A conceptual design decision approach by integrating rough Bayesian network and game theory under uncertain behavior selections
Introduction
Within a conceptual design project, the product function is decomposed into simple design problems, and then, the functions satisfying the low-level problems are designed to seek and combine the principle solutions, followed by the selection of the conceptual schemes (CSs) (Zheng et al., 2018). Due to the diversity of product behaviors and the mapping of principle solutions to sub-functions, conceptual design decision is clearly a complicated design task (Jing et al., 2021c; Gu et al., 2015). The function-behavior-structure (FBS) model (Khadilkar and Cash, 2020, Zhu et al., 2020), a mainstream design framework supporting multi-level knowledge mapping, has been proposed to address the design challenge for the best scheme decision. And behavior serves as a carrier to describe the relationship between specified design requirements and principle solutions, how to select appropriate behavior plays a critical role in the design process, directly affecting the quality, cost, and delivery performance of the terminal CS (Wang et al., 2019). It is obvious that the FBS-based design process is essentially a decision problem, and thus analyzing the intrinsic relation of product functions and providing feasible behavior selections in the early design phase is one of the goals in concept decision research (Jing et al., 2018, Cao et al., 2021).
Conventionally, conceptual design decision-making involves acquisition of fuzzy evaluation information and aggregate these assessments of schemes objectively (Pahl & Beitz, 2013). Extant studies have focused more on the quantitative assessments of the terminal CSs, which are ambiguous and imprecise. To address the inherent uncertainty of decision-makers (DMs) in the evaluation process of schemes, many mathematical models have been developed to improve the reliability of decision results, such as fuzzy sets (Shidpour et al., 2016, Ma et al., 2020, Mohebbi et al., 2018), intuitionistic fuzzy sets (IFS) (Geng et al., 2010, Büyüközkan and Güleryüz, 2016), rough sets (Qi et al., 2020, Jing et al., 2021), and others (Hayat et al., 2018, Ying et al., 2018). The basic idea of these methods is to transform the subjective linguistic terms into quantitative values, and describe the uncertainty of evaluation values by determining the boundary of interval values. These mathematical models focused on modeling the uncertainties of the terminal CS, but ignored the impact of design factors (i.e., sub-function interaction, behavior selection) behind the decision process on the scheme value. In practice, the problem becomes significant when converting the uncertain information into certainty scheme value since the behavior’s partial belief (how certain is the selection of behavior for each sub-function) needs to be incorporated in the conversion process. To the most of our knowledge, behavior selection preferences derived from users are not considered during the evaluation process, which is still a gap existed between the decision results and the actual product design phase.
After modeling the uncertainty of DMs’ assessments, how to aggregate these assessments from DMs is critical. For this reason, many multi-criteria decision-making (MCDM) methods have been proposed to calculate the criteria weights, such as analytic hierarchy process (AHP) (Saaty, 1996), best worst method (BWM) (Song et al., 2021), and entropy weight method (Chen, 2021). Some studies are proposed to select an optimal CS that close to the ideal solution, such as technique for order of preference by similarity to ideal solution (TOPSIS) (Ma et al., 2020), višekriterijumska optimizacija i kompromisno rešenje (VIKOR) (Tiwari et al., 2016). To adapt the complicated decision environment, integrated decision approaches such as rough set-based interval-valued intuitionistic fuzzy (Jing et al., 2021c), vague set-TOPSIS (Geng, Chu, & Zhang, 2010) and rough-quality function deployment (Fang, Li, & Song, 2020), have been developed to accomplish these decision tasks. Previous studies focus on the utility ranking of the terminal CSs in FBS model, and the principle solution only considers the implementation of a single sub-function, ignoring the limitation of sub-function interactivity on the selection direction of the principle solutions. Before selecting the principle solutions, it is required to filter a suitable behavior that satisfies each sub-function, and its behavior selections depends on the selected behavior of other sub-functions. Then the optimal CS with maximized overall benefits will be obtained by coordination of the behavior selections to meet the expectations of each sub-function in a balanced manner, and that is similar to non-cooperative games essentially (Yang and Xu, 2021, Jing et al., 2021). Thus, it is necessary to solve the trade-off of sub-functions around the uncertain behavior selections (e.g., a probability distribution of the behavior being selected is [T:40%, F:60%]) which formed by fuzzy preferences in design process.
In view of the foregoing arguments, this paper is motivated by two gaps of the state-of-the-art in conceptual design decision-making.
- (1)
Few studies focused on the influence of uncertain beliefs about behavior selection in the early design stages on the scheme decision results, and the probabilistic reasoning for implementing the optimal principle solution in the FBS-based design process.
- (2)
In the principle solutions screening process, the propagation influence of the interactive sub-functions on the terminal CS performance is ignored, resulting in the scheme utility obtained by the traditional MCDM method failing to resolve the conflict expectation of sub-functions.
To tackle this issue, a rough Bayesian game decision approach for conceptual design utilizing probability model in the FBS model is proposed. This paper has below objectives: 1) a Bayesian network (BN)-based probabilistic model is used to transform the FBS-based design decision problem into the interactive sub-functions game process based on uncertain behavior selections; 2) the conditional probabilities of the sub-functions’ behaviors in BN model as beliefs are used to realize the optimal scheme decision for the principle solutions; 3) in order to effectively quantify the subjective judgments of DMs, rough set theory is used to obtain the conditional probability of sub-functions under uncertain behaviors to support the construction of the sub-function BN model. Inspired by this research, by updating the sub-function BN can provide a causal inference interface that gives an intuitive risk probability assessment of various schemes in the industry, a better design service to predict the feasibility of concept recommendations under diverse design preferences.
The remaining of this research paper is organized as follows. Section 2 outlines the related literature review. Section 3 describes the proposed non-cooperative game conceptual scheme decision approach based on Bayesian network model. In Section 4, the tree climbing and trimming (TCT) machine is taken as a case study to verify the proposed approach, followed by results comparison and sensitivity analysis discussion. The paper ends with a conclusion and future work section.
Section snippets
Related works
In this section, the related literature examined includes product functional modeling and conceptual design decision approach. Simultaneous, the research gaps identified from the literature review incorporating uncertain scheme decision are described at the end of this section.
Methodology
Fig. 1 is the framework of the proposed rough Bayesian game decision approach for conceptual design, and the details are presented as below:
Step 1: Construct a product functional model, use the input and output relationships between sub-functions to complete DSM clustering and importance calculation of sub-functions, and effectively extract the core sub-functions.
Step 2: Take the behavior that implements the core sub-functions as the state nodes to construct the sub-functions BN model.
Case study
To satisfy the market’s growing demand for wood, it is necessary to develop a tree climbing and trimming (TCT) machine to prune the fast-growing forest efficiently. Then, in order to maximize customers’ design preferences and functions under uncertainty, a case study of the TCT machine will be employed to verify the proposed decision model, and an optimal design concept is selected.
Conclusions and future work
To overcome the issue of uncertain beliefs about behavior selections and sub-functions interaction in the FBS design process, the Bayesian network model and the game theory are adopted in our scheme decision approach. By constructing probability distributions of behavior selections based on BN model, probabilistic reasoning of sub-functions around uncertain behavior is driven to achieve the principle solution game decision based on probabilistic representations. Furthermore, the probability
CRediT authorship contribution statement
Liting Jing: Methodology, Supervision, Funding acquisition. Qizhi Li: Writing – original draft. Junfeng Ma: Writing – review & editing. Jing Xie: Investigation. Xiang Peng: Conceptualization. Jiquan Li: Formal analysis. Shaofei Jiang: Project administration.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors, Liting Jing, Qizhi Li, Xiang Peng, Jiquan Li, Shaofei Jiang was supported by the National Natural Science Foundation of China [under grant number 52105282], the Zhejiang Provincial Natural Science Foundation of China [under grant numbers LQ22E050011 and LY20E050020], the China Postdoctoral Science Foundation [under grant number 2021M702893], the Key R & D Program of Zhejiang Province [No.: 2021C01086], and the 111 Project (No.: D16004).
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