Abstract
In this paper we develop a method for finding all efficient extreme points for multiple objective linear programs. Simple characterizations of the efficiency of an edge incident to a nondegenerate or a degenerate efficient vertex are given. These characterizations form the basis of an algorithm for enumerating all efficient vertices. The algorithm appears to have definite computational advantages over other methods. Some illustrative examples are included.
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References
S.M. Belenson and K.C. Kapur, “An algorithm for solving multicriterion linear problems with examples”,Operational Research Quarterly 24 (1973) 65–77.
R. Benayoun, J. de Montgolfier, J. Tergeny and O. Larichev, “Linear programming with multiple objective functions: step method (STEM)”,Mathematical Programming 1 (1971) 366–375.
A. Charnes and W. Cooper,Management models and industrial applications of linear programming (Wiley, New York, 1961).
J.L. Cochrane and M. Zeleny, eds.,Multiple criteria decision making (University of South Carolina Press, 1973).
J.G. Ecker and I.A. Kouada, “Finding efficient points for linear multiple objective programs”,Mathematical Programming 8 (1975) 375–377.
J.G. Ecker, Nancy S. Hegner and I.A. Kouada, “Generating all maximal efficient faces for multiple objective linear programs”, to appear.
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 method for determining the set of all efficient solutions to a linear vector maximum problem”, Rept. 75/13, Institut für Wirtschaftswissenschaften, Aachen, Germany (1975).
A.M. Geoffrion, J.S. Dyer and A. Feinberg, “An interactive approach for multicriterion optimization with an application to the operation of an academic department”,Management Science 19 (1972) 357–368.
T.C. Koopmans, ed.,Activity analysis of production and allocation (Wiley, New York, 1951).
I.A. Kouada, “Linear vector maximization”, Dissertation, Rensselaer Polytechnic Institute Troy, New York (1975).
J. Philip, “Algorithms for the vector maximization problem”,Mathematical Programming 2 (1972) 207–278.
B. Roy, “Problems and methods with multiple objective functions”,Mathematical Programming 1 (1971) 239–266.
S. Zionts and J. Wallenius, “An interactive programming method for solving the multiple criteria problem”,Management Science 22 (1976) 652–663.
P.L. Yu and M. Zeleny, “The set of all nondominated solutions in linear cases and a multicriteria simplex method”,Journal of Mathematical Analysis and Application 49 (1975) 430–468.
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Ecker, J.G., Kouada, I.A. Finding all efficient extreme points for multiple objective linear programs. Mathematical Programming 14, 249–261 (1978). https://doi.org/10.1007/BF01588968
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DOI: https://doi.org/10.1007/BF01588968