The Best Allocation Resource Size (BARS) problem: a bi-objective integer programming model to balance cost and complexity in resource selection

This paper introduces a variant of the assortment problem, i.e., the Best Allocation Resource Size (BARS) problem, conceived as a multi-objective nonlinear optimization model that jointly minimizes total production-related costs and input complexity by selecting the optimal subset of resource sizes, explicitly balancing cost-efficiency with scrap minimization. Unlike traditional assortment problems, the BARS adopts a manufacturer-led perspective, handling multiple characteristics of the inputs, i.e., it is n-dimensional, and it integrates multiple cost dimensions - including setup, scrap, safety stock, and non-linear purchase costs - under realistic manufacturing constraints, thus offering a flexible, producer-oriented decision-support tool adaptable to diverse industrial contexts. The problem formulation does not incorporate an ex-ante budget constraint, allowing the solution space to be retrospectively filtered by applying cost thresholds - either on aggregate expenditure or individual cost components - thus enabling targeted post-hoc analyses of feasible item sets in line with strategic objectives, including scenarios that allow for controlled increases in spending. The model proposed is solved using an adapted ε-constraint method that, unlike a generic scalarization, is provably exact for bi-objective nonlinear integer programs: a second-order cone reformulation of the safety-stock term allows to determine properly calibrated ε values, ensuring recovery of the complete Pareto front. The efficacy of the proposed model is demonstrated through validation in three real-world case studies spanning different industrial sectors. The developed model serves as a strategic decision tool for management, facilitating an understanding of the costs consequences associated with different sets of selected raw materials.