Constraints, mechanisms, and strategies for industry-education integration in vocational education: An empirical study

PLOS One | https://doi.org/10.1371/journal.pone.0339158 December 18, 2025 8 / 18 to analyze the constraining mechanisms affecting IEI development in vocational education. With reference to existing research, the implementation adheres to a five-step procedure illustrated in Fig 2: (1) defining the conceptual system and its corresponding element set, (2) establishing the structural self-interaction matrix, (3) computing the reachability matrix, (4) implementing level partition, and (5) establishing the ISM-based hierarchical framework. (1) Defining the conceptual system and element set. The conceptual system refers to the research problem and system boundaries that will be addressed using ISM, whereas the element set consists of specific influencing factors under investigation. Aligned with the research objectives, this study defines the conceptual system as the constraining mech- anisms for the development of IEI in vocational education, which includes both encompassing constraining factors and their corresponding pathways. The element set consists of the 12 constraining factors identified in Section 2.2. (2) Establishing the structural self-interaction matrix. The structural self-interaction matrix quantifies the pairwise influence relationships among system elements. In this study, the matrix was constructed based on expert interview results out- lined in Section 2.3.1. (3) Calculating the reachability matrix. The reachability matrix is used to explicitly delineate the presence of influence

relationships among system elements, encompassing both direct and indirect effects. This matrix is derived through Boolean algebraic operations performed on the structural self-interaction matrix, with its computational implementation realized through MATLAB programming. (4) Implementing level division. Hierarchical partitioning was conducted by using the reachability matrix to determine the tier of each factor within the system. The computational procedure consisted of four main steps: (a) Identifying connected components: This task aims to group elements that have direct or indirect influence relation- ships into independent sub-systems. It helps eliminate isolated elements (if any) that have no connection to the core constraint system, ensuring the subsequent analysis focuses only on mutually related factors that truly affect IEI development. Fig 2. Implementation steps of ISM. https://doi.org/10.1371/journal.pone.0339158.g002