We consider a service region planning problem faced by a crowdsourced delivery platform where drivers are commuters who are willing to deviate from their original routes to a make a delivery in exchange for a compensation. The availability of drivers and service demand are uncertain, and it is difficult to estimate their probability distributions due to the absence of service history. To mitigate the effects of data ambiguity, we propose a two-stage distributionally robust optimization (DRO) model. The first stage selects which nodes to offer service in, while the second stage matches drivers and orders after uncertainty is realized. We derive an exact reformulation of the DRO model and develop a monolithic approximation based on a convex relaxation of the subproblem. We further strengthen the proposed approximation by a set of valid inequalities inspired by the linearization reformulation technique. The benefit of the proposed approach for service region design in crowdsourced delivery is demonstrated through numerical experiments.

Facility disruptions can severely impact the reliability and operational cost of many service systems. In this paper, we consider a reliable uncapacitated facility location problem, in which opened facilities may suffer random failure due to, e.g., natural disasters. In reality, past observations of such failure are often limited, making it challenging to estimate the probability distribution. In this paper, we assume that this distribution is ambiguous and it lives in a Wasserstein ball centered at an empirical distribution. Accordingly, we consider a two-stage model that minimizes the fixed location cost plus the expected assignment cost, with regard to the worst-case probability distribution in the Wasserstein ball. This approach implicitly considers many more failure distributions than the empirical one, leading to a ``distributionally robust'' solution with better out-of-sample performance. Although two-stage distributionally robust models are NP-hard to solve in general, we exploit the problem structure to derive a compact reformulation based on a mixed-integer linear program. In addition, this derivation produces the worst-case failure distribution of the facilities. To solve this model, we design two branch-and-cut algorithms, both of which significantly outperform commercial solvers in computation. Finally, we conduct extensive numerical studies on standard test instances in the literature. The results demonstrate the effectiveness and potential value of our approach.

We study a novel class of fractional distributionally robust optimization, namely P-DRO, which maximizes the worst-case probability of exceeding of a reward-risk ratio. We consider two types of ambiguity sets, moment-based and statistical distance-based and two types of support, uncertain probabilities and continuum of realizations. Under specific conditions of the reward-risk functions, we develop new convex reformulations for each type of ambiguity sets.

We propose a novel polyhedral uncertainty set for robust optimization. This set imposes bounds one the difference of pairs of uncertain parameters. Intuitively, this is because some uncertain parameters may not differ by more than some given absolute amounts. Such bounds may be dictated by the underlying physics of the problem and may be easily expressed by domain experts. Compared with other methods that may model the relationship of uncertain parameters by linear correlation, smooth uncertainty may only require bounds on the difference of uncertain parameters. Although not required to define our sets, we derive probability guarantees that improve as a function of the magnitude of positive correlation between uncertain parameter values. Finally, we develop compact reformulation techniques for specially structured problems and a row generation algorithm.