A Crowd-Intelligence-Driven, Multi-Attribute Decision-Making Approach for Product Form Design in the Cloud Environment

Abstract

In the traditional decision-making process for product form design, designers and experts often prioritize schemes based on their own knowledge and experience. This approach can lead to an oversight of user preferences, ultimately affecting decision outcomes. In contrast, crowd-intelligence-driven, multi-attribute decision-making for product form design in the cloud environment builds upon traditional approaches by leveraging the vast and diverse expertise of individuals on cloud platforms, engaging participants from various fields and roles in the decision-making process to enhance comprehensiveness and accuracy. To address the issue of a single decision-maker and limited user participation in the decision-making process for product form design schemes in the cloud environment, a multi-attribute decision-making method integrating expert knowledge and user preferences is proposed. This method aims to select a product form design scheme that optimally balances expert and user satisfaction. Initially, the Pythagorean Hesitant Fuzzy Set (PHFS) is used to quantify qualitative product attributes and to establish a comprehensive multi-attribute evaluation system. In the aspect of expert decision-making, a gray correlation coefficient decision matrix based on expert knowledge is established and the overall score of the base alternative is calculated by the ViseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method and the Improved Osculating Value method. In terms of user decision-making, weights are determined by calculating the similarity between user evaluation matrices, and the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) is used to calculate scores for product form designs based on user preferences. Ultimately, optimal selection is achieved by aggregating the aforementioned expert evaluation values and user preference values. The method’s effectiveness and feasibility are confirmed through a case study of coffee machine product form design schemes.

1. Introduction

Product form design is the process of integrating design art, art, and engineering technology to complete a proposal, the essence of which is that designers, according to the requirements, rely on personal experience and artistic inspiration to deconstruct and reorganize a variety of product form attributes, such as shape, color, material, and texture, in order to generate new design schemes [1,2]. With the transformation of Internet technology and the service model, in order to obtain more innovative design schemes for product forms, enterprises outsource product form design tasks to non-specific masses in the form of free and voluntary outsourcing, in the model of combining the objective ability of computers with the subjective ability of the group intelligence. Moreover, through the cloud environment, enterprises gain technical support, and they utilize service platforms for online product form design, so as to obtain a large number of product form design schemes with innovative potential and differences [3,4,5]. In the cloud environment, the designers involved in the product form design task are diverse and different in terms of knowledge background and product design ideas; therefore, a large number of product form design alternatives are obtained with different styles and diversified trends in terms of design purpose and inspiration presentation [6,7]. From there, product form design is the result of a combination of multiple attributes from these design alternatives, since it is difficult for these attributes to remain entirely independent, and complex, ambiguous interrelationships can often be found between them. Therefore, in this stage of bringing together design alternatives, more balanced and comprehensive designs can be achieved, though this involves making decisions about the desired performance of the product in terms of various attributes.

Multi-attribute decision-making on the product form effectively ranks alternative schemes by considering multiple attributes, thereby assisting designers in selecting the optimal choices. This approach has become a central component of the product form design process. In the traditional approach to product form design decision-making, the selection of the optimal scheme is typically carried out by designers or domain experts based on their individual knowledge and experience. However, the product form design decision-making model in the cloud environment breaks the limitations of geography, time, crowd, etc. Through the massive, multi-role, multi-discipline group intelligence, individuals can participate in the process of product form design decision-making together. This model helps enterprises to integrate resources and improve the efficiency and quality of decision-making in product form design programs. Furthermore, intense market competition has driven product design to shift from being designer-oriented to user-oriented. Therefore, considering user preferences in the decision-making process is crucial for improving the product’s market acceptance [8]. However, due to the influence of various complex factors such as educational background, living environment, age, profession, and economic status, users differ in their understanding, needs, satisfaction, and emotional responses to the same product form [9]. These differences may also be affected by emotions, the environment, or external information, leading users to make different choices regarding product form design schemes. Additionally, there are certain differences in perceptions of product form design between users and designers or experts. If only the opinions of designers and experts are considered, the final selected scheme may not entirely align with user preferences. In seeking to mitigate the potential for this mismatch, the crowd-intelligence-driven decision-making model for product form design in the cloud environment brings together a diverse public from various backgrounds and roles in order to leverage their knowledge. This model facilitates effective convergence in the design process and enhances both the comprehensiveness and scientific rigor of decision-making [10]. Therefore, effectively integrating crowd intelligence to make objective decisions about product form attributes is of significant practical relevance for designing and implementing product form schemes in the cloud environment.

There are various strands of research into product design scheme decisions. Pei et al. proposed a multi-attribute decision-making method for product concept design based on Concept Hierarchy Development (CHD) to address the issue of a lack of interconnection between attributes in product concept design proposals. This method divides the design proposals into hierarchies and constructs specific design feature comparison matrices within each level. Furthermore, it combines Pythagorean Fuzzy Sets (PFS) and Prospect Theory (PT) to scientifically allocate weights to experts and design criteria, thereby achieving an objective and rational multi-attribute decision-making process for product concept design [11]. Jin et al. addressed the issue of an insufficient consideration of user preferences in the multi-attribute evaluation process of product concept design proposals by proposing a model that integrates user preferences and designer perceptions based on Rough Numbers and the ViseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method. This model employs Shannon Entropy to analyze customer preferences and determine the weights of design criteria, while integrating customer preferences with designers’ understanding of design standards to construct the ideal scheme. The IR-VIKOR model is then used to calculate and rank the rough evaluation metrics of each design proposal to obtain the most rational design scheme [12]. Chen et al. addressed the decision-making challenges in crowdsourced product design within the cloud environment by transforming the multi-objective group decision problem into a multi-objective optimization problem. They used the Analytic Hierarchy Process (AHP) to quantify qualitative and quantitative evaluation objectives and their weights, then applied multi-objective optimization methods to solve the model. The effectiveness of their method was validated through the decision-making process for the crowdsourced design of a medical analgesic pump [13]. Sun et al. proposed a group multi-attribute decision-making method based on Prospect Theory (PT) to address the issue of unknown attribute weights in hesitant fuzzy environments. This method employs hesitant fuzzy sets to represent the value assignments made by decision experts for all attributes of each scheme. The attribute weights are determined using an improved entropy weight method. Subsequently, the hesitant fuzzy decision matrix is converted into a prospect decision matrix based on positive and negative ideal schemes. The final ranking of the schemes is achieved by replacing the closeness coefficient with the benefit–loss ratio [14]. Fan et al. tackled the issue of inadequate consideration of diverse user needs in cloud-based design schemes, proposing an optimal decision-making method integrating Quality Function Deployment (QFD) and Rough Set Theory [15]. Dou et al. developed a decision-making strategy for product appearance based on user emotional satisfaction, enhancing product appeal by integrating emotional satisfaction models into new product designs [16]. Fan addressed the issue of optimal design selection in the cloud environment by establishing an evaluation objective system from four dimensions: economic, social, environmental, and cultural. The weights of the indicators were determined using Intuitionistic Fuzzy Numbers and the Fuzzy Analytic Network Process (Fuzzy-ANP). Finally, Fuzzy-QFD was employed to rank the design schemes in the cloud environment [17]. Jing et al. tackled the issues of subjectivity, uncertainty, and heterogeneity in decision semantics during the conceptual design phase by proposing a decision-making method that incorporates multi-granularity heterogeneous evaluation semantics under uncertainty. This method transforms multi-granularity heterogeneous semantics into a cloud model decision matrix and constructs an advantage matrix using random numbers based on the cloud model distribution. Finally, the VIKOR model is employed to select the optimal scheme [18]. Cheng et al. proposed a quantitative evaluation method based on Rough Sets to address the issue of ambiguity in user emotional expression. This method reduces uncertainty in the evaluation process to identify product features and user experiences that require improvement. The effectiveness of the proposed method was demonstrated through a case study involving a wireless vacuum cleaner [19]. Yu et al. proposed a multi-criteria decision model for product design schemes based on TOPSIS-MOGA to address the issues of decision continuity and integration throughout the design phase. This model ensures continuity in the decision-making process by integrating decision criteria and employs the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), utilizing hesitant fuzzy linguistic term sets and the entropy weight method to select the optimal conceptual design scheme [20]. Wu et al. aimed to reduce reliance on subjective judgments of designers in product design decision-making by introducing a data-driven method that employs an ideal scheme ranking method, the entropy weight method for weighting decision criteria, and the Borda count method for aggregating decision information [21]. Kong et al. addressed the generation and decision-making of green design schemes throughout the product lifecycle, proposing a multi-attribute decision-making method based on Composite Weighting, TOPSIS, and gray relation analysis, optimizing green design schemes [22]. Yang et al. discussed the effective integration of Multi-Phase Product Design Decision-Making, proposing a nonlinear fusion method that converts decision-makers’ linguistic judgments into three-parameter interval gray number scales, using the interval Analytic Hierarchy Process to calculate weights and support the optimal selection of schemes through the Plant Growth Simulation Algorithm (PGSA) [23]. Liu et al. proposed a conceptual design evaluation method based on Z-numbers to address the issues of uncertainty and subjectivity in the decision-making process. They employed the fuzzy Analytic Hierarchy Process (Z-AHP) under Z-number conditions to determine indicator weights and ultimately evaluated design schemes using the Z-Technique for Order of Preference by Similarity to Ideal Solution (Z-TOPSIS), thereby optimizing the conceptual design schemes [24]. Yao et al. proposed a systematic design decision-making method that utilizes PFA theory as a data collection tool and employs Structural Equation Modeling (SEM) to validate the model. They then integrated path load normalization with the Analytic Hierarchy Process (AHP) for a comprehensive evaluation of product schemes, using a machine tool product design as a case study to verify the reliability of the method [25]. Li et al. utilized Interval-valued Intuitionistic Fuzzy Numbers (IVIFNs) for attribute evaluation and developed an aggregation operator to construct the Fuzzy Synthetic Decision Matrix (FSDM) using the Karnik–Mendel algorithm. They then combined interval-valued intuitionistic fuzzy cross-entropy with the TOPSIS framework to rank the candidate schemes [26].

The above studies provide multiple approaches to the decision-making issues in product design schemes. However, existing research primarily focuses on the expression and processing of evaluative information from decision-makers and the integration of multi-stage decision data. Few studies integrate user preferences with expert evaluations in product design decision-making. Therefore, there remains significant potential for improvement in enhancing user participation and innovating the product form design decision-making model. Although the open nature of the cloud environment effectively harnesses crowd intelligence, it also adds complexity and variability to the decision-making process. When faced with the challenge of different user preferences and needs and other uncertainties, it is even more difficult for ordinary users who do not have professional design knowledge to make clear and objective decision-making results, leading to decision-making results with a large degree of uncertainty and ambiguity. In light of this, this paper proposes a crowd-intelligence-driven decision-making method for product form design in the cloud environment. This method integrates the subjective preferences of the user group with the objective evaluations of the design team in the decision-making process. On the one hand, it broadens the scale of decision-makers and sources of information, enhancing the objectivity and accuracy of the outcomes. On the other hand, it is crucial for guiding the direction of product form design and assessing the feasibility of improvement plans.

2. Crowd Intelligence Service Model for Product Form Design in the Cloud Environment

2.1. Crowd Intelligence Product Form Design in the Cloud Environment

Crowd intelligence product form design in the cloud environment refers to collaborative design innovation activities conducted within a new economic model. This model leverages the open internet to attract and engage interdisciplinary experts and the public, fostering both competition and cooperation [27]. By constructing a cloud platform and implementing diverse incentive mechanisms, this model attracts external resources, integrates and optimizes distributed public knowledge across the network, and generates a wide range of innovative problem-solving approaches and numerous potential product form design schemes. As a result, this model has emerged as the leading paradigm for product design among many enterprises [28,29,30]. The group of participants in product form design activities within the cloud environment is no longer limited to professional designers; it also includes those with product demand, industry practitioners, experts from various fields, enterprises, and government representatives. This shift reflects the multiple heterogeneous characteristics of the cloud environment. At the same time, this asynchronous collaboration model also fosters innovation in design schemes [31].

The product form design cloud platform studied in this paper primarily offers product form design services. The platform allows designers providing various design services to virtually gather in the cloud environment and enables centralized management and control of these designers. At the same time, based on the task requirements published by the platform, each designer engages in creative brainstorming according to their personal experiences, integrating attributes such as shape, color, and material to develop diverse and differentiated product form schemes, and the optimal scheme is then obtained by incorporating multidisciplinary and multi-actor population knowledge to make accurate and rational design decisions. Throughout this process, direct communication between the designers is unnecessary, nor is there a requirement to be aware of other members’ specific information. The entire design process, including task decomposition, allocation, and scheduling, is managed by the cloud platform, which ultimately delivers the final solution to the users, as illustrated in Figure 1.

2.2. Crowd Intelligence Decision-Making Issues in Product Form Design within the Cloud Environment

Due to the diversity, differences, and the vast number of members in the cloud environment [6], after the designers have completed the target program design by integrating multiple attributes based on their own design creativity and experience, a large number of alternative product form designs with different styles and varying quality will be generated, thus posing the problem of deciding on the final product form. Consequently, accurate and rational design decisions are especially important for the subsequent advancement of the product program [32]. The cloud environment expands the approach to product form design decision-making by incorporating the knowledge and experiences of multiple groups into the decision-making process. The public can provide their evaluations of different design options, and this evaluative information adds to the objective evaluations from experts with other angles incorporating users’ needs and emotions. In the actual design process, the decision-making for product form design schemes relies not only on designers’ judgments but also significantly considers users’ subjective preferences. In this process, the decision is based on evaluations of both designers and users, which hold equal importance [12]. Here, it is important to effectively utilize the advantages of crowd intelligence and integrated group evaluation information in order to select a design scheme that not only conforms to user’s preferences but also ensures high feasibility and practicality based on the knowledge of experts.

Accordingly, this paper will focus on the problem of product form design decision-making for cloud environments. This kind of decision-making problem arises as the subjective preference, generated after the cognitive processing of the product form scheme by different users, varies from person to person and is influenced by a variety of factors, such as personal characteristics, which may lead to ambiguities and discrepancies in the decision-making results. The existing decision-making methods are less likely to consider the fuzzy and hesitant preference information of user groups. To overcome that limitation, this paper proposes a crowd-intelligence-driven product form design decision-making method in the cloud environment. This method incorporates user group preferences and expert evaluations into the decision-making process, optimizing the product form design scheme to achieve the most balanced satisfaction between the design group and the user group. At the same time, designers can utilize the evaluation information as a basis for optimizing the product form design scheme, thereby establishing a group intelligence-driven product form design decision-making mode in the cloud environment. This approach also fosters innovation in user participation within the design scheme’s decision-making method.

3. Multi-Attribute Evaluation System for Product Form Design

3.1. 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 Zadeh, 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.

Let $X$ be a domain, and let M be a PHFS defined based on $X$.

$$\mathrm{M}=\left\{<x,{\mathsf{\Gamma }}_{\mathrm{M}}\left(x\right),{\mathsf{\psi }}_{\mathrm{M}}\left(x\right)>|x\in \mathrm{X}\right\}$$

(1)

In Equation (1), $X$ is the set of all product form design alternatives. M is used to denote the ambiguity and uncertainty in the decision-making process for a specific attribute $x,$ where $x$ is one of the attributes of a product form design alternative scheme. Both ${\Gamma }_M\left(x\right)$ and ${\psi }_M\left(x\right)$ are non-empty finite subsets of $\left[0, 1\right]$. ${\Gamma }_M\left(x\right)$ denotes the degree of affiliation of $x$ to M, and it reflects the extent to which the decision-maker believes that the product form design attribute x satisfies or exceeds the evaluation criteria. ${\psi }_M\left(x\right)$ denotes the degree of non-affiliation of $x$ to M, and it reflects the extent to which the decision-maker believes that the product form design attribute x is unsatisfactory or inferior to the evaluation criteria. This set satisfies the condition: $\forall x\in X,$ ${\mu }_M\in {\Gamma }_M\left(x\right),$ $v_M\in {\Psi }_M\left(x\right),$ ${\mu }_M\in \left[\mathrm{0,1}\right],$ $v_M\in \left[\mathrm{0,1}\right],$ $0\le {\mu }_M^2\left(x\right)+v_M^2\left(x\right)\le$ 1, ${\mu }_M^{+}=max\left\{{\mu }_M\right\},$ ${\mu }_A^{+}=max\left\{{\mu }_A\right\}$.

Definition 2.

Let the PHFN be denoted as $\alpha =〈{\Gamma }_{\alpha }, {\Psi }_{\alpha }〉,$ which represents the Pythagorean hesitant fuzzy evaluation linguistic value. The scoring function for each attribute of the alternative schemes is defined as follows:

$${\mathrm{S}}_{\mathsf{\alpha }}={\displaystyle \frac{1}{\left|{\mathsf{\Gamma }}_{\mathsf{\alpha }}\right|}}{\sum }_{\mathsf{\mu }\in {\mathsf{\Gamma }}_{\mathsf{\alpha }}}{\mathsf{\mu }}^2-{\displaystyle \frac{1}{\left|{\mathsf{\psi }}_{\mathsf{\alpha }}\right|}}{\sum }_{v\in {\mathsf{\psi }}_{\mathsf{\alpha }}}{v}^2$$

(2)

In Equation (2), $S_{\alpha }$ is a score function of $\alpha$ that represents the composite score of an attribute in an alternative. $\left|{\Gamma }_{\alpha }\right|$ and $\left|{\psi }_{\alpha }\right|,$ respectively, represent the number of elements in ${\Gamma }_{\alpha }$ and ${\psi }_{\alpha },$ derived from the designers’ and users’ ratings of the attribute.

Definition 3.

Let ${\alpha }_1$ and ${\alpha }_2$ be the evaluative information from two different decision-makers. The Euclidean distance between ${\alpha }_1$ and ${\alpha }_2$ is defined as follows:

$$d({\alpha }_1,{\alpha }_2)=\sqrt{{\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{l\left(\Gamma \right)}}\sum _{h=1}^{l\left(\Gamma \right)}{\left({\mu }_{{\alpha }_{1h}}^2-{\mu }_{{\alpha }_{2h}}^2\right)}^2+{\displaystyle \frac{1}{l\left(\psi \right)}}\sum _{h=1}^{l\left(\psi \right)}{\left(v_{{\alpha }_{1h}}^2-v_{{\alpha }_{2h}}^2\right)}^2+{\displaystyle \frac{1}{l\left(\prod \right)}}\sum _{h=1}^{l\left(\psi \right)}{\left({\pi }_{{\alpha }_{1h}}^2-{\pi }_{{\alpha }_{2h}}^2\right)}^2\right]}$$

(3)

In Equation (3), $l\left(\Gamma \right)$ and $l\left(\psi \right)$ represent the number of elements in the membership and non-membership sets, respectively, and $l\left(\prod \right)$ represents the number of elements in the hesitation set, where the degree of hesitation ${\pi }_{\alpha }$ is the following:

$${\pi }_{\alpha }=\sqrt{1-\left({\displaystyle \frac{1}{\left|{\Gamma }_{\alpha }\right|}}\sum _{\mu \in {\Gamma }_{\alpha }}{\mu }^2+{\displaystyle \frac{1}{\left|{\psi }_{\alpha }\right|}}\sum _{v\in {\Gamma }_{\alpha }}{v}^2\right)}$$

(4)

3.2. Multi-Attribute Evaluation System for Product Form Design

In the process of product form design in the cloud environment, assume there are m alternative scheme sets denoted as $P=\left\{P_1, P_2, P_3,\dots , P_m\right\},$ $n$ attributes represented as $F=\left\{F_1, F_2, F_3,\dots , F_n\right\},$ $u$ sets of expert groups denoted as $D=\left[D_1, D_2, D_3,\dots ,D_u\right],$ and $q$ user groups represented as $U=\left[U_1, U_2, U_3,\dots , U_q\right]$. Experts $D_u$ and users $U_q$ evaluate and assign values to the satisfaction and dissatisfaction of attribute $P_i$ corresponding to scheme $F_j$ in the form of Pythagorean hesitant fuzzy numbers denoted as ${\alpha }_{ij}=〈{\Gamma }_{ij}, {\psi }_{ij}〉$.

They construct the Pythagorean hesitant fuzzy expert individual evaluation matrix $C={\left(c_{ij}^u\right)}_{m\times n}$ and user individual evaluation matrix $R={\left(r_{ij}^q\right)}_{m\times n},$ where $c_{ij}^u$ and $r_{ij}^q,$ respectively, represent the assignment to the $j$th attribute of scheme $P_i$ by the $u$th expert and the $q$th user. Upon standardizing the evaluation matrices based on the Pythagorean Hesitant Fuzzy Sets, the normalized individual decision matrices $\overline{C}$ and $\overline{R}$ are obtained. By integrating the Pythagorean hesitant fuzzy aggregation operator, the attributes of each scheme are integrated to obtain the comprehensive attribute value of the scheme. Finally, the scoring function value for each attribute is calculated, which determines the final score. The multi-attribute evaluation system based on Pythagorean hesitant fuzzy set is shown in

Table 1. Multi-attribute evaluation system based on Pythagorean Hesitant Fuzzy Sets.

		Attributes
Evaluators		${\mathbf{F}}_1$	${\mathbf{F}}_2$	…	${\mathbf{F}}_{\mathbf{n}}$
Experts	${\mathrm{D}}_1$	$〈{\Gamma }_{11}^{\mathrm{D}},{\psi }_{11}^{\mathrm{D}}〉$	$〈{\Gamma }_{12}^{\mathrm{D}},{\psi }_{12}^{\mathrm{D}}〉$	…	$〈{\Gamma }_{1n}^{\mathrm{D}},{\psi }_{1n}^{\mathrm{D}}〉$
	${\mathrm{D}}_2$	$〈{\Gamma }_{21}^{\mathrm{D}},{\psi }_{21}^{\mathrm{D}}〉$	$〈{\Gamma }_{22}^{\mathrm{D}},{\psi }_{22}^{\mathrm{D}}〉$	…	$〈{\Gamma }_{2n}^{\mathrm{D}},{\psi }_{2n}^{\mathrm{D}}〉$
	…	…	…	$〈{\Gamma }_{ij}^{\mathrm{D}},{\psi }_{\mathrm{i}\mathrm{j}}^{\mathrm{D}}〉$	…
	${\mathrm{D}}_{\mathrm{u}}$	$〈{\Gamma }_{q1}^{\mathrm{D}},{\psi }_{q1}^{\mathrm{D}}〉$	$〈{\Gamma }_{q2}^{\mathrm{D}},{\psi }_{q2}^{\mathrm{D}}〉$	…	$〈{\Gamma }_{un}^{\mathrm{D}},{\psi }_{un}^{\mathrm{D}}〉$
Users	${\mathrm{U}}_1$	$〈{\Gamma }_{11}^{\mathrm{u}},{\psi }_{11}^{\mathrm{u}}〉$		…	$〈{\Gamma }_{1n}^{\mathrm{U}},{\psi }_{1n}^{\mathrm{U}}〉$
	${\mathrm{U}}_2$	$〈{\Gamma }_{21}^{\mathrm{u}},{\psi }_{21}^{\mathrm{u}}〉$		…	$〈{\Gamma }_{2n}^{\mathrm{U}},{\psi }_{2n}^{\mathrm{U}}〉$
	…	…	…	$〈{\Gamma }_{ij}^{\mathrm{U}},{\psi }_{ij}^{\mathrm{U}}〉$	…
	${\mathrm{U}}_{\mathrm{q}}$	$〈{\Gamma }_{q1}^{\mathrm{u}},{\psi }_{q1}^{\mathrm{u}}〉$		…	$〈{\Gamma }_{un}^{\mathrm{U}},{\psi }_{un}^{\mathrm{U}}〉$

.

4. Crowd-Intelligence-Driven, Multi-Attribute Decision-Making Model for Product Form Design in the Cloud Environment

To enhance the accuracy and objectivity of decision-making outcomes in the cloud environment, this paper proposes a crowd-intelligence-driven, multi-attribute decision-making model for product form design. By integrating evaluation data informed by expert knowledge and user preferences, this model optimizes the selection of product form design alternatives, thereby facilitating more comprehensive and effective decision-making.

In terms of expert knowledge-based decision making for product form, experts must not only evaluate the alternatives holistically but also engage in a comparative analysis of attributes to provide more objective decision support. The gray relation analysis (GRA) method can effectively respond to the differences between each attribute of the alternatives and the ideal scheme by quantifying the correlation among multiple attributes of the product form. However, there are defects in its overall evaluation of the system solution [40]. The VIKOR method is a decision-making approach that integrates multiple attributes to determine the optimal scheme based on the combined score of the scheme across each attribute [41]. The Improved Osculating Value method, which transforms multiple decision attributes into a composite value that can measure the advantages and disadvantages of a program as a whole, is more prominent in terms of simplicity and flexibility [42]. By integrating the GRA, VIKOR method, and the Improved Osculating Value method, the decision-making process can comprehensively consider the correlations among the attributes of the product form design alternatives, visually portraying the proximity of the alternatives to the ideal solution, thereby facilitating multi-faceted assessment. Therefore, this paper constructs an expert’s preliminary decision matrix based on PHFS, and the GRA method is employed to construct a gray relational coefficient decision matrix, which calculates the attribute weights by considering the degree of association, thereby quantifying the differences and correlations among various alternative schemes. Finally, the scheme evaluation value, based on expert knowledge, is determined using the VIKOR method combined with a modified osculating method.

When it comes to decision-making for product form schemes based on user preferences, user preference reflects the user’s needs, interests, and preferences for the product form, and the ultimate goal of the enterprise is to better satisfy the user preferences. Taking user preferences into account in the decision-making process of product design is helpful to improve the overall satisfaction of users [43]. Consequently, this paper incorporates user preferences into the decision-making process by constructing a user-preference-based decision model for product form schemes to improve user’s acceptance of the schemes. It is difficult for users to give specific evaluation information directly because they do not have a good knowledge of the product domain in which they want to make decisions. Therefore, this paper adopts PHFS to construct the decision matrix of user preference, and then calculates the similarity between the decision matrix of user evaluation through the Hamming distance to obtain the weights of different users, thereby making the weight determination more scientific. Finally, the TOPSIS method is introduced to consider the closeness of the alternatives to the ideal solution, to derive the preference information value. The advantage of this approach is that it does not require users to be very familiar with the domain; they can simply rate the attributes according to their own preferences.

The specific decision-making process is shown in Figure 2:

4.1. Decision-Making for the Product Form Design Scheme Based on Expert Knowledge

Step 1. Construct an evaluation matrix based on expert knowledge.

① The set of alternatives $P$ and the set of attributes $F$ are identified.

② Experts evaluate each scheme’s performance on various attributes using Pythagorean hesitant fuzzy numbers, creating an initial $m\times n$ decision matrix $C={\left(c_{ij}\right)}_{m\times n}$.

$$\mathrm{C}=\left[\begin{array}{cc}{c}_{11}& \begin{array}{ccc}{c}_{12}& \cdots & c_{1n}\end{array}\\ c_{21}& \begin{array}{ccc}{c}_{22}& \cdots & c_{2n}\end{array}\\ \begin{array}{c}⋮\\ c_{m1}\end{array}& \begin{array}{c}\begin{array}{ccc}⋮& & ⋮\end{array}\\ \begin{array}{ccc}{c}_{m2}& \cdots & c_{mn}\end{array}\end{array}\end{array}\right]$$

(5)

③ Normalize matrix $\overline{\mathrm{C}}$ to refine the evaluations.

$$\overline{\mathrm{C}}=\left[\begin{array}{cc}\overline{{\mathrm{c}}_{11}}& \begin{array}{ccc}\overline{{\mathrm{c}}_{12}}& \cdots & \overline{{\mathrm{c}}_{1\mathrm{n}}}\end{array}\\ \overline{{\mathrm{c}}_{21}}& \begin{array}{ccc}\overline{{\mathrm{c}}_{21}}& \cdots & \overline{{\mathrm{c}}_{2\mathrm{n}}}\end{array}\\ \begin{array}{c}⋮\\ \overline{{\mathrm{c}}_{\mathrm{m}1}}\end{array}& \begin{array}{c}\begin{array}{ccc}⋮& & ⋮\end{array}\\ \begin{array}{ccc}\overline{{\mathrm{c}}_{\mathrm{m}2}}& \cdots & \overline{{\mathrm{c}}_{\mathrm{m}\mathrm{n}}}\end{array}\end{array}\end{array}\right]$$

(6)

④ The overall score for each attribute of the program is derived by applying the PHFWG to integrate the expert evaluation matrix $C^{\prime }$.

Step 2. Establish a gray relational coefficient decision matrix and determine the weights of attributes.

① For each attribute in the decision matrix, its gray correlation coefficient with the ideal sequence is calculated by the GRA method. All the calculated gray correlation coefficients are combined into a gray correlation coefficient decision matrix $\xi$. Calculate the correlation between attributes to determine the importance of each attribute to the overall scheme and the degree of influence between attributes. The formula is as follows:

$$\left\{\begin{array}{c}{\mathrm{P}\mathrm{I}\mathrm{S}}_x^{+}=max\left({\mathrm{c}}_{i\mathrm{j}}\right)\\ {\mathrm{N}\mathrm{I}\mathrm{S}}_x^{-}=min\left({\mathrm{c}}_{i\mathrm{j}}\right)\end{array}\right.$$

(7)

In Equation (7), ${PIS}_x^{+}$ and ${NIS}_x^{-}$ denote the positive and negative ideal solutions for the jth attribute of the ith scheme, respectively.

$${\mathsf{\xi }}_{ij}={\displaystyle \frac{min min\left|{\mathrm{C}}_0\left(j\right)-\mathrm{c}(i,j)\right|+\rho max max\left|{\mathrm{C}}_0\left(j\right)-\mathrm{c}(i,j)\right|}{\left|{\mathrm{C}}_0\left(j\right)-\mathrm{c}(i,j)\right|+\rho max max\left|{\mathrm{C}}_0\left(j\right)-\mathrm{x}(i,j)\right|}}$$

(8)

In Equation (8), ${\xi }_{ij}$ represents the gray relational coefficient between the $j$th attribute of the $i$th scheme and the ideal solution$, C_0\left(j\right)$ is the value of the $j$th attribute in the ideal solution$, x(i,j)$ is the value of the $j$th attribute for the $i$th scheme in the matrix, $\rho$ is the resolution coefficient, typically set at 0.5 [44], and $min min$ and $max max,$ respectively, represent the min min and max max operations, taken over all $i$ and$j$.

② Form the gray relational coefficient decision matrix $\xi$ and calculate attribute correlations $r_j$.

$$\mathsf{\xi }=\left[\begin{array}{ccc}{\mathsf{\xi }}_{11}& {\mathsf{\xi }}_{12}& \begin{array}{cc}\cdots & {\mathsf{\xi }}_{1\mathrm{n}}\end{array}\\ {\mathsf{\xi }}_{21}& {\mathsf{\xi }}_{22}& \begin{array}{cc}\cdots & {\mathsf{\xi }}_{2\mathrm{n}}\end{array}\\ \begin{array}{c}⋮\\ {\mathsf{\xi }}_{\mathrm{m}1}\end{array}& \begin{array}{c}⋮\\ {\mathsf{\xi }}_{\mathrm{m}2}\end{array}& \begin{array}{c}\begin{array}{cc}& ⋮\end{array}\\ \begin{array}{cc}\cdots & {\mathsf{\xi }}_{\mathrm{m}\mathrm{n}}\end{array}\end{array}\end{array}\right]$$

(9)

In Equation (9), $\xi$ is the gray correlation coefficient decision matrix.

$$r_j={\displaystyle \frac{1}{m(\mathrm{n}-1)}}\sum _{i=1}^m\sum _{k=1, k\ne i}^n\left({\mathsf{\xi }}_{ij}\right)$$

(10)

In Equation (10), $r_j$ is the gray correlation of the attribute.

By normalization, the attribute weights $w_j$ are as follows:

$$w_j=r_j/\sum _{j=1}^m{r}_j$$

(11)

Step 3. Obtain the expert evaluation values.

The VIKOR method was first used to determine the group utility value $S_i,$ individual regret $R_i$ value, and benefit ratio $Q_i$ for each scheme. Then, the Improved Closeness Method was applied to enhance the variability between the schemes, in order to more accurately assess the degrees of program strengths and weaknesses [45]. A more comprehensive and balanced decision can be made by combining the two approaches.

① Calculate the group utility value $S_i,$ individual regret value $R_i,$ and benefit ratio $Q_i$ for each scheme to evaluate proximity to the ideal scheme.

$$\left\{\begin{array}{c}{S}_i={\displaystyle \sum _{j=1}^{\mathrm{b}}}{w}_j·{\displaystyle \frac{\left({\mathrm{P}\mathrm{I}\mathrm{S}}_x^{+}-{\mathrm{c}}_{i\mathrm{j}}\right)}{\left({\mathrm{P}\mathrm{I}\mathrm{S}}_x^{+}-{\mathrm{N}\mathrm{I}\mathrm{S}}_x^{-}\right)}}\\ R_i={\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathrm{j}}\left[w_j·{\displaystyle \frac{\left({\mathrm{P}\mathrm{I}\mathrm{S}}_x^{+}-{\mathrm{c}}_{i\mathrm{j}}\right)}{\left({\mathrm{P}\mathrm{I}\mathrm{S}}_x^{+}-{\mathrm{N}\mathrm{I}\mathrm{S}}_x^{-}\right)}}\right]\\ Q_{\mathrm{i}}=v{\displaystyle \frac{({\mathrm{S}}_i-\mathrm{m}\mathrm{i}\mathrm{n}{\mathrm{S}}_i)}{({\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{S}}_i-\mathrm{m}\mathrm{i}\mathrm{n}{\mathrm{S}}_i)}}+(1-\beta ){\displaystyle \frac{({\mathrm{R}}_i-\mathrm{m}\mathrm{i}\mathrm{n}{\mathrm{R}}_i)}{({\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{R}}_i-\mathrm{m}\mathrm{i}\mathrm{n}{\mathrm{R}}_i)}}\end{array}\right.$$

(12)

In Equation (12), $\beta$ is the decision coefficient, generally set at 0.5 [42]. Use $Q_i$ to measure the proximity to the ideal solution to determine the optimal scheme, with smaller values indicating better schemes.

② Calculate the osculating value.

$${NIS}_x^{*}=2·{NIS}_x^{-}-{PIS}_x^{+}$$

(13)

$$\left\{\begin{array}{c}{D}_i^{+}=\sqrt{{\displaystyle \sum _{j=1}^b}{w}_j·{\left(c_{ij}-{PIS}_x^{+}\right)}^2}\\ D_i^{*}=\sqrt{{\displaystyle \sum _{j=1}^b}{w}_j·{\left(c_{ij}-{NIS}_x^{*}\right)}^2}\end{array}\right.$$

(14)

$${OV}_i={\displaystyle \frac{D_i^{+}}{D^{+}}}-{\displaystyle \frac{D_i^{*}}{D^{*}}}$$

(15)

In Equations (13)–(15), ${NIS}_x^{*}$ represents the virtual worst-case ideal solution, $D_i^{+}$ and $D_i^{*}$ are, respectively, the weighted Euclidean distances of ${PIS}_x^{+}$ and ${NIS}_x^{*},$ and ${OV}_i$ represents the osculating value of the product form scheme, with smaller values indicating a better performance of the scheme.

③ Aggregate evaluation value.

$${\mathsf{\sigma }}_{\mathrm{i}}=\mathsf{\lambda }{\mathrm{Q}}_{\mathrm{i}}^{\mathrm{*}}+\left(1-\mathsf{\lambda }\right){\mathrm{O}\mathrm{V}}_{\mathrm{i}}^{\mathrm{*}}$$

(16)

In Equation (16), $Q_i^{*}$ and ${OV}_i^{*}$ are the normalized values of the benefit ratio and the osculating value, respectively. $\lambda$ is the allocation coefficient, generally set at λ = 0.5.

4.2. Decision-Making for the Product Form Design Scheme Based on User Preferences

Step 1. Construct an evaluation matrix based on user preferences.

① Each user assigns a preference rating $r_{ij}$ to the attribute $F_j$ of scheme $P_i$ using PHFNs, reflecting their degree of preference.

② Compile these evaluations into an individual user evaluation matrix $R={\left(r_{ij}\right)}_{m\times n},$ with the normalized matrix being $\overline{R}={\left(\overline{r_{ij}}\right)}_{m\times n}$.

Step 2. Calculate user weights.

Due to variations in the needs and priorities of different users, conflicts in evaluation data may occur, complicating the accurate reflection of user satisfaction. Consequently, this paper determines user weights based on the similarity of hesitant Pythagorean fuzzy matrices [46]. The core idea revolves around computing the similarity (SIM) between the evaluation matrices of each user, utilizing the Hamming distance, in order to obtain the support (SUP) for that user’s preference towards other users. Higher similarity indicates greater support, suggesting that the evaluation provided by that user is more influential, i.e., user weight $\gamma$. Accordingly, users with greater influence are assigned higher weights. The formula is as follows:

$$SIM\left(q,l\right)={\displaystyle \frac{1}{m}}\sum _{i=1}^m\sum _{j=1}^n{w}_{j}s\left({\alpha }_{ij}^q,{\alpha }_{ij}^l\right)$$

(17)

$$SIM\left({\alpha }_{ij}^q,{\alpha }_{ij}^l\right)=\left\{\begin{array}{c}0.5 ,{\alpha }_1={\alpha }_2{=\alpha }_3 \\ {\displaystyle \frac{d\left({{\alpha }_{ij}^q}^{\prime },{\left({{\alpha }_{ij}^l}^{\prime }\right)}^c\right)}{d\left({{\alpha }_{ij}^q}^{\prime },{{\alpha }_{ij}^l}^{\prime }\right)+d\left({{\alpha }_{ij}^q}^{\prime },{\left({{\alpha }_{ij}^l}^{\prime }\right)}^c\right)}},others\end{array}\right.$$

(18)

$$SIM=\left[\begin{array}{ccc}{SIM}_{11}& {SIM}_{12}& \begin{array}{cc}\cdots & {SIM}_{1q}\end{array}\\ {SIM}_{21}& {SIM}_{22}& \begin{array}{cc}\cdots & {SIM}_{2q}\end{array}\\ \begin{array}{c}⋮\\ {SIM}_{q1}\end{array}& \begin{array}{c}⋮\\ {SIM}_{q2}\end{array}& \begin{array}{c}\begin{array}{cc}& ⋮\end{array}\\ \begin{array}{cc}\cdots & {SIM}_{qq}\end{array}\end{array}\end{array}\right]$$

(19)

In Equations (17)–(19), $SIM\left(q,l\right)$ represents the similarity between the evaluation matrices of users $U_q$ and $U_l,$ $SIM\left({\alpha }_{ij}^q,{\alpha }_{ij}^l\right)$ is the similarity between fuzzy numbers ${\alpha }_{ij}^q$ and ${\alpha }_{ij}^l,$ also known as the Hamming distance, $w_j$ is the attribute weight, and $SIM$ is the similarity matrix of all users.

$${SUP}_q=\sum _{\begin{array}{c}L\\ l=1,l\ne q\end{array}}^m{SIM}_{ql}$$

(20)

$${\gamma }_q={\displaystyle \frac{{SUP}_q}{\sum _{k=1}^m{SUP}_k}}$$

(21)

In Equations (20)–(21), ${SUP}_q$ and ${\gamma }_q$ are the support and weight of user $U_q,$ respectively.

Step 3. Calculate the user preference values for the alternative schemes.

① Aggregate the normalized decision matrix and user weight information to obtain the Pythagorean hesitant fuzzy weighted scores for each attribute of the alternative schemes.

② Map the distance of each scheme to the positive and negative ideal schemes according to Pythagorean hesitant fuzzy distance mapping.

$$\left\{\begin{array}{c}{d}^{+}\left(P_i\right)=\sqrt{{\displaystyle \sum _{j=1}^n}{w}_j\left[{\left(s_{ij}^{+}-s_{ij}\right)}^2+{\left(t_{ij}^{+}-t_{ij}\right)}^2\right]}\\ d^{-}\left(P_i\right)=\sqrt{{\displaystyle \sum _{j=1}^n}{w}_j\left[{\left(s_{ij}^{-}-s_{ij}\right)}^2+{\left(t_{ij}^{-}-t_{ij}\right)}^2\right]}\end{array}\right.$$

(22)

In Equation (22), $d^{+}\left(P_i\right)$ and $d^{-}\left(P_i\right)$ are the distances from scheme $P_i$ to the positive and negative ideal schemes, respectively; $s_{ij}^{+}$ and $t_{ij}^{+}$ represent the satisfaction and dissatisfaction of the jth attribute in the positive ideal scheme, respectively; $s_{ij}^{-}$ and $t_{ij}^{-}$ represent the satisfaction and dissatisfaction of the jth attribute in the negative ideal scheme, respectively; and $w_j$ is the weight of the attribute.

③ Calculate the degree of superiority or inferiority for each scenario based on the ratio of the distances between each scheme and the positive and negative ideal schemes.

$${\tau }_i={\displaystyle \frac{d^{-}\left(P_i\right)}{d^{+}\left(P_i\right)+d^{-}\left(P_i\right)}}$$

(23)

In Equation (23), ${\tau }_i$ represents the final score of the scheme based on user preferences, with higher values indicating a closer to optimal scheme.

4.3. Crowd Intelligence Scheme Optimization

To ensure that the decision-making outcomes accurately reflect both the objectivity of the expert group and the varying preference levels of different user groups, this study employed the Deviation Minimization method to merge the evaluations of experts and users in the final decision-making process.

$$\left\{\begin{array}{c}\mathit{min}\eta =\mathit{min}{\displaystyle \sum _{g=1}^2}{\displaystyle \sum _{h=1}^n}{(w_h^{*}-{\overline{w}}_g)}^2\\ w_h^{*}=\phi w_{1h}+(1-\phi )w_{2h}\\ {\overline{w}}_g={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^n}{w}_{gh}\end{array}\right.$$

(24)

In Equation (24), the final decision result is obtained by ranking the alternatives based on the composite score $\mathit{min}\eta$. The higher the value, the more the scheme satisfies the experts and users, meaning it is the optimal scheme.

5. The Case Study Results and Analysis

Taking the coffee machine product form design solution as an example, the method proposed in this paper was used to make a decision on the set of alternatives. The entire case process simulated the cloud environment, where decision-making members complete an evaluation of the product form design scheme in a network-connected environment. The specific decision-making process is described below:

5.1. Constructing a Multi-Attribute Evaluation System for Alternative Schemes

In response to corporate requirements for product form design, eight alternative design schemes, denoted as $\mathrm{P}=\left[{\mathrm{P}}_1,{\mathrm{P}}_2,\dots ,{\mathrm{P}}_8\right],$ were proposed, as illustrated in Figure 3. Reflecting corporate needs, five attributes, labeled $\mathrm{F}=\left[{\mathrm{F}}_1,{\mathrm{F}}_2,{\mathrm{F}}_3,{\mathrm{F}}_4,{\mathrm{F}}_5\right],$ were identified: shape $\left({\mathrm{F}}_1\right),$ color $\left({\mathrm{F}}_2\right),$ material $\left({\mathrm{F}}_3\right),$ interactivity $\left({\mathrm{F}}_4\right),$ and ergonomics $\left({\mathrm{F}}_5\right)$.

Among these attributes, shape evaluates each scheme’s alignment with user requirements for simplicity, beauty, and fashion in appearance, as well as how well it integrates into modern home design. Color assesses whether the color design of each scheme not only enhances the overall aesthetics of the product but also reflects its function and purpose, particularly through the use of colors that prompt alertness. Material considers the durability of each scheme’s materials and whether their combination with surface treatment techniques can elevate the product’s texture and grade. Interactivity measures how easily users can operate the machine and make coffee, facilitated by intuitive button layouts, clear displays, and user-friendly procedures. Ergonomics evaluates each scheme for its conformity to user habits and operational convenience during use.

The expert decision group $\mathrm{D}$ and the user decision group $\mathrm{U}$ each consisted of 10 members, represented as $\mathrm{D}=\left[{\mathrm{D}}_1,{\mathrm{D}}_2,\dots ,{\mathrm{D}}_{10}\right]$ and $\mathrm{U}=\left[{\mathrm{U}}_1,{\mathrm{U}}_2,\dots ,{\mathrm{U}}_{10}\right],$ respectively. According to the proposed crowd-intelligence-driven, multi-attribute decision-making method for product form design, the alternative schemes were optimized with the objective of selecting the design scheme that achieved the most balanced satisfaction between experts and users.

5.2. Multi-Attribute Decision-Making Process for Product Form Design Driven by Crowd Intelligence in the Cloud Environment

Step 1: The expert group evaluated and assigned values to the attributes corresponding to the alternatives, utilizing PHFNs based on 0–1 rating values transformed by 11 levels of semantic information evaluation. They then construct the hesitant Pythagorean fuzzy evaluation matrix based on the given decision-making information, taking ${\mathrm{P}}_1$ as an example, as shown in

Table 2. Pythagorean hesitant fuzzy expert evaluation of the initial matrix.

${\mathbf{P}}_1$	${\mathbf{F}}_1$	${\mathbf{F}}_2$	${\mathbf{F}}_3$	${\mathbf{F}}_4$	${\mathbf{F}}_5$
${\mathrm{D}}_1$	(0.5,0.6,0.7) (0.3,0.4)	(0.5,0.6) (0.3,0.4)	(0.5) (0.5)	(0.6,0.7) (0.4)	(0.5,0.7) (0.4,0.5)
${\mathrm{D}}_2$	(0.7,0.8) (0.2,0.3)	(0.7,0.8) (0.2,0.3)	(0.7,0.8) (0.2,0.3)	(0.4,0.5) (0.5,0.6)	(0.4,0.5) (0.5,0.6)
${\mathrm{D}}_3$	(0.5,0.6) (0.4,0.5)	(0.4,0.5) (0.5,0.6)	(0.5,0.6) (0.4,0.5)	(0.4,0.5) (0.5,0.6)	(0.3,0.4) (0.6,0.7)
${\mathrm{D}}_4$	(0.6,0.7) (0.4,0.5)	(0.7,0.8) (0.2,0.3)	(0.8,0.9) (0.2,0.3)	(0.5,0.6) (0.4,0.5)	(0.6) (0.4,0.5)
${\mathrm{D}}_5$	(0.5,0.6) (0.4,0.5)	(0.4,0.5) (0.5,0.6)	(0.5,0.6) (0.4,0.5)	(0.5,0.7) (0.3,0.4)	(0.5,0.7) (0.4,0.5)
${\mathrm{D}}_6$	(0.5,0.6) (0.4,0.5)	(0.8,0.9) (0.1,0.2)	(0.7,0.8) (0.3,0.4)	(0.6,0.7) (0.4,0.5)	(0.6,0.7) (0.4,0.5)
${\mathrm{D}}_7$	(0.5,0.6) (0.4,0.5)	(0.5,0.6) (0.3,0.4)	(0.5,0.6) (0.3,0.5)	(0.7,0.8) (0.2,0.3)	(0.6,0.8) (0.2,0.4)
${\mathrm{D}}_8$	(0.5,0.6,0.7) (0.2,0.3)	(0.7,0.8) (0.2,0.4)	(0.7,0.9) (0.2,0.3)	(0.7,0.9) (0.2,0.3)	(0.6,0.7) (0.4,0.5)
${\mathrm{D}}_9$	(0.5,0.6) (0.4,0.5)	(0.5,0.6) (0.3,0.4)	(0.5,0.6) (0.2,0.4)	(0.6,0.7) (0.4,0.5)	(0.5,0.6) (0.3,0.4)
${\mathrm{D}}_{10}$	(0.6,0.7) (0.4,0.6)	(0.6,0.7) (0.2,0.3)	(0.7,0.8) (0.3,0.5)	(0.6,0.7) (0.3,0.5)	(0.6,0.8) (0.2,0.3)

.

Step 2: Using a single PHFN $〈\left(\mathrm{0.5,0.6}\right)\left(\mathrm{0.3,0.4}\right)〉$ as an example, we standardized it to $〈\left(\mathrm{0.5,0.55,0.6}\right)\left(\mathrm{0.3,0.4}\right)〉,$ thus obtaining the entire standardized Pythagorean hesitant fuzzy expert evaluation matrix.

Step 3: Then, in accordance with Equation (2), we aggregated the membership degree and non-membership degree values from the expert evaluations for each attribute using the PHFWG, as shown in

Table 3. Score of attributes for each alternative scheme based on expert knowledge.

	${\mathbf{P}}_1$	${\mathbf{P}}_2$	${\mathbf{P}}_3$	${\mathbf{P}}_4$	${\mathbf{P}}_5$	${\mathbf{P}}_6$	${\mathbf{P}}_7$	${\mathbf{P}}_8$
${\mathrm{F}}_1$	0.224288	0.497183	0.184793	0.321727	0.341332	0.590379	0.640773	0.473467
${\mathrm{F}}_2$	0.356094	0.484521	0.198694	0.491200	0.463439	0.410996	0.612154	0.283782
${\mathrm{F}}_3$	0.374902	0.409930	0.415529	0.347796	0.272320	0.413849	0.533784	0.578542
${\mathrm{F}}_4$	0.271011	0.380614	0.387004	0.378025	0.433941	0.413410	0.365342	0.515332
${\mathrm{F}}_5$	0.202519	0.342664	0.533862	0.316413	0.514885	0.489927	0.315685	0.377816

.

Step 4: In accordance with Equations (7)–(9), we calculated the gray relational coefficients between each attribute in the expert decision matrix and the ideal scheme. This process resulted in the formation of the gray relational coefficient decision matrix.

$$\mathsf{\xi }=\left[\begin{array}{ccccc}0.353761& 0.471005& 0.528207& 0.482712& 0.407610\\ 0.613569& 0.641101& 0.574858& 0.628577& 0.543885\\ 0.333333& 0.355429& 0.583090& 0.639851& 1.000000\\ 0.416773& 0.653372& 0.496996& 0.624123& 0.511832\\ 0.432265& 0.605221& 0.426778& 0.736923& 0.923160\\ 0.818977& 0.531262& 0.580595& 0.691064& 0.838431\\ 1.000000& 1.000000& 0.835902& 0.603181& 0.510097\\ 0.576758& 0.409787& 1.000000& 1.000000& 0.593669\end{array}\right]$$

Step 5: We calculated the correlation weights for the attributes of each scheme using Equations (10) and (11), as shown in

Table 4. Associated weights of attributes for each scheme.

${\mathbf{F}}_1$	${\mathbf{F}}_2$	${\mathbf{F}}_3$	${\mathbf{F}}_4$	${\mathbf{F}}_5$
0.177493	0.183588	0.201573	0.220597	0.216750

.

Step 6: Using Equations (12)–(15), we calculated the group utility value ${\mathbf{S}}_{\mathbf{i}},$ individual regret value ${\mathit{R}}_{\mathit{i}},$ benefit ratio ${\mathit{Q}}_{\mathit{i}},$ and closeness value ${\mathit{O}\mathit{V}}_{\mathit{i}}$ for each scheme. Finally, we derived the comprehensive score for each scheme based on Equation (16), as shown in

Table 5. Comprehensive score of schemes based on expert knowledge.

	${\mathit{S}}_{\mathit{i}}$	${\mathit{R}}_{\mathit{i}}$	${\mathit{Q}}_{\mathit{i}}$	${\mathit{O}\mathit{V}}_{\mathit{i}}$	${\mathit{\sigma }}_{\mathit{i}}$
${\mathrm{P}}_1$	0.847211	0.220597	1.000000	1.000000	1.000000
${\mathrm{P}}_2$	0.470265	0.125073	0.224982	0.238228	0.220334
${\mathrm{P}}_3$	0.584252	0.183588	0.591397	0.830444	0.704978
${\mathrm{P}}_4$	0.596007	0.151890	0.461016	0.547013	0.496176
${\mathrm{P}}_5$	0.470068	0.201573	0.565749	0.435473	0.494296
${\mathrm{P}}_6$	0.338111	0.108410	0.028264	0.059884	0.029942
${\mathrm{P}}_7$	0.307608	0.142721	0.152920	0.000000	0.064141
${\mathrm{P}}_8$	0.313009	0.145806	0.171674	0.284489	0.216035

.

Step 7: Consistent with the expert group in Step 1, the user evaluation information was aggregated and standardized. After obtaining the standardized evaluation matrix, the similarity between the evaluation matrices of each user was calculated using Equations (3), (4), (17) and (18) to obtain the user similarity matrix.

$$SIM=\left[\begin{array}{cccccccccc}\begin{array}{c}1\\ 0.6352\\ 0.6631\\ 0.6680\\ 0.6693\\ 0.6576\\ 0.6743\\ 0.6464\\ 0.6544\\ 0.6804\end{array}& \begin{array}{c}0.6356\\ 1\\ 0.7054\\ 0.7138\\ 0.7396\\ 0.7144\\ 0.6673\\ 0.6800\\ 0.7149\\ 0.7119\end{array}& \begin{array}{c}0.6630\\ 0.7054\\ 1\\ 0.7696\\ 0.6888\\ 0.7228\\ 0.7251\\ 0.6845\\ 0.7359\\ 0.7430\end{array}& \begin{array}{c}0.6680\\ 0.7138\\ 0.7696\\ 1\\ 0.7466\\ 0.7329\\ 0.7570\\ 0.7028\\ 0.7155\\ 0.7572\end{array}& \begin{array}{c}0.6695\\ 0.7396\\ 0.6888\\ 0.7466\\ 1\\ 0.7431\\ 0.7167\\ 0.7329\\ 0.7264\\ 0.7746\end{array}& \begin{array}{c}0.6578\\ 0.7144\\ 0.7228\\ 0.7239\\ 0.7431\\ 1\\ 0.6862\\ 0.7190\\ 0.7040\\ 0.7313\end{array}& \begin{array}{c}0.6743\\ 0.6673\\ 0.7251\\ 0.7570\\ 0.7167\\ 0.6862\\ 1\\ 0.7097\\ 0.7052\\ 0.7579\end{array}& \begin{array}{c}0.6464\\ 0.6800\\ 0.6845\\ 0.7082\\ 0.7329\\ 0.7109\\ 0.7079\\ 1\\ 0.7092\\ 0.7147\end{array}& \begin{array}{c}0.6544\\ 0.7194\\ 0.7359\\ 0.7155\\ 0.7264\\ 0.7040\\ 0.7052\\ 0.7092\\ 1\\ 0.7315\end{array}& \begin{array}{c}0.6804\\ 0.7119\\ 0.7403\\ 0.7570\\ 0.7745\\ 0.7313\\ 0.7579\\ 0.7147\\ 0.7315\\ 1\end{array}\end{array}\right]$$

Step 8: We obtained each user’s support $\mathrm{s}\mathrm{u}\mathrm{p}$ and user weight $\mathsf{\gamma }$ using Equations (20) and (21), as shown in

Table 6. User support.

${\mathit{U}}_1$	${\mathit{U}}_2$	${\mathit{U}}_2$	${\mathit{U}}_4$	${\mathit{U}}_5$	${\mathit{U}}_6$	${\mathit{U}}_7$	${\mathit{U}}_8$	${\mathit{U}}_9$	${\mathit{U}}_{10}$
5.9487	6.2829	6.4354	6.5596	6.5382	6.4025	6.3994	6.3046	6.3970	6.5996

and

Table 7. User weights.

${\mathit{U}}_1$	${\mathit{U}}_2$	${\mathit{U}}_2$	${\mathit{U}}_4$	${\mathit{U}}_5$
0.093141	0.098373	0.100761	0.102706	0.102371
${\mathit{U}}_6$	${\mathit{U}}_7$	${\mathit{U}}_8$	${\mathit{U}}_9$	${\mathit{U}}_{10}$
0.100246	0.100197	0.098713	0.100160	0.103332

.

Step 9: The score function for each attribute was calculated by applying user weights to the normalized decision matrix, as shown in

Table 8. Score of attributes for each alternative scheme based on user preferences.

	${\mathbf{P}}_1$	${\mathbf{P}}_2$	${\mathbf{P}}_3$	${\mathbf{P}}_4$	${\mathbf{P}}_5$	${\mathbf{P}}_6$	${\mathbf{P}}_7$	${\mathbf{P}}_8$
${\mathrm{F}}_1$	0.373414	0.473689	0.147769	0.357678	0.348689	0.574266	0.335775	0.557677
${\mathrm{F}}_2$	0.426050	0.402978	0.197844	0.454809	0.489687	0.598626	0.357815	0.453676
${\mathrm{F}}_3$	0.193873	0.436209	0.270683	0.398721	0.199942	0.535557	0.528005	0.433870
${\mathrm{F}}_4$	0.314353	0.304394	0.290424	0.318713	0.341915	0.495648	0.450992	0.467468
${\mathrm{F}}_5$	0.336141	0.424044	0.401942	0.274257	0.286089	0.550080	0.419141	0.425946

.

Step 10: We calculated the scheme score $\mathsf{\tau }$ based on user preferences according to Equations (22) and (23), as shown in

Table 9. Comprehensive score of schemes based on user preferences.

${\mathbf{P}}_1$	${\mathbf{P}}_2$	${\mathbf{P}}_3$	${\mathbf{P}}_4$	${\mathbf{P}}_5$	${\mathbf{P}}_6$	${\mathbf{P}}_7$	${\mathbf{P}}_8$
0.255068	0.596740	0.175258	0.497878	0.265392	1.000000	0.715532	0.752909

.

Step 11: After that, we normalized ${\mathit{\sigma }}_{\mathit{i}}$ and ${\mathit{\tau }}_{\mathit{i}}$. Using a deviation minimization optimization function, we determined the optimal values of $\mathit{\phi },$ which resulted in expert weights of 0.52 and user weights of 0.42. We calculated the comprehensive score of the scheme according to Equation (24), as shown in

Table 10. Comprehensive score of crowd-intelligence-driven product form design schemes.

	${\mathsf{\sigma }}_{\mathbf{i}}$	${\mathit{\tau }}_{\mathit{i}}$	Comprehensive Score
${\mathrm{P}}_1$	0.000000	0.096770	0.046450
${\mathrm{P}}_2$	0.803731	0.511047	0.663243
${\mathrm{P}}_3$	0.304128	0.000000	0.158147
${\mathrm{P}}_4$	0.519375	0.391177	0.457840
${\mathrm{P}}_5$	0.521313	0.109288	0.323541
${\mathrm{P}}_6$	1.000000	1.000000	1.00000
${\mathrm{P}}_7$	0.964745	0.655082	0.816107
${\mathrm{P}}_8$	0.808163	0.700402	0.756438

.

The final ranking of the schemes was ${\mathbf{P}}_6$→${\mathbf{P}}_7$→${\mathbf{P}}_8$→${\mathbf{P}}_2$→${\mathbf{P}}_4$→${\mathbf{P}}_5$→${\mathbf{P}}_3$→${\mathbf{P}}_1,$ where ${\mathbf{P}}_6$ is the product form design scheme that best balanced the satisfaction of experts and users.

5.3. Case Analysis

Using the multi-attribute decision-making method for product form design proposed in this paper, which integrates expert knowledge and user preference, the weights of the expert group and the user group were determined to be 0.52 and 0.48 respectively, through the Deviation Minimization method. The final weighting resulted in the order of proximity of the alternatives to the idealized scheme in this case as follows: ${\mathbf{P}}_6$→${\mathbf{P}}_7$→${\mathbf{P}}_8$→${\mathbf{P}}_2$→${\mathbf{P}}_4$→${\mathbf{P}}_5$→${\mathbf{P}}_3$→${\mathbf{P}}_1$.

As can be seen in Table 5, although the maximum group utility values ${\mathbf{S}}_{\mathbf{i}}$ of ${\mathbf{P}}_1$ are all greater than the rest of the schemes in the expert-knowledge-based decision-making process, its minimum individual regret value ${\mathbf{R}}_{\mathbf{i}}$ is also the largest, indicating that this scheme has the largest gap between it and the best scheme as well as the worst performance under each criterion. ${\mathbf{P}}_6$ has the smallest individual regret value ${\mathit{R}}_{\mathit{i}},$ indicating that this scheme has the smallest gap with the best scheme and performs relatively well under each criterion, with the least compromise across them. The ratio value ${\mathit{Q}}_{\mathit{i}}$ measures the degree of balance between ${\mathit{R}}_{\mathit{i}}$ and ${\mathit{Q}}_{\mathit{i}}$. ${\mathbf{P}}_6$ has the smallest benefit ratio, indicating a superior balance between the attributes. Therefore, ${\mathbf{P}}_6$ is the best scheme, scoring 1 after the normalization process. From Table 9, it is evident that ${\mathbf{P}}_6$ has the maximum performance scores under each attribute in user-preference-based decision making. Therefore, the scores of each attribute for ${\mathbf{P}}_6$ are defined as the positive ideal solution in the TOPSIS calculation, resulting in a final scheme score of 1. Therefore, an analysis of the calculations reveals that the combined scores for ${\mathbf{P}}_6$ are greater than those of the other scheme, regardless of the decision-making weights of the groups.

However, in the actual decision-making process, to mitigate the impact of differing weights between experts and user groups on scheme ordering, this study employed three methods of fusion: the entropy weight method, variance method, and chi-square method. These methods were used to verify the scores from both sides. The calculated weights for the expert group and user group were, respectively, 0.6394 and 0.3606 for the entropy weight method, 0.5057 and 0.4943 for the variance method, and 0.4451 and 0.5549 for the chi-square method. The final scheme sorting obtained using these methods demonstrated consistency in scheme ordering. This further proves the robustness and reliability of the method in selecting the optimal product form design solution. However, in practical applications, it is still necessary to select appropriate weight coefficients according to the actual situation.

5.4. Case Discussion

The method proposed in this paper was compared with an expert decision-making method that combines the traditional Fuzzy Set and linear superposition methods. The expert decision-making method yielded a ranking result of ${\mathbf{P}}_7$→${\mathbf{P}}_6$→${\mathbf{P}}_8$→${\mathbf{P}}_2$→${\mathbf{P}}_5$→${\mathbf{P}}_4$→${\mathbf{P}}_3$→${\mathbf{P}}_1$. The final scenarios obtained from the two decision-making methods were not the same, so a comparative analysis of ${\mathbf{P}}_6$ and ${\mathbf{P}}_7$ was needed.

In the expert decision-making method, ${\mathbf{P}}_7$ had a composite score of 0.6372 and ${\mathbf{P}}_6$ had a composite score of 0.6205. ${\mathbf{P}}_7$ outperformed ${\mathbf{P}}_6$ in terms of overall evaluation score. However, from the analysis of each evaluation attribute (with relative weightings of ${\mathbf{P}}_7$ in the order ${\mathit{w}}_4$→${\mathit{w}}_5$→${\mathit{w}}_3$→${\mathit{w}}_2$→${\mathit{w}}_1),$ it could be observed that ${\mathbf{P}}_7$ was significantly better than ${\mathbf{P}}_6$ only in terms of the shapes of the attributes. The color and material attributes were relatively close between the two schemes. In contrast, regarding the relative weight of interactivity and ergonomics, ${\mathbf{P}}_6$ outperformed ${\mathbf{P}}_7$. Therefore, ${\mathbf{P}}_6$ was superior to ${\mathbf{P}}_7$ from the point of view of a comprehensive comparison of all attributes.

Meanwhile, the small difference between the scores of ${\mathbf{P}}_6$ and ${\mathbf{P}}_7$ was attributed to overlooking the decision-maker’s hesitancy during the decision-making process, which led to an absence of certain evaluative information. This also highlights the distinction in the processing of linguistic decision information between traditional Fuzzy Sets and the PHFS. The method proposed in this paper considers the hesitant and ambiguous psychological behavioral characteristics of decision-makers during the decision-making process, particularly in ambiguous decision-making environments where decision-makers may not be fully rational. By retaining more initial decision-making information, this method enables decision-making results to be more objective and better aligned with actual decision-making needs. When dealing with decision language information on product form attributes, the advantage of PHFS lies in its ability to reduce the problem of missing information during the assembly of evaluation information, thereby helping to mitigate product development risks stemming from decision errors.

6. Conclusions

This paper has addressed the inadequacy of current multi-attribute decision-making methods for product form design in considering both the subjective preferences of users and the objective evaluations of designers, particularly in the cloud environment, where research on hesitant fuzzy evaluation information for different groups is limited. It has introduced the crowd intelligence decision-making method, which integrates user preferences with designers’ professional knowledge in the cloud environment. This method utilizes a Pythagorean Hesitant Fuzzy linguistic term Set to articulate the evaluative information of users and designers, effectively resolving issues of uncertainties and fuzziness in this information and enhancing the objectivity of decision outcomes. Moreover, in the user decision-making process, the user weights are derived by taking into account the support for user preferences. The similarity between individual user evaluation matrices is calculated directly, thereby avoiding a loss of information and leading to more accurate decision-making results. An illustrative analysis of the decision-making process for a coffee machine’s form demonstrated the practicality and feasibility of this method, providing robust decision-making support for product form design.

Although there are some deficiencies in the proposed methods, the overall train of thought is feasible. The expert decision-making process is based on an ideal model, which assumes that all experts possess the necessary knowledge to make relatively objective evaluations, without accounting for variations in knowledge levels among different experts. Consequently, future research will focus on refining the subdivision of different subject weights, enhancing and perfecting the methods proposed in this article, and broadening the scope of the research subjects.