New Ranking of Generalized Quadrilateral Shape Fuzzy Number Using Centroid Technique

1  Introduction

One definition of a fuzzy number (FN) is a real number whose fuzzy set (FS) is both normal and convex. The FN approach is fantastic since it allows for the combination of subjective and objective information. Ranking fuzzy numbers (RFNs) is necessary for many fuzzy application systems, including those dealing with linguistic decision-making, risk management, Industrial engineering, optimization, medicine, artificial intelligence etc. We shall investigate the ordering of quadrilateral fuzzy number generalizations (GQFNs). For a problem with four points and two levels of representational depth, GQFNs can be used. The ranking of FNs is arguably the most essential part, and there are many different ways to do it using techniques from data analysis, optimization, and other disciplines. In the future, it may be used for problems in decision-making, data analysis, optimization, engineering, and technology. Decisions can be aided by RFN measurements (DMP). Since 1976, many systems for categorizing ranking events have been created. Zadeh [1] is credited with coming up with the concept of an FS. An original FRA method for handling risk uncertainty is described by Lee et al. [2]. Fuzzy integers of varying sizes and variances form the basis of the process. Fuzzy numbers are crucial to conveying uncertain values. In this research, we propose many mathematical operations on generalized quadrilateral fuzzy numbers (GQFNs) using the defuzzification technique. Additionally, several features of the defuzzification method are presented, along with relevant numerical examples to back up the features. In the beginning, Jain [3] predicted the positions of FNs for the DMP. Dubois et al. [4]. This article employs a fuzzification approach to allow conventional algebraic operations to be applied to fuzzy numbers. The work of Mizumoto et al. [5,6]. When it comes to the difficulty of making decisions, several approaches have been developed, such as the Pythagorean fuzzy Dombi aggregation operators, the Extended TOPSIS technique, and the Intuitionistic Fuzzy Rough Frank Aggregation Operator-Based EDAS Method.

2  Literature Review

By comparing the rankings obtained by using the FRA problem with the positive, negative, and centroid of the generalized numbers, Ming Chen et al. [7] suggested a hypothesis. Using the counter example of Lee and Chen, Singh [8] advocated a new way of RFN based on the expectation of centroid predictions of varying heights and demonstrated that the sorting method is erroneous. In [9], Pathinathan et al. organized a pentagonal QFN using mathematical computation. Using the mathematical function CEFM, Stephen Dinagar et al. [10] presented a GQF transportation problem. In their essay “defuzzification of QFN and arithmetic operation of CEFM,” authors Dinagar et al. examined the characteristics of CEFM. There is a GQF optimum solution to Assignment issues, and it was proposed by Stephen Dinagar et al. [11]. The GQF economic inventory model, which takes into account backorders, was created by Stephen Dinagar et al. [12,13]. Fuzzy Assignment Tricky makes use of a technique proposed by Thiruppathi et al. [14,15] called Ranking of Hexagonal Fuzzy Using Centroids. Using the center of gravity, Barazandeh et al. [16] calculated the score of a generalized fuzzy trapezoidal number (GFTN) with a range of left and right heights. A better system of rankings was created by Jiang et al. [17]. The strengths and disadvantages of current fuzzy ranking systems are investigated, and the elements of FN regions, including the upsides and downsides, and the spreads, are discussed. Hajjari [18] offered a fresh magnitude method for sorting trapezoidal fuzzy numbers (TFNs) using lowest and maximum points and the value of FNs, and he compared and approved the benefits of the supplied strategy comparison. In this study, Rezvani [19] presented a method for using Euclidean distance and central centroids. Results from the proposed method are checked against those of other popular approaches to ensure they are reliable. Yu Vincent et al. [20] proposes a novel method of rating FNs that displays their updated left, right and total integral values. After that, we utilize the median value ranking approach to identify FNs that have area compensation. This article by Ming wang et al. [21] discusses an optional ranking method for FNs based on positive and negative ideal foci, and calls it “area ranking.” Predicting the efficacy and modesty of non-preemptive priority fuzzy queues necessitates the use of performance measurements, and Vinnarasi et al. presented the Bell-shaped fuzzy number using the centroid of centroids method [22]. To rate items, only a select few authors have turned to generalized quadrilateral fuzzy numbers. The suggested technique, however, is unique among existing methods and provides a straightforward identification ranking strategy for fuzzy numbers in a quadrilateral form.

The work is coordinated as follows Section 3 provides a fundamental depiction of a GQFN. The method proposed for ordering the integers that can be found in any generalized quadrilateral is discussed in Section 4. (RGQNs). Equivalent Properties in the Classical Period It has been proven that fuzzy arithmetic operations do exist. The sixth section provides an overview of one of the problems with Lee and Chen’s RA has been highlighted. In Section 7, we see the implementation of mathematical operations on GQFNs. The Algorithm for Comparing Two Fuzzy Quantities is discussed in Section 8. Existing approaches are compared to the new GQSFN-CT methods in Section 9. This is the tenth section. This section offers guidance on how to use RA to the FRA problem. The final section summarises the entire paper.

Note: Throughout the paper the following abbreviations are used

3  Preliminaries

3.1 Fuzzy Set

Let Q(A)={(x,μA(X)/xϵQ(A),μA(X)ϵ(0,1)} First element x belongs to set Q (A), second element μA(X) belongs to the interval [0, 1], is called membership function. Fig. 1 represent the membership function of Generalized quadrilateral fuzzy numbers.

Figure 1: Generalizations of quadrilateral fuzzy number

3.2 Definition: [9–11]

A FN that is GQFNs is defined as Q˘=[qa1,qa2,qa3,qa4;𝓌1,𝓌2] where qa1<qa2<qa3<qa4 the membership function is defined by

μQ˘(x)={𝓌1(x−qa1qa2−qa1)ifqa1≤x≤qa2(x−qa2)𝓌2+(qa3−x)𝓌1(qa3−qa2)ifqa2≤x≤qa3𝓌2(x−qa4qa3−qa4)ifqa3≤x≤qa40otherwise (1)

When 𝓌1=𝓌2=𝓌 , the GQFN is turned into the trapezoidal fuzzy number (TFN) and when qa1=qa2 or qa3=qa4 the GQFN is formed into the TFN. The trapezoid is the conjecture of the quadrilateral, because the flat line is not parallel in this case, a trapezoidal is inappropriate, as shown by the graphical representation provided by the aforementioned FN. If, however, the two horizontal lines here are brought to equality, we get a trapezoid. These lines may also be referred to as QFNs due to the topographical character of the particular FN in question.

4  Ranking Approach

A technique suggested for ranking generalized fuzzy number (RGFN) of quadrilateral form. ( 𝓌1<𝓌2) . Three figures were created by vertically dividing the GQFN. Draw the horizontal line parallel to 𝓌1 , now we get three triangles namely triangle ABE, Triangle CDG and triangle EFG. Fig. 2 represent the ranking approach of Generalized quadrilateral fuzzy numbers. Fig. 3 represent the ranking of Generalized quadrilateral fuzzy numbers.

Figure 2: Diagram for Ranking GQFN

Figure 3: Diagram for Ranking GQFN

Q˘=[qa1,qa2,qa3,qa4;𝓌1,𝓌2] Here heights 𝓌1<𝓌2

The centroid point triangle ABE, Triangle CDG and triangle EFG respectively,

Q(G1)=(qa1+2qa23,𝓌13)

Q(G2)=(2qa3+qa43,𝓌23)

Q(G3)=(qa2+2qa33,2𝓌1+𝓌23)

Subsequently, connect the points Q(G1),Q(G2) and Q(G3) . They come together to build a triangle. The centre of the triangle (CoT) formed by the vertices Q(G1),Q(G2) and Q(G3) is now

Q(G)=(qa1+3qa2+4qa3+qa49,3𝓌1+2𝓌29) (2)

The GQFN ranking function Q˘=[qa1,qa2,qa3,qa4;𝓌1,𝓌2] translates the set of all FNs to a set of real numbers. It is well-defined as follows

R(Q)=qa1+3qa2+4qa3+qa49×3𝓌1+2𝓌29

GQFN Q(R) assigns real numbers for each FN in Q(R) given in the following new ranking GQSFN-CT method,

Q˘1=[qa1,qa2,qa3,qa4;𝓌1,𝓌2]∈Q(R),𝓌1<𝓌2

R(Q)=qa1+3qa2+4qa3+qa49×3𝓌1+2𝓌29 (3)

Ranking approach method

A technique suggested for RGFN of quadrilateral form. 𝓌1>𝓌2

Three fig were created by vertically dividing the GQFN. Draw the horizontal line parallel to 𝓌1 , now we get three triangles namely triangle ABE, Triangle CDG and triangle EFG.

Q˘=[qa1,qa2,qa3,qa4;𝓌1𝓌2]Here heights𝓌1>𝓌2

The centroid point triangle ABE, Triangle CDG and triangle EFG respectively

Q(G1)=(qa1+2qa23,𝓌13)

Q(G2)=(2qa3+qa43,𝓌23)

Q(G3)=(2qa2+qa33,2𝓌1+𝓌23)

Subsequently, connect the points Q(G1),Q(G2) and Q(G3) . They come together to create a triangle. The CoT formed by the vertices Q(G1),Q(G2) and Q(G3) is now

Q(G)=(qa1+4qa2+3qa3+qa49,3𝓌1+2𝓌29) (4)

The GQFN ranking function Q˘=[qa1,qa2,qa3,qa4;𝓌1,𝓌2] . translates the set of all FNs to a set of real numbers. It is defined as follows

R(Q)=qa1+4qa2+3qa3+qa49×3𝓌1+2𝓌29

GQFN Q(R) assigns real numbers for each FN in Q(R) given in the following new ranking GQSFN-CT method,

Q˘1=[qa1,qa2,qa3,qa4;𝓌1,𝓌2]∈Q(R)𝓌1>𝓌2

R(Q)=qa1+4qa2+3qa3+qa49×3𝓌1+2𝓌29 (5)

5  Properties of Classical Equivalent Fuzzy Arithmetic Operation; [10–12]

Property: 5.1

Let Q˘1 and Q˘2 be two GQFN and R(Q˘1) and R(Q˘2) be the respective fuzzy ranking values (FRVs). Using Classical equivalent Fuzzy mean arithmetic operation. FRV of sum of GQFNS Q˘1,Q˘2 is the RS-FN of Q˘1,Q˘2 .

i.e.,R(Q˘1+Q˘2)=R(Q˘1)+R(Q˘2) (6)

Property: 5.2

Let Q˘1,Q˘2 and Q˘3 be three GQFN and R(Q˘1),R(Q˘2) and R(Q˘3) be the respective FRVs. Using Classical equivalent Fuzzy mean arithmetic operation. FRV of sum of GQFNS Q˘1,Q˘2,Q˘3 is the RS-FN of Q˘1,Q˘2,Q˘3 .

i.e.R(Q˘1+Q˘2+Q˘3)=R(Q˘1)+R(Q˘2)+R(Q˘3) (7)

Proof:

Q˘1=[qa1,qa2,qa3,qa4;ℓ1,ℓ2]

Q˘2=[qb1,qb2,qb3,qb4;𝓂1,𝓂2]

Q˘3=[qc1,qc2,qc3,qc4;𝓃1,𝓃2]

𝓌l=3ℓ1+2ℓ2

𝓌𝓂=3𝓂1+2𝓂2

𝓌𝓃=3𝓃1+2𝓃2

Q˘1+Q˘2+Q˘3=[3(qa1𝓌l+qb1𝓌𝓂+qc1𝓌𝓃)𝓌l+𝓌𝓂+𝓌𝓃,3(qa2𝓌l+qb2𝓌𝓂+qc2𝓌𝓃)𝓌l+𝓌𝓂+𝓌𝓃3(qa3𝓌l+qb3𝓌𝓂+qc3𝓌𝓃)𝓌l+𝓌𝓂+𝓌𝓃,3(qa4𝓌l+qb4𝓌𝓂+qc4𝓌𝓃)𝓌l+𝓌𝓂+𝓌𝓃;(ℓ1+𝓂1+𝓃1)3,(ℓ2+𝓂2+𝓃2)3]

R(Q˘1+Q˘2+Q˘3)=3(qa1𝓌l+qb1𝓌𝓂+qc1𝓌𝓃)𝓌l+𝓌𝓂+𝓌𝓃+3×3(qa2𝓌l+qb2𝓌𝓂+qc2𝓌𝓃)𝓌l+𝓌𝓂+𝓌𝓃9

+4×3(qa3𝓌l+qb3𝓌𝓂+qc3𝓌𝓃)𝓌l+𝓌𝓂+𝓌𝓃+3(qa4𝓌l+qb4𝓌𝓂+qc4𝓌𝓃)𝓌l+𝓌𝓂+𝓌𝓃9

×(3(ℓ1+𝓂1+𝓃1)3+2(ℓ2+𝓂2+𝓃2)39)

=3qa1𝓌l+3qb1𝓌𝓂+3qc1𝓌𝓃𝓌l+𝓌𝓂+𝓌𝓃+9qa2𝓌l+9qb2𝓌𝓂+9qc2𝓌𝓃𝓌l+𝓌𝓂+𝓌𝓃9

+12qa3𝓌l+12qb3𝓌𝓂+12qc3𝓌𝓃𝓌l+𝓌𝓂+𝓌𝓃+3qa4𝓌l+3qb4𝓌𝓂+3qc4𝓌𝓃𝓌l+𝓌𝓂+𝓌𝓃9

×((3ℓ1+3𝓂1+3𝓃1)3+(2ℓ2+2𝓂2+2𝓃2)39)

=3(qa1𝓌l+3qa2𝓌l+4qa3𝓌l+3qa4𝓌l)+(qb1𝓌𝓂+3qb2𝓌𝓂+4qb3𝓌𝓂+qb4𝓌m)9(𝓌l+𝓌𝓂+𝓌𝓃)

+((qc1𝓌𝓃+3qc2𝓌𝓃+4qc3𝓌𝓃+3qc4𝓌𝓃))9(𝓌l+𝓌𝓂+𝓌𝓃)×((3ℓ1+2ℓ2)+(3𝓂1+2𝓂2)+(3𝓃1+2𝓃2)39)

=[(3(qa1+3qa2+4qa3+3qa4)𝓌l+(qb1+3qb2+4qb3+qb4)𝓌𝓂+(qc1+3qc2+4qc3+qc4)𝓌𝓃𝓌l+𝓌𝓂+𝓌𝓃9)×(𝓌+𝓌m+𝓌n39)]

=[((qa1+3qa2+4qa3+3qa4)𝓌l+(qb1+3qb2+4qb3+qb4)𝓌𝓂+(qc1+3qc2+4qc3+qc4)𝓌𝓃9)×(19)]

=(qa1+3qa2+4qa3+3qa4)9×𝓌19+(qb1+3qb2+4qb3+qb4)9×𝓌m9+(qc1+3qc2+4qc3+qc4)9×𝓌n9

=(qa1+3qa2+4qa3+3qa4)9×3ℓ1+2ℓ29+(qb1+3qb2+4qb3+qb4)9×3𝓂1+2𝓂29

+(qc1+3qc2+4qc3+qc4)9×3𝓃1+2𝓃29

=R(Q˘1)+R(Q˘2)+R(Q˘3)

Property: 5.3

Let Q˘1,Q˘2,Q˘3…Q˘n be three GQFN and R(Q˘1),R(Q˘2) …, R(Q˘n) be the respective FRVs. Using Classical equivalent Fuzzy mean arithmetic operation. FRV of sum of GQFNS Q˘1,Q˘2,Q˘3…Q˘n is the ranking summation of the fuzzy data (RS-FD) of Q˘1,Q˘2,Q˘3…Q˘n .

i.e.,R(Q˘1+Q˘2+Q˘3+…,+Q˘n)=R(Q˘1)+R(Q˘2)+R(Q˘3)+…,+R(Q˘n) (8)

Similar to the above proof

6  A Shortcoming of Lee and Chen’s Ra

Wang and Keere proposed the following acceptable conditions for ranking function validation.

Q˘1=[qa1,qa2,qa3,qa4;ℓ1,ℓ2] and Q˘2=[qb1,qb2,qb3,qb4;𝓂1,𝓂2] are two normal FS then Q1>Q2⇒(Q1⊕Q3)>(Q1⊕Q3)

Q1<Q2⇒(Q1⊕Q3)<(Q1⊕Q3)

Q1:Q2⇒(Q1⊕Q3):(Q1⊕Q3) (9)

where Q˘3=[qc1,qc2,qc3,qc4;𝓃1,𝓃2] is normal FS. Where

(n1,n2)≤(min(ℓ1,𝓂1)min(ℓ2,𝓂2)

Example 1. [12] Let Q˘1=(1,2,3,4;0.6,0.4) , Q˘2=(0,3,4,5;0.4,0.2) and, Q˘3=(1,3,4,5;0.4,0.2) GQFNS, Then then according to Lee and Chen’s RA Q˘1<Q˘2⇏Q˘1⊕Q˘3>Q˘2⊕Q˘3

Example 2. [12] Let Q˘1=(2,5,6,7;0.6,0.4) , Q˘2=(3,4,5,6;0.8,0.6) and, Q˘3=(4,3,7,8;0.6,0.4) GQFNS, Then then according to Lee and Chen’s RA Q˘1>Q˘2⇏Q˘1⊕Q˘3<Q˘2⊕Q˘3

Our new RA has been implemented. Mathematical operations on GQFNS have been proposed as follows [9–11].

7  Numerical Examples

Example-1

Based on Classical equivalent Fuzzy mean mathematical operations on GQFNS.

Q˘1=[3,4,5,7;0.2,0.4]⇒R(Q˘1)=0.73

Q˘2=[7,8,10,11;0.3,0.6]⇒R(Q˘2)=2.13

Addition

Q˘1+Q˘2=[10.80,12.80,16,18.80;.025,0.50]R(Q˘1+Q˘2)=2.86R(Q˘1)+R(Q˘1)=0.73+2.13=2.86∴ R(Q˘1+Q˘2)=R(Q˘1)+R(Q˘1) (10)

Example-2

Based on Classical equivalent Fuzzy mean mathematical operations on GQFNS.

Q˘1=[8,11,13,15;0.2,0.4]⇒R(Q˘1)=1.87

Q˘2=[6,8,10,12;0.3,0.6]⇒R(Q˘2)=2.13

Addition

Q˘1+Q˘2=[13.60,18.40,22.40,26.40.025,0.50]R(Q˘1+Q˘2)=3.99R(Q˘1)+R(Q˘1)=1.87+2.13=3.99∴R(Q˘1+Q˘2)=R(Q˘1)+R(Q˘1) (11)

Example-3

Based on Classical equivalent Fuzzy mean mathematical operations on GQFNS.

Q˘1=[3,4,5,7;0.2,0.4]⇒R(Q˘1)=0.73

Q˘2=[7,8,10,11;0.3,0.6]⇒R(Q˘2)=2.13

Subtraction

Q˘1−Q˘2=[−6.00,−6.40,−8.00,−7.60;.025,0.50]R(Q˘1−Q˘2)=−1.40R(Q˘1)−R(Q˘1)=0.73−2.13=−1.40∴ R(Q˘1−Q˘2)=R(Q˘1)−R(Q˘1) (12)

Example-4

Using arithmetic operation on GQFNS based on CEFM

Q˘1=[8,11,13,15;0.2,0.4]⇒R(Q˘1)=1.87

Q˘2=[6,8,10,12;0.3,0.6]⇒R(Q˘2)=2.13

Subtraction

Q˘1−Q˘2=[−0.80,−0.80,−1.60,−2.40;025,0.50]R(Q˘1−Q˘2)=−0.26R(Q˘1)−R(Q˘1)=1.87−2.13=−0.26∴ R(Q˘1−Q˘2)=R(Q˘1)−R(Q˘1) (13)

Example-5

Based on Classical equivalent Fuzzy mean mathematical operations on GQFNS.

Q˘1=[8,11,13,15;0.2,0.4]⇒R(Q˘1)=1.87

Q˘2=[6,8,10,12;0.3,0.6]⇒R(Q˘2)=2.13

Multiplication if R(Q˘1)>0

Q˘1×Q˘2=[13.61,18.71,22.11,25.51;025,0.50]R(Q˘1×Q˘2)=3.97R(Q˘1)×R(Q˘1)=1.87×2.13=3.97∴ R(Q˘1×Q˘2)=R(Q˘1)×R(Q˘1) (14)

Example-6

Based on Classical equivalent Fuzzy mean mathematical operations on GQFNS.

Q˘1=[8,11,13,15;0.2,0.4]⇒R(Q˘1)=1.87

Q˘2=[6,8,10,12;0.3,0.6]⇒R(Q˘2)=2.13

Multiplication if R(Q˘1)>0

Q˘1Q˘2=[3.0104.1394.8925.645;0250.50]R(Q˘1Q˘2)=.88R(Q˘1)R(Q˘1)=1.87×2.13=3.97∴R(Q˘1Q˘2)=R(Q˘1)R(Q˘1) (15)

8  Algorithm for Comparing Two Fuzzy Quantities Q˘1 and Q˘2

Step 1: when heights w1<w2 . Use formulas (2) to calculate the Q(G)=(x,y)

Step 2: when heights w1>w2 Use formulas (4) to calculate the Q(G)=(x,y)

Step 3: when heights w1<w2 . Use step (1) and formulas (3) to calculate the R(Q˘1)andR(Q˘2)

Step 4: when heights w1>w2 Use step (2) and formulas (5) to calculate the R(Q˘1)andR(Q˘2)

Compare R(Q˘1)andR(Q˘2)

If, (Q˘1)<R(Q˘2) , then Q˘1<Q˘2

If, R(Q˘1)>R(Q˘2) , then Q˘1>Q˘2

If, R(Q˘1)≅R(Q˘2) , then Q˘1≅Q˘2

Table 1 specifies, the comparison between the proposed methods and existing benchmark methods tested by various researchers in different times.

9  An Application of RFN

In this section, we will appertain the suggested ranking technique to FRA. Let’s assume that there are three manufactories C1,C2 and C3 to produce the components A1,A2 and A3 respectively. Each module Ai is formed of three sub-modules Ai1,Ai2 and Ai3 , where 1≤i≤3 . Given Table 2 shows the extent of loss Wik and the chances of failure Rik of the sub-module Aik made by manufactory Ci , where 1≤i≤3 and 1≤k≤3 . In the accompanying, the proposed New GQSFN-CT FRA manner to manage with the FRA problem is shown as follows: