# A Novel Multiple-Criteria Decision-Making Approach Based on Picture Fuzzy Sets

Authors: Hanen Karamti; Muhammad Sarwar Sindhu; Muhammad Ahsan; Imran Siddique; Ibrahim Mekawy; Hamiden Abd El-Wahed Khalifa
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Experts are using picture fuzzy sets (PFSs) in their probes to resolve the uncertain and vague information during the process of decision making because PFSs describe human attitudes naturally. Divergence measure (DM) plays a dominant role in discriminating between two distributions of probability and extracting consequences from that discrimination. In the present work, a novel picture fuzzy divergence measure (PF-DM) is developed between two PFSs. Some of the suggested measure’s important qualities are also discussed with particular situations to validate it. Based on the suggested PF-DM, a multiple-criteria decision-making (MCDM) model is established to grab the fuzzy information. The suggested measure’s performance is compared to that of various existing measures in the literature. An MCDM model has been proven for the usefulness of the suggested technique in dealing with real-life scenarios in the context of dengue sickness and pattern identification. Validation of the suggested MCDM model has been further investigated using validity testing. To improve the generated model, a thorough comparison with several current methodologies has been carried out while taking the time complexity (TC) factor into account.


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## Body

## 1. Introduction

Handling vagueuncertain information in real-life situations has trouble. Therefore, various techniques, such as the theory of fuzzy sets (FSs), have been examined to address the ambiguity and uncertainty found in the real world. Zadeh [1] introduced the concept of FSs in 1965 to tackle the imprecise, unclear data and applied them in decision-making complications by considering diverse perspectives. Although the FS theory has been applied to a variety of sectors, some circumstances in actual life appear to be beyond its capabilities. To capture such occurrences, Atanassov [2] suggested intuitionistic fuzzy sets (IFSs). IFSs are a generalization of FSs made up of two degrees: membership degree (MD) and non-membership degree (NMD), with 0≤MD+NMD≤1. Innumerable investigators contributed their work in this extension, for example, based on the Hamming distance, Szmidt and Kacprzyk [3] developed a similarity measure (SM) between IFSs. Xu and Xia [4] categorized SM and the geometric distance of IFSs for group decision-making problems. Several scholars have begun research on Atanassov’s intuitionistic fuzzy theory due to its elasticity and effective applications. Furthermore, stimulating solicitations in specific areas such as risk analysis, image processing, and medical diagnosis have been established in [5, 6].This cosmos is full of uncertainty, inconsistency, and ambiguity. In reality, the great majority of the concepts we discuss include ambiguity instead of exact information. Addressing fear or uncertainties is a major difficulty in many professions, including sociology, natural science, engineering, medicine, and economics. Recently, a lot of authors have expressed a significant interest in depicting ambiguity. Traditional hypotheses, such as FS [1], vague sets [7], and rough sets [8], are well known and play a significant role in demonstrating uncertainty. Molodtsov [9] defined soft sets as an entirely new scientific instrument for coping with uncertainty.Regardless, IFSs have been used in many fields of life. However, there are a variety of situations in which IFSs cannot be used. For example, casting a vote and dealing with issues such as yes, neutral, rejection, and refusal, which restricts the use of IFSs. Cuong [10] developed a generalization of FS called picture fuzzy (PF) sets (PFSs) to address such issues. PFSs have considered three factors from their inception: neutral membership degree (NMD), acceptance membership degree (AMD), and refusal membership degree (RMD). The requirement 0≤AMD+NMD+RMD≤1 was met by these three features. From its beginning to the current day, PFSs have been verified for treating unclear circumstances in real-life issues, for example, decision making, pattern recognition, clustering, etc.Based on PFSs, numerous researchers have worked in different fields, for instance, Thong [11] put forward an algorithm for fuzzy clustering that depicts the benefits of PFSs, Wei [12] developed the PF cross-entropy model for MCDM problems, Ashraf et al. [13] explained several procedures to MCDM issues under PF framework, and Sindhu et al. [14] implemented PFSs to select the best alternative. Recently, Zhao et al. [15] presented a dynamic distance measure of PFS depending on a PF point operator to resolve the real-life problem. Duong and Thao established a dissimilarity measure on PFSs and applied it MCDM problem to choose the optimal option. Verma and Rohtagi [16] presented the similarity measures based on PFSs to diagnose diseases and pattern recognition. Gocer [17] created a group decision-making technique based on interval-valued PFSs and then used it to determine a sustainable supply chain strategy. Sustainable supply chain strategy is a complex MCDM issue involving multiple parameters that may be contradictory at the same time. Luo et al. [18] proposed the SM based on relationship matrix and showed that it satisfied all the conditions of SM and then implemented the established SM to resolve the MCDM problem.DM is useful for discriminating between two probability distributions and extracting conclusions from that discrimination. With the use of HFSs [19] and IFSs [20], many academics have worked and developed to answer decision-making difficulties. DM has also been utilized in MCDM problems by several experts [21, 22]. The PFS has been regarded as an extension of the FS and IFSs. As a result of the findings of the study on FS measures, the IFSs for the PFSs are likewise natural and necessary. This is the motivation behind our research into the DM of PFSs, both in theory and in practice. The major aims and contributions of this study are as follows:(i)
A novel PF divergence measure (PF-DM) is proposed along with its properties.(ii)
PF-DM is developed to measure the degree of fuzziness.(iii)
Two examples, one related to medical diagnosis and the other to pattern recognition, are used to reveal the strength and reliability of the newly developed PF-DM.Since the inception of the current MCDM theory in the early1960s, MCDM has been developing to aid decision making to meet various applications [23]. Its importance in numerous application fields has grown significantly, particularly when new techniques are developed and existing ones are improved. MCDM approaches can provide powerful tools for allocating resources inside difficult operations, as well as in complicated exercises such as picking alternatives, weighing criteria, and establishing priorities. MCDM strategies are techniques and ways that attempt to find an optimal solution when there are several competing criteria. The selection of the most appealing DM goals from a wide number of available options while taking into account a variety of physical and abstract features or attributes [24] is at the heart of MCDM techniques. MCDM is a discipline that supports decision makers in making the best decision possible from a set of options based on several factors [25]. Several MCDM approaches have been created in the current decade [26] and used in many fields, such as supplier selection [27, 28] and project management development [29]. Sindhu et al. [30, 31] proposed a TOPSIS-based assessment approach for PFSs to address MCDM problems.The remainder of the paper is organized as follows. Some basic and related knowledge is penned in Section2. Novel PF-DM and its properties are discussed comprehensively in Section 3. An MCDM model based on PF-DM is established to resolve the uncertain information. Section 5 comprises two MCDM problems related to medical diagnosis and pattern recognition. In Section 6, the validity of the suggested model is investigated. A comprehensive comparison based on TC is analyzed in Section 6.3. Finally, Section 7 points out some conclusions.


## 2. Basic Concepts

Some basic ideas connected to the present work like FSs, IFSs, and PFSs and their operational laws are briefly penned in this section.Definition 1 (see [1]).
Let Y be a discourse set such thatY=y1,y2,…,yn; then, a FSℱoverYcan be illustrated as(1)ℱ=y,μℱy|y∈Y,whereμℱy:X⟶0,1is a MD so thaty∈Ytoℱ.Definition 2 (see [2]).
Suppose thatYis a fixed set, and an IFSℐonYis illustrated as(2)ℐ=y,αℐy,βℐy|y∈Y,whereαℐyandβℐy∈0,1are called the MD and NMD ofy∈Ytoℐ, respectively, with the condition:0≤αℐy+βℐy≤1, for ally∈Y.For ally∈Y,ηIyis known as hesitancy degree ofy∈ℐwhereηIy=1−αℐy−βℐy.Definition 3 (see [10]).
A PFS onY=y1,y2,…,ynis penned as follows:(3)P=y,αPy,ηPy,βPy|y∈Y,where αPy, βPy, and ηPy∈0,1 are called the MD, neutral degree, and NMD of y∈Y to the set P; subsequently, αPy, ηPy, and βPy fulfil the condition: 0≤αPy+ηPy+βPy≤1, for all y∈Y. Also, ηPy=1−αPy−ηPy−βPy; then, ηPy is supposed to represent a degree of refusal membership of y∈Y in P. For simplicity, PFS over a fixed set Y is read as PFS (Y).Definition 4 (see [10]).
LetP1=αP1y,ηP1y,βP1y, P2=αP2y,ηP2y,βP2y, andP=αPy,ηPy,βpybe three PFSs onY; then, some arithmetic operations are described as follows:(1)
P1⊆P2iffαP1y≤αP2yandηP1y≤ηP2yfor ally∈Y.(2)
P1=P2iffP1⊆P2andP2⊆P1.(3)
P1⊔P2=y,maxαP1y,αP2y,minηP1y,ηP2y,minβP1y,βP2y|y∈Y.(4)
P1⊓P2=y,maxαP1y,αP2y,minηP1y,ηP2y,minβP1y,βP2y|y∈Y.(5)
PC=y,βPy,ηPy,αPy|y∈Y.


## 3. Novel Divergence Measure for PFSs

In the present work, a novel PF-DM is proposed.(i)
DmP1,P2≥0.(ii)
DmP1,P2=0⇔P1=P2.Definition 5.
LetP1,P2∈PFSs. The function D: PFS×PFS R is called a PF-divergence measure (PF-DM) denoted byDmP1,P2if it satisfiesDefinition 6.
LetP1,P2∈PFSs. Then, a PF-DM is defined by(4)DmP1,P2=1n∑i=1n16−1P16yi+P26yi66−P1yi+P2yi6,where Pyi=αpyi,ηpyi,βpyi and Pkyi=αpkyi,ηpkyi,βpkyi,k=6.Theorem 1.
LetP1,P2∈PFSs. Then, the measureDMP1,P2is valid PF-divergence measure.Proof.
(i)
The following inequality holds for any two real numbersa,b∈0,1:a6+b6/66≥a+b/6.Since0≤P1yi,P2yi≤1.⇒P16yi+P26yi/66≥P1yi+P2yi/6.⇒P16yi+P26yi/66−P1yi+P2yi/6≥0.⇒DmP1,P2≥0.(ii)
Suppose thatP1=P2; therefore, α1yi=α2yi,η1yi=η2yi,β1yi=β2yi, ∀i=1,2,…,n. Then, (4) becomes(5)DmP1,P2=1n∑i=1n16−1P16yi+P26yi66−P1yi+P2yi6=0.
Conversely, assumeDmP1,P2=0.(6)DmP1,P2=1n∑i=1n16−1P16yi+P26yi66−P1yi+P2yi6=0.
This is possible if and only ifα1yi=α2yi,η1yi=η2yi,β1yi=β2yi, ∀i=1,2,…,n.
This proves thatP1=P2.
Since the measureDmP1,P2 holds the postulates of Definition 5, DmP1,P2 is an appropriate divergence measure for PFSs.Theorem 2.
LetP1,P2,3∈PFSs (Y); then, the measureDmP1,P2 satisfies the following axioms:iDmP1,P2=DmP2,P1.ii∀P1⊆P2⊆P3, we haveaDmP1,P2≤DmP1,P3.bDmP2,3≤DmP1,P3.Proof.
(i)
It is easy to understand.(ii)
a Let P1⊆P2⊆P3; then, for any 0≤i≤n,α1yi≤α2yi≤α3yi,η1yi≤η2yi≤η3yi,andβ1yi≤β2yi≤β3yi, and we obtainDmP1,P2=1/n∑i=1n1/6−1P16yi+P26yi/66−P2yi+P2yi/6≤1/n∑i=1n1/6−1P16yi+P36yi/66−P1yi+P3yi/6=DmP1,P3.Hence,DmP1,P2≤DmP1,P3. Similarly, we can verify axiom b.Theorem 3.
LetP1,P2∈PFSs. Then, forDmP1,P2, we get(1)
DmP1ℭ,P2ℭ=DmP1,P2,(2)
DmP1ℭ,P2=DmP1,P2ℭ,(3)
∀P1⊆P2orP2⊆P1, we get(i)
DmP1∩P2,P2=DMP1,P1∪P2≤DmP1,P2,(ii)
DmP1∩P2,P1∪P2=DmP1,P2.Proof.
(1)
SinceP1=α1yi,η1yi,β1yi, P2=α2yi,η2yi,β2yi,P1ℭ=β1yi,η1yi,α1yi, P2ℭ=β2yi,η2yi,α2yi; then,DmP1ℭ,P2ℭ=1/n∑i=1n1/6−1β16yi+η16yi+α16yi+β26yi+η26yi+α26yi/66−β1yi+η1yi+α1yi+β2yi+η2yi+α2yi/6,=1/n∑i=1n1/6−1α16yi+η16yi+β16yi+α26yi+η26yi+β26yi/66−α1yi+η1yi+β1yi+α2yi+η2yi+β2yi/6⇒DmP1ℭ,P2ℭ=DmP1,P2.(2)
DmP1ℭ,P2=1/n∑i=1n1/6−1β16yi+η16yi+α16yi+α26yi+η26yi+β26yi/66−β1yi+η1yi+α1yi+α2yi+η2yi+β2yi/6,=1/n∑i=1n1/6−1α16yi+η16yi+β16yi+β26yi+η26yi+α26yi/66−α1yi+η1yi+β1yi+β2yi+η2yi+α2yi/6,(ii)⇒DmP1ℭ,P2=DmP1,P2ℭ.(3)
(i) TakeP1⊆P2; then, DmP1∩P2,P2=DmP1,P2, also, DmP1,P1∪P2=DmP1,P2.Now ifP2⊆P1, then DmP1∩P2,P2=DmP2,P2=0; also, DmP1,P1∪P2=DmP1,P1=0.That is, ifP1⊆P2 or P2⊆P1, we get DmP1∩P2,P2=DmP1,P1∪P2≤DmP1,P2.(ii) Its proof is simple.


## 4. MCDM Process Based on PF-DM

In this section, a decision-making process based on PF-DM has been provided, followed by practical examples to demonstrate the concept. Suppose a collection ofn alternatives represented by A=A1,A2,…,An that has been investigated by the decision maker by considering the collection of m different criteria Q=Q1,Q2,…,Qm having weight vectors is W=w1,w2,…,wmT so that ∑j=1mwj=1. Consider the decision maker assessing the rating of an alternative Ai,1≤i≤n on the criteria Gj,1≤j≤m in the form of PFSs. To arrive at the best solution, MCDM takes the following actions.Step 1. Gather data in the form of PF decision matrix. Pc=pijn×m.Pc=αP11,ηP11,βP11αP12,ηP12,βP12…αP1m,ηP1m,βP1mαP21,ηP21,βP11αP22,ηP22,βP22…αP2m,ηP2m,βP2m⋮⋮⋱⋮αPn1,ηPn1,βPn1αPn2,ηPn2,βPn2…αPnm,ηPnm,βPnm.Step 2. Evaluate the ideal solution N=minpijn×m.Step 3. Based on (4), compute the PF-DM.Step 4. Arrange the values of PF-DM from lower order to top order.Step 5. Based on Step 4, choose the best alternative.


## 5. Solution of Problems Based on PF-DM

The authors used the proposed MCDM model to recognize the pattern and breakout of dengue disease in this section.Step 1. Picture fuzzy (PF) decision matrix Pc=pij5×5 is denoted in Table 1.Step 2. The ideal solution N = {(y1,0.1000, 0.1000, 0.1000), (y2,0,0.1000, 0.3000), (y3,0.3000, 0.1000, 0.1000), (y4, 0.2000, 0.1000,0.1000), (y5,0.1000,0.1000, 0.2000)}.Step 3. The PF-DM values among the symptoms B1, B2, B3,B4, and B5 and the ideal solution N areDmB1,N=0.0629, DmB2,N=0.1006,DmB3,N=0.0824,DmB4,N=0.0866, DmB5,N=0.0558.Step 4. Arrange the values of PF-DM from lower order to top order such that B5≺B1≺B3≺B4≺B2.Step 5. The most frequent symptom is B5 which can be seen in Figure 1.Table 1
PF decision matrix.
SymptomsB1{(y1, 0.1, 0.4, 0.3), (y2, 0.4, 0.1, 0.3), (y3, 0.3, 0.2, 0.4),(y4, 0.2, 0.3, 0.5), (y5, 0.5, 0.1, 0.3)}B2{(y1, 0.3, 0.3, 0.3), (y2, 0.0, 0.2, 0.7), (y3, 0.6, 0.1, 0.1),(y4, 0.4, 0.4, 0.2), (y5, 0.1, 0.5, 0.3)}B3{(y1, 0.5, 0.1, 0.3), (y2, 0.3, 0.4, 0.3), (y3, 0.6, 0.1, 0.3),(y4, 0.4, 0.4, 0.1), (y5, 0.1, 0.5, 0.2)}B4{(y1, 0.6, 0.2, 0.1), (y2, 0.3, 0.2, 0.4), (y3, 0.4, 0.2, 0.3),(y4, 0.6, 0.1, 0.1), (y5, 0.4, 0.2, 0.2)}B5{(y1, 0.1, 0.4, 0.3), (y2, 0.4, 0.1, 0.3), (y3, 0.3, 0.2, 0.4),(y4, 0.2, 0.3, 0.4), (y5, 0.5, 0.1, 0.3)}Figure 1
Ranking of symptoms by PF-DM.Example 1.
In the context of dengue disease, it is critical to provide a productive path in crisis response in order to avoid additional misfortunes and save people’s lives. As a result of such a life-threatening situation, health professionals must respond quickly in order to effectively control the situation and prevent further deaths. There are five major symptoms, namely, aching muscles and jointsB1, body rash that can disappear and then reappear B2, high fever B3, intense headache B4, and vomiting and feeling nauseous B5, that can be investigated to diagnose this disease. All symptoms, B1, B2, B3,B4, and B5, are penned in terms of PFSs in Table 1.An unknown pattern is classified into a specific pattern using numerous fuzzy information parameters such as distance, similarity, accuracy, and others. For pattern recognition, we employ the proposed PF-DM. A pattern recognition problem is phrased as follows in the picture fuzzy environment.(i)
Step 1. PF decision matrix Pc=pij3×5 is denoted in Table 2.Step 2. An unknown pattern represented by N is:N¯=0.1000,0.1000,0.1000,0.2000,0.3000,0.1000,0.2000,0.1000,0.0,0.1000,0.0,0.2000,0.3000,0.1000,0.1000Step 3. The PF-DM values among the alternatives B1, B2, and B3 and the unknown pattern N¯ areDmB1,N¯=0.2953, DmB2,N¯=0.3303, DmB3,N¯=0.2495.Step 4. Arrange the values of PF-DM from lower order to top order such that B3≺B1≺B2.Step 5. Hence, B3 is an optimal pattern. Figure 2 illustrates the ranking order of the alternatives.Table 2
PF decision matrix.
PatternsB1{(y1, 0.4, 0.3, 0.1), (y2,0.5, 0.3, 0.2), (y3,0.4, 0.3, 0.0), (y4,0.7, 0.0, 0.2), (y5,0.6, 0.1, 0.1)}B2{(y1, 0.7,0.1, 0.1), (y2, 0.2, 0.3, 0.4), (y3,0.2, 0.1, 0.5), (y4,0.1, 0.5, 0.2), (y5,0.3, 0.3, 0.3)}B3{(y1, 0.1, 0.3, 0.4), (y2,0.4, 0.3, 0.1), (y3,0.3, 0.4, 0.2), (y4,0.2, 0.5, 0.3), (y5,0.5, 0.3, 0.1)}Example 2.
SupposeA=B1,B2,B3are some well-known patterns distinguished by PFSs in the universal setY=Y1,Y2,Y3,Y4,Y5which are given in the table below.Because determining which option is the best match for a particular choice issue is impractical in real life, Wang and Triantaphyllou [32] developed testing criteria to examine the validity of MCDM approaches, which are as follows.Figure 2
Ranking of alternatives by PF-DM.


## 6. Validity Testing of Criteria for the Proposed MCDM Model

Criterion 1.
An effective MCDM approach should not change the indication of the best option when replacing a non-optimal option with a worse option without changing the relative relevance of any decision criterion.Criterion 2.
The transitive property should be followed by an effective MCDM approach.Criterion 3.
When an MCDM problem is broken down into smaller problems and the same MCDM method is used to assess the options on the smaller problems, the combined ranking of the alternatives should equal the undecomposed problem’s original rating. These testing criteria are being used to evaluate the proposed PF-DM MCDM model’s validity.


### 6.1. Testing Criterion 1

Decision matrix provided in Table3 is attained by altering the AMD and RMD of the alternative B1 (non-favorable) and B2 (inferior) in the original decision matrix to check the efficacy of the suggested model under test criterion 1 of Example 1.Table 3
PF decision matrix.
SymptomsB1{(y1, 0.3, 0.4, 0.1), (y2, 0.3, 0.1, 0.4), (y3, 0.4 0.2, 0.3),(y4, 0.5, 0.3, 0.2), (y5, 0.3, 0.1, 0.5)}B2{(y1,0.3, 0.3, 0.3), (y2, 0.70, 0.2, 0.0), (y3, 0.1, 0.1, 0.6),(y4, 0.2, 0.4, 0.4), (y5, 0.3, 0.5, 0.1)}B3{(y1, 0.5, 0.1, 0.3), (y2, 0.3, 0.4, 0.3), (y3, 0.6, 0.1, 0.3),(y4, 0.4, 0.4, 0.1), (y5, 0.1, 0.5, 0.2)}B4{(y1, 0.6, 0.2, 0.1), (y2, 0.3, 0.2, 0.4), (y3, 0.4, 0.2, 0.3),(y4, 0.6, 0.1, 0.1), (y5, 0.4, 0.2, 0.2)}B5{(y1, 0.1, 0.4, 0.3), (y2, 0.4, 0.1, 0.3), (y3, 0.3, 0.2, 0.4),(y4, 0.2, 0.3, 0.4), (y5, 0.5, 0.1, 0.3)}The PF-DM values among the symptomsB1, B2, B3,B4, and B5 and the ideal solution N are DmB1,N=0.0660, DmB2,N=0.1037, DmB3,N=0.0855, DmB4,N=0.0896, and DmB5,N=0.0590. As a result, the proposed technique supported test criterion 1 provided in [32].


### 6.2. Testing Criteria 2 and 3

To validate the suggested approach, use test criterion 2 and test criterion 3. The MCDM problem presented in Example1 has been divided into five smaller subsets such as B1,B2,B3,B4, B1,B2,B3,B5,B1,B3,B4,B5, B1,B2,B3,B5, and B2,B3,B4,,B5. We may obtain the following arrangements for each subset using the suggested MCDM model: B1≺B3≺B4≺B2, B5≺B1≺B3≺B2, B5≺B3≺B4≺B1, B5≺B1≺B3≺B1, and B5≺B3≺B4≺B2, respectively. By aggregating all these arrangements, overall ranking B5≺B1≺B3≺B4≺B2 is obtained which is identical to the original one. Thereby, the suggested MCDM model is validated under the test criteria 2 and 3 presented in [32].


### 6.3. Comparison Based on Time Complexity

In order to strengthen the results obtained from the proposed MCDM model, TC analysis is performed in this section. TC is the time required to execute an algorithm to reach the final result. TC is measured among the proposed and the existing techniques presented by Kadian and Kumar [6], Thao et al. [33], Wang et al. [34], and Wei et al. [35]. The executing time of each technique is evaluated with the help of MATLAB which is presented in Table 4, and its graphical view is drawn in Figure 3. From Table 4, our approach takes less time as compared to others, and hence the proposed MCDM model is more effective and resolves the issues rapidly.Table 4
TC among the proposed and existing techniques.
TechniquesExecuting timeProposed model0.3513 secondsKadian and Kumar [6]0.4133 secondsThao et al. [33]0.3910 secondsWang et al. [34]0.4402 secondsWei et al. [35]0.4000 secondsFigure 3
Graphical view of TC analysis.


## 6.1. Testing Criterion 1

Decision matrix provided in Table3 is attained by altering the AMD and RMD of the alternative B1 (non-favorable) and B2 (inferior) in the original decision matrix to check the efficacy of the suggested model under test criterion 1 of Example 1.Table 3
PF decision matrix.
SymptomsB1{(y1, 0.3, 0.4, 0.1), (y2, 0.3, 0.1, 0.4), (y3, 0.4 0.2, 0.3),(y4, 0.5, 0.3, 0.2), (y5, 0.3, 0.1, 0.5)}B2{(y1,0.3, 0.3, 0.3), (y2, 0.70, 0.2, 0.0), (y3, 0.1, 0.1, 0.6),(y4, 0.2, 0.4, 0.4), (y5, 0.3, 0.5, 0.1)}B3{(y1, 0.5, 0.1, 0.3), (y2, 0.3, 0.4, 0.3), (y3, 0.6, 0.1, 0.3),(y4, 0.4, 0.4, 0.1), (y5, 0.1, 0.5, 0.2)}B4{(y1, 0.6, 0.2, 0.1), (y2, 0.3, 0.2, 0.4), (y3, 0.4, 0.2, 0.3),(y4, 0.6, 0.1, 0.1), (y5, 0.4, 0.2, 0.2)}B5{(y1, 0.1, 0.4, 0.3), (y2, 0.4, 0.1, 0.3), (y3, 0.3, 0.2, 0.4),(y4, 0.2, 0.3, 0.4), (y5, 0.5, 0.1, 0.3)}The PF-DM values among the symptomsB1, B2, B3,B4, and B5 and the ideal solution N are DmB1,N=0.0660, DmB2,N=0.1037, DmB3,N=0.0855, DmB4,N=0.0896, and DmB5,N=0.0590. As a result, the proposed technique supported test criterion 1 provided in [32].


## 6.2. Testing Criteria 2 and 3

To validate the suggested approach, use test criterion 2 and test criterion 3. The MCDM problem presented in Example1 has been divided into five smaller subsets such as B1,B2,B3,B4, B1,B2,B3,B5,B1,B3,B4,B5, B1,B2,B3,B5, and B2,B3,B4,,B5. We may obtain the following arrangements for each subset using the suggested MCDM model: B1≺B3≺B4≺B2, B5≺B1≺B3≺B2, B5≺B3≺B4≺B1, B5≺B1≺B3≺B1, and B5≺B3≺B4≺B2, respectively. By aggregating all these arrangements, overall ranking B5≺B1≺B3≺B4≺B2 is obtained which is identical to the original one. Thereby, the suggested MCDM model is validated under the test criteria 2 and 3 presented in [32].


## 6.3. Comparison Based on Time Complexity

In order to strengthen the results obtained from the proposed MCDM model, TC analysis is performed in this section. TC is the time required to execute an algorithm to reach the final result. TC is measured among the proposed and the existing techniques presented by Kadian and Kumar [6], Thao et al. [33], Wang et al. [34], and Wei et al. [35]. The executing time of each technique is evaluated with the help of MATLAB which is presented in Table 4, and its graphical view is drawn in Figure 3. From Table 4, our approach takes less time as compared to others, and hence the proposed MCDM model is more effective and resolves the issues rapidly.Table 4
TC among the proposed and existing techniques.
TechniquesExecuting timeProposed model0.3513 secondsKadian and Kumar [6]0.4133 secondsThao et al. [33]0.3910 secondsWang et al. [34]0.4402 secondsWei et al. [35]0.4000 secondsFigure 3
Graphical view of TC analysis.


## 7. Conclusions

A PF-DM is proposed in this study, along with proof of its validity, and some of its features are studied. Existing DMs for PFSs are less feasible than the suggested DM. The suggested divergence metric surpasses other current similarities and DMs by illustrating real challenges in medical analysis and pattern recognition. It can be seen that in pattern recognition, the suggested divergence metric can assign an unknown pattern to one of the previously recognized patterns. Diseases are diagnosed in medical analysis based on their symptoms, which are represented by visual fuzzy numbers. In addition, pattern recognition employs the proposed divergence measure. It holds in both circumstances.As a result, the proposed DM is useful for pattern identification as well as medical diagnostics. In addition, the following are the study’s long-term goals. (i) It is possible to develop a parametric generalized divergence measure. (ii) PFS can be enhanced by including a third parameter, the degree of refusal. (iii) The suggested divergence metric may be expanded and implemented in an image fuzzy soft set environment. (iv) The suggested DMs may be used to a wide range of real-world problems [36].AbbreviationsDescriptionDMDivergence measureFSsFuzzy setsIFSsIntuitionistic fuzzy setsMDMembership degreeMCDMMultiple-criteria decision makingNMDNon-membership degreePFSsPicture fuzzy setsPF-DMPicture fuzzy divergence measureSMSimilarity measure



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*Source: 2537513-2022-02-27.xml*