Using OMOPSO & MMOPSO algorithms on water distribution systems design.

The particle swarm optimization, PSO, was firstly developed by Kennedy & Eberhart (1995) to solve simple single-objective continuous optimization problems. This algorithm depends mainly on imitating the concept of swarm intelligence which is extensively found in birds flocks or fish schools. The algorithm initiates a prespecified number of flying particles that are led by a leader which moves towards the area of the global optimum solution with the most reasonable direction and flying speed. The algorithm keeps iterating while improving the positions and flying speeds for all the particles by following the particles’ leader. Eventually, the algorithm may reach the global optimum or may fall in local optima (if any). Coello Coello & Lechuga (2002) extended the usage of the PSO algorithm to adapt with multi-objective optimization problems by providing an external archive to store the set of non-dominated solutions that may be found during the search. The original multi-objective particle swarm optimization, OMOPSO, algorithm usually suffers from some difficulties when it is applied in the problem of designing water distribution systems (WDSs) since this problem is a combinatorial discrete NP-hard optimization problem (Montalvo et al., 2010; Surco et al., 2017). The algorithm frequently falls in local minima during the search and fails to capture the whole set of non-dominated feasible solutions after the end of search especially in medium to large search space problems. Therefore, a trial is made to improve the performance of the OMOPSO in designing WDSs by developing a modified multi-objective particle swarm optimization, MMOPSO. The modifications include using some strategies to improve the overall convergence and diversity of the final non-dominated feasible set of solutions such as; the self-adaptive PSO parameters strategy, the selection strategy of the external archive members, the regeneration-on-collision strategy, and the adaptive population size strategy. In the subsequent sections, the pseudo-codes of the OMOPSO and MMOPSO are presented, the explanations of using both algorithms are provided, and an example of using both algorithms is illustrated.

粒子群优化算法（Particle Swarm Optimization, PSO）由Kennedy与Eberhart于1995年首次提出，旨在求解简单的单目标连续优化问题。该算法核心借鉴了群智能概念——这一现象广泛存在于鸟群、鱼群等生物群体中。算法初始化预设数量的飞行粒子，由一个领航粒子引导，以最合理的方向与飞行速度向全局最优解区域移动。算法通过迭代不断更新所有粒子的位置与飞行速度，始终跟随领航粒子的轨迹。最终，算法或可收敛至全局最优解，或陷入局部最优（若存在）。
Coello Coello与Lechuga（2002）通过引入外部存档存储搜索过程中发现的非支配解集合，将PSO算法拓展至多目标优化问题场景。原始多目标粒子群优化（Original Multi-Objective Particle Swarm Optimization, OMOPSO）算法在应用于配水系统（Water Distribution Systems, WDSs）设计问题时往往存在诸多局限：该问题属于组合离散NP难优化问题（Montalvo等人，2010；Surco等人，2017），算法在搜索过程中极易陷入局部极小值，且在中等至大规模搜索空间场景下，往往无法在搜索结束后获取完整的非支配可行解集合。
为此，本研究提出一种改进型多目标粒子群优化（Modified Multi-Objective Particle Swarm Optimization, MMOPSO）算法，以提升OMOPSO在配水系统设计任务中的性能。所采用的改进策略包括：自适应PSO参数策略、外部存档成员选择策略、碰撞再生策略以及自适应种群规模策略，旨在优化最终非支配可行解集合的整体收敛性与多样性。
后续章节将分别给出OMOPSO与MMOPSO的伪代码，阐述二者的应用方法，并辅以算法应用实例进行说明。