Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Numbers

Neutrosophic Sets and Systems, Vol. 97, 2026 University of New Mexico Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Numbers Madineh Farnam 1, Gholam Hassan Shirdel 2, Majid Darehmiraki 3 1 Department of Electrical Engineering, Shohadaye Hoveizeh Campus of Technology, Shahid Chamran University of Ahvaz, Dasht-e Azadegan, Khuzestan, Iran; [email protected] 2 Department of Mathematic, Faculty of Basic Sciences, University of Qom, Qom, Iran; [email protected] 3 Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Behbahan, Khouzestan, Iran; [email protected] * Correspondence: [email protected]; Tel.: (optional; include country code) Abstract: Clustering, as one of the most basic data mining strategies, has a prominent role in various fields of application, especially decision making in management systems. Experts can make and implement decisions according to the features of each cluster by examining the characteristics, nature, and essence of the data that are in the same cluster. Since neutrosophic sets in various application fields have inspired tremendous research endeavors to model problems from varius of aspects, the main goal of this research is to combine hierarchical clustering with neutrosophic trapezoidal fuzzy data. For this purpose, a new distance measure is first introduced to calculate the difference between neutrosophic trapezoidal fuzzy data. In the following, while proving some characteristics of the measure, the hierarchical clustering algorithm based on the new distance measure with neutrosophic trapezoidal fuzzy data is explained. By extending continuous fuzzy data from an explanatory example in fuzzy literature, the effectiveness and efficiency of the proposed algorithm are tested in MATLAB software. Although the resulting dendrogram provides appropriate clustering to the decision maker, two criteria gap, and silhouette, are also used to determine the optimal number of clusters. The hybrid process developed in this research can not only be used in the study areas of clustering but also makes it possible to propose the optimal number of clusters for neutrosophic trapezoidal fuzzy data. Keywords: clustering; agglomerative hierarchical clustering; neutrosophic set; neutrosophic trapezoidal fuzzy number; distance measure 1. Introduction Clustering methods are recognized as efficient tools for extracting, analyzing, and identifying hidden patterns from existing data. The concept of clusters in clustering involves grouping together data or objects with the highest similarity (or least distance) within the group and the greatest dissimilarity (or least similarity) with members of other clusters. Therefore, measures of distance and similarity in the data space are crucial for the final clustering outcome. Various methods, including partitional and hierarchical clustering algorithms, are available for deterministic data, many of which are detailed in [1]. Neutrosophic Sets and Systems, Vol. 97, 2026 311 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num While numerous techniques for deterministic data clustering have been proposed theoretically, there has been relatively less research on non-deterministic data in this domain. With the introduction of fuzzy numbers by Zadeh [2] and their extensions, particularly Neutrosophic fuzzy numbers [3], researchers have become interested in their applications in scientific and engineering problems such as decision science, pattern recognition, medicine, networking, and cluster analysis. For example, Ye and Smarandache (2016) [4] developed measures such as Cosine, Jaccard, and Dice for multi-criteria decision-making scenarios using single-valued neutrosophic sets. Khalifa and Kumar (2020) [5] proposed a framework for the neutrosophic assignment problem under interval-valued trapezoidal neutrosophic numbers, transforming it into an interval-valued assignment problem using the score function and considering ordinal relationships based on decision makers' preferences to optimize the interval objective function. Additionally, Zulqarnain et al. [6] advocated for the use of fuzzy TOPSIS as a practical methodology for neutrosophic fuzzy TOPSIS in the context of supplier selection in the manufacturing sector. Haq et al. [7] developed a fuzzy multi-objective framework to solve complex multi-site hybrid supply chain problems under neutrosophic data, focusing on optimizing transportation cost and delivery time simultaneously. Poonia and Bajaj [8] devised an architecture of exponential-function-based similarity measures and weighted types for neutrosophic collections, applying these measures to pattern recognition and decision-making problems to demonstrate their effectiveness and computational manageability. The Kruskal approach for finding the minimum spanning tree in an undirected network with trapezoidal neutrosophic edge weights was introduced by Patro et al. in 2017 [9]. Subsequently, Broumi et al. [10] proposed hybrid structures and methods for addressing medical diagnosis issues under neutrosophic information to expedite disease identification and treatment. Recently, Jdid and Smarandache (2023) [11] developed a comprehensive model for optimal agricultural land distribution using neutrosophic science, asserting that neutrosophic values can more accurately represent environmental fluctuations, natural factors affecting production, and price fluctuations, all of which impact final profitability. Broumi et al. (2023) [12] formulated the shortest route problem based on the interval set of a Fermatean neutrosophic setting and proposed a new score function for its solution. Rosli et al. (2023) [13] visualized the neutrosophic Bézier curve (NBC) of the quartic version, demonstrating the approximation of the neutrosophic control point to the NBC and introducing its mathematical construction and logical properties. Research in the field of clustering methods with neutrosophic numbers, particularly continuous numbers, is much scarcer compared to other neutrosophic problems. Ye (2014) [14] proposed an extended distance measure for single-valued neutrosophic sets (SVNSs) and introduced two similarity measures based on this distance measure, leading to a practical clustering algorithm for SVN data. Long et al. [15] utilized the neutrosophic association matrix to propose a clustering approach, constructing a neutrosophic equivalent matrix and then deriving the lambda-cutting matrix. Zhang et al. [16] are among the authors who formulated a clustering model using neutrosophic datasets, considering both original clusters and noise clusters representing real and false memberships, respectively. They defined uncertainty, factoring in data density, and employed Lagrangian multipliers to further optimize the cost function under the neutrosophic dataset. Neutrosophic Sets and Systems, Vol. 97, 2026 312 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num Hierarchical clustering stands out among clustering methods for providing an optimal display of clusters through dendrograms, from the highest to the lowest possible number of clusters [17]. Despite being considered computationally intensive for extensive data, hierarchical clustering offers diverse cluster numbers, catering to expert preferences in data segmentation. Its advantages, including simplicity in utilizing distance and similarity criteria, accessibility, and flexibility, have sparked interest among researchers in expanding such algorithms. Consequently, this study aims to develop an agglomerative hierarchical clustering algorithm using trapezoidal fuzzy neutrosophic numbers. Geva (1999) [18] pioneered a fuzzy hierarchical algorithm, while Ghasemigol et al. (2010) [19] introduced a hierarchical clustering approach for fuzzy data clustering, incorporating dendrogram drawing and evaluating its performance against noisy data. Vandhana and Anuradha [20] employed fuzzy hierarchical clustering and neutrosophic fuzzy hierarchical clustering to identify dengue-affected foci in Sri Lanka, revealing clusters with varying risk levels of dengue attack. Elhassouny [21] outlined the steps of the DIANA hierarchical clustering algorithm using singlevalued neutrosophic sets (SVNs). In light of the indisputable role played by distance measures in data clustering methodologies, it becomes imperative to spearhead further research aimed at introducing innovative distance metrics tailored for neutrosophic trapezoidal fuzzy numbers. Although some clustering methods have been applied to different types of neutrosophic numbers, research specifically investigating hierarchical clustering on neutrosophic trapezoidal fuzzy numbers remains scarce. Thus, the primary motivation of this research is to introduce a distance measure between neutrosophic trapezoidal fuzzy numbers based on a novel structure capable of effectively facilitating hierarchical clustering. To illustrate this, we implemented the developed hierarchical clustering method on a set of neutrosophic trapezoidal fuzzy numbers. Furthermore, in the comparative analysis, while evaluating the performance of the proposed algorithm against existing measures, we also assess its performance based on a linear combination of these measures. To proceed, the remainder of this paper is structured as follows: In Sect. 2, we gather some preliminaries and notations regarding neutrosophic sets and neutrosophic trapezoidal fuzzy numbers (NTraFNs). In Sect. 3, we introduce the novel structure and its main properties. In Sect. 4, we present the agglomerative hierarchical clustering method based on the proposed distance within the NTraFNs dataset. We provide illustrative examples, discuss the determination of the optimal number of clusters, and conduct sensitivity analyses. Furthermore, in Sect. 5, we conduct a comparative study and discuss the advantages of the suggested approach. Finally, Sect. 6 includes some concluding remarks and outlines prospects for further research. 2. Preliminaries In this section, we will recall the preliminary concepts about Neutrosophic sets and neutrosophic trapezoidal fuzzy numbers (NTraFNs) and notations that will be used often in the rest of the paper. Definition 1 (Samarandache 1999). Let a crisp set 𝑀 be fixed. A neutrosophic set 𝑁  in 𝑀 is an object of the following representation [3]: 𝑁  ={⟨𝑥,𝜇𝑁 (𝑥),π𝑁 (𝑥),𝜌𝑁 (𝑥)⟩| 0≤𝜇𝑁 (𝑥),π𝑁 (𝑥),𝜌𝑁 (𝑥)≤1,𝑥∈𝑀}, (1) Where functions 𝜇𝑁 ,π𝑁 ,𝜌𝑁 :M→[0,1] are the degree of truth-membership, indeterminacymembership, and falsity-membership of the 𝑥∈𝑀 to 𝑁 . Furthermore, for every 𝑥∈𝑀, 0≤ 𝜇𝑁 (𝑥)+π𝑁 (𝑥)+𝜌𝑁 (𝑥)≤3. Neutrosophic Sets and Systems, Vol. 97, 2026 313 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num Definition 2 [22]. Let a crisp set 𝑀 be fixed, 𝑛=⟨𝜇𝑁 (𝑥),π𝑁 (𝑥),𝜌𝑁 (𝑥)⟩ is a neutrosophic fuzzy number in the set of 𝑀. Then its truth membership function is defined as 𝜇𝑛(𝑥)= { 𝜇𝑛 𝑙(𝑥), 𝑎1≤𝑥≤𝑎2 1, 𝑎2≤𝑥≤𝑎3 μ𝑛 𝑟(𝑥), 𝑎3≤𝑥≤𝑎4 0, 𝑜.𝑤. (2) The membership function of indeterminacy is defined as 𝜋𝑛(𝑥)= { π𝑛 𝑙(𝑥), 𝑏1≤𝑥≤𝑏2 0, 𝑏2≤𝑥≤𝑏3 π𝑛 𝑟(𝑥), 𝑏3≤𝑥≤𝑏4 1, 𝑜.𝑤. (3) Furthermore, the falsehood membership function is defined as 𝜌𝑛(𝑥)= { 𝜌𝑛𝑙(𝑥), 𝑐1≤𝑥≤𝑐2 0, 𝑐2≤𝑥≤𝑐3 𝜌𝑛𝑟(𝑥), 𝑐3≤𝑥≤𝑐4 1, 𝑜.𝑤, (4) where 0≤𝜇𝑛(𝑥),π𝑛(𝑥),𝜌𝑛(𝑥)≤1 and 0≤𝜇𝑛(𝑥)+π𝑛(𝑥)+𝜌𝑛(𝑥)≤3. Definition 3 [3]. Let 𝑛1and 𝑛2 be two neutrosophic fuzzy numbers given as 𝑛1= ⟨𝜇𝑛1(𝑥),π𝑛1(𝑥),𝜌𝑛1(𝑥)⟩ and 𝑛2=⟨𝜇𝑛2(𝑥),π𝑛2(𝑥),𝜌𝑛2(𝑥)⟩. Then, 𝑛1⊆𝑛2 if and only if 𝜇𝑛1(𝑥)≤𝜇𝑛2(𝑥),π𝑛1(𝑥)≥π𝑛2(𝑥),𝜌𝑛1(𝑥)≥𝜌𝑛2(𝑥) ,𝑓𝑜𝑟 every 𝑥∈M Definition 4 [22]. Let a crisp set 𝑀 be fixed, 𝑛=⟨(𝑎1,𝑎2,𝑎3,𝑎4),(𝑏1,𝑏2,𝑏3,𝑏4),(𝑐1,𝑐2,𝑐3,𝑐4)⟩ is a neutrosophic trapezoidal fuzzy number of 𝑀. Then its truth-membership function is defined as 𝜇𝑛(𝑢)= { (𝑥−𝑎1) 𝑎2−𝑎1, 𝑎1≤𝑥<𝑎2 1, 𝑎2≤𝑥≤𝑎3 (𝑎4−𝑥) 𝑎4−𝑎3, 𝑎3≤𝑥<𝑎4 0, 𝑜.𝑤. (5) The membership function of indeterminacy is defined as π𝑛(𝑥)= { (𝑏2−𝑥) 𝑏2−𝑏1, 𝑏1≤𝑥<𝑏2 0, 𝑏2≤𝑥<𝑏3 (𝑥−𝑏3) 𝑏4−𝑏3, 𝑏3≤𝑥<𝑏4 0, 𝑜.𝑤, (6) Furthermore, the falsehood membership function is defined as 𝜌𝑛(𝑥)= { (𝑐2−𝑥) 𝑐2−𝑐1, 𝑐1≤𝑥<𝑐2 0, 𝑐2≤𝑥<𝑐3 (𝑥−𝑐3) 𝑐4−𝑐3, 𝑐3≤𝑥<𝑐4 0, 𝑜.𝑤, (7) Neutrosophic Sets and Systems, Vol. 97, 2026 314 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num Where 0≤𝜇𝑛(𝑥),π𝑛(𝑥),𝜌𝑛(𝑥)≤1 and 0≤𝜇𝑛(𝑥)+π𝑛(𝑥)+𝜌𝑛(𝑥)≤3. 3. Proposed distance measure for neutrosophic trapezoidal fuzzy numbers In this section, we will conduct a comprehensive survey of the novel distance measure, its structure, and its properties. 3.1. Structure Here, we will to design a meaningful scheme to model the distance measure between two neutrosophic trapezoidal fuzzy numbers. The primary idea of this subsection is based on the ranking method provided by Abbasi and Darehmiraki [23]. Assume 𝑛𝑖=⟨(𝑎1𝑛𝑖,𝑎2𝑛𝑖,𝑎3𝑛𝑖,𝑎4𝑛𝑖),(𝑏1𝑛𝑖,𝑏2𝑛𝑖,𝑏3𝑛𝑖,𝑏4𝑛𝑖),(𝑐1𝑛𝑖,𝑐2𝑛𝑖,𝑐3𝑛𝑖,𝑐4𝑛𝑖)⟩, be a neutrosophic trapezoidal fuzzy number. The left and right line formulas of the truth-membership function, the complement indeterminacy-membership function, and the complement falsitymembership function can be considered for 𝑛𝑖 to the vertical line 𝜔=0, respectively, as follows: [𝜎𝑙𝑛 𝑖(ω),𝜎𝑟𝑛 𝑖(ω)]=[(𝑎1𝑛𝑖)+𝜔(𝑎2𝑛𝑖−𝑎1𝑛𝑖),(𝑎4𝑛𝑖)+ω(𝑎3𝑛𝑖−𝑎4𝑛𝑖)], (8) [𝜏𝑙𝑛 𝑖(ω),𝜏𝑟𝑛 𝑖(ω)]=[(𝑏1𝑛𝑖)+ω(𝑏2𝑛𝑖−𝑏1𝑛𝑖),(𝑏4𝑛𝑖)+ω(𝑏3𝑛𝑖−𝑏4𝑛𝑖)], (9) [𝜑𝑙𝑛 𝑖(ω),𝜑𝑟𝑛 𝑖(ω)]=[(𝑐1𝑛𝑖)+ω(𝑐2𝑛𝑖−𝑐1𝑛𝑖),(𝑐4𝑛𝑖)+ω(𝑐3𝑛𝑖−𝑐4𝑛𝑖)]. (10) In this sense, the area of the left and right half of the truth-membership interval respective to the 𝜔=0 can state: 𝒮𝑙𝜇(𝑛𝑖)=∫ ((𝑎1𝑛𝑖)+𝜔(𝑎2𝑛𝑖−𝑎1𝑛𝑖))𝑑𝜔 1 0=(𝑎1𝑛𝑖)+(𝑎2𝑛𝑖−𝑎1𝑛𝑖) 2, (11) 𝒮𝑟𝜇(𝑛𝑖)=∫ ((𝑎4𝑛𝑖)+ω(𝑎3𝑛𝑖−𝑎4𝑛𝑖))𝑑𝜔 1 0=(𝑎4𝑛𝑖)+(𝑎3𝑛𝑖−𝑎4𝑛𝑖) 2. (12) The area of the left and right half of the indeterminacy-membership interval respective to the 𝜔=0 can state: 𝒮𝑙𝜋(𝑛𝑖)=∫ (𝑏1𝑛𝑖)+𝜔(𝑏2𝑛𝑖−𝑏1𝑛𝑖))𝑑𝜔 1 0=(𝑏1𝑛𝑖)+(𝑏2𝑛𝑖−𝑏1𝑛𝑖) 2, (13) 𝒮𝑟π(𝑛𝑖)=∫ ((𝑏4𝑛𝑖)+ω(𝑏3𝑛𝑖−𝑏4𝑛𝑖))𝑑𝜔 1 0=(𝑏4𝑛𝑖)+(𝑏3𝑛𝑖−𝑏4𝑛𝑖) 2. (14) Furthermore, the area of the left and right half of the falsity-membership interval respective to the 𝜔=0 can state: 𝒮𝑙𝜌(𝑛𝑖)=∫ ((𝑐1𝑛𝑖)+𝜔(𝑐2𝑛𝑖−𝑐1𝑛𝑖))𝑑𝜔 1 0= (𝑐1𝑛𝑖)+(𝑐2𝑛𝑖−𝑐1𝑛𝑖) 2, (15) 𝒮𝑟𝜌(𝑛𝑖)=∫ ((𝑐4𝑛𝑖)+ω(𝑐3𝑛𝑖−𝑐4𝑛𝑖))𝑑𝜔 1 0=(𝑐4𝑛𝑖)+(𝑐3𝑛𝑖−𝑐4𝑛𝑖) 2. (16) Based on the relations 11-16, the score functions corresponding to the truth-membership function, indeterminacy-membership function, and falsity-membership function are: ℛ𝜇(𝑛𝑖)=1 2{𝒮𝑙𝜇(𝑛𝑖)+𝒮𝑟𝜇(𝑛𝑖)}=1 2{(𝑏1𝑛𝑖)+(𝑏2𝑛𝑖−𝑏1𝑛𝑖) 2+(𝑎4𝑛𝑖)+(𝑎3𝑛𝑖−𝑎4𝑛𝑖) 2}, (17) ℛπ(𝑛𝑖)=1 2{𝒮𝑙π(𝑛𝑖)+𝒮𝑟π(𝑛𝑖)}=1 2{(𝑎1𝑛𝑖)+(𝑎2𝑛𝑖−𝑎1𝑛𝑖) 2+(𝑏4𝑛𝑖)+(𝑏3𝑛𝑖−𝑏4𝑛𝑖) 2}, (18) ℛ𝜌(𝑛𝑖)=1 2{𝒮𝑙𝜌(𝑛𝑖)+𝒮𝑟𝜌(𝑛𝑖)}=1 2{(𝑐1𝑛𝑖)+(𝑐2𝑛𝑖−𝑐1𝑛𝑖) 2+(𝑐4𝑛𝑖)+(𝑐3𝑛𝑖−𝑐4𝑛𝑖) 2}. (19) Neutrosophic Sets and Systems, Vol. 97, 2026 315 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num Let 𝑛𝑖=⟨(𝑎1𝑛𝑖,𝑎2𝑛𝑖,𝑎3𝑛𝑖,𝑎4𝑛𝑖),(𝑏1𝑛𝑖,𝑏2𝑛𝑖,𝑏3𝑛𝑖,𝑏4𝑛𝑖),(𝑐1𝑛𝑖,𝑐2𝑛𝑖,𝑐3𝑛𝑖,𝑐4𝑛𝑖)⟩ and 𝑛𝑗=⟨(𝑎1𝑛𝑗,𝑎2𝑛𝑗,𝑎3𝑛𝑗,𝑎4𝑛𝑗),(𝑏1𝑛𝑗,𝑏2𝑛𝑗,𝑏3𝑛𝑗,𝑏4𝑛𝑗),(𝑐1𝑛𝑗,𝑐2𝑛𝑗,𝑐3𝑛𝑗,𝑐4𝑛𝑗)⟩ are two neutrosophic trapezoidal fuzzy numbers. Then our proposed distance measure defines as 𝐷𝑖𝑠(𝑛𝑖,𝑛𝑗)= 1 √3{ (ℛ𝜇(𝑛𝑖)−ℛ𝜇(𝑛𝑗))2+(ℛπ(𝑛𝑖)−ℛπ(𝑛𝑗))2+(ℛ𝜌(𝑛𝑖)−ℛ𝜌(𝑛𝑗))2 }12 ⁄. (20) 3.2. Theorems and properties Now, the validity and reasonable structure of the suggested distance measure are proven according to the following theorem. Theorem1: Show the proposed distance measure in equation (20) for all 𝑛1, 𝑛2 and 𝑛3 of a dataset, including the NTraFNs are satisfied following principles: 1) 𝐷𝑖𝑠(𝑛1,𝑛2) ϵ [0,1] 2) 𝑛1 =𝑛2⟹𝐷𝑖𝑠(𝑛1,𝑛2)=0 3) 𝐷𝑖𝑠(𝑛1,𝑛2)=𝐷𝑖𝑠(𝑛2,𝑛1) 4) If 𝑛1⊆𝑛2⊆𝑛3⟹𝐷𝑖𝑠(𝑛1,𝑛2)≤𝐷𝑖𝑠(𝑛1,𝑛3) and 𝐷𝑖𝑠(𝑛1,𝑛2)≤𝐷𝑖𝑠(𝑛1,𝑛3). Proof: 1. Since 0≤(ℛ𝜇(𝑛1)−ℛ𝜇(𝑛2))2≤1,0≤(ℛπ(𝑛1)−ℛπ(𝑛2))2≤1,0≤(ℛ𝜌(𝑛1)−ℛ𝜌(𝑛2))2≤1, Certainly, we have: 0≤ 𝐷𝑖𝑠(𝑛1,𝑛2)≤1 2. If 𝑛1=𝑛2, then ℛ𝜇(𝑛1)=ℛ𝜇(𝑛2), ℛπ(𝑛1)=ℛπ(𝑛2), ℛ𝜌(𝑛1)=ℛ𝜌(𝑛2) So, 𝐷𝑖𝑠(𝑛1,𝑛2)=0. 3. For every 𝑛1 and 𝑛2, Since the each of the expressions in (20) is positive value. We have: 𝐷𝑖𝑠(𝑛1,𝑛2)=1 √3{ (ℛ𝜇(𝑛1)−ℛ𝜇(𝑛2))2+(ℛπ(𝑛1)−ℛπ(𝑛2))2+(ℛ𝜌(𝑛1)−ℛ𝜌(𝑛2))2 }12 ⁄ = 1 √3{ (ℛ𝜇(𝑛2)−ℛ𝜇(𝑛1))2+(ℛπ(𝑛2)−ℛπ(𝑛1))2+(ℛ𝜌(𝑛2)−ℛ𝜌(𝑛1))2 }12 ⁄ = 𝐷𝑖𝑠(𝑛2,𝑛1). 4. If 𝑛1⊆𝑛2⊆𝑛3, then: (ℛ𝜇(𝑛1)−ℛ𝜇(𝑛2))≤(ℛ𝜇(𝑛1)−ℛ𝜇(𝑛3)), (ℛπ(𝑛1)−ℛπ(𝑛2))≤(ℛπ(𝑛1)−ℛπ(𝑛3)), (ℛ𝜌(𝑛1)−ℛ𝜌(𝑛2))≤(ℛ𝜌(𝑛1)−ℛ𝜌(𝑛3)). Hence 1 √3{ (ℛ𝜇(𝑛1)−ℛ𝜇(𝑛2))2+(ℛπ(𝑛1)−ℛπ(𝑛2))2+(ℛ𝜌(𝑛1)−ℛ𝜌(𝑛2))2 }12 ⁄≤ 1 √3{ (ℛ𝜇(𝑛1)−ℛ𝜇(𝑛3))2+(ℛπ(𝑛1)−ℛπ(𝑛3))2+(ℛ𝜌(𝑛1)−ℛ𝜌(𝑛3))2 }12 ⁄, Therefore 𝐷𝑖𝑠(𝑛1,𝑛2)≤ 𝐷𝑖𝑠(𝑛1,𝑛3). Similarly, 𝐷𝑖𝑠(𝑛2,𝑛3)≤𝐷𝑖𝑠(𝑛1,𝑛3) also holds. Property 1: If 𝑛1=⟨(𝑎,𝑎,𝑎,𝑎),(𝑎,𝑎,𝑎,𝑎),(𝑎,𝑎,𝑎,𝑎)⟩ and 𝑛2=⟨(𝑏,𝑏,𝑏,𝑏),(𝑏,𝑏,𝑏,𝑏),(𝑏,𝑏,𝑏,𝑏)⟩ then, 𝐷𝑖𝑠(𝑛1,𝑛2)=|𝑎−𝑏|. Property 2: If 𝑛1=⟨(𝑎,𝑎,𝑎,𝑎),(𝑎,𝑎,𝑎,𝑎),(𝑎,𝑎,𝑎,𝑎)⟩ and 𝑛2=⟨(0,0,0,0),(0,0,0,0),(0,0,0,0)⟩ then, 𝐷𝑖𝑠(𝑛1,𝑛2)=𝑎. Neutrosophic Sets and Systems, Vol. 97, 2026 316 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num Property 3: If 𝑛1=⟨(𝑎,𝑎,𝑎,𝑎),(𝑎,𝑎,𝑎,𝑎),(𝑎,𝑎,𝑎,𝑎)⟩ and 𝑛2=⟨(1,1,1,1),(1,1,1,1),(1,1,1,1)⟩ then, 𝐷𝑖𝑠(𝑛1,𝑛2)=|1−𝑎|. Example 1: Consider the following TrNFNs. 𝑛1=⟨(0,0,0,0),(0,0,0,0),(0,0,0,0)⟩,𝑛2= ⟨(1,1,1,1),(1,1,1,1),(1,1,1,1)⟩, 𝑛3=⟨(0.5,0.5,0.5,0.5),(0.5,0.5,0.5,0.5),(0.5,0.5,0.5,0.5)⟩, 𝑛4= ⟨(0.8,0.8,0.8,0.8),(0.8,0.8,0.8,0.8),(0.8,0.8,0.8,0.8)⟩. Distance measures; 𝐷𝑖𝑠(𝑛𝑖,𝑛𝑗) for every 𝑖,𝑗=1,2,3,4 calculated in Table 1. Table 1. Distance measures of Ex 1. 𝒏 𝟒 𝒏 𝟑 𝒏 𝟐 𝒏 𝟏 𝑫𝒊𝒔(𝒏 𝒊,𝒏 𝒋) 0.8000 0.5000 1.0000 0 𝑛1 0.2000 0.5000 0 1.0000 𝑛2 0.3000 0 0.5000 0.5000 𝑛3 0 0.3000 0.2000 0.8000 𝑛4 The results of Table 1 show the logical performance related to the properties of the proposed distance measure. 4. Agglomerative hierarchical clustering (AHC) under neutrosophic trapezoidal fuzzy numbers Hierarchical clustering methods that use a dendrogram tree to depict the status of data relative to each other at different levels are known as the most widely used methods in clustering. Therefore, while providing a proper picture of the data placement in the clusters, it reveals a proper picture of the sub-clusters at each stage. Two general views, bottom-up and top-down, are known as the main approaches in hierarchical clustering. In the implementation mechanism of this method, based on the first approach, each data point is assigned to a cluster. Then the pairs of clusters that are closest to each other are combined. How to combine the clusters with spare parts will be influential in determining the final shape of the cluster. Some of these criteria to determine the distance between two clusters are: Single-link method: The similarity or closeness between two clusters is measured based on the smallest distance between the members of the two clusters. In this case, for two clusters, we have: 𝐷𝑖𝑠(𝐶𝑖,𝐶𝑗)= 𝑚𝑖𝑛 𝑎∈𝐶𝑖 , 𝑏∈𝐶𝑗𝐷𝑖𝑠(𝑎,𝑏) (21) Complete-link method: The similarity or closeness between two clusters is measured based on the highest distance between the members of the two clusters. In this case, for two clusters, we have: 𝐷𝑖𝑠(Ci,C𝑗)= max 𝑎∈Ci , 𝑏∈C𝑗𝐷𝑖𝑠(a,b) (22) Average-link method: The similarity or closeness between two clusters is measured based on the unweighted average distance between the members of the two clusters. In this case, for two clusters, we have: 𝐷𝑖𝑠(Ci,C𝑗)= 1 |Ci||Cj|∑ 𝐷𝑖𝑠(a,b) 𝑎∈Ci , 𝑏∈C𝑗 (23) Our goal in this part is to present a hybrid algorithm based on the first approach. The details of the second approach can be found in [24]. 4.1. AHC algorithem under neutrosophic trapezoidal fuzzy numbers Neutrosophic Sets and Systems, Vol. 97, 2026 317 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num Clustering is the separation of a heterogeneous group of data into several clusters, with the goal of maximizing the inter-cluster difference and minimizing the intra-cluster difference. In the AHC algorithm, each data point is placed in a separate cluster at first. Then, according to the dispersion of the data, the two closest clusters (according to the selected distance comparison) are merged. This process continues until all the data are in one cluster. The steps of the AHC procedure under the NtraFN dataset are described as follows: Step 1: Record and identify the elements of NtraFN dataset 𝐷𝑁={𝑛1,𝑛2,…,𝑛𝑚} Where 𝑛𝑖=⟨(𝑎1𝑛𝑖,𝑎2𝑛𝑖,𝑎3𝑛𝑖,𝑎4𝑛𝑖),(𝑏1𝑛𝑖,𝑏2𝑛𝑖,𝑏3𝑛𝑖,𝑏4𝑛𝑖),(𝑐1𝑛𝑖,𝑐2𝑛𝑖,𝑐3𝑛𝑖,𝑐4𝑛𝑖)⟩and 𝑚 denote the number of data. Step 2: Set 𝛿= max 𝑙=1,2,3,4 𝑖=1,2,…,𝑚{𝑎𝑙𝑛𝑖,𝑏𝑙𝑛𝑖,𝑐𝑙𝑛𝑖}, the normalized NTraF-dataset denote by 𝐷 𝑁 which each element obtain as 𝑛𝑖𝑛𝑜𝑟=𝑛𝑖/𝛿=⟨(𝑎1𝑛𝑖,𝑎2𝑛𝑖,𝑎3𝑛𝑖,𝑎4𝑛𝑖),(𝑏1𝑛𝑖,𝑏2𝑛𝑖,𝑏3𝑛𝑖,𝑏4𝑛𝑖),(𝑐1𝑛𝑖,𝑐2𝑛𝑖,𝑐3𝑛𝑖,𝑐4𝑛𝑖)⟩/𝛿. Step 3: Transform 𝐷 𝑁 to 𝐷 𝑁 𝑅 such that: 𝐷 𝑁 𝑅={ℛ(𝑛1𝑛𝑜𝑟),ℛ(𝑛2 𝑛𝑜𝑟),…,ℛ(𝑛𝑚 𝑛𝑜𝑟)}. And ℛ(𝑛𝑖𝑛𝑜𝑟)=⟨ℛ𝜇(𝑛𝑖𝑛𝑜𝑟),ℛπ(𝑛𝑖𝑛𝑜𝑟),ℛ𝜌(𝑛𝑖𝑛𝑜𝑟)⟩ for every 𝑖𝜖{1,2,…,𝑚}. Step 4: Place each element of 𝐷 𝑁 𝑅 in a separate cluster. So, the number of clusters is 𝑘=𝑚. Step 5: Form the Neutrosophic distance matrix, between clusters according to equation (20) as follows [𝐷𝑖𝑠(𝑛𝑖,𝑛𝑗)]𝑚∗𝑚 Step 6: To choose the shortest distance between two clusters for merging, select (fix), and use one of the criteria presented in equations 21, 22 and 23. Step 7: Put 𝑘=𝑚−1. Then, repeat steps 5 to 7 until we reach k=1. Note: In big data, for the termination condition, k=t can be considered as the optimal number of clusters (1≤t≤m). 4.2. Illustrative example In this part, the experimental study is conducted using the previously suggested method. A typical example includes collecting trapezoidal fuzzy numbers, which is used for clustering methods in [19]. We, have 𝐷𝐹={𝑓󰆻1,𝑓󰆻2,𝑓󰆻3,𝑓󰆻4,𝑓󰆻5,𝑓󰆻6}={⟨(1.0,1.5,3.5,4.5)⟩,⟨(1.5,2.5,2.5,3.5)⟩,⟨(6.0,6.5,7.5,8.5)⟩, ⟨(7.0,8.0,8.0,9.0)⟩,⟨(8.0,8.5,8.5,9.0)⟩,⟨(3.5,4.5,4.5,5.5)⟩}. Due to the intrinsic limitations of NTraF data, Without making a noticeable change in the distribution of fuzzy data from 𝐷𝐹, by applying the following equation 𝑓󰆻𝑖=⟨(𝑎1𝑛𝑖,𝑎2𝑛𝑖,𝑎3𝑛𝑖,𝑎4𝑛𝑖)⟩ so 𝑛𝑖=⟨(𝑎1𝑛𝑖,𝑎2𝑛𝑖,𝑎3𝑛𝑖,𝑎4𝑛𝑖),(𝑎1𝑛𝑖−0.1,𝑎2𝑛𝑖−0.1,𝑎3𝑛𝑖−0.1,𝑎4𝑛𝑖−0.1), (𝑎1𝑛𝑖+0.1,𝑎2𝑛𝑖+0.1,𝑎3𝑛𝑖+0.1,𝑎4𝑛𝑖+0.1) ⟩ (24) We can construct the following NTraF-dataset as follows 𝐷𝑁={𝑛1,𝑛2,𝑛3,𝑛4,𝑛5,𝑛6} ={⟨(1.0,1.5,3.5,4.5),(0.9,1.4,3.4,4.4),(1.1,1.6,3.6,4.6)⟩,⟨(1.5,2.5,2.5,3.5),(1.4,2.4,2.4,3.4),(1.6,2.6,2.6,3.6)⟩, ⟨(6.0,6.5,7.5,8.5),(5.9,6.4,7.4,8.4),(6.1,6.6,7.6,8.6)⟩,⟨(7.0,8.0,8.0,9.0),(6.9,7.9,7.9,8.9),(7.1,8.1,8.1,9.1)⟩, Neutrosophic Sets and Systems, Vol. 97, 2026 318 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num ⟨(8.0,8.5,8.5,9.0),(7.9,8.4,8.4,8.9),(8.1,8.6,8.6,9.1)⟩,⟨(3.5,4.5,4.5,5.5),(3.4 ,4.4,4.4,5.4),(3.6,4.6,4.6,5.6)⟩} In Figure 1 a, b, the elements of 𝐷𝐹 and 𝐷𝑁 is represented. From step 2, since max 𝑙=1,2,3,4 𝑖=1,2,…,6{𝑎𝑙𝑛𝑖,𝑏𝑙𝑛𝑖,𝑐𝑙𝑛𝑖}=9.1 , the normalized NTraF-dataset is obtained as 𝐷 𝑁= {⟨(0.1099,0.1648,0.3846,0.4945),(0.0989,0.1538,0.3736,0.4835),(0.1209,0.1758,0.3956,0.5055)⟩, ⟨(0.1648,0.2747,0.2747,0.3846),(0.1538,0.2637,0.2637,0.3736),(0.1758,0.2857,0.2857,0.3956)⟩, ⟨(0.6593,0.7143,0.8242,0.9341),(0.6484,0.7033,0.8132,0.9231),(0.6703,0.7253,0.8352,0.9451)⟩, ⟨(0.7692,0.8791,0.8791,0.9890),(0.7582,0.8681,0.8681,0.9780),(0.7802,0.8901,0.8901,1.0000)⟩, ⟨(0.8791,0.9341,0.9341,0.9890),(0.8681,0.9231,0.9231,0.9780),(0.8901,0.9451,0.9451,1.0000)⟩, ⟨(0.3846,0.4945,0.4945,0.6044),(0.3736,0.4835,0.4835,0.5934),(0.3956,0.5055,0.5055,0.6154)⟩}. For each element of the 𝐷 𝑁, we calculate the triple ranks corresponding to the truth-membership, indeterminacy-membership, and falsity-membership functions. Triple-ranks replace the previous elements of 𝐷 𝑁. The new set is named 𝐷 𝑁 𝑅 with the following representation. 𝐷 𝑁 𝑅={⟨ 0.2885 ,0.3187 ,0.2995⟩,⟨0.2747 ,0.2363,0.2857⟩,⟨0.7830,0.7857,0.7940⟩, ⟨0.8791,0.8407 ,0.8901⟩,⟨0.9341,0.9093 ,0.9451⟩,⟨0.4945 ,0.4560,0.5055⟩}. From step 4, and equation 20 the distance measure matrix is obtained, which is shown in Table 2. Table 2. Distance measure matrix of illustrative example (k=6). 𝑛6 𝑛5 𝑛4 𝑛3 𝑛2 𝑛1 𝐷𝑖𝑠(𝑛𝑖,𝑛𝑗) 0.1860 0.6287 0.5687 0.4855 0.0489 0 𝑛1 0.2198 0.6640 0.6044 0.5223 0 0.0489 𝑛2 0.3028 0.1425 0.0847 0 0.5223 0.4855 𝑛3 0.3864 0.0599 0 0.0847 0.6044 0.5687 𝑛4 0 . 4442 0 0 . 0599 0 . 1425 0.6640 0.6287 𝑛5 0 0 . 4442 0 . 3846 0 . 3028 0.2198 0.1860 𝑛6 The distance results of Table 2 show 𝐷𝑖𝑠(𝑛1,𝑛2)=0.0489 has the smallest value among all distances. So, 𝑛1 and 𝑛2 are placed together in one cluster. The information of the new distance Figure 1. a) The elements of 𝐷𝐹 (TraF-dataset), b) The elements of 𝐷𝑁 (NTraF-dataset). Neutrosophic Sets and Systems, Vol. 97, 2026 325 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num From the information in Table 12, it can be concluded that the hierarchical clustering method based on the 𝐷𝑖𝑠𝐵(𝑛1,𝑛2) and combination of distance measures has the same output as the proposed method in subsection 4.2. In addition, the results of clustering based on the 𝐷𝑖𝑠𝑡𝐻(𝑛1,𝑛2) and 𝐷𝑖𝑠𝑡𝐸(𝑛1,𝑛2) are different from other approaches only in cluster 5. 5.2. Managerial implications and advantages Clustering serves as a conduit for encapsulating latent knowledge within the final categories. Each cluster's output embodies characteristics closely intertwined within the cluster's data, offering invaluable insights that can inform organizational actions such as marketing strategies, customer relationship enhancement, resource allocation, and investment decisions. This act of data labeling facilitates precise managerial decisions by elucidating internal data patterns. By delving into the intricate relationships within the data, managers gain access to more reasoned options, thereby enhancing organizational performance. Additionally, decision-makers can harness the strengths of various distance metrics simultaneously by amalgamating the output of the distance matrix, thus yielding more accurate results. For instance, in the context of customer data, appropriate clustering enables organizations to devise effective promotional and incentive strategies tailored to each cluster, fostering customer retention, satisfaction, and acquisition. Among the myriad management advantages offered by hierarchical clustering methods, one standout feature is the decision-maker's autonomy in determining the final number of clusters. This flexibility allows for adjustments in resource allocation based on fluctuating resource availability (be it human, financial, or temporal), enabling organizations to tailor their goals accordingly. As previously highlighted, existing distance metrics may not fully capture all the nuances inherent in NTraFNs. Hence, in Section 3, we introduce a novel distance measure based on vertical axis levels. Some key advantages of this distance measure include: - Intuitive and logical interpretation of NTraFN dimensions. - Theorems and properties outlined in subsection 3.2 affirm the accuracy of the proposed distance measure. - Simplicity in calculations facilitates its applicability in diverse distanceor similarity-based problems. - Combined with existing distance measures, the proposed measure yields consistent clustering outputs, offering decision-makers confidence in their analytical endeavors. 6. Conclusions In this study, we introduced a novel and efficacious conceptual distance measure for comparing NTraFNs. The validity of the proposed distance measure structure was substantiated through the verification of its underlying principles and logical properties. Furthermore, the performance of the clustering algorithm, predicated on this distance measure, underscores the practical utility of our research. Among the limitations encountered in this study is the paucity of a comprehensive database of NTraFNs, which hampers our ability to tackle problems with larger dimensions. Nonetheless, the dearth of research pertaining to data mining under NTraFNs, coupled with the significance of information theory, underscores the importance and continued exploration of this field. Several avenues for future research warrant consideration: Neutrosophic Sets and Systems, Vol. 97, 2026 326 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num - Developing similarity measures and entropy metrics based on the proposed distance measure between NTraFNs. - Generalizing the conceptual framework to encompass other distance, similarity, and entropy measures. - Integrating the suggested distance measure with alternative clustering methodologies. - Employing the proposed distance measure to address challenges associated with largerdimensional problems and other distance-based predicaments. - These avenues not only hold promise for advancing our understanding of NTraFNs but also offer opportunities to enhance the efficacy and versatility of clustering methodologies in diverse problem domains. References 1. Gan, G.; Ma, C.; Wu, J. Data clustering: theory, algorithms, and applications, 2nd ed.; Society for Industrial and Applied Mathematics: 3600 Market Street, 6th Floor Philadelphia, USA, 2020. 2. Zadeh, L.A. Fuzzy sets. Information and control 1965, 8, 338-353. 3. Smarandache, RF. A unifying field in logics. In: Neutrosophy. Neutrosophic probability, set and logic. American Research Press, Rehoboth 1999, 7-8. 4. Ye, J.; Smarandache, F. Similarity measure of refined single-valued neutrosophic sets and its multicriteria decision making method. Neutrosophic Sets and Systems 2016, 12, 41–44. 5. Khalifa, H.A.E.W. and Kumar, P., A novel method for neutrosophic assignment problem by using intervalvalued trapezoidal neutrosophic number (Vol. 36). Infinite Study 2020. 6. Zulqarnain, R. M.; Xin, X. L.; Saeed, M.; Smarandache, F.; Ahmad, N. Generalized neutrosophic TOPSIS to solve multi-criteria decision-making problems. Neutrosophic Sets and Systems 2020, 38, 276-292. 7. Haq, A.; Gupta, S.; Ahmed, A. A multi-criteria fuzzy neutrosophic decision-making model for solving the supply chain network problem. Neutrosophic Sets and Systems 2021, 46, 50-66. 8. Poonia, M.; Bajaj, R.K. On Measures of Similarity for Neutrosophic Sets with Applications in Classification and Evaluation Processes. Neutrosophic Sets and Systems 2021, 39, 86-100. 9. Patro, Santanu & Said, Broumi & Smarandache, Florentin & Talea, Mohamed & Bakali, Assia. (2017). Minimum Spanning tree problem with single valued Trapezoidal neutrosophic number. IEEE Technically Sponsored Computing Conference 2018. At: London, UK. 10. Broumi, S.; Dhar, M.; Bakhouyi, A.; Bakali, A.; Talea, M. Medical diagnosis problems based on neutrosophic sets and their hybrid structures: A survey. Neutrosophic Sets and Systems 2022, 49, 1-18. 11. Jdid, Maissam & Smarandache, Florentin. (2023). Optimal Agricultural Land Use: An Efficient Neutrosophic Linear Programming Method. Neutrosophic Systems with Applications. 10. 53-59. 10.61356/j.nswa.2023.76. 12. Broumi, S.; Prabha, S.; Krishna.; Uluçay, Vakkas. (2023). Interval-Valued Fermatean Neutrosophic Shortest Path Problem via Score Function. Neutrosophic Systems with Applications. 11. 1-10. 10.61356/j.nswa.2023.83. 13. Rosli, Siti Nur & Zulkifly, Mohammad Izat Emir. (2023). 3-Dimensional Quartic Bézier Curve Approximation Model by Using Neutrosophic Approach. Neutrosophic Systems with Applications. 11. 11-21. 10.61356/j.nswa.2023.78. 14. Ye, J. Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. Journal of Intelligent Systems 2014, 23, 379-389. 15. Long, H.V.; Ali, M.; Khan, M.; Tu, D.N. A novel approach for fuzzy clustering based on neutrosophic association matrix. Computers and Industrial Engineering 2019, 127, 687-697. 16. Zhang, D.; Ma, Y.; Dai, X.; Qiao, Y. Clustering algorithm based on data indeterminacy in neutrosophic set. Neutrosophic Sets and Systems 2022, 51, 556-569. 17. Johnson, S.C.; Hierarchical clustering schemes, Psychometrika 1967, 32, 241–254. 18. Geva, A.B. Hierarchical unsupervised fuzzy clustering, IEEE Transactions on Fuzzy Systems 1999, 7, 723-733. 19. Ghasemigol, M.; Sadoghi Yazdi, H.; Monsefi, R. A new hierarchical clustering algorithm on fuzzy data (FHCA). International Journal of Computer and Electrical Engineering 2010, 2, 134-140. Neutrosophic Sets and Systems, Vol. 97, 2026 327 Madineh Farnam, Gholam Hassan Shirdel, Majid Darehmiraki, Agglomerative Hierarchical Clustering Method under Neutrosophic Trapezoidal Fuzzy Num 20. Vandhana, S.; Anuradha, J. Neutrosophic fuzzy hierarchical clustering for dengue analysis in Sri Lanka. Neutrosophic Sets and Systems 2020, 31, 179-199. 21. Elhassouny, A. Neutrosophic Logic-based DIANA Clustering algorithm. Neutrosophic Sets and Systems 2023, 55, 498-509. 22. Biswas, P.; Pramanik, S.; Giri, B.C. A new methodology for neutrosophic multi-attribute decision making with unknown weight information. Neutrosophic Sets and Systems 2014, 3, 42–52. 23. Abbasi Shureshjani. R.; Darehmiraki, M. A new parametric method for ranking fuzzy numbers. Indagationes Mathematicae 2013, 24, 518–529. 24. Laros, D. Discovering knowledge in data: An introduction to data mining, 2nd ed.; Wiley, New York, America, 2005. 25. Charrad, M.; Ghazzali, N.; Boiteau, V.; Niknafs, A. NbClust: An R package for determining the relevant number of clusters in a data set. Journal of Statistical Software 2014, 61, 1-36. 26. Kaufman, L.; Rousseeuw, P. Finding Groups in Data: An Introduction to Cluster Analysis, Wiley, New York, America, 1990. 27. Tan, Ruipu & Zhang, Wende & Chen, Shengqun. (2019). Decision-Making Method Based on Grey Relation Analysis and Trapezoidal Fuzzy Neutrosophic Numbers under Double Incomplete Information and Its Application in Typhoon Disaster Assessment. IEEE Access. PP. 1-1. 10.1109/ACCESS.2019.2962330. 28. Broumi, Said; Malayalan Lathamaheswari; Ruipu Tan; Deivanayagampillai Nagarajan; Talea Mohamed; Florentin Smarandache; and Assia Bakali. "A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids." Neutrosophic Sets and Systems 35, 1 (2020). https://digitalrepository.unm.edu/nss_journal/vol35/iss1/27 Received: April 22, 2025. Accepted: Sep 30, 2025