We address a multi-period facility location problem with two customer segments having distinct service requirements. While customers in one segment receive preferred service, customers in the other segment accept delayed deliveries as long as lateness does not exceed a pre-specified threshold. The objective is to define a schedule for facility deployment and capacity scalability that satisfies all customer demands at minimum cost. Facilities can have their capacities adjusted over the planning horizon through incrementally increasing or reducing the number of modular units they hold. These two features, capacity expansion and capacity contraction, can help substantially improve the flexibility in responding to demand changes. Future customer demands are assumed to be unknown. We propose two different frameworks for planning capacity decisions and present a two-stage stochastic model for each one of them. While in the first model decisions related to capacity scalability are modeled as first-stage decisions, in the second model, capacity adjustments are deferred to the second stage. We develop the extensive forms of the associated stochastic programs for the case of demand uncertainty being captured by a finite set of scenarios. Additional inequalities are proposed to enhance the original formulations. An extensive computational study with randomly generated instances shows that the proposed enhancements are very useful. Specifically, 97.5% of the instances can be solved to optimality in much shorter computing times. Important insights are also provided into the impact of the two different frameworks for planning capacity adjustments on the facility network configuration and its total cost.

我们解决了具有不同服务要求的两个客户群的多时期设施选址问题。当一个细分市场的客户获得首选服务时，另一个细分市场的客户接受延迟交付，只要延迟不超过预先指定的阈值。目标是定义设施部署和容量可扩展性的时间表，以最低成本满足所有客户需求。设施可以通过逐步增加或减少其拥有的模块化单元的数量，在规划范围内调整其容量。产能扩张和产能收缩这两个特征，可以帮助大幅提高应对需求变化的灵活性。假设未来的客户需求是未知的。我们为规划容量决策提出了两种不同的框架，并为每个框架提供了一个两阶段随机模型。在第一个模型中，与容量可扩展性相关的决策被建模为第一阶段决策，而在第二个模型中，容量调整被推迟到第二阶段。我们针对由一组有限场景捕获的需求不确定性的情况开发了相关随机程序的广泛形式。提出了额外的不等式来增强原始公式。对随机生成的实例进行的广泛计算研究表明，所提出的增强功能非常有用。具体来说，97.5% 的实例可以在更短的计算时间内得到最优解。