Annals of Operations Research

, Volume 147, Issue 1, pp 317–341 | Cite as

A decision-theoretic approach to robust optimization in multivalued graphs

  • Patrice Perny
  • Olivier Spanjaard
  • Louis-Xavier Storme


This paper is devoted to the search of robust solutions in finite graphs when costs depend on scenarios. We first point out similarities between robust optimization and multiobjective optimization. Then, we present axiomatic requirements for preference compatibility with the intuitive idea of robustness in a multiple scenarios decision context. This leads us to propose the Lorenz dominance rule as a basis for robustness analysis. Then, after presenting complexity results about the determination of Lorenz optima, we show how the search can be embedded in algorithms designed to enumerate k best solutions. Then, we apply it in order to enumerate Lorenz optimal spanning trees and paths. We investigate possible refinements of Lorenz dominance and we propose an axiomatic justification of OWA operators in this context. Finally, the results of numerical experiments on randomly generated graphs are provided. They show the numerical efficiency of the suggested approach.


Robust optimization Multicriteria optimization Lorenz optima k best solutions Minimum spanning tree Shortest path 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bellman, R. (1954). “Some Applications of the Theory of Dynamic Programming—A Review.” Journal of the Operational Research Society of America, 2(3), 275–288.Google Scholar
  2. Brucker, P. and H. Hamacher. (1989). “k-Optimal Solution sets for Some Polynomially Solvable Scheduling Problems.” European Journal of Operational Research, 41, 194–202.CrossRefGoogle Scholar
  3. Cayley, A. (1889). “A Theorem on Trees.” Quaterly Journal of Mathematics, 23, 376–378.Google Scholar
  4. Chegireddy, C. and H. Hamacher. (1987). “Algorithms for Finding k-best Perfect Matchings.” Discrete Applied Mathematics, 18, 155–165.CrossRefGoogle Scholar
  5. Chong, K.M. (1976). “An Induction Theorem for Rearrangements.” Candadian Journal of Mathematics, 28, 154–160.Google Scholar
  6. Climaco, J. and E. Martins. (1982). “A Bicriterion Shortest Path Algorithm.” European Journal of Operational Research, 11, 399–404.CrossRefGoogle Scholar
  7. D. Dubois and Ph. Fortemps (2005) “Selecting Preferred Solutions in the Minimax Approach to Dynamic Programming Problems Under Flexible Constraints” European Journal of Operational Research, 160(3), 582–598.CrossRefGoogle Scholar
  8. Ehrgott, M. and A. Skriver. (2003). “Solving Biobjective Combinatorial Max-Ordering Problems by Ranking Methods and a Two-Phase Approach.” European Journal of Operational Research, 147, 657–664.CrossRefGoogle Scholar
  9. Emelichev, V. and V. Perepelitsa. (1988). “Multiobjective Problems on the Spanning Trees of a Graph.” Soviet Math. Dokl., 37(1), 114–117.Google Scholar
  10. Eppstein, D. (1998). “Finding the k Shortest Paths.” SIAM Journal on Computing, 28(2), 652–673.CrossRefGoogle Scholar
  11. Fishburn, P. (1970). Utility Theory for Decision Making. Wiley.Google Scholar
  12. Gabow, H. (1977). “Two Algorithms for Generating Weighted Spanning Trees in Order.” SIAM Journal on Computing, 6(1), 139–150.CrossRefGoogle Scholar
  13. Garey, M. and D. Johnson. (1979). Computers and Intractability. W.H. Freeman and company.Google Scholar
  14. Gupta, S. and J. Rosenhead. (1968). “Robustness in Sequential Investment Decisions.” Management Science, 15(2), 18–29.Google Scholar
  15. Hamacher, H. (1995). “A Note on k-best Network Flows.” Annals of Operations Research, 57, 65–72.CrossRefGoogle Scholar
  16. Hamacher, H. and G. Ruhe. (1994). “On Spanning Tree Problems with Multiple Objectives.” Annals of Operations Research, 52, 209–230.CrossRefGoogle Scholar
  17. Hansen, P. (1980). “Bicriterion Path Problems.” In G. Fandel and T. Gal (Eds.), Multicriteria Decision Making.Google Scholar
  18. Hardy, G.H., J.E. Littlewood, and G. Pólya. (1934). Inequalities. Cambridge University Press.Google Scholar
  19. Herstein, I. and J. Milnor. (1953). “An Axiomatic Approach to Measurable Utility.” Econometrica, 21, 291–297.CrossRefGoogle Scholar
  20. R. Hites, Y. De Smet, N. Risse, M. Salazar-Neumann and P. Vincke (2006) “About the Applicability of MCDA to Some Robustness Problems” European Journal of Operational Research, 174(1), 322–332.CrossRefGoogle Scholar
  21. Jensen, N. (1967). “An Introduction to Bernoullian Utility Theory.” Swedish Journal of Economics, 69, 163–183.CrossRefGoogle Scholar
  22. O. Karasan, M. Pinar and H. Yaman (2001) “The Robust Shortest Path Problem with Interval Data” Bilkent University, Ankara, Turkey.Google Scholar
  23. Kostreva, M. and W. Ogryczak. (1999). “Linear Optimization with Multiple Equitable Criteria.” RAIRO Operations Research, 33, 275–297.CrossRefGoogle Scholar
  24. Kostreva, M., W. Ogryczak, and A. Wierzbicki. (2004). “Equitable Aggregations and Multiple Criteria Analysis.” European Journal of Operational Research, 158(2), 362–377.CrossRefGoogle Scholar
  25. Kouvelis, P. and G. Yu. (1997). Robust Discrete Optimization and its Applications. Kluwer Academic Publisher.Google Scholar
  26. Kruskal, J. (1956). “On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem.” In Proc. Am. Math. Soc.Google Scholar
  27. Marshall, W. and I. Olkin. (1979). Inequalities: Theory of Majorization and its Applications. London: Academic Press.Google Scholar
  28. R. Montemanni and L. M. Gambardella (2004) “An Exact Algorithm for the Robust Shortest Path Problem with Interval Data” Computers and Operations Research, 31(10) 1667–1680.CrossRefGoogle Scholar
  29. Murthy, I. and S. Her. (1992). “Solving Min-Max Shortest-Path Problems on a Network.” Naval Research Logistics, 39, 669–683.CrossRefGoogle Scholar
  30. Ogryczak, W. (2000). “Inequality Measures and Equitable Approaches to Location Problems.” European Journal of Operational Research, 122, 374–391.CrossRefGoogle Scholar
  31. Perny, P. and O. Spanjaard. (2003). “An Axiomatic Approach to Robustness in Search Problems with Multiple Scenarios.” In Proceedings of the 19th Conference on Uncertainty in Artificial Intelligence. pp. 469–476, Acapulco, Mexico.Google Scholar
  32. Rosenhead, J., M. Elton, and S. Gupta. (1972). “Robustness and Optimality as Criteria for Strategic Decisions.” Operational Research Quaterly, 23(4), 413–430.CrossRefGoogle Scholar
  33. Roy, B. (1996). Multicriteria Methodology for Decision Aiding. Kluwer Academic Publisher.Google Scholar
  34. Roy, B. (1998). “A Missing Link in OR-DA: Robustness Analysis.” Foundations of Computing and Decision Sciences, 23(3), 141–160.Google Scholar
  35. Roy, B. (2002). “Robustesse de quoi et vis-à-vis de quoi mais aussi robustesse pourquoi en aide à la décision.” Newsletter of the European Working Group Multicriteria Aid for Decisions, 3(6).Google Scholar
  36. Sayin, S. and P. Kouvelis. (2002). “Robustness and Efficiency: A Study of the Relationship and an Algorithm for the Bicriteria Discrete Optimization Problem.” Working Paper, Olin School of Business, Washington University.Google Scholar
  37. Sen, A. (1997). On Economic Inequality. Clarendon Press, expanded ed.Google Scholar
  38. Shorrocks, A. (1983). “Ranking Income Distributions.” Economica, 50, 3–17.CrossRefGoogle Scholar
  39. Stewart, B.S. and C.C. White III. (1991). “Multiobjective A*.” Journal of the Association for Computing Machinery, 38(4), 775–814.Google Scholar
  40. Vincke, P. (1999a). “Robust and Neutral Methods for Aggregating Preferences into an Outranking Relation.” European Journal of Operational Research, 112(2), 405–412.CrossRefGoogle Scholar
  41. Vincke, P. (1999b). “Robust Solutions and Methods in Decision-Aid.” Journal of Multicriteria Decision Analysis, 8, 181–187.CrossRefGoogle Scholar
  42. von Neumann, J. and O. Morgenstern. (1947). Theory of Games and Economic Behavior. 2nd Ed. Princeton University Press.Google Scholar
  43. Warburton, A. (1985). “Worst Case Analysis of Greedy and Related Heuristics for Some Min-Max Combinatorial Optimization Problems.” Mathematical Programming, 33, 234–241.CrossRefGoogle Scholar
  44. Weymark, J. (1981). “Generalized Gini Inequality Indices.” Mathematical Social Sciences, 1, 409–430.CrossRefGoogle Scholar
  45. Yaari, M. (1987). “The Dual Theory of Choice Under Risk.” Econometrica, 55, 95–115.CrossRefGoogle Scholar
  46. Yager, R. (1988). “On Ordered Weighted Averaging Aggregation Operators in Multicriteria Decision Making.” In IEEE Trans. Systems, Man and Cybern. vol. 18, pp. 183–190.Google Scholar
  47. Yaman, H., O. Karaşan, and M. Pinar. (2001). “The Robust Spanning Tree Problem with Interval Data.” Operations Research Letters, 29, 31–40.CrossRefGoogle Scholar
  48. Yu, G. and J. Yang. (1998). “On the Robust Shortest Path Problem.” Computers and Operations Research, 25(6), 457–468.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LCC 2006

Authors and Affiliations

  • Patrice Perny
    • 1
  • Olivier Spanjaard
    • 1
  • Louis-Xavier Storme
    • 1
  1. 1.LIP6 - University of Paris VIParisFrance

Personalised recommendations