Modern probability theory, whose foundation is based on the axioms set forth by Kolmogorov, is currently the major tool for performance analysis in stochastic systems. While it offers insights in understanding such systems, probability theory, in contrast to optimization, has not been developed with computational tractability as an objective when the dimension increases. Correspondingly, some of its major areas of application remain unsolved when the underlying systems become multidimensional: Queueing networks, auction design in multi-item, multi-bidder auctions, network information theory, pricing multi-dimensional options, among others. We propose a new approach to analyze stochastic systems based on robust optimization. The key idea is to replace the Kolmogorov axioms and the concept of random variables as primitives of probability theory, with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem. In addition, we observe that several desired system properties such as incentive compatibility and individual rationality in auction design are naturally expressed in the language of robust optimization. In this way, the performance analysis questions become highly structured optimization problems (linear, semidefinite, mixed integer) for which there exist efficient, practical algorithms that are capable of solving problems in high dimensions. We demonstrate that the proposed approach achieves computationally tractable methods for (a) analyzing queueing networks, (b) designing multi-item, multi-bidder auctions with budget constraints, and (c) pricing multi-dimensional options.

We propose an alternative approach for studying queues based on robust optimization. We model the uncertainty in the arrivals and services via polyhedral uncertainty sets which are inspired from the limit laws of probability. Using the generalized central limit theorem, this framework allows to model heavy-tailed behavior characterized by bursts of rapidly occurring arrivals and long service times. We take a worst-case approach and obtain closed form upper bounds on the system time in a multi-server queue. These expressions provide qualitative insights which mirror the conclusions obtained in the probabilistic setting for light-tailed arrivals and services and generalize them to the case of heavy-tailed behavior. We also develop a calculus for analyzing a network of queues based on the following key principle: (a) the departure from a queue, (b) the superposition, and (c) the thinning of arrival processes have the same uncertainty set representation as the original arrival processes. The proposed approach (a) yields results with error percentages in single digits relative to simulation, and (b) is to a large extent insensitive to the number of servers per queue, network size, degree of feedback, traffic intensity, and somewhat sensitive to the degree of diversity of external arrival distributions in the network.

In this paper, we revisit the auction design problem for multi-item auctions with budget constrained buyers by introducing a robust optimization approach to model (a) concepts such as incentive compatibility and individual rationality that are naturally expressed in the language of robust optimization and (b) the auctioneer’s beliefs on the buyers' valuations of the items. Rather than using probability distributions (the classical probabilistic approach) or an adversarial model to model valuations, we introduce an uncertainty set based model for these valuations. We construct these uncertainty sets to incorporate historical information available to the auctioneer in a way that is consistent with limit theorems of probability theory or knowledge of the probability distribution. In this setting, we formulate the auction design problem as a robust optimization problem and provide a characterization of the optimal solution as an auction with reservation prices, thus extending the work of Myerson [Myerson RB (1981) Optimal auction design. Math. Oper. Res. 6(1):58–73] from single item without budget constraints to multiple items with budgets, potentially correlated valuations, and uncertain budgets. Unlike the Myerson auction where the reservation prices do not depend on the item, the reservation prices in our approach are a function of both the bidder and the item. We propose an algorithm for calculating the reservation prices by solving a bilinear optimization problem that, although theoretically difficult in general, is numerically tractable for the polyhedral uncertainty sets we consider. Moreover, this bilinear optimization problem reduces to a linear optimization problem for auctions without budget constraints and the auction becomes the classical second price auction. We report computational evidence that suggests the proposed approach (a) is numerically tractable for large scale auction design problems with the polyhedral uncertainty sets we consider, (b) leads to improved revenue compared to the classical probabilistic approach when the true distributions are different from the assumed ones, and (c) leads to higher revenue when correlations in the buyers' valuations are explicitly modeled.

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Problem definition: We consider two problems faced by an operating room (OR) manager: (1) how many baseline (core) staff to hire for OR suites, and (2) how to schedule surgery requests that arrive one by one. The OR manager has access to historical case count and case length data, and needs to balance the fixed cost of baseline staff and variable cost of overtime, while satisfying surgeons’ preferences. Academic/practical relevance: ORs are costly to operate and generate about 70% of hospitals’ revenues from surgical operations and subsequent hospitalizations. Because hospitals are increasingly under pressure to reduce costs, it is important to make staffing and scheduling decisions in an optimal manner. Also, hospitals need to leverage data when developing algorithmic solutions, and model tradeoffs between staffing costs and surgeons’ preferences. We present a methodology for doing so, and test it on real data from a hospital. Methodology: We propose a new criterion called the robust competitive ratio for designing online algorithms. Using this criterion and a robust optimization approach to model the uncertainty in case mix and case lengths, we develop tractable optimization formulations to solve the staffing and scheduling problems. Results: For the staffing problem, we show that algorithms belonging to the class of interval classification algorithms achieve the best robust competitive ratio, and develop a tractable approach for calculating the optimal parameters of our proposed algorithm. For the scheduling phase, which occurs one or two days before each surgery day, we demonstrate how a robust optimization framework may be used to find implementable schedules while taking into account surgeons’ preferences such as back-to-back and same-OR scheduling of cases. We also perform numerical experiments with real and synthetic data, which show that our approach can significantly reduce total staffing cost. Managerial implications: We present algorithms that are easy to implement and tractable. These algorithms also allow the OR manager to specify the size of the uncertainty set and to control overtime costs while meeting surgeons’ preferences.