Abstract
We develop an interactive algorithm for biobjective integer programs that finds the most preferred solution of a decision maker whose preferences are consistent with a quasiconvex preference function to be minimized. During the algorithm, preference information is elicited from the decision maker. Based on this preference information and the properties of the underlying quasiconvex preference function, the algorithm reduces the search region and converges to the most preferred solution progressively. Finding the most preferred solution requires searching both supported and unsupported nondominated points, where the latter is harder. We develop theory to further restrict the region where unsupported nondominated points may lie. We demonstrate the algorithm on the generalized biobjective traveling salesperson problem where there are multiple efficient edges between node pairs and show its performance on a number of randomly generated instances.
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Bérubé, J. F., Gendreau, M., & Potvin, J. Y. (2009). An exact \(\upvarepsilon \)-constraint method for Bi-objective combinatorial optimization problems—Application to the traveling salesman problem with profits. European Journal of Operational Research, 194(1), 39–50.
Chankong, V., & Haimes, Y. Y. (1983). Multiobjective decision making: Theory and methodology. New York: North-Holland.
Ehrgott, M. (2005). Multicriteria optimization (Vol. 2). Berlin: Springer.
Foo, J. L., Knutzon, J., Kalivarapu, V., Olver, J., & Winer, E. (2009). Path planning of unmanned aerial vehicles using B-splines and particle swarm optimization. Journal of Aerospace Computing, Information and Communication, 6(4), 271–290.
Hansen, M. P. (2000). Use of substitute scalarizing functions to guide a local search based heuristic: The case of moTSP. Journal of Heuristics, 6, 419–431.
Jaszkiewicz, A., & Zielniewicz, P. (2009). Pareto memetic algorithm with path relinking for bi-objective traveling salesperson problem. European Journal of Operational Research, 193(3), 885–890.
Jozefowiez, N., Glover, F., & Laguna, M. (2008). Multi-objective meta-heuristics for the traveling salesman problem with profits. Journal of Mathematical Modelling and Algorithms, 7(2), 177–195.
Karademir, S. (2008). A genetic algorithm for the biobjective traveling salesman problem with profits. M.S. thesis, Department of Industrial Engineering, Middle East Technical University.
Korhonen, P., Wallenius, J., & Zionts, S. (1984). Solving the discrete multiple criteria problem using convex cones. Management Science, 30(1), 1336–1345.
Köksalan, M. M., & Sagala, P. N. S. (1995). Interactive approaches for discrete alternative multiple criteria decision making with monotone utility functions. Management Science, 41, 1158–1171.
Laporte, G. (1992). The traveling salesman problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59(2), 231–247.
Lokman, B., Köksalan, M., Korhonen, P. J., & Wallenius, J. (2014). An interactive algorithm to find the most preferred solution of multi-objective integer programs. Annals of operations research. doi:10.1007/s10479-014-1545-2
Lust, T., & Teghem, J. (2010). The multiobjective traveling salesman problem: A survey and a new approach. Advances in Multi-Objective Nature Inspired Computing Studies in Computational Intelligence, 272, 119–141.
Miller, C. E., Tucker, A. W., & Zemlin, R. A. (1960). Integer programming formulation of traveling salesman problems. Journal of the Association for Computing Machinery, 7(4), 326–329.
Müller-Hannemann, M., & Weihe, K. (2006). On the cardinality of the Pareto set in bicriteria shortest path problems. Annals of Operations Research, 147, 269–286.
Özpeynirci, Ö., & Köksalan, M. (2009). Multiobjective traveling salesperson problem on halin graphs. European Journal of Operational Research, 196, 155–161.
Özpeynirci, Ö., & Köksalan, M. (2010). Pyramidal tours and multiple objectives. Journal of Global Optimization, 48(4), 569–582.
Paquete, L., & Stützle, T. (2003). A two-phase local search for the biobjective traveling salesman problem. Proceedings of the 2nd international conference on evolutionary multi-criterion optimization (EMO 2003), LNCS 2632, (pp. 479–493). New York: Springer.
Przybylski, A., Gandibleux, X., & Ehrgott, M. (2008). Two phase algorithms for the Bi-objective assignment problem. European Journal of Operations Research, 185, 509–533.
Ramesh, R., Karwan, M. H., & Zionts, S. (1990). An interactive method for bicriteria integer programming. IEEE Transactions on Systems, Man, and Cybernetics, 20, 395–403.
Steuer, R. E. (1986). Multiple criteria optimization: Theory, computation, and application. New York: Wiley.
Tezcaner, D., & Köksalan, M. (2011). An interactive algorithm for multi-objective route planning. Journal of Optimization Theory and Applications, 150(2), 379–394.
Tuyttens, D., Teghem, J., Fortemps, Ph, & Van Nieuwenhuyse, K. (2000). Performance of the MOSA method for the bicriteria assignment problem. Journal of Heuristics, 6, 295–310.
Waldock, A., & Corne, D. W. (2012). Exploiting prior information in multi-objective route planning. Parallel Problem Solving from Nature XII Lecture Notes in Computer Science, 7492, 11–21.
Wu, P. P., Campbell, D., & Merz, T. (2009). On-board multi-objective mission planning for unmanned aerial vehicles. Aerospace conference IEEE.
Zheng, C., Ding, M., & Zhou, C. (2003). Real-time route planning for unmanned air vehicle with an evolutionary algorithm. International Journal of Pattern Recognition and Artificial Intelligence, 17(1), 68–81.
Zionts, S. (1981). A multiple criteria method for choosing among discrete alternatives. European Journal of Operational Research, 7(1), 143–147.
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Tezcaner Öztürk, D., Köksalan, M. An interactive approach for biobjective integer programs under quasiconvex preference functions. Ann Oper Res 244, 677–696 (2016). https://doi.org/10.1007/s10479-016-2149-9
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DOI: https://doi.org/10.1007/s10479-016-2149-9