An Explainable and Interpretable Composite Indicator Based on Decision Rules

This paper proposes a novel framework for constructing explainable and interpretable composite indicators by applying the Dominance-based Rough Set Approach to induce transparent if-then decision rules that clarify the rationale behind classifications or scores across various MCDA scenarios, while also handling missing values and efficiently generating minimal rules for continuous indicators.

The Gist

Imagine you are trying to grade a student's performance. In the old way, you might take their math score, add it to their science score, multiply by a "weight" for how important science is, and then add a "bonus" for art. You get a final number, say 87.5.

But here's the problem: Why 87.5?
If you ask the teacher, they might say, "Well, the math was weighted at 0.4 and science at 0.6, and the formula is complex." To a parent or the student, this feels like a black box. You see the input (grades) and the output (score), but the magic happening inside is hidden.

This paper proposes a new way to build these "grades" (called Composite Indicators) that turns the black box into a glass box. Instead of a mysterious math formula, the grade is explained by simple, clear rules, like a recipe or a flowchart.

The Core Idea: "If This, Then That"

The authors suggest replacing complex math formulas with Decision Rules. Think of these rules as traffic signs or simple logic gates.

Instead of saying, "Your score is 87.5 because of this weighted average," the new system says:

"IF your Math score is above 80 AND your Science score is above 70, THEN you get a 'Good' grade."

This is much easier to understand. It's like a doctor diagnosing a patient. They don't just give a number; they say, "Because your fever is high and your cough is dry, you likely have the flu."

The Four Scenarios: How It Works in Real Life

The paper shows how this works in four different situations, using some great real-world examples:

1. The Medical Scale (The Glasgow Coma Scale)

• The Old Way: Doctors add up points for eye opening, verbal response, and movement. If the total is 7, the patient is "Severe." But why is 7 severe? Is it because the eyes didn't open? Or because they can't speak? The total score hides the specific reason.
• The New Way: The system generates rules like: "If the patient cannot open their eyes AND cannot speak, THEN they are Severe."
• The Analogy: It's like a detective solving a crime. Instead of just saying "The suspect is guilty because the math adds up," the detective says, "The suspect is guilty because they were at the scene AND had the weapon."

2. The "Obscure" National Index (Human Development Index)

• The Old Way: Countries are ranked by a complex formula involving life expectancy, schooling, and income. It's hard to tell exactly which factor pushed a country into "High Development" or "Low Development."
• The New Way: The system creates rules like: "If a country's life expectancy is over 73 years AND schooling is over 12 years, THEN it is 'Very High Development'."
• The Analogy: Imagine a club with a secret handshake. The old way just tells you if you're in or out. The new way tells you exactly what the handshake is: "You get in if you can say the password AND do the wave."

3. Learning from the Boss (Decision Maker Preferences)

• The Scenario: A boss looks at 50 stocks and says, "I like Stock A, I hate Stock B, and Stock C is okay." They don't give a formula; they just give their gut feeling.
• The New Way: The computer looks at the boss's choices and reverse-engineers the rules. It figures out: "Ah, the boss likes stocks where the profit margin is high AND the risk is low."
• The Analogy: It's like a cooking apprentice watching a master chef. The apprentice doesn't know the recipe, but by watching the master add salt only when the soup is sour, the apprentice learns the rule: "If soup is sour, add salt."

4. Explaining a Complex Computer Model

• The Scenario: Sometimes, a super-complex AI model (like a "black box" algorithm) gives a score.
• The New Way: The system takes that complex score and translates it back into simple rules for humans to understand.
• The Analogy: It's like a translator. The computer speaks "Math," and the human speaks "English." This method translates the computer's math into a story the human can understand.

Handling the Messy Stuff: Missing Data and Contradictions

Real life is messy. Sometimes data is missing (like a student who didn't take the math test), or the rules might seem to contradict each other.

• Missing Data: Imagine a student is missing a math score. The old way might throw the student out of the calculation. The new way says: "We don't know the math score, but if their Science score is high enough, they still qualify for the 'Good' category, regardless of what the math score turns out to be." It's like saying, "Even if we don't know the full story, we have enough evidence to make a safe decision."
• Contradictions: Sometimes a student might fit a rule for "Good" and a rule for "Bad" at the same time. The paper introduces a smart "referee" (an algorithm) that picks the best set of rules that don't fight each other, ensuring the final decision is fair and consistent.

Why This Matters

The authors are essentially saying: "Stop hiding behind complex math."

In a world where AI and data drive decisions (from who gets a loan to which country gets aid), people have a right to know why a decision was made.

• Transparency: You can see the logic.
• Fairness: You can check if the rules are biased.
• Trust: You can trust the result because you understand the reasoning.

The Big Takeaway

Think of this paper as a translator between the cold, hard logic of data and the warm, fuzzy logic of human understanding. It turns a mysterious "Score of 87.5" into a clear, honest story: "You got this score because you did X, Y, and Z."

It's not just about getting a grade; it's about understanding the story behind the grade.

Technical Summary

1. Problem Statement

Composite Indicators (CIs) are widely used to aggregate multiple criteria into a single score for ranking or classifying units (e.g., countries, hospitals, stocks). However, traditional CI construction faces several critical challenges:

• Black Box Nature: Conventional methods often rely on complex aggregation functions (weighted sums, geometric means) and arbitrary weighting schemes, making it difficult to explain why a specific unit received a certain score.
• Lack of Transparency: The "hubris" of using many indicators often obscures the actual drivers of the evaluation.
• Compensatory vs. Non-Compensatory: Standard aggregation often assumes compensatory logic (a high score in one area can offset a low score in another), which may not align with decision-makers' (DM) preferences.
• Data Limitations: Real-world datasets often contain missing values or qualitative/ordinal data, which standard aggregation methods struggle to handle without imputation (which introduces bias).
• AI Explainability Gap: While AI models are often "black boxes," there is a growing demand for "glass box" systems that provide ante-hoc interpretability (understanding the model's logic before output) and post-hoc explainability (justifying specific outputs).

The paper proposes a novel framework to construct CIs that are inherently explainable and interpretable by replacing numerical aggregation with if-then decision rules.

2. Methodology

The core of the proposed methodology is the Dominance-based Rough Set Approach (DRSA). Unlike standard Rough Set Theory, DRSA handles preference-ordered data (ordinal scales) and dominance relations (if unit $a$ is better than unit $b$ on all criteria, $a$ should be assigned to a class at least as good as $b$).

Key Components:

1. Decision Rules: The CI is represented by two types of rules:

   • "At-least" rules ($r^{\geq}$): "If criteria $g_1 \geq q_1$ and $g_2 \geq q_2$, then the unit is assigned to at least Class $t$."
   • "At-most" rules ($r^{\leq}$): "If criteria $g_1 \leq q_1$ and $g_2 \leq q_2$, then the unit is assigned to at most Class $t$."
   • These rules are expressed in natural language, ensuring human interpretability.

2. Four Application Scenarios:

   • Scenario (i) Ante-hoc: Explaining classifications derived from summing ordinal codes (e.g., Glasgow Coma Scale).
   • Scenario (ii) Post-hoc: Explaining an existing "opaque" numerical CI (e.g., Human Development Index) by inducing rules from the resulting classifications.
   • Scenario (iii) Ante-hoc Construction: Constructing a CI directly from a DM's classifications of reference units.
   • Scenario (iv) Post-hoc Explanation: Explaining results from other MCDA methods (e.g., ELECTRE-Score).

3. Rule Induction Algorithm (Algorithm 1 & 2):

   • The authors propose a new exhaustive algorithm to induce all minimal, non-contradictory decision rules in a single run.
   • Unlike heuristic approaches (like DOMLEM) that find a small set of rules, this algorithm guarantees the generation of the complete set of minimal rules consistent with the data.
   • It treats continuous scores as ordered classes, allowing the method to handle both discrete and continuous CIs.

4. Handling Contradictions and New Units (Algorithm 3):

   • When classifying new units, it is possible for a unit to satisfy an "at-least" rule for a high class and an "at-most" rule for a lower class (a contradiction, e.g., $s^-(a) > s^+(a)$).
   • The authors formulate a Mixed-Integer Linear Programming (MILP) problem to select a maximal subset of rules that ensures a non-contradictory classification for all units (both reference and new) while minimizing changes to the original DM preferences.
   • A second MILP step selects the minimal subset of these rules required to explain the final classification, ensuring parsimony.

5. Handling Missing Values (Algorithm 4):

   • The framework extends to datasets with missing values without imputation.
   • A rule is considered satisfied by a unit if the unit's known values satisfy the condition, and the missing values are assumed to be compatible with the rule. This preserves the original data structure and avoids bias.

3. Key Contributions

• Decision-Rule-Based CI Framework: The first methodological proposal to construct composite indicators entirely based on logical decision rules rather than numerical aggregation functions.
• Ante-hoc and Post-hoc Integration: The framework supports both constructing CIs from scratch based on preferences (ante-hoc) and explaining existing opaque indicators (post-hoc).
• Exhaustive Minimal Rule Induction: A novel algorithm that efficiently generates all minimal rules in one run, overcoming the limitations of greedy heuristics.
• Contradiction Management: A robust procedure using MILP to resolve classification contradictions when applying rules to new data, ensuring logical consistency.
• Missing Value Robustness: A natural extension of DRSA to handle missing data without imputation, enhancing practical applicability in real-world scenarios.
• Continuous CI Support: The ability to treat every distinct numerical score as an ordered class, making the method applicable to continuous composite indicators.

4. Results and Illustrative Examples

The paper validates the methodology through four distinct case studies:

1. Glasgow Coma Scale (Medical): Demonstrates how decision rules can explain severity classifications derived from summing ordinal test scores, providing clear "cause-effect" scenarios for patient triage.
2. Human Development Index (HDI): Shows how to explain the classification of 193 countries into development quartiles. The induced rules reveal specific thresholds (e.g., "If Expected Years of Schooling $\geq$ 11.018 and GNI $\geq$ 3157, then at least Medium Development") that drive the classification, offering transparency missing in the standard geometric mean aggregation.
3. Stock Portfolio Selection: Illustrates the construction of a CI from a DM's classifications of reference stocks. The method successfully classifies new stocks and resolves contradictions where a stock might be "at least Class 3" but "at most Class 1" by adjusting the rule set via MILP.
4. Hotel Location Selection (ELECTRE-Score): Demonstrates the post-hoc explanation of an existing MCDA method, translating complex outranking scores into simple, understandable rules.
5. PISA Data (Missing Values): Shows the application of the method on student data with missing values. The rules are induced without imputing missing data, and new students with missing values are classified consistently based on their known attributes.

5. Significance and Impact

• Transparency and Accountability: By shifting from "black box" scoring to "glass box" rule-based reasoning, the methodology addresses the "Mind the assumptions" and "Mind the framing" principles of Saltelli et al. (2020). Stakeholders can understand the exact conditions leading to a classification.
• Avoidance of Weighting Controversies: The method does not require explicit weight assignment or compensatory aggregation, eliminating debates regarding the relative importance of criteria.
• AI Interpretability: It bridges the gap between MCDA and Explainable AI (XAI), offering a model that is inherently interpretable (ante-hoc) rather than relying on post-hoc approximation.
• Practical Applicability: The ability to handle missing values, ordinal data, and continuous scores makes the framework highly versatile for socio-economic, environmental, and healthcare applications where data is often imperfect.
• Scalability: While the rule induction is exponential in the number of criteria, the authors note that for typical CI applications (usually <20 indicators), the algorithm runs efficiently on standard hardware. For larger datasets, the "explainability on demand" feature (showing only rules satisfied by a specific unit) ensures the output remains intelligible.

In conclusion, the paper presents a paradigm shift in Composite Indicator construction, prioritizing interpretability, logical consistency, and transparency over mathematical aggregation complexity.