Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1472 https://internationalpubls.com An Innovative Distance Measure for Picture Fuzzy Sets and Applications in Medical Diagnosis and Pattern Analysis Satpal Sing𝐑𝟏,𝟐*, Satish Kumar 𝟏 1Department of Mathematics, Maharishi Markandeshwar (Deemed to be University), Mullana-Ambala, Haryana, India, 133207. 2Department of Mathematics, IIHS, Kurukshetra University, Kurukshetra, Haryana, India, 136119. *Corresponding author email: satpal.iihs@kuk.ac.in Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this paper, we proposed a novel distance measure and applied on picture fuzzy sets which have the various applications in uncertain problems that cannot be solved easily by fuzzy set, intuitionistic fuzzy sets, fermatean fuzzy sets, Pythagorean fuzzy sets, ortho-pair fuzzy sets easily. We can explore these sets with the help of problems in the field of personnel selection, pattern analysis, medical diagnosis, or human voting etc. This field type of problem required the answer in the form of yes, no, refusal and abstain. The distance measures play an important role for comparing the picture fuzzy sets. If we take a look in the background of literature then we find a lot of works have been done on distance measures for picture fuzzy sets. Unreasonable results in most of the problems have been found, all of these distance measures. So, we are suggesting a new distance measures for picture fuzzy sets in this paper which is more effective and useful compare to the already existed distance measures. We also illustrating its importance, classification and medical problems and view on performance with existed measures. Keywords: Distance measure, fuzzy set, picture fuzzy set, intuitionistic fuzzy set, medical diagnosis, pattern analysis. 1. Introduction Zadeh introduced the theory of fuzzy sets (FSs), emphasizing the notion of degrees of membership and non-membership. The applications of fuzziness has been extended to learning theory, algorithms, formal languages, automata, probability theory and it deals the imprecise, uncertainties and vagueness problems in an efficient way [1], [2]. It is also explained that fuzzy parametric function and nonfuzzy mappings are treated as special classes of fuzzy functions and mappings. Specially in situation of real life, due to imprecise nature of medical issues and uncertain information collected for decision making needs the concept of fuzzy[3], [4], [5], [6], [7]. In fuzzy set theory (FST), each element is given a membership value that ranges between 0 and 1. However sometimes it is not possible to assign for a membership function, therefore, it is more practical to assign an interval value. Fuzzy sets theory (FST) has been applied in different fields such as transmission system, medical diagnosis, facial pattern recognition, cluster point and decisions making process etc. The element’s value in a fuzzy set (FSs) cannot be chosen independently, hence the concept of intuitionistic fuzzy sets (IFSs) was introduced by Atanassov. [8]. Atanassov described intuitionistic fuzzy sets as having every element assigned a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1473 https://internationalpubls.com membership value and non-membership value, both of which lies in the closed interval [a, b] where 0≀ π‘Ž, 𝑏 ≀ 1 and their sum must be less than or equal to one. This restriction of the sum of membership and non-membership values bound the scope of (IFSs) and opened the way for Yager [5] to the extended concept of Pythagorean fuzzy sets (PFSs). In a Pythagorean fuzzy set (PFSs), every member has a membership and non-membership value lies in the closed interval a, b] where 0≀ π‘Ž, 𝑏 ≀1 its property their square sum equal or less than to 1. Although Pythagorean fuzzy sets (PFSs) are much quicker compare to fuzzy sets (FSs) and Intuitionistic fuzzy sets (IFSs) as they cannot solve the conditions in which the square sum of membership grades exceeds one. Then, the next concept of generalized ortho pair fuzzy sets was introduced by Yager [6], and he named it q-rung orthopair fuzzy set (q-ROPFSs). Q-ring ortho-pair fuzzy set (q-ROPFSs) is characterized by each member having a membership value and a non-membership value both of which lies in the closed interval [0,1] using the power sum equal or less than 1. Indeed, these extensions of fuzzy set (FSs) are insufficient to consider for the neutrality degree of a member which play a vital role in various decision-making problems, often referred as voting, personnel selection, pattern analysis medical diagnosis and cluster point, etc. To fill this gap, picture fuzzy set (PFSs) was an extension of fuzzy set (FSs), which was suggested by Coung and Kreinovich [9], [10]. In a picture, fuzzy set (PFSs), every element is represented by three values referred by a degree by positive membership, a degree by negative membership, and a degree of neutral membership. In a picture fuzzy set, every member having membership value, a non-membership value and a neutrality value lies between the closed interval [0,1] with all sums equal or less than 1. Coung[9] also defined the picture fuzzy sets (PFSs) properties or operations. After that, Son[11], [12], [13] applied picture fuzzy sets (PIFSs) to solve clustering problems. Nguyen et al.[14] applied picture fuzzy sets into geographic data clustering applications. Some applications of picture fuzzy sets in databases and studies on some fuzzy logic operators for picture fuzzy sets were introduced by [13], [15]. Atanassov [16] has presented new results on intuitionistic fuzzy sets (IFSs), opening the door for further work. In another study, Atanassov [17] introduced new types of operations for intuitionistic fuzzy sets (IFSs). Later, Atanassov [18] defined several operators for interval-valued intuitionistic fuzzy sets, which are various applicable in various application to solve the real life problems. Atanassov[19] pro Meredith’s axiom, axiom, which is valid for the intuitionistic fuzzy propositional calculus. Atanassov and Gargov have generalized the notion of intuitionistic fuzzy sets in the spirit of ordinary interval- valued fuzzy sets (IVFSs). In this paper, basic preliminaries of (IVIFSs) theory are determined. The relation of equivalencebetween two picture fuzzy sets and their applications in clustering were discussed in [20]. Additionally, [21] introduced methods for comparing two picture fuzzy sets (PFSs), as well as distance and dissimilarity measure operators for picture fuzzy sets (PFSs). In this section we will discuss about the distance measures. Some distance measures in picture fuzzy sets(PIFSs) are introduced by Dutta [22] and Son[22] introduced few generalized distance measures using (PIFSs) for application in Clustering analysis. Joshi [23], [24], [25], [26] dealing with comparative study of distance measures on picture fuzzy sets. In strategic decision-making, Wei [27] explored similarity measures for picture fuzzy sets (PIFSs) using cosine and cotangent functions. Peng [28] introduces an algorithm for picture fuzzy sets (PIFSs) and applying this algorithm into the decision- making process[29], [30], [31]. Wei [32] contributes cosine, weighted cosine, weighted set-theoretic similarity measures for (PIFSs), describing their applications in pattern recognition problems. Picture Fuzzy Set (PFSs) serves as an extension of conventional fuzzy sets (FS) and intuitionistic fuzzy sets Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1474 https://internationalpubls.com (IFSs) introduced by Atanassov in 1986. Within the framework pf PFSs, elements exhibit positive, negative, neutral and refusal degrees, each elucidating the diverse levels of importance attributed to a member within a given set [33]. This extension provides a nuanced representation that captures a richer spectrum of information, allowing for a more comprehensive characterization of membership relationships[34]. Several significant factors motivated us to conduct this research which are following as: ❖ There is several (PIFSs) distance measure that do not satisfy all the necessary fundamental conditions ❖ Many existing (PIFSs) distance measures deliver unreasonable results when computing distances between different (PIFSs). ❖ Existing distance measures for (PIFSs) unable to identifying unknown patterns in problems related to pattern recognition. ❖ In view of these factors, this paper proposes a novel distance measure for (PIFSs) which explores its application in classification and medical problems ❖ This paper contributes including and introducing an innovative distance measure for (PIFSs) along with its properties. ❖ Proving, the proposed measure through numerical problems, that the proposed measure showing the limitations of existing distance measures. ❖ Illustrating how the proposed measure can be utilized in pattern analysis, medical diagnosis and comparing its efficiency to the existing measures. The paper is divided mainly into the 6 section that follows. Preliminary is given in Section 2. In Section 3 a carefully reviewed covered the existed measures (PIFSs). In Section 4, a new distance measure for (PIFSs) is proposed together with its properties. Numerical problems are used to comparing this proposed measure with the previous ones. Section 5 discusses how the proposed measure is applied to medical and classification problems. Finally, Section 6 concludes the paper along with some suggestion for future study. 2. Preliminaries Definition 2.1 Let π‘ˆ be an universal set having elements such as π‘Žπ‘–, then we define a fuzzy set (FS) 𝐹1 in π‘ˆ as 𝐹1 = {(π‘Žπ‘— , πœ‡πΉ1 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ}, where πœ‡πΉ1 (π‘Žπ‘—) is the degree of membership of π‘Žπ‘– in π‘ˆ such that 0 ≀ πœ‡πΉ1 (π‘Žπ‘—) ≀ 1, here membership degree 0 denotes no presence and membership, degree 1 means the element is completely part of the set, while values between 0 and 1 indicate the element is only partially included in the set. [35]. Definition 2.2 In fuzzy set (FS) theory, intuitionistic fuzzy set (IFSs) consider more than one uncertainty membership and non-membership degrees. The intuitionistic fuzzy set (IFSs) [35] 𝐹1 𝑖𝑛 π‘ˆ may be referred as: 𝐹1 = {(π‘Žπ‘— , πœ‡πΉ1 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ}, where πœ‡πΉ1 (π‘Žπ‘—) and 𝜈𝐹1 (π‘Žπ‘—) denotes the degree of membership and non-membership functions of an element π‘Žπ‘— in π‘ˆ such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1475 https://internationalpubls.com 0 ≀ πœ‡πΉ1 (π‘Žπ‘—) + 𝜈𝐹1 (π‘Žπ‘—) ≀ 1. Also, the intuitionistic fuzzy index known as hesitancy degree of π‘Žπ‘– 𝑖𝑛 π‘ˆ is given by πœ‡πΉ1 (π‘Žπ‘—) + 𝜈𝐹1 (π‘Žπ‘—) + πœ‹πΉ1 (π‘Žπ‘—) = 1. Definition 2.3 For two intuitionistic fuzzy sets (IFSs) [8], [35], [36] 𝐹1π‘Žπ‘›π‘‘ 𝐹2 in π‘ˆ with πœ‡πΉ1 (π‘Žπ‘—) π‘Žπ‘›π‘‘ 𝜈𝐹1 (π‘Žπ‘—) are the degree of membership and non-membership of the elements in set 𝐹1, and πœ‡πΉ2 (π‘Žπ‘—) π‘Žπ‘›π‘‘ 𝜈𝐹2 (π‘Žπ‘—) are the membership and non-membership degrees of the elements in set 𝐹2., then the following operations are hold. a) 𝐹1 βŠ† 𝐹2 𝑖𝑓𝑓 πœ‡πΉ1 (π‘Žπ‘—) ≀ πœ‡πΉ2 (π‘Žπ‘—) π‘Žπ‘›π‘‘ 𝜈𝐹1 (π‘Žπ‘—) β‰₯ 𝜈𝐹2 (π‘Žπ‘—), b) 𝐹1 = 𝐹2 𝑖𝑓𝑓 𝐹1 βŠ† 𝐹2 π‘Žπ‘›π‘‘ 𝐹2 βŠ† 𝐹1, c) 𝐹1 βˆͺ 𝐹2 = {π‘Žπ‘— , π‘šπ‘Žπ‘₯ (πœ‡πΉ1 (π‘Žπ‘—), πœ‡πΉ2 (π‘Žπ‘—)) , π‘šπ‘–π‘› (𝜈𝐹1 (π‘Žπ‘—), 𝜈𝐹2 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ}, d) 𝐹1 ∩ 𝐹2 = {π‘Žπ‘— , π‘šπ‘–π‘› (πœ‡πΉ1 (π‘Žπ‘—), πœ‡πΉ2 (π‘Žπ‘—)) , π‘šπ‘Žπ‘₯ (𝜈𝐹1 (π‘Žπ‘—), 𝜈𝐹2 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ}, e) (𝐹1)𝑐 = {(π‘Žπ‘— , πœ‡πΉ1 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—) |π‘Žπ‘— ∈ π‘ˆ}, f) 𝐹1. 𝐹2 = {π‘Žπ‘— , πœ‡πΉ1 (π‘Žπ‘—). πœ‡πΉ2 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—) + 𝜈𝐹2 (π‘Žπ‘—) βˆ’ 𝜈𝐹1 (π‘Žπ‘—). 𝜈𝐹2 (π‘Žπ‘—)}. Definition 2.4 A picture fuzzy set (PFS) [9], [10] 𝐹1 ∈ π‘ˆ is defined as 𝐹1 = {(π‘Žπ‘— , πœ‡πΉ1 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—), 𝛾𝐹1 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ}, where πœ‡πΉ1 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—) and 𝛾𝐹1 (π‘Žπ‘—) denotes the degree of membership (positive), non-membership (negative) and neutrality respectively of π‘Žπ‘— ∈ π‘ˆ holds the following conditions 0 ≀ πœ‡πΉ1 (π‘Žπ‘—) + 𝜈𝐹1 (π‘Žπ‘—) + 𝛾𝐹1 (π‘Žπ‘—) ≀ 1 and refusal degree πœ‘πΉ1 = 1 βˆ’ πœ‡πΉ1 (π‘Žπ‘—) βˆ’ 𝜈𝐹1 (π‘Žπ‘—) βˆ’ 𝛾𝐹1 (π‘Žπ‘—) for all π‘Žπ‘— ∈ π‘ˆ. Definition 2.5 For two picture fuzzy set (PFS) [9], [10] 𝐹1π‘Žπ‘›π‘‘ 𝐹2 ∈ π‘ˆ following operations holds a) 𝐹1 βŠ† 𝐹2 𝑖𝑓𝑓 πœ‡πΉ1 (π‘Žπ‘—) ≀ πœ‡πΉ2 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—) β‰₯ 𝜈𝐹2 (π‘Žπ‘—) π‘Žπ‘›π‘‘ 𝛾𝐹1 (π‘Žπ‘—) ≀ 𝛾𝐹2 (π‘Žπ‘—), b) 𝐹1 = 𝐹2 𝑖𝑓𝑓 𝐹1 βŠ† 𝐹2 π‘Žπ‘›π‘‘ 𝐹2 βŠ† 𝐹1, c) 𝐹1 βˆͺ 𝐹2 = {π‘Žπ‘— , π‘šπ‘Žπ‘₯ (πœ‡πΉ1 (π‘Žπ‘—), πœ‡πΉ2 (π‘Žπ‘—)) , π‘šπ‘–π‘› (𝜈𝐹1 (π‘Žπ‘—), 𝜈𝐹2 (π‘Žπ‘—)) , π‘šπ‘–π‘› (𝛾𝐹1 (π‘Žπ‘—), 𝛾𝐹2 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ}, d) 𝐹1 ∩ 𝐹2 = {π‘Žπ‘— , π‘šπ‘–π‘› (πœ‡πΉ1 (π‘Žπ‘—), πœ‡πΉ2 (π‘Žπ‘—)) , π‘šπ‘Žπ‘₯ (𝜈𝐹1 (π‘Žπ‘—), 𝜈𝐹2 (π‘Žπ‘—)) , π‘šπ‘–π‘› (𝛾𝐹1 (π‘Žπ‘—), 𝛾𝐹2 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ}, e) (𝐹1)𝑐 = {(π‘Žπ‘— , πœ‡πΉ1 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—), 𝛾𝐹1 (π‘Žπ‘—) |π‘Žπ‘— ∈ π‘ˆ}, Definition 2.6 A picture fuzzy set (PFSs) [32], [33] then measure of distance of a function π‘š: 𝑃𝐹𝑆(π‘ˆ) Γ— 𝑃𝐹𝑆(π‘ˆ) β†’ [0, 1]such that π‘š1. 0 ≀ π‘š(𝐹1, 𝐹2) ≀ 1 π‘š2. π‘š(𝐹1, 𝐹2) = π‘š(𝐹2, 𝐹1) π‘š3. π‘š(𝐹1, 𝐹2) = 0 𝑖𝑓𝑓 𝐹1 = 𝐹2 π‘š 4. π‘š(𝐹1, 𝐹2) ≀ π‘š(𝐹1, 𝐹3) π‘Žπ‘›π‘‘ π‘š(𝐹2, 𝐹3) ≀ π‘š(𝐹1, 𝐹3), where 𝐹1 βŠ† 𝐹2 βŠ† 𝐹3. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1476 https://internationalpubls.com 3. Existing distance measure This section provides a summary of the existing measures for Picture Fuzzy Sets (PIFSs), as described in the paper [37]. There are several distance measures that uses a various range of methodologies and technique. These measures have been applied to quantify the dissimilarity or similarity between multiple fields. These measures play a central role in diverse applications, contributing to the fields such as data analysis, pattern recognition, classification, medical diagnostic and cluster theory. Within the current context of distance measure, researchers as well as users are always exploring and improving methods to increase their efficiency and applications in various areas. Hamming Distance 𝑀𝑑1 (𝐹1, 𝐹2) = 1 4𝑛 βˆ‘ (βˆ†πœ‡π‘— + βˆ†πœˆπ‘— + βˆ†π›Ύπ‘— + |πœ‘πΉ1 (π‘Žπ‘—) βˆ’ πœ‘πΉ2 (π‘Žπ‘—)|)𝑛 𝑗=1 Euclidean Distance 𝑀𝑑2 (𝐹1, 𝐹2) = [ 1 4𝑛 βˆ‘ ((βˆ†πœ‡π‘—) 2 + (βˆ†πœˆπ‘—) 2 + (βˆ†π›Ύπ‘—) 2 + |πœ‘πΉ1 (π‘Žπ‘—) βˆ’ πœ‘πΉ2 (π‘Žπ‘—)| 2 )𝑛 𝑗=1 ] 1 2 Hausdorff Distance 𝑀𝑑3 (𝐹1, 𝐹2) = 1 4𝑛 βˆ‘ π‘šπ‘Žπ‘₯ (βˆ†πœ‡π‘— , βˆ†πœˆπ‘—, βˆ†π›Ύπ‘—, |πœ‘πΉ1 (π‘Žπ‘—) βˆ’ πœ‘πΉ2 (π‘Žπ‘—)|)𝑛 𝑗=1 𝑀𝑑4 (𝐹1, 𝐹2) = [ 1 4𝑛 βˆ‘ π‘šπ‘Žπ‘₯ ((βˆ†πœ‡π‘—) 2 , (βˆ†πœˆπ‘—) 2 , (βˆ†π›Ύπ‘—) 2 , |πœ‘πΉ1 (π‘Žπ‘—) βˆ’ πœ‘πΉ2 (π‘Žπ‘—)| 2 )𝑛 𝑗=1 ] 1 2 Where, βˆ†πœ‡π‘— = |πœ‡πΉ1 (π‘Žπ‘—) βˆ’ πœ‡πΉ2 (π‘Žπ‘—)| βˆ†πœˆπ‘— = |𝜈𝐹1 (π‘Žπ‘—) βˆ’ 𝜈𝐹2 (π‘Žπ‘—)| βˆ†π›Ύπ‘— = |𝛾𝐹1 (π‘Žπ‘—) βˆ’ 𝛾(π‘Žπ‘—)| βˆ†π›Ώπ‘— = |πœ‹πΉ1 (π‘Žπ‘—) βˆ’ πœ‹πΉ2 (π‘Žπ‘—)| for all 1 ≀ 𝑗 ≀ 𝑛, πœ‘π‘— 𝐹1 = |πœ‡πΉ1 (π‘Žπ‘—) + 𝜈𝐹1 (π‘Žπ‘—) + 𝛾𝐹1 (π‘Žπ‘—)| πœ‘π‘— 𝐹2 = |πœ‡πΉ2 (π‘Žπ‘—) + 𝜈𝐹2 (π‘Žπ‘—) + 𝛾𝐹2 (π‘Žπ‘—)| for all 1 ≀ 𝑗 ≀ 𝑛, The generalized normalized Hausdorff distance based on picture fuzzy set follws as: 𝑀𝑑 (𝐹1, 𝐹2) = [ 1 4𝑛 βˆ‘ π‘šπ‘Žπ‘₯ ((βˆ†πœ‡π‘—) πœ† + (βˆ†πœˆπ‘—) πœ† + (βˆ†π›Ύπ‘—) πœ† + |πœ‘πΉ1 (π‘Žπ‘—) βˆ’ πœ‘πΉ2 (π‘Žπ‘—)| πœ† )𝑛 𝑗=1 ] 1 πœ† where πœ† > 0. Son[33] mentioned the generalized picture distance measures Hausdorff-Hamming and Hausdorff Euclidean used these measures in clustering analysis for (πœ† = 1) and (πœ† = 2) given as: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1477 https://internationalpubls.com 𝑀𝑑𝑑 = [βˆ‘ ( βˆ†πœ‡π‘— πœ†+βˆ†πœˆπ‘— πœ†+βˆ†π›Ύπ‘— πœ† 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘— πœ†,βˆ†πœˆπ‘— πœ†,βˆ†π›Ύπ‘— πœ†))𝑛 𝑗=1 ] 1 πœ† [βˆ‘ ( βˆ†πœ‡π‘— πœ†+βˆ†πœˆπ‘— πœ†+βˆ†π›Ύπ‘— πœ† 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘— πœ†,βˆ†πœˆπ‘— πœ†,βˆ†π›Ύπ‘— πœ†))𝑛 𝑗=1 ] 1 πœ† +[π‘šπ‘Žπ‘₯ 𝑗 (πœ‘ 𝑗 𝐹1 ,πœ‘ 𝑗 𝐹2)+ βˆ‘ |πœ‘ 𝑗 𝐹1βˆ’πœ‘ 𝑗 𝐹2|𝑛 𝑗=1 ] 1 πœ† +1 For πœ† = 1 𝑀𝑑5 = [βˆ‘ ( βˆ†πœ‡π‘—+βˆ†πœˆπ‘—+βˆ†π›Ύπ‘— 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘—,βˆ†πœˆπ‘—,βˆ†π›Ύπ‘—))𝑛 𝑗=1 ] [βˆ‘ ( βˆ†πœ‡π‘—+βˆ†πœˆπ‘—+βˆ†π›Ύπ‘— 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘—,βˆ†πœˆπ‘—,βˆ†π›Ύπ‘—))𝑛 𝑗=1 ] +[π‘šπ‘Žπ‘₯ 𝑗 (πœ‘ 𝑗 𝐹1 ,πœ‘ 𝑗 𝐹2)+ βˆ‘ |πœ‘ 𝑗 𝐹1βˆ’πœ‘ 𝑗 𝐹2|𝑛 𝑗=1 ] +1 For πœ† = 2 𝑀𝑑6 = [βˆ‘ ( βˆ†πœ‡π‘— 2+βˆ†πœˆπ‘— 2+βˆ†π›Ύπ‘— 2 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘— 2,βˆ†πœˆπ‘— 2,βˆ†π›Ύπ‘— 2))𝑛 𝑗=1 ] 1 2 [βˆ‘ ( βˆ†πœ‡π‘— 2+βˆ†πœˆπ‘— 2+βˆ†π›Ύπ‘— 2 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘— 2,βˆ†πœˆπ‘— 2,βˆ†π›Ύπ‘— 2))𝑛 𝑗=1 ] 1 2 +[π‘šπ‘Žπ‘₯ 𝑗 (πœ‘ 𝑗 𝐹1 ,πœ‘ 𝑗 𝐹2)+ βˆ‘ |πœ‘ 𝑗 𝐹1βˆ’πœ‘ 𝑗 𝐹2|𝑛 𝑗=1 ] 1 2 +1 Son[33] also discussed the extended normalized picture Hausdorff-Hamming and the Hausdorff Euclidean taken as: 𝑀𝑑𝑑𝑑 (𝐹1, 𝐹2) = [ 1 𝑛 βˆ‘ ( βˆ†πœ‡π‘— πœ†+βˆ†πœˆπ‘— πœ†+βˆ†π›Ύπ‘— πœ† 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘— πœ†,βˆ†πœˆπ‘— πœ†,βˆ†π›Ύπ‘— πœ†))𝑛 𝑗=1 ] 1 πœ† [ 1 𝑛 βˆ‘ ( βˆ†πœ‡π‘— πœ†+βˆ†πœˆπ‘— πœ†+βˆ†π›Ύπ‘— πœ† 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘— πœ†,βˆ†πœˆπ‘— πœ†,βˆ†π›Ύπ‘— πœ†))𝑛 𝑗=1 ] 1 πœ† +[π‘šπ‘Žπ‘₯ 𝑗 (πœ‘ 𝑗 𝐹1 ,πœ‘ 𝑗 𝐹2)+ 1 𝑛 βˆ‘ |πœ‘ 𝑗 𝐹1βˆ’πœ‘ 𝑗 𝐹2| πœ† 𝑛 𝑗=1 ] 1 πœ† +1 For πœ† = 1 𝑀𝑑7 (𝐹1, 𝐹2) = 1 𝑛 [βˆ‘ ( βˆ†πœ‡π‘—+βˆ†πœˆπ‘—+βˆ†π›Ύπ‘— 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘—,βˆ†πœˆπ‘—,βˆ†π›Ύπ‘—))𝑛 𝑗=1 ] 1 𝑛 [βˆ‘ ( βˆ†πœ‡π‘—+βˆ†πœˆπ‘—+βˆ†π›Ύπ‘— 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘—,βˆ†πœˆπ‘—,βˆ†π›Ύπ‘—))𝑛 𝑗=1 ] +[π‘šπ‘Žπ‘₯ 𝑗 (πœ‘ 𝑗 𝐹1 ,πœ‘ 𝑗 𝐹2)+ 1 𝑛 βˆ‘ |πœ‘ 𝑗 𝐹1βˆ’πœ‘ 𝑗 𝐹2|𝑛 𝑗=1 ] +1 For πœ† = 2 𝑀𝑑8 (𝐹1, 𝐹2) = [ 1 𝑛 βˆ‘ ( βˆ†πœ‡π‘— 2+βˆ†πœˆπ‘— 2+βˆ†π›Ύπ‘— 2 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘— 2,βˆ†πœˆπ‘— 2,βˆ†π›Ύπ‘— 2))𝑛 𝑗=1 ] 1 2 [ 1 𝑛 βˆ‘ ( βˆ†πœ‡π‘— 2+βˆ†πœˆπ‘— 2+βˆ†π›Ύπ‘— 2 3 +π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘— 2,βˆ†πœˆπ‘— 2,βˆ†π›Ύπ‘— 2))𝑛 𝑗=1 ] 1 2 +[π‘šπ‘Žπ‘₯ 𝑗 (πœ‘ 𝑗 𝐹1 ,πœ‘ 𝑗 𝐹2)+ 1 𝑛 βˆ‘ |πœ‘ 𝑗 𝐹1βˆ’πœ‘ 𝑗 𝐹2|𝑛 𝑗=1 ] 1 2 +1 Din and Thao [21] in his article applied the distance measures in pattern analysis which are given below, 𝑀𝑑9 (𝐹1, 𝐹2) = 1 3𝑛 βˆ‘ (βˆ†πœ‡π‘— + βˆ†πœˆπ‘— + βˆ†π›Ύπ‘—)𝑛 𝑗=1 𝑀𝑑10 (𝐹1, 𝐹2) = 1 𝑛 [βˆ‘ (βˆ†πœ‡π‘— 2 + βˆ†πœˆπ‘— 2 + βˆ†π›Ύπ‘— 2)𝑛 𝑗=1 ] 1 2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1478 https://internationalpubls.com 𝑀𝑑11 (𝐹1, 𝐹2) = 1 𝑛 βˆ‘ π‘šπ‘Žπ‘₯ (βˆ†πœ‡π‘—, βˆ†πœˆπ‘—, βˆ†π›Ύπ‘—)𝑛 𝑗=1 𝑀𝑑12 (𝐹1, 𝐹2) = 1 𝑛 [βˆ‘ π‘šπ‘Žπ‘₯ (βˆ†πœ‡π‘— 2, βˆ†πœˆπ‘— 2, βˆ†π›Ύπ‘— 2)𝑛 𝑗=1 ] 1 2 Dutta[22] proposed the new distance measures given below and applied these distance measures in the field of medical diagnosis. Hamming Distance Measure: 𝑀𝑑13 (𝐹1, 𝐹2) = 1 2 βˆ‘ (βˆ†πœ‡π‘— + βˆ†πœˆπ‘— + βˆ†π›Ύπ‘— + βˆ†π›Ώπ‘—)𝑛 𝑗=1 Normalized Hamming Distance Measure: 𝑀𝑑14 (𝐹1, 𝐹2) = 1 2𝑛 βˆ‘ (βˆ†πœ‡π‘— + βˆ†πœˆπ‘— + βˆ†π›Ύπ‘— + βˆ†π›Ώπ‘—)𝑛 𝑗=1 Euclidean Distance Measure: 𝑀𝑑15 (𝐹1, 𝐹2) = [ 1 2 βˆ‘ (βˆ†πœ‡π‘— 2 + βˆ†πœˆπ‘— 2 + βˆ†π›Ύπ‘— 2 + βˆ†π›Ώπ‘— 2)𝑛 𝑗=1 ] 1 2 Normalized Euclidean Distance: 𝑀𝑑16 (𝐹1, 𝐹2) = [ 1 2𝑛 βˆ‘ (βˆ†πœ‡π‘— 2 + βˆ†πœˆπ‘— 2 + βˆ†π›Ύπ‘— 2 + βˆ†π›Ώπ‘— 2)𝑛 𝑗=1 ] 1 2 The Measure given by Dutta [22] as the extension of measure given by Wang and Xing [38] 𝑀𝑑17 (𝐹1, 𝐹2) = 1 𝑛 βˆ‘ [ (βˆ†πœ‡π‘—+βˆ†πœˆπ‘—+βˆ†π›Ύπ‘—+βˆ†π›Ώπ‘—) 4 + π‘šπ‘Žπ‘₯(βˆ†πœ‡π‘—,βˆ†πœˆπ‘—,βˆ†π›Ύπ‘—,βˆ†π›Ώπ‘— ) 2 ]𝑛 𝑗=1 The measure given by Ganie [3] can be defined as 𝑀𝐺(𝐹1, 𝐹2) = 1 4𝑛 βˆ‘ [|tanβˆ’1 πœ‡πΉ1 (π‘Žπ‘—) βˆ’ tanβˆ’1 πœ‡πΉ2 (π‘Žπ‘—) | + |tanβˆ’1 𝜈𝐹1 (π‘Žπ‘—) βˆ’ tanβˆ’1 𝜈𝐹2 (π‘Žπ‘—)| +𝑛 𝑗=1 |tanβˆ’1 𝛾𝐹1 (π‘Žπ‘—) βˆ’ tanβˆ’1 𝛾𝐹2 (π‘Žπ‘—)| + |tanβˆ’1 πœ‘πΉ1 (π‘Žπ‘—) βˆ’ tanβˆ’1 πœ‘πΉ2 (π‘Žπ‘—)|] 4. Proposed distance measure In this section, we proposed a novel distance measure for Picture Fuzzy Sets (PIFSs). Proposed distance measure and their properties are discussed in the section 4.1, 4.2 and 4.3. Our aim is to provide a novel measure for determining the distance between Picture Fuzzy Sets (PIFSs). 4.1 A novel distance measure for PFSs Let 𝐹1 and 𝐹2 be any two IFSs 𝐹1 = {(π‘Žπ‘— , πœ‡πΉ1 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—), 𝛾𝐹1 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ} and 𝐹2 = {(π‘Žπ‘— , πœ‡πΉ2 (π‘Žπ‘—), 𝜈𝐹2 (π‘Žπ‘—), 𝛾𝐹2 (π‘Žπ‘—)) |π‘Žπ‘— ∈ π‘ˆ} on U,where πœ‡πΉ1 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—) and 𝛾𝐹1 (π‘Žπ‘—) denotes the membership (positive), non-membership (negative) and neutrality respectively of π‘Žπ‘— ∈ π‘ˆ holds the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1479 https://internationalpubls.com following conditions 0 ≀ πœ‡πΉ1 (π‘Žπ‘—) + 𝜈𝐹1 (π‘Žπ‘—) + 𝛾𝐹1 (π‘Žπ‘—) ≀ 1 and refusal degree πœ‘πΉ1 = 1 βˆ’ πœ‡πΉ1 (π‘Žπ‘—) βˆ’ 𝜈𝐹1 (π‘Žπ‘—) βˆ’ 𝛾𝐹1 (π‘Žπ‘—) for all π‘Žπ‘— ∈ π‘ˆ.then a PIFSs measure of distance is given by 𝑀𝑆𝐡(𝐹1, 𝐹2) = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 Theorem 4.1 The PIFSs measure of distance 𝑀𝑆𝐡(𝐹1, 𝐹2) = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ1(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 is an effective distance measure for PIFSs. Proof: To prove this measure, we will show that the proposed picture fuzzy distance measure 𝑀𝑆𝐡 for any two sets 𝐹1 and 𝐹2 hold the conditions 𝑀1 to 𝑀4 in definition 2.6. π‘΄πŸ. As 0 ≀ πœ‡πΉ1 (π‘Žπ‘—), πœ‡πΉ2 (π‘Žπ‘—) ≀ 1 βˆ€ 1 ≀ 𝑗 ≀ 𝑛, so that we have 0 ≀ π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—), π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—) ≀ 1, and so 0 ≀ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| ≀ 1. Similarly, we have 0 ≀ |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ1(π‘Žπ‘—)| ≀ 1, 0 ≀ |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—)| ≀ 1, and 0 ≀ |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ≀ 1. Hence, we get 0 ≀ 𝑀𝑆𝐡(𝐹1, 𝐹2) ≀ 1 βˆ€ 1 ≀ 𝑗 ≀ 𝑛. π‘΄πŸ. To prove, 𝑀𝑆𝐡(𝐹1, 𝐹2) = 𝑀𝑆𝐡(𝐹2, 𝐹1) we have, 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ2(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ1(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ2 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ1(π‘Žπ‘—)| ]𝑛 𝑗=1 = 𝑀𝑆𝐡(𝐹2, 𝐹1) π‘΄πŸ‘. Let 𝐹1 = 𝐹2 then πœ‡πΉ1 (π‘Žπ‘—) = πœ‡πΉ2 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—) = 𝜈𝐹2 (π‘Žπ‘—), 𝛾𝐹1 (π‘Žπ‘—) = 𝛾𝐹2 (π‘Žπ‘—) π‘Žπ‘›π‘‘ πœ‘πΉ1 (π‘Žπ‘—) = πœ‘πΉ2 (π‘Žπ‘—) βˆ€ 1 ≀ 𝑗 ≀ 𝑛. Then we have 𝑀𝑆𝐡(𝐹1, 𝐹2) = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ2(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ2 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ] = 0𝑛 𝑗=1 π‘΄πŸ’ Let 𝐹1 βŠ† 𝐹2 βŠ† 𝐹3, then πœ‡πΉ1 (π‘Žπ‘—) ≀ πœ‡πΉ2 (π‘Žπ‘—) ≀ πœ‡πΉ3 (π‘Žπ‘—), 𝜈𝐹1 (π‘Žπ‘—) β‰₯ 𝜈𝐹2 (π‘Žπ‘—) β‰₯ 𝜈𝐹3 (π‘Žπ‘—) π‘Žπ‘›π‘‘ 𝛾𝐹1 (π‘Žπ‘—) ≀ 𝛾𝐹2 (π‘Žπ‘—) ≀ 𝛾𝐹3 (π‘Žπ‘—)βˆ€ 1 ≀ 𝑗 ≀ 𝑛. Thus, we have |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| ≀ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ3(π‘Žπ‘—)|, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1480 https://internationalpubls.com |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| ≀ |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ3(π‘Žπ‘—)|, |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| ≀ |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ3(π‘Žπ‘—)| π‘Žπ‘›π‘‘ |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ≀ |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ3(π‘Žπ‘—)| Therefore, 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 ≀ 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ3(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ3(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ3(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ3(π‘Žπ‘—)| ]𝑛 𝑗=1 . So picture fuzzy distance measure 𝑀𝑆𝐡(𝐹1, 𝐹2) ≀ 𝑀𝑆𝐡(𝐹1, 𝐹3), also we can see that 𝑀𝑆𝐡(𝐹2, 𝐹3) ≀ 𝑀𝑆𝐡(𝐹1, 𝐹3) which shows picture fuzzy distance measure 𝑀𝑆𝐡 is an effective PIFSs measure of distance. Theorem 4.3 The PIFSs measure of distance 𝑀𝑆𝐡(𝐹1, 𝐹2) = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 satisfies the following properties. π‘Ž) 𝑀𝑆𝐡(𝐹1 𝑐, 𝐹2 𝑐) = 𝑀𝑆𝐡(𝐹1, 𝐹2) βˆ€ 𝐹1, 𝐹2 ∈ PIFSs X. 𝑏) 𝑀𝑆𝐡(𝐹1, 𝐹2 𝑐) = 𝑀𝑆𝐡( 𝐹1 𝑐, 𝐹2) βˆ€ 𝐹1, 𝐹2 ∈ PIFSs X. 𝑐) 𝑀𝑆𝐡(𝐹1, 𝐹1 𝑐) = 0 iff πœ‡πΉ1 (π‘Žπ‘—) = 𝜈𝐹1 (π‘Žπ‘—) βˆ€ 1 ≀ 𝑗 ≀ 𝑛. Proof: We have a) 𝑀𝑆𝐡(𝐹1 𝑐, 𝐹2 𝑐) = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 = 𝑀𝑆𝐡(𝐹1, 𝐹2). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1481 https://internationalpubls.com b) = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)| ]𝑛 𝑗=1 = 𝑀𝑆𝐡(𝐹1 𝑐, 𝐹2) c) 𝑀𝑆𝐡(𝐹1, 𝐹1 𝑐) = 0 ⟺ 1 4𝑛 βˆ‘ [|π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΉ2(π‘Žπ‘—)| +𝑛 𝑗=1 |π‘’βˆ’π›ΎπΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΉ2(π‘Žπ‘—)| + |π‘’βˆ’πœ‘πΉ1 (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΉ2(π‘Žπ‘—)|] = 0 ⟺ |π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—)| = 0 βˆ€ 𝑗 ⟺ π‘’βˆ’πœ‡πΉ1(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΉ2(π‘Žπ‘—) = 0 βˆ€ 𝑗 ⟺ πœ‡πΉ1 (π‘Žπ‘—) = 𝜈𝐹1 (π‘Žπ‘—) βˆ€ 𝑗. 4.4 Experiments and Analysis We will compare the results of pre-existing measure defined in section 3 with proposed measure in section 4, in this section. We will use particular test sets to justify the rationality of proposed measure. Validating the picture fuzzy set measure with axiomatic condition is the key objective of any distance measure. If any picture fuzzy set is violating one or more axiomatic requirement of similarity measure, then we say that it produces counterintuitive situation. In addition, several well-known picture fuzzy sets are unable to accurately differentiate between various picture fuzzy set pairs. For example, in Problem 4.1, Problem 4.2 and Problem 4.3 we can see that picture fuzzy set produce equal distance measure for different sets. This situation is absurd and goes against the both counterintuitive and intuition. 4.5 Superiority Analysis Here, we have compared the suggested and existed measures with numerical problems to show the distance between various Picture Fuzzy Sets (PIFSs). Problem 4.1 Consider six different types of PIFSs with every type containing two different PIFSs as given below: Type-(a): {𝐹1 = {(0.28, 0.55, 0.1)}, 𝐹2 = {(0.6, 0.27, 0.1)}} Type-(b): {𝐹1 = {(0.28, 0.54, 0.1)}, 𝐹2 = {(0, 0.87, 0.1)}} Type-(c): {𝐹1 = {(0, 0, 0)}, 𝐹2 = {(0.6, 0.4, 0)}} Type-(d): {𝐹1 = {(0, 0, 0)}, 𝐹2 = {(0.5, 0.5, 0)}} Type-(e): {𝐹1 = {(0.2, 0.5, 0.3)}, 𝐹2 = {(0.4, 0.4, 0.2)}} Type-(f): {𝐹1 = {(0.1, 0.5, 0.1)}, 𝐹2 = {(0.2, 0.4, 0.2)}} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1482 https://internationalpubls.com Table 1: Distance between two picture fuzzy sets calculated by existing and proposed distance measure related to problem 1 Measures Type-(a) Type-(b) Type-(c) Type-(d) Type-(e) Type-(f) 𝑀𝑑1 0.1600 0.1600 0.5000 0.5000 0.1000 0.1000 𝑀𝑑2 0.2135 0.2135 0.6164 0.6124 0.1225 0.1000 𝑀𝑑3 0.0800 0.0800 0.2500 0.2500 0.0500 0.0250 𝑀𝑑4 0.1600 0.1600 0.5000 0.5000 0.1000 0.0500 𝑀𝑑5 0.2055 0.2055 0.2373 0.2174 0.1429 0.0952 𝑀𝑑6 0.1675 0.1675 0.2322 0.2110 0.1091 0.0677 𝑀𝑑7 0.2055 0.2055 0.2373 0.2174 0.1429 0.0952 𝑀𝑑8 0.1675 0.1675 0.2322 0.2110 0.1091 0.0677 𝑀𝑑9 0.2000 0.2000 0.3333 0.3333 0.1333 0.1000 𝑀𝑑10 0.4252 0.4252 0.7211 0.7071 0.2449 0.1732 𝑀𝑑11 0.3200 0.3200 0.6000 0.5000 0.2000 0.1000 𝑀𝑑12 0.3200 0.3200 0.6000 0.5000 0.2000 0.1000 𝑀𝑑13 0.3200 0.3200 1.0000 1.0000 0.2000 0.2000 𝑀𝑑14 0.3200 0.3200 1.0000 1.0000 0.2000 0.2000 𝑀𝑑15 0.3020 0.3020 0.8718 0.8660 0.1732 0.1414 𝑀𝑑16 0.3020 0.3020 0.8718 0.8660 0.1732 0.1414 𝑀𝑑17 0.0640 0.0640 0.2000 0.2000 0.0400 0.0300 𝑀𝐺 0.1366 0.1315 0.4266 0.4282 0.0901 0.0932 𝑴𝑺𝑩 0.1079 0.1101 0.3532 0.3548 0.0725 0.0785 Bold numerical values display the unreasonable results i) The distance measures for PIFSs derived from 𝑀𝑑1 , 𝑀𝑑13and 𝑀𝑑14 display identical distances for distinct fuzzy sets in three pairs {Type-(a) and Type-(b)}, {Type-(c) and Type- (d)}, and {Type-(e) and Type-(f)}. ii) The distance measure for PIFSs obtained from 𝑀𝑑2 , 𝑀𝑑5 , 𝑀𝑑6 , 𝑀𝑑7 , 𝑀𝑑8 , 𝑀𝑑9 , 𝑀𝑑10 , 𝑀𝑑11 , 𝑀𝑑12 , 𝑀𝑑15 and 𝑀𝑑16 display identical distances for two distinct types, namely {Type-(a) and Type-(b)} iii) The distance measures for PIFSs derived from. 𝑀𝑑2 , 𝑀𝑑5 , 𝑀𝑑6 , 𝑀𝑑7 , 𝑀𝑑8 , 𝑀𝑑9 , 𝑀𝑑10 , 𝑀𝑑11 , 𝑀𝑑12 , 𝑀𝑑15 and 𝑀𝑑16 display identical distances for two different types in {Type-(a) and Type-(b)}, {Type-(c) and Type-(d)} and {Type-(e) and Type-(f)}. iv) The PIFSs distance measures from 𝑀𝑑17 display the same distance for two different PIFSs sets {Type -(a) and Type-(b)}, {Type-(c) and Type-(d)}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1483 https://internationalpubls.com v) The distance measures derived from 𝑀𝑑13 and , 𝑀𝑑14 for PIFSs display an identical distance of β€œ1” in both Type-(c) and Type-(d). However, the distances between these PIFSs are not complement each other for distinct picture fuzzy sets. vi) The proposed distance measure 𝑀𝑆𝐡 computes the distance for all distinct types, avoiding any similarity and gives more accurate results compared to existing measure. Figure1: Comparison of proposed and existing measures The result of comparative analysis of proposed measure for this problem is discussed in Table 1 and Fig. 1, respectively. Problem 4.2 Consider the six different types of PIFSs with every type containing two different PIFSs as given below: Type-(a): {𝐹1 = {(0.2, 0.5, 0.3)}, 𝐹2 = {(0.4, 0.4, 0.2)}} Type-(b): {𝐹1 = {(0.2, 0.5, 0.3)}, 𝐹2 = {(0.1, 0.4, 0.5)}} Type-(c): {𝐹1 = {(0, 0, 0)}, 𝐹2 = {(0.4, 0, 0.6)}} Type-(d): {𝐹1 = {(0, 0, 0)}, 𝐹2 = {(0.5, 0, 0.5)}} Type-(e): {𝐹1 = {(0.2, 0, 0.8)}, 𝐹2 = {(0.3, 0, 0.7)}} Type-(f): {𝐹1 = {(0.1, 0.6, 0.3)}, 𝐹2 = {(0.3, 0.3, 0.4)}} Table 2: Distance between two picture fuzzy sets calculated by existing and proposed distance measure related to problem 2 Measures Type-(a) Type-(b) Type-(c) Type-(d) Type-(e) Type-(f) 𝑀𝑑1 0.1000 0.1000 0.4500 0.5000 0.0500 0.1500 𝑀𝑑2 0.1225 0.1225 0.5523 0.6124 0.0707 0.1871 𝑀𝑑3 0.0500 0.0500 0.2250 0.2500 0.0250 0.0750 𝑀𝑑4 0.1000 0.1000 0.4500 0.5000 0.0500 0.1500 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1484 https://internationalpubls.com 𝑀𝑑5 0.1429 0.1429 0.2222 0.2174 0.0769 0.2000 𝑀𝑑6 0.1091 0.1091 0.2098 0.2110 0.0606 0.1560 𝑀𝑑7 0.1429 0.1429 0.2222 0.2174 0.0769 0.2000 𝑀𝑑8 0.1091 0.1091 0.2098 0.2110 0.0606 0.1560 𝑀𝑑9 0.1333 0.1333 0.3000 0.3333 0.0667 0.2000 𝑀𝑑10 0.2449 0.2449 0.6403 0.7071 0.1414 0.3742 𝑀𝑑11 0.2000 0.2000 0.5000 0.5000 0.1000 0.3000 𝑀𝑑12 0.2000 0.2000 0.5000 0.5000 0.1000 0.3000 𝑀𝑑13 0.2000 0.2000 0.9000 1.0000 0.1000 0.3000 𝑀𝑑14 0.2000 0.2000 0.9000 1.0000 0.1000 0.3000 𝑀𝑑15 0.1732 0.1732 0.7810 0.8660 0.1000 0.2646 𝑀𝑑16 0.1732 0.1732 0.7810 0.8660 0.1000 0.2646 𝑀𝑑17 0.0400 0.0400 0.1800 0.2000 0.0200 0.0600 𝑀𝐺 0.0901 0.0883 0.3825 0.4282 0.0395 0.1325 𝑀𝑆𝐡 0.0725 0.0710 0.3150 0.3548 0.0313 0.1066 Bold numerical values show unreasonable results i) The distance measures for PIFSs derived from 𝑀𝑑11 and 𝑀𝑑12 displays identical distances for distinct fuzzy sets in three pairs {Type-(a) and Type-(b)}, {Type-(c) and Type-(d)}, and {Type-(e) and Type-(f)}. ii) 𝑀𝑑1 , 𝑀𝑑2 , 𝑀𝑑3 , 𝑀𝑑4 , 𝑀𝑑5 , 𝑀𝑑6 , 𝑀𝑑7 , 𝑀𝑑8 , 𝑀𝑑9 , 𝑀𝑑10 , 𝑀𝑑13 , 𝑀𝑑14 , 𝑀𝑑15 , 𝑀𝑑16 , 𝑀𝑑17 continuously display an identical distance measurement in both Type-(a) and Type-(b) iii) The distance measures derived from 𝑀𝑑13 and 𝑀𝑑14 for PIFSs display an identical distance of "1" in both Type-(c) and Type-(d). However, the distances between these PIFSs are not complement of each other for distinct picture fuzzy sets. iv) The proposed distance measure 𝑀𝑆𝐡 computes the distance for all distinct types, avoiding any similarity and gives more accurate results compared to existing measures. Figure 2: Comparison of proposed and existing measures. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1485 https://internationalpubls.com The result of comparative analysis of proposed measure for this problem is discussed in Table 2 and Fig. 2, respectively. Problem 4.3 Consider five different types of PIFSs with every type containing two different PIFSs as given below: Type-(a): {𝐹1 = {(0, 0, 0)}, 𝐹2 = {(1, 0, 0)}} Type-(b): {𝐹1 = {(0.5, 0.1, 0.1)}, 𝐹2 = {(0.4, 0.2, 0.1)}} Type-(c): {𝐹1 = {(0.4, 0.3, 0.1)}, 𝐹2 = {(0.3, 0.4, 0.1)}} Type-(d): {𝐹1 = {(0.5, 0.2, 0.1)}, 𝐹2 = {(0.4, 0.2, 0.1)}} Type-(e): {𝐹1 = {(0, 0, 0)}, 𝐹2 = {(0.5, 0.5, 0)}} Type-(f): {𝐹1 = {(0.2, 0, 0.8)}, 𝐹2 = {(0.3, 0, 0.7)}} Table 3: Distance between two picture fuzzy sets calculated by existing and proposed distance measure related to problem 3 Measures Type-(a) Type-(b) Type-(c) Type-(d) Type-(e) Type-(f) 𝑀𝑑1 0.0500 0.0500 0.0500 0.0500 0.5000 0.0500 𝑀𝑑2 0.0707 0.0707 0.0707 0.0707 0.6124 0.0707 𝑀𝑑3 0.0250 0.0250 0.0250 0.0250 0.2500 0.0250 𝑀𝑑4 0.0500 0.0500 0.0500 0.0500 0.5000 0.0500 𝑀𝑑5 0.1000 0.0893 0.0847 0.0656 0.2174 0.0769 𝑀𝑑6 0.0739 0.0657 0.0638 0.0559 0.2110 0.0606 𝑀𝑑7 0.1000 0.0893 0.0847 0.0656 0.2174 0.0769 𝑀𝑑8 0.0739 0.0657 0.0638 0.0559 0.2110 0.0606 𝑀𝑑9 0.0333 0.0667 0.0667 0.0333 0.3333 0.0667 𝑀𝑑10 0.1000 0.1414 0.1414 0.1000 0.7071 0.1414 𝑀𝑑11 0.1000 0.1000 0.1000 0.1000 0.5000 0.1000 𝑀𝑑12 0.1000 0.1000 0.1000 0.1000 0.5000 0.1000 𝑀𝑑13 0.1000 0.1000 0.1000 0.1000 1.0000 0.1000 𝑀𝑑14 0.1000 0.1000 0.1000 0.1000 1.0000 0.1000 𝑀𝑑15 0.1000 0.1000 0.1000 0.1000 0.8660 0.1000 𝑀𝑑16 0.1000 0.1000 0.1000 0.1000 0.8660 0.1000 𝑀𝑑17 0.0200 0.0200 0.0200 0.0200 0.2000 0.0200 𝑀𝐺 0.0381 0.0452 0.0445 0.0443 0.4282 0.0395 𝑀𝑆𝐡 0.0335 0.0375 0.0352 0.0354 0.3548 0.0313 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1486 https://internationalpubls.com i) The PIFSs distance measure in 𝑀𝑑10 continuously display identical distances for two distinct fuzzy sets in various types, namely {Type -(a) and Type-(d)} and {Type-(c), Type- (c) and Type-(f)}. ii) 𝑀𝑑1 , 𝑀𝑑2 , 𝑀𝑑3 , 𝑀𝑑4 , 𝑀𝑑11 , 𝑀𝑑12 , 𝑀𝑑13 , 𝑀𝑑14 , 𝑀𝑑15 , 𝑀𝑑16 , 𝑀𝑑17 display an identical distance in all types {Type -(a) to Type-(d) and Type-(f)}. iii) The proposed distance measure 𝑀𝑆𝐡 computes the distance through various types without any similarity, and it avoids generating unreasonable results. It shows superior outcomes over existing measure. Figure 3: Comparison of proposed and existing measures. The result of comparative analysis of proposed measure for this problem is discussed in Table 3 and Fig. 3, respectively. 5. Medical Diagnosis (Decision) Medical diagnosis [22] is the process to finding the disease or symptoms. The information needed for medical diagnosis is very difficult and gathered from a person’s medical history and physical examination. Medical records contain uncertain information gathered, which can be useful for decision-making using fuzzy logic in everyday problems. In medical diagnosis (decision) many applications of FS theory and extension have been applied. It can be noted that in medical diagnosis some indications may be neutral impact on disease. We can see by an example that degree of neutrality for headache and temperature to the disease chest and stomach. Accordingly, indications chest and stomach pain have neutral impact on the disease to typhoid, malaria and viral fever. So, it becomes very important to consider degree of neutrality. Regarding neutrality PFSs in medical diagnosis is essential for its practical application in healthcare sector. How we can use PFSs in medical problems can be seen in below example. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1487 https://internationalpubls.com 5.1 Significance of distance measures in medical diagnosis Sometimes diagnosis is very challenging, as there are lots of symptoms and sign are nonspecific. For example, we can see that tiredness of body, is a sign of many disorders and not clearly seen what is wrong. It is very difficult to identifying the relationship between disease and patients. Measures has an important role in medical diagnosis, it helps us how to relate the relationship between disease and patients and disease and symptoms. The small distance between the relationship of disease and patients shows the better relationship while large distance shows weak relation. So, it can be seen that if the distance between disease and patient is small then it can be concluded that patient has the disease. 5.2 Methodology In this section, we will discuss the algorithmic steps detailing how to apply the medical diagnosis into problem-solving. We are defining disease within the set οΏ½ΜƒοΏ½, providing patient information in set οΏ½ΜƒοΏ½, and symptoms of both patients and disease in set οΏ½ΜƒοΏ½. The algorithm steps for applying the Picture Fuzzy Set Relationship (PFSR) to a medical problem are as follows: Step 1: Using medical knowledge to construct a set Containing both patient symptoms and disease symptoms. Step 2: Defining the positive, negative, and neutral degrees within the dataset. Step 3: Employing the proposed measure to compute the distance between the disease and patients. Step 4: Identifying the shortest distance between the patient and disease, indicating the patient's most probably suffering from the disease. Step 5: If the medical expert finds the result unsatisfactory, then reconstruct the Picture Fuzzy Set Relationship (PFSR) and reapply Steps 3 and 4 for further results. Problem 5.1 Picture fuzzy relation between patient and symptoms are given in Table 4, and relation between disease and symptoms are given in Table 5. Here, �̃�𝑖, �̃�𝑖 and �̃�𝑖 for 𝑖 = 1 to 5 represents disease, symptoms and patient respectively. Here, (πœ‡, 𝜈, 𝛾) represents the positive, negative and neutral membership degree of the PFSs. Table 4: Relation between symptoms of the patients PFSR οΏ½ΜƒοΏ½1 οΏ½ΜƒοΏ½2 οΏ½ΜƒοΏ½3 οΏ½ΜƒοΏ½4 οΏ½ΜƒοΏ½5 οΏ½ΜƒοΏ½1 (0.1,0.2, 0.4) (0.2, 0.3, 0.5) (0.8, 0.1, 0) (0.6, 0.2, 0.1) (0.8, 0.1, 0) οΏ½ΜƒοΏ½2 (0.9, 0.1, 0) (0.3, 0.2, 0.4) (0.1, 0.6, 0.2) (0.2, 0.5, 0.2) (0.1, 0.6, 0.2) οΏ½ΜƒοΏ½3 (0.1, 0.5, 0.2) (0.1, 0.4, 0.3) (0.7, 0.2, 0.1) (0.3, 0.2, 0.4) (0.7, 0.2, 0.1) οΏ½ΜƒοΏ½4 (0.3, 0.5, 0.1) (0.1, 0.7, 0.2) (0.114, 0.3, 0.2) (0.3, 0.4, 0.2) (0.4, 0.3, 0.2) Table 5: Relation between symptoms of the disease PFSR οΏ½ΜƒοΏ½1 οΏ½ΜƒοΏ½2 οΏ½ΜƒοΏ½3 οΏ½ΜƒοΏ½4 οΏ½ΜƒοΏ½5 οΏ½ΜƒοΏ½1 (0.2, 0.5, 0.3) (0.8, 0.1, 0) (0.1, 0.5, 0.3) (0.2, 0.4, 0.35) (0, 0.35, 0.5) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1488 https://internationalpubls.com οΏ½ΜƒοΏ½2 (0.8, 0, 0) (0.2, 0.3, 0.35) (0.1, 0.5, 0.3) (0.2, 0.3, 0.4) (0.2, 0.3, 0.4) οΏ½ΜƒοΏ½3 (0.2, 0.4, 0.3) (0.1, 0.6, 0.2) (0.3, 0.3, 0.4) (0.2, 0.3, 0.35) (0.6, 0.1, 0.2) οΏ½ΜƒοΏ½4 (0, 0.5, 0.4) (0.1, 0.5, 0.3) (0.7, 0, 0) (0.7, 0, 0.1 ) (0.2, 0.35, 0.4) οΏ½ΜƒοΏ½5 (0.1, 0.5, 0.35) (0.1, 0.5, 0.25) (0.4, 0, 0) (0.4, 0.2, 0.3) (0.3, 0.4, 0.2) Table 6: Calculated distances between disease and patients using existing and proposed measure PFSR οΏ½ΜƒοΏ½1 οΏ½ΜƒοΏ½2 οΏ½ΜƒοΏ½3 οΏ½ΜƒοΏ½4 οΏ½ΜƒοΏ½5 𝑀𝐺 𝑀𝑆𝐡 𝑀𝐺 𝑀𝑆𝐡 𝑀𝐺 𝑀𝑆𝐡 𝑀𝐺 𝑀𝑆𝐡 𝑀𝐺 𝑀𝑆𝐡 οΏ½ΜƒοΏ½1 0.2722 0.2258 0.2260 0.1870 0.1692 0.1404 0.1482 0.1246 0.1624 0.1339 οΏ½ΜƒοΏ½2 0.1557 0.1272 0.0846 0.0699 0.1946 0.1594 0.2586 0.2146 0.2285 0.1891 οΏ½ΜƒοΏ½3 0.2168 0.1811 0.1831 0.1515 0.1033 0.0855 0.1464 0.1240 0.1111 0.0920 οΏ½ΜƒοΏ½4 0.1670 0.1392 0.1515 0.1245 0.0817 0.0680 0.1590 0.1340 0.1153 0.0979 Bold numerical values show the disease by which patient is suffering. Table 6, displays the correlation between the patients and their respective disease with bold values, using relationship between symptoms of patient and relationship between symptoms of disease given in Table 4 and Table 5 respectively. Particularly, the Table 6 indicates that patient οΏ½ΜƒοΏ½1 is affected with disease οΏ½ΜƒοΏ½4, patient οΏ½ΜƒοΏ½2 has disease οΏ½ΜƒοΏ½2 patient οΏ½ΜƒοΏ½3 is suffering from disease οΏ½ΜƒοΏ½3 and patient οΏ½ΜƒοΏ½4 also has disease οΏ½ΜƒοΏ½3. Also, proposed measure 𝑀𝑆𝐡 shows the superior value compare to 𝑀𝐺 . Figure 4: Comparisons of distance of the disease with the patients using existing measure 𝑀𝐺 and proposed measure 𝑀𝑆𝐡. The result of comparative analysis of proposed measure for this problem is discussed in Table 6 and Fig. 4, respectively. Problem 5.2 Picture fuzzy relation between patient and symptoms are given in Table 7, and relation between disease and symptoms are given in Table 8. Table 7: Relation between symptoms of the patients PFSR οΏ½ΜƒοΏ½1 οΏ½ΜƒοΏ½2 οΏ½ΜƒοΏ½3 οΏ½ΜƒοΏ½4 οΏ½ΜƒοΏ½5 οΏ½ΜƒοΏ½1 (0.1, 0.5, 0.2) (0.1, 0.5, 0.3) (0.1, 0.6, 0.2) (0.3, 0.3, 0.2) (0.8, 0.1, 0) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1489 https://internationalpubls.com οΏ½ΜƒοΏ½2 (0.2, 0.4, 0.3) (0.2, 0.3, 0.4) (0.3, 0.2, 0.3) (0.7, 0, 0.1) (0.4, 0.2, 0.3) οΏ½ΜƒοΏ½3 (0.3, 0.4, 0.2) (0.2, 0.3, 0.4) (0.6, 0.2, 0.1) (0.2, 0.3, 0.4) (0.5, 0.2, 0.1) οΏ½ΜƒοΏ½4 (0.2, 0.3, 0.3) (0.2, 0.6, 0.2) (0.4, 0.2, 0.3) (0.2, 0.3, 0.3) (0.2, 0.6, 0.1) Table 8: Relation between symptoms of the disease PFSR οΏ½ΜƒοΏ½1 οΏ½ΜƒοΏ½2 οΏ½ΜƒοΏ½3 οΏ½ΜƒοΏ½4 οΏ½ΜƒοΏ½5 οΏ½ΜƒοΏ½1 (0.1, 0.2, 0.3) (0.5, 0.2, 0.3) (0.4, 0.2, 0.4) (0.1, 0.2, 0.6) (0.5, 0.5, 0) οΏ½ΜƒοΏ½2 (0.5, 0.1, 0.3) (0.1, 0.5, 0.3) (0.1, 0.2, 0.5) (0.3, 0.2, 0.4) (0.8, 0.1, 0) οΏ½ΜƒοΏ½3 (0.3, 0.2, 0.4) (0.1, 0.6, 0.2) (0.2, 0.2, 0.3) (0.1, 0, 0.9) (0.2, 0.5, 0.2) οΏ½ΜƒοΏ½4 (0.4, 0.2, 0.3) (0.7, 0.1, 0.2) (0.5, 0.1, 0.2) (0.4, 0.3, 0.1) (0.1, 0.5, 0.2) οΏ½ΜƒοΏ½5 (0.1, 0.2, 0.5) (0.2, 0.4, 0.3) (0.3, 0.5, 0.1) (0.2, 0.7, 0.1) (0.5, 0.2, 0.1) Table 9: Calculated distances between disease and patients using existing and proposed measure PFSR οΏ½ΜƒοΏ½1 οΏ½ΜƒοΏ½2 οΏ½ΜƒοΏ½3 οΏ½ΜƒοΏ½4 οΏ½ΜƒοΏ½5 𝑀𝐺 𝑀𝑆𝐡 𝑀𝐺 𝑀𝑆𝐡 𝑀𝐺 𝑀𝑆𝐡 𝑀𝐺 𝑀𝑆𝐡 𝑀𝐺 𝑀𝑆𝐡 οΏ½ΜƒοΏ½1 0.1791 0.1470 0.1001 0.0821 0.1946 0.1605 0.2066 0.1704 0.1147 0.0948 οΏ½ΜƒοΏ½2 0.1737 0.1455 0.1424 0.1174 0.1485 0.1203 0.1439 0.1184 0.1439 0.1189 οΏ½ΜƒοΏ½3 0.1377 0.1153 0.1251 0.1029 0.1596 0.1311 0.1457 0.1204 0.1066 0.0865 οΏ½ΜƒοΏ½4 0.1174 0.0970 0.1434 0.1184 0.1075 0.0905 0.1173 0.0966 0.1340 0.1099 Bold numerical values show the disease by which patient is suffering. Table 9, displays the correlation between the patients and their respective disease with bold values, using relationship between symptoms of patient and relationship between symptoms of disease given in Table 7 and Table 8 respectively. Particularly, the Table 9 indicates that patient οΏ½ΜƒοΏ½1 is affected with disease οΏ½ΜƒοΏ½2, patient οΏ½ΜƒοΏ½2 has disease οΏ½ΜƒοΏ½2 patient οΏ½ΜƒοΏ½3 is suffering from disease οΏ½ΜƒοΏ½5 and patient οΏ½ΜƒοΏ½4 also has disease οΏ½ΜƒοΏ½3. Also, proposed measure 𝑀𝑆𝐡 shows the superior value compare to 𝑀𝐺 . Figure 5: Comparisons of distance of the disease with the patients using existing measure 𝑀𝐺 and proposed measure 𝑀𝑆𝐡. The result of comparative analysis of proposed measure for this problem is discussed in Table 9 and Fig. 5, respectively. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1490 https://internationalpubls.com 5.3 Pattern analysis The classical problem was expected to solve using an artificial neuronal network which is a case of pattern classification, i.e., handwritten characters[39], [40], [41]. Pixels represented the object of reality; frequency patterns represent a linguistic signal and a sound also came under the concept of pattern. The first machine for pattern classification was developed by Frank Rosenblatt at Cornell University, New York in 1957 and 1958. Here we apply the proposed measure and compare the performance with the existing measures with the help of examples which shows the superiority of proposed measure. 5.4 Algorithm based on stated measure Consider 𝐴 = {π‘Ž1, π‘Ž2, π‘Ž3 … . . , π‘Žπ‘›} be finite universal set and assume the n patterns 𝐡 = {𝐹1, 𝐹2, 𝐹3 … , 𝐹𝑛} which can be expressed by (PIFSs) as 𝐹𝑗 = {(π‘Žπ‘– , πœ‡πΉπ‘— (π‘Žπ‘–), πœˆπΉπ‘— (π‘Žπ‘–), 𝛾𝐹𝑗 (π‘Žπ‘–) |π‘Žπ‘– ∈ 𝐴)} and π‘˜ test sample 𝐢 = {𝐢1, 𝐢2, 𝐢3 … πΆπ‘˜} which can be expressed by (PIFSs) as 𝐢𝑝 = {(π‘Žπ‘– , πœ‡πΉπ‘— (π‘Žπ‘–), πœˆπΉπ‘— (π‘Žπ‘–), 𝛾𝐹𝑗 (π‘Žπ‘–) |π‘Žπ‘– ∈ 𝐴)} the steps on pattern can be follows as Step 1 Find the measure between the given pattern 𝐹𝑗 and test sample πΆπ‘˜, by using the new (PIFSs) method 𝑀𝑆𝐡(𝐹𝑗 , πΆπ‘˜) = 1 4𝑛 βˆ‘ [ |π‘’βˆ’πœ‡πΉπ‘—(π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‡πΆπ‘˜(π‘Žπ‘—)| + |π‘’βˆ’πœˆπΉπ‘—(π‘Žπ‘—) βˆ’ π‘’βˆ’πœˆπΆπ‘˜(π‘Žπ‘—)| + |π‘’βˆ’π›ΎπΉπ‘—(π‘Žπ‘—) βˆ’ π‘’βˆ’π›ΎπΆπ‘˜(π‘Žπ‘—)| + |𝑒 βˆ’πœ‘πΉπ‘— (π‘Žπ‘—) βˆ’ π‘’βˆ’πœ‘πΆπ‘˜(π‘Žπ‘—)| ]𝑛 𝑗=1 Step 2 Pick the minimum value between the PIFSs 𝐹𝑗 and πΆπ‘˜ calculating in the step-1 Symbolically 𝑀𝑆𝐡(𝐹𝛼, πΆπ‘˜) = π‘šπ‘–π‘› 1≀𝑗≀𝑛 𝑀𝑆𝐡(𝐹𝑗 , πΆπ‘˜) Step 3 Now, text sample πΆπ‘˜ is assigned for the pattern 𝐹𝛼, where 𝛼 = π‘Žπ‘Ÿπ‘” π‘šπ‘–π‘› 1≀𝑗≀𝑛 𝑀𝑆𝐡(𝐹𝑗 , πΆπ‘˜) Problem 5.3 Consider 𝐹1, 𝐹2, 𝐹3 and 𝐹 be the four-pattern taken in the form of PIFSs as: 𝐹1 = {(0.4, 0.3, 0), (0.6, 0.1, 0.1), (0.4, 0.3, 0.1), (0.7, 0, 0.2), (0.5, 0.3, 0.2)} 𝐹2 = {(0.2, 0.1, 0.5), (0.3, 0.3, 0.3), (0.7, 0.1, 0.1), (0.1, 0.5, 0.2), (0.2, 0.3, 0.4)} 𝐹3 = {(0.3, 0.4, 0.2), (0.5, 0.3, 0.1), (0.1, 0.3, 0.4), (0.2, 0.5, 0.3), (0.4, 0.3, 0.1)} 𝐹 = {(0.4, 0.3, 0.2), (0.4,0.2,0.2), (0.6, 0.2, 0.1), (0.7, 0.1, 0), (0.3, 0.4, 0.2)} We need to check the measure between 𝐹𝑗 and 𝐹 where 𝑗 takes values 1, 2 and 3. The pattern 𝐹 is categorized under 𝐹𝑗 where 1 ≀ 𝑗 ≀ 3, if the measure between 𝐹 and 𝐹𝑗 is smallest. The calculated measure values between 𝐹 and 𝐹𝑗 where 1 ≀ 𝑗 ≀ 3, using the proposed and existing measure, displayed in table 10 and graphically represented in Figure 6. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1491 https://internationalpubls.com Table 10: Computed values of distance measures with existing and proposed measure 𝑀𝑆𝐡 Measures (𝐹1, 𝐹) (𝐹2, 𝐹) (𝐹3, 𝐹) Results 𝑀𝑑1 0.1000 0.1500 0.1700 𝐹1 𝑀𝑑2 0.1265 0.2074 0.2236 𝐹1 𝑀𝑑3 0.0500 0.0650 0.0650 Unable to classify 𝑀𝑑4 0.1000 0.1597 0.1628 𝐹1 𝑀𝑑5 0.3750 0.5154 0.4755 𝐹1 𝑀𝑑6 0.3491 0.4011 0.3880 𝐹1 𝑀𝑑7 0.1250 0.1872 0.1775 𝐹1 𝑀𝑑8 0.1054 0.1725 0.1686 𝐹1 𝑀𝑑9 0.1000 0.1867 0.1933 𝐹1 𝑀𝑑10 0.1000 0.1833 0.1929 𝐹1 𝑀𝑑11 0.2000 0.2600 0.2600 Unable to classify 𝑀𝑑12 0.0894 0.1428 0.1456 𝐹1 𝑀𝑑13 1.0000 1.5000 1.7000 𝐹1 𝑀𝑑14 0.2000 0.3000 0.3400 𝐹1 𝑀𝑑15 0.4000 0.6557 0.7071 𝐹1 𝑀𝑑16 0.8944 1.4663 1.5811 𝐹1 𝑀𝑑17 0.0400 0.0400 0.0200 Unable to classify 𝑀𝐺 0.0926 0.1360 0.1552 𝐹1 𝑀𝑆𝐡 0.0794 0.1123 0.1292 𝐹1 Table 10 illustrates that F is closest to 𝐹1 as proved by all distance measures and 𝑀𝑑3 𝑀𝑑11 and 𝑀𝑑17 suggests an unreasonable result. Our proposed distance measure 𝑀𝑆𝐡 continuously provides better results and closely with 𝐹1. It can be noticed that F is close to 𝐹1 according to all distance measures, and the proposed measure verify this result. Consequently, F is grouped with 𝐹1. Figure 6 Pattern analysis of proposed and existing measures. 0 0.5 1 Pattern analysis of proposed and existing measures (F1,F) (F2,F) (F3,F) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1492 https://internationalpubls.com The result of comparative analysis of proposed measure for this problem is discussed in Table 10 and Fig. 6, respectively. Problem 5.4 Consider 𝐹1, 𝐹2, 𝐹3 and 𝐹 be the four-pattern taken in the form of PFSs as: 𝐹1 = {(0.4, 0, 0.5), (0.7, 0, 0.3), (0.5, 0, 0.3), (0.7, 0, 0.3), (0.2, 0.2, 0.6)} 𝐹2 = {(0.2, 0, 0.7), (0.7, 0, 0.3), (0.5, 0, 0.2), (0.7, 0, 0.3), (0.1, 0.5, 0.4)} 𝐹3 = {(0.4, 0, 0.6), (0.7, 0, 0.2), (0.5,0, 0.4), (0.7, 0, 0.2 ), (0.3, 0.4, 0.3)} 𝐹 = {(0.3, 0, 0.6), (0.7, 0, 0.3), (0.4, 0, 0.3), (0.7, 0, 0.3), (0.5, 0.1, 0.4)} We need to find the measure between 𝐹𝑗 and 𝐹 where 𝑗 takes values 1, 2 and 3. The pattern 𝐹 is classified with 𝐹𝑗 where 1 ≀ 𝑗 ≀ 3, if the measure between 𝐹 and 𝐹𝑗 is smallest. The calculated measure values between 𝐹 and 𝐹𝑗 where 1 ≀ 𝑗 ≀ 3, using the proposed and existing measure, are given in the table 11 and graphically represented in Figure 7. Table 11: Computed values of distance with existing measure and proposed measure 𝑀𝑆𝐡 Measure (𝐹1, 𝐹) (𝐹2, 𝐹) (𝐹3, 𝐹) Results 𝑀𝑑1 0.0500 0.0600 0.0800 𝐹1 𝑀𝑑2 0.0949 0.1342 0.1140 𝐹1 𝑀𝑑3 0.0250 0.0300 0.0400 𝐹1 𝑀𝑑4 0.0742 0.0949 0.0894 𝐹1 𝑀𝑑5 0.2759 0.3333 0.2991 𝐹1 𝑀𝑑6 0.2810 0.3056 0.3112 𝐹1 𝑀𝑑7 0.0734 0.0909 0.0922 𝐹1 𝑀𝑑8 0.0836 0.1091 0.0887 𝐹1 𝑀𝑑9 0.0600 0.0800 0.0733 𝐹1 𝑀𝑑10 0.0825 0.1200 0.0872 𝐹1 𝑀𝑑11 0.1000 0.1200 0.1400 𝐹1 𝑀𝑑12 0.0663 0.0849 0.0721 𝐹1 𝑀𝑑13 0.5000 0.6000 0.8000 𝐹1 𝑀𝑑14 0.1000 0.1200 0.1600 𝐹1 𝑀𝑑15 0.3000 0.4243 0.3606 𝐹1 𝑀𝑑16 0.6708 0.9487 0.8062 𝐹1 𝑀𝑑17 0.0100 0.0100 0.0200 Unable to classify 𝑀𝐺 0.0433 0.0535 0.0741 𝐹1 𝑀𝑆𝐡 0.0345 0.0434 0.0625 𝐹1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1493 https://internationalpubls.com Table 11 illustrates that F is closest to 𝐹1 as proved by all distance measures and 𝑀𝑑17 shows an unreasonable result. Our proposed distance measure 𝑀𝑆𝐡 continuously provides better results and very closely to 𝐹1. It is observed that F is close to 𝐹1 calculating by all distance measures, and the proposed measure verify this result. Consequently, F is classified with 𝐹1. Figure 7: Pattern analysis of proposed and existing measures. The result of comparative analysis of proposed measure for this problem is discussed in Table 11 and Fig. 7, respectively. 6. Conclusion This paper introduces a new method to measure distances for picture fuzzy sets and demonstrates its efficiency in identifying the relationships and differences between fuzzy elements. Through various comparative analysis with existing measures, it consistently shows superior performance, and provide more accurate and meaningful results. Future efforts can be focused on refining the algorithm for advanced computational efficiency and exploring its applicability in various field such as pattern analysis, machine learning, artificial intelligence etc. Additionally, advanced techniques such as machine learning or optimization methods may contribute to improvement of the proposed distance measure. Further studies can also explore real-world applications and validate the measure's effectiveness in solving complex problems in areas like pattern recognition, decision-making, and image processing. Conflict of interest The authors confirm that they have no conflict of interest. Data availability statement All the data supporting this study's findings are provided in the article. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Pattern analysis of proposed and existing measures (F1,F) (F2,F) (F3,F) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1494 https://internationalpubls.com Funding There is no funding for this study. Refrences [1] L. A. Zadeh, β€œFuzzy sets,” Inf. Control, vol. 8, no. 3, pp. 338–353, Jun. 1965, doi: 10.1016/S0019- 9958(65)90241-X. [2] E. T. Lee and L. A. Zadeh, β€œNote on fuzzy languages,” Inf. Sci., vol. 1, no. 4, pp. 421–434, 1969. [3] A. Ganie, β€œA picture fuzzy distance measure and its application to pattern recognition problems,” Iran. J. Fuzzy Syst., vol. 20, pp. 71–85, Jan. 2023, doi: 10.22111/IJFS.2023.7347. [4] J. Bajaj and S. Kumar, β€œA new intuitionistic fuzzy correlation coefficient approach with applications in multi-criteria decision-making,” Decis. Anal. 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