PROPERTIES OF QUADRASOPHIC FUZZY SET AND ITS APPLICATIONS Текст научной статьи по специальности «Медицинские технологии»

PROPERTIES OF QUADRASOPHIC FUZZY SET AND

ITS APPLICATIONS

G. Aruna 1, 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

Fuzzy set theory is a distinctive way of approaching ambiguous information. In this artifact, we introduce a new extension of fuzzy set known as Quadrasophic Fuzzy set and its properties. The Quadrasophic Fuzzy set has four parameters. The attributes and operations of the Quadrasophic Fuzzy sets are defined with pertinent examples. The arithmetic aggregator operators with a redefined level of 0.5 are introduced. The theorems of aggregator operators of Quadrasophic Fuzzy sets are explained using mathematical formulations. Suitable results and examples are provided to enlighten the proposed method. The arithmetic aggregator operators of the proposed method have been used in decision- making to get the optimal solution with supplementary statistics. Additionally, the selection of appropriate fertilizer in farming is demonstrated using the operators of the suggested model. A decision making approach is also used to develop the proposed method in order to identify the ideal solution. An illustration is provided to examine the unique feature of the proposed method to resolve the decision-making problems with a perfect solution.

Keywords: Quadrasophic Fuzzy set, Operations, Aggregated operator, Decision making , QF-VIKOR

1. Introduction

The most crucial factor in analyzing the vagueness state is the fuzzy set. The theory of fuzzy sets and its numerous extensions help in solving decision-making issues and are useful in real-world circumstances that involve uncertain data. To address the newly introduced non-membership parameter the Intuitionistic fuzzy set was introduced by Atanassov [1] . R. Yager [7] develops the idea of the Pythagorean fuzzy set (PFS) in order to address the IFS deficiency. To tackle a bipolar environment, Zhang [8] proposed the concept of a bipolar fuzzy set. Cuong proposed the idea of a Picture fuzzy set to manage the neutral environment. The Neurosophic fuzzy set was a concept that Florentin Smarandache [2] presented to handle three new parameters .

There is a great deal of unclear information available today. Poor choices can cause a person to suffer in life. A successful existence depends on making decisions. Every field relies heavily on decisions to move forward in the right direction. For the examination of decision-making, fuzzy set theory is useful. The authors devised a variety of strategies to find the MCDM problem's ideal answer. Some of the operators used in MCDM problems comprise the fuzzy weighted technique, arithmetic and geometric operators, power operators, Yager's operator, Dombi's operator, weighted ordered arithmetic and geometric, and hybrid operator. To obtain precise outcomes, authors today use cutting-edge techniques to tackle decision-making problems. And also, in order to obtain the results in an appropriate manner, several authors have incorporated the current models into new methodologies. Also, various extensions of the fuzzy set are used in decision-making environments to produce the best results. The use of MDCM

is widespread, including applications in areas like business management, operations research, neural-network, and medical science.

C. Jana et al. [5] analyzed the role of bipolar fuzzy Dombi aggregation operators in the multi-attribute decision-making process. K. Mohana and R. Jansi [6] developed the weighted arithmetic operator and applied it to the MCDM problem in a bipolar-Pythagorean environment. The VIKOR technique [10] for decision-making was employed by several authors.

A new idea in fuzzy set design, known as a Quadrasophic fuzzy set (QFS), employs two new parameters. Restricted membership values are a QFS exclusive feature that helps it effectively manage uncertain situations. QFS has four parameters: positive membership functions, restricted positive, restricted negative, and negative. The system's advantages and disadvantages are revealed by the restricted value. As a result, QFS consists of techniques with built-in values that make it simpler to understand the situation and deliver accurate results.

In this artifact, Section 2 provides the preliminary definitions. Basic operations of QFS and results are given in Section 3, certain properties, operations and theorems of QFS are presented in Section 4 with pertinent examples. The comparison study with other current models is presented in Section 5 to corroborate our findings. Section 6 illustrates the use of QFS to analyze fertilizer in agricultural decision-making. The novel application of the QF-VIKOR technique in decisionmaking is shown in Section 7 and Section 8 supplies the conclusion with its scope for further research.

2. Preliminaries

Intuitionistic Fuzzy sets [4]: An intuitionistic fuzzy set I in the non-empty set X is defined as

I = {(x, p(x), v( x); x € X) } where the function p( x), v( x) : X ^ [ 0,1] represents the membership degree and the non- membership degree value with the condition 0 < p( x) + v( x) < 1. The value n( x) = 1 — p( x) — v( x) is named as the degree of indeterminacy Vx € X.

Pythagorean Fuzzy set [7] : A Pythagorean fuzzy set P, is defined as P = {(x, p ( x), v( x); x € X} where the function p (x), v (x):X ^ [ 0,1] represents the membership degree and the non- membership degree value with the condition 0 < pp (x) + vp (x) < 1. The value

nP (x) = ^J1 — ( pp (x) + vp (x)) is named as the degree of indeterminacy for Vx € X.

Picture fuzzy sets [3]: A picture fuzzy set (PFS) on the universe U is defined as P = {(x, p( x), n(x), v( x)) } where p( x) € [0,1] is the positive membership degree of x in P, n(x) € [ 0,1] is the neutral membership degree and v( x) € [ 0,1] is the negative membership degree.

Bipolar fuzzy sets [8] : Let X be the non-empty fuzzy set. The bipolar fuzzy set B = {B —, B+ } is defined in X. B+ € [ 0,1] is the satisfaction degree of x in B, B— € [ —1,0] is the satisfaction degree of the implicit counter property of x in B.

Properties of Pythagorean fuzzy set [7]: Let PF1, PF2 € PFS( x) then the result as follows,

PF1 + PF2 = {x, ^( pPF1 ( x) ) 2 + ( Ppf2 ( x) ) 2 — ( pPf1 ( x) ) 2 ( Ppf2 ( x) ) 2, vPF1 (x) vPF2 (x) }, Vx € X.

3. QUADRASOPHIC Fuzzy SET

Quadrasophic fuzzy set: The Quadrasophic fuzzy set ( QFS) on the universal set X is defined as

Q = {(x, n(x), Xn (x), Xy(x), y( x)) |x € X}

where y(x) : X ^ [ 0,1] is the degree of high positive membership of x in Q, n( x) : X ^ [ -1,0] is the degree of high negative membership of x in Q, Xy(x) : X ^ [ 0,0.5] is the degree of restricted positive membership of x in Q,Xn (x) : X ^ [ -0.5,0] is the degree of restricted negative membership of x in Q. And it satisfies the following condition: for all x € X, —1 <

y( x) + n( x) < 1, —0.5 < X < 0.5 and 0 < y2 + n2 + X2 < 3 where X =Length of (Xy, Xn). Let QFS( x) denotes the collection of all Quadrasophic fuzzy set on X.

3.1. Operations of Quadrasophic Fuzzy Set

Intersection of QFS : The intersection of two quadrasophic fuzzy set Q1 and Q2 in QFS is defined as:

Q1 n Q2 = {max( nQ1 (x), nQ2(x)), max( XnQi (x), XVQi (x)),

min( XyQ1 (x), XFQ2 (x)), min( yQ1 (x), yQ2(x)) }Vx € X.

Union of QFS: The union of Q1 and Q2 in Quadrasophic fuzzy set is defined as:

Q1 U Q2 = {min( nQ1 (x), nq2(x)), min( XnQi (x),XVQ2 (x)),

max( XyQ^ (x), XyQ2 (x)), max( yQ1 (x), yQ2 (x)) }Vx € X.

Subset: Let Q1, Q2 € Q defined on the non - empty set X then Q! is the subset of Q2 denoted

by Q1 C Q2, if for each x € X; nQ1 (x) > nQ2 (x), Xvq1 (x) > Xvq2 (x), XyQ1 (x) < XyQ2 (x) , Uq1 (x) < yQ2(x).

Complement of QFS: The complement of the set Q1 € Q in X is represented as Qf and is defined as Qf = (nC, Xf, Xf, yC), where nC = —1 — n, Xf = —0.5 — Xn, Xf = 0.5 — Xy and yC = 1 — y.

Equal Set: Let Q1, Q2 € Q be defined on the non empty set X then Q! is equal set to Q2 denoted by Q1 = Q2 if for each x € X;

nQ1(x) = nQ2(x), XnQ1 (x) = Xna2 (x), XyQ1 (x) = XyQ2 (x), yQ1(x) = yQ2(x).

Distance Metric of QFS : The normalized Hamming distance between any QFS set Q1, Q2 € Q( x) is defined as,

dQh( Q1, q2 )= E[ nQ1(xi)) — (nQ2 (xi)) + (XnQ1 (xi)) — (XnQl (xi))'

1

i=1

+ ' '

( XyQ1 (xi)) 2 — ( XyQ2 (xi))) 2 + ( yQ1 ( xi)) 2 — yQ2 ( xi)) 2] .

The normalized Euclidean distance between any QFS set Q1, Q2 € Q( x) is defined as,

/"1 n

dQh ( Q1, Q2) =J 2^7 E [ nQ1 (xr))2 — (no2 (xr))2]2 + [](XnQ1 (xt))2 — (XnQ2 (xt))2] 2 V i=1

Proposition 1. The QFS is not the simplification of bipolar fuzzy set.

Proof. Let Qi = (n0i, -n0i,A^o1, ) be the set in QFS and B1 = {b-, b+ } be the set in BFS. Although positive ( b+) and negative (b-) degree of membership exists in BFS, there is no restricted level. It does not give the information about the partial belongingness or level of influential. In QFS, separate parameter is fixed to trace the level of restricted. Hence, QFS differs from BFS. ■

Remark 1. If both membership grade of positive and negative restricted value is zero. Then, QFS is equal to BFS.

4. Certain Properties of Quadrasophic Fuzzy Set

In this segment, the operations and Quadrasophic fuzzy set weighted arithmetic operator ( QFWA) is defined and proved with theorems and illustrated with model. Algebraic Sum : Let Q1, Q2 G Q

i) If Qi = (7Q1, Q1, AhQ1, ^Q1) and Q2 = (no2, -nQ2, A^o2, Vo2) be any QFS, then the algebraic sum is defined as:

Qi © O2 = {-( -noi) ( -n02), -( -Ano1) ( -Ano2),

^0.5AJqi + 0.5A2Q2 - (0.5A2Qi) (a^), V^li+^L"^^}

Then ,

Qi © O2 = {-( -n0i) ( -no2), -( -Anoi) ( -Ano2),

((i - (i - 0.5A^Qi) + (i - (i - 0.5A2HQ2) - (i - (i - 0.5Aloi).(i - (i - 0.5A^)) 2, (( i - ( i - pQi) + ( i - ( i - PO2) - ( i - ( i - pQi).( i - ( i - PO2)) 2 }

ii) For a > 0,

aQi = {-( -noi)a,-( -AnOi) a, ( i - ( i - 0.5A2Qi)a) l ( i - ( i - ^) a) 2 }.

Theorem 1. If Qi = (noi, -nQi, A^q1, poi) and O2 = (no2, -nQ2, Afq2, po2) be any two set in QFS defined in the non-empty set X and for a1, a2 > 0 the results follow:

i) Qi © 02 = 02 © Qi

Qi © 02 = {-(-noi) (-no2), -(-Anoi) (-Ano2),

V0.5A'oi + 0.5A'o2 - (°.5A'oi) (0.5A'o2), V^Oi + ^2 - 4^2}

= {-( -n02) ( -noi), -( -Ano2) ( -Anoi),

V°.5AIq2 + °.5A^0i - (°.5A'o2) (°.5A'oi), V^02 + ^Oi - ^2^0i} = 02 © Qi

ii) a( Qi © 02) = (aQi © aQ2)

a( Qi © q2) = a{-( -n0i) ( nO2), -( -Anoi) ( -Ano2),

(i - (i - ( °.5A2qi + 0.5A2O2 - (0.5A2Qi x 0.5A^)))) 2,

(1 - (1 - (^ + Vq2 - ^O1.^02)))1} = {-( -noi)a( -n02)a, -( -Ano1)a( -AnQ2)a, (i - (i - ( °.5A2qi + O.5A202 - (0.5A2Qi X 0.5-^)))a) I ( i - ( i - (pQi + p2Q2 - p0i.p2Q2)) a) 2 }

(aQ1 © aQ2)= { —( —nQl)a, — ( —XnQi) (1 — (1 — ) a) 2, (1 — (1 — y^)a) 2 } © { —( —nQ2)", — ( —XnQ2) a, (1 — (1 — 0.5X2yQ2) a) I (1 — (1 — yQ2)a)1} = { — ( —nQl)a ( —no2) a, — ( —XnQ1) a ( — XnQ2) a,

(1 — (1 — 0.5X^1)a (1 — 0.5X2yQ2)a) I (1 — (1 — yh)a (1 — yQ2)a)1} = { — ( —nQl)a ( —no2) a, — ( —XnQ1) a ( — XnQ2) a, (1 — (1 — (0.5X2yQ1 + 0.5X2yQ2 — (0.5X2yQ1 x 0.5X2yQ2)))a) I (1 — (1 — ( yh + yQ2 — y2Q1.y2Q2)))a)1}

Hence, a( Q1 © Q2) = ( aQ1 © aQ2) iii) a1 Q1 © a2 Q1 = («1 © U2) Q1

11 «1Q1 © «2 Q1 = *1{( —nQ1), ( —XnQ1), (1 — (1 — 0.5X^21)) z, (1 — (1 — ( y2Q1)) 2}

© «2 {( — nQ1), ( —XnQ1), (1 — (1 — 0.5X2yQ1)) K (1 — (1 — ( y2Q1)) 2 }

= { — ( — nQ1)a1 ( — nQ1) "2, — ( —XnQ1)a1 ( —XnQ1)a2,

(1 — (1 — 0.5X^21) a1 (1 — 0.5X^21) a2) 2, (1 — (1 — y2Qt)a1 (1 — y2Q1) *2)1}

= { —( — nQ1) "1+a2, —( —XnQ1) a1+"2,

(1 — (1 — 0.5X2yQ1) a1 +a2) i, (1 — (1 — yh) a1+a2)1}

11

(«1 © «2) Q1 = («1 © a2) {( — nQ1), ( —XnQ1), (1 — (1 — 0.5X2VQ1)) (1 — (1 — ( yQ))) 2 } = { —( —no,)X1+X2, —( —XVQ1) a1+a2,

(1 — (1 — 0.5X2yQ1) a1+a2)1, (1 — (1 — yh) a1+a2) 2 }

Hence, «1Q1 © a2 Q1 = (© «2) Q1

Example 1: Consider Q1 = ( —0.5, —0.2,0.4,0.7) and Q2 = ( —0.6, —0.4,0.5,0.8)

i) Since, —0.5 > —0.6, —0.2 > —0.4, 0.4 < 0.5, 0.7 < 0.8

Q1 C Q2.

Assume n = 0.5 then —0.25 > —0.3, —0.1 > —0.2, 0.2 < 0.25, 0.35 < 0.4 nQ1 C nQ2( for any positive integer n).

ii) The union of Q1 and Q2 is: Q1 U Q2 = ( —0.6, —0.4,0.5,0.8) Assume n = 0.5 then n( Q1 U Q2) = ( —0.3, —0.2,0.25,0.4) nQ1 = ( —0.25, —0.1,0.2,0.35), nQ2 = ( —0.3, —0.2,0.25,0.4)

nQ1 U nQ2 = ( —0.3, —0.2,0.25,0.4) Thus, n( Q1 U Q2) = nQ1 U nQ2.

iii) The intersection of Q1 and Q2 is: Q1 n Q2 = ( —0.5, —0.2,0.4,0.7) If n = 0.5, then n( Q1 n Q2) = ( —0.25, —0.1,0.2,0.35)

nQ1 n nQ2 = ( —0.25, —0.1,0.2,0.35) Thus, n( Q1 n Q2) = nQ1 n nQ2.

Theorem 2. Let Q1 = (no1, Xvq1, XyQ1, yo1) and Q2 = (no2, Xno2, Xfq2, yo2) be any two QFS set defined in the non-empty set X and for n > 0 the results follow:

i) If Q1 C Q2 then nQ1 C nQ2.

ii) n( Q1 U Q2) = nQ1 U nQ2.

iii) n( Q1 n Q2) = nQ1 n nQ2.

Proof. The proof is obvious by Example 1.

QFWA Operator: Let Qs = (ys,AVs,A„s, „s) (s = 1,2,...,n) be the set of QFS. Then QFWA (Quadrasophic Fuzzy Weighted Arithmetic operator) with respect to «¿=( a1, a2,..., an) is the weight vector, where a G [ 0,1] such that T!i=1 %i = 1 is a function defined from Qn ^ Q. Then QFWA is defined as:

QFWA( Q1, Q2.....Qn) = ®'n=1ttiQi

= ( «1Q1 + U2Q2 +-----+ UnQn)

Theorem 3. Let Qs = (ys, AVs, A„s, „s) (s = 1,2,3..., n) be the set of QFS defined in the nonempty set X. Then QFWA operator of QFS is defined as

ss

QFWA( Q1, Q2.....Qs) = ( - n( -nQk) Kk, - n( ) Kk,

k=1 k=1 s 1 s 1

(1 - n ( 1 - 0.5A2 ) ak ) 2, (1 - n ( 1 - „Qk ) Kk ) 2 ) (1)

k=1 k k=1

Proof. The proof follows the method of mathematical induction. Assume that n = 2 then, QFWA( Q1, Q2) = «1Q1 ® a.2Q2 = <*1( VQ1, Anq1, A^, „Q1) ® Vq2, A^, A„q2, „Q2)

«1Q1 © a2Q1 = «!{( -VQ1), ( -A,q1 ), (1 - (1 - 0.5A2q1 ))1, (1 - (1 - ( ^)) 2 }

© a2{( -VQ1), ( -A^), (1 - (1 - 0.5A2Q1)) 2, (1 - (1 - ( F2Q1)) 2 } = {- ( -7Q1)a1 ( -7Q1) "2, - ( -AnQ1) a1 ( -A^ )%

([ 1 - (1 - 0.5A2Q1)a1 ] +[ 1 - (1 -0.5A2Q2)a2]

- [ 1 - (1 - 0.5A2Q1)a1 ][ 1 - (1 - 0.5A2Q2)a2 ]) 2, ([ 1 - (1 - 4)a1 ] +[ 1 - (1 - F2Q2 )a2 ]

- [ 1 - (1 - 4)a1)[ 1 - (1 - „Q2 )a2 ]) 2 }

2 2 1 = {- n ( -nQk) ak, - n ( -AnQk)ak, (1 - (1 - 0.5A2 ) a1) (1 - 0.5A2 )a2) k=1 k=1 k 1 2 ,2 ) «1 ( 1 „2 ) «2) 1'

(1 - (1 - „Q1)a1 (1 - „q2 )a2)2}

2 2 2 1

«1Q1 © a2Q2 = {- n( -nQk) Kk, - n( -AnQi)Kk, (1 - n(1 - 0.5A„ ) ak) 2,

k=1 k=1 k=1

21 (1 - n (1 - „2Qk) ak) 2 } k=1

For n = s assume Equation 1 is true. Thus, the result follows:

ss

QFWA( Q1, Q2.....Qs) = ( - n( -nQk) Kk, - n( -AnQk)Kk,

k=1 k=1 k s 1 s 1

(1 - n (1 - 0.5A2)ak) 1, (1 - n (1 - „2Qk) ak) 2)

k=1 k k=1

For n = s + 1,

QFWA( Qi, O2.....Qs+i) = ©k=i( akQk) © (as+iQs+i)

s s

= ( - n (-nok)ak, - n (-v)ak, k=1 k=1

s 1 s 2

(i - n( i - 0.5A2 ) ak) 2, ( i - n( i - Vok) ak) 2))

k=i k k=i

© (( -nOs+i)^(-Anos+1)^ (i - (i - °.5A2Qs+i) «S+1) 2, (i - ( i - pQs+i) «S+1)1)

s

0FWA( Qi.....Qs+i) = (- n (-nok)ak © (-nos+i) as+1,

k=1

s

- n ( -AnQk)ak © ( Anos+i) ^

k=1

s 1

(i - n( i - °.5A2Qk) ak)1 © ( i - ( i - 0.5A*) as+1) 2,

k=1

(i - n (i - ^Ok ) ak) 1 © ( i - ( i - ^+1) as+1) 1)

k=1

s+1 s+1

= {( - n( -nok)ak, - n(-AnQk)ak, k=1 k=1 s 1 1

( i - n( i - °.5A2Qk)ak)1 + ( i - ( i - 0.5A2qs+i ) as+1)1 k=1

s 1 1

- ( i - n( i - 0.5A2Ok)ak)1 ( i - ( i - 0.5A2qs+i) «S+1 ]) 2,

k=1

s 1 1

( i - n ( i - ^Ok ) ak ) 2 + ( i - ( i - ^+1) as+1) 1 k=1

s 1 1

- (i - n ( i - A)ak) 1 (i - (i - ^+1) as+1) 1 }

k=1

s+1 s+1

QFWA( Qi, O2.....Qs) = ( - n( -nOk)ak, - n( -Anok) ak,

k=1 k=1 s+1 1 s+1 1

(i - n( i - O.5A2 ) ak) 2, (i - n(i - pQk)ak) 2)

k=1 k k=1

Hence 1 is true for n = s + 1. By, the method of mathematical induction we conclude 1 is true for any n > °. ■

Example 2: Let Qk = (nk, -nk, A^k, ^), k = 1,2,3,4 be any four Quadrasophic fuzzy set.Consider Q2 = ( -0.7, -0.3,0.3,0.5) , Q2 = ( -0.5, -0.2,025,0.7) , Q3=( -0.8, -0.4,0.3,0.7), and Q4 = ( -0.6, -0.4,0.5,0.9). We assume the weight vector is a = (0.2,0.3,0.4, °.i). Then

QFWA(Qi, O2, Q3, O4) = ©4=i(akQk)

= °.2( -0.7, -0.3,0.3,0.5) + °.3( -0.5, -0.2,025,0.7) + °.4( -0.8, -0.4,0.3,0.7) + °.1( -0.6, -0.4,0.5,0.9) = ( -0.657, -0.307,0.221,0.708)

Theorem 4 (Idem-potency Property). If Qk = (nk, -nk, A^k, ^), k=1,2,3,..., n be the set of QFS and Qk = Q Vk. Then QFWA(Qi, O2, O3.....On) = (3.

Proof. Consider

QFWA(Qi, Q2, Q3.....Qn ) = ©Li( OtQk )

n n

QFWA( Qi, Q2.....Q„) = ( - П( —nQk) "k, - П( ~4k ) "k,

k=1 k=1 n i n i

(i — П( i — 0.5A2Qk )ak) i, (i — П( i — Uok) ak) 2) ( since Qk = Q)

= (( —nQ ), ( — A№k ), ( i — ( i — 0.5A2Q )) 2,

( i — ( i — Uq )) 2 ) QFWA(Qi, Q2, Q3.....Qn ) = Q.

■

Theorem 5 (Monotonicity). Let Qk and Qk be the pair of sets in QFS. If Qk < Qk Vk, then

QFWA( Qi, Q2, Q3.....Qn ) = QFWA( Qi, Q2, q3.....Qn ).

Proof. If Qk, Qk G QFS and Qk < Qk Vk, then

n n n .

( — П( —nok)ak, — П( — 4k )ak, (i — П( i — 0.5A2UQk ) ak) ^

k=i k=i k=i k n . n n

(i — П( i — uQk)ak) i ) < ( — П( —nQk)ak, — П( —С )ak,

k=i k=i k=i k n , n ,

(i — П( i — ( 0.5Auok )2)ak) i, ( i — П( i — ( UQk) 2) ak) i )

QFWA( Qi, O2, Q3.....Qn) = QFWA( Qi, Q2, Q3.....Qn).

■

Example 3: Consider Q2 = ( -0.7, -0.3,0.4,0.8) and Q2 = ( -0.6, -0.2,0.5,0.7) be two sets in QFS. The weight vector is a = (0.5,0.5).

Solution: Consider (s) for smaller and (l) for larger value of membership. We know that,

2 22

n(-nok(s))ak < n(-nok)ak < n(-nok( Jak

k=1 (s) k=1 k=1 (()

= ( -0.7) °.5( -0.7) °.5 < ( -0.7) °.5 ( -0.6) °.5 < ( -0.6) °.5 ( -0.6) °.5

Also,

—0.7006 < —0.648i < —0.600.

2 22

П ( —AnQk())ak < П ( —AnQk )ak < П ( —AnQk( )ak k=i k( s) k=i k k=i k( г)

= -0.299 < -0.244 < -0.i99

Also,

2 22

( i — П( i — 0.5A2 ) ak ) 2 < ( i — П ( i — 0.5A2 )ak ) 2 < ( i — П ( i — 0.5A2 ) ak ) 2

k=i k( s) k=i k k=i k( 0

= 0.282 < 0.320 < 0.353

Also,

2 1 2 1 2 1

(1 - n(1 - A )ak) 1 < (1 - n(1 - „Qk)ak) 1 < (1 - n(1 - A )ak) 1

k=1 (s) k=1 k=1 (i)

= 0.700 < 0.756 < 0.800

2 2 2 1 2 1 {-n( -no,s))^ - n( -V )^ (1 - n(1 - 0.5A„Qk )ak)1, (1 - n(1 - „Qk( )) ak)1} k=1 ( ) k=1 k( ) k=1 k( ) k=1 ( ) 2 2 2 1 2 1

< {- n( -nQk) ak), - n( -Anok)ak, (1 - n(1 - 0.5A„ ) ak) 1, (1 - n(1 - „Qk) ak) 1}

k=1 k=1 k k=1 k k=1 2 2 2 1 2 1

< {- n( -VQ,l)) ak, - n( -AnQk )ak, (1 - n(1 - 0.5A„Qk ) ak) ^ (1 - n(1 - „Qk(|))ak)1}

k=1 (l) k=1 (l) k=1 k( l) k=1 (l)

And thus, the set of smaller membership grade is lesser than actual membership grade which is lesser than the larger membership grade.

Table 1: Comparative study between QFS and Bipolar environment

Type of operator Environment Results

Dombi Weighted [5] Proposed method Bipolar Fuzzy environment Quadrasophic Fuzzy Set Q4 > Qi > Q2 > Qs > Q3 0.706 > 0.694 > 0.661 > 0.600 > 0.583 Q4 > Qi > Q2 > Qs > Q3 0.134 > 0.124 > 0.121 > 0.057 > 0.017

Table 2: Comparative study between QFS and Bipolar Pythagorean environment

Type of operator Environment Results

Weighted Average[6] Bipolar Pythagorean Q1 > Q2 > Q3 > Q4

environment 0.383 > 0.189 > -0.031 > -0.102

Proposed method Quadrasophic Fuzzy set Q1 > Q2 > Q4 > Q3

0.256 > 0.084 > -0.045 > -0.052

Table 3: Types, ratio and duration of fertilizer

Fertilizer types Composition in ratio Duration

f1 120:40:40 Short

f2 150:60:50 Medium

f3 150:80:50 Long

Theorem 6 (Bounded property). Consider Qk = (Vk, Ank, A„k, „k) , k=1,2,3,..., n be the set of QFS. If k k

Q( s) = min( ^ Q3.....Qn ) =( Vk( s), AVk( s) , A„k( s) , „k( s)) and

Q( i) = max( Q1, Q2, Q3,..., Qn ) =( Vk( i), AVk( l) , A„k( l) , „k( i))) .

Then Q(s) < QFWA( Q1, Q2, Q3.....Qn) < Q(i).

Proof. By the example 3, the proof is obvious.

n n

(- n (-/Qk( ), - n (-4)ak,

k=1 (s) k=1 k( s)

n . n .

(1 - n(1 - 0.5A2 ) ) 1, (1 - n(1 - A, ,) ) 2)

k=i k( s) k=1 (s)

n n n 1 n 1

< ( - n ( -nQk ) ak, - n ( ) , ( 1 - n ( 1 - 0.5A2 ) <*k ) 2, ( 1 - n ( 1 - A ) ) 2 )

k=1 k=1 k=1 k k=1

nn

< (- n (-nQk(,))ak, - n (-V l))ak,

k=1 (l) k=1 (l)

n 1 n 1

(1 - n (1 - 0.5A2 ) ) 1, (1 - n (1 - ) ak) 2)

k=1 k( l) k=1 (l)

» Q(S) < QFWA( Q1, Q2, Q3.....Qn) < Q(i).

Table 4: Decision Matrix of Quadrasophic Fuzzy Set

Fertilizer S1 S2 S3

/1 (-0.8,-0.4,0.3,0.6 ) (-0.7,-0.3,0.5,0.8) (-0.6,-0.2,0.3,0.7)

f2 (-0.7,-0.3,0.2,0.4) (-0.5,-0.3,0.5,0.7) (-0.4,-0.2,0.4,0.8)

f3 (-0.4,-0.2,0.4,0.8) (-0.7,-0.3,0.4,0.8) (-0.6,-0.3,0.3,0.6)

5. Comparative study

In this segment, to corroborate our method, a comparison study is performed to prove that the proposed method generates better and more accurate results than the other existing methodologies. The Table 1 and 2 gives the study of comparison.

Table 5: Normalized decision matrix of QFS

Fertilizer S1 S2 S3

/1 (-0.2,-0.1,0.2,0.4 ) (-0.7,-0.3,0.5,0.8) (-0.6,-0.2,0.3,0.7)

f2 (-0.3,-0.2,0.3,0.6) (-0.5,-0.3,0.5,0.7) (-0.4,-0.2,0.4,0.8)

/3 (-0.6,-0.3,0.1,0.2) (-0.7,-0.3,0.4,0.8) (-0.6,-0.3,0.3,0.6)

Table 6: QFWA value of QFS

Fertilizer QFWA value

/1 (-0.4519,-0.183,0.25,0.683)

/2 (-0.3923,-0.225,0.28,0.724)

/3 (-0.628,-0.299,0.2096,0.6259)

6. Application of Quadrasophic Fuzzy Set in agricultural field

The QFS has a special function that allows the difficulties to be solved perfectly. In this

segment, we design a technique to deal with the decision-making assessment based on the QFWA

Table 7: Decision Matrix of Quadrasophic Fuzzy Set

M1 M2 M3

Bi (-0.7,-0.4,0.2,0.6 ) (-0.6,-0.3,0.4,0.7) (-0.5,-0.3,0.1,0.3)

B2 (-0.5,-0.2,0.1,0.3) (-0.3,-0.2,0.4,0.8) (-0.2,-0.1,0.3,0.7)

B3 (-0.3,-0.1,0.4,0.7) (-0.6,-0.3,0.2,0.7) (-0.5,-0.3,0.2,0.5)

Table 8: Score Value of Quadrasophic Fuzzy Set

M1 M2 M3

B1 -0.1 0.06 -0.13

B2 -0.1 0.23 0.23

B3 0.23 0 -0.03

operator and score function of QFS.

Crop yields are significantly influenced by fertilizers. Several types of fertilizers are applied to the soil to improve cultivation. The two primary types of fertilizer used in agriculture are organic and inorganic materials. For various purposes during the cultivation process, three primary plant nutrients: nitrogen, potassium, and phosphorus are used in different ratios. Suppose that the farmer wants to get better yields from the cultivation in medium term. The following Table 3 lists the type, ratio, and length of time employed in the cultivation for better results. Let F = (fi,f2,f3} be the set of alternative fertilizers used on agricultural land.

Membership grade of QFS in Agriculture :

n — refers to the side effects of using fertilizer Xn —refers to the rate of soil pollution — refers to the rate of soil potential ц— refers to the rate of fertilizer absorption level

Let s = (si, s2, s3 } be the group of criteria satisfied by the fertilizers used in the cultivation land. where, si— to upgrade the soil fertility, s2— to promote the cultivation and s3— healthy crop. Step 1: The Table 4 provides the QFS decision matrix.

Step 2: The normalized QFS decision matrix is presented in the Table 5 by considering si as the cost factor.

Step 3: Assume that a = (0.3,0.3,0.4) is the weight vector of each criteria.

Step 4: Use the QFWA operator to find the aggregated value. The Table 6 presents the QFWA

value of a normalized QFS decision matrix.

Step 5: Apply the formula sv( Q) = (x) +n(x)+Xn(x) to find the score value of QFSDM.

QSk QRk QQk

Figure 1: Values of QSk, QRk, and QQk

Table 9: QFS positive and negative ideal solutions

( *) sl * S( -* sl

Ml (-0.3,-0.1,0.4,0.7) (-0.7,-0.4,0.1,0.3)

M2 (-0.3,-0.2,0.4,0.8) (-0.6,-0.3,0.2,0.7)

M3 (-0.2,-0.1,0.3,0.7) (-0.5,-0.3,0.1,0.3)

Table 10: Value of OSk, ORk, QQk

_QSk QRk QQk

B1 0.5324 0.3028 0.6378

B2 0.5802 0.3000 0.5

B3 0.51421 0.3000 0

Ranking B3, B1, B2 B3 = B2, B1 B3, B2, B1

sv' f1) = 0.0993, sv' f2) = 0.1289, and sv' f3) = -0.0306. Step 6: Rank the score value: sv' f2) > sv' f1) > sv' f3).

The fertilizer f2 is therefore the ideal option for the best production in a medium period of time. Evidently, a medium-duration cultivation technique uses the nutrient ratio 150:60:50. The application gives sufficient justification for QFS to be able to determine the most appropriate decision.

7. A novel MCDM QF-VIKOR technique

A VIKOR technique focuses on ranking options and identifying compromises that are closest to the ideal answer. To define and identify the optimal solution of the Quadrasophic Fuzzy Set, we employ the QF-VIKOR technique in this segment.Consider a businessman who wants to boost profits from their investment in the share marketing sector. Let M = {M1, M2, M3 } be the set of investment criteria that leads to the alternative businesses B1, B2, and B3. Step 1: The QFS decision-making matrix is shown in Table 7. Step 2:The QFS Score matrix is shown in Table 8.

Step 3:The Table 9 shows the values of the QFS ideal solutions, both positive and negative.

Step 4: Assume that the weight vector for each criterion is a = {0.3,0.4,0.3}

Step 5:The values of QSk, QRk, QQk are provided in the Table 10 using the distance metric 3.1.

Rank the alternatives as well. where, [9] QSk = E"=1 a d(% f- , QRk = maxi a d(s(lt) f-j

d' Si ,si ) d' Si ,si )

QQk = + '1-QS.TQ-QR;" and QS( •) = m,nQS„ QS( -) = maxQSi, QR< •) =

minQRi, and QR' *) = maxQRi, Qfi e [ 0,1].

Step 6: Figure 1 shows that B3 is the minimum value. We evaluate how effectively the compromise solution of QQk accepts B3 and B2. QQ' B2) - QQ' B3) > nL1 0.5 > ^ = 0.5 > 0.5 Similarly for QSk and QRk .

Hence, the greatest alternative is B3, whereas B2 and B3 are the compromise solutions. Therefore, the ideal firm to invest in for increased profit is B3.

8. Conclusion

The definitions, characteristics, and some Quadrasophic Fuzzy set operations are defined in this artifact. Theorems and results are also demonstrated using pertinent examples and remarks. To validate our technique, comparative research was conducted in several fuzzy environments. The use of Quadrasophic Fuzzy set in the field of agriculture to identify appropriate

fertilizer has been explored. Further,the QF-VIKOR approach is introduced with its new decisionmaking application. The use of Quadrasophic Fuzzy set is grounded in the notion that it also works in artificial intelligence, neural-networks, and medicine. Other aggregation operators and their uses in various fields will be examined in further work. Due to its unique parameter classification, Quadrasophic Fuzzy set will also be functional in many domains, including corporate management and psychology.

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