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 9 / 18 (b) Performing condensation operations: This step simplifies the complex network of element relationships by merging elements within the same connected component into a single “condensed node” if they have identical reachabil- ity and antecedent sets. Its purpose is to reduce redundant calculations and clarify the hierarchical structure by removing duplicate or highly overlapping element relationships. (c) Executing edge reduction: This involves removing transitive edges (i.e., indirect influence relationships that can be inferred through intermediate elements) from the network. It streamlines the relationship structure, avoiding over-complication of the model and making the direct causal links between elements more prominent. (d) Conducting level partitioning. Specifically, for each factor, the reachability set, antecedent set, and intersection set were computed from the reachability matrix, which facilitated the assignment of tiers based on predefined rules (i.e., an element is assigned to a level if its reachability set intersects with its antecedent set exactly at the reach- ability set itself). (5) Developing the ISM-based hierarchical framework. The final hierarchical framework was established by integrating level division results with the reachability matrix, thereby ensuring a systematic and comprehensive structure. Rep- resented through nodes (denoting elements) and directed edges (indicating influence relationships), the framework

adheres to the principle of simplification by removing redundant transitive paths. This visualization enhances model clarity and facilitates intuitive understanding of systemic structures. 3. Results and discussion 3.1. The structural self-interaction matrix Based on the interview results, the structural self-interaction matrix established in this study is shown in Table 3. “V” indicates that the constraint Si in the row had a direct impact on the constraint Sj in the corresponding column, such as “S1 (The completeness of policies)” had a direct impact on “S4 (The level of attention given by universities)”; “A” means that the constraint Sj in the column had a direct impact on the constraint Si in the corresponding row, such as “S3 (The alignment of talent cultivation quality)” had a direct impact on “S2 (The willingness of enterprises to cooperate)”; and “O” Table 3. The structural self-interaction matrix. Constraint factorsa S12 S11 S10 S9 S8 S7 S6 S5 S4 S3 S2 S1 O O O V O O O O V O O S2 A A A O A A O O O A S3 O A A O O A A O O S4 V O O O O O V V S5 O A A O O O O S6 O O O O O O S7 O O A O O S8 O O A O S9 O O A S10 V A S11 V aS1 (The completeness of policies), S2 (The willingness of enterprises to cooperate), S3 (The alignment of talent cultivation quality), S4 (The level of atten- tion given by universities), S5 (The degree of teacher involvement), S6 (The adequacy of hardware and software facilities), S7 (The differences in talent

cultivation philosophies), S8 (The achievement of school-enterprise mutual interests), S9 (The clarity of school-enterprise responsibilities and rights), S10 (The effectiveness of school-enterprise communication), S11 (The closeness of prior university-enterprise collaboration), S12 (The fluidity of school- enterprise cooperation channels). https://doi.org/10.1371/journal.pone.0339158.t003