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9
QUADRASOPHIC FUZZY MATRIX AND ITS
APPLICATION
G. Aruna1, J. Jesintha Rosline2 •
1,2 PG and Research Department of Mathematics, Auxilium College (Autonomous) , Affiliated to Thiruvalluvar University, Vellore Dist., Tamil Nadu, India [email protected] , [email protected]
Abstract
Quadrasophic Fuzzy Set is one of the generalizations of Fuzzy set theory. In this artifact, a definition of the Quadrasophic Fuzzy Algebra and its characteristics are provided. The definition of a Quadrasophic Fuzzy Matrix is explored with the aid of Quadrasophic Fuzzy Algebra. The binary operators of Fuzzy Matrices are used to describe various kinds and specific operations on Quadrasophic Fuzzy Matrices. The theorems and results of Quadrasophic Fuzzy Matrix are demonstrated with pertinent examples and proofs. Additionally, the illustration of the identification of paddy illnesses is analyzed with the tool of Quadrasophic Fuzzy Matrix in the decision-making process.
Keywords: Quadrasophic Fuzzy Set, Quadrasophic Fuzzy Algebra, Quadrasophic Fuzzy Matrix, Operations, Plant illness identification
1. Introduction
The ability to recognize both the advantages and disadvantages of a problem is crucial in problem-solving. In recent years, we have considered other elements while making judgements, like the environment and peer pressure. The bipolar fuzzy set addresses both the problems in positive and negative aspects. However, the environment's impact on the problem is crucial to manage the unclear scenario. The current fuzzy set extension is insufficient to account for the environment's influence or the rate at which the issue is becoming more prevalent. With the unique characteristics of positive reluctant membership and negative reluctant membership functions, QFS facilitates solving such issues.
One of the generalizations of fuzzy set theory is the Quadrasophic Fuzzy Set (QFS) [1]. Many industries, including medicine, business, decision-making, agriculture, etc., rely heavily on QFS. Any sort of work has both a positive and a negative aspect.These aspects might change due to influential variables, producing different kinds of outcomes. QFS considers the impact of influential factors as the reluctant positive side, i.e., the partial positive side, and the reluctant negative side [1]. Let's take an example where someone is driving from City A to City B. Positive membership indicates that the traveller has reached City B; negative membership indicates that there has been a delay in reaching the destination because of traffic or road construction; these variations in positive and negative membership values are therefore regarded as positive and negative reluctant memberships. The QFS quantifies the value of positive membership in the interval [0,1], negative membership in the interval [-1,0], negative reluctant ranges in [-0.5,0], and postive reluctant ranges in [0,0.5] [2].
In numerous distinct domains, matrices are significant. The traditional matrix is insufficient to cover unpredictable circumstances. The ambiguous situations make it possible to create
the idea of a fuzzy matrix to depict the actual situation. Thomason discussed the convergence of power in a fuzzy matrix in 1977 [3]. Fuzzy matrices were first conceptualized by Hashimoto. Several authors established and validated numerous results, including determinant, trace, Nil potency, adjoint matrices, adjacent matrices, Idem potency, features of transitive and closure, and power series convergence in fuzzy matrices as well as in their different expansions [4], [5], [6].
Cho conducted research on the regularity characteristics and rankings of fuzzy matrices in 1999 [7]. Pal applied numerous matrix concepts to the fuzzy sets. Pal investigated and established Interval-valued Intuitionistic fuzzy matrices as well as the Intuitionistic fuzzy determinant [5], [8]. The notion of a matrix was applied to triangular fuzzy numbers by A.K. Shyamal et al. [9]. They also developed triangular fuzzy matrices and addressed their specific types of properties. Pal (2016) developed four binary fuzzy operators, expanded the idea of fuzzy matrices with fuzzy rows and columns, and applied them to image processing [10]. In 2016, Mondal and Pal investigated the eigen vectors and eigen values of bipolar fuzzy matrices [11]. In 2019, Pal and Sanjib defined and introduced the concept of bipolar fuzzy matrices, their power convergence, and their applications[12]. Dogra and Pal introduced the picture fuzzy matrix with the determinant and adjoint of its unique restricted square picture fuzzy matrices [13]. Fuzzy matrices were created and used in a variety of fields to handle uncertain situations. Numerous authors, including Sanchez, Pal, Muthuraji, Nagoor Gani, and Meenakshi, have examined and applied various results on fuzzy intuitionistic matrix, fuzzy interval valued matrix, triangular fuzzy matrix, bipolar fuzzy matrix, picture fuzzy matrix, neutrosophic fuzzy number, and fuzzy soft matrix in the field of decision making [14],[15], [16], [17], [18]. Inspired by these inventions, we define the Quadrasophic Fuzzy Matrix (QFM) in this artifact. Also, we explore the application of Quadrasophic Fuzzy Matrix in the detection of plant illnesses.
In this artifact, Section 2 provides an overview of QFS's fundamental definitions and functions. In Section 3, we defined the operations of QFS, and Section 4 presented the introduction to Quadrasophic Fuzzy Algebra. The Quadrasophic Fuzzy Matrix is defined in Section 5 and includes relevant examples along with its operations and propositions. In Section 6, the identification of plant diseases serves as an example of the use of the Quadrasophic Fuzzy Matrix in decision-making. The conclusion and the scope of future investigation potential are presented in Section 7.
2. Preliminaries
Quadrasophic fuzzy set [1]: The Quadrasophic fuzzy set q on the set U is defined as
q = {( ^ nq ( x) , \ ( x) , \Fq ( x) , Vq ( x) ) |x G U}
where the degree of positive membership grade is Vq (x) : U ^ [ 0,1], the degree of negative membership is nq(x) : U ^ [ -1,0] the degree of restricted positive membership is Apq (x) : U ^ [ 0,0.5] the degree of restricted negative membership is Anq (x) : U ^ [ -0.5,0] And the condition follows: -1 < pq(x) + nq(x) < 1, -0.5 < Aq < 0.5 and 0 < + rf + tf < 3 for all x G U, such that Aq = Length of (hp. , Xnq).
Intersection operation of QFS [2]: Let qa, qb G QFS, then for all x G X the intersection of qa and qb is represented as:
q« n qb =( nqa ( x) V nqb ( x) , Anqa (x) V \b (x) , AVqa (x) A AVqb (x) , Vqa ( x) A Vqb ( x) )
Union operation of QFS [2]: Let qa, qb G QFS, then for all x G X the union of q« and qb is
represented as:
q« U qb =( nq« ( x) A nqb ( x) , Anq« (x) A Anqb (x) , AVq« (x) V AVqb (x) , Vq« ( x) V Vqb ( x) )
Complement[2]: Let q € QFS, then the complement of q is described as qc = ( —1 — nq, -0.5 —
,°.5 — A^q ,1 — Vq ).
Score- valued function [2]: The score- valued function score( q) of QFS is represented as
Sc0re( q) = (x)+S(x) +nq(x)+Anq(x) , for all score( q) € [ —1,1].
The 0 operator: [6] For u, v € [0,1], the 0 operator can be defined as :
{u, if u > v „
0, if u < v.
3. Certain Operations on Quadrasophic Fuzzy Sets
Operations on Quadrasophic Fuzzy Sets: Let qi = ( -A^, A^, pqi) and q2 = ( -^q2,-A7,2, A^2, Vq2) be two Quadrasophic Fuzzy sets in the set QF.
Disjunction: qi + q2 = ( nqi,-Anqy V^ Vqi) + ( ^^ ^ Vq2) = {- max(nqi, nq2) , -max ( Anqi , A7q2 ) , max ( A^qi , AHq2 ) , max ( Vqi, Vq2 ) }
= {-( nqi V Vq2 ) , -( Anqi V AVq2 ) , ( AHqi V AVq2 ) , ( Vqi V Vq2 ) }
Parallel conjunction: qi ■ q2 = ( nqi , -AVq^ AHq1, Vqi ) + ( -nq2, ^q^ ^ Vq2 )
= {-min( nqi, n?2 ) , -min( Anq1 , Anq2 ) ,
min( AFq^ AFq2 ) , min( ^ ^ ) }
= {-( nqi A nq2 ) , -( Anq1 A A7q2 ) , ( AHq1 A AFq2 ) , ( Vqi A Vq2 ) }
Serial conjunction: qi 0 q2 =( nqi , -AVq^ AHq1, Vqi ) + ( -nq2, ^q^ hq^ Vq2 )
= {-(( nqi A Vq2 ) V ( Vqi A ^2 )) } {-(( Anq1 A AFq2 ) V ( AFq1 A A^ )) },
{Anq1 A AVq2 ) V ( AVq1 A AFq2 ) },
{-(( nqi A Vq2 ) V ( Vqi A Vq2 )) } Negation of qi: - ql = - ( nqi,-Anq1, AVq1, Vqi)
= (-Vqi,-AVq1, Anq1, nqi )
Complement of qi: ^ q1 = ^ ( -AnqV A-^ Vqi)
= ( -1 + nqi, -0.5 + Anq1, 0.5 - AVq1 , 1 - Vqi )
QFS 0 operator:
qi 0 q2
'-nqi, for -( nqi > ^
Anqi, for -( AVq1 > AVq2 )
AVqi, for ( % > AVq2 )
Fqi, for ( Vqi > Vq2)
0, otherwise.
Zero element of QFS: The zero element of QFS is represented by 0q and is described as 0q = (0,0,0,0).
Unit element of QFS: The unit element of QFS is represented by iq and is described as iq = ( -1, -0.5,0.5,1).
4. Quadrasophic Fuzzy Algebra
A Quadrasophic Fuzzy Algebra is a mathematical structure (QF, +,.) with two binary operations (+ and .) defined on QFS that fulfill the following principle.
Theorem 1. Consider qi = ( -nqv -Anq1, AVq1, Vqi) q2 = ( -nq2, -Anq2, AVq2, Vq2) q3 = ( -nqз, -Anq3, AVq3, Vq3) belongs to QFS. Then the results as follows:
i. Law of Idempotent: qi + qi = qi, qi.qi = qi
ii. Law of Commutativity: q1 + q2 = q2 + q1, q1.q2 = q2.q1
iii. Law of Associativity: qi + (q2 + q3) = (qi + q2) + q3, qi.(q2.q3) = (qi.q2 ).q3
iv. Law of Absorption: q1 + (q1.q2) = q1, q1.(q1 + q2) = q1
v. Law of Distributivity: qi.(q2 + q3) = (qi.q2) + (qi.q3), qi + (q2.q3) = (qi + q2).(qi + q3)
vi. Universal Law: qi + 0q = qi, qi + iq = iq, qi.0q = 0q, qi.iq = qi
Proof. i qi + qi = ( -{nqi V nqi, }, -{Ann V AVq1 } {AVq1 V AVq1 }, {Vqi V Vqi }) = ( -nqi,-Anq1, AVq1, Vqi ) = q1
Similarly, qi ■ qi = ( -{nqi A nqi, }, -{Anq1 A Anq1 } {AVq1 A AVq1 } {Vqi A Vqi}) = ( -nqi, -Anq1, AVq1, Vqi ) = q1
ti. qi + q2 =( -{nqi V nq2, } -{Anq1 V Anq2 }, {AVq1 V AVq2 }, {Vqi V Vq2 })
= ( -nqi, -Anq2, AVq1, Vq2 )
= ( -{nq2 V nqi , }, -{Anq2 V Anq1 } {AVq2 V AVq1 }, {Vq2 V Vqi }) = ( -nq2, -Anq1 , AVq2, Vqi )
= q2 + qi
Similarly, qi ■ q2 = q2 ■ qi.
m. qi + (q2 + q3) =( -Anq1, avq^ Vqi) +(-{nq2 V nqз,},-{Anq2 V Anq3}
{AVq2 V AVq3 } {Vq2 V Vq3 })
= -m«x( nqi , {nq2 V nq3 }) , -m«x( Anq1 , {A7q2 V AVq3 }) ,
m«x( AVq1 , { AVq2 V AVq3 } ) , m«x( Vqi, { Vq2 V Vq3 } )
= -( nqi V {nq2 V nq3 }) , -( V V {A7q2 V \3 }) ,
( AVq1 V {AVq2 V AVq3 }) , ( Vqi V {Vq2 V Vq3 })
( qi + q2) + q3 =( -{nqi V nq2, }, -{AVq1 V Anq2 } {AVq1 V AVq2 } {Vqi V Vq2 }) + ( -nq3, -Anq3, AVq3, Vq3 )
= -m«x( {nqi V nq2 }, nq3 ) , -m«x( {Anq1 V Anq2 }, Anq3 ) , m«x( {AVq2 V AVq3 } AVq1 ) , m«x( Vql, { Vq2 V Vq3 })
= - ( nqi V {nq2 V nq3 }) , - ( V V {AVq2 V A7q3 } ) ,
( AVq1 V {AVq2 V AVq3 }) , ( Vqi V {Vq2 V Vq3 })
Hence, q1 + (q2 + q3) = (q1 + q2) + q3 In similar way, qi • (q2 • q3) = (q1 • q2) • q3
iv. q1 + (q1 • q2) =( — nq^ —Anq1, %) +( — {nq1 A nq2, } —{Anq1 A Anq2}
{AVq1 A AVq2 } {Vq1 A Vq2 })
= — ( nq1 V {nq1 A nqi }) , — ( Anq1 V {AVq1 A AVq2 } ) ,
( AVq1 V {AVq1 A AVq2 }) , ( % V W A Vq2 }) = ( —nq1, —Anq1, AFq1, Vq1 ) = q1
Similarly, q1.( q1 + q2) = q1
v. q1 • ( q2 + q3) =( — n^ —Anq^ ^ % ) • ( —{nq2 V nqз, }, —{Anq2 V Anq3 } {AVq2 V AVq3 } {%2 V Vq3 })
= —( nq1 A{nq2 V nq3 }) , —( Anq1 A {AVq2 V Anq3 }) , ( AVq1 A {AFq2 V AVq3 }) , ( % A {Vq2 V Vq3 })
= —( {nq1 A nq2 }V{nq1 A ^3 }) , ( {AVq1 A AVq2 } V {AVq1 A Aq3 }) , ( {AVq1 A AVq2 }V{AVq1 A AVq3 }) , ( W A Vq2 } V W A Vq3 }) ( q1 • q2) + ( q1 • q3) =( — {^1 A ^2 } —{Anq1 A K2 }, {AVq1 A AFq2 } W A Vq2 })
+ ( —{nq1 A Vq3 } —{Anq1 A Anq3 } {AVq1 A AVq3 }, W A Vq3 }) = —( {nq1 A nq2 }V{nq1 A ^3 }) , —( {AVq1 A AVq2 } V {AVq1 A Anq3 }) , ( {AVq1 A AVq2 }V{AVq1 A AVq3 }) , ( W A Vq2 } V W A Vq3 })
Thus, q1 • (q2 + q3) = (q1 • q2) + (q1 • q3) Similarly, q1 + (q2 • q3) = (q1 + q2) • (q1 + q3)
vi. q1 + 0q =( —nqu —Anq1 , AVq1 , % ) + ( 0, 0, 0, 0)
= ( {nq1 V 0}, —{Anq1 V 0}, {A^ V 0}, V 0})
= ( — nq1, — Anq1, AFq1, Vq1 ) = q1
q1 + iq =( — nq1 , — Anq1 , hq^ % ) + ( —1 —0.5, 0.5, 1)
= ( {nq1 V —1}, —{Anq1 V —0.5}, {Avq1 V 0.5}, V 1}) = ( —1, —0.5,0.5,1)
= iq
Similarly, we can prove q1 • 0q = 0q and q1.iq = q1. ■ Thus, ( QFS, +, •) is Quadrasophic Fuzzy Algebra (QFA).
Example 2: Let us consider three Quadrasophic Fuzzy Sets q1=( —0.4, —0.1,0.3,0.5), q2 =
( -0.5, -0.3,0.4,0.7) and q3 = ( -0.7, -0.4,0.2,0.6)
i.q1 + (q1 ■ q2) =q1 + ( -0.4, -0.1,0.3,0.5) = ( -0.4, -0.1,0.3,0.5)
= q1
ii.q1 ■ (q1 + q2) =q1.( -0.5, -0.3,0.4,0.7)
= (-0.4, -0.1,0.3,0.5)
= qi.
iii. q1 + (q2 ■ q3) =q1 + ( -0.5, -0.3,0.2,0.6) = ( -0.5, -0.3,0.3,0.6) iv.( q1 + q2) ■ (q1 + q3) =( -0.5, -0.3,0.4,0.7) ■ ( -0.7,-0.4,0.3,0.6)
= (-0.5, -0.3,0.3,0.6)
Hence, qi + (q2 ■ q3) = (qi + q2) ■ (qi + q3)
5. Quadrasophic Fuzzy Matrix
Let Qrxs be a Quadrasophic Fuzzy Matrix, where
Qr
qii qi2 ■ ■ ■ qis q2i q22 ■ ■ ■ q2s
qri qr2 ■ ■ ■ qrs
Here, qj = ( -nq¡j,-AVqi., A^,,, Vq¡j), (nqij, Vqij) G [0,1] represents the positive and negative grade, and (Anq,,, A^ ,) G [0,0.5] represents the restricted positive and restricted negative grade of qij. A Quadrasophic Fuzzy Matrix (QFM) is the matrix defined over the Quadrasophic Fuzzy Algebra.
Zero Quadrasophic Matrix: The zero Quadrasophic Fuzzy Matrix 0qs of order s x s is the matrix where all the elements are 0q = ( 0, 0, 0, 0) .
Identity Quadrasophic Matrix: The identity Quadrasophic Matrix Iqs of order s x s is the matrix where all the diagonal entries are iq = ( -1, -0.5,0.5,1) and all the other entries are
0q = ( 0,0,0,0) .
Remark 1. Mrs indicates the set of all QFM's of order r x s, while Ms indicates the set of square QFM's of order s x s.
5.1. Operations on Quadrasophic Fuzzy Matrix
Binary opertors of QFM: Let Q1 = (q1ij) and Q2 = (q2ij) be two QFMs of order r x s. Then the following operations as follows:
Qi + Q2 = (qiij + q2ij) rxs
= {-m«x( nqij , nq2ij ) , -m«x( Anq1ij , Anq2ij ) , m«x( AVq1ij , AVq2ij ) , m«x( Vqij, Vq2ij ) }
Qi ■ q2 = (qiij ■ q2ij) rxs
= {-min( nqiij, nq2ij ) ,-min( Anq1ij , AVq2ij ) , min( AVq1ij, AVq2ij ) , min( Vqiij, Vq2ij ) }
© and ® operators of QFM: Let Q1 = (qiij) rxs and Q2 = (q2j) sxt be two QFMs. Then the following operations as follows:
s
Qi © Q2 = ( ^ qiii ■ q2ij) rxt i=i
= ( -m«xs=i ( min {n qm , nq2lj }) , ( -m«xs=1( min{Anq1il , Anq2lj }) ,
m«x\=1 ( min { AVqia , AVq2l. } ) , m«x\=i ( min { Vqil , Vq2lj } ) )
s
Qi ® Q2 = (n qui + q2j) rxt i=i
= ( -mins=i ( m«x {n qm , nq2lj }) , ( -mins=1( m«x{Anqw , Anq2lj }) , mins=1 ( m«x{AVqia , AVq } ) , mins=i ( m«x{Vqiil, Vq2lj }) )
Theorem 2. Let Qi, Q2, and Q3 be three Quadrasophic Fuzzy matrices, and then we have the following results:
i. Qi + Q2 = Q2 + Qi, Qi ■ Q2 = Q2 ■ Qi
ii. Qi + (Q2 + Q3) = (Qi + Q2) + Q3, Qi ■ (Q2 ■ Q3) = ( Qi ■ Q2) ■ Q3
iii. Qi ■ (Q2 + Q3) = (Q1.Q2) + (Qi ■ Q3) , Qi + (Q2 ■ Q3) = (Qi + Q2) ■ (Qi + Q3)
iv. Qi + 0q = 0q + Qi = Qi, Qi ■ 0q = 0q ■ Qi = 0q, where 0q is the zero QFM. Proof. To prove the results, let us consider the following example:
Qi
Q2
(-0.3,-0.1,0.3,0.5) (-0.4,-0.4,0.5,0.7)
(-0.2,-0.1,0.4,0.7) (-0.7,-0.3,0.4,0.8)
(-0.4,-0.3,0.4,0.7) (-0.6,-0.2,0.5,0.8)
(-0.6,-0.2,0.3,0.9) (-0.7,-0.4,0.3,0.5)
(i.) Qi + Q2
{-m«x(0.3,0.4), -m«x(0.1,0.3), {-m«x(0.4,0.6), -m«x(0.4,0.2),
m«x(0.3,0.4), m«x(0.5,0.7)} m«x(0.5,0.5), max (0.7,0.8)}
{-m«x(0.2,0.6), -m«x(0.1,0.2), {-m«x(0.7,0.7), -m«x(0.3,0.4), m«x(0.4,0.3), m«x(0.7,0.9)} m«x(0.4,0.3), max (0.8,0.5)}
Qi + Q2
(-0.4, -0.3,0.4,0.7) (-0.6, -0.4,0.5,0.8) (-0.6, -0.2,0.4,0.9) (-0.7, -0.4,0.4,0.8)
Qi + Q2
{-m«x(0.4,0.3), -m«x(0.3,0.1),
m«x(0.4,0.3), max (0.7,0.5)} {-m«x(0.2,0.6), -m«x(0.1,0.2), m«x(0.4,0.3), max (0.7,0.9)}
{-m«x(0.6,0.4), -m«x(0.2,0.4),
m«x(0.5,0.5), m«x(0.8,0.7)} {-m«x(0.7,0.7), -m«x(0.4,0.3), m«x(0.3,0.4), m«x(0.5,0.8)}
Q2 + Qi
(-0.4, -0.3,0.4,0.7) (-0.6, -0.4,0.5,0.8) (-0.6, -0.2,0.4,0.9) (-0.7, -0.4,0.4,0.8)
Hence, Q1 + Q2 = Q2 + Q1
Q1 • Q2
Q2 • Q1
{ —min(0.3,0.4), —min(0.1,0.3), {—min(0.4,0.6), — min (0.4,0.2),
min (0.3,0.4), min (0.5,0.7)} min(0.5,0.5), min(0.7,0.8)} { —min(0.2,0.6), —min (0.1,0.2), {—min (0.7,0.7), —min (0.3,0.4),
min (0.4,0.3), min (0.7,0.9)} min(0.4,0.3), min (0.8,0.5)}
Q1 • Q2
(—0.3, —0.1,0.3,0.5) (—0.4, —0.3,0.5,0.7) (—0.2, —0.1,0.3,0.7) (—0.7, —0.3,0.3,0.5)]
{ —min(0.4,0.3), — min (0.3,0.1), {—min (0.6,0.4), —min (0.4,0.3),
min (0.4,0.3), min (0.7,0.5)} min (0.3,0.4), min(0.5,0.8)} { —min(0.6,0.2), — min (0.2,0.1), {—min (0.7,0.7), —min (0.4,0.3),
min (0.3,0.4), min (0.9,0.7)} min (0.3,0.4), min (0.5,0.8)}
Q2 • Q1
Q1 • Q2 = Q2 • Q1, Thus, (i.) is proved. (ii.) The proof is obvious. (iii.)
(—0.3, —0.1,0.3,0.5) (—0.4, —0.3,0.5,0.7) (—0.2, —0.1,0.3,0.7) (—0.7, —0.3,0.3,0.5)]
Q1 • ( Q2 + Q3) =
(—0.3, —0.1,0.3,0.5) (—0.4, —0.3,0.5,0.7) (—0.2, —0.1,0.3,0.7) (—0.7, —0.3,0.4,0.8)
( Q1 • Q2) + ( Q1 • Q3)
(—0.3, —0.1, (—0.4, —0.3,'
0.3,0.5) 0.5,0.7)
(—0.2, —0.1, (—0.7, —0.3,
0.3,0.7) 0.3,0.5)
+
(—0.3, —0.1, (—0.4, —0.3,
0.3,0.5) 0.2,0.7)
(—0.2, —0.1, (—0.7, —0.3,
0.2,0.5) 0.4,0.8)
( Q1 • Q2) + ( Q1 • Q3)
(—0.3, —0.1,0.3,0.5) (—0.4, —0.3,0.5,0.7) (—0.2, —0.1,0.3,0.7) (—0.7, —0.3,0.4,0.8)
Hence, Q1 • ( Q2 + Q3) = ( Q1 • Q2) + ( Q1 • Q3).
Consider, Q1 + ( Q2 • Q3)
(—0.4, —0.3,0.3,0.6) (—0.6, —0.4,0.5,0.7) (—0.5, —0.2,0.4,0.7) (—0.7, —0.4,0.4,0.8)
( Q1 + Q2) • ( Q1 + Q3)
(—0.4, —0.3, (—0.6, —0.4,'
0.4,0.7) 0.5,0.8)
(—0.6, —0.2, (—0.7, —0.4,
0.4,0.9) 0.4,0.8)
+
(—0.7, —0.4, (—0.6, —0.4,'
0.3,0.6) 0.5,0.7)
(—0.5, —0.3, (—0.8, —0.4,
0.4,0.7) 0.4,0.9)
( Q1 + Q2) • ( Q1 + Q3) = Hence, Q1 + ( Q2 • Q3) = ( Q1 + Q2) • ( Q1 + Q3) Thus, (iii.) is proved.
(—0.4, —0.3,0.3,0.6) (—0.6, —0.4,0.5,0.7) (—0.5, —0.2,0.4,0.7) (—0.7, —0.4,0.4,0.8)
(iv).Q1 + 0q =
{—max(0.3,0), —max(0.1,0), {—max(0.4,0), —max(0.4,0),
max(0.3,0), max(0.5,0)} max(0.5,0), max(0.7,0)}
{—max(0.2,0), —max(0.1,0), {—max(0.7,0), —max(0.3,0), max(0.4,0), max(0.7,0)} max(0.4,0), max(0.8,0)}
(—0.3,0.1,0.3,0.5) (—0.4, —0.4,0.5,0.7) (—0.2, —0.1,0.4,0.7) (—0.7, —0.3,0.4,0.8)
Q1
0q + Qi
Consider, Qu ■ 0q
(-0.3,0.1,0.3,0.5) (-0.4, -0.4,0.5,0.7) "I = Q (-0.2, -0.1,0.4,0.7) (-0.7, -0.3,0.4,0.8) J = Qi
{-min(0.3,0), -min(0.1,0), {-min(0.4,0), -min(0.4,0),
min (0.3,0), min (0.5,0)} min (0.5,0), min (0.7,0)}
{-min (0.2,0), - min (0.1,0), {-min(0.7,0), -min(0.3,0), min(0.4,0), min(0.7,0)} min(0.4,0), min(0.8,0)}
(0,0,0,0) (0,0,0,0)
(0,0,0,0) (0,0,0,0)
0q
Hence, Qi ■ 0q = 0q ■ Qi = 0q
Thus, (i.), (ii.) , (iii.) and (iv.) are easily verified when Qi, Q2, Q3 are of same order. ■
Theorem 3. Let Qi, Q2, and Q3 be three Quadrasophic Fuzzy matrices, and then we have the following results:
i. Qi © (Q2 © Q3) = (Qi © Q2) © Q3
ii.Qi ® (Q2 ® Q3) = (Qi ® Q2) ® Q3
iii. Qi © iq = iq © Qi = iq
iv.Qi ® iq = iq ® Qi = iq
v.Qi © Q2 = Q2 © Qi
vi.Qi ® Q2 = Q2 ® Qi
vii.Qi © (Q2 + Q3) = (Qi © Q2) + (Qi ® Q3)
viii.Qi ® (Q2 ■ Q3) = (Qi ® Q2) ■ (Qi ® Q3)
Proof. By using the definitions (i.),(ii.), (iii.) and (iv.) are easily verified. (v.) Let Qi = (qiij) rxs and Q2 = (q2ij) sxt be two QFMs. Then Qi © Q2 = (qi2ij) rxt exist. But Q2 © Qu does not exist. Suppose Qu and Q2 are square matrices. Then by using previous example we can easily verify the result.
Qi © Q2
Q2 © Qi
(-0.4, -0.2,0.3,0.7) (-0.4, -0.4,0.3,0.5)
(-0.6, -0.2,0.4,0.7) (-0.7, -0.3,0.4,0.7)
(-0.3, -0.1,0.4,0.7) (-0.6, -0.3,0.4,0.8)
(-0.3, -0.1,0.3,0.5) (-0.7, -0.3,0.3,0.7)
Hence, Qi © Q2 = Q2 © Qi
In similar way, (vi.) Qu ® Q2 = Q2 ® Qu inequality holds. (vii.) By computing,
Qi © ( Q2 + Q3) =
(-0.4, -0.2,0.3,0.7) (-0.4, -0.4,0.4,0.7) (-0.6, -0.2,0.4,0.7) (-0.7, -0.3,0.4,0.7)
(Qi © Q2) + (Qi © Q3)
(-0.4,-0.2, (-0.4, -0.4,"
0.3,0.7) 0.3,0.5)
(-0.6,-0.2, (-0.7, -0.3,
0.4,0.7) 0.4,0.7)
+
(-0.4, -0.3,
0.3,0.5) (-0.5, -0.3, 0.3,0.6)
(-0.4,-0.4,
0.4,0.7) (-0.7,-0.3, 0.4,0.8)
(-0.4, -0.3,0.3,0.7) (-0.4, -0.4,0.4,0.7) (-0.5, -0.3,0.4,0.8) (-0.7, -0.3,0.4,0.8)
Hence, Qi © ( Q2 + Q3) = ( Qi © Q2) + ( Qi © Q3)
Similarly, (viii.) Qi ® ( Q2 ■ Q3) = ( Qi ® Q2) • ( Qi ® Q3). holds.
6. Identification of Plant diseases by employing a Quadrasophic Fuzzy
Matrix
The implementation of the Fuzzy Matrix framework uses several techniques to diagnose a wide range of uncertain problems. A Fuzzy Matrix is a crucial tool for decision-making analysis. In India, the primary factor influencing employment and economic standing is agriculture. Approximately 60% of India's rural community works in agriculture [19]. Moreover, the backbone of the Indian economy is agriculture [19]. The damage that natural catastrophes, plant diseases, and climate change have caused to agriculture. To increase the pace of agricultural production, we might focus on plant diseases among these elements. Insects, bacteria, viruses, fungi, and protozoa can all have an impact on plants. In Tamil Nadu, the most important crops farmed are rice, oilseeds, pulses, sugarcane, cotton, and millets. Tamil Nadu is the third-largest producer of paddy in India among these crops. The main staple food in Tamil Nadu is paddy.
In this section, a new ranking mechanism for the Quadrasophic Fuzzy Matrix is proposed and applied to the detection of agricultural plant disease.
Let us consider four paddy plants p = {p1, p2, p3, p4} that are afflicted with bacterial infections, with indications represented by 5 = {ii,i2,i3,i4} are presented in Table 1. Let к = {11,12,13, } indicates the collection of paddy illnesses.
Table 1: Notations of paddy illness and indicators
Notation
Description
11 Spindle shaped spots with brown margin and grey centre.
12 yellowing of leaves and wilting
13 Irregular greyish brown water -soaked lesions on flag leaf sheath ¿4 Oval or cylindrical dark brown spots with a yellow halo
¡1 Rice blast
¡2 Bacterial leaf blight of rice
¡3 Sheath rot of rice
¡4 Rice brown spot
Procedure:
Step 1: Construct the Quadrasophic Fuzzy Matrix Qi, illustrating the association between indicators and illnesses of the paddy, and Q2, which illustrates the association between illnesses and paddy.
Qi
Q2
(-0.7, -0.3,0.2,0.6) (-0.6, -0.4,0.3,0.8) (-0.5, -0.2,0.3,0.8) (-0.6, -0.2,0.4,0.9)
0.4,0.6) 0.4,0.9) 0.3,0.9)
0.3,0.8) 0.4,0.9) 0.3,0.8) 0.5,0.9)
(-0.5 (-0.6 (-0.9
(-0.6 (-0.8 (-0.9 (-0.3
-0.2,0.3,0.7) -0.3,0.4,0.8) -0.4,0.3,0.7)
-0.3,0.3,0.7) -0.4,0.2,0.4) -0.3,0.2,0.5) -0.1,0.2,0.4)
(-0.7, -0.3,0.2,0.5) (-0.8, -0.4,0.3,0.7) (-0.3, -0.2,0.4,0.8)
(-0.4, -0.2,0.3,0.6) (-0.9, -0.4,0.3,0.7) (-0.7, -0.2,0.2,0.6) (-0.2, -0.1,0.4,0.8)
(-0.7, -0.3,0.2,0.8) (-0.3, -0.2,0.3,0.7) (-0.7, -0.3,0.4,0.9)
(-0.3, -0.1,0.2,0.5) (-0.3, -0.1,0.4,0.7) (-0.3, -0.2,0.4,0.7) (-0.3, -0.2,0.4,0.9)
(-0.9 (-0.7 (-0.7
(-0.6 (-0.5 (-0.6 (-0.4
-0.3, -0.3, -0.3,
-0.2, -0.3, -0.3, 0.2,
Step 2: Generate the complement matrices Q1 and Q2>.
Q1
(-0.3, -0.2,0.3,0.4) (-0.4, -0.1,0.2,0.2)
(-0.5, -0.3,0.2,0.3) (-0.3, -0.2,0.3,0.5)
(-0.4, -0.2,0.1,0.2) (-0.2, -0.1,0.2,0.3)
(-0.1, -0.1,0.2,0.3) (-0.7, -0.3,0.1,0.2)
(-0.5, -0.3,0.2,0.5)
(-0.3, -0.2,0.3,0.2)
(-0.7, -0.3,0.2,0.3)
(-0.3, -0.2,0.1,0.1)
(-0.4, -0.3,0.1,0.1)
(-0.1, -0.2,0.1,0.4) (-0.3, -0.2,0.1,0.1) (-0.3, -0.2,0.2,0.4)
Q2
Q3
Q4
(- 0.4, -0.2,0.2,0.3)
(- 0.2, -0.1,0.3,0.6)
(- 0.1, -0.2,0.3,0.5)
(- -0.7, -0.4,0.3,0.6)
Construct Q3 = Q2 0
(- 0.5, -0.2,0.3,0.7)
(- 0.6, -0.3,0.2,0.6)
(- 0.6, -0.2,0.2,0.6)
(- 0.5, -0.2,0.2,0.6)
Construct Q4 = Q2 0
(- 0.4, -0.2,0.2,0.5)
(- 0.3, -0.2,0.1,0.3)
(- 0.3, -0.2,0.1,0.3)
(- 0.6, -0.3,0.1,0.2)
(-0.6, -0.3,0.2,0.4) (-0.7, -0.4,0.3,0.5) (-0.4, -0.3,0.2,0.2)
(-0.1, -0.1,0.2,0.3) (-0.7, -0.4,0.1,0.3) (-0.5, -0.2,0.1,0.l)
(-0.3, -0.3,0.3,0.4) (-0.7, -0.3,0.1,0.3) (-0.4, -0.2,0.2,0.2)
(-0.8, -0.4,0.1,0.2) (-0.7, -0.3,0.1,0.1) (-0.6, -0.3,0.0,0.1)
Q2 0 Q1, which describes the relationship between Q1 and Q2.
(-0.6, -0.2,0.3,0.6) (-0.3, -0.2,0.3,0.7)
( -0.5, -0.3,0.3,0.7) ( -0.3, -0.2,0.3,0.5)
(-0.6, -0.3,0.2,0.6) (-0.3, -0.2,0.2,0.5)
(-0.4, -0.2,0.3,0.8) (-0.3, -0.2,0.3,0.5)
(-0.6, -0.3,0.3,0.6) (-0.7, -0.3,0.4,0.7) (-0.7, -0.3,0.3,0.6) (-0.6, -0.2,0.4,0.8)
Q2 0 Q1, which provides the relation between Q1 and Qc2.
(-0.4, -0.2,0.2,0.2) (-0.4, -0.3,0.2,0.2)
(-0.3,-0.1,0.1,0.2) (-0.3,-0.2,0.1,0.1)
(-0.3,-0.2,0.2,0.2) (-0.3,-0.2,0.2,0.2)
(-0.7, -0.3,0.1,0.2) (-0.6, -0.3,0.1,0.1)
(-0.4, -0.3,0.2,0.3)
(-0.1, -0.2,0.1,0.3) (-0.3, -0.2,0.1,0.3) (-0.6, -0.3,0.1,0.2)
Step 5: By using the definition of QFS 0 operator, compute Q3 0 Q4.
Q5 = (Q3 0 Q4)
(-0.5,0, (-0.6,0, (0,0, (-0.6,0,
0.3,0.7) 0.3,0.6) 0, 0.7) 0.3,0.6)
(-0.6, -0.3, (-0.5, -0.3, (0,0, (-0.7, -0.3,
0.2,0.6) 0.3,0.7) 0.3,0.5) 0.4,0.7)
(-0.6,0, (-0.6, -0.3, (0,0, (-0.7, -0.3,
0.2,0.6) 0,0.6) 0, 0.5) 0.3,0.6)
(0,0, (0,0, (0,0, (0,0,
0.2,0.6) 0.3,0.8) 0.3,0.5) 0.4,0.8)
0.166 0.1 0.233 0.1
-0.03 0.066 0.266 0.033
0 -0.1 0.166 -0.033
0.266 0.366 0.266 0.4
Step 6: By using the score function, compute the scores of Q5.
Score (Q5 )
The score values indicated the phase of plant disease. The impact of plant illnesses is identified by the lowest value.
1. i2 and i4 illnesses may have an impact on p^The impact of illnesses on p1 can be mitigated by applying appropriate manure or fertilizer, as every value falls within the restricted positive range.
2. p2 is influenced by i1's illness. Due to its negative restricted range, the condition of p2 can be effectively managed to prevent the plant from rotting.
3. p3 is affected by illnesses i2 and i4. Because of its negatively restricted range, p3 can be most effectively regulated in order to prevent the plant from rotting. However, yield has not increased.
4. The illnesses i1 and i3 afflict p4, with each value falling within a restricted positive range, allowing for the application of appropriate manure or fertilizer to prevent all the illnesses. Hence, Quadrasophic Fuzzy Matrix is a powerful tool for estimating uncertainty in agriculture, aiding in the identification of illness stages in targeted plant and improving production levels.
7. Conclusion
Certain operations of the Quadrasophic Fuzzy Set are defined in this artifact. The properties of Quadrasophic Fuzzy Algebra are defined. The development of the Quadrasophic Fuzzy Matrix and its properties are facilitated by the Quadrasophic Fuzzy Algebra. Propositions and theorems of the Quadrasophic Fuzzy Matrix are explained through relevant cases. Further, a new ranking
technique utilizing the Quadrasophic Fuzzy Matrix tool is proposed and implemented to identify paddy ailments in the farming industry. The conceptual framework of the Quadrasophic Fuzzy Matrix can be used in decision-making, business statistics, medicine, and operations research. The forthcoming investigation will explore the features of the Quadrasophic Fuzzy Matrix and their uses in various disciplines.
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