ISM Interpretive Structural Modeling: Calculating Reachability Matrix from Adjacency Matrix

ISM (Interpretive Structural Modeling) is an effective methodology for analyzing complex system structures, which reveals hierarchical relationships between elements through adjacency matrices and reachability matrices. This process is particularly crucial for understanding the overall system architecture and interdependencies between elements.

### Adjacency Matrix vs Reachability Matrix The adjacency matrix is a matrix representation of direct relationships between system elements, where rows and columns represent system elements, and matrix values indicate whether direct relationships exist between elements. The reachability matrix is derived from the adjacency matrix by computing the transitive closure, reflecting both direct and indirect reachability between elements.

### Calculation Methods for Reachability Matrix Adjacency Matrix Initialization: First define the system's adjacency matrix A, where A[i][j] = 1 indicates element i directly affects element j, otherwise 0. In code implementation, this is typically represented as a 2D array or matrix data structure. Matrix Power Operations: Compute powers of A (such as A², A³,…) until the matrix stabilizes, i.e., A^(k) = A^(k+1). This step, implemented through iterative matrix multiplication, identifies all possible indirect relationships. Identity Matrix Addition: Add the adjacency matrix with the identity matrix (I + A), then perform Boolean operations to ensure each element is at least reachable to itself. This can be coded using element-wise OR operations. Reachability Matrix Determination: The final reachability matrix M is obtained through Boolean power operations (typically using Warshall's algorithm), where M[i][j] = 1 indicates element i can reach element j. The algorithm implementation involves nested loops for efficient path computation.

### Level Partitioning Algorithm Level partitioning is the core step of ISM, used to divide system elements into different levels and reveal the system's hierarchical structure. Reachability Set and Antecedent Set: For each element, compute its reachability set (all elements it can reach) and antecedent set (all elements that can reach it). These sets can be derived from the reachability matrix using row and column extraction operations. Intersection Determination: If the intersection of an element's reachability set and antecedent set equals its antecedent set, the element belongs to the current highest level. This condition check can be implemented using set comparison operations. Level Extraction: Place qualifying elements into the current level, remove them from the matrix, and repeat the process until all elements are partitioned. The algorithm requires iterative processing with complexity O(n²) for n elements.

Through ISM's level partitioning, we can clearly understand the system's hierarchical structure, identify key influential elements, and provide basis for system optimization. This method is widely applied in engineering management, social sciences, and other domains.