Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-24T15:00:49.042Z Has data issue: false hasContentIssue false
  • English
  • Français

Minmax regret combinatorial optimization problems: an AlgorithmicPerspective

Published online by Cambridge University Press:  24 August 2011

Alfredo Candia-Véjar ,
Eduardo Álvarez-Miranda  and
Nelson Maculan
Alfredo Candia-Véjar
Affiliation:
Modeling and Industrial Management Department, Universidad de Talca, Curicó, Chile. acandia@utalca.cl
Eduardo Álvarez-Miranda
Affiliation:
Modeling and Industrial Management Department, Universidad de Talca, Curicó, Chile. acandia@utalca.cl Operations Management Master Program, Universidad de Talca, Curicó, Chile.
Nelson Maculan
Affiliation:
COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil.
Get access

Abstract

Uncertainty in optimization is not a new ingredient. Diverse models considering uncertainty have been developed over the last 40 years. In our paper we essentially discuss a particular uncertainty model associated with combinatorial optimization problems, developed in the 90's and broadly studied in the past years. This approach named minmax regret (in particular our emphasis is on the robust deviation criteria) is different from the classical approach for handling uncertainty, stochastic approach, where uncertainty is modeled by assumed probability distributions over the space of all possible scenarios and the objective is to find a solution with good probabilistic performance. In the minmax regret (MMR) approach, the set of all possible scenarios is described deterministically, and the search is for a solution that performs reasonably well for all scenarios, i.e., that has the best worst-case performance. In this paper we discuss the computational complexity of some classic combinatorial optimization problems using MMR approach, analyze the design of several algorithms for these problems, suggest the study of some specific research problems in this attractive area, and also discuss some applications using this model.

Keywords

Minmax regret modelcombinatorial optimizationexact algorithms and heuristicsrobust optimization.
Type
Research Article
Information
RAIRO - Operations Research , Volume 45 , Issue 2 , April 2011 , pp. 101 - 129
Copyright
© EDP Sciences, ROADEF, SMAI, 2011

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

Aissi, H., Bazgan, C. and Vanderpooten, D., Complexity of the minmax and minmax regret assignment problem. Oper. Res. Lett. 33 (2005) 634–640. CrossRef
Aissi, H., Bazgan, C. and Vanderpooten, D., Approximation of min-max and min-max regret versions of some combinatorial optimization problems. Eur. J. Oper. Res. 179 (2007) 281–290. CrossRef
Aissi, H., Bazgan, C. and Vanderpooten, D., Approximation complexity of min-max (regret) versions of shortest path, spanning tree, and knapsack. Lect. Notes Comput. Sci. 3669 (2005) 862–873. CrossRef
Aissi, H., Bazgan, C. and Vanderpooten, D., Min-max and min-max regret versions of some combinatorial optimization problems: a survey. Eur. J. Oper. Res. 197 (2009) 427–438. CrossRef
M.A. Aloulou, R. Kalai and D. Vanderpooten, Minmax regret 1-center problem on a network with a discrete set of scenarios. Lamsade technical Report No. 132, LAMSADE, Université Paris-Dauphine, Cahier du LAMSADE (2006).
E. Alvarez-Miranda and A. Candia-Vejar, Robust Shortest Path: Models, Algorithms and Comparisons, Proceedings of the VI ALIO/EURO Workshop on Applied Combinatorial Optimization. Buenos Aires, Argentina (2008).
I. Aron and P. Van Hentenryck, A constraint satisfaction approach to the robust spanning tree with interval data, Proceedings of the International Conference on Uncertainty in Artificial Intelligence UAI (2002) 18–25.
I. Aron and P. Van Hentenryck, On the complexity of the robust spanning tree problem with interval data, Oper. Res. Lett. 32 (2004) 36–40.
Assavapokee, T., Realff, M.J. and Ammons, J.C., Min-max Regret robust optimization approach on interval data uncertainty. J. Optim. Theory Appl. 137 (2008) 297–316. CrossRef
Averbakh, I., On the complexity of a class of combinatorial optimization problems with uncertainty. Math. Program. Ser. A 90 (2001) 263–272. CrossRef
Averbakh, I., Complexity of robust single facility location problems on networks with uncertain edge lengths. Discr. App. Math. 127 (2003) 505–522. CrossRef
Averbakh, I., Minmax regret linear resource allocation problems. Oper. Res. Lett. 32 (2004) 174–180. CrossRef
Averbakh, I., Computing and minimizing the relative regret in combinatorial optimization with interval data. Discr. Optim. 2 (2005) 273–287. CrossRef
Averbakh, I. and Bereg, S., Facility location problems with uncertainty on the plane. Discret. Optim. 2 (2005) 3–34. CrossRef
Averbakh, I. and Berman, O., Minmax regret median location on a network under uncertainty. ORSA J. Comput. 12 (2000) 104–110. CrossRef
Averbakh, I. and Berman, O., Algorithms for the robust 1-center problem on a tree. Eur. J. Oper. Res. 123 (2000) 292–302. CrossRef
Averbakh, I. and Lebedev, V., Interval data regret network optimization problems. Discr. App. Math. 138 (2004) 289–301. CrossRef
Averbakh, I. and Lebedev, V., On the complexity of minmax regret linear programming. Eur. J. Oper. Res. 160 (2005) 227–231. CrossRef
Beasley, J.E., OR-Library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41 (1990) 1069–1072. CrossRef
Bertsimas, D. and Sim, M., Robust discrete optimization and network flows. Math. Program. Ser. B 98 (2003) 49–71. CrossRef
Bertsimas, D. and Sim, M., The price of robustness. Oper. Res. 52 (2004) 35–53. CrossRef
L. Bianchi, M. Dorigo, L. Gambardella and W. Gutjahr, Metaheuristics in Stochastic Combinatorial Optimization: a Survey. IDSIA Technical Report, IDSIA-08-06 (2006), Natural Computing 8 (2009) 239–287.
Burkard, R.E. and Dollani, H., A note on the robust 1-center problem on trees. Disc. Appl. Math. 138 (2004) 289–301.
A. Candia-Véjar and E. Álvarez-Miranda, On a class of interval data minmax regret CO problems. 2007 (2007) 123–128.
Chang, N. and Davila, E., Siting and routing assessment for solid waste management under uncertainty using the grey min-max regret criterion. Environ. Manag. 38 (2006) 654–672. CrossRef
Chang, N. and Davila, E., Minimax regret optimization analysis for a regional solid waste management system. Waste Manag. 27 (2007) 820–832. CrossRef
X. Chen, J. Hu and X. Hu, On the minimum risk-sum path problem, ESCAPE'07 Proceedings, Lect. Notes Comput. Sci. 4614 (2007) 175–185.
X. Chen, J. Hu and X. Hu, The minimum risk spanning tree problem, COCOA'07 Proceedings, Lecture Notes in Computer Science 4616 (2007) 81–90.
Chen, X., Hu, J. and Hu, X., A new model for path planning with interval data. Comput. Oper. Res. 39 (2009) 1893–1899. CrossRef
Conde, E., An improved algorithm for selecting p items with uncertain return according to the minmax-regret criterion. Math. Program. Ser. A 100 (2001) 345–353. CrossRef
Conde, E., On the complexity of the continuous unbounded knapsack problem with uncertain coefficients. Oper. Res. Lett. 33 (2005) 481–485. CrossRef
Conde, E., Minmax regret location-allocation problem on a network under uncertainty. Eur. J. Oper. Res. 179 (2007) 1025–1039. CrossRef
Conde, E., A branch and Bound algorithm for minimum spanning arborescences. J. Glob. Optim. 37 (2007) 467–480. CrossRef
Conde, E., A note on the minmax regret centdian location on trees. Oper. Res. Lett. 36 (2008) 271–275. CrossRef
Conde, E., A 2-approximation for minmax regret problems via a mid-point scenario optimal solution. Oper. Res. Lett. 38 (2010) 326–327. CrossRef
Conde, E. and Candia, A., Minimax regret spanning arborescences under uncertain costs. Eur. J. Oper. Res. 182 (2007) 561–577. CrossRef
Dantzig, G.B., Fulkerson, D.R. and Johnson, S.M., Solutions of a large scale traveling salesman problem. Oper. Res. 2 (1954) 393–410.
Deineko, V. and Woeginger, G., On the robust assigment problem under fixed number of cost scenarios. Oper. Res. Lett. 34 (2006) 175–179. CrossRef
Demir, M., Tansel, B. and Scheuenstuhl, G., Tree Network 1-median location with interval data: a parameter space-based approach. IIE Trans. 37 (2005) 429–439. CrossRef
Hites, R., De Smeta, Y., Risse, N., Salazar-Neumann, N. and Vincke, P., About the applicability of MCDA to some robustness problems. Eur. J. Oper. Res. 174 (2006) 322–332. CrossRef
O. Karasan, M. Pinar and H. Yaman, The Robust Shortest Path Problem with Interval Data. Technical Report Bilkent University (2001), revised (2004).
A. Kasperski, Discrete Optimization with Interval Data: Minmax Regret and Fuzzy Approach. Studies in Fuzzines and Soft Computing, Springer (2008).
Kasperski, A. and Zieliński, P., An approximation algorithm for interval data minmax regret combinatorial optimization problems. Inf. Process. Lett. 97 (2006) 177–180. CrossRef
Kasperski, A. and Zieliński, P., The robust shortest path problem in series-parallel multidigraphs with interval data. Oper. Res. Lett. 34 (2006) 69–76. CrossRef
Kasperski, A. and Zieliński, P., On combinatorial optimization problems on matroids with uncertain weights. Eur. J. Oper. Res. 177 (2007) 851–864. CrossRef
Kazakci, A., Rozakis, S. and Vanderpooten, D., Energy crop supply in France: a min-max regret approach. J. Oper. Res. Soc. 58 (2007) 1470–1479. CrossRef
P. Kouvelis and G. Yu, Robust discrete optimization and its applications. Kluwer Academic Publishers (1997).
Loulou, R. and Kanudia, A., Minimax regret strategies for greenhouse gas abatement: methodology and application. Oper. Res. Lett. 25 (1999) 219–230. CrossRef
T.L. Magnanti and L. Wolsey, optimal trees, network models, in Handbook in Operations research and management science 7, edited by M.O. Ball et al., North-Holland, Amsterdam (1997) 503–615.
Mausser, H.E. and Laguna, M., A new mixed integer formulation for the maximum regret problem. Int. Trans. Oper. Res. 5 (1998) 389–403. CrossRef
Mausser, H.E. and Laguna, M., Minimising the maximum relative regret for linear programmes with interval objective function coefficents. J. Oper. Res. Soc. 50 (1999) 1063–1070. CrossRef
Mausser, H.E. and Laguna, M., A heuristic to minmimax absolute regret for linear programs with interval objective function. Eur. J. Oper. Res. 117 (1999) 157–174. CrossRef
Montemanni, R., Benders de, Acomposition approach for the robust spanning tree problem with interval data. Eur. J. Oper. Res. 174 (2006) 1479–1490. CrossRef
Montemanni, R., Gambardella, L.M. and Donati, A.V., A branch and bound algorithm for the robust shortest path with interval data. Oper. Res. Lett. 32 (2004) 225–232. CrossRef
Montemanni, R. and Gambardella, L.M., An exact algorithm for the robust shortest path problem with interval data. Comput. Oper. Res. 31 (2004) 1667–1680. CrossRef
Montemanni, R. and Gambardella, L.M., A branch and bound algorithm for the robust spanning tree with interval data. Eur. J. Oper. Res. 161 (2005) 771–779. CrossRef
R. Montemanni and L.M. Gambardella, The robust shortest path problem with interval data via Benders decomposition, 40R 3 (2005) 315–328.
R. Montemanni, J. Barta and L.M. Gambardella, Heuristic and preprocessing techniques for the robust traveling salesman problem with interval data. Technical Report IDSIA-01-06 (2006).
Montemanni, R., Barta, J. and Gambardella, L.M., The robust traveling salesman problem with interval data. Transp. Sci. 41 (2007) 366–381. CrossRef
Y. Nikulin, Robustness in combinatorial optimization and scheduling theory: An annotated bibliography. Christian-Albrechts University in Kiel, Working paper (2005).
Nikulin, Y., Simulated annealing algorithm for the robust spanning tree problem. J. Heurist. 14 (2008) 391–402. CrossRef
Y. Nikulin, Solving the robust shortest path problem with interval data using probabilistic metaheuristic approach. N597 CAU (2005) (with A. Drexl).
Pereira, J. and Averbakh, I., Exact and heuristic algorithms for the interval data robust assignment problem. Comput. Oper. Res. 38 (2011) 1153–1163 CrossRef
Salazar-Neumann, M., The robust minimum spanning tree problem: Compact and convex uncertainty. Oper. Res. Lett. 35 (2007) 17–22. CrossRef
Snyder, L.V., Facility location under uncertainty: A review. IIE Trans. 38 (2006) 547–564. CrossRef
Yaman, H., Karasan, O. and Pinar, M., The robust spanning tree with interval data. Oper. Res. Lett. 29 (2001) 31–40. CrossRef
Zieliński, P., The computational complexity of the relative robust shortest path problem with interval data. Eur. J. Oper. Res. 158 (2004) 570–576. CrossRef