A Multi-Item Partial Backordering and Lost Sales Economic Order Quantity Model with Cross-Selling Correlated Demand considering Budget and Greenhouse Gas Emission Constraints

Abstract

In today’s sustainability-driven retail environment, inventory decisions must simultaneously account for cross-product demand interactions, financial restrictions, and environmental limits. This study develops a multi-item Economic Order Quantity model that assumes correlated demand arising from cross-selling relationships, incorporates partial backordering and lost sales, and operates under strict procurement-budget and greenhouse-gas emission constraints. The objective is to maximize total profit by jointly determining optimal order quantities and inventory cycles while satisfying demand requirements and adhering to environmental and financial limitations. Association-rule mining is employed to identify demand-correlation structures, allowing the model to represent realistic customer purchasing behavior. The problem is formulated as a nonlinear integer program and analytically shown to be NP-hard, motivating the use of a tailored Genetic Algorithm featuring tournament and elitist selection, single-point crossover, and uniform mutation. Benchmarking against GAMS demonstrates that although exact optimization yields slightly superior results for small instances, the GA achieves near-optimal objective values with up to 99.3% lower CPU time in large-scale settings, confirming its scalability. Sensitivity analyses indicate that budget increases and higher margins for fast-selling products significantly enhance profitability, whereas tighter environmental constraints reduce it. The results also reveal that reducing independent demand for low-margin, high-volume items can improve overall profit by enabling greater allocation toward high-margin, cross-sold products. Overall, the study offers a rigorous and scalable decision-support framework that integrates economic, operational, and environmental objectives in correlated-demand inventory systems.