Skip to main content
SpringerLink
Log in

On Some Fractional Programming Models Occurring in Minimum-Risk Problems

  • Conference paper
Generalized Convexity and Fractional Programming with Economic Applications

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 345))

Abstract

This paper deals with an extension of the minimum-risk criterion considered by B. Bereanu [5, 6, 7] and A. Charnes and W. W. Cooper [11] in the linear case to the nonlinear mathematical programming case. In what follows the minimum-risk criterion is applied to some special classes of nonlinear problems as are, for instance, linear Tchebysheff problems, bottlenech transportation problems, max-min (linear or linear fractional) problems with linked constraints, max-min bilinear programming problems. It is shown that, under certain hypotheses, these stochastic problems are equivalent to deterministic fractional problems. The last section examines the vectorial minimum-risk problem. The ideas discussed in this paper string together the developments given by the authors in [33–38, 43–45].

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. pp.161–167 (Russian); translated as Automat. Remote Control 41 (1980), n°7, part. 2, (1981) pp. 1017–1022.

    Google Scholar 

  2. pp. 157–170 (Russian); translated as Automat. Remote Control 42, n° 10, (1982) pp. 1409–1419.

    Google Scholar 

  3. Benson H.P., Horin T.L., The Vector Maximization Problem; Proper Efficiency and Stability. SIAM J.AppLMath. 32 (1977) 1, pp. 64 - 72.

    Google Scholar 

  4. Bereanu B., On Stochastic Linear Programming. I: Distribution Problems. A Single Random Variable. Rev.Roumaine Math. Pures Appi. 8 (4), (1963), pp.683–697.

    Google Scholar 

  5. Bereanu B., Solutii cu Risc Minim in Programarea Liniara. Studii de Statistica. Lucrarile celei de a 3-a Consfatuiri Stiintifice de Statistica 26–30 Dec. (1963). Also in: An. Univ. Bucuresti, Mat. Mec., 13, (1964), pp. 121–140.

    Google Scholar 

  6. Bereanu B., Programme de Risque Minimal en Programmation Linfoire Stochastique. C.R. Acad. Sci. Paris, 259, 5 (1964), pp. 981-983.

    Google Scholar 

  7. Bereanu B., Distribution Problems and Minimum-RM Solution in Stochastic Programming. In: Colloquium on Applications of Mathematics to Economics, Budapest (1963), pp.37–42, A. Prékopa (ed.), Publishing House of the Hungarian Academy of Sciences, Budapest 1965.

    Google Scholar 

  8. Bereanu B., Some Numerical Methods in Stochastic Linear Programming Under Risk and Uncertainty. In: Stochastic Programming, M. A. H. Dempster (ed.) Academic Press, (1980), pp. 169–205.

    Google Scholar 

  9. Bergthaller C., A Quadratic Equivalent for the Minimum-Risk Problem. Rev. Roumaine Math. Pures Appl. 15, 1, (1970) pp. 17–23.

    Google Scholar 

  10. Bitran G.R., Linear Multiple Objective Program with Zçro-One Variables. Math. Programming 13, (1977), pp.121–139.

    Google Scholar 

  11. Charnes A., Cooper W. W., Deterministic Equivalents for Optimising and Satisfying under Chance Constraints. Operations Research, 11, (1963), pp. 18–39.

    Article  Google Scholar 

  12. Choo E.U., Atkins D.R., Bicriteria Linear Fractional Programming. J.O.T.A. 36 (2), (1982), pp. 203–220.

    Article  Google Scholar 

  13. Cook W.D., Kirby M.JJL, Mehndiratta S.L., A Linear Fractional Max-Min Problem. Operations Res. 23 (3), (1975), pp. 511–521.

    Article  Google Scholar 

  14. Danskin J.M., The Theory of Max-Min. Springer, Berlin 1967.

    Google Scholar 

  15. Dinkelbach W., On Nonlinear Fractional Programming. Management Sri. 13 (7), (1967), pp.492–498.

    Google Scholar 

  16. Falk J.E., A Linear Mai-Min Problem. Math. Programming, 5, (1973), pp.169–188.

    Google Scholar 

  17. Gondraii M., Minouz M., Graphes et Algorithmes. Eyrolles, Paris, 1979.

    Google Scholar 

  18. Gupta B., Finding the Set of all Efficient Solutions for the linear Fractional Multiobjective Program with Zero-One Variables. Opsearch 18 (4), (1982), pp.204–214.

    Google Scholar 

  19. Hendrix G.C., Stedry A., The Elementary Redundancy-Optimization Problem: A Case Study in Probabilistic Multiple-Goal Programming. Operations Res. 22, (1974), pp. 639–653.

    Article  Google Scholar 

  20. Kaplan R., Soden J., On the Objective Function for the Sequential P-Model of Chance-Constrained Programming. Operations Res. 19 (1), (1971), pp. 105–114.

    Article  Google Scholar 

  21. Konno H., Bilinear Programming. Part. II. Applications of Bilinear Programming. Technical Report n°7 1–10, Operations Research House, Stanford, California, 1971.

    Google Scholar 

  22. Leciercq J.P., Résolution de Programmes Linéaires Stochastique par des Techniques Multicritères. Thèse de doctorat Faculté des Sciences de l’Université de Namur, 1979.

    Google Scholar 

  23. Leciercq J.P., La Programmation linéaire Stochastique; une Approache Multicritère. Partie I-ére. Formulation, Partie Ii-ème; Un Algorithme Interactif Adapté aux Distributions Multinormales. Cahiers Centre Etudes Rech. Opér. 23, (1), (1981) pp.31–41; ibidem 23(2), (1981), pp.121–132.

    Google Scholar 

  24. Leciercq J.P., Stochastic Programming; an Interactive Multicriteria Approach. European J. Oper. Res. 10 (1), (1982), pp.33–41.

    Google Scholar 

  25. Owen G., Teoria jocurilor (traducere din lb. engleza) Tehnica, Bucuresti, 1972.

    Google Scholar 

  26. Stancu-Minasian I.M., Nota Asupra Programdrii Stochastice eu mai Multe Functii-Obiectiv. Avind Vectorul c Aleator. Stud.Cerc.Mat.27(4), (1975), pp.453–459.

    Google Scholar 

  27. Functii Obiectiv. Stud.Cerc.Mat. 28 (1976), 5, pp. 617–623; ibidem 28 (1976), 6, pp. 723–734.

    Google Scholar 

  28. Stancu-Minasian I.M.L., Criterii Multiple in Programarea Stohastica. Doctoral thesis. Centrul de Statistica Matematica, Bucuresti, 1976.

    Google Scholar 

  29. Stancu-Minasian I.M., Asupra Probiemei lui Kataoka. stud. Cere. Mat. 28,1, (1976), pp.95–111.

    Google Scholar 

  30. Stancu-Minasian I.M., On the Multiple Minimum Risk Problem. Bull. Math. Soc.Sci. Math. R.S. Roumanie (N.S.) 23 (71), 4, (1979), pp.427–437.

    Google Scholar 

  31. Stancu-Minasian I.M., Stochastic Programming with Multiple Objective Functions. Ed. Academiei, Bucaresti and D. Reidel Publishing Company, Dordrecht, Boston, Lancester, Tokio, 1984.

    Google Scholar 

  32. Stancu-Minasian LML, Patkar V.N., A Note on Nonlinear Fractional Max-Min Problem. Nat. Acad. Sci. Letters (India) 8 (2), (1985), pp. 39–41.

    Google Scholar 

  33. Stancu-Minasian I.M., Tigan St., The Minimum Risk Approach to Special Problems of Mathematical Programming. The Distribution Function of the Optimal Value. Rev. Anal. Numer. Théor. Approx. 13 (2), (1984), pp. 175–187.

    Google Scholar 

  34. Stancu-Minasian I.M., Tigan St., The Vectorial Minimum-Risk Problem. Proceedings of the Colloquium on Approximation and Optimization, Cluj-Napoca, October 25–27, (1984), pp. 321–328.

    Google Scholar 

  35. Stancu-Minasian I.M., Tigan St., The Minimum-Risk Approach to the Bottleneck Transportation Problem. Itinerant Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca, (1985), pp. 203–208.

    Google Scholar 

  36. Stancu-Minasian I.M., Tigan St., The Minimum Risk Approach to Max-Min Bilinear Programming. An. Stiint. Univ. “Al. I.Cuza” Iasi Sect. I-a Mat.(N.S.) Tome XXXI, s. II. b, (1985), pp. 205–209.

    Google Scholar 

  37. Stancu Minasian I.M., Tigan St., The Stochastic linear-Fractional Max-Min Problem. Itinerant Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca, (1987), pp. 275–280.

    Google Scholar 

  38. Stancu-Minasian I.M., Tigan St., Criteriul Riscului Minim in Programarea Stohastica. Lucrarile Sesiunii Stiintifice a Centrului de Calcul al Universitatii Bucuresti, 20–21 februarie 1987, pp. 392–397.

    Google Scholar 

  39. Tigan St., Sur un Problème d Affectation. Mathematica 11(34), 1, (1969), pp.163–166.

    Google Scholar 

  40. Tigan St., Sur le Problème de Programmation Vectorielle Fractionnaire. Math.-Rev. Analyse Numér. Théor. Approx. 4, (1975), pp. 99–103.

    Google Scholar 

  41. Tigan St., On ttre Mai-Mia Nonlinear Fractional Problem. Rev. Anal. Numér. Théor. Approx. 9(2), (1980), pp. 283–288.

    Google Scholar 

  42. Tigan St., A Parametria Method for Max-Min Nonlinear Fractional Problems. Itinerant Seminar on Functional Equations, Approximation and Convexity 19–21 mai 1983, Cluj-Napoca, pp. 175–184.

    Google Scholar 

  43. Tigan St., Stancu-Minasian I.M., Criteriul Riscului Minim Pentru Problema Cebisev. Lucrarile celui de al IV-lea simpozion “Modelarea cibernetica a proceselor de productie” 26–28 mai 1983, ASE-Bucuresti, vol. I, pp. 338–342.

    Google Scholar 

  44. Tigan St., Stancu-Minasian I.M., The Stochastic Bottleneck Transportation Problem. Rev. Anal. Numér. Théor, Approx. 12 (2), (1985), pp. 153–158.

    Google Scholar 

  45. Tigan St., Stancu Minasian I.M., The Stochastic Max-Min Problem. Cahiers Centre Etudes Rech. Opér. 27 (3–4), (1985), pp. 247–254.

    Google Scholar 

  46. Weber R., Pseudomonotonic Multiobjective Programming. Cahiers Centre Etudes Rech. Opér. 25, (1983), pp. 115–128.

    Google Scholar 

  47. Techiali U., A Stochastic Bottleneck Assignment Problem. Management Sei., 14(11), (1968), pp. 732–734.

    Google Scholar 

  48. Yechiftli U., A Note on a Stochastic Production Maximizing Transportation Problem. Naval Res. Logist. Quart., 18 (3), (1971), pp. 429–431.

    Google Scholar 

  49. Zoutendijk G., Method of Feasible Directions. Elsevier, Amsterdam, 1960.

    Google Scholar 

Download references

Authors
  1. I. M. Stancu-Minasian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Tigan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Statistics and Applied Mathematics, University of Pisa, Via Ridolfi 10, I-56100, Pisa, Italy

    Alberto Cambini  & Laura Martein  & 

  2. Institute of Quantitative Methods, Bocconi University, Via Sarfatti 25, I-20136, Milano, Italy

    Erio Castagnoli

  3. Institute of Mathematics, University of Verona, Via dell’Artigliere 10, I-37129, Verona, Italy

    Piera Mazzoleni

  4. Graduate School of Management, University of California, 92521, Riverside, California, USA

    Siegfried Schaible

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Stancu-Minasian, I.M., Tigan, S. (1990). On Some Fractional Programming Models Occurring in Minimum-Risk Problems. In: Cambini, A., Castagnoli, E., Martein, L., Mazzoleni, P., Schaible, S. (eds) Generalized Convexity and Fractional Programming with Economic Applications. Lecture Notes in Economics and Mathematical Systems, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46709-7_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-46709-7_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52673-5

  • Online ISBN: 978-3-642-46709-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation