Coordination of lot-sizing and pricing decisions with cross-price demand and discrete menu of prices

This paper addresses a dynamic multi-product coordinated lot-sizing and pricing problem faced by a profit-maximizing manufacturer with limited production capacity. The problem is formulated as a mixed-integer nonlinear programming (MINLP) model, where capacity constraints, setup costs, and dynamic demand functions incorporating substitution and complementarity effects among products are considered. Unlike previous studies, a discrete pricing menu is introduced, providing a more realistic pricing framework. To handle the non-linearity and non-convexity of the model, a big M-based linear reformulation is proposed, and a fix-and-relax heuristic is developed to decompose the planning horizon and improve computational efficiency. A benchmark dataset inspired by the literature is used to analyze different demand scenarios and capacity constraints, demonstrating the impact of pricing decisions on production planning. Computational experiments show that the integration of pricing and lot-sizing significantly enhances decision-making, while the proposed heuristic efficiently finds near-optimal solutions within one minute for instances with up to 32 periods. These findings provide valuable managerial insights into the relationship between pricing, production decisions, and capacity constraints in dynamic manufacturing environments.