Advances in Cone-Based Preference Modeling for Decision Making with Multiple Criteria
DOI:
https://doi.org/10.7494/dmms.2007.1.2.153Keywords:
cones, preferences, nondominated solutions, Pareto solutions, multiple criteria, decision makingAbstract
Decision making with multiple criteria requires preferences elicited from the decision maker to determine a solution set. Models of preferences, that follow upon the concept of nondominated solutions introduced by Yu (1974), are presented and compared within a unified framework of cones. Polyhedral and nonpolyhedral, convex and nonconvex, translated, and variable cones are used to model different types of preferences. Common mathematical properties of the preferences are discussed. The impact of using these preferences in decision making is emphasized.References
Baatar D. & Wiecek M.M. (2006). Advancing equitability in multiobjective programming. Computers & Mathematics with Applications, 52(1–2), pp. 225–234. DOI: https://doi.org/10.1016/j.camwa.2006.08.014.
Bergstresser K., Charnes A. & Yu P.L. (1976). Generalization of domination structures and nondominated solutions in multicriteria decision making. Journal of Optimization Theory and Applications, 18(1), pp. 3–13. DOI: https://doi.org/10.1007/BF00933790.
Berman V.P. & Naumov G.E. (1989). Preference relation and interval value tradeoffs in criterion space. Automation and Remote Control, 50(3), pp. 398–410.
Blouin V.Y., Hunt B.J. & Wiecek M.M. (2007). Relative importance of criteria in configuration design of vehicles. Technical Report. Clemson, SC: Department of Mathematical Sciences, Clemson University.
Cambini A., Luc D.T. & Martein L. (2003). Order-preserving transformations and applications. Journal of Optimization Theory and Applications, 118(2), pp. 275–293. DOI: https://doi.org/10.1023/A:1025495204834.
Chen G.Y., Huang X. & Yang X. (2005). Vector Optimization: Set-Valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, 541. Berlin–Heidelberg: Springer. DOI: https://doi.org/10.1007/3-540-28445-1.
Chen G.Y. & Yang X.Q. (2002). Characterizations of variable domination structures via nonlinear scalarization. Journal of Optimization Theory and Applications, 112(1), pp. 97–110. DOI: https://doi.org/10.1023/A:1013044529035.
Chew K.L. (1979). Domination structures in abstract spaces. In: Proceedings of the First Franco-Southeast Asian Mathematical Conference, Vol. II, pp. 190–204.
Doležal J. (1976). Hierarchical solution concept for static and multistage decision problems with two objectives. Kybernetika, 12(5), pp. 363–385.
Ehrgott M. & Wiecek M.M. (2005). Multiobjective programming. In: Figueira J., Greco S. & Ehrgott M. (Eds.), Multiple Criteria Decision Analysis: State of the Art Surveys, pp. 667–708. New York: Springer. DOI: https://doi.org/10.1007/0-387-23081-5_17.
Engau A. (2007). Domination and Decomposition in Multiobjective Programming [PhD Thesis]. Clemson, SC: Clemson University.
Engau A. & Wiecek M.M. (2007). Cone characterizations of approximate solutions in real vector optimization. Journal of Optimization Theory and Applications, 134(3), pp. 499–513. DOI: https://doi.org/10.1007/s10957-007-9235-8.
Engau A. & Wiecek M.M. (2007). Generating ε-efficient solutions in multiobjective programming. European Journal of Operational Research, 177(3), pp. 1566–1579. DOI: https://doi.org/10.1016/j.ejor.2005.10.023.
Engau A. & Wiecek M.M. (2007). Exact generation of epsilon-efficient solutions in multiple objective programming. OR Spectrum, 29(2), pp. 335–350. DOI: https://doi.org/10.1007/s00291-006-0044-5.
Engau A. & Wiecek M.M. (2007). 2D decision-making for multicriteria design optimization. Structural and Multidisciplinary Optimization, 34(4), pp. 301–315. DOI: https://doi.org/10.1007/s00158-006-0078-y.
Figueira J., Greco S. & Ehrgott M. (Eds.) (2005). Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science, 78. New York: Springer. DOI: https://doi.org/10.1007/b100605.
Hazen G.B. & Morin T.L. (1983). Optimality conditions in nonconical multiple-objective programming. Journal of Optimization Theory and Applications, 40(1), pp. 25–60. DOI: https://doi.org/10.1007/BF00934630.
Hunt B.J. (2004). Multiobjective Programming with Convex Cones: Methodology and Applications [PhD Thesis]. Clemson, SC: Clemson University.
Hunt B.J., Blouin V.Y. & Wiecek M.M. (2007). Modeling relative importance of design criteria with a modified Pareto preference. Journal of Mechanical Design, 129(9), pp. 907–914. DOI: https://doi.org/10.1115/1.2747634.
Hunt B.J. & Wiecek M.M. (2003). Cones to aid decision making in multicriteria programming. In: Tanino T., Tanaka T. & Inuiguchi M. (Eds.), Multi-Objective Programming and Goal Programming, pp. 153–158. Berlin–Heidelberg: Springer. DOI: https://doi.org/10.1007/978-3-540-36510-5_20.
Hunt B.J., Wiecek M.M. & Fadel G. (2004). Matrices as preference modeling tools in bi-criteria engineering design. In: Proceedings of the 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, NY, August 30–September 1, 2004, Paper 2004-4320. DOI: https://doi.org/10.2514/6.2004-4320.
Karasakal E. & Michalowski W. (2003). Incorporating wealth information into a multiple criteria decision making model. European Journal of Operational Research, 150(1), pp. 204–219. DOI: https://doi.org/10.1016/S0377-2217(02)00468-X.
Kostreva M.M. & Ogryczak W. (1999). Linear optimization with multiple equitable criteria. RAIRO – Operations Research, 33(3), pp. 275–297. DOI: https://doi.org/10.1051/ro:1999112.
Kostreva M.M., Ogryczak W. & Wierzbicki A. (2004). Equitable aggregations and multiple criteria analysis. European Journal of Operational Research, 158(2), pp. 362–377. DOI: https://doi.org/10.1016/j.ejor.2003.06.010.
Kutateladze S.S. (1979). Convex ε-programming. Soviet Mathematics Doklady, 20(2), pp. 391–393.
Lin S.A.Y. (1976). A comparison of Pareto optimality and domination structure. Metroeconomica, 28(1–3), pp. 62–74. DOI: https://doi.org/10.1111/j.1467-999X.1976.tb00549.x.
Loridan P. (1984). ε-Solutions in vector minimization problems. Journal of Optimization Theory and Applications, 43(2), pp. 265–276. DOI: https://doi.org/10.1007/BF00936165.
Luenberger D.G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons.
Men’shikova O.R. & Podinovskii V.V. (1988). Constructing the preference relation and the core in multicriterion problems with inhomogeneous criteria ordered by importance. USSR Computational Mathematics and Mathematical Physics, 28(3), pp. 15–22. DOI: https://doi.org/10.1016/0041-5553(88)90172-3.
Noghin V.D. (1997). Relative importance of criteria: A quantitative approach. Journal of Multi-Criteria Decision Analysis, 6(6), pp. 355–363. DOI: https://doi.org/10.1002/(SICI)1099-1360(199711)6:6%3C355::AID-MCDA174%3E3.0.CO;2-O.
Noghin V.D. & Tolstykh I.V. (2000). Using quantitative information on the relative importance of criteria for decision making. Computational Mathematics and Mathematical Physics, 40(11), pp. 1529–1536.
Ogryczak W. (2000). Multiple criteria linear programming model for portfolio selection. Annals of Operations Research, 97, pp. 143–162. DOI: https://doi.org/10.1023/A:1018980308807.
Ogryczak W. (2000). Inequality measures and equitable approaches to location problems. European Journal of Operational Research, 122(2), pp. 374–391. DOI: https://doi.org/10.1016/S0377-2217(99)00240-4.
Ogryczak W., Wierzbicki A. & Milewski M. (2008). A multi-criteria approach to fair and efficient bandwidth allocation. Omega, 36(3), pp. 451–463. DOI: https://doi.org/10.1016/j.omega.2005.12.005.
Podinovskii V.V. (1977). Two-criterial problems with unequally rated criteria. Engineering Cybernetics, 5, pp. 44–59.
Podinovskii V.V. (1978). Importance coefficients of criteria in decision making problems: Serial or ordinal importance coefficients. Automation and Remote Control, 39(10), pp. 1514–1524.
Podinovskii V.V. (1994). Criteria importance theory. Mathematical Social Sciences, 27(3), pp. 237–252. DOI: https://doi.org/10.1016/0165-4896(93)00737-F.
Podinovskii V.V. (2000). Quantitative importance of criteria. Automation and Remote Control, 61(5), pp. 817–828.
Roy B. & Mousseau V. (1996). A theoretical framework for analysing the notion of relative importance of criteria. Journal of Multi-Criteria Decision Analysis, 5(2), pp. 145–159. DOI: https://doi.org/10.1002/(SICI)1099-1360(199606)5:2%3C145::AID-MCDA99%3E3.0.CO;2-5.
Sawaragi Y., Nakayama H. & Tanino T. (1985). Theory of Multiobjective Optimization. Mathematics in Science and Engineering, 176. Orlando: Academic Press. DOI: https://doi.org/10.1016/S0076-5392(09)X6017-0.
Singh V.K. (2007). Equitable Efficiency in Multiple Criteria Optimization [PhD Thesis]. Clemson, SC: Clemson University.
Takeda E. & Nishida T. (1980). Multiple criteria decision problems with fuzzy domination structures. Fuzzy Sets and Systems, 3(2), pp. 123–136. DOI: https://doi.org/10.1016/0165-0114(80)90050-0.
Wei Q., Yan H., Ma J. & Fan Z. (2000). A compromise weight for multi-criteria group decision making with individual preference. Journal of the Operational Research Society, 51(5), pp. 625–634. DOI: https://doi.org/10.1057/palgrave.jors.2600911.
Weidner P. (2001). Problems in scalarizing multicriteria approaches. In: Köksalan M. & Zionts S. (Eds.), Multiple Criteria Decision Making in the New Millennium, pp. 199–209. Berlin–Heidelberg: Springer. DOI: https://doi.org/10.1007/978-3-642-56680-6_18.
Weidner P. (2003). Tradeoff directions and dominance sets. In: Tanino T., Tanaka T. & Inuiguchi M. (Eds.), Multi-Objective Programming and Goal Programming, pp. 275–280. Berlin–Heidelberg: Springer. DOI: https://doi.org/10.1007/978-3-540-36510-5_40.
Wendell R.E. & Lee D.N. (1977). Efficiency in multiple objective optimization problems. Mathematical Programming, 12(1), pp. 406–414. DOI: https://doi.org/10.1007/BF01593807.
White D.J. (1986). Epsilon efficiency. Journal of Optimization Theory and Applications, 49(2), pp. 319–337. DOI: https://doi.org/10.1007/BF00940762.
Wiecek M.M. (2007). Multi-scenario multi-objective optimization for engineering design. In: Deb K., Chakraborty P., Iyengar N.G.R. & Gupta S.K. (Eds.), Advances in Computational Optimization and Its Applications, pp. 170–174. India: Universities Press.
Wu H.C. (2004). A solution concept for fuzzy multiobjective programming problems based on convex cones. Journal of Optimization Theory and Applications, 121(2), pp. 397–417. DOI: https://doi.org/10.1023/B:JOTA.0000037411.25509.6A.
Ying M.Q. (1983). The set of cone-extreme points and the grouping hierarchy problem. Journal of Systems Science and Mathematical Sciences, 3(2), pp. 125–138.
Yu P.L. (1974). Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. Journal of Optimization Theory and Applications, 14(3), pp. 319–377. DOI: https://doi.org/10.1007/BF00932614.
Yu P.L. (1985). Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions. New York: Plenum Press. DOI: https://doi.org/10.1007/978-1-4684-8395-6.
