On the Value of Risk-Averse Multistage Stochastic Programming in Capacity Planning

TL;DR

This paper investigates the benefits of multistage risk-averse stochastic programming in capacity planning under uncertainty, providing bounds, guidelines, and algorithms to compare multistage and two-stage approaches.

Contribution

It introduces analytical bounds and approximation algorithms for risk-averse multistage capacity planning, and offers insights on when to prefer multistage over two-stage models.

Findings

Gaps increase with higher uncertainty variability.

Gaps decrease with increased risk aversion.

Stagewise-dependent scenarios lead to larger gaps.

Abstract

We consider a risk-averse stochastic capacity planning problem under uncertain demand in each period. Using a scenario tree representation of the uncertainty, we formulate a multistage stochastic integer program to adjust the capacity expansion plan dynamically as more information on the uncertainty is revealed. Specifically, in each stage, a decision maker optimizes capacity acquisition and resource allocation to minimize certain risk measures of maintenance and operational cost. We compare it with a two-stage approach that determines the capacity acquisition for all the periods up front. Using expected conditional risk measures (ECRMs), we derive a tight lower bound and an upper bound for the gaps between the optimal objective values of risk-averse multistage models and their two-stage counterparts. Based on these derived bounds, we present general guidelines on when to solve risk-averse two-stage or multistage models. Furthermore, we propose approximation algorithms to solve the two models more efficiently, which are asymptotically optimal under an expanding market assumption. We conduct numerical studies using randomly generated and real-world instances with diverse sizes, to demonstrate the tightness of the analytical bounds and efficacy of the approximation algorithms. We find that the gaps between risk-averse multistage and two-stage models increase as the variability of the uncertain parameters increases and decrease as the decision maker becomes more risk-averse. Moreover, stagewise-dependent scenario tree attains much higher gaps than stagewise-independent counterpart, while the latter produces tighter analytical bounds.