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.

翻译：本文讨论了允许缺货回补的单物品批量问题。时间范围被划分为若干计划期，每个计划期具有固定和可变生产成本，以及具有特定需求的未来交付期，期间可能产生库存持有成本和缺货回补成本。每个计划期给定一个共同的名义交货期。实际交货期可能在一定程度上偏离名义值，且其确切值在计划阶段未知。我们假设交货期仅取整数值，且不允许分批生产订单。此外，禁止订单交叉，即较早下达的订单不得晚于较晚下达的订单到达。定义了可能的交货期情景的预算不确定性集，其中预算允许我们控制交货期的不确定性程度。本文展示了如何构建从最乐观（最佳交货期情景发生）到最悲观（最差交货期情景发生）的一系列生产计划。为计算这些计划，采用了R*准则，该准则推广了鲁棒优化中常用的保守鲁棒最小-最大准则。研究了该问题的计算复杂度。提出了多项式、伪多项式时间算法以及混合整数规划公式来求解一般问题及其特殊情况。提供的计算测试结果表明，使用R*准则可以显著扩大候选生产计划的集合。