## 1 Introduction
Reference [16] proposed a methodological framework for the scalarization of multiple optimization objective functions, wherein five distinct objective functions--comprising electricity procurement costs, natural gas acquisition expenses, battery operational parameters, and hydrogen production expenditures--were systematically consolidated into a unified cost function formulation. An optimal scheduling method of an integrated wind-photovoltaic-hydrogen system was proposed, incorporating an improved discrete step-size transformation. In reference [17], the net present cost (NPC) and cost of energy (COE) of a hybrid renewable system comprising photovoltaic, wind turbine, fuel cell, and electrolyzer components were calculated. The system configuration was optimized through the implementation of three distinct swarm intelligence algorithms. Prior research documented in references [18; 19] established methodological frameworks employing hierarchical analysis and intuitionistic fuzzy number approaches to ascertain the relative weights among multiple optimization objective functions. This approach converts the multi-objective optimization problem into a single optimization model by summation. Another option is to utilize swarm intelligence optimization techniques to identify a non-dominated solution set under multiple objectives, which can be assessed and merged with the initial solution to reach the final recommended outcome [20]. For instance, an enhanced sea squirt swarm intelligence algorithm has been proposed to effectively improve the economy and energy utilization of a wind-hydrogen storage system [21]. The non-dominated sorting genetic algorithm II (NSGA II) was used to optimize and determine the best energy management strategy and capacity allocation scheme for a grid-connected wind power green hydrogen system. The two objective functions were to enhance the economic efficiency and improve energy utilization [22; 23]. To determine the optimal configuration of the capacity of the hydrogen production system in an off-grid state, references [24; 25; 26] employed the NSGA II method to solve several different perspectives including those relating to the economy of hydrogen production, the probability of power supply loss, and the potential for excess energy. The NSGA II method is employed to identify the optimal capacity configuration for off-grid hydrogen production systems, taking into account several key factors such as the economic viability of hydrogen production, the risk of power supply disruptions, and the possibility of excess energy generation [24; 25; 26]. The resulting Pareto frontier is more uniform and covers a broader range compared with those produced by previous methodologies. To enhance the design of a novel PV-pumped storage system for hydrogen storage, both the NSGA-II and MOPSO methods were utilized to improve the system cost and efficiency [27]. Finally, it is also possible to implement a multi-stage or multi-layer structure to address a multi-objective optimization problem [28]. In a multi-objective optimization of wind-hydrogen production systems, an iterative procedure approach was proposed to find Pareto-optimal solutions [29]. In an off-grid hydrogen generation system, a methodology for day-ahead scheduling was proposed to optimize the capacity allocation based on the multi-state transitions of alkaline electrolyzers that utilized a particle swarm optimization (PSO) algorithm [30]. The existing literature has not addressed the complementary characteristics of different types of electrolyzers, particularly the complementarity between AWE and PEME regarding hydrogen production cost and efficiency. A multi-objective planning problem of a hybrid electrolyzer is discussed in a stand-alone wind-photovoltaic-battery system. Two key issues urgently need to be addressed in this optimization problem. A mathematical model that describes a hybrid electrolyzer's optimal composition and proportions and a reasonable recommendation method for a non-dominated solution set in a multi-objective optimization problem.