Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T04:41:30.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Lexicographic α-robustness: an application to the 1-median problem

Published online by Cambridge University Press:  27 April 2010

R. Kalaï ,
M. A. Aloulou ,
Ph. Vallin  and
D. Vanderpooten
R. Kalaï
Affiliation:
Rouen Business School, 1 rue du Maréchal Juin, BP 215 – 76825 Mont Saint Aignan Cedex, France; rkj@rouenbs.fr
M. A. Aloulou
Affiliation:
LAMSADE, Université Paris-Dauphine, Pl. du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France.
Ph. Vallin
Affiliation:
LAMSADE, Université Paris-Dauphine, Pl. du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France.
D. Vanderpooten
Affiliation:
LAMSADE, Université Paris-Dauphine, Pl. du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France.
Get access

Abstract

In the last decade, several robustness approaches have beendeveloped to deal with uncertainty. In decision problems, andparticularly in location problems, the most used robustnessapproach rely either on maximal cost or on maximal regretcriteria. However, it is well known that these criteria are tooconservative. In this paper, we present a new robustness approach,called lexicographic α-robustness, which compensatesfor the drawbacks of criteria based on the worst case. We applythis approach to the 1-median location problem under uncertaintyon node weights and we give a specific algorithm to determinerobust solutions in the case of a tree. We also show that thisalgorithm can be extended to the case of a general network.

Keywords

Robustness1-median location problemminmax cost/regret
Type
Research Article
Information
RAIRO - Operations Research , Volume 44 , Issue 2 , April 2010 , pp. 119 - 138
Copyright
© EDP Sciences, ROADEF, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Averbakh, I., Complexity of robust single facility location problems on networks with uncertain edge length. Discrete Appl. Math. 127 (2003) 505–522. CrossRef
Averbakh, I. and Berman, O., Minmax regret median location on a network under uncertainty. Informs J. Comput. 12 (2000) 104–110. CrossRef
Averbakh, I. and Berman, O., An improved algorithm for the minmax regret median problem on a tree. Networks 41 (2003) 97–103. CrossRef
D. Bertsimas and M. Sim, Robust discrete optimization and network flows. Math. Program., Ser. B 98 (2003) 49–71.
Brodal, G.S., Georgiadis, L. and Katriel, I., An O(n log n) version of the Averbakh-Berman algorithm for the robust median on a tree. Oper. Res. Lett. 36 (2008) 14–18. CrossRef
Burkard, R.E. and Dollani, H., Robust location problems with pos/neg weights on a tree. Networks 38 (2001) 102–113. CrossRef
Chen, B. and Lin, C.S., Min-max regret robust 1-median location on a tree. Networks 31 (1998) 93–103. 3.0.CO;2-E>CrossRefPubMed
M.S. Daskin, Network and Discrete Location: Models, Algorithms and Applications. Wiley (1995).
Daskin, M.S., Hesse, S.M. and Revelle, C.S., α-reliable p-minimax regret: a new model for strategic facility location modeling. Location Science 5 (1997) 227–246. CrossRef
Edelsbrunner, H. and Guibas, L.J., Topologically sweeping an arrangement. J. Comput. Syst. Sci. 38 (1989) 165–194. Corrigendum in 42 (1991) 249–251. CrossRef
Fishburn Le, P.C.xicographic orders, utilities and decision rules: a survey. Manage. Sci. 20 (1974) 1442–1471. CrossRef
M. Grabisch and P. Perny Agrégation multicritère, in Logique Floue, principes, aide à la décision, edited by B. Bouchon-Meunier and C. Marsala. Hermès-Lavoisier (2003) 81–120.
Hakimi, S.L., Optimum locations of switching centers and the absolute centers and medians of a graph. Oper. Res. 12 (1964) 450–459. CrossRef
Harries, C., Correspondence to what? coherence to what? what is good scenario-based decision making? Technol. Forecast. Soc. Change 70 (2003) 797–817. CrossRef
R. Kalaï, Une nouvelle approche de robustesse: application à quelques problèmes d'optimisation. Ph.D. thesis, Université Paris-Dauphine, France (2006).
R. Kalaï and D. Vanderpooten Lexicographic α-robust knapsack problem: Complexity results, in International Conference on Service Systems & Service Management (IEEE SSSM'06), Troyes, France, October (2006).
Kouvelis, P., Kurawarwala, A.A. and Gutiérrez Al, G.J.gorithms for robust single and multiple period layout planning for manufacturing systems. Eur. J. Oper. Res. 63 (1992) 287–303. CrossRef
P. Kouvelis, G. Vairaktarakis and G. Yu, Robust 1-median location on a tree in the presence of demand and transportation cost uncertainty. Working paper 93/94-3-4, Department of Management Science and Information Systems, University of Texas at Austin (1994).
P. Kouvelis and G. Yu, Robust Discrete Optimization and its Applications. Kluwer Academic Publishers (1997).
H. Moulin, Social welfare orderings, in Axioms of cooperative decision making, Cambridge University Press (1988) pp. 30–60.
Mulvey, J.M., Vanderbei, R.J. and Zenios Robust, S.A. optimization of large-scale systems. Oper. Res. 43 (1995) 264–281. CrossRef
Ogryczak, W., On the lexicographic minimax approach to location problems. Eur. J. Oper. Res. 100 (1997) 566–585. CrossRef
Ogryczak, W., Multiple criteria optimization and decisions under risk. Control and Cybernetics 31 (2002) 975–1003.
Perny, P., Spanjaard, O. and Storme, L.X., A decision-theoretic approach to robust optimization in multivalued graphs. Ann. Oper. Res. 147 (2006) 317–341. CrossRef
Pomerol, J.-C., Scenario development and practical decision making under uncertainty. Decis. Support Syst. 31 (2001) 197–204. CrossRef
Puerto, J., Rodriguez-Chia, A.M. and Tamir, A., Minimax regret single-facility ordered median location problems on networks. Informs J. Comput. 21 (2009) 77–87. CrossRef
E. Rafalin, D. Souvaine and I. Streinu Topological sweep in degenerate cases, in Proc. of the 4th international workshop on Algorithm Engineering and Experiments, ALENEX02, in Lect. Notes Comput. Sci. 2409, Springer-Verlag (2002) 155–156.
Roy, B., A missing link in OR-DA: robustness analysis. Found. Comput. Decis. Sci. 23 (1998) 141–160.
Ph, B. Roy. Vincke, Relational systems of preference with one or more pseudo-criteria: some new concepts and results. Manage. Sci. 30 (1984) 1323–1335.
J.-R. Sack and J. Urrutia (Eds.), Handbook of Computational Geometry. Elsevier Sciences (2000).
Schoemaker, P.J.H., Multiple scenario development: Its conceptual and behavioral foundation. Strateg. Manage. J. 14 (1993) 193–213. CrossRef
Snyder, L.V. and Daskin, M.S., Stochastic p-robust location problems. IIE Trans. 38 (2006) 971–985. CrossRef
Vairaktarakis, G.L. and Kouvelis, P., Incorporating dynamic aspects and uncertainty in 1-median location problems. Nav. Res. Logist. 46 (1999) 147–168. 3.0.CO;2-4>CrossRef
Vincke, P., Robust solutions and methods in decision-aid. J. Multi-Crit. Decis. Anal. 8 (1999) 181–187. 3.0.CO;2-P>CrossRef
H. Yaman, O.E. Karasan and M.C. Pinar, Restricted robust optimization for maximization over uniform matroid with interval data uncertainty. Technical report, Bilkent University, Department of Industrial Engineering, Bilkent, Turkey (2005).