[1] Ben-Tal A. and Nemirovski A. (1998). Robust convex optimization. Mathematics of Operations Research 23, 769 - 805. Ben-Tal and Nemirovski address the over conservatism of robust solutions by allowing the uncertainty sets for the data to be ellipsoids, and propose efficient algorithms to solve convex optimization problems under data uncertainty.
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- [15] Bertsimas D. and Sim M. (2003). Robust discrete optimization and network flows. Mathematical Programming 98, 49-71. Additionally to the results of [14], the authors propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network.
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[16] Bertsimas D. and Sim M. (2004). The price of robustness. Operations Research 52, 35 - 53.
- [18] Bertsimas D. and Sim M. (2004). Robust discrete optimization under ellipsoidal uncertainty sets. Working Paper. Operations Research Center, MIT. It is probably the first attempt to investigate robust discrete optimization under ellipsoidal uncertainty sets. It is shown that the robust counterpart of a discrete optimization problem with correlated objective function data is NP-hard even though the original problem is polynomially solvable. For uncorrelated and identically distributed data, it is proved that the robust counterpart retains the complexity of the original problem. A generalization of the robust discrete optimization approach proposed earlier is given which presents the tradeoff between robustness and optimality.
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- [19] Bertsimas D., Pachamanova D. and Sim M. (2004). Robust linear optimization under general norms. Operations Research Letters 32, 510 - 516. The explicit characterization of the robust counterpart of a linear programming problem with uncertainty set is described by an arbitrary norm. This approach encompasses several approaches from the literature and provides guarantees for constraint violation under probabilistic models that allow arbitrary dependencies in the distribution of the uncertain coefficients.
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[21] Burkard R. and Dollani. H. (2002). A note on the robust 1-center problem on trees. Annals of Operations Research 110, 69 - 82. In this paper the authors consider different aspects of robust 1-median problems on a tree network with uncertain or dynamically changing edge lengths and vertex weights which can also take negative values. The dynamic nature of a parameter is modeled by a linear function of time. A linear algorithm is designed for the absolute dynamic robust 1-median problem on a tree.
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[31] Montemanni R. and Gambardella L. (2005). A branch and bound aIgorithm for the robust spanning tree problem with interval data. European Journal of OperationsI Research 161, 771 - 779.
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- [34] Montemanni R. (2005). A Benders decomposition approach for the robust spanning tree problem with interval data. European Journal of Operational Research (to appear).
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- [36] Montemanni R., Barta j. and Gambardella L. (2005). The robust traveling salesman problem with interval data. Technical Report IDSIA-20-05. The authors present a new extension to the basic problem, where travel times are specified as a ränge of possible values. The robust deviation criterion is applied to drive optimization over the interval data problem so obtained. Some interesting theoretical properties of the new optimization problems are identified and presented, together with a new mathematical formulation and some exact algorithms. The exact methods are compared from an experimental point of view. The methodology proposed can be used to attack the robust counterpart of other NP-hard combinatorial optimization problems too.
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- [39] Pinar M. (2004). A note on robust 0-1 optimization with uncertain cost coefficients. 40R 2, 309 - 316. Based on the approach of Bertsimas and Sim [15], [18] to robust optimization in the presence of data uncertainty, an easily computable bound on the probability that robust Solution gives and objective function value worse than the robust objective function value, under the assumption that only cost coefficients are subject to uncertainty.
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- [4] Ben-Tal A., El Ghaoui L. and Nemirovski A. (2000). Robust semidefinite programming. In Saigal R., Vandenberghe L. and Wolkowicz H. editors, Handbook of Semidefinite programming and applications, Kluwer Academic Publishers, Waterloo.
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[43] Yu G. (1996). On the max-min knapsack problem with robust optimization applications. Operations Research 44, 407 - 415.
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- [47] Ballestin F. and Leus R. (2006). Metaheuristics for stable scheduling on a Single machine. Working Paper. A Single machine scheduling problem is considered. Two metaheuristics for solving an approximate formulation of the model that assumes that exactly one job is disrupted during schedule execution are proposed. Uncertainty is modelled for job duration and primal objective is to minimize deviation between planned and actual job starting times.
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- [48] Davenport A. and Beck J. (2000). A survey of techniques for scheduling with uncertainty. Working Paper, Toronto. (http://www.eil.utoronto.ca/profiles/chris/chris.papers.html) This paper surveys some robust scheduling techniques that have been appeared during the last decade since the well-known survey by McKay et al. about the state-of-the-art in job-shop scheduling was written. Many new approaches are discussed. redundancybased techniques (a reservation of extra time and resources for unexpected events), probabilistic techniques (they do not construct a robust schedule, but we have the possibility to measure the probability of uncertainty and, moreover, to construct an optimal schedule so to maximize), different on-line and off-line approaches as well as rescheduling techniques.
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- [49] Jensen M. (2001). Robust and flexible scheduling with evolutionary computation. Ph.D. thesis, University of Aarhus, Department of Computer Science. This thesis presents two fundamentally different approaches for scheduling Job shops facing machine breakdowns. The first method is called neighbourhood based robustness and is based on an idea of minimizing the cost of a neighbourhood of schedules. The scheduling algorithm attempts to find a small set of schedules with an acceptable level of Performance. The other method for stochastic scheduling uses the idea of co-evolution to create schedules with a guaranteed worst-case Performance for a known set of scenarios. The method is demonstrated to improve worst-case Performance of the schedules when compared to ordinary scheduling; it substantially reduces running time when compared to a more Standard approach explicitly considering all scenarios.
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[5] Ben-Tal A., Nemirovski A. and Roos C. (2002). Robust solutions to uncertain quadratic and conic-quadratic problems. SIAM Journal on Optimization 13, 535 - 560.
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- [51] Lambrechts O., Demeulemeester E. and Herroelen W. (2006). Proactive and reactive strategies for resource-constrained project scheduling with uncertain resource availability. Working Paper. The goal of the paper is to build a robust schedule that meets the project due date and minimizes the schedule instability cost, defined as the expected weighted sum of the absolute deviations between the planned and actually realized activity starting times during project execution. The authors describe how stochastic resource breakdowns can be modeled, which reaction is recommended when a resource infeasibility occurs due to a breakdown and how one can protect the initial schedule from the adverse effects of Potential breakdowns.
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- [52] Leus R. (2003). The generation of stable project plans. PhD thesis. Department of applied economics, Katholieke Universiteit Leuven, Belgium.
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- [58] Sörensen K. (2001). Tabu search for robust solutions. In Proceedings of the 4th Metaheuristics International Conference, Porto, Portugal, 707 - 712. The robust tabu search firstly introduced in this paper is a new and original technique based on ideas taken from theory of robust optimization for continuous mathematical functions.
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- [59] Tsutsui S. and Ghosh A. (1997). Genetic algorithms with a robust Solution searching scheme. IEEE Transactions on Evolutionary Computation 1, 201 - 208.
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- [6] Ben-Tal A., Goryashko A., Guslitzer E. and Nemirovski A. (2004). Adjustable robust solutions of uncertain linear programs. Mathematical Programming 99, 351 -376. A number of important formulations as well as applications are introduced in [1] - [6] and some other papers of these authors. A detailed analysis of the robust optimization framework in linear programming is provided.
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[61] Van de Vonder S., Demeulemeester E., Herroelen W. and Leus R. (2005). The use of buffers in project management. The trade-off between stability and makespan. International Journal of Production Economics 97, 227 - 240. The case of stochastic activity duration is considered. The concept of buffers is introduced to protect an optimal schedule from disruption. There are also some interesting papers devoted to the concept of super solutions for constraint programming: Super solutions are a mechanism to provide robustness. They are solutions in which, if a small number of variables lose their values, one can guarantee to be able to repair the Solution with only a few changes.
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- [65] Mehta S. and Uzsoy R. (1998). Predictible scheduling of a single machine subject to breakdowns. Int. J. Computer Integrated Manufacturing 12, 15 - 38. Finally, we would like to mention the text book [66] (written and available in German only), which contains a comprehensive description of a large variety of robustness concepts.
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[66] Scholl, A. Robust scheduling and Optimization: Basics, Concepts and Methodology, Physics, Heidelberg, 2001 (in German: Robuste Planung und Optimierung: Grundlagen, Konzepte und Methoden, Experimentelle Untersuchungen. Physica-Verlag, Heidelberg). Part iV. Robustness in Economics Additionally, we give a short list of references on robustness and stability in economics, namely in portfolio optimization, supply chain management, master production scheduling, lot-sizing and monetary policy.
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- [69] Ben-Tal A., Margalit T. and Nemirovski A. (2000). Robust modelling of multi-stage portfolio problems. In K. Roos, T. Terlaky, and S. Zhang, editors, High Performance Optimization, Kluwer Academic Publisher, Dordrecht, 303 - 328.
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[72] Bertsimas D. and Thiele A. (2006). A robust optimization approach to inventory theory. Operations Research 54, 150 - 168.
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[74] El Ghaoui L., Oks M. and Oustry F. (2003). Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research 51, 543 -556.
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[77] Giannoni M. (2002). Does model uncertainty justify caution? Robust optimal monetary policy in a forward-looking model. Macroeconomic Dynamics 6, 111 - 144.
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[79] Kimms, A. (1998). Stability measures for rolling schedules with applications to capacity expansion planning, master production scheduling, and lot sizing. OMEGA 26, 355 - 366.
[8] El Ghaoui L., Oustry F. and Lebret H. (1998). Robust solutions to uncertain semidefinite programs. SIAM Journal Optimization 9, 33 - 52. El Ghaoui et al. derived in [7] - [8] results similar to [1] - [4], In particular, they deal with robust reformulation of optimization model by adapting robust control techniques under the assumption that the coefficient matrix data may vary inside ellipsoidal uncertainty set. The robust counterpart of some important problems are either exactly or approximately tractable problems that are efficiently solvable with interior point methods. However, the difficulty of the robust problems increases.
[80] Levin A. and Williams J. (2003). Robust monetary policy with competing reference models. Working Paper, Federal Reserve Bank of San Francisco.
[81] Lütgens F. and Sturm J. (2002). Robust option modelling. Technical Report, University of Maastricht.
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- [83] Schyder L. (2005). Facility location under uncertainty: a review. Technical Report, 04T-015, Department of Industrial and Systems Engineering, Lehigh University. USA This paper reviews the literature on stochastic and robust facility location models. It illustrates the rieh variety of approaches for optimization under uncertainty and their application to facility location problem.
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[84] Yu C. and Li H. (2000). A robust optimization model for stochastic logistic problems. International Journal of Production Economics 64, 385 - 397.
[85] Yu G. (1997). Robust economic order quantity models. European Journal of Operational Research 100, 482 - 493.
[9] Mulvey J., Vanderbei R. and Zenios S. (1995). Robust optimization of large-scale systems. Operations Research 43, 264 - 281. Mulvey et al. present an approach that integrates goal programming formulations with scenario-based description of the problem data. They use penalty functions to develop robust models to hedge against the worst possible scenario.
- A comparison between multicriteria and robustness frameworks, IS-MG 2003/16, Universite Libre de Bruxelles. A parallelism between multicriteria optimization and robustness concepts is established. New problems like multicriteria multiscenarios problem and multicriteria evaluation of robustness are discussed.
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- A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. The authors propose another approach which is different from previous ones in order to control the level of conservatism in the Solution. This approach has the advantage that it leads to a linear optimization model and it can be directly applied to discrete optimization problems (it was done later in [14]).
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- • Averbakh I. (2003). Complexity of robust single-facility location problems on networks with uncertain lengths of edges. Discrete Applied Mathematics 127, 505 -522.
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- • Averbakh I. (2004). Minmax regret linear resource allocation problems. Opera tions Research Letters 32, 174 - 180.
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• Averbakh I. and German O. (2000). Algorithms for the robust 1-center problem on a tree. European Journal of OperationsI Research 123, 292 - 302.
- • Averbakh I. and German O. (2000). Minmax regret median location on a network under uncertainty. INFORMS Journal on Computing 12, 104 - 110.
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- • Averbakh I. and German O. (2003). An improved algorithm for the minmax regret median problem on a tree. Networks 41, 97 - 103.
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• Averbakh I. and Lebedev V. (2005). On the complexity of minmax regret linear programming. European Journal of Operational Research 160, 227 - 231. • Lebedev V. and Averbakh I. Complexity of minimizing the total flow time with interval data and minmax regret criterion. (to appear in Discrete Applied Mathematics). • Averbakh I. The minmax relative regret median problem on networks. (to appear in INFORMS Journal on Computing). • Averbakh I. The minmax regret permutation flow shop problem with two Jobs, (to appear in European Journal of Operational Research).
- Kluwer Academic Publishers, Norwell, M.A. A comprehensive treatment of the State of the art (up to 1997) in robust discrete optimization and extensive references are presented in this work. However, there still are more open problems than solved ones. Most of the known results correspond to scenario-represented models of uncertainty, i.e. where there exists a finite number of possible scenarios each of which is given explicitly by listing the corresponding values of parameters. It is shown that most classical polynomially solvable combinatorial optimization problems loose this nice property and become NP-hard in a robust version with scenario-represented uncertainty (to appear).
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The following paper are closely related to Averbakh's approach: • Averbakh I. (2000). Minmax regret solutions for minimax optimization problems with uncertainty. Operations Research Letters 27, 57 - 65.