Interpretive Structural Modeling (ISM): A Practical Guide to Attribution Analysis

Today we introduce a practical yet relatively uncommon mathematical model used to solve attribution problems, such as determining responsibility when an incident occurs.

Interpretive Structural Modeling (ISM) provides an effective, quantitative method for clarifying relationships among factors and assigning responsibility.

Overview of the ISM Model

ISM is a technique for analyzing relationships among multiple factors in a system. By performing a series of matrix operations, it decomposes complex systems and builds a multi‑level structural model that reveals both direct and indirect relationships, helping to understand internal structures and mechanisms.

The basic steps are:

First, identify the factors . For example, in an autonomous driving system the factors might include sensor data acquisition, environment perception, path planning, vehicle control, user input, and system monitoring.

Next, construct the relation matrix to capture direct influences (1 for influence, 0 for none).

Then, build the reachable matrix by applying Boolean algebra to the relation matrix, uncovering indirect influences.

After that, perform hierarchical division to separate factors into foundational, intermediate, and outcome levels.

Finally, draw the structural diagram that visualizes the hierarchy, akin to a system family tree.

Mathematical Model and Computation Process

1. Determining Factors and Relation Matrix

Assume there are n factors, denoted F1…Fn. The initial relation matrix R has entries r_{ij}=1 if factor i directly influences factor j, otherwise 0.

2. Constructing the Reachable Matrix

By repeatedly calculating powers of the relation matrix using Boolean algebra, we obtain the reachable matrix that indicates direct or indirect influence.

The Boolean power iteration continues until the matrix stabilizes.

3. Hierarchical Division

Using the reachable matrix, we define reachable sets and predecessor sets for each factor. Factors whose reachable set intersected with the predecessor set is empty belong to the current level; they are removed and the process repeats until all factors are assigned.

4. Building the ISM Model

Based on the hierarchical results, we draw the ISM model, clearly showing factor relationships and levels.

Case Study

We analyze responsibility attribution in a software development project with factors: Requirement Analysis (F1), System Design (F2), Coding (F3), Testing (F4), Project Management (F5), User Feedback (F6).

1. Determining the Relation Matrix

Expert opinion or historical data yields the following relation matrix, where a "1" indicates a direct influence.

F1 directly influences F2.

F2 does not directly influence F4.

Other entries follow similarly.

2. Constructing the Reachable Matrix

By computing Boolean powers of the relation matrix until convergence, we obtain the reachable matrix that captures all direct and indirect influences among the factors.

3. Hierarchical Division

Using the reachable matrix, we extract hierarchical levels. The extraction process is illustrated below:

First extraction:

Second extraction:

Third extraction:

Fourth extraction:

Resulting hierarchical layers:

Level 1: F5 (Project Management), F6 (User Feedback)

Level 2: F4 (Testing)

Level 3: F3 (Coding)

Level 4: F2 (System Design)

Level 5: F1 (Requirement Analysis)

4. Building the ISM Model

Combining the hierarchical layers with the original direct influence matrix, we draw the ISM model:

The diagram shows how the lowest layer (F1) influences F2, which influences F3, then F4, and finally F5 and F6.

Interpretive Structural Modeling (ISM) is an effective method for clarifying factor relationships and assigning responsibility in complex systems. By constructing relation and reachable matrices, performing hierarchical division, and visualizing the structure, practitioners can quickly locate and analyze responsible factors when problems arise.

References: [1] Scientific Platform Serving for Statistics Professional 2021. SPSSPRO. (Version 1.0.11) Retrieved from https://www.spsspro.com. [2] 王敬敏, 康俊杰. 基于解释结构模型的能源需求影响因素分析[J]. 中国电力, 2017, 50(9):6.