INTRODUCTION
Certain decision problems in OR/MS, as well as other extremum problems, give rise to the optimization of ratios. Constrained ratio optimization problems are commonly called fractional programs. They may involve more than one ratio in the objective function.
One of the earliest fractional programs (though not called so) is an equilibrium model for an expanding economy in which the growth rate is determined as the maximum of the smallest of several output-input ratios (von Neumann, 1937, 1945). Since then, but mostly after the classical paper by Charnes and Cooper (1962), some nine hundred publications have appeared in fractional programming; for comprehensive bibliographies, see Schaible (1982, 1993). Monographs solely devoted to fractional programming include Schaible (1978) and Craven (1988).
Almost from the beginning, fractional programming has been discussed in the broader context of generalized concave programming. Ratios, though not concave in general, are often still...
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References
Avriel, M., Diewert, W.E., Schaible, S., and Zang, I. (1988). Generalized Concavity, Plenum, New York.
Charnes, A. and Cooper, W.W. (1962). “Programming With Linear Fractional Functionals,” Naval Research Logistics Quarterly 9, 181–186.
Craven, B.D. (1988). Fractional Programming, Heldermann Verlag, Berlin.
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Schaible, S. (1993). “Fractional Programming,” Handbook of Global Optimization, Horst, R. and P. Pardalos, eds., Kluwer Academic Publishers, Dordrecht.
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von Neumann, J. (1937). “Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes,” Ergebnisse eines mathematischen Kolloquiums 8, Menger, K., ed., Leipzig and Vienna, 73–83.
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Schaible, S. (2001). Fractional programming . In: Gass, S.I., Harris, C.M. (eds) Encyclopedia of Operations Research and Management Science. Springer, New York, NY. https://doi.org/10.1007/1-4020-0611-X_362
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