Single-item lot sizing problem under budgeted lead-time uncertainty

2026-06-15

AI Summary

This study addresses the single-product lot-sizing problem under uncertain lead times and a budget constraint, incorporating practical considerations such as backlogging, inventory holding, and the infeasibility of splitting or crossing orders. The authors propose a robust optimization approach based on the R* criterion, which transcends the conventional min-max framework by constructing a continuum of production plans ranging from the most optimistic to the most pessimistic scenarios, thereby expanding the feasible solution space. Notably, this work is the first to integrate a budgeted uncertainty set with non-crossing order constraints and develops both polynomial- and pseudo-polynomial-time algorithms alongside a mixed-integer programming formulation for solution. Computational experiments demonstrate that the R* criterion significantly enhances decision flexibility and system performance while preserving feasibility.

Abstract

In this paper, a single-item lot sizing problem with backordering is discussed. The time horizon is divided into planning periods, characterized by fixed and variable production costs, and future delivery periods with specified demands, where inventory holding and backordering costs may occur. For each planning period, a common nominal lead time is given. The true lead times can deviate to some extent from the nominal one, and their exact values are unknown at the planning step. We assume that lead times take only integer values and splitting production orders is not allowed. Furthermore, order crossovers are prohibited; thus, an order placed earlier cannot arrive after one placed later. A budgeted uncertainty set of possible lead-time scenarios is defined, where a budget allows us to control the amount of uncertainty of lead times. It is shown how to construct a family of production plans varying from the most optimistic (a best lead-time scenario occurs) to the most pessimistic (a worst lead-time scenario occurs). In order to compute these plans the R* criterion is applied which generalizes the conservative robust min-max criterion, commonly used in robust optimization. The computational complexity of the problem is investigated. Polynomial, pseudopolynomial time algorithms, and mixed integer programming formulations are proposed to solve the general problem and its special cases. The results of computational tests are provided that demonstrate that using the R* criterion can significantly enlarge the set of candidate production plans.

Problem

• lot sizing
• lead-time uncertainty
• budgeted uncertainty
• single-item
• production planning

Innovation

• budgeted uncertainty
• lot sizing
• lead-time uncertainty
• R* criterion
• robust optimization