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...
- Avriel, M., Diewert, W.E., Schaible, S., and Zang, I. (1988). Generalized Concavity, Plenum, New York.Google Scholar
- Charnes, A. and Cooper, W.W. (1962). “Programming With Linear Fractional Functionals,” Naval Research Logistics Quarterly 9, 181–186.Google Scholar
- Craven, B.D. (1988). Fractional Programming, Heldermann Verlag, Berlin.Google Scholar
- Dinkelbach, W. (1967). “On Nonlinear Fractional Programming,” Management Science 13, 492–498.Google Scholar
- Martos, B. (1975). Nonlinear Programming: Theory and Methods, North-Holland, Amsterdam.Google Scholar
- Schaible, S. (1976). “Duality in Fractional Programming: A Unified Approach,” Operations Research 24, 452–461.Google Scholar
- Schaible, S. (1978). Analyse und Anwendungen von Quotientenprogrammen, Mathematical Systems in Economics 42, Hain-Verlag, Meisenheim.Google Scholar
- Schaible, S. (1982). “Bibliography in Fractional Programming,” Zeitschrift fur Operations Research 26, 211–241.Google Scholar
- Schaible, S. (1990). “Multi-Ratio Fractional Programming — Analysis and Applications,” Proceedings of 13th Annual Conference of Associazione per la Matematica Applicata alle Scienze Economiche e Sociali, Verona/Italy, September 1989, Mazzoleni, P., ed., Pitagora Editrice, Bologna, 47–86.Google Scholar
- Schaible, S. (1993). “Fractional Programming,” Handbook of Global Optimization, Horst, R. and P. Pardalos, eds., Kluwer Academic Publishers, Dordrecht.Google Scholar
- Schaible, S. and Ibaraki, T. (1983). “Fractional Programming,” European Jl. Operational Research 12, 325–338.Google Scholar
- 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. Google Scholar
- von Neumann, J. (1945). “A Model of General Economic Equilibrium,” Review Economic Studies 13, 1–9.Google Scholar