## 1 Introduction
However, since the number of student types is often unknown beforehand, unsupervised machine learning methods are necessary for clustering. Among the various clustering algorithms, the k-means algorithm [25; 26; 27] is widely used for analyzing student consumption behavior due to its speed, ease of understanding, and efficiency [28]. However, traditional k-means has limitations, including a lack of flexibility with fixed weights, dependency on initial cluster centers, and the requirement for a pre-specified number of clusters (K). These limitations restrict the algorithm's applicability. In recent years, various meta-heuristic algorithms have been proposed to optimize clustering algorithms, overcoming the shortcomings of k-means and achieving better results. Some of these meta-heuristics include the monarch butterfly optimization (MBO) algorithm [29], the slime mold algorithm (SMA) based on the feeding behavior of slime molds [30], the moth swarm algorithm (MSA) inspired by moth navigation toward moonlight [31; 32], and the Harris hawks optimization (HHO) algorithm that simulates predatory behavior [33]. The firefly algorithm, as a meta-heuristic clustering intelligence algorithm [34], has garnered increasing attention from researchers since its inception. It boasts advantages such as a simple principle, clear process, and ease of implementation, and has been successfully applied in fields like image processing, computer networks,and engineering design [35; 36; 37]. [PERSON] proposed a fuzzy clustering method based on the firefly algorithm to enhance clustering accuracy [38]. [PERSON] applied the firefly algorithm to determine the insulation levels of DC transmission lines, improving the insulation of UHV DC lines [39]. [PERSON] developed a traffic signal timing optimization method based on the firefly algorithm, effectively reducing emissions at intersections [40]. However, the firefly algorithm typically relies on previous memory during its search process. To address the issue of local optima, the introduction of fractional-order memory--based on the properties of memory and genetic processes--has been suggested. This paper employs a fractional-order firefly-optimized k-means clustering algorithm to enhance clustering performance effectively. The main contributions and innovations of this paper are as follows:  
(1) The proposed methodology involves the utilization of a clustering algorithm for the analysis of students' spending patterns. In comparison with the conventional statistical behavior analysis model, this clustering-based approach has the capacity to reduce the impact of human factors and thereby facilitate the more objective identification of students in need of financial aid. (2) We proposed three models: the overall consumption model, the living diet model, and the average consumption model for morning, lunch, and dinner. These models seek to evaluate students' consumption levels by leveraging data from the university's digital system, thereby offering a more comprehensive representation of their consumption patterns. (3) In order to address the issues inherent to the k-means algorithm, namely the challenges associated with determining the optimal number of student clusters and the random allocation of cluster centers, a methodology is proposed that utilizes the elbow rule and the Calinski-Harabasz index to determine the number of clusters. Concurrently, the fractional-order firefly algorithm is employed to select the initial cluster centers. This approach has been demonstrated to be effective in addressing the issue of the algorithm's sensitivity to the initial value and its susceptibility to local optimization. In comparison with the conventional k-means algorithm, the proposed method facilitates more rational clustering. (4) Based on the aforementioned models, we conduct a comprehensive analysis of students' consumption characteristics. The experimental findings have significant practical implications, assisting college administrative personnel in enhancing their operational efficiency.