This study introduces an advanced framework for picture fuzzy linear programming problems (PFLPP), deploying picture fuzzy numbers (PFNs) to articulate diverse parameters. Integral to this approach are the three cardinal membership functions: membership, neutral, and non-membership, each contributing distinctly to the formation of the PFLPP. Emphasis is placed on employing these degrees to formulate the PFLPP in its most unadulterated form. Furthermore, the research delineates a novel optimization model, tailored specifically for the resolution of the PFLPP. A meticulous case study, accompanied by a numerical example, is presented, demonstrating the efficacy and robustness of the proposed methodology. The study culminates in a comprehensive discussion of the findings, highlighting pivotal insights and delineating potential avenues for future inquiry. This exploration not only advances the theoretical underpinnings of picture fuzzy sets but also offers practical implications for the application of linear programming in complex decision-making scenarios.

This study, rooted in extension theory and the principles of knowledge engineering, explores and formulates a novel method for generating sports protective gear designs. Given the critical role of sports protective gear in safeguarding athletes from injuries, coupled with escalating demands for product quality, the aim is to uncover a more effective approach to innovative design. This method involves formalizing modeling of various elements in the design process and representing this information in the elemental form of knowledge engineering. Through the related analysis, divergent analysis, as well as permutation and conduction transformations of these elements, innovative design schemes for sports protective gear are generated. This process not only optimizes design schemes in depth but also ventures into new design methods and processes. The objective is to offer a novel perspective in integrating extension theory and knowledge engineering in the design of sports protective gear, aspiring to provide more effective strategies to enhance existing design workflows. The goal of this new design method is to produce sports protective gear that is both practical and innovative, thereby enhancing the safety and enjoyment of athletes.