[论文解读] On the Value of Multistage Risk-Averse Stochastic Facility Location With or Without Prioritization

Xianjun Yu, Siqian Shen|arXiv (Cornell University)|May 23, 2021

Facility Location and Emergency Management被引用 4

一句话总结

本文提出了一种在不确定需求和预算下，带有与不带优先级的多阶段风险规避随机设施选址模型，使用期望条件风险度量（ECRMs）推导出多阶段与两阶段解决方案之间差距的紧致下界。该研究引入了近似算法与优先级切割以提高计算效率，表明随着不确定性和阶段间依赖性的增加，差距也随之增大，而下界保持紧致，且在市场扩张条件下，算法具有渐近最优性。

ABSTRACT

We consider a multiperiod stochastic capacitated facility location problem under uncertain demand and budget in each period. Using a scenario tree representation of the uncertainties, we formulate a multistage stochastic integer program to dynamically locate facilities in each period and compare it with a two-stage approach that determines the facility locations up front. In the multistage model, in each stage, a decision maker optimizes facility locations and recourse flows from open facilities to demand sites, to minimize certain risk measures of the cost associated with current facility location and shipment decisions. When the budget is also uncertain, a popular modeling framework is to prioritize the candidate sites. In the two-stage model, the priority list is decided in advance and fixed through all periods, while in the multistage model, the priority list can change adaptively. In each period, the decision maker follows the priority list to open facilities according to the realized budget, and optimizes recourse flows given the realized demand. Using expected conditional risk measures (ECRMs), we derive tight lower bounds for the gaps between the optimal objective values of risk-averse multistage models and their two-stage counterparts in both settings with and without prioritization. Moreover, we propose two approximation algorithms to efficiently solve risk-averse two-stage and multistage models without prioritization, which are asymptotically optimal under an expanding market assumption. We also design a set of super-valid inequalities for risk-averse two-stage and multistage stochastic programs with prioritization to reduce the computational time. We conduct numerical studies using both randomly generated and real-world instances with diverse sizes, to demonstrate the tightness of the analytical bounds and efficacy of the approximation algorithms and prioritization cuts.

研究动机与目标

• 使用多阶段随机整数规划建模在不确定需求与预算下的多期容量设施选址问题。
• 利用一致风险度量比较多阶段与两阶段框架在风险规避成本优化方面的表现。
• 研究在不确定性下，动态设施选址决策与固定前期决策之间的差异影响。
• 为求解风险规避的两阶段与多阶段模型，开发计算高效的近似算法与优先级切割方法。
• 分析不确定性结构（如阶段间依赖性）对多阶段相对于两阶段决策价值的影响。

提出的方法

• 使用场景树表示需求与预算的阶段依赖性不确定性，构建多阶段随机整数规划模型。
• 采用期望条件风险度量（ECRMs）建模跨阶段的风险规避决策，最小化风险规避总成本。
• 利用ECRMs推导出在有无优先级情况下的最优多阶段与两阶段解决方案之间差距的紧致解析下界。
• 提出两种在市场扩张假设下对风险规避两阶段与多阶段模型渐近最优的近似算法。
• 引入超有效不等式（优先级切割）以加强公式表达，减少具有固定或自适应优先级列表模型的计算时间。
• 通过真实世界与合成实例的数值实验，验证下界、算法性能与切割方法的有效性。

实验结果

研究问题

• RQ1 在不确定需求与预算下，多阶段风险规避设施选址相对于两阶段解决方案的价值如何？
• RQ2 最优多阶段与两阶段解决方案之间的理论差距是多少？所推导的下界有多紧？
• RQ3 近似算法在求解风险规避两阶段与多阶段模型时表现如何，特别是在市场扩张假设下？
• RQ4 优先级切割在具有固定或自适应优先级列表的风险规避模型中，能在多大程度上减少计算时间？
• RQ5 不确定性结构（如阶段间依赖性与独立性）如何影响设施选址中动态决策的价值？

主要发现

• 最优多阶段与两阶段解决方案之间的差距随着不确定参数（如需求与预算）的波动性增加而增大。
• 阶段间依赖的场景树产生的差距显著高于阶段间独立的场景树，凸显了动态适应的价值。
• 利用ECRMs推导出的解析下界非常紧，在许多测试案例中能准确恢复真实差距。
• 所提出的近似算法在真实网络实例中达到最低为1.05的近似比，表现出强劲的实证性能。
• 优先级切割显著减少了计算时间与最优性差距，尤其在具有固定或自适应优先级列表的模型中效果明显。
• 最优设施选址列表随需求模式变化而变化，但高需求区域（如加利福尼亚州）始终位列前六名，归因于需求集中。