Skip to main content
SpringerLink
Log in

Fractional programming

  • Papers
  • Series A: Theory
  • Published:
Zeitschrift für Operations-Research Aims and scope Submit manuscript

Abstract

Following a comprehensive bibliography recently published in this journal, we review major results in fractional programming. The emphasis is on structural properties of fractional programs and their algorithmic implications. We limit the discussion to those types of ratio optimization problems for which we see a significant interest in the applications. It is attempted to provide a theoretical framework for future research in this area.

Zusammenfassung

Im Anschluß an die kürzliche Veröffentlichung einer umfassenden Bibliographie zur Quotientenprogrammierung in dieser Zeitschrift wird in der vorliegenden Arbeit ein überblick über wichtige Ergebnisse auf diesem Gebiet der nichtlinearen Programmierung gegeben. Es stehen dabei strukturelle Eigenschaften von Quotientenprogrammen sowie deren Bedeutung für Lösungsverfahren im Mittelpunkt der Untersuchung. Die Diskussion beschränkt sich auf solche Quotientenprogramme, die für die Anwendungen von größerem Interesse sind. Es wird in der Arbeit versucht einen theoretischen Rahmen zu entwickeln, der für weitere Untersuchungen zur Quotientenprogrammierung hilfreich sein kann.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Avriel, M.: Nonlinear Programming: Analysis and Methods. Englewood Cliffs 1976.

  • Avriel, M., et al.: Introduction to Concave and Generalized Concave Functions. Generalized Concavity in Optimization and Economics. Ed. by S. Schaible and W.T. Ziemba. New York 1981, 21–50.

  • Bitran, G.R.: Experiments with Linear Fractional Problems. Naval Research Logistics Quarterly26, 1979, 689–693.

    Article  MATH  Google Scholar 

  • Bitran, G.R., andT.L. Magnanti: Duality and Sensitivity Analysis for Fractional Programs. Operations Research24, 1976, 675–699.

    Article  MathSciNet  MATH  Google Scholar 

  • Cambini, A.: An Algorithm for a Special Class of Fractional Programs. Generalized Concavity in Optimization and Economics. Ed. by S. Schaible and W.T. Ziemba. New York 1981, 491–508.

  • Chandra, S., andM. Chandramohan: A Branch and Bound Method for Integer Nonlinear Fractional Programs. Zeitschrift für Angewandte Mathematik und Mechanik60, 1980, 735–737.

    Article  MathSciNet  MATH  Google Scholar 

  • Charnes, A., andW. W. Cooper: Programming with Linear Fractional Functionals. Naval Research Logistics Quarterly9, 1962, 181–186.

    Article  MathSciNet  MATH  Google Scholar 

  • —: Deterministic Equivalents for Optimizing and Satisficing Under Chance Constraints. Operations Research11, 1963, 18–39.

    Article  MathSciNet  MATH  Google Scholar 

  • —: Goal Programming and Multi-objective Optimization, Part I. European Journal of Operational Research1, 1977, 39–54.

    Article  MathSciNet  MATH  Google Scholar 

  • Choo, E.U.: Multicriteria Linear Fractional Programming. Ph.D. Thesis, University of British Columbia, Vancouver 1980.

    Google Scholar 

  • Craven, B.D.: Mathematical Programming and Control Theory. London 1978.

  • -: Duality for Generalized Convex Fractional Programs. Generalized Concavity in Optimization and Economics. Ed. by S. Schaible and W.T. Ziemba. New York 1981, 473–489.

  • Crouzeix, J.P., J.A. Ferland, andS. Schaible: Duality in Generalized Linear Fractional Programming. Technical Report No. 399, Département d'informatique et de recherche opérationelle, Université de Montréal, 1981.

  • Deumlich, R., andK.H. Elster: Duality Theorems and Optimality Conditions for Nonconvex Optimization Problems. Mathematische Operationenforschung und Statistik, Ser. Optimization11, 1980, 181–219.

    MathSciNet  MATH  Google Scholar 

  • Dinkelbach, W.: On Nonlinear Fractional Programming. Management Science13, 1967, 492–498.

    Article  MathSciNet  MATH  Google Scholar 

  • Flachs, J., andM. Pollatschek: Equivalence Between a Generalized Fenchel Duality Theorem and a Saddle-Point Theorem for Fractional Programs. Journal of Optimization Theory and Applications37, 1982, 23–32.

    Article  MathSciNet  MATH  Google Scholar 

  • Frank, M., andP. Wolfe: An Algorithm for Quadratic Programming. Naval Research Logistics Quarterly3, 1956, 95–110.

    Article  MathSciNet  Google Scholar 

  • Geoffrion, A.M.: Solving Bi-criterion Mathematical Programs. Operations Research15, 1967, 39–54.

    Article  MathSciNet  MATH  Google Scholar 

  • —: Duality in Nonlinear Programming: A Simplified Applications-Oriented Development. SIAM Review13, 1971, 1–37.

    Article  MathSciNet  MATH  Google Scholar 

  • Gill, P.E., andW. Murray: Numerical Methods for Constrained Optimization. London 1974.

  • Granot, D., andF. Granot: On Integer and Mixed Integer Fractional Programming Problems. Annals of Discrete Mathematics1, 1977, 221–231.

    Article  MathSciNet  MATH  Google Scholar 

  • Grunspan, M.: Fractional Programming: A Survey. Technical Report No. 50, Department of Industrial and Systems Engineering, University of Florida, Gainesville 1971.

    Google Scholar 

  • Hodgson, T.J., andT.J. Lowe: Production Lot Sizing with Material Handling Cost Considerations. AIIE Transactions, 1982 (forthcoming).

  • Ibaraki, T.: Solving Mathematical Programming Problems with Fractional Objective Functions. Generalized Concavity in Optimization and Economics. Ed. by S. Schaible and W.T. Ziemba. New York 1981, 441–472.

  • —: Parametric Approaches to Fractional Programs. Department of Applied Mathematics and Physics, Kyoto University, Kyoto 1982.

    MATH  Google Scholar 

  • Ibaraki, T., et al.: Algorithms for Quadratic Fractional Programming Problems. Journal of the Operations Research Society of Japan19, 1976, 174–191.

    MathSciNet  MATH  Google Scholar 

  • Jagannathan, R.: On Some Properties of Programming Problems in Parametric Form Pertaining to Fractional Programming. Management Science12, 1966, 609–615.

    Article  MathSciNet  MATH  Google Scholar 

  • Jagannathan, R., andS. Schaible: Duality in Generalized Fractional Programming via Farkas' Lemma. Journal of Optimization Theory and Applications, to appear.

  • Kornbluth, J.S.H., andR.E. Steuer: Multiple Objective Linear Fractional Programming. Management Science27, 1981a, 1024–1039.

    Article  MATH  Google Scholar 

  • —: Goal Programming with Linear Fractional Criteria. European Journal of Operational Research8, 1981b, 58–65.

    Article  MathSciNet  MATH  Google Scholar 

  • Mangasarian, O.L.: Nonlinear Programming. New York 1969a.

  • —: Nonlinear Fractional Programming. Journal of the Operations Research Society of Japan12, 1969b, 1–10.

    MathSciNet  MATH  Google Scholar 

  • Martos, B.: Hyperbolic Programming. Naval Research Logistics Quarterly11, 1964, 135–155.

    Article  MathSciNet  MATH  Google Scholar 

  • Originally published in Math. Institute of Hungarian Academy of Science5, 1960, 383–406 (Hungarian).

  • -: Nonlinear Programming: Theory and Methods. Amsterdam 1975.

  • Meister, B., andW. Oettli: On the Capacity of a Discrete, Constant Channel. Information and Control11, 1967, 341–351.

    Article  MATH  Google Scholar 

  • Mond, B., andT. Weir: Generalized Concavity and Duality. Generalized Concavity in Optimization and Economics. Ed. by S. Schaible and W.T. Ziemba. New York 1981, 263–279.

  • Neumann, J. von: über ein ökonomisches Gleichgewichtssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergebnisse eines Mathematischen Kolloquiums. Hrsg. v. K. Menger. Leipzig-Wien 1937, Heft 8, 73–83.

  • —: A Model of General Economic Equilibrium. Review of Economic Studies13, 1945, 1–9.

    Article  Google Scholar 

  • Ohlson, J.A., andW.T. Ziemba: Portfolio Selection in a Log-normal Market when the Investor has a Power Utility Function. Journal of Financial and Quantitative Analysis11, 1976, 57–71.

    Article  Google Scholar 

  • Passy, U.: Pseudo-duality and Non-convex Programming. Generalized Concavity in Optimization and Economics. Ed. by S. Schaible and W.T. Ziemba. New York 1981, 239–261.

  • Ritter, K.: A Parametric Method for Solving Certain Nonconcave Maximization Problems. Journal of Computer and System Sciences1, 1967, 44–54.

    Article  MathSciNet  MATH  Google Scholar 

  • Schaible, S.: Beiträge zur quasikonvexen Programmierung. Doctoral Dissertation, Köln 1971.

  • —: Fractional Programming: Transformations, Duality and Algorithmic Aspects. Technical Report 73-9, Department of Operations Research, Stanford University, Stanford 1973.

    Google Scholar 

  • —: Maximization of Quasiconcave Quotients and Products of Finitely Many Functionals. Cahiers du Centre d'Etudes de Recherche Opérationelle16, 1974, 45–53.

    MathSciNet  MATH  Google Scholar 

  • —: Fractional Programming. I, Duality. Management Science22, 1976a, 858–867.

    Article  MathSciNet  MATH  Google Scholar 

  • —: Fractional Programming. II, On Dinkelbach's Algorithm. Management Science22, 1976b, 868–873.

    Article  MathSciNet  MATH  Google Scholar 

  • —: Duality in Fractional Programming: A Unified Approach. Operations Research24, 1976c, 452–461.

    Article  MathSciNet  MATH  Google Scholar 

  • —: Minimization of Ratios. J. of Optimization Theory and Applications19, 1976d, 347–352.

    Article  MathSciNet  MATH  Google Scholar 

  • —: On the Sum of a Linear and Linear-fractional Function. Naval Research Logistics Quarterly24, 1977, 691–693.

    Article  MATH  Google Scholar 

  • -: Analyse und Anwendungen von Quotientenprogrammen. Meisenheim 1978.

  • -: A Survey of Fractional Programming. Generalized Concavity in Optimization and Economics. Ed. by S. Schaible and W.T. Ziemba. New York 1981, 417–440.

  • -: Bibliography in Fractional Programming. Zeitschrift für Operations Research26 (7), 1982a.

  • —: Simultaneous Optimization of Absolute and Relative Terms. Faculty of Business Administration and Commerce, University of Alberta, Edmonton 1982b.

    MATH  Google Scholar 

  • -: Bicriteria Quasiconcave Programs. Cahiers du Centre d'Etudes de Recherche Opérationelle25, 1983 (forthcoming).

  • Schaible, S., andT. Ibaraki: Invited Review: Fractional Programming. European J. of Operational Research, 1983 (forthcoming).

  • Schaible, S., andT.J. Lowe: A Note on a Material Control Problem. Krannert Graduate School of Management, Purdue University, West Lafayette 1982.

    Google Scholar 

  • Schaible, S., andW.T. Ziemba: On the Concavity of the Sum of Lognormals is Lognormal Approximation in Portfolio Theory. Graduate School of Management, University of California, Los Angeles, 1982.

    Google Scholar 

  • Schaible, S., andW.T. Ziemba (eds.): Generalized Concavity in Optimization and Economics. New York 1981.

  • Wagner, H.M., andJ.S.C. Yuan: Algorithmic Equivalence in Linear Fractional Programming. Management Science14, 1968, 301–306.

    Article  MathSciNet  MATH  Google Scholar 

  • Warburton, A.R.: Topics of Multiple Criteria Optimization. Ph.D. Thesis, University of British Columbia, Vancouver 1981.

    Google Scholar 

  • Weber, R.: Pseudomonotonic Multiobjective Programming. Discussion Paper, Department of Economics, University of Saarland, Saarbrücken 1982.

    Google Scholar 

  • Zangwill, W.I.: Nonlinear Programming: A Unified Approach. Englewood Cliffs 1969.

Download references

Author information

Authors and Affiliations

  1. Department of Finance and Management Science, Faculty of Business, University of Alberta, T6G 2G1, Edmonton, Canada

    S. Schaible

Authors
  1. S. Schaible
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Invited survey.

This research was supported by J.D. Muir Fund, Faculty of Business, University of Alberta.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schaible, S. Fractional programming. Zeitschrift für Operations Research 27, 39–54 (1983). https://doi.org/10.1007/BF01916898

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01916898

Keywords

Search

Navigation