Uncovering latent consensus in heterogeneous populations: The Mixture Linear Ordering Problem

TL;DR

This paper introduces a new approach to identify latent groups with distinct preferences within heterogeneous populations using a mixture linear ordering model, combining exact and heuristic methods.

Contribution

It extends the classical linear ordering problem by modeling multiple latent preference groups and develops mixed-integer programming formulations and heuristics for practical solutions.

Findings

Exact methods effectively recover underlying groups in synthetic data.

Heuristic approaches provide high-quality solutions faster, sometimes outperforming exact methods.

The geometric reformulation offers new insights into the linear ordering polytope.

Abstract

The classical linear ordering problem seeks a single ranking representing a given preference matrix. While suitable for homogeneous populations, it fails when observed preferences arise from several latent groups with distinct ranking patterns. To address this limitation, we introduce an extension partitioning the population into latent groups, each characterized by its own linear order, relative size, and preference structure. The observed matrix is then explained as the aggregate outcome of these group-specific preferences. We develop mixed-integer programming formulations, including a compact reformulation yielding a geometric interpretation within the linear ordering polytope. Because exact solutions become computationally demanding for larger instances, we propose a multi-start alternating-direction matheuristic iteratively updating group rankings and weights. Computational experiments on synthetically generated instances, matching sizes typical in preference aggregation scenarios, demonstrate the effectiveness of the exact approach in successfully recovering the underlying groups. Furthermore, the proposed heuristic delivers high-quality solutions in substantially shorter times, occasionally improving upon the exact method's best incumbent in difficult instances within the imposed time limit.