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 10 / 18 denotes that there was no direct impact between the constraint Si in the row and the constraint Sj in the corresponding col- umn, such as “S1 (The completeness of policies)” and “S11 (The closeness of prior university-enterprise collaboration)”. 3.2. The reachability matrix The reachability matrix, calculated based on the structural self-interaction matrix in this study, is shown in Table 4. In the reachability matrix, “1” indicates that the constraint Si in the row directly or indirectly influences the constraint Sj in the corresponding column, such as “S1 (The completeness of policies)” had affected “S2 (The willingness of enterprises to cooperate)”, “S3 (The alignment of talent cultivation quality)”, “S4 (The level of attention given by universities)”, “S5 (The degree of teacher involvement)”, “S6 (The adequacy of hardware and software facilities)”, “S9 (The clarity of school- enterprise responsibilities and rights)”, and “S12 (The fluidity of school-enterprise cooperation channels)”; “0” indicates that the constraint Si in the row does not directly or indirectly influence the constraint Sj in the corresponding column, such as “S1 (The completeness of policies)” had not affected “S7 (The differences in talent cultivation philosophies)”, “S8 (The achievement of school-enterprise mutual interests)”, “S10 (The effectiveness of school-enterprise communication)”, and “S11 (The closeness of prior university-enterprise collaboration)”. 3.3. The level partition

Based on the reachability matrix, this study systematically calculated the reachability, previous, and intersec- tion sets of each constraint, and determined the level partition of each constraint according to the calculation rule L1={Si|R(Si)∩Q(Si)=R(Si), i = 0,1,2,…, k}, as shown in Table 5. The results indicated that the 12 constraints are divided into five levels. Specifically, the constraints “S2 (The willingness of enterprises to cooperate),” “S5 (The degree of teacher involvement),” and “S9 (The clarity of school-enterprise responsibilities and rights)” were classified into Level I; “S3 (The alignment of talent cultivation quality),” “S8 (The achievement of school-enterprise mutual interests),” and “S12 (The fluidity of school-enterprise cooperation channels)” were located at Level II; “S6 (The adequacy of hardware and software facil- ities)” and “S7 (The differences in talent cultivation philosophies)” belong to Level III; “S4 (The level of attention given by Table 4. The reachability matrix. Constraint factors S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 Driving powera S1 1 1 1 1 1 1 0 0 1 0 0 1 8 S2 0 1 0 0 0 0 0 0 0 0 0 0 1 S3 0 1 1 0 0 0 0 0 0 0 0 0 2 S4 0 1 1 1 1 1 0 0 0 0 0 1 6 S5 0 0 0 0 1 0 0 0 0 0 0 0 1 S6 0 1 1 0 0 1 0 0 0 0 0 0 3 S7 0 1 1 0 0 0 1 0 0 0 0 0 3 S8 0 1 0 0 0 0 0 1 0 0 0 0 2 S9 0 0 0 0 0 0 0 0 1 0 0 0 1 S10 0 1 1 0 1 0 1 1 1 1 0 1 8 S11 0 1 1 0 1 0 1 1 1 1 1 1 9

S12 0 1 0 0 0 0 0 0 0 0 0 1 2 Dependence powerb 1 10 7 2 5 3 3 3 4 2 1 5 46 aDriving force refers to the extent to which a constraint influences other constraints. The greater the driving force, the more constraints it influences. bDependence power refers to the extent to which a constraint is influenced by other constraints. The greater the dependence power, the more con- straints it is influenced by. https://doi.org/10.1371/journal.pone.0339158.t004