Adaptive Wasserstein Robust Optimization for Procurement and Location Design

Abstract

Individuals and organizations frequently face dynamic decision-making under uncertainty. Each decision must consider both its immediate effects and future ramifications. Adaptive distributionally robust optimization (ADRO) is a powerful tool for tackling multi-stage dynamic optimization problems. However, a key challenge in ADRO is developing a statistically and computationally efficient framework that effectively extrapolates path-dependent uncertain parameters while maintaining tractability. This paper addresses this challenge in the context of two-stage option procurement and multi-stage hub location problems, leveraging the Wasserstein distributionally robust optimization framework. Specifically, we extend modeling approaches and solution methods for two-stage distributionally robust optimization and explore techniques for extracting useful information from time-series data to better characterize multi-stage uncertainties, thereby establishing a time-series predictive multi-stage distributionally robust optimization paradigm.

Firstly, we study a procurement problem in which the buyer procures from suppliers with option contracts and/or spot market with a random price, to satisfy the random demand. To address randomness and distributional ambiguity of price and demand, we adopt the minimax regret criterion based on a Wasserstein distributionally robust optimization approach. We identify the regret-optimal portfolio of options and derive optimality conditions of the minimax regret model under three cases: one option contract, fixed spot price, and high ambiguity level. These optimality conditions extend the trade-off between overorder and underorder regrets, as established in the literature for the minimax regret criterion. From an optimization perspective, we derive tractable reformulations for the three representative cases and devise an efficient cutting-plane algorithm to address the general case. We then conclude several managerial insights: (i) the regret-optimal portfolio well balances, at each cumulative reservation position, the regrets of ordering too much and ordering too little; (ii) under the minimax regret criterion, contracts with a lower critical overorder-underorder ratio are prioritized; and (iii) numerical experiments demonstrate that the regret-optimal decision outperforms alternative data-driven decisions in view of both out-of-sample regret and cost.

Secondly, we study the (un)capacitated multi-period hub location problem with uncertain periodic demands. With a distributionally robust approach that considers time series, we build a model driven by budgets on periodic costs. In particular, we construct a nested ambiguity set that characterizes uncertain periodic demands via a general multivariate time-series model, and to ensure stable periodic costs, we propose to constrain each expected periodic cost within a budget while optimizing the robustness level by maximizing the size of the nested ambiguity set. Statistically, the nested ambiguity set ensures that the model's solution enjoys finite-sample performance guarantees, under certain regularity conditions on the underlying VAR(p) or VARMA(p,q) process of the stochastic demand. Operationally, we show that our budget-driven model in the uncapacitated case essentially optimizes a "Sharpe Ratio"-type criterion over the worst case among all periods, and we discuss how cost budgets would affect the optimal robustness level. Computationally, the uncapacitated model can be efficiently solved via a bisection search algorithm that solves (in each iteration) a mixed-integer conic program, while the capacitated model can be approximated by using decision rules. Finally, numerical experiments demonstrate the attractiveness and competitiveness of our proposed model.

Date of Award	6 Nov 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Shuming Wang (External Supervisor), Zhi CHEN (Supervisor) & Guangwu LIU (Supervisor)