## 1 Introduction
Following this, [PERSON] proposed a Pythagorean hesitant fuzzy cross-impact Bonferroni mean operator [14], and [PERSON] introduced an MCDM method based on the Pythagorean hesitant fuzzy Maclaurin symmetric mean operator [15]. Generally, in MCDM problems, the degree to which one option or indicator is better than another is often represented by a value between 0 and 1. For instance, if one decision maker gives a rating of 0.5, another 0.7, and a third 0.8, this can be depicted as a hesitant fuzzy element (HFE) \(\{0.5,0.7,0.8\}\). In such cases, the HFE can more accurately reflect the decision group's evaluation of the option or indicator compared to interval-valued fuzzy numbers or intuitionistic fuzzy numbers. The objective of MCDM problems can be understood as selecting the best option from a set of alternatives grounded on one or several criteria, or ranking these alternatives. Many MCDM methods have been proposed by scholars, founded on a compromise of programming principles initiated by [PERSON] [16]. For example, [PERSON] and others proposed the TOPSIS method [17], [PERSON] introduced the VIKOR method [18], and [PERSON] and others developed the PROMETHEE method [19]. [PERSON] and others carried out comparative analyses of the VIKOR approach alongside various MCDM techniques such as TOPSIS,PROMETHEE, and ELECTRE, finding that VIKOR has distinct advantages in dealing with MCDM problems, especially those with conflicting objectives [20]. VIKOR has seen extensive application across multiple domains, including but not limited to design, mechanical engineering, manufacturing, material selection, and new product development. [PERSON] and others applied the VIKOR method to the selection of ergonomic office chairs [21]. [PERSON] and others used VIKOR for selecting magnetic materials [22]. [PERSON] and others analyzed and compared TOPSIS method, VIKOR method, and the Ashby method for material selection in micro-electromechanical system (MEMS) devices [23]. [PERSON] and others utilized methods such as fuzzy VIKOR method, fuzzy TOPSIS method, and COPRAS-G method for the selection of casting robots [24]. [PERSON] and others utilized the interval type-2 fuzzy VIKOR method for robot selection and compared it with several other MCDM methods [25]. [PERSON] and others proposed an MCDM method integrating rough-number analytic hierarchy process (AHP) and rough-number VIKOR, applied to design concept evaluation [26]. [PERSON] proposed an MCDM method founded on DEMATEL, ANP, and VIKOR, utilized for the selection of machining processes, workpiece materials, and features of workpiece shape [27]. In MCDM problems, the comparative significance of various evaluation indicators toward the decision-making goal varies, hence assigning scientific and reasonable weights to each indicator is a crucial consideration. Both TOPSIS and VIKOR methods ground their assessment on the proximity of alternatives to an optimal solution. However, traditional TOPSIS and VIKOR methods overlook the significance of these distances in relative terms. Approaches for ascertaining weights can be categorized into subjective and objective weighting methods. Subjective methods rely on expert experience and personal opinions to determine the relative weights of each evaluation indicator, while objective methods are based on objective experimental data. [PERSON] and others proposed a VIKOR method grounded on the AHP, employing AHP principles to ascertain the relative weights of each indicator [28]. [PERSON] and others used fuzzy AHP to determine indicator weights and applied fuzzy VIKOR, fuzzy TOPSIS, and COPRAS-G methods to manage the choice of casting robots [24]. [PERSON] and others proposed a weighting method based on rough-number-integrating AHP, applied to design concept evaluation [26]. [PERSON] and others used DEMATEL and ANP to determine the weights of evaluation indicators for processing technology, material selection, and workpiece shape characteristics [27].