## 1 Introduction
Several kinds of modified position and velocity updating models have been constructed and presented to enhance the local and global optimization capacities of PSO, such as the crossover operator [43; 44], which includes the multi-crossover operator [45] and the vertical crossover operator [46], and the mutation operator [47; 48]. Furthermore, improved PSO integrates with various learning strategies that have been designed and developed to alter the exemplar particle in the population, including adaptive comprehensive learning strategies [49; 50], adaptive learning strategies [51; 52], and reinforcement learning strategies [53; 54]. Some studies on PSO have utilized the multi-population strategy to split the whole population of particles into multiple smaller subpopulations to avoid prematurity [55; 56]. Communication between different subpopulations is allowed to realize information exchange and sharing [57; 58]. In order to utilize the advantages of both approaches, various variants of PSO have been proposed that combine PSO with other heuristic algorithms or optimization strategies to enhance the performance of PSO in solving engineering application problems [59; 60; 61]. Based on the above studies, MAs based on TPs provide a new way of enhancing the performance of PSO, which distinguishes them from previous improvements to PSO. Furthermore, the combination of PSO and TPs is also a new attempt to overcome limitations,such as incomplete basic operations and implementation complexities in the original model. The properties of distributed and parallel processing in P systems are introduced to address the issue of exponential complexity increase in PSOs with problem scale enlargement. Under the computing framework of TPs, the particle population is divided into multiple sub-populations with a network structure, and communication between different sub-populations is achieved through the symport/antiport rules of TPs. Unlike previous related works about MAs, this paper modifies the evolutionary mechanism of PSO to better suit the computing framework of MTP. Each object in the elementary membrane has two evolution rules for updating velocity, which can be selected randomly to avoid premature convergence. Additionally, two evolution rules based on the classic PSO and crossover operator are defined to update the position of objects in different elementary membranes in order to balance the exploration and exploitation of the system. The exchange of information based on global best practices in a single direction is achieved through the specific communication mechanism of MTP, which distinguishes it from the related works on MAs based on TPs. This work focuses on the combination of extended TPs with PSO for solving clustering problems. A novel variant of MIEAs or MAs with SNS, called ECPSO-MTP, is proposed and designed by integrating MTP and an improved PSO based on environmental factors and crossover operators. The proposed ECPSO-ECP comprises the evolutionary mechanism of modified PSO and the computation framework of MTP to establish a highly efficient P system for solving clustering problems. In ECPSO-MTP, two types of evolution rules are defined and introduced to rewrite and modify objects in different elementary membranes based on environmental factors and crossover operators. The environmental factors utilize the partitioning information to randomly replace the global best of objects and attend to the evolution process to improve the clustering performance of ECPSO-MTP. The crossover operator is achieved by hybridizing the global best of elementary membranes to reject randomness. The information exchange and resource sharing between different elementary membranes in ECPSO-MTP are accomplished through the evolutional support with promoters for objects of MTP, including forward and backward communication rules. Furthermore, eight test clustering and image datasets are employed in comparison experiments to verify the effectiveness of ECPSO-MTP compared with other existing approaches.