
[(max{
a c Iij 2Tijb Tijc 2Iij 2Tija Tijc , b }, min{ a a c c , c Tij + Tij Tij + Tij Iij + Iij
b c 2Iij Iij
=
4.9
2Fija Fijc 2Fijb Fijc , })] Fija + Fijc Fijb + Fijc
b + Ic Iij ij a c Iij 2Fija Fijc 2Tija Tijc 2Iij [( a , a + I c , F a + F c )] Tij + Tijc Iij ij ij ij
∪[( =
}, min{
2Tijb Tijc
Tijb
+
Tijc
,
b c Iij 2Iij
b Iij
+
c Iij
,
2Fijb Fijc
Fijb + Fijc
)]
(A C) ∪ (B C)
Proposition (Idempotent law)
a Let A = [(Tija , Iij , Fija )] ∈ N SMm×n . Then, (i) A w A = A (ii) A w A = A (iii) A w A = A.
Proof. For all i, j and w1 , w2 > 0 we have, a a w1 Tij +w2 Tij w1 I a +w2 I a w1 F a +w2 F a , wij1 +w2 ij , wij1 +w2 ij , )] w1 +w2
2 · Fija · min{Fijb , Fijc }
=
=
)]
(i) A w A = [( a [(Tija , Iij , Fija )] = A.
A (B ∪ C) a , Fija )] [(Tija , Iij b c [(max{Tijb , Tijc }, min{Iij , Iij }, min{Fijb , Fijc })]
=
a b c , Iij } · Iij 2 · max{Tija , Tijb } · Tijc 2 · min{Iij , , b , Ic } + Ic max{Tijb , Tijc } + Tijc min{Iij ij ij
2 · min{Fija , Fijb } · Fijc
(A B) ∪ (A C)
(A ∪ B) C a b , Iij }, min{Fija , Fijb })] = [(max{Tija , Tijb }, min{Iij c , Fijc )] [(Tijc , Iij a , Ib } · Ic , = [( max{Tija , Tijb } · Tijc , min{Iij ij ij min{Fija , Fijb } · Fijc , )] a · Ic , = [(max{ Tija · Tijc , Tijb · Tijc }, min{ Iij ij b · I c }, min{ F a · F c , Iij Fijb · Fijc })] ij ij ij a · Ic , Fija · Fijc )] = [( Tija · Tijc , Iij ij b · Ic , Fijb · Fijc )] ∪[( Tijb · Tijc , Iij ij
[(
(A ∪ B) C a b , Iij }, min{Fija , Fijb })] [(max{Tija , Tijb }, min{Iij c , Fijc )] [(Tijc , Iij
=
(ii) A w A = [( (w1 +w2 ) (Tija )w1 · (Tija )w2 , a )w1 · (I a )w2 , (w1 +w2 ) (F a )w1 · (F a )w2 )] (w1 +w2 ) (Iij ij ij ij a a = [( (w1 +w2 ) (Tij )w1 +w2 , (w1 +w2 ) (Iij )w1 +w2 , a (w1 +w2 ) (Fija )w1 +w2 )] = [(Tija , Iij , Fija )] = A. (iii) A w A a [(Tija , Iij , Fija )]
5
5.1
= A.
=
[(
w1 +w2 w1 w + T a2 Ta ij ij
,
w1 +w2 w1 w + I a2 Ia ij ij
,
w1 +w2 w1 w + F a2 Fa ij ij
)]
=
Neutrosophic soft matrix theory in decision making (score function algorithm) Definition
a 1. Let A = [aij ]m×n be an NSM where aij = (Tija , Iij , Fija ). Then the value of the matrix A is denoted by V (A) and is defined a a a ]m×n where vij = Tija − Iij − Fija , ∀i, j. as : V (A) = [vij 2. The score of two NSMs A and B is defined as S(A, B) = a b [sij ]m×n where sij = vij + vij . So, S(A, B) = V (A) + V (B). 3. The total score for each object in U is Σnj=1 sij .
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making
Neutrosophic Sets and Systems, 18/2017
5.2
Properties of Score Function
Value matrices are classical real matrices which follow all properties of classical real matrices. The score function is basically a real matrix in classical sense derived from two or more value matrices. So score functions obey all properties of real matrices.
5.3
Methodology
12
N = {fN (e1 ), fN (e2 ), fN (e3 ), fN (e4 ), fN (e5 )} where fN (e1 )
=
fN (e2 )
=
fN (e3 )
=
{< s1 , (0.6, 0.4, 0.5) >, < s2 , (0.7, 0.4, 0.2) >, < s3 , (0.9, 0.4, 0.2) >} {< s1 , (0.5, 0.5, 0.6) >, < s2 , (0.8, 0.5, 0.1) >, < s3 , (0.6, 0.7, 0.5) >} {< s1 , (0.7, 0.3, 0.4) >, < s2 , (0.8, 0.5, 0.3) >, < s3 , (0.5, 0.6, 0.7) >} {< s1 , (0.7, 0.5, 0.3) >, < s2 , (0.6, 0.7, 0.5) >, < s3 , (0.5, 0.5, 0.5) >} {< s1 , (0.6, 0.4, 0.6) >, < s2 , (0.6, 0.3, 0.7) >, < s3 , (0.8, 0.3, 0.3) >}}
Suppose, N number of decision makers wish to select an ob- fN (e4 ) = ject jointly from m number of objects i.e., universal set U with respect to n number of features i.e., parametric set E. Each deci- fN (e5 ) = sion maker forms an NSS over (U, E) and corresponding to each NSS, each get an NSM of order m × n. It needs to compute the value matrix corresponding to each matrix. Then the score matrix P = {fP (e1 ), fP (e2 ), fP (e3 ), fP (e4 ), fP (e5 )} where and finally, the total score of each object will be calculated. fP (e1 ) = {< s1 , (0.8, 0.3, 0.3) >, < s2 , (0.8, 0.5, 0.3) >, < s3 , (1.0, 0.4, 0.2) >} 5.3.1 Algorithm fP (e2 ) = {< s1 , (0.6, 0.4, 0.5) >, < s2 , (0.7, 0.6, 0.2) >, Step 1 : Construct the NSMs from the given NSSs. < s3 , (0.8, 0.5, 0.4) >} Step 2 : Calculate the value matrices of corresponding NSMs. fP (e3 ) = {< s1 , (0.8, 0.4, 0.1) >, < s2 , (0.7, 0.5, 0.5) >, Step 3 : Compute the score matrix from value matrices and the < s3 , (0.6, 0.7, 0.3) >} total score for each object in U . (e ) = {< s1 , (0.6, 0.6, 0.2) >, < s2 , (0.8, 0.6, 0.4) >, f P 4 Step 4 : Find the object of maximum score and it is the optimal solution. < s3 , (0.7, 0.3, 0.6) >} Step 5 : If score is maximum for more than one object, then find f (e ) = {< s , (0.8, 0.4, 0.2) >, < s , (0.6, 0.4, 0.3) >, P 5 1 2 Σnj=1 (sij )k , k ≥ 2 successively. Choose the object of maximum < s3 , (0.7, 0.5, 0.4) >}} score and hereby the optimal solution.
5.3.2
Case study 1 (application in class room)
The above three NSSs are represented by the NSMs A, B and C, respectively, as following :
⎛ (.7, .2, .6) (.4, .6, .7) (.5, .5, .3) (.6, .6, .5) Three students {s1 , s2 , s3 } from class - x in a school have been shortened to win the best student award in an academic session. ⎝ (.6, .3, .5) (.7, .6, .3) (.7, .4, .4) (.5, .8, .6) (.8, .3, .5) (.5, .5, .4) (.6, .4, .6) (.4, .7, .4) A team of three teachers {T1 , T2 , T3 } has been formed by the Head Master of that school for this purpose. Final selection is ⎛ (.6, .4, .5) (.5, .5, .6) (.7, .3, .4) (.7, .5, .3) based on the set of parameters {e1 , e2 , e3 , e4 , e5 } indicating the ⎝ (.7, .4, .2) (.8, .5, .1) (.8, .5, .3) (.6, .7, .5) quality of student, participation in school cultural programme, (.9, .4, .2) (.6, .7, .5) (.5, .6, .7) (.5, .5, .5) class room interactions, maintenance of discipline in class room, ⎛ daily attendance, respectively. Teachers have given their valuable (.8, .3, .3) (.6, .4, .5) (.8, .4, .1) (.6, .6, .2) opinions by the following NSSs separately i.e., first NSS given by ⎝ (.8, .5, .4) (.7, .6, .2) (.7, .5, .5) (.8, .6, .4) first teacher and so on. (1, .4, .2) (.8, .5, .4) (.6, .7, .3) (.7, .3, .6) M = {fM (e1 ), fM (e2 ), fM (e3 ), fM (e4 ), fM (e5 )} where Then the corresponding value matrices are : ⎛ −.1 −.9 −.3 −.5 0.1 fM (e1 ) = {< s1 , (0.7, 0.2, 0.6) >, < s2 , (0.6, 0.3, 0.5) >, V (A) = ⎝ −.2 −.2 −.1 −.9 −.1 < s3 , (0.8, 0.3, 0.5) >} 0.0 −.4 −.4 −.7 0.6 fM (e2 ) = {< s1 , (0.4, 0.6, 0.7) >, < s2 , (0.7, 0.6, 0.3) >, ⎛ −.3 −.6 0.0 −.1 −.4 < s3 , (0.5, 0.5, 0.4) >} V (B) = ⎝ 0.1 0.2 0.0 −.6 −.4 fM (e3 ) = {< s1 , (0.5, 0.5, 0.3) >, < s2 , (0.7, 0.4, 0.4) >, 0.3 −.6 −.8 −.5 0.2 ⎛ < s3 , (0.6, 0.4, 0.6) >} 0.2 −.3 0.3 −.2 0.2 fM (e4 ) = {< s1 , (0.6, 0.6, 0.5) >, < s2 , (0.5, 0.8, 0.6) >, V (C) = ⎝ −.1 −.1 −.3 −.2 −.1 < s3 , (0.4, 0.7, 0.4) >} 0.4 −.1 −.4 −.2 −.2
⎞ (.8, .3, .4) (.7, .2, .6) ⎠ (.9, .1, .2) ⎞ (.6, .4, .6) (.6, .3, .7) ⎠ (.8, .3, .3) ⎞ (.8, .4, .2) (.6, .4, .3) ⎠ (.7, .5, .4) ⎞ ⎠ ⎞ ⎠ ⎞ ⎠
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making
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Neutrosophic Sets and Systems, 18/2017
⎛
The score matrix is : ⎛
⎜ ⎜ A=⎜ ⎜ ⎝
⎞
−0.2 −1.8 00.0 −0.8 −0.1 S(A, B, C) = ⎝ −0.2 −0.1 −0.4 −1.7 −0.6 ⎠ 00.7 −1.1 −1.6 −1.4 00.6 ⎛
⎛
⎞
⎜ ⎜ B=⎜ ⎜ ⎝
−2.9 and the total score = ⎝ −3.0 ⎠ −2.8 Hence, the student s3 will be selected for the best student award from class-x in that academic session.
5.3.3
fN1 (a)
=
fN1 (b)
=
fN1 (c)
=
{< D, (0.9, 0.4, 0.5) >, < M, (0.8, 0.5, 0.4) >, < K, (0.7, 0.6, 0.6) >, < C, (0.6, 0.4, 0.7) >, < B, (0.5, 0.3, 0.8) >} {< D, (0.8, 0.5, 0.5) >, < M, (0.9, 0.3, 0.3) >, < K, (0.7, 0.6, 0.5) >, < C, (0.6, 0.7, 0.8) >, < B, (0.6, 0.8, 0.5) >} {< D, (0.7, 0.5, 0.4) >, < M, (0.9, 0.3, 0.2) >, < K, (0.5, 0.6, 0.7) >, < C, (0.7, 0.4, 0.6) >, < B, (0.6, 0.3, 0.4) >}
N2 = {fN2 (a), fN2 (b), fN2 (c)} where fN2 (a)
=
fN2 (b)
=
fN2 (c)
=
{< D, (1.0, 0.5, 0.4) >, < M, (0.9, 0.4, 0.5) >, < K, (0.7, 0.7, 0.5) >, < C, (0.6, 0.5, 0.3) >, < B, (0.6, 0.7, 0.4) >} {< D, (0.9, 0.4, 0.5) >, < M, (0.9, 0.2, 0.3) >, < K, (0.8, 0.5, 0.4) >, < C, (0.7, 0.7, 0.6) >,
(0.8, 0.5, 0.5) (0.9, 0.3, 0.3) (0.7, 0.6, 0.5) (0.6, 0.7, 0.8) (0.6, 0.8, 0.5)
(0.7, 0.5, 0.4) (0.9, 0.3, 0.2) (0.5, 0.6, 0.7) (0.7, 0.4, 0.6) (0.6, 0.3, 0.4)
(1.0, 0.5, 0.4) (0.9, 0.4, 0.5) (0.9, 0.4, 0.5) (0.9, 0.2, 0.3) (0.7, 0.7, 0.5) (0.8, 0.5, 0.4) (0.6, 0.5, 0.3) (0.7, 0.7, 0.6) (0.6, 0.7, 0.4) (0.6, 0.8, 0.7)
(0.8, 0.3, 0.2) (0.9, 0.2, 0.1) (0.4, 0.5, 0.6) (0.5, 0.6, 0.6) (0.7, 0.4, 0.3)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
Then the corresponding value matrices are : ⎛ ⎞ 0.0 −.2 −.2 ⎜ −.1 0.3 0.4 ⎟ ⎜ ⎟ ⎟ V (A) = ⎜ ⎜ −.5 −.4 −.8 ⎟ ⎝ −.5 −.9 −.3 ⎠ −.6 −.7 −.1 ⎛ ⎞ 0.1 0.0 0.3 ⎜ 0.0 0.4 0.6 ⎟ ⎜ ⎟ ⎟ V (B) = ⎜ ⎜ −.5 −.1 −.7 ⎟ ⎝ −.2 −.6 −.7 ⎠ −.5 −.9 0.0
Case study 2 (application in security management)
An important discussion on internal security management has been arranged by the order of Home Minister. Two officers have mate in that discussion to analyse and arrange the security management in five mega-cities e.g., Delhi(D), Mumbai(M), Kolkata(K), Chennai(C), Bengaluru(B). The priority of management is given to the cities based on the set of parameters {a, b, c} indicating their geographical position(e.g., having international boarder line, having sea coast etc ), population density, past history of terrorist attack, respectively. Following NSSs refer the opinions of two officers individually regarding that matter. N1 = {fN1 (a), fN1 (b), fN1 (c)} where
(0.9, 0.4, 0.5) (0.8, 0.5, 0.4) (0.7, 0.6, 0.6) (0.6, 0.4, 0.7) (0.5, 0.3, 0.8)
The score matrix and the total score for selection are : ⎛ ⎞ 00.1 −0.2 00.1 ⎜ −0.1 00.7 01.0 ⎟ ⎜ ⎟ ⎟ S(A, B) = ⎜ ⎜ −1.0 −.5 −1.5 ⎟ ⎝ −0.7 −1.5 −1.0 ⎠ −1.1 −1.6 −0.1 ⎛ ⎞ 00.0 ⎜ 01.6 ⎟ ⎟ ⎜ ⎟ Total score = ⎜ ⎜ −3.0 ⎟ ⎝ −3.2 ⎠ −2.8 Hence, the priority of security management should be given in descending order to Mumbai, Delhi, Bangaluru, Kolkata and Chennai.
6
Conclusion
< B, (0.6, 0.8, 0.7) >} {< D, (0.8, 0.3, 0.2) >, < M, (0.9, 0.2, 0.1) >,
In this paper, some definitions regarding neutrosophic soft matrices have been brought and some new operators have been included, illustrated by suitable examples. Moreover, application of neutrosophic soft matrix theory in decision making problems have been made. We expect, this paper will promote the future study on different algorithms in several other decision making problems.
< K, (0.4, 0.5, 0.6) >, < C, (0.5, 0.6, 0.6) >, < B, (0.7, 0.4, 0.3) >}
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Received: October 2, 2017. Accepted: October 25, 2017.
Tuhin Bera, Nirmal Kumar Mahapatra, Neutrosophic Soft Matrix and its application to Decision Making