## 2 Methodology
### Pythagorean Hesitant Fuzzy VIKOR Method
#### 2.3.3 The Basic Idea of the Pythagorean Hesitant Fuzzy VIKOR Method
\[DQ=\frac{1}{m-1} \tag{34}\]  
Condition 2: The alternative with the smallest \(Q_{i}\) value, after sorting in ascending order, must also rank first in either the \(S_{i}\) or \(R_{i}\) values when these are sorted in ascending order. If both Condition 1 and Condition 2 cannot be satisfied simultaneously: (a) If Condition 1 is not met, the compromise solution includes both \(X^{(1)},X^{(2)},\cdots X^{(m)}\), where the value of m is determined by Equation (35): \[Q(X^{(m)})-Q(X^{(1)})\geq DQ\] (35) (b) If only Condition 2 is not satisfied, then both the alternatives \(X^{(1)}\) and \(X^{(2)}\) (the ones with the smallest and the second smallest \(Q_{i}\) values, respectively) are considered as optimal solutions. In this case, both \(X^{(1)}\) and \(X^{(2)}\) are the compromise solutions. This methodological framework provides a systematic approach for handling decision-making problems with multiple criteria and uncertain assessments, offering a balance between the maximum group benefit and the minimum individual regret. Safety Risk Assessment Method for Quayside Container Cranes Based on BWM-Pythagorean Hesitant Fuzzy VIKOR Method  
In this section, we will explore how to apply the BWM-Pythagorean hesitant fuzzy VIKOR method to address the safety risk assessment of quayside container cranes. The safety condition assessment of quayside container cranes is a complex MCDM problem, involving multiple risk indicators affecting the overall safety of the equipment. It necessitates the involvement of experts in the relevant field for analysis. Due to the inherent fuzziness and the limitations of testing equipment precision and subjective variability among evaluators, the safety risk assessment of gantry cranes can be viewed as a multi-attribute decision-making problem represented by hesitant fuzzy elements (HFEs). Often, decision makers may find it difficult to precisely express an evaluation value for a particular criterion. Different situations and different evaluators may have varying evaluation values for the same criterion, making it challenging to set a standard evaluation criterion applicable to all scenarios. Hence, it is reasonable to use hesitant fuzzy elements (HFEs) to represent the evaluation values of each criterion. For determining the safety status of gantry cranes, we aim to obtain a safety risk level for the crane under evaluation, thereby providing guidance for the work and management of port enterprises and dock managers. Using the VIKOR method, we can obtain a ranking of the alternative solutions. Thus, by sorting the evaluation data of the crane under evaluation along with the boundary data of each safety risk level, we can determine the safety risk level of the crane under evaluation. Generally, the safety risk levels of gantry crane equipment are divided into five levels, ranging from level one (excellent) to level five (very poor), indicating increasing levels of safety risk. With five levels, there are six boundary divisions. By sorting these six division boundaries along with the Pythagorean hesitant fuzzy evaluation set of the crane under evaluation using the principles of the VIKOR method, we can obtain the safety risk level of the crane under evaluation. The weights of each evaluation indicator represent their relative importance. As the safety risk assessment indicators for gantry cranes are mostly qualitative and difficult to quantify, and the few quantitative indicators are also subject to the adverse experimental environment of the dock and the precision of testing equipment, there is inherent fuzziness and uncertainty. Therefore, we choose the BWM method to weigh each evaluation indicator; the specific weighting method is discussed in Section 2.2. In the traditional VIKOR method, we sort the alternatives based on a specific distance measure from the ideal solution. Thus, in the Pythagorean hesitant fuzzy VIKOR method, we need to determine the ideal solution and the distance measure method.