Abstract
Scalarization of fuzzy multiobjective programming problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Since the set of all fuzzy numbers can be embedded into a normed space, this motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to evaluate the a multiobjective programming problem. Two solution concepts are proposed in this paper by considering different convex cones.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manage. Sci. 17, 141–164 (1970)
Slowiński, R. (ed.): Fuzzy Sets in Decision Analysis, Operations Research and Statistics. Kluwer Academic, Dordrecht (1998)
Delgado, M., Kacprzyk, J., Verdegay, J.-L., Vila, M.A. (eds.): Fuzzy Optimization: Recent Advances. Physica-Verlag, Heidelberg (1994)
Zimmermann, H.-J.: Fuzzy Set Theory—And Its Applications, 3rd edn. Kluwer Academic, Dordrecht (1996)
Lai, Y.-J., Hwang, C.L.: Fuzzy Mathematical Programming: Methods and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 394. Springer, Berlin (1992)
Lai, Y.-J., Hwang, C.L.: Fuzzy Multiple Objective Decision Making: Methods and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 404. Springer, Berlin (1994)
Wu, H.-C.: A solution concept for fuzzy multiobjective programming problems based on convex cones. J. Optim. Theory Appl. 121, 397–417 (2004)
Benson, H.P.: Existence of efficient solutions for vector maximization problems. J. Optim. Theory Appl. 26, 569–580 (1978)
Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)
Benson, H.P.: Efficiency and proper efficiency in vector maximization with respect to cones. J. Math. Anal. Appl. 93, 273–289 (1983)
Borwein, J.M.: Proper efficient points for maximization with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)
Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–377 (1974)
Wu, H.-C.: An (α,β)-optimal solution concept in fuzzy optimization problems. Optimization 53, 203–221 (2004)
Puri, M.L., Ralescu, D.A.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91, 552–558 (1983)
Kaleva, O.: The calculus of fuzzy valued functions. Appl. Math. Lett. 3, 55–59 (1990)
Zadeh, L.A.: The concept of linguistic variable and its application to approximate reasoning, I. Inf. Sci. 8, 199–249 (1975)
Zadeh, L.A.: The concept of linguistic variable and its application to approximate reasoning, II. Inf. Sci. 8, 301–357 (1975)
Zadeh, L.A.: The concept of linguistic variable and its application to approximate reasoning, III. Inf. Sci. 9, 43–80 (1975)
Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)
Jahn, J.: Mathematical Vector Optimization in Partially Ordered Linear Spaces. Peter Lang, Frankfurt am Main (1986)
Negoita, C.V., Ralescu, D.A.: Applications of Fuzzy Sets to Systems Analysis. Wiley, New York (1975)
Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods and applications. SIAM Rev. 35, 380–429 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H.P. Benson.
Rights and permissions
About this article
Cite this article
Wu, H.C. Solutions of Fuzzy Multiobjective Programming Problems Based on the Concept of Scalarization. J Optim Theory Appl 139, 361–378 (2008). https://doi.org/10.1007/s10957-008-9419-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-008-9419-x