Abstract
Bilevel decision making and optimization problems are commonly framed as leader-follower problems, where the leader desires to optimize her own decision taking the decisions of the follower into account. These problems are known as Stackelberg problems in the domain of game theory, and as bilevel problems in the domain of mathematical programming. In a number of practical scenarios, both the leaders and the followers might be faced with multiple criteria bringing bilevel multi-criteria decision making aspects into the problem. In such cases, the Pareto-optimal frontier of the leader is influenced by the decision structure of the follower facing multiple objectives. In this paper, we analyze this effect by modeling the lower level decision maker using value functions. We study the problem using test cases and propose an algorithm that can be used to solve such problems.
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References
Bard, J.F.: Coordination of multi-divisional firm through two levels of management. Omega 11(5), 457–465 (1983)
Brotcorne, L., Labbe, M., Marcotte, P., Savard, G.: A bilevel model for toll optimization on a multicommodity transportation network. Transportation Science 35(4), 345–358 (2001)
Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002)
Deb, K., Sinha, A.: Constructing test problems for bilevel evolutionary multi-objective optimization. In: 2009 IEEE Congress on Evolutionary Computation (CEC-2009), pp. 1153–1160. IEEE Press (2009)
Deb, K., Sinha, A.: An evolutionary approach for bilevel multi-objective problems. In: Shi, Y., Wang, S., Peng, Y., Li, J., Zeng, Y. (eds.) MCDM 2009. CCIS, vol. 35, pp. 17–24. Springer, Heidelberg (2009)
Deb, K., Sinha, A.: Solving bilevel multi-objective optimization problems using evolutionary algorithms. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 110–124. Springer, Heidelberg (2009)
Deb, K., Sinha, A.: An efficient and accurate solution methodology for bilevel multi-objective programming problems using a hybrid evolutionary-local-search algorithm. Evolutionary Computation Journal 18(3), 403–449 (2010)
Eichfelder, G.: Solving nonlinear multiobjective bilevel optimization problems with coupled upper level constraints. Technical Report Preprint No. 320, Preprint-Series of the Institute of Applied Mathematics, Univ. Erlangen-Nurnberg, Germany (2007)
Eichfelder, G.: Multiobjective bilevel optimization. Math. Program. 123(2), 419–449 (2010)
Fudenberg, D., Tirole, J.: Game theory. MIT Press (1993)
Halter, W., Mostaghim, S.: Bilevel optimization of multi-component chemical systems using particle swarm optimization. In: Proceedings of World Congress on Computational Intelligence (WCCI-2006), pp. 1240–1247 (2006)
Kirjner-Neto, C., Polak, E., Der Kiureghian, A.: An outer approximations approach to reliability-based optimal design of structures. Journal of Optimization Theory and Applications 98(1), 1–16 (1998)
Linnala, M., Madetoja, E., Ruotsalainen, H., Hämäläinen, J.: Bi-level optimization for a dynamic multiobjective problem. Engineering Optimization 44(2), 195–207 (2012)
Migdalas, A.: Bilevel programming in traffic planning: Models, methods and challenge. Journal of Global Optimization 7(4), 381–405 (1995)
Pieume, C.O., Fotso, L.P., Siarry, P.: Solving bilevel programming problems with multicriteria optimization techniques. OPSEARCH 46(2), 169–183 (2009)
Pramanik, S., Dey, P.P.: Bi-level multi-objective programming problem with fuzzy parameters. International Journal of Computer Applications 30(10), 13–20 (2011), Published by Foundation of Computer Science, New York, USA
Ruuska, S., Miettinen, K.: Constructing evolutionary algorithms for bilevel multiobjective optimization. In: 2012 IEEE Congress on Evolutionary Computation (CEC), pp. 1–7 (June 2012)
Shi, X., Xia, H.S.: Model and interactive algorithm of bi-level multi-objective decision-making with multiple interconnected decision makers. Journal of Multi-Criteria Decision Analysis 10(1), 27–34 (2001)
Sinha, A.: Bilevel multi-objective optimization problem solving using progressively interactive EMO. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds.) EMO 2011. LNCS, vol. 6576, pp. 269–284. Springer, Heidelberg (2011)
Sinha, A., Deb, K.: Towards understanding evolutionary bilevel multi-objective optimization algorithm. In: IFAC Workshop on Control Applications of Optimization (IFAC-2009), vol. 7. Elsevier (2009)
Sinha, A., Malo, P., Deb, K.: Efficient evolutionary algorithm for single-objective bilevel optimization. CoRR, abs/1303.3901 (2013)
Sinha, A., Malo, P., Deb, K.: An improved bilevel evolutionary algorithm based on quadratic approximations. In: 2014 IEEE Congress on Evolutionary Computation (CEC-2014), pp. 1870–1877. IEEE Press (2014)
Sinha, A., Malo, P., Frantsev, A., Deb, K.: Multi-objective stackelberg game between a regulating authority and a mining company: A case study in environmental economics. In: 2013 IEEE Congress on Evolutionary Computation (CEC-2013). IEEE Press (2013)
Sinha, A., Malo, P., Frantsev, A., Deb, K.: Finding optimal strategies in a multi-period multi-leader-follower stackelberg game using an evolutionary algorithm. Computers & Operations Research 41, 374–385 (2014)
Sinha, A., Srinivasan, A., Deb, K.: A population-based, parent centric procedure for constrained real-parameter optimization. In: 2006 IEEE Congress on Evolutionary Computation (CEC-2006), pp. 239–245. IEEE Press (2006)
Smith, W.R., Missen, R.W.: Chemical Reaction Equilibrium Analysis: Theory and Algorithms. John Wiley & Sons, New York (1982)
Sun, H., Gao, Z., Jianjun, W.: A bi-level programming model and solution algorithm for the location of logistics distribution centers. Applied Mathematical Modelling 32(4), 610–616 (2008)
Wang, F. J., Periaux, J.: Multi-point optimization using gas and Nash/Stackelberg games for high lift multi-airfoil design in aerodynamics. In: Proceedings of the 2001 Congress on Evolutionary Computation (CEC-2001), pp. 552–559 (2001)
Yin, Y.: Genetic algorithm based approach for bilevel programming models. Journal of Transportation Engineering 126(2), 115–120 (2000)
Zhang, T., Hu, T., Zheng, Y., Guo, X.: An improved particle swarm optimization for solving bilevel multiobjective programming problem. Journal of Applied Mathematics (2012)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance Assessment of Multiobjective Optimizers: An Analysis and Review. IEEE Transactions on Evolutionary Computation 7(2), 117–132 (2003)
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Sinha, A., Malo, P., Deb, K. (2015). Towards Understanding Bilevel Multi-objective Optimization with Deterministic Lower Level Decisions. In: Gaspar-Cunha, A., Henggeler Antunes, C., Coello, C. (eds) Evolutionary Multi-Criterion Optimization. EMO 2015. Lecture Notes in Computer Science(), vol 9018. Springer, Cham. https://doi.org/10.1007/978-3-319-15934-8_29
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DOI: https://doi.org/10.1007/978-3-319-15934-8_29
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