Optimal order quantities of a multi-period inventory of compatible products

Abstract

Effectively managing perishable inventory that consists of a heterogeneous assortment of mutually compatible products presents a complex and computationally challenging problem in operations research and supply chain fields. This particular problem arises when certain products can serve as substitutes for others, an operation commonly observed in maintenance parts, electronics, pharmaceuticals, batteries, and various other industries. Unlike single-period/single-product inventory models, where closed form solutions like the classical newsvendor are available in the literature, the compatibility between multiple products across multiple planning periods has remained largely unexplored in previous research.

Thus, the objective of our study is to find the optimal order quantities of different product types in an inventory system to minimize shortages and expiration, taking into account the challenging aspects of perishability and the existence of compatible substitutes. To rigorously address the complexities of this inventory system, a mixed-integer linear programming (MILP) framework is developed to encapsulate the intricate structural and operational characteristics of the problem. The proposed model is designed to accommodate both deterministic and stochastic scenarios, enabling the derivation of exact and approximate solutions through the integration of advanced optimization and simulation-based methodologies.

For large-scale instances, we recognize the computational challenges that arise when using standard solvers. Consequently, a metaheuristic algorithm is developed, which is specifically designed to reduce the computational time required to solve big instances. By tackling the interplay between perishability, compatibility, and multi periods, our study pushes the boundaries of existing research and presents innovative solution to real inventory where product substitution is allowed.

The findings showed that less shortages and expiration result in the case of compatible products. Compatibility also results in less order quantities. For a single inventory period, compatibility demonstrates higher effect on reducing the expected shortage and spoilage. For the case of full compatibility where any product can serve as a substitute for the others, closed form solution can be found for single period planning, while in multiple period planning, MILP optimization is required to address the problem. Finally, when compatibility is not present, it will lead to independent demand, thereby making the classical newsvendor problem applicable to this special case, but even in this case, optimization will be required for multi-period.

The proposed model exhibits broad applicability across a wide spectrum of industrial and medical applications, including but not limited to management of spare parts inventory, pharmaceutical supply chains, perishable food products such as dairy and ready-to-eat meals, the automotive sector, and systems involving modular subassemblies.