## 1 Introduction  
With the advancement of socio-economic development, the scale of industrial production and logistics transportation has been expanding significantly. As an indispensable component within this sector, quayside container cranes have garnered widespread attention. Concurrently, with the ongoing research and development in new technologies, the crane manufacturing industry is progressively incorporating an array of innovative processes, materials, and technological applications. The research findings of [PERSON] [1] and [PERSON] [2] demonstrate substantial potential and prospects for application in future crane manufacturing endeavors. Despite these innovations in manufacturing techniques, the safety concerns associated with quayside container cranes remain a critical issue for port enterprises. This is primarily due to harsh working conditions, challenges in maintenance, prolonged service lifetimes, high manufacturing costs, and the severe implications of accidents. As such, ensuring the safety of these cranes continues to be a focal point of concern that demands rigorous attention and strategic planning to mitigate risks effectively. The safety risk evaluation of quayside container cranes constitutes a type of multi-criteria decision-making (MCDM) problem, which involves a wide array of both qualitative and quantitative indicators. It is determined by various factors affecting the overall safety status of the equipment, collectively defining the safety condition of the equipment under evaluation. In traditional MCDM methods, all evaluation indicators are assumed to have precise values. However, in the field related to gantry cranes, obtaining accurate data is often challenging due to the harsh dock environment, the limitations of measuring instrument precision, and the presence of numerous qualitative indicators. To address these issues, an increasing number of scholars have recognized the necessity of incorporating factors such as uncertainty and fuzziness into MCDM methods [3]. In 1965, the concept of fuzzy sets was pioneered by [PERSON], effectively addressing issues characterized by uncertainty and fuzziness [4]. This theory has been widely applied by scholars across various fields. Building upon fuzzy set theory, researchers from different countries have expanded it into various forms for application in multiple areas. [PERSON] and others discussed the differences in construction, classification, and application between intuitionistic fuzzy sets and interval-valued fuzzy sets [5]. In intuitionistic fuzzy sets, both membership and non-membership degrees are considered, but their sum is required to be less than or equal to 1. In contrast, the membership function within interval-valued fuzzy sets is denoted by a range. [PERSON] suggested an approach to expand interval type-2 similarity measures to encompass general type-2 fuzzy sets [6]. [PERSON] introduced hesitant fuzzy sets (HFSs) on the basis of traditional fuzzy sets, offering a new approach to address uncertainty and fuzziness [7]. An HFS has greater advantages over other fuzzy-set-based methods in handling fuzziness, especially in decision-making scenarios where different decision makers may have divergent views on the same evaluation criterion and find it hard to reach a consensus. [PERSON] introduced Pythagorean fuzzy sets [8], in which the combined squares of membership and non-membership degrees are required to not exceed 1, offering a broader membership space compared to intuitionistic and interval intuitionistic fuzzy sets, thus offering enhanced benefits in addressing indeterminacy and vagueness [9; 10]. [PERSON] proposed a variety of Pythagorean fuzzy Einstein-combining operators and utilized them in MCDM [11]. [PERSON] addressed the limitations of the generalized Bonferroni mean by proposing a new dual generalized Bonferroni mean for multi-attribute group decision making (MGDM) [12]. [PERSON] introduced a Pythagorean fuzzy VIKOR method grounded on the distance index [13].