Abstract
We study the lexicographic centre of multiple objective optimization. Analysing the lexicographic-order properties yields the result that, if the multiple objective programming’s lexicographic centre is not empty, then it is a subset of all efficient solutions. It exists if the image set of multiple objective programming is bounded below and closed. The multiple objective linear programming’s lexicographic centre is nonempty if and only if there exists an efficient solution to the multiple objective linear programming. We propose a polynomial-time algorithm to determine whether there is an efficient solution to multiple objective linear programming, and we solve the multiple objective linear programming’s lexicographic centre by calculating at most the same number of dual linear programs as the number of objective functions and a system of linear inequalities.
Similar content being viewed by others
References
Ogryczak, W.: On the lexicographic minimax approach to location problems. Eur. J. Oper. Res. 100, 566–585 (1997)
Luss, H.: On equitable resource allocation problems: a lexicographic minimax approach. Oper. Res. 47(3), 182–187 (1999)
Shmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17, 1163–1170 (1969)
Maschler, M., Peleg, B., Shapley, L.S.: Geometric properties of the kernel, nucleolus and related solution concepts. Math. Oper. Res. 4, 303–338 (1979)
Dragan, I.: A procedure for finding the nucleolus of a cooperative n person game. Zeitschrift für Oper. Res. 25(5), 119–131 (1981)
Benoît, J.-P.: The nucleolus is contested-garment-consistent: a direct proof. J. Econ. Theory 77, 192–196 (1997)
Potters, Jos A.M., Reijnierse, J.H., Ansing, M.: Computing the nucleolus by solving a prolonged simplex algorithm. Math. Oper. Res 21, 757–768 (1996)
Meertens, M., Potters, Jos A.M.: The nucleolus of trees with revenues. Math. Method Oper. Res. 64(2), 363–382 (2006)
Potters, Jos A.M., Reijnierse, H., Biswas, A.: The nucleolus of balanced simple flow networks. Game Econ. Behav. 54(1), 205–225 (2006)
Maschler, M., Potters, Jos A.M., Reijnierse, H.: The nucleolus of a standard tree game revisited: a study of its monotonicity and computational properties. Int. J. Game Theory 39(1–2), 89–104 (2010)
Marchi, E., Oviedo, J.A.: Lexicographic optimality in the multiple objective linear programming: the nucleolar solution. Eur. J. Oper. Res. 57(3), 355–359 (1992)
Kostreva, M.M., Ogryczak, W., Wierzbicki, A.: Equitable aggregations and multiple criteria analysis. Eur. J. Oper. Res. 158, 362–377 (2004)
Ogryczak, W., Sliwinski, T., Wierzbicki, A.: Fair resource allocation schemes and network dimensioning problems. J. Telecommun. Inf. Technol. 3, 34–42 (2003)
Ogryczak, W., Śliwiński, T.: On direct methods for lexicographic min–max optimization. Lect. Notes Comput. Sci. 3982, 802–811 (2006)
Radunović, B., Le Boudec, J.-Y.: A unified framework for max–min and min–max fairness with applications. IEEE/ACM Trans. Netw. 15(5), 1073–1083 (2007)
Klein, R.S., Luss, H., Smith, D.R.: A lexicographic minimax algorithm for multiperiod resource allocation. Math. Program. 55, 213–234 (1992)
Tomaszewski, A.: A polynomial algorithm for solving a general max–min fairness problem. Eur. Trans. Telecommun. 16, 233–240 (2005)
Ogryczak, W., Pióro, M., Tomaszewski, A.: Telecommunications network design and max–min optimization problem. J. Telecommun. Inf. Technol. 3, 43–56 (2005)
Ehrgott, M., Holder, A., Reese, J.: Beam selection in radiotherapy design. Linear Algebra Appl. 428, 1272–1312 (2008)
Pióro, M.: Fair routing and related optimization problems. In: Proceedings of the 15th International Conference on Advanced Computing and Communications, Guwahati, Assam, pp. 229–235. IEEE Computer Society (2007)
Pióro, M., Nilsson, P., Kubilinskas, E., Fodor, G.: On efficient max-min fair routing algorithms. In: Proceedings of the Eighth IEEE International Symposium on Computers and Communication (ISCC’03), Kemer - Antalya, Turkey, pp. 365–372 vol. 1 (2003)
Sun, M.: Some issues in measuring and reporting solution quality of interactive multiple objective programming procedures. Eur. J. Oper. Res. 162, 468–483 (2005)
Ogryczak, W., Wierzbicki, A.: On multi-criteria approaches to bandwidth allocation. Control Cybern. 33(3), 427–448 (2004)
Ogryczak, W., Wierzbicki, A., Milewskia, M.: A multi-criteria approach to fair and efficient bandwidth allocation. Omega 36, 451–463 (2008)
Hoang, D.T., Vitter, J.S.: Efficient Algorithms for MPEG Video Compression. Wiley, New York, NY (2001)
Dragan, I.: A game theoretic approach for solving the multiobjective linear programming problems. Lib. Math. 30, 149–158 (2010)
Behringer, F.A.: A simplex based algorithm for the lexicographically extended linear maxmin problem. Eur. J. Oper. Res. 7, 274–283 (1981)
Pourkarimi, L., Zarepisheh, M.: A dual-based algorithm for solving lexicographic multiple objective programs. Eur. J. Oper. Res. 176, 1348–1356 (2007)
Klee, V., Minty, G.J.: How Good is the Simplex Algorithm? Inequalities-III. Academic Press, New York (1972)
Friedmann, O., Hansen, T., Zwick, U.: Subexponential lower bounds for randomized pivoting rules for the simplex algorithm. In: STOC’11 Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pp. 283–292. ACM, San Jose, California, New York (2011)
Behringer, F.A.: Linear multiobjective maxmin optimization and some Pareto and lexmaxmin extensions. OR Spektrum 8, 25–32 (1986)
Khorram, E., Zarepisheh, M., Ghaznavi-ghosoni, B.A.: Sensitivity analysis on the priority of the objective functions in lexicographic multiple objective linear programs. Eur. J. Oper. Res. 207, 1162–1168 (2010)
Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming, 3rd edn. Springer, New York, NY (2010)
Acknowledgments
We thank Dr. Irinel C. Dragan for his helpful comments and suggestions
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Irinel C. Dragan.
Rights and permissions
About this article
Cite this article
Jiangao, Z., Yang, S. On the Lexicographic Centre of Multiple Objective Optimization. J Optim Theory Appl 168, 600–614 (2016). https://doi.org/10.1007/s10957-015-0810-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-015-0810-0
Keywords
- Multiple objective programming
- Lexicographic order
- Lexicographic centre
- Efficient solution
- Image set
- \(\theta \)-image set