{"title":"Stability Analysis for a Multicriteria Problem with Linear Criteria and Parameterized Principle of Optimality \u201cfrom Lexicographic to Slater\u201c","authors":"Yury Nikulin","volume":29,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":299,"pagesEnd":304,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/7443","abstract":"
A multicriteria linear programming problem with integer variables and parameterized optimality principle "from lexicographic to Slater" is considered. A situation in which initial coefficients of penalty cost functions are not fixed but may be potentially a subject to variations is studied. For any efficient solution, appropriate measures of the quality are introduced which incorporate information about variations of penalty cost function coefficients. These measures correspond to the so-called stability and accuracy functions defined earlier for efficient solutions of a generic multicriteria combinatorial optimization problem with Pareto and lexicographic optimality principles. Various properties of such functions are studied and maximum norms of perturbations for which an efficient solution preserves the property of being efficient are calculated.<\/p>\r\n","references":"[1] M. Ehrgott, Multicriteria optimization, Springer, Berlin, 2005.\r\n[2] V. Emelichev, E. Girlich, Y. Nikulin and D. Podkopaev, (2002). \"Stability\r\nand regularization of vector problems of integer linear programming\".\r\nOptimization 51, 645 - 676.\r\n[3] V. Emelichev, A. Platonov, (2006). \"About one discrete analog of\r\nHausdorff semi-continuity of suitable mapping in a vector combinatorial\r\nproblem with a parametric principle of optimality (\"from Slater to\r\nlexicographic\")\". Revue dAnalyse Numerique et de Theorie de lApproximation\r\n35.\r\n[4] H. Greenberg, (1998). \"An annotated bibliography for post-solution\r\nanalysis in mixed integer and combinatorial optimization.\" In D.\r\nWoodruff (ed.) Advances in computational and stochastic optimization,\r\nLogic programming and heuristic search, 97 - 148. Dordrecht: Kluwer\r\nAcademic Publishers.\r\n[5] M. Libura, (1999). \"On accuracy of solution for combinatorial optimization\r\nproblems with perturbed coefficients of the objective function\".\r\nAnnals of Operation Research 86, 53 - 62.\r\n[6] M. Libura, (2000). \"Quality of solutions for perturbed combinatorial\r\noptimization problems\". Control and Cybernetics 29, 199 - 219.\r\n[7] M. Libura and Y. Nikulin, (2006). \"Stability and accuracy functions\r\nin multicriteria linear combinatorial optimization problems\". Annals of\r\nOperations Research 147, 255 - 267.\r\n[8] M. Libura, (2007). \"On the robustness of optimal solutions for combinatorial\r\noptimization problems\". Research Report 2\/2007, Systems\r\nResearch Institute, Polish Academy of Sciences.\r\n[9] M. Libura, (2008). \"Robustness tolerances for combinatorial optimization\r\nproblems\". Research Report 4\/2008, Systems Research Institute,\r\nPolish Academy of Sciences.\r\n[10] K. Miettinen, Nonlinear Multiobjective Optimization (Kluwer Academic\r\nPublishers, Boston, 1999).\r\n[11] Y. Nikulin, (2008). \"Stability and accuracy functions in a coalition game\r\nwith bans, linear payoffs and antagonistic strategies\". (to appear in\r\nAnnals of Operations Research)\r\n[12] Y. Nikulin and M.M. M\u252c\u00bfakel\u252c\u00bfa, (2008). \"Quantitative measures of solution\r\nrobustness in a parametrized multicriteria zero-one linear programming\r\nproblem\". Turku Center for Computer Science, Technical Report 917.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 29, 2009"}