## 1 Introduction
These studies collectively concentrate on the analysis and application of single-valued data under hybrid or modified fuzzy C-means clustering algorithms; that is, these papers all focused on single-valued data analysis and application. [PERSON] and [PERSON] introduced an enhanced FCM clustering algorithm [29] known as the improved fuzzy partitions fuzzy C-means clustering (IFPFCM) algorithm. The IFPFCM algorithm encourages the assignment of crisp membership degrees by incorporating a'reward' term for membership degrees close to zero or one. This modification of the objective function used in fuzzy clustering results in distinct membership functions, facilitating rapid convergence and improved performance. A large amount of data can be expressed in intervals to simplify the amount of data and contain hidden data. Interval data represent a range of values between two bounds denoted as \([a_{l},b_{r}]\). The \(a_{l}\) and \(b_{r}\) are called the starting point and ending point of an interval, respectively. In general, interval data can represent the analysis of data in a specific area, i.e., user data mining [30]. Interval data definition and distance measures were quickly discussed, as were their applications in clustering [31; 32]. [PERSON] et al. [33] (2006) proposed interval data clustering with applications. In 2007, [PERSON] [34] introduced a fuzzy C-means clustering method tailored for symbolic interval data (SID). This marked a traditional application of the fuzzy C-means algorithm to address SID. Subsequently, [PERSON] et al. [35] proposed the interval possibilistic FCM (IPFCM) clustering method for SID, while [PERSON] et al. [36] developed a rough IPF clustering algorithm. The latter utilized derived formulas for fuzzy membership degrees (FMDs) and possibilistic membership degrees (PMDs) to improve clustering in SID, especially when dealing with noisy data and overlapping clusters. However, these methods involved complex mathematics and led to lengthy computations. The motivation behind this paper was to extend the IPFCM approach with IIFPFCM, emphasizing clear membership, which results in rapid convergence but may be susceptible to outliers. To address this limitation, we also introduced modifications in the city block distance calculation to mitigate these challenges. Regarding the advantages of the proposed IIFPFCM, we used Figures 1 and 2 with IFCM with Euclidean distance (IFCME) and IIFPFCM with Euclidean distance (IIFPFCME) algorithms on the data involving five groups to show that the division of membership in IIFPFCME is more crisply. The use of this concept sped up the convergence of IIFPFCME in this paper. Hence, the contributions in this paper are two fuzzy clustering approaches called the new interval fuzzy partitions fuzzy C-means clustering algorithms that were proposed. The proposed algorithms independently combined Euclidean distance and city block distance to achieve fast convergence. The proposed methods showed a faster convergence speed and better performance results compared to the traditional interval fuzzy c-means clustering method in SIDA. Additionally, symbolic interval data often present a challenge of large and small group division, which was also addressed using the proposed methods. Additionally, the IFPFCM algorithm was introduced to handle outliers through interval distance measurement. The experimental comparative analysis revealed that the IFPFCM clustering algorithm with the city block distance measure was suitable for dealing with SIDA with outliers. Finally, this study tested nine groups of data, and the statistical results demonstrated that the proposed algorithm outperformed the traditional method in terms of convergence and efficiency. Figure 1: (**a**) IFCME clustering results and fuzzy membership, and (**b**) relationship between data and degree of membership. Figure 2: (**a**) IIFPFCME clustering results and fuzzy membership, and (**b**) relationship between data and degree of membership. The organization of the rest of this paper is as follows.