Frontiers in Computing and Intelligent Systems ISSN: 2832-6024 | Vol. 14, No. 1, 2025 70 Adaptive Clustering‐based Multi‐modal Multi‐Objective Location Optimization Algorithm Jiaying Wang *, Bingyuan Wu and Yinxing Chen College of Computer Science, Guangdong University of Science and Technology, Dongguan Guangdong 523083, China * Corresponding author: Jiaying Wang Abstract: To address the issues of insufficient solution diversity, imbalanced multi-objective trade-offs, and poor adaptability to complex constraints in traditional facility location optimization, this paper proposes an Adaptive Clustering-based Multi- modal Multi-Objective Location Optimization Algorithm (ACM-MOLA). The algorithm adopts a multi-modal multi-objective optimization framework, integrating adaptive clustering and local search mechanisms: it dynamically adjusts the number of clusters to achieve refined exploration of the solution space, employs non-dominated sorting and crowding distance to maintain the Pareto optimal frontier, and enhances the local optimization accuracy of high-quality solutions through local search. Comparative results with other multi-modal multi-objective algorithms demonstrate that ACM-MOLA outperforms in terms of both the quantity of Pareto solutions and solution diversity, providing decision-makers with location options that better cover diverse needs, thereby validating the algorithm's effectiveness and practicality. Keywords: Facility Location; Multi-modal Multi-objective Optimization; Adaptive Clustering; Non-dominated Sorting; Local Search. 1. Introduction The site selection problem is a core decision-making problem in the fields of supply chain and urban planning. Its essence is to achieve collaborative optimization of multiple objectives such as transportation costs, construction costs, and service quality while meeting demand coverage and resource constraints [1]. With the acceleration of urbanization and the increase in supply chain complexity, traditional location selection algorithms are facing problems such as lack of solution diversity, imbalanced multi-objective trade-offs, and poor constraint adaptability[2]. Designing a good location selection algorithm is of great practical significance for reducing supply chain operating costs and improving urban resource utilization. The development of multi-objective location selection algorithms can be divided into three generations: the first generation is represented by weighted sum method, which transforms multi-objective into single objective, but the subjectivity of weight setting is strong [3]. The second generation, with NSGA-II and MOEA/D as the core, maintains the Pareto frontier through non dominated sorting or decomposition strategies, significantly improving the diversity of understanding [4, 5]. However, there is still a deficiency of insufficient exploration of multi-modal regions. The third generation is oriented towards multi-modal multi- objective optimization (MMO), focusing on discovering high-quality solutions distributed in different regions. Zhang et al. [6] proposed a clustering based multi-modal algorithm, which divides the solution space through K-means, but the number of clusters is fixed and difficult to adapt to dynamic iterative processes. At present, the combination of multi- modal exploration and clustering strategies is not closely integrated in existing research, and a fixed number of clusters is difficult to adapt to the dynamic changes in the solution space during the iteration process; The synergy between local search and global exploration is insufficient, and most algorithms only perform local optimization after global iteration, which can easily miss opportunities for fine improvement of high-quality solutions. Based on this, this article proposes a multi-modal multi- objective optimization algorithm for adaptive clustering, which dynamically adjusts the number of clusters with the iteration progress using an adaptive clustering mechanism to achieve dynamic partitioning of the solution space and exploration of multi-modal regions; Build a three-layer optimization framework consisting of global iteration, clustering partitioning, and local enhancement, combined with non dominated sorting and crowded distance maintenance to maintain the Pareto front, effectively identifying different modal regions and searching between multiple local optimal solutions. 2. Related Work 2.1. Theoretical Basis of Multi-Modal Multi- Objective Optimization Algorithm In practical applications, real-world problems often involve not only multiple conflicting objectives but also multi-modal objective functions. This requires algorithms to not only balance these objectives and identify a set of Pareto-optimal solutions, but also possess robust search capabilities to explore complex search spaces and discover multiple local optima. Such algorithms, designed to ensure solution diversity and comprehensiveness, are known as multi-modal multi-objective optimization algorithms [7]. Consider the logistics distribution center location selection problem as an example. This issue involves multiple objectives such as minimizing transportation costs and optimizing service quality. These objectives often have interdependent relationships. For instance, reducing transportation costs might lead to choosing locations closer to suppliers, but this could extend delivery times and compromise service quality. Multi-modal multi-objective optimization algorithms can effectively balance and select among multiple local optimal solutions through strategic 71 search methods. This process identifies a set of Pareto- optimal solutions that achieve equilibrium across multiple objectives while covering different modal areas. These solutions provide decision-makers with diverse options, enabling them to flexibly choose the most suitable location plans based on actual needs and preferences. 2.2. Location Modeling Suppose a logistics company plans to establish a new warehouse in a specific region to meet the storage and distribution needs of multiple neighboring cities. When making location decisions, it is essential to identify sites with minimized costs and maximized market demand to ensure optimal warehouse operations. To address the location optimization challenge, this paper constructs a mathematical model for multi-modal multi-objective location optimization as follows: Let the candidate site set be 𝐢 𝑐 , 𝑐 , … , 𝑐 . Select k sites as facility locations, with decision variables π‘₯ π‘₯ , π‘₯ , … , π‘₯ where π‘₯ ∈{0,1} . π‘₯ 1 indicates selecting site 𝑐 , while π‘₯ 0 indicates non-selection. The constraint is βˆ‘ π‘₯ π‘˜. Minimize transportation costs: 𝑓 π‘₯ π‘šπ‘–π‘› βˆ‘ 𝑑 𝑀 , where 𝑑 is the distance between the warehouse and the i-th demand city, and 𝑀 is the demand volume of goods from the i-th demand city. Minimize construction costs: 𝑓 π‘₯ π‘šπ‘–π‘›πΆ , where 𝐢 represents the warehouse construction cost. Market demand maximization: 𝑓 π‘₯ π‘šπ‘Žπ‘₯ βˆ‘ 𝑠 π‘ž , where 𝑠 denotes the service level of the warehouse for the i- th demand city, and π‘ž represents the market demand scale of the i-th demand city. In this model, there are multiple local optimal solutions in the decision space, and the objective functions are conflicting, which is in line with the characteristics of multi-modal multi- objective optimization problems. 3. ACM-MOLA 3.1. Suppose a Logistic Adaptive Clustering Strategy Traditional clustering-based multi-modal multi-objective optimization algorithms typically employ fixed cluster quantities, which may struggle to effectively address complex and dynamic facility location problems. The proposed algorithm introduces an adaptive clustering strategy that dynamically adjusts cluster numbers based on population distribution and search progress. During the initial search phase, when solution space exploration is limited, a larger cluster quantity is adopted to extensively explore the solution space and identify potential multi-modal regions. In the mid- search phase, cluster quantities are gradually adjusted according to solution density distribution and objective function value variations within the population. If solutions in a cluster become overly concentrated with similar objective function values, the cluster quantity can be appropriately reduced to avoid getting trapped in local optima. By implementing this adaptive clustering mechanism, the algorithm demonstrates enhanced flexibility in adapting to facility location problem complexities while improving search efficiency. The adaptive adjustment mechanism for cluster quantity (denoted as k) is defined as follows: kt+1=max(kmin, min(kmax, kt+Ξ”k)) (1) 3.2. A Local Search Strategy Based on Gradient Descent To enhance the algorithm's search capability in local regions and improve the discovery of local optimal solutions, the improved algorithm introduces a local search mechanism within each cluster. After clustering the population, the algorithm applies a gradient descent-based local search method to optimize individuals within each cluster. The local search update formula is: π‘₯ π‘₯ Ξ·βˆ™βˆ‡f(π‘₯ ) (2) where Ξ· is the learning rate and βˆ‡f(π‘₯ ) denotes the gradient of the function f at point π‘₯ . During the search process, the algorithm simultaneously considers the impact of multiple objective functions by applying weighted integration to combine them. This ensures that while optimizing transportation costs, other objectives are not excessively compromised. The local search mechanism enables fine-tuning of solutions within each cluster, thereby improving solution quality and enhancing the overall performance of the algorithm. 3.3. Algorithm Flow Input: candidate site areas, population size 𝑁, maximum evaluation attempts π‘€π‘Žπ‘₯𝐹𝐸𝑠 , external archive capacity A , local search radius r, crossover probability pc , mutation probability pm. Output: the set of Pareto optimal solutions in the external Archive. 1.Generate an initial population P of size 𝑁 randomly and initialize the external archive Archive=βˆ…,and set the iteration count 𝐹𝐸𝑠 𝑁 2.while (FEs