Abstract
An algorithm for finding the whole efficient set of a multiobjective linear program is proposed. From the set of efficient edges incident to a vertex, a characterization of maximal efficient faces containing the vertex is given. By means of the lexicographic selection rule of Dantzig, Orden and Wolfe, a connectedness property of the set of dual optimal bases associated to a degenerate vertex is proved. An application of this to the problem of enumerating all the efficient edges incident to a degenerate vertex is proposed. Our method is illustrated with numerical examples and comparisons with Armand—Malivert's algorithm show that this new algorithm uses less computer time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.V. Aho, J.E. Hopcroft and J.D. Ullman,The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, MA, 1974).
P. Armand and C. Malivert, “Determination of the efficient set in multiobjective linear programming,”Journal of Optimization Theory and Applications 70 (1991) 467–489.
M.L. Balinski, “On the graph structure of convex polyhedra inn-space,”Pacific Journal of Mathematics 11 (1961) 431–434.
H.P. Benson, “Finding an initial efficient extreme point for a linear multiple objective program,”Journal of the Operational Research Society 32 (1981) 495–498.
A. Brondsted,An Introduction to Convex Polytopes (Springer, New York, 1983).
G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).
J.G. Ecker and N.S. Hegner, “On computing an initial efficient extreme point,”Journal of the Operational Research Society 29 (1978) 1005–1007.
J.G. Ecker, N.S. Hegner and I.A. Kouada, “Generating all maximal efficient faces for multiple objective linear programs,”Journal of Optimisation Theory and Applications 30 (1980) 353–381.
J.G. Ecker and I.A. Kouada, “Finding efficient points for linear multiple objective programs,”Mathematical Programming 8 (1975) 375–377.
J.G. Ecker and I.A. Kouada, “Finding all efficient extreme points for multiple objective linear programs,”Mathematical Programming 14 (1978) 249–261.
J.G. Ecker and N.E. Shoemaker, “Selecting subsets from the set of nondominated vectors in multiple objective linear programming”,SIAM Journal on Control and Optimization 19 (1981) 505–515.
J.P. Evans and R.E. Steuer, “A revised simplex method for linear multiple objective programs,”Mathematical Programming 5 (1973) 54–72.
T. Gal, “A general method for determining the set of all efficient solutions to a linear vector maximum problem,”European Journal of Operational Research 1 (1977) 307–322.
T. Gal, “On the structure of the set bases of a degenerate point,”Journal of Optimisation Theory and Applications 45 (1985) 577–589.
B. Grünbaum,Convex Polytopes (Wiley-Interscience, London, 1967).
G. Hadley,Linear Programming (Addison-Wesley, Reading, MA, 1963).
R. Hartley, “Survey of algorithms for vector optimization problems,” in: S. French, R. Hartley, L.C. Thomas and D.J. White, eds.,Multi-objective Decision Making (Academic Press, London, 1983) pp. 1–34.
H. Isermann, “The enumeration of the set of all efficient solutions for a linear multiple objective program,”Operational Research Quarterly 28 (1977) 711–725.
H.J. Kruse, “Degeneracy graphs and the neighbourhood problem,”Lecture Notes in Economics and Mathematical Systems No. 260 (Springer, Berlin, 1986).
K.G. Murty, “Faces of polyhedron,”Mathematical Programming Study 24 (1985) 30–42.
J. Philip, “Algorithms for the vector maximization problem,”Mathematical Programming 2 (1973) 207–229.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
R.E. Steuer,Multiple Criteria Optimization: Theory, Computation, and Application (Wiley, New York, 1986).
P.L. Yu,Multiple-Criteria Decision Making (Plenum, New York, 1985).
P.L. Yu and M. Zeleny, “The set of all non-dominated solutions in linear cases and a multicriteria simplex method,”Journal of Mathematical Analysis and Applications 49 (1975) 430–468.
M. Zeleny, “Linear multi-objective programming,”Lecture Notes in Economics and Mathematical Systems No. 95 (Springer, Berlin, 1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Armand, P. Finding all maximal efficient faces in multiobjective linear programming. Mathematical Programming 61, 357–375 (1993). https://doi.org/10.1007/BF01582157
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582157