Sort-Inverse Update for MR-Sort Models

• Sort-Inverse Update is a method for inferring MR-Sort model parameters from labeled data, effectively handling non-monotone criteria like single-peaked or single-valley preferences.
• Recent mixed-integer programming formulations optimize weights, thresholds, and interval parameters, achieving classification accuracies of around 93% in empirical studies.
• Real-world validations, such as pre-anesthesia data analysis, demonstrate the approach’s capability to recover interpretable preference structures and accurate model assignments.

The Sort-Inverse Update, commonly referred to in the literature as the Inverse MR-Sort (Inv-MR-Sort) problem, pertains to the computation of parameters for Majority Rule Sorting (MR-Sort) models so as to fit a labeled dataset. MR-Sort models assign alternatives evaluated over multiple criteria to one of a predefined set of ordered categories. The Inv-MR-Sort challenge is central to preference learning in multiple criteria decision analysis, particularly when learning from data where monotonicity on the criteria does not necessarily hold. Recent advancements include mixed-integer programming (MIP)-based methodologies that allow the learning of non-monotone, including single-peaked or single-valley, preference structures alongside the standard MR-Sort parameters (Minoungou et al., 2021).

1. Fundamentals of the MR-Sort Model

MR-Sort operates over $n$ criteria, with each criterion $i$ defined over a real-valued scale $X_i$. Alternatives $x = (x_1, ..., x_n)$ are assigned to one of $p$ ordered categories $C^1 \prec C^2 \prec ... \prec C^p$, bounded by $(p-1)$ profiles $b^h = (b^h_1, ..., b^h_n)$. Each criterion's contribution is encoded through non-negative weights $w_i$ (subject to $\sum w_i = 1$) and a majority threshold $\lambda \in [0.5, 1]$.

In the classic monotone MR-Sort, criteria possess a direction of preference: "gain" (non-decreasing utility) or "cost" (non-increasing utility). Approved sets $\mathcal{R}_i^h$ for each criterion and profile are determined by thresholding, and an alternative's assignment depends on its aggregate weighted approval exceeding the threshold $\lambda$ at some frontier, and not at the next (Minoungou et al., 2021).

2. Inverse MR-Sort Problem Definition

The Inv-MR-Sort problem is learning from example assignments to infer the MR-Sort parameters $(w, b^h, \lambda)$ so as to match or best approximate the training labels. The learning set consists of alternatives $a^j \in X$ with known assignments $c(a^j) \in \{1, ..., p\}$. Decision variables include the weights, frontiers, threshold, and binary classification indicators $\gamma_j$, with linear constraints enforcing MR-Sort assignment consistency for each instance. The objective is typically to maximize the total number of correctly assigned alternatives, $\max \sum_j \gamma_j$ (Minoungou et al., 2021).

3. Non-Monotone Extension: Single-Peaked and Single-Valley Criteria

The extension to non-monotone cases addresses criteria whose preferences are not strictly increasing or decreasing. Single-peakedness is formalized by a peak $p_i$ on $X_i$ such that utility decreases when deviating from $p_i$, and single-valleyness by a trough where utility increases away from the minimum. Approved sets thus become intervals or their complements rather than half-lines. Each criterion is equipped with a binary type variable $\sigma_i$ indicating single-peaked (1) or single-valley (0).

Approved set representation at the frontier for each criterion $i$ involves parameters $b_i^\perp$ (midpoint) and $b_i$ (half-width), forming $\{x_i : |x_i - b_i^\perp| \leq b_i\}$ for single-peaked, and its complement for single-valley. The extension accommodates mixed monotone/non-monotone criteria in a unified framework (Minoungou et al., 2021).

4. Mixed-Integer Programming Formulation

The unified MIP formulation for two-category (binary) MR-Sort with non-monotone criteria involves:

• Continuous variables: weights $w_i$, threshold $\lambda$, interval parameters $b_i^\perp$, $b_i$, absolute value linearizations $\alpha_i^{j+}, \alpha_i^{j-}$, and "earned weights" $c_{ij}$.
• Binary variables: criterion type $\sigma_i$, absolute value disambiguators $\beta_i^j$, set membership $\delta_{ij}$, and correctness indicators $\gamma_j$.
• Constraints linearize absolute values, enforce approved set membership, link "earned weight" to criterion approval, encode the MR-Sort assignment rule, and bound intervals.
• The objective remains maximization of correctly classified examples.

This MIP is solved using branch-and-bound and cutting planes in a state-of-the-art solver (IBM-CPLEX). Computational experiments indicate median solution times from seconds to several minutes depending on problem size, with generalization accuracy $CAg \approx 0.93$ on average over a test set of 10,000 alternatives (Minoungou et al., 2021).

5. Algorithmic Performance and Empirical Insights

Empirical results reveal computational feasibility up to $n = 9$ criteria and $|L| = 200$ with up to $q = 4$ criteria of unknown preference direction, under a 1-hour timeout. About 70% of instances are solved to optimality for $n \leq 7, q \leq 2$. The Preference-Direction Restoration (PDR)—the probability of correctly identifying a criterion’s type—diminishes with the number of criteria and with extremely small criterion weights, with PDR ≈ 75% for medium weights and ≈ 44% for very small weights.

6. Real-World Validation: Case Study on Pre-Anesthesia Data

A substantial application involves 898 patients evaluated over 14 medical indicators (ASA data) categorized as "risky" or "not risky." Among the most relevant criteria, glycemia is clinically single-peaked (risk higher for hypo- or hyper-glycemia). The learned model can restore up to 99.4% of assignments but initially infers glycemia as a cost criterion due to confounding "normal" data. When excluding normal glycemia cases from the risky category, the single-peaked nature is correctly recovered, matching clinical standards. This demonstrates that Inv-MR-Sort with non-monotone extension can recover both parameter values and the shape of criterion preferences, contingent on an informative learning set (Minoungou et al., 2021).

7. Significance and Theoretical Implications

The ability to learn MR-Sort models from non-monotone data extends the practical and theoretical scope of preference learning in multiple criteria decision analysis. The approach accommodates real-world domains where criterion utility is not monotonic, such as medicine and risk analysis, and provides interpretable frontiers and weights. A plausible implication is that the informativeness and representativeness of the labeled data are critical for parameter identifiability and meaningful restoration of underlying preference structures (Minoungou et al., 2021).