  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.
