EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6045 ISSN 1307-5543 – ejpam.com Published by New York Business Global Applications of Weighted Tangent Similarity Measure of Picture Hesitant Fuzzy Sets Noura Awad Al Qarni1,2,∗, Noura Omair Alshehri1, Rania Saeed Alghamdi1 1 Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, Jeddah 23218, Saudi Arabia 2 Department of Mathematics, College of Science, University of Bisha, Bisha 61922, Saudi Arabia Abstract. Picture hesitant fuzzy sets (PHFSs) provide a powerful framework for modeling uncer- tainty by incorporating the degrees of membership, non-membership, neutrality, and refusal. This paper introduces a novel similarity measure, the Weighted Tangent Similarity Measure (ωTSM), designed to improve sensitivity to subtle variations and to capture nonlinear interactions among PHFS components. The proposed measure is theoretically established and empirically validated through two real-world decision-making problems: medical diagnosis and building material clas- sification. In both cases, ωTSM effectively distinguishes between closely related alternatives and consistently outperforms existing methods. The comparative analysis highlights its robustness, flexibility, and practical value in environments characterized by ambiguity and hesitation. Overall, this study advances fuzzy decision-making models and provides a foundation for future applications in knowledge representation and intelligent systems. 2020 Mathematics Subject Classifications: 03E72, 68T37, 92C50 Key Words and Phrases: Picture hesitant fuzzy sets, Tangent similarity measure, Weighted tangent similarity measure, Medical diagnosis, Building material recognition 1. Introduction In 1965, Zadeh [1] introduced the concept of fuzzy sets (FS), marking a pivotal advance- ment in the mathematical modeling of uncertainty. In this framework, the membership degree of an element in a fuzzy set is defined by a characteristic function mapping to the unit interval [0, 1], enabling a flexible representation of vagueness. Later, Atanassov [2] extended this foundation by proposing intuitionistic fuzzy sets (IFS), which incorporate both membership and non-membership degrees, thereby enhancing the capacity to repre- sent and analyze imprecise information. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6045 Email addresses: nalgarni0063.stu@uj.edu.sa (N.A. Al Qarni), noal-shehri@uj.edu.sa (N.O. Alshehri), rsaalghamdi@uj.edu.sa (R.S. Alghamdi), noqarni@ub.edu.sa (N.A. Al Qarni) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 2 of 15 Building on these developments, Cuong and Kreinovich [3, 4] introduced picture fuzzy sets (PFS), which extend FS and IFS by incorporating additional degrees of neutrality and refusal alongside membership and non-membership. This richer representation allows for more nuanced modeling of human evaluations, particularly in contexts where responses may fall into categories such as “yes,” “no,” “neutral,” or “refusal.” Concurrently, Torra [5] proposed hesitant fuzzy sets (HFS), in which the membership degree of an element is characterized by a set of possible values within [0, 1]. Combining these two concepts, Wang et al. [6] developed picture hesitant fuzzy sets (PHFS), providing a flexible and comprehensive framework for handling uncertainty in complex decision-making scenarios. Similarity measures play a crucial role in numerous fuzzy set applications, including decision-making, pattern recognition, and medical diagnosis. Several similarity measures for FS have been proposed by researchers such as Pappis and Karacapilidis [7], Chen [8], and Lohrmann [9]. Within the IFS framework, Dengfeng and Chuntian [10] introduced similarity measures that have been successfully applied to pattern recognition. Never- theless, conventional approaches often encounter limitations when faced with increasingly complex and uncertain data. To address these challenges, recent studies by Wei [11] and Zhang et al. [12] proposed similarity measures for PFS and HFS based on cosine and grey similarity functions, achieving promising results in applications such as material classifi- cation and medical diagnosis. More recently, Alshehri et al. [13] developed new PHFS distance measures with ap- plications in medical diagnosis. Li [14] introduced distance and similarity measures for PFS. Mostafa [15] proposed a novel PFS distance measure ρ∗ together with a proof of the triangular inequality. Cao et al. [16] presented advanced PFS distance measures for multi- criteria group decision-making in healthcare. In addition, Palanikumar et al. [17] designed complex Pythagorean normal interval-valued fuzzy aggregation operators for solving med- ical diagnosis problems, while Palanikumar et al. [18] explored various distance measures between generalized Diophantine fuzzy sets with applications in multi-criteria decision making. Collectively, these studies highlight the continuing evolution of distance and sim- ilarity measures in fuzzy set theory and their growing practical significance. Despite these advances, existing similarity measures such as the picture hesitant fuzzy weighted hybrid vector similarity measure (PHFWHVSM) [19] and other weighted sim- ilarity methods [20] continue to face challenges in capturing the complex relationships among membership, non-membership, and neutrality degrees. This difficulty becomes more pronounced in environments characterized by nonlinear data. A key limitation of such approaches is their reliance on linear assumptions, which restricts their ability to represent the true behavior of PHFS. In addition, these measures often exhibit low sensi- tivity to subtle variations in fuzzy parameters, making it difficult to distinguish between closely related PHFS instances. Another drawback is computational inefficiency, as many of these methods require significant processing resources, limiting their scalability in large- scale decision-making problems. N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 3 of 15 To address these shortcomings, this study introduces a novel similarity measure called the Weighted Tangent Similarity Measure (ωTSM). This measure leverages the nonlinear properties of the tangent function to enhance sensitivity to small variations and more ac- curately model the intricate relationships among fuzzy components. By exploiting these characteristics, the proposed measure provides a robust analytical framework for pro- cessing complex and ambiguous information. Its effectiveness is demonstrated through rigorous theoretical validation and comprehensive comparative experiments, demonstrat- ing improvements in both computational efficiency and classification accuracy. The practicality of ωTSM is further illustrated through real-world case studies involv- ing medical diagnosis and building material classification. These applications confirm its ability to accurately associate symptoms with diseases and to improve material identifi- cation. By overcoming the limitations of conventional methods, the proposed approach contributes to advances in diverse domains, including clustering, image segmentation, and decision-making under uncertainty. The main contributions of this study in the context of PHFS are summarized as follows: (i) Introduction and validation of the Weighted Tangent Similarity Measure (ωTSM), which effectively evaluates the membership, neutrality, and non-membership degrees in PHFS while improving sensitivity to subtle variations. (ii) Resolution of key limitations in existing similarity measures, particularly in medical diagnosis, by demonstrating the method’s ability to accurately associate symptoms with diseases under uncertain conditions. (iii) Development of detailed case studies using PHFS-based data in both medical and material classification tasks, highlighting the robustness, adaptability, and practical significance of the proposed measure. (iv) Execution of a sensitivity analysis to evaluate the stability and consistency of ωTSM across various datasets and decision-making scenarios. The remainder of this paper is organized as follows. Section 2 presents the necessary definitions and background on PHFS. Section 3 introduces the proposed similarity measure and discusses its theoretical properties. Section 4 provides real-world case studies in medical diagnosis and material classification. Finally, Section 5 concludes the paper and outlines directions for future research. 2. Preliminaries Definition 1 [1] Let X be a non-empty set. A fuzzy set E on X can be represented as E = {⟨x, µE(x)⟩ | x ∈ X} , where µE : X → [0, 1]. (1) N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 4 of 15 Definition 2 [2] An intuitionistic fuzzy set (IFS) E on a universal set X is defined as E = {⟨x, µE(x), νE(x)⟩ | x ∈ X} , (2) where µE : X → [0, 1] and νE : X → [0, 1] represent the degree of membership and the degree of non-membership of the element x ∈ X, respectively. These functions satisfy the condition 0 ≤ µE(x) + νE(x) ≤ 1, ∀x ∈ X. Definition 3 [3] A picture fuzzy set (PFS) E on a universal set X is defined as E = {⟨x, µE(x), ηE(x), νE(x)⟩ | x ∈ X} , (3) where µE(x) ∈ [0, 1] denotes the membership degree of x in E, ηE(x) ∈ [0, 1] denotes the neutrality degree, and νE(x) ∈ [0, 1] denotes the non-membership degree. These values satisfy 0 ≤ µE(x) + ηE(x) + νE(x) ≤ 1, ∀x ∈ X. (4) The refusal degree is given by πE(x) = 1− (µE(x) + ηE(x) + νE(x)) , ∀x ∈ X. (5) Definition 4 [5] A hesitant fuzzy set (HFS) E on a universal set X is defined as E = {⟨x,Eh(x)⟩ | x ∈ X} , (6) where Eh(x) ⊆ [0, 1] represents a set of possible membership degrees of the element x, reflecting hesitation in assigning a precise value. Given hesitant fuzzy elements (HFEs) Eh(x), Eh1(x), and Eh2(x), several operations have been introduced by Xia and Xu [21], and Liao et al. [22], including: E− h (x) = minEh(x), E+ h (x) = maxEh(x), (7) Ec h(x) = ⋃ γ∈Eh(x) {1− γ}, (8) Eλ h(x) = ⋃ γ∈Eh(x) {γλ}, λ > 0, (9) λEh(x) = ⋃ γ∈Eh(x) {1− (1− γ)λ}, λ > 0, (10) Eh1(x) ∪ Eh2(x) = ⋃ γ1∈Eh1 (x),γ2∈Eh2 (x) {max(γ1, γ2)}, (11) N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 5 of 15 Eh1(x) ∩ Eh2(x) = ⋃ γ1∈Eh1 (x),γ2∈Eh2 (x) {min(γ1, γ2)}, (12) Eh1(x)⊕ Eh2(x) = ⋃ γ1∈Eh1 (x),γ2∈Eh2 (x) {γ1 + γ2 − γ1γ2}, (13) Eh1(x)⊗ Eh2(x) = ⋃ γ1∈Eh1 (x),γ2∈Eh2 (x) {γ1 · γ2}, (14) Definition 5 [19] A picture hesitant fuzzy set (PHFS) E on a universal set X is defined as E = {⟨x, µiE(x), ηiE(x), νiE(x)⟩ | x ∈ X} , i = 1, 2, 3, . . . , z, (15) where µiE(x), ηiE(x), and νiE(x) are finite subsets of [0, 1]. The corresponding picture hesitant fuzzy number(PHFN) is denoted as ( µiEp(x), ηiEp(x), νiEp(x) ) , and the refusal degree is expressed as ρiEp(x) = 1− ( µiEp(x) + ηiEp(x) + νiEp(x) ) . (16) The following operations on PHFNs are defined: 1. Union: EP∪FP = {〈 x,max ( µiE(x), µiF (x) ) ,min ( ηiE(x), ηiF (x) ) ,min ( νiE(x), νiF (x) )〉 ∣∣∣x ∈ X } . (17) 2. Intersection: EP∩FP = {〈 x,min ( µiE(x), µiF (x) ) ,max ( ηiE(x), ηiF (x) ) ,max ( νiE(x), νiF (x) )〉 ∣∣∣x ∈ X } . (18) 3. Complement: Ec P = {⟨x, νiE(x), ηiE(x), µiE(x)⟩ | x ∈ X}. (19) 3. Weighted Tangent Similarity Measures for Picture Hesitant Fuzzy Sets Definition 6 Let E = {⟨x, µiE(x), ηiE(x), νiE(x)⟩ | x ∈ X} , F = {⟨x, µiF (x), ηiF (x), νiF (x)⟩ | x ∈ X} be two picture hesitant fuzzy sets defined on the universal set X. Then, the tangent similarity measure between E and F is defined as: TSM (E,F ) = 1− 1 m ( tan ( π 12 ) m∑ j=1 [ n∑ i=1 ( |µiE(xj)− µiF (xj)| N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 6 of 15 + |ηiE(xj)− ηiF (xj)|+ |νiE(xj)− νiF (xj)| )]) (20) Theorem 1. Let E, F , and G be three picture hesitant fuzzy sets (PHFS). The tangent similarity measure TSM satisfies the following properties: 1. 0 ≤ TSM (E,F ) ≤ 1 2. TSM (E,F ) = TSM (F,E) 3. TSM (E,F ) = 1 if and only if E = F 4. If E ⊆ F ⊆ G, then TSM (E,F ) ≥ TSM (E,G) and TSM (F,G) ≥ TSM (E,G) Proof. 1. As the membership,non-membership,and neutral degrees of picture hesitant fuzzy sets are all defined within the interval [0, 1], and the value of the tangent function used in the similarity measure is normalized to lie within [0, 1], the similarity measure based on the tangent function is also bounded in [0, 1]. Hence, 0 ≤ TSM (E,F ) ≤ 1. 2. From the definition 6, we have TSM (E,F ) = 1− 1 m ( tan ( π 12 ) m∑ j=1 [ n∑ i=1 ( |µiE(xj)− µiF (xj)| + |ηiE(xj)− ηiF (xj)|+ |νiE(xj)− νiF (xj)| )]) = 1− 1 m ( tan ( π 12 ) m∑ j=1 [ n∑ i=1 ( |µiF (xj)− µiE(xj)| + |ηiF (xj)− ηiE(xj)|+ |νiF (xj)− νiE(xj)| )]) = TSM (F,E) Therefore, TSM (E,F ) = TSM (F,E) 3.Let E = F , then we have µiE(xj) = µiF (xj), ηiE(xj) = ηiF (xj), νiE(xj) = νiF (xj), this implies that |µiE(xj)− µiF (xj)| = 0, |ηiE(xj)− ηiF (xj)| = 0, |νiE(xj)− νiF (xj)| = 0. N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 7 of 15 Therefore, TSM (E,F ) = 1− 1 m  m∑ j=1 tan(0)  = 1. Conversely, suppose that TSM (E,F ) = 1. This implies that |µiE(xj)− µiF (xj)| = 0, |ηiE(xj)− ηiF (xj)| = 0, |νiE(xj)− νiF (xj)| = 0. Hence, µiE(xj) = µiF (xj), ηiE(xj) = ηiF (xj), νiE(xj) = νiF (xj). Therefore, E = F . 4. Since E ⊆ F ⊆ G, for every xj ∈ X we have µiE(xj) ≤ µiF (xj) ≤ µiG(xj), ηiE(xj) ≤ ηiF (xj) ≤ ηiG(xj), νiE(xj) ≥ νiF (xj) ≥ νiG(xj). Hence, |µiE(xj)− µiF (xj)| ≤ |µiE(xj)− µiG(xj)| , |ηiE(xj)− ηiF (xj)| ≤ |ηiE(xj)− ηiG(xj)| , |νiE(xj)− νiF (xj)| ≤ |νiE(xj)− νiG(xj)| . TSM (E,F ) = 1− 1 m ( m∑ j=1 tan ( π 12 )[ n∑ i=1 ( |µiE(xj)− µiF (xj)|+ |ηiE(xj)− ηiF (xj)| + |νiE(xj)− νiF (xj)| )]) ≥ 1− 1 m ( m∑ j=1 tan ( π 12 )[ n∑ i=1 ( |µiE(xj)− µiG(xj)|+ |ηiE(xj)− ηiG(xj)| + |νiE(xj)− νiG(xj)| )]) = TSM (E,G). This implies that TSM (E,F ) ≥ TSM (E,G). Similarly, we can prove that TSM (F,G) ≥ TSM (E,G). Therefore, the proof is complete. N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 8 of 15 Definition 7 Let E = {⟨x, µiE(x), ηiE(x), νiE(x)⟩ | x ∈ X} , F = {⟨x, µiF (x), ηiF (x), νiF (x)⟩ | x ∈ X} be two picture hesitant fuzzy sets defined on the universal set X. Then, the weighted tangent similarity measure between E and F is defined as: ωTSM (E,F ) = 1− 1 m ( tan ( π 12 ) m∑ j=1 ωj [ n∑ i=1 ( |µiE(xj)− µiF (xj)| + |ηiE(xj)− ηiF (xj)|+ |νiE(xj)− νiF (xj)| )]) , where m∑ j=1 ωj = 1 (21) Theorem 2 Let E , F and G be three picture hesitant fuzzy set PHFS on X then: 1. 0 ⩽ ωTSM (E,F ) ⩽ 1 2. ωTSM (E,F ) = ωTSM (F,E) 3. ωTSM (E,F ) = 1 iff E = F 4. If E ⊆ F ⊆ G, then ωTSM (E,F ) ≥ ωTSM (E,G) and ωTSM (F,G) ≥ ωTSM (E,G). Proof : The proof is similar to Theorem 1 The rationale for adopting the tangent function in both applications presented in this study is grounded in its mathematical characteristics and practical utility. Its ability to model nonlinear relationships and amplify subtle differences, particularly near the origin, enhances the discriminative capacity of the proposed measure, which is essential in distin- guishing closely related PHFS elements. This sensitivity is particularly valuable in real- world scenarios such as medical diagnosis and building material classification, where minor variations can lead to significantly different outcomes. The integration of expert-defined weights further strengthens the context-awareness and accuracy of the decision-making process. 4. Applications of the Weighted Tangent Similarity Measure Medical Diagnosis Medical diagnosis plays a crucial role in healthcare, where physicians must analyze a wide variety of clinical symptoms and complex medical data to identify underlying diseases ac- curately. However, the inherent uncertainty and imprecision in medical information often limit the reliability of traditional diagnostic techniques. To address this issue, the pro- posed Weighted Tangent Similarity Measure (ωTSM) under the Picture Hesitant Fuzzy Set N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 9 of 15 (PHFS) environment provides a robust mathematical framework for quantifying the simi- larity between a patient’s symptoms and the known symptom patterns of specific diseases. This approach incorporates three essential components membership degree, neutrality de- gree, and non-membership degree which collectively enhance the accuracy, consistency, and structure of the diagnostic process. Example 1: Consider a physician evaluating a group of patients represented as: P = {James,Harry, Jak,Oliver}. The potential diseases under consideration are given by: D = {Viral Fever,Malaria,Typhoid,Stomach Problem,Chest Problem}. These diseases are diagnosed based on a set of clinical symptoms defined as: V = {Fever,Cough, Shortness of Breath,Chest Pain,Headache}. In the proposed method, the diagnostic process involves determining the degrees of mem- bership, neutrality, and non-membership for each patient according to their exhibited symptoms. The similarity between each patient’s symptom profile and the characteristic symptom patterns of the diseases is computed using Definition (7). Based on the resulting similarity scores, a ranking of potential diagnoses is generated. This ranking enables informed clinical decision-making by identifying the disease with the highest similarity score as the most probable diagnosis. The method effectively distinguishes among competing diagnoses while accounting for uncertainty in patient data. Table 1: Patients Data and Symptoms Fever Cough Shortness of breath Chest pain Headache James ( {0.25,0.24} {0.30,0.28} {0.40,0.37} ) ( {0.30,0.28} {0.15,0.12} {0.45,0.50} ) ( {0.32,0.28} {0.40,0.35} {0.20,0.15} ) ( {0.25,0.20} {0.35,0.33} {0.35,0.30} ) ( {0.55,0.50} {0.05,0.0} {0.35,0.40} ) Harry ( {0.40,0.38} {0.30,0.28} {0.25,0.22} ) ( {0.38,0.35} {0.08,0.0} {0.45,0.40} ) ( {0.15,0.12} {0.55,0.50} {0.25,0.20} ) ( {0.30,0.35} {0.20,0.20} {0.45,0.40} ) ( {0.20,0.18} {0.25,0.22} {0.45,0.45} ) Jak ( {0.40,0.35} {0.35,0.28} {0.20,0.20} ) ( {0.45,0.40} {0.25,0.20} {0.25,0.25} ) ( {0.33,0.30} {0.25,0.20} {0.35,0.30} ) ( {0.45,0.50} {0.15,0.10} {0.30,0.30} ) ( {0.48,0.50} {0.05,0.0} {0.40,0.45} ) Oliver ( {0.18,0.15} {0.25,0.20} {0.50,0.45} ) ( {0.35,0.30} {0.20,0.18} {0.40,0.45} ) ( {0.38,0.30} {0.30,0.25} {0.30,0.35} ) ( {0.45,0.40} {0.30,0.20} {0.25,0.30} ) ( {0.38,0.35} {0.10,0.08} {0.45,0.50} ) N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 10 of 15 Table 2: Diseases Data and Symptoms Fever Cough Shortness of breath Chest pain Headache Viral fever ( {0.36,0.36} {0.35,0.28} {0.28,0.35} ) ( {0.31,0.31} {0.31,0.38} {0.38,0.31} ) ( {0.40,0.41} {0.41,0.17} {0.17,0.40} ) ( {0.47,0.20} {0.20,0.33} {0.33,0.47} ) ( {0.59,0.19} {0.19,0.20} {0.20,0.59} ) Malaria ( {0.29,0.43} {0.41,0.29} {0.29,0.28} ) ( {0.37,0.31} {0.31,0.31} {0.31,0.37} ) ( {0.14,0.50} {0.50,0.36} {0.35,0.13} ) ( {0.28,0.22} {0.22,0.49} {0.49,0.27} ) ( {0.41,0.32} {0.32,0.27} {0.27,0.40} ) Typhoid ( {0.40,0.27} {0.27,0.33} {0.33,0.40} ) ( {0.26,0.26} {0.26,0.47} {0.47,0.26} ) ( {0.38,0.31} {0.31,0.31} {0.31,0.38} ) ( {0.63,0.07} {0.07,0.29} {0.29,0.63} ) ( {0.43,0.21} {0.21,0.36} {0.36,0.43} ) Stomach problem ( {0.24,0.29} {0.29,0.47} {0.47,0.23} ) ( {0.53,0.06} {0.06,0.40} {0.40,0.52} ) ( {0.31,0.46} {0.46,0.23} {0.23,0.30} ) ( {0.29,0.29} {0.29,0.41} {0.41,0.29} ) ( {0.38,0.38} {0.38,0.23} {0.23,0.38} ) Chest problem ( {0.57,0.14} {0.14,0.29} {0.29,0.57} ) ( {0.39,0.11} {0.11,0.50} {0.50,0.39} ) ( {0.23,0.46} {0.46,0.31} {0.31,0.23} ) ( {0.50,0.20} {0.20,0.10} {0.10,0.50} ) ( {0.20,0.47} {0.47,0.33} {0.33,0.20} ) The weight distribution for each criterion in the proposed tangent similarity measure was determined by a domain expert (e.g., a doctor or medical practitioner) as follows: ω1 = 0.225, ω2 = 0.195, ω3 = 0.200, ω4 = 0.190, and ω5 = 0.190. These weights reflect the relative importance of each clinical symptom in the similarity assessment process. Table 3 displays the computed weighted tangent similarity scores, with the highest score in each row indicating the most likely diagnosis for the corresponding patient. Based on these results, James is most likely diagnosed with Stomach Problem, while Harry is predicted to have Malaria. Similarly, Jak is most closely associated with Viral Fever, and Oliver with Typhoid. Table 3: Computed Weighted Tangent Similarity Scores Between Patients and Diseases Weighted Tangent Similarity Measure Viral Fever Malaria Typhoid Stomach Problem Chest Problem James 0.96393 0.96379 0.95921 0.9656 0.94963 Harry 0.95835 0.9659 0.95421 0.95699 0.95181 Jak 0.96341 0.9576 0.95917 0.95212 0.9453 Oliver 0.96198 0.95445 0.96267 0.96141 0.95156 Table 4: Comparative Analysis Methods James Harry Jak Oliver HWP [19] Stomach Problem Malaria Viral Fever Viral Fever W 1 PHFS [20] Stomach Problem Malaria Viral Fever Viral Fever W 2 PHFS [20] Viral Fever Malaria Viral Fever Viral Fever W 3 PHFS [20] Chest Problem Viral Fever Stomach Problem Chest Problem ωTSM Stomach Problem Malaria Viral Fever Typhoid To assess the effectiveness of the proposed weighted tangent similarity measure ωTSM , we conducted a comparative analysis against several established methods, including PH- FWHVSM [19], and the weighted cosine, set-theoretic, and grey similarity measures [20]. As summarized in Table 4, the results demonstrate both consistency and contrast across cases. N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 11 of 15 For James and Harry, the proposed measure yields diagnoses that align with those from other leading methods, supporting its reliability in typical scenarios. Notably, for Jak, our method correctly identifies Viral Fever, in agreement with most techniques, while avoiding less accurate alternatives such as Stomach Problem. For Oliver, the proposed measure uniquely predicts Typhoid, capturing distinctions that other methods favoring Viral Fever or Chest Problem appear to miss. These findings highlight the superior discriminative capability of ωTSM , particularly in cases with overlapping symptoms and hesitant information. Its sensitivity to subtle vari- ations ensures more nuanced and accurate classifications under the PHFS environment. Figure 1: Comparison of similarity measures in disease diagnosis Building Material Recognition In this study, we classify an unknown building material using the weighted tangent sim- ilarity measure under the picture hesitant fuzzy set (PHFS) environment. This approach leverages expert knowledge of known building materials to construct a reference database. The similarity between the unknown material and each known material is computed using the weighted tangent similarity measure. The unknown material is then assigned to the category of the known material with the highest similarity score. The complete procedure is summarized below: (i) Collect data for both known and unknown building materials, represented as picture hesitant fuzzy numbers (PHFNs). (ii) Compute the weighted tangent similarity measure between the unknown material and each known material. (iii) Rank the similarity scores and classify the unknown material based on the highest score. Example 2: [20] Let Ei(1 ≤ i ≤ 4) represent four building materials: brick, stone, muddy, and steel. Let X = {x1, x2, x3, x4} be the space of attributes with the weights ω1 = 0.27, N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 12 of 15 ω2 = 0.33, ω3 = 0.11, and ω4 = 0.29. The details of the unknown and known building materials are shown in Table 4. We classified the unknown material as follows: Table 5: Information of unknown and known building materials x1 x2 x3 x4 E1 ({0.0, 0.1}, {0.0, 0.3}, {0.0, 0.5}) ({0.1}, {0.2, 0.4}, {0.3}) ({0.2, 0.4}, {0.0, 0.5}, {0.0, 0.01}) ({0.2, 0.3}, {0.0, 0.2}, {0.0, 0.3}) E2 ({0.0, 0.2}, {0.1}, {0.0, 0.4}) ({0.0, 0.1}, {0.3}, {0.0}) ({0.0, 0.1}, {0.2, 0.4}, {0.0, 0.2}) ({0.2, 0.5}, {0.0, 0.2}, {0.0, 0.1}) E3 ({0.22}, {0.23, 0.27}, {0.0}) ({0.1}, {0.2, 0.11}, {0.3}) ({0.0, 0.5}, {0.1, 0.3}, {0.0, 0.2}) ({0.0, 0.17}, {0.54, 0.63}, {0.1, 0.2}) E4 ({0.42, 0.47}, {0.0, 0.53}, {0.0}) ({0.0, 0.15}, {0.0, 0.71}, {0.14}) ({0.1}, {0.0, 0.3}, {0.4, 0.5}) ({0.0, 0.1}, {0.02, 0.6}, {0.3, 0.35}) E ({0.1, 0.2}, {0.2, 0.3}, {0.3, 0.4}) ({0.1, 0.2}, {0.0, 0.1}, {0.0, 0.4}) ({0.0, 0.2}, {0.4}, {0.3}) ({1.00}, {0.00}, {0.00}) By applying the definition (7), we obtain: ωTSM (E,E1) = 0.910382, ωTSM (E,E2) = 0.925676, ωTSM (E,E3) = 0.875248, ωTSM (E,E4) = 0.863091 Table 6: Comparative Analysis of Similarity Measures Similarity Measure (E, E1) (E, E2) (E, E3) (E, E4) Ranking (E, Ei) HWP [19] 0.5898 0.5186 0.4423 0.3774 (E1) ≻ (E2) ≻ (E3) ≻ (E4) W 1 PHFS [20] 0.8423 0.8147 0.6418 0.3978 (E1) ≻ (E2) ≻ (E3) ≻ (E4) W 2 PHFS [20] 0.6178 0.5563 0.5022 0.2880 (E1) ≻ (E2) ≻ (E3) ≻ (E4) W 3 PHFS [20] 0.8739 0.9101 0.8728 0.7489 (E2) ≻ (E1) ≻ (E3) ≻ (E4) ωTSM 0.9104 0.9257 0.8752 0.8631 (E2) ≻ (E1) ≻ (E3) ≻ (E4) Table 6 presents a comparative analysis of various similarity measures applied to the classification task. Conventional measures such as HWP , W 1 PHFS , and W 2 PHFS consistently identify E1 as the material most similar to the unknown sample E. In contrast, both W 3 PHFS and the proposed measure ωTSM yield a different outcome, ranking E2 as the most similar. This divergence highlights the underlying methodological differences between the measures. Traditional approaches typically employ linear aggregations of membership, non-membership, and abstinence values. While computationally straightforward, such linear models may overlook subtle yet meaningful inter- actions inherent in Picture Hesitant Fuzzy Sets (PHFS). The proposed ωTSM measure, by contrast, introduces a weighted tangent-based transformation, which captures nonlinear relationships and amplifies small variations in the in- put data. This transformation enables a more refined and sensitive evaluation of similarity, particularly in cases involving hesitant or ambiguous information. Moreover, the proposed measure demonstrates greater robustness and consis- tency across multiple comparisons. Unlike static linear models that treat all components uniformly, ωTSM incorporates adaptive weighting, allowing it to N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 13 of 15 respond dynamically to the structural complexity of PHFS data. This adapt- ability enhances its discriminative power and reduces the likelihood of misclas- sification, especially in borderline cases. From an application standpoint, these results underscore the practical value of ωTSM in domains such as material science, medical diagnosis, and pattern recognition. By accurately modeling nonlinear dependencies and accounting for hesitation, the proposed measure offers a reliable and theoretically sound framework for similarity assessment in high-stakes decision-making environ- ments. 4.1. Advantages (i) The enhanced sensitivity of the proposed measure enables it to effectively capture subtle variations in membership, non-membership, neutrality, and refusal levels. This enables more accurate differentiation between closely related cases, rendering it exceptionally beneficial in applications such as medical diagnosis and material classification. (ii) The proposed similarity measure effectively models ambiguous and hesitant informa- tion, in contrast to conventional approaches that rely on linear assumptions. This feature enhances its suitability for decision-making in uncertain contexts, where pre- cise differentiation is crucial. (iii) Conventional measures, such as theWeighted Cosine Similarity Measure, Set-Theoretic Similarity Measure, PHFWHVSM, and Weighted Grey Similarity, often struggle to encapsulate the intricate, nonlinear interdependencies among features. The pro- posed approach addresses this limitation by effectively modeling these relationships in fuzzy data, ensuring more reliable and precise results. 5. Conclusion This study proposed two novel similarity measures: the Tangent Similarity Measure (TSM) and the Weighted Tangent Similarity Measure (ωTSM), which are specifically de- signed for Picture Hesitant Fuzzy Sets (PHFS). These measures aim to overcome key limitations of traditional linear similarity models by incorporating the nonlinear charac- teristics of the tangent function. This approach improves sensitivity to subtle differences and captures complex relationships among membership, non-membership, and neutrality degrees. Theoretical validation confirmed that both TSM and ωTSM satisfy the fundamental properties required for similarity measures. Empirical evaluations, which were conducted on two real-world decision-making problems, namely medical diagnosis and building ma- terial classification, demonstrated that ωTSM provides significantly higher accuracy and better discriminative performance compared to existing similarity measures. N. A. Al Qarni, N. O. Alshehri, R. S. Alghamdi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6045 14 of 15 These results confirm that ωTSM is capable of reliably distinguishing between closely related alternatives in uncertain and hesitant environments. Furthermore, its flexibility and adaptability make it a promising tool for a wide range of practical applications. Nonetheless, the proposed methodology is not without limitations. First, the reliance on expert-defined weights may introduce subjectivity in some contexts. Second, while ωTSM’s heightened sensitivity is generally advantageous, it may also increase vulnerability to noise in certain datasets. 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