[Paper Review] 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 Management4 citations

TL;DR

This paper proposes a multistage risk-averse stochastic facility location model with and without prioritization under uncertain demand and budget, using expected conditional risk measures (ECRMs) to derive tight lower bounds on the gap between multistage and two-stage solutions. It introduces approximation algorithms for efficiency and prioritization cuts to reduce computation, showing that gaps increase with uncertainty and stagewise dependence, while bounds remain tight and algorithms are asymptotically optimal under expanding markets.

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.

Motivation & Objective

• To model multiperiod capacitated facility location under uncertain demand and budget using a multistage stochastic integer program.
• To compare multistage and two-stage frameworks in terms of risk-averse cost optimization using coherent risk measures.
• To investigate the impact of dynamic facility location decisions versus fixed upfront decisions under uncertainty.
• To develop computationally efficient methods—approximation algorithms and prioritization cuts—for solving risk-averse two-stage and multistage models.
• To analyze how uncertainty structure (e.g., stagewise dependence) affects the value of multistage over two-stage decision-making.

Proposed method

• Formulates a multistage stochastic integer program using a scenario tree to represent stagewise-dependent uncertainty in demand and budget.
• Employs expected conditional risk measures (ECRMs) to model risk-averse decision-making across stages, minimizing risk-averse total cost.
• Derives tight analytical lower bounds on the gap between optimal multistage and two-stage solutions using ECRMs, both with and without prioritization.
• Proposes two asymptotically optimal approximation algorithms for risk-averse two-stage and multistage models under an expanding market assumption.
• Introduces super-valid inequalities (prioritization cuts) to strengthen the formulation and reduce computational time in models with fixed or adaptive priority lists.
• Employs numerical experiments on real-world and synthetic instances to validate bounds, algorithm performance, and cut effectiveness.

Experimental results

Research questions

• RQ1 How does the value of multistage risk-averse facility location compare to two-stage solutions under uncertain demand and budget?
• RQ2 What is the theoretical gap between optimal multistage and two-stage solutions, and how tight are the derived lower bounds?
• RQ3 How do approximation algorithms perform in solving risk-averse two-stage and multistage models, especially under expanding market assumptions?
• RQ4 To what extent do prioritization cuts reduce computational time in risk-averse models with fixed or adaptive priority lists?
• RQ5 How does the structure of uncertainty (e.g., stagewise dependence vs. independence) affect the value of dynamic decision-making in facility location?

Key findings

• The gap between optimal multistage and two-stage solutions increases with higher variations in uncertain parameters such as demand and budget.
• Stagewise dependent scenario trees yield significantly higher gaps than stagewise independent ones, highlighting the value of dynamic adaptation.
• The analytical lower bounds derived using ECRMs are tight and recover the true gaps in many test cases.
• The proposed approximation algorithms achieve approximation ratios as low as 1.05 in real-world network instances, demonstrating strong empirical performance.
• Prioritization cuts significantly reduce computational time and optimality gaps, especially in models with fixed or adaptive priority lists.
• The optimal facility site list changes across demand patterns, but high-demand regions like California consistently rank in the top six due to demand concentration.