Solving Bicriterion Mathematical Programs

Arthur M. Geoffrion
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Arthur M. Geoffrion: University of California, Los Angeles, California

Operations Research, 1967, vol. 15, issue 1, 39-54

Abstract: It often happens in applications of mathematical programming that there are two incommensurate objective functions to be extremized, rather than just one. One thus encounters bicriterion programs of the form of equation (1), \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mbox{maximize}_{x \epsilon X}\ h[f_{1}(x),f_{2}(x)],$$\end{document} where h is an increasing utility function, preferably quasiconcave, defined on outcomes of the concave objective functions f 1 and f 2 , and x is a decision n -vector constrained to the convex set X . It is shown how such programs can be numerically solved if a parametric programming algorithm is available for the parametric subproblem \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mbox{maximize}_{x \epsilon X}\ \alpha f_{1}(x) + (l-\alpha)f_{2}(x).\quad(0\leq \alpha \leq 1)$$\end{document} A natural byproduct of the calculations is a relevant portion of the “tradeoff curve” between f 1 and f 2 . Outlines of several algorithms for solving equation (1) under various special assumptions and a numerical example are presented to illustrate the application of the theory developed herein. A useful extension is presented that permits nonlinear scale changes to be made on the f ı .

Date: 1967
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