×

zbMATH — the first resource for mathematics

Variable preference modeling with ideal-symmetric convex cones. (English) Zbl 1166.90014
The paper addresses the problem of modeling preferences in multiple criteria decision making and multiobjective programming. Variable domination structures, where the dominated set of any point \(y\) is modeled by an ideal-symmetric convex cone \(D(y)\) that contains the nonnegative orthant, are used for this purpose. Well known results for multicriteria optimization with constant domination structures are generalized to this case. The results include results on weighted sum scalarization, necessary and sufficient conditions for nondominated points and further results for problems where the nondominated set \(N(Y,\mathbb{R}_\geq^m)\) is \(\mathbb{R}_\geq^m\)-convex or \(N(Y,\mathbb{R}_\geq^m)\) is \(\mathbb{R}_\geq^m\)-concave and \(\mathbb{R}_\geq^m\)-compact. The paper also contains some examples.

MSC:
90C29 Multi-objective and goal programming
90B50 Management decision making, including multiple objectives
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Programming. 95(1, Ser. B), 3–51. ISMP 2000, Part 3 (Atlanta, GA) (2003) · Zbl 1153.90522
[2] Benson H.P. (1979). An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71(1): 232–241 · Zbl 0418.90081 · doi:10.1016/0022-247X(79)90226-9
[3] Benson H.P. (1983). Efficiency and proper efficiency in vector maximization with respect to cones. J. Math. Anal. Appl. 93(1): 273–289 · Zbl 0519.90080 · doi:10.1016/0022-247X(83)90230-5
[4] Bergstresser K., Charnes A. and Yu P.-L. (1976). Generalization of domination structures and nondominated solutions in multicriteria decision making. J. Optim. Theory Appl. 18(1): 3–13 · Zbl 0298.90003 · doi:10.1007/BF00933790
[5] Bergstresser K. and Yu P.-L. (1977). Domination structures and multicriteria problems in N-person games. Theory Decis. 8(1): 5–48 · Zbl 0401.90117 · doi:10.1007/BF00133085
[6] Borwein J. (1977). Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15(1): 57–63 · Zbl 0369.90096 · doi:10.1137/0315004
[7] Chen G.Y. and Yang X.Q. (2002). Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112(1): 97–110 · Zbl 0988.49005 · doi:10.1023/A:1013044529035
[8] Chen G.Y., Yang X.Q. and Yu H. (2005). A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Global Optim. 32(4): 451–466 · Zbl 1130.90413 · doi:10.1007/s10898-003-2683-2
[9] Chew, K.L.: Domination structures in abstract spaces. In: Proceedings of the First Franco-Southeast Asian Mathematical Conference (Singapore, 1979), vol. II, Special Issue b, pp. 190–204 (1979) · Zbl 0481.90083
[10] Coladas Uría L. (1981). Nondominated solutions in multiobjective problems. Trabajos Estadíst. Investigación Oper. 32(1): 3–12 · Zbl 0548.62016 · doi:10.1007/BF03021686
[11] Corley H.W. (1980). A new scalar equivalence for Pareto optimization. IEEE Trans. Automat. Control 25(4): 829–830 · Zbl 0443.90099 · doi:10.1109/TAC.1980.1102401
[12] Corley H.W. (1981). Duality theory for maximizations with respect to cones. J. Math. Anal. Appl 84(2): 560–568 · Zbl 0474.90081 · doi:10.1016/0022-247X(81)90188-8
[13] Corley H.W. (1985). On optimality conditions for maximizations with respect to cones. J. Optim. Theory Appl. 46(1): 67–78 · Zbl 0542.90088 · doi:10.1007/BF00938760
[14] Edgeworth F.Y. (1881). Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. P. Keagan, London · Zbl 0005.17402
[15] Ehrgott M. and Tenfelde-Podehl D. (2003). Computation of ideal and Nadir values and implications for their use in MCDM methods. European J. Oper. Res. 151(1): 119–139 · Zbl 1043.90039 · doi:10.1016/S0377-2217(02)00595-7
[16] Ehrgott, M., Wiecek, M.M.: Multiobjective programming. In: Figueira, J., Greco, S., Ehrgott, M. (eds) Multiple Criteria Decision Analysis: State of the Art Surveys, vol. 78, pp. 667–722. International Series in Operations Research & Management Science. Springer, New York (2005) · Zbl 1072.90031
[17] (2000). Preferences and Decisions under Incomplete Knowledge, volume 51 of Studies in Fuzziness and Soft Computing. Physica-Verlag, Heidelberg
[18] Fujita T. (1996). Characterization of the solutions of multiobjective linear programming with a general dominated cone. Bull. Inform. Cybernet. 28(1): 23–29 · Zbl 0863.90123
[19] Gass S. and Saaty T. (1955). The computational algorithm for the parametric objective function. Naval Res. Logist. Quart. 2: 39–45 · doi:10.1002/nav.3800020106
[20] Geoffrion A.M. (1968). Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22: 618–630 · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1
[21] Guddat J., Guerra Vasquez F., Tammer K. and Wendler K. (1985). Multiobjective and Stochastic Optimization Based on Parametric Optimization. Mathematical Research, 26. Akademie-Verlag, Berlin · Zbl 0583.90055
[22] Henig M.I. (1982). Proper efficiency with respect to cones. J. Optim. Theory Appl. 36(3): 387–407 · Zbl 0452.90073 · doi:10.1007/BF00934353
[23] Henig M.I. (1990). Value functions, domination cones and proper efficiency in multicriteria optimization. Math. Programming, 46(2, (Ser. A)): 205–217 · Zbl 0691.90081 · doi:10.1007/BF01585738
[24] Hsia W.-S. and Lee T.Y. (1988). Lagrangian function and duality theory in multiobjective programming with set functions. J. Optim. Theory Appl. 57(2): 239–251 · Zbl 0619.90072 · doi:10.1007/BF00938538
[25] Hunt, B.J.: Multiobjective Programming with Convex Cones: Methodology and Applications. PhD thesis, Clemson University, Clemson, South Carolina, USA. Margaret M. Wiecek, advisor. (2004)
[26] Hunt B.J. and Wiecek M.M. (2003). Cones to aid decision making in multicriteria programming. In: Tanino, T., Tanaka, T. and Inuiguchi, M. (eds) Multi-Objective Programming and Goal Programming, pp 153–158. Springer-Verlag, Berlin · Zbl 1165.90496
[27] Hyers D.H., Isac G. and Rassias T.M. (1997). Topics in Nonlinear Analysis & Applications. World Scientific Publishing Co. Inc., River Edge · Zbl 0878.47040
[28] Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability,1950, pp. 481–492, University of California Press, Berkeley and Los Angeles (1951)
[29] Lin S.A.Y. (1976). A comparison of Pareto optimality and domination structure. Metroeconomica 28(1–3): 62–74 · Zbl 0417.90007 · doi:10.1111/j.1467-999X.1976.tb00549.x
[30] Naccache P.H. (1978). Connectedness of the set of nondominated outcomes in multicriteria optimization. J. Optim. Theory Appl. 25(3): 459–467 · Zbl 0363.90108 · doi:10.1007/BF00932907
[31] Noghin V.D. (1997). Relative importance of criteria: a quantitative approach. J. Multi-Criteria Decision Analysis 6: 355–363 · Zbl 1078.90546 · doi:10.1002/(SICI)1099-1360(199711)6:6<355::AID-MCDA174>3.0.CO;2-O
[32] Öztürk, M., Tsoukiàs, A., Vincke, P.: Preference modelling. In Multiple criteria decision analysis. State of the art surveys, volume 78 of International Series in Operations Research & Management Science, pp. 29–71. Springer, Berlin (2005) · Zbl 1072.90018
[33] Pareto V. (1896). Cours d’Économie Politique. Rouge, Lausanne
[34] Ramesh R., Karwan M.H. and Zionts S. (1988). Theory of convex cones in multicriteria decision making. Ann. Oper. Res. 16(1–4): 131–147 · Zbl 0692.90095 · doi:10.1007/BF02283741
[35] Ramesh R., Karwan M.H. and Zionts S. (1989). Preference structure representation using convex cones in multicriteria integer programming. Management Sci. 35(9): 1092–1105 · Zbl 0683.90089 · doi:10.1287/mnsc.35.9.1092
[36] Rockafellar R.T. (1970). Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton · Zbl 0193.18401
[37] Sawaragi Y., Nakayama H. and Tanino T. (1985). Theory of Multiobjective Optimization, volume 176 of Mathematics in Science and Engineering. Academic Press Inc., Orlando · Zbl 0566.90053
[38] (1998). Fuzzy Sets in Decision Analysis, Operations Research and Statistics, volume 1 of The Handbooks of Fuzzy Sets Series. Kluwer Academic Publishers, Boston
[39] Takeda E. and Nishida T. (1980). Multiple criteria decision problems with fuzzy domination structures. Fuzzy Sets Systems 3(2): 123–136 · Zbl 0429.90073 · doi:10.1016/0165-0114(80)90050-0
[40] Tamura K. and Miura S. (1979). Necessary and sufficient conditions for local and global nondominated solutions in decision problems with multi-objectives. J. Optim. Theory Appl. 28(4): 501–523 · Zbl 0387.90095 · doi:10.1007/BF00932220
[41] Tanino T., Nakayama H. and Sawaragi Y. (1980). Domination structures and their parametrization in group decision making. Syst. Control 24(12): 807–815
[42] Tanino T. and Sawaragi Y. (1978). Stability of the solution set in problems of finding cone extreme points. Syst. Control 22(11): 693–700
[43] Tanino T. and Sawaragi Y. (1979). Duality theory in multiobjective programming. J. Optim. Theory Appl. 27(4): 509–529 · Zbl 0378.90100 · doi:10.1007/BF00933437
[44] Tanino T. and Sawaragi Y. (1980). Stability of nondominated solutions in multicriteria decision-making. J. Optim. Theory Appl. 30(2): 229–253 · Zbl 0414.90085 · doi:10.1007/BF00934497
[45] Weidner, P.: Domination sets and optimality conditions in vector optimization theory. Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau 31(2), 133–146. (in German) (1985) · Zbl 0582.90086
[46] Weidner, P.: Tradeoff directions and dominance sets. In: Multi-objective programming and goal programming, Adv. Soft Comput., pp. 275–280. Springer, Berlin (2003) · Zbl 1140.90313
[47] Wendell R.E. and Lee D.N. (1977). Efficiency in multiple objective optimization problems. Math. Programming 12(3): 406–414 · Zbl 0362.90092 · doi:10.1007/BF01593807
[48] Wu H.C. (2004). A solution concept for fuzzy multiobjective programming problems based on convex cones. J. Optim. Theory Appl. 121(2): 397–417 · Zbl 1076.90070 · doi:10.1023/B:JOTA.0000037411.25509.6a
[49] Yamamoto Y. (2002). Optimization over the efficient set: overview. J. Global Optim. 22(1–4): 285–317 · Zbl 1045.90061 · doi:10.1023/A:1013875600711
[50] Yu P.-L. (1973). Introduction to domination structures in multicriteria decision problems. In: Cochrane, J.L. and Zeleny, M. (eds) Multiple Criteria Decision Making, pp 249–261. University of South Carolina Press, Columbia
[51] Yu P.-L. (1974). Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14: 319–377 · Zbl 0268.90057 · doi:10.1007/BF00932614
[52] Yu, P.-L.: Domination structures and nondominated solutions. In: Multicriteria Decision Making, pp. 227–280. Internat. Centre for Mech Sci (CISM) Courses and Lectures, No. 211. Springer, Vienna (1975)
[53] Yu P.-L. (1985). Multiple-criteria Decision Making. Concepts, Techniques and Extensions Mathematical Concepts and Methods in Science and Engineering, 30. Plenum Press, New York-London
[54] Yu P.-L. and Leitmann G. (1974). Compromise solutions, domination structures and Salukvadze’s solution. J. Optim. Theory Appl. 13: 362–378 · Zbl 0362.90111 · doi:10.1007/BF00934871
[55] Yun Y.B., Nakayama H. and Tanino T. (2004). A generalized model for data envelopment analysis. European J. Oper. Res. 157(1): 87–105 · Zbl 1106.90350 · doi:10.1016/S0377-2217(03)00140-1
[56] Zadeh, L.: Optimality and nonscalar-valued performance criteria. IEEE Trans Auto. Cont., AC-8:1 (1963)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.