## 3 Multi-Attribute Evaluation System for Product Form Design  
### Multi-Attribute Evaluation Method Based on PHFS  
In the actual product form design decision-making process, the decision-makers have multi-disciplinary and multi-role characteristics, and their psychological and personalized preferences are characterized by uncertainty and ambiguity. While decision-makers can remain objective and rational in the evaluation process, the raw data for evaluations are derived from the subjective experience of decision-makers, which makes it difficult to provide relatively accurate evaluation information on different product form design attributes [33]. This uncertainty and ambiguity are primarily reflected in the following: product form design encompasses multiple factors, including shape, color, ergonomics, material selection, and so forth. These factors are interrelated and influence each other, necessitating a comprehensive consideration of multiple aspects during the evaluation process. Secondly, the decision-making process of product form design is often influenced by the subjective feelings of decision-makers, and different decision-makers may establish evaluation criteria based on varying aesthetic concepts, usage experiences, or cultural backgrounds. Decision-makers often prefer to use linguistic variables such as \"good\", \"very good\", and \"very bad\" to assign an evaluation value to their decisions. Alternatively, they may be unable to provide a specific value but only a vague range, and their evaluation may vacillate between several possible values. This subjectivity makes it difficult to standardize evaluation criteria, thus creating hesitancy and ambiguity. In response to the above issues, the Fuzzy Sets (FS) theory, proposed by [PERSON], addresses the issue of ambiguity by employing fuzzy numbers in lieu of specific assessment values provided by the decision-maker [34]. However, it does not fully capture the hesitancy that arises when the decision-maker evaluates each attribute of the product form. Intuitionistic Fuzzy Sets (IFS) consider the decision-maker's indecision based on fuzzy sets, but when the sum of the decision-maker's satisfaction scores for the attributes of the program exceeds 1, the Intuitionistic Fuzzy Sets become inapplicable [35]. Hesitant Fuzzy Sets (HFS) overcome the limitations of fuzzy sets by enabling decision-makers to assign multiple possible and non-repeating scoring values to the performance of each attribute of the product's morphological design, thereby enabling a more comprehensive expression of the hesitancy and uncertainty in the decision-making process [36]. The PHFS, which is developed from the integration of the Pythagorean Fuzzy Set and the Hesitation Fuzzy Set, aptly describes the hesitancy that arises among decision-makers in terms of both satisfaction and dissatisfaction. Simultaneously, it allows the sum of the decision-makers'satisfaction and dissatisfaction with each attribute of the product's form design scheme to exceed 1, a phenomenon that cannot be characterized by any other approach [37; 38]. Therefore, this paper adopts PHFS for evaluation, which not only overcomes the limitations of traditional fuzzy sets but also effectively reflects the hesitancy and vagueness of the decision-makers. Consequently, it is able to more objectively describe the uncertainty and fuzzy phenomena associated with attributes and inter-attributes in the product morphology design scheme [39]. The evaluation process presented in this paper involved domain experts and users jointly scoring the alternatives and evaluating attribute contents in the cloud environment. They provided their evaluation of each alternative relative to each attribute using Pythagorean Hesitant Fuzzy Numbers (PHFNs), and the group evaluation scores were then weighted using the Pythagorean Hesitant Fuzzy Weighting Geometric (PHFWG). The evaluation based on Pythagorean Hesitant Fuzzy Sets [37] was performed as follows:  
**Definition 1**.