Unravelling Error Patterns in Linear Programming: A Hierarchical Bayesian Approach with Small-Sample Insights

Abstract

This study examines cognitive error patterns in students’ linear programming (LP) formulations using a hierarchical Bayesian approach tailored for small-sample contexts. The research was conducted in the Mathematics Education program at Universitas Pekalongan, involving 12 undergraduate students who completed a structured LP problem-solving task. Errors were categorized into five theoretically grounded types: constraint omission, objective misidentification, sign reversal, variable mislabeling, and infeasibility misjudgment. A Bayesian hierarchical multinomial logistic model was employed to capture individual differences and group-level error structures, using weakly informative priors and Hamiltonian Monte Carlo sampling (30,000 iterations). Posterior estimates indicated dominant error trends in constraint omission and sign reversal among participants, with between-student variability accounting for 43% of the total variance. Posterior predictive checks and leave-one-out cross-validation confirmed model adequacy. These findings illustrate the diagnostic value of Bayesian modeling in identifying systematic misconceptions in LP learning, even with limited data, and provide actionable insights for personalized instructional support in mathematical problem-solving contexts