Abstract
The topic of numerical methods in multicriteria optimization lends itself to many interpretations, and, of course, much has been written on the subject. This chapter will not be a survey of the numerical techniques of multicriteria optimization (MCO). Those interested in such a work should see, for example, the book of Hwang and Masud (Ref. 1). Nor will this chapter contain comparisons of the numerical efficiency of a variety of MCO algorithms. Instead, this work will be on my views and experience in numerically analyzing “real” linear MCO problems and the mathematics necessary for such an analysis. Of course, “real” means MCO problems as I have encountered them in applications. Imaginary (or unreal) must therefore refer to all the rest.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hwang, C.-L., and Masud, A. S. Md., Multiple Objective Decision Making— Methods and Applications, Lecture Notes in Economics and Mathematical Systems, No. 164, Springer-Verlag, New York, 1979.
Dauer, J. P., and Krueger, R. J., Multiobjective Screening Model for Water Resources Planning, Proceedings of the First International Conference on Mathematical Modeling, Vol. IV (X. J. R. Avula, ed.), St. Louis, Missouri, pp. 2203–2211, 1977.
Zeleny, M., Linear Multiobjective Programming, Springer-Verlag, New York, 1974.
Philip, J., Algorithms for the Vector Maximization Problem, Mathematical Programming, 2, 207–229, 1972.
Mangasarian, O. L., Nonlinear Programming, McGraw-Hill, New York, 1969.
Kuhn, H. W., and Tucker, A. W., Nonlinear Programming, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, pp. 481–492, 1950.
Charnes, A., and Cooper, W. W., Management Models and Industrial Applications of Linear Programming, Volume I, Wiley, New York, 1969.
Evans, J. P., and Steuer, R. E., A Revised Simplex Method for Linear Multiple Objective Programs, Mathematical Programming, 5, 54–72, 1973.
Dauer, J. P., Analysis of the Objective Space in Multiple Objective Linear Programming, Journal of Mathematical Analysis and Applications, 126, 579–593, 1987.
Bod, P., Linear Optimization with Several Simultaneously Given Objective Functions (in Hungarian), Mathematical Institute of the Hungarian Academy of Sciences, Vol. 8, pp. 541–556, 1963.
Gal, T., A General Method for Determining the Set of All Efficient Solutions to a Linear Vector Maximum Problem, European Journal of Operational Research, 1, 307–322, 1977.
Ecker, J. G., and Kouada, I. A., Finding Efficient Points for Linear Multiple Objective Programs, Mathematical Programming, 8, 375–377, 1975.
Ecker, J. G., and Kouada, I. A., Finding All Efficient Extreme Points for Multiple Objective Linear Programs, Mathematical Programming, 14, 249–261, 1978.
Isermann, H., The Enumeration of the Set of All Efficient Solutions for a Linear Multiple Objective Program, Operational Research Quarterly, 28, 711–725, 1977.
Ecker, J. G., Hegner, N. S., and Kouada, I. A., Generating All Maximal Efficient Faces for Multiple Objective Linear Programs, Journal of Optimization Theory and Applications, 30, 353–381, 1980.
Dauer, J. P., and Liu, Y. H., Solving Multiple Objective Linear Programs in Objective Space, European Journal of Operational Research, to appear.
Dauer, J. P., and Krueger, R. J., A Multiobjective Optimization Model for Water Resources Planning, Applied Mathematical Modelling, 4, 171–175, 1980.
Dantzig, G. B., Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963.
El-Abyad, A. M., Geometric Analysis of the Objective Space in Linear Multiple Objective Programming, University of Nebraska-Lincoln, Lincoln, Nebraska, Ph.D. thesis, 1986.
Dauer, J. P., An Equivalence Result for Solutions of Multiobjective Linear Programs, Computers and Operations Research, 7, 33–39, 1980.
Cohon, J. L., and Marks, D. H., Multiobjective Screening Models and Water Resources Investment, Water Resources Research, 9, 208–220, 1973.
Haimes, Y. Y., Wismer, D. A., and Lasdon, L. S., On Bicriterion Formulation of the Integrated System Identification and System Optimization, IEEE Transactions on Systems, Man and Cybernetics, SMC-1, 296–297, 1971.
Dauer, J. P., and Stadler, W., A Survey of Vector Optimization in Infinite-Dimensional Spaces, Part II, Journal of Optimization Theory and Applications, 51, 205–241, 1986.
Haimes, Y. Y., Multiobjective Analysis in the Maumee River Basin: a Case Study on Level-B Planning, Case Western Reserve University, Cleveland, Ohio, 1977.
Philip, J., An Algorithm for Combined Quadratic and Multiobjective Programming, Multiple Criteria Decision Making, Proceedings of a Conference, Jouyen-Josas, France, 1975 (H. Thiriez and S. Zionts, eds.), Springer Lecture Notes in Economics and Mathematical Systems, Vol. 130, pp. 35–52, 1976.
Duesing, E. C., Polyhedral Convex Sets and the Economic Analysis of Production, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, Ph.D. thesis, 1978.
Sebo, D., Multiple Objective Linear Programming in Objective Space, University of Nebraska-Lincoln, Lincoln, Nebraska, Ph.D. thesis, 1981.
Dauer, J. P., and El-Abyad, A.M., Algorithms for Constructing the Faces of a Finitely Generated Cone, University of Nebraska-Lincoln, Lincoln, Nebraska, Technical Report, 1986.
Gal, T., and Leberling, H., Redundant Objective Functions in Linear Vector Maximum Problems and Their Determination, European Journal of Operational Research, 1, 176–184, 1977.
Wets, R. J. B., and Witzgall, C, Algorithms for Frames and Linearity Spaces of Cones, Journal of Research of the National Bureau of Standards, 71B, 1–7,1967.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer Science+Business Media New York
About this chapter
Cite this chapter
Dauer, J.P. (1988). Numerically Analyzing Linear Multicriteria Optimization Problems. In: Stadler, W. (eds) Multicriteria Optimization in Engineering and in the Sciences. Mathematical Concepts and Methods in Science and Engineering, vol 37. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-3734-6_2
Download citation
DOI: https://doi.org/10.1007/978-1-4899-3734-6_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-3736-0
Online ISBN: 978-1-4899-3734-6
eBook Packages: Springer Book Archive