A physical model approach to order lot sizing

Abstract

The growing need for companies to reduce costs and maximize profits has led to an increased focus on logistics activities. Among these, inventory management plays a crucial role in minimizing organizational expenses by optimizing the storage and transportation of materials. In this context, this study introduces an optimization model for the lot-sizing problem based on a physical system approach. By establishing that the material supply problem is isomorphic to a one-dimensional mechanical system of point particles connected by elastic elements, we leverage this analogy to derive cost optimization conditions naturally and obtain an exact solution. This approach determines lot sizes that minimize the combined ordering and inventory holding costs in a significantly shorter time, eliminating the need for heuristic methods. The optimal lot sizes are defined in terms of the parameter $ γ= 2C_O / C_H $, which represents the relationship between the ordering cost per order ($ C_O $) and the holding cost per period for the material required in one period ($ C_H $). This parameter fully dictates the system's behavior: when $ γ\leq 1 $, the optimal strategy is to place one order per period, whereas for $ γ> 1 $, the number of orders $ N $ is reduced relative to the planning horizon $ M $, meaning $ N < M $. By formulating the total cost function in terms of the intensive variable $ N/M $, we consolidate the entire optimization problem into a single function of $ γ$. This eliminates the need for complex algorithms, enabling faster and more precise purchasing decisions. The proposed model was validated through a real-world case study and benchmarked against classical algorithms, demonstrating superior cost optimization and reduced execution time. These findings underscore the potential of this approach for improving material lot-sizing strategies.