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Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 110 (16 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
A Robust Optimization Perspective Of Stochastic Programming
, 2005
"... In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for bounded random variables known as the forward and backward deviations. These deviation measures capture distributional asymmetry and lead to better approximations of c ..."
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Cited by 51 (12 self)
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In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for bounded random variables known as the forward and backward deviations. These deviation measures capture distributional asymmetry and lead to better approximations of chance constraints. We also propose a tractable robust optimization approach for obtaining robust solutions to a class of stochastic linear optimization problems where the risk of infeasibility can be tolerated as a tradeoff to improve upon the objective value. An attractive feature of the framework is the computational scalability to multiperiod models. We show an application of the framework for solving a project management problem with uncertain activity completion time.
Approximation algorithms for stochastic inventory control models
 Math. Oper. Res
, 2007
"... We consider two classical stochastic inventory control models, the periodicreview stochastic inventory control problem and the stochastic lotsizing problem. The goal is to coordinate a sequence of orders of a single commodity, aiming to supply stochastic demands over a discrete, finite horizon wit ..."
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Cited by 19 (5 self)
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We consider two classical stochastic inventory control models, the periodicreview stochastic inventory control problem and the stochastic lotsizing problem. The goal is to coordinate a sequence of orders of a single commodity, aiming to supply stochastic demands over a discrete, finite horizon with minimum expected overall ordering, holding and backlogging costs. In this paper, we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated and nonstationary (timedependent) stochastic demands. This problem arises in many domains and has many practical applications in supply chain management. Our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel costbalancing techniques. Specifically, in each period, we balance the expected cost of over ordering (i.e, costs incurred by excess inventory) against the expected cost of under ordering (i.e., costs incurred by not satisfying demand on time). This leads to what we believe to be the first computationally efficient policies with constant worstcase performance guarantees for a general class of important stochastic inventory models. That is, there exists a constant C such that, for any instance of the problem, the expected cost of the policy is at most C times the expected cost of an
Robust and DataDriven Optimization: Modern DecisionMaking Under Uncertainty
, 2006
"... Traditional models of decisionmaking under uncertainty assume perfect information, i.e., accurate values for the system parameters and specific probability distributions for the random variables. However, such precise knowledge is rarely available in practice, and a strategy based on erroneous inp ..."
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Cited by 18 (0 self)
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Traditional models of decisionmaking under uncertainty assume perfect information, i.e., accurate values for the system parameters and specific probability distributions for the random variables. However, such precise knowledge is rarely available in practice, and a strategy based on erroneous inputs might be infeasible or exhibit poor performance when implemented. The purpose of this tutorial is to present a mathematical framework that is wellsuited to the limited information available in reallife problems and captures the decisionmaker’s attitude towards uncertainty; the proposed approach builds upon recent developments in robust and datadriven optimization. In robust optimization, random variables are modeled as uncertain parameters belonging to a convex uncertainty set and the decisionmaker protects the system against the worst case within that set. Datadriven optimization uses observations of the random variables as direct inputs to the mathematical programming problems. The first part of the tutorial describes the robust optimization paradigm in detail in singlestage and multistage problems. In the second part, we address the issue of constructing uncertainty sets using historical realizations of the random variables and investigate the connection between convex sets, in particular polyhedra, and a specific class of risk measures.
Provably nearoptimal samplingbased policies for stochastic inventory control models
 Proceedings, 38th Annual ACM Symposium on Theory of Computing
, 2006
"... In this paper, we consider two fundamental inventory models, the singleperiod newsvendor problem and its multiperiod extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true ..."
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Cited by 16 (2 self)
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In this paper, we consider two fundamental inventory models, the singleperiod newsvendor problem and its multiperiod extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions. Under the assumption that the demand distributions are given explicitly, these models are wellstudied and relatively straightforward to solve. However, in most reallife scenarios, the true demand distributions are not available or they are too complex to work with. Thus, a samplingdriven algorithmic framework is very attractive, both in practice and in theory. We shall describe how to compute samplingbased policies, that is, policies that are computed based only on observed samples of the demands without any access to, or assumptions on, the true demand distributions. Moreover, we establish bounds on the number of samples required to guarantee that with high probability, the expected cost of the samplingbased policies is arbitrarily close (i.e., with arbitrarily small relative error) compared to the expected cost of the optimal policies which have full access to the demand distributions. The bounds that we develop are general, easy to compute and do not depend at all on the specific demand distributions.
A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing
 IEEE Transactions on Circuits and SystemsI
, 2004
"... A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is th ..."
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Cited by 15 (4 self)
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A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is the minimum time required to complete all activities. In a stochastic activity network (SAN), the durations of the activities and the makespan are random variables. The analysis of SANs is quite involved, but can be carried out numerically by Monte Carlo analysis. This paper concerns the optimization of a SAN, i.e., the choice of some design variables that affect the probability distributions of the activity durations. We concentrate on the problem of minimizing a quantile (e.g., 95%) of the makespan, subject to constraints on the variables. This problem has many applications, ranging from project management to digital integrated circuit (IC) sizing (the latter being our motivation). While there are effective methods for optimizing DANs, the SAN optimization problem is much more difficult; the few existing methods cannot handle largescale problems.
Risk aversion in inventory management
 The Logic of Logistics: Theory, Algorithms, and Applications for Logistics Management, 2nd
, 2004
"... Traditional inventory models focus on risk neutral decision makers, i.e., characterizing replenishment strategies that maximize expected total profit, or equivalently, minimize expected total cost over a planning horizon. In this paper, we propose a general framework for incorporating risk aversion ..."
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Cited by 14 (3 self)
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Traditional inventory models focus on risk neutral decision makers, i.e., characterizing replenishment strategies that maximize expected total profit, or equivalently, minimize expected total cost over a planning horizon. In this paper, we propose a general framework for incorporating risk aversion in multiperiod inventory models as well as multiperiod models that coordinate inventory and pricing strategies. In each case, we characterize the optimal policy for various measures of risk that have been commonly used in the finance literature. In particular, we show that the structure of the optimal policy for a decision maker with exponential utility function is almost identical to the structure of the optimal risk neutral inventory (and pricing) policies. Computational results demonstrate the importance of this approach not only to risk averse decision makers, but also to risk neutral decision makers with limited information on the demand distribution. 1
Adaptive datadriven inventory control policies based on KaplanMeier estimator
, 2009
"... Using the wellknown productlimit form of the KaplanMeier estimator from statistics, we propose a new class of nonparametric adaptive datadriven policies for stochastic inventory control problems. We focus on the distributionfree newsvendor model with censored demands. The assumption is that the ..."
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Cited by 14 (2 self)
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Using the wellknown productlimit form of the KaplanMeier estimator from statistics, we propose a new class of nonparametric adaptive datadriven policies for stochastic inventory control problems. We focus on the distributionfree newsvendor model with censored demands. The assumption is that the demand distribution is not known and there is only sales data available. We study the theoretical performance of the new policies and show that for discrete demand distributions they converge almost surely to the set of optimal solutions. Computational experiments suggest that the new policies converge for general demand distributions, not necessarily discreet, and demonstrate that they are significantly more robust than previously known policies. As a byproduct of the theoretical analysis, we obtain new results on the asymptotic consistency of the KaplanMeier estimator for discrete random variables that extend existing work in statistics. To the best of our knowledge, this is the first application of the KaplanMeier estimator within an adaptive optimization algorithm, in particular, the first application to stochastic inventory control models. We believe that this work will lead to additional applications in other domains.
Robust capacity expansion of network flows
, 2005
"... We consider the question of deciding capacity expansions for a network flow problem that is subject to demand and travel time uncertainty. We introduce a robust optimization based approach to obtain a capacity expansion solution that is insensitive to this uncertainty. We show that solving for a rob ..."
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Cited by 13 (2 self)
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We consider the question of deciding capacity expansions for a network flow problem that is subject to demand and travel time uncertainty. We introduce a robust optimization based approach to obtain a capacity expansion solution that is insensitive to this uncertainty. We show that solving for a robust solution is a computationally tractable problem for general uncertainty sets and under reasonable conditions for network flow applications. For example, the robust problem is tractable for a multicommodity flow problem with a single source and sink per commodity and uncertain demand and travel time represented by bounded convex sets. Preliminary computational results show that the robust solution is attractive, as it can reduce the worst case cost by more than 20%, while incurring on a 5 % loss in optimality when compared to the optimal solution of a representative scenario.