Abstract
An introduction to ratio optimization problems is provided which covers various applications as well as major theoretical and algorithmic developments. In addition to an extensive treatment of single-ratio fractional programming, three types of multi-ratio fractional programs are discussed: maximization of the smallest of several ratios, maximization of a sum of ratios and multi-objective fractional programs. Earlier as well as recent developments are discussed and open problems are identified. The article concludes with a comprehensive, up-to-date bibliography in fractional programming. Well over one thousand articles have appeared in more than thirty years of increasingly intensive research in fractional programming. The bibliography includes all references from the beginning until late 1993 to the extent they are known to the author at this time.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Bibliography for Fractional Programming
Abadie, J.M. and Williams, A.C. (1963), Dual and Parametric Methods in Decomposition, in Graves, R. and Wolfe, P., (eds.), Recent Advances in Mathematical Programming, McGraw–Hill, New York, 149–158.
Abrham, J. and Buie, R.N. (1980), Duality in Continuous Fractional Programming, Utilitas Mathematica 17, 35–44.
Abrham, J. and Luthra, S. (1977), Comparison of Duality Models in Fractional Linear Programming, Zeitschrift für Operations Research 21, 125–130.
Aggarwal, R.C. (1969), A New Approach to Planning and Programming in Agriculture Linear Fractional Functionals Programming, Indian J. of Agriculture Economics 24, 24–25.
Aggarwal, S. and Bhatia, D. (1991), Duality for Nondifferentiable Nonlinear Programming Problems: A Feasible Direction Approach, Opsearch 28 (4), 282–290.
Aggarwal, S., Bhatia, D. and Lau, N. (1991), Duality in Multiple Right–Hand Choice Linear Fractional Problems, J. of Information and Optimization Sciences 12 (1), 13–24.
Aggarwal, S.P. (1966), Stability of the Solution to a Linear Fractional Functional Programming Problem, Zeitschrift für Angewandte Mathematik und Mechanik 46, 343–349.
Aggarwal, S.P. (1967), Contributions to the Theory of Fractional Programming, PhD. Thesis, Delhi University.
Aggarwal, S.P. (1968), A Note on Quasiconvex Programming, Metrika 12, 97105.
Aggarwal, S.P. (1968), Parametric Linear Fractional Functionals Programming, Metrika 12, 1968, 106–114.
Aggarwal, S.P. (1968), Standard Error Fractional Functional Programming, Istanbul Universitesi Fen Fakultesi Mecmuasi, A Serisi 30, 45–51.
Aggarwal, S.P. (1970), Analyses of the Solution to a Linear Fractional Functionals Programming, Metrika 16, 9–26.
Aggarwal, S.P. (1972), Indefinite Quadratic Fractional Programming, Ekonomicko–Matematicky Obzor 8, 191–199.
Aggarwal, S.P. (1972), Quadratic Fractional Functionals Programming with Non–Linear Constraints, Ricerca Operativa 2, 51–53. [Italian]
Aggarwal, S.P. (1972), Transportation Technique for Quadratic Fractional Programming, Revue Belge de Statistique d’ Informatique et de Recherche Operationelle 12 (2), 3–7.
Aggarwal, S.P. (1972), Variation in Parameters of Quadratic Fractional Functional Programming, Revue Belge de Statistique d’ Informatique et de Recherche Operationelle 11 (4), 3–12.
Aggarwal, S.P. (1973), Indefinite Quadratic Fractional Programming with a Quadratic Constraint, Cahiers du Centre d’Etudes de Recherche Operationelle 15, 405–410.
Aggarwal, S.P. (1973), Quadratic Fractional Functionals Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 15, 157–165.
Aggarwal, S.P. (1973), Upper Bounds and Quadratic Fractional Functional Programming, Revue Belge de Statistique d’ Informatique et de Recherche Operationelle 12 (4), 17–21.
Aggarwal, S.P. and Arora, S. (1974), A Special Class of Non–Linear Fractional Functional Programming Problems, SCIMA Journal of Management Science and Applied Cybernetics 3, 30–39.
Aggarwal, S.P. and Parkash, O. (1978), Duality in General Linear Fractional Functional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 20, 75–81.
Aggarwal, S.P. and Patkar, V.N. (1978), Dual of a Linear Fractional Program Through Geometric Programming, Portugaliae Mathematica 37 (1–2), 8186.
Aggarwal, S.P. and Saxena, P.C. (1974), Duality Theorems for Fractional Functional Programming with Quadratic Constraint, Ekonomicko Matematicky Obzor 10, 86–92.
Aggarwal, S.P. and Saxena, P.C. (1975), Duality Theorems for Non–Linear Fractional Programs, Zeitschrift für Angewandte Mathematik und Mechanik 55, 523–525.
Aggarwal, S.P. and Saxena, P.C. (1976), The Decomposition Method for Linear Programming Problems with Linear and Fractional Target Functions, Przeglad Statyst 23, 211–219.
Aggarwal, S.P. and Saxena, P.C. (1979), A Class of Fractional Functional Programming Problems, New Zealand Operational Research 7, 79–90.
Aggarwal, S.P. and Sharma, I.C. (1970), Maximization of the Transmission Rate of a Discrete Constant Channel, Unternehmensforschung 14, 152155.
Aggarwal, S.P. and Swamp, K. (1966), Fractional Functionals Programming with a Quadratic Constraint, Operations Research 14, 950–956.
Aggarwal, V., Aneja, Y.P. and Nair, K.P.K. (1981), Ratio Rewards in Networks, RAIRO Rech. Oper. 15 (2), 129–138.
Aggarwal, V., Chandrasekaran R. and Nair, K.P.K. (1977), Markov Ratio Decision Processes, Journal of Optimization Theory and Applications 21, 27–37.
Aggarwal, V., Chandrasekaran, R. and Nair, K.P.K. (1980), Discounted Stochastic Ratio Games, SIAM J. Algebraic Discrete Methods 1 (2), 201210.
Aggarwal, V., Nair, K.P.K. and Chandrasekaran, R. (1980), Non Terminating Stochastic Ratio Game, RAIRO Rech. Oper. 14 (1), 21–30.
Agrawal, S.C. (1974), A Primal Integer Programming Algorithm with Parabolic Constraints, Opsearch 11, 59–80.
Agrawal, S.C. (1975), On Integer Solutions to Linear Fractional Functional Programming Problems, Acta Ciencia Indica 1, 203–208.
Agrawal, S.C. (1976), On Integer Solutions to Linear Fractional Functionals by a Branch and Bound Technique, Acta Ciencia Indica 2, 75–78.
Agrawal, S.C. (1977), An Alternative Method on Integer Solutions to Linear Fractional Functionals by a Branch and Bound Technique, Zeitschrift far Angewandte Mathematik and Mechanik 57, 52–53.
Agrawal, S.C. and Chand, M. (1978), On Integer Solutions to Complementary Programming Problems with Linear Fractional Objective Function by a Branch and Bound Technique, Acta Ciencia Indica 4, 283–289.
Agrawal, S.C. and Chand, M. (1979), On Intersection Cuts in Fractional Interval Integer Programming, Acta Ciencia Indica 5, 140–142.
Agrawal, S.C. and Chand, M. (1979/80), On Mixed Integer Solutions to Complementary Programming Problems with Linear Fractional Objective Function, Alighar Bull. Math. 9, 21–30.
Agrawal, S.C. and Chand, M. (1980), A Note on the Sum of Linear and Fractional Interval Programming, Revista de Informatica e Investigacion Operativa 20, 33–36.
Agrawal, S.C. and Chand, M. (1980), On Integer Solutions to Complementary Programming Problems with Linear Fractional Objective Functions, Ricerca Operativa 10 (13), 19–30.
Agrawal, S.C. and Chand, M. (1981), A Note on Integer Solutions to Linear Fractional Interval Programming Problems by a Branch & Bound Technique, Naval Research Logistics Quarterly 28 (4), 671–677.
Agrawal, S.C. and Chand, M. (1985), An Alternative Method on Integer Solutions to Complementary Programming Problems with Linear Fractional Objective Function by a Branch and Bound Technique, Acta Ciencia Indica Mathematics 11 (3), 213–221.
Agrawal, S.C. and Verma, R.K. (1980), p–Variables Replacement in Linear Fractional Functional Programming, Acta Ciencia Indica 6, 95–103.
Agrawal, S.C. and Verma, R.K. (1981), A Suboptimization Method for the Sum of Linear and Linear Fractional Interval Programming Problems, Acta Ciencia Indica Mathematics 7 (1–4), 14–24.
Agrawal, S.C. and Verma, R.K. (1981), On Explicit Solutions for Linear Fractional Interval Programming Problems, Revista de Informatica e Investigacion Operativa 11, 97–114.
Agrawal, S.C. and Verma, R.K. (1983), On the Solutions of the Sum of Linear and Linear Fractional Interval Programming Problems, Pure and Applied Mathematica Sciences 17 (1–2), 73–81.
Agrawal, U., Swarup, K. and Garg, K.C. (1984), Goal Programming Problem with Linear Fractional Objective Function, Cahiers du Centre d’Etudes de Recherche Opérationelle 26, 33–41.
Ahuja, R.K. (1988), Minimum Cost–Reliability Ratio Path Problem, Computers and Operations Research 15 (1), 83–89.
Ahuja, R.K., Bata, J.L. and Gupta, S.K. (1983), Combinatorial Optimization with Rational Objective Functions: A Communication, Mathematics of Operations Research 8 (2), 314.
Ali, A.I. (1991), Strict vs Weak Ordinal Relations for Multipliers in Data Envelopement Analysis, Management Science 37 (6), 733–738.
Almogy, Y. and Levin, O. (1970), Parametric Analysis of a Multi–stage Stochastic Shipping Problem, in Lawrence, J., (ed.), Operational Research ‘69, Tavistock Publications, London, 359–370.
Almogy, Y. and Levin, O. (1971), A Class of Fractional Programming Problems, Operations Research 19, 57–67.
Almogy, Y. and Levin, O. (1971), The Fractional Fixed–Charge Problem, Naval Research Logistics Quarterly 18, 307–315.
Anand, P. (1971), Dual and Parametric Methods in Decomposition for Linear Fractional Programs, Studia Scientarium Mathematicarum Hungarica 6, 267–275.
Anand, P. (1973), Decomposition Procedure for Linear Fractional Programs with Upper Bounds, Zeitschrift für Angewandte Mathematik and Mechanik 53, 635–636.
Anand, P. and Swarup, K. ( 1970), The Procedure for Local Separable Programming, Zeitschrift für Angewandte Mathematik und Mechanik 50, 320–321.
Aneja, Y.P. and Nair, K.P.K. (1984), Ratio Dynamic Program, Operations Research Letters 3, 167–173.
Anstreicher, K M. (1986), A Monotonic Projective Algorithm for Fractional Linear Programming, Algorithmica 1 (4), 483–498.
Anstreicher, K M. (1989), A Combined Phase I–Phase H Projective Algorithm for Linear Programming, Mathematical Programming 43 (2), 209–223.
Anzai, Y. (1974), On Integer Fractional Programming, Journal of the Operations Research Society of Japan 17, 49–66.
Arbuzova, N.I. (1968), Interdependence of the Stochastic e–Stabilities of Linear and Linear Fractional Programming Problems of a Special Form, Ekonom. i Mat. Metody 4, 108–110. [Russian]
Arisawa, S. and Elmaghraby, S.E. (1972), Optimal Time–Cost Trade Offs in GERT–Networks, Management Science 18, 589–599.
Armstrong, R. et al. (1980), Effective Solution of Non–Convex Multi–Objective Ratio Goal Problems, Research Report CCS 390, Center for Cybernetic Studies, University of Texas, Austin.
Armstrong, R., Charnes, A. and Haksever, C. (1987), Successive Linear Programming for Ratio Goal Problems, European J. of Operational Research 32 (3), 426–434.
Armstrong, R., Charnes, A. and Haksever, C. (1988), Implementation of Successive Linear Programming Algorithms for Non–Convex Goal Programming, Computers and Operations Research 15, 37–49.
Arora, S. R., Puri, M.C. and Swarup, K. (1977), Cutting Plane Technique for Set Covering Problem with Linear Fractional Functional, Zeitschrift für Angewandte Mathematik und Mechanik 57, 597–602.
Arora, S.R. (1977), A Note on Fractional Fixed Charge Problems, New Zealand Operational Research 5, 66–71.
Arora, S.R. (1977), A Set Partitioning Problem with Linear Fractional Objective Function, Indian Journal of Pure and Applied Mathematics 8, 961–968.
Arora, S.R. and Aggarwal, S.P. (1977), Dynamic Programming Approach to Linear Fractional Functional Programming, Revue Belge de Statistique d’Informatique et de Recherche Operationelle 17, 10–23.
Arora, S.R. and Aggarwal, S.P. (1977), Linear Fractional Functionals Programming with a Parameter in an Activity Vector, Economic Computation and Economic Cybernetics Studies and Research 3, 37–56.
Arora, S.R. and Puri, M.C. (1977), Enumeration Technique for the Set Covering Problem with a Linear Fractional Functional as the Objective Function, Zeitschrift für Angewandte Mathematik und Mechanik 57, 181186.
Arora, S.R., Puri, M.C. and Swamp, K. (1977), The Set Covering Problem with Linear Fractional Functionals, Indian Journal of Pure and Applied Mathematics 8, 578–588.
Arsham, H. and Kahn, A.B. (1990), A Complete Algorithm for Linear Fractional Programs, Computers and Mathematics with Applications 20 (7), 11–23.
Artjuhin, A.V. (1973), An Algorithm for the Solution of the Distribution Problem of Parametric Fractional Linear Programming, lzdat. “Him”, Frunze, 30–36. [Russian]
Artjuhin, A.V. (1973), Some Applications of Parametric Fractional Linear Programming, lzdat., “Ilim”, Frunze, 37–54. [Russian]
Ashton, D.J. and Atkins, D.R. (1979), Multi–Criteria Programming for Financial Planning, Journal of the Operational Research Society 30, 259270.
Averbakh, I.L. (1990), An Additive Method for Optimization of Two–Stage Stochastic Systems with Discrete Variables, Izvestiya Akademii Nauk SSR. Tekhnicheskaya Kibernetika 1, 162–166. [Russian]; Translation: Soviet Journal of Computer and Systems Sciences 28 (4), 161–165.
Averochkin, V.A., Veranov, P.Y. and Tokolov, V.S. (1987), A Recursive Filter which Maximizes the Output Signal–to–Noise Ratio, Soy. J. Commun. Technol. Electron. (USA) 32 (8), 115–118. Translation of Radiotekh Elektron [USSR]
Avriel, M. (1976), Nonlinear Programming: Analysis and Methods, Prentice–Hall, Englewood–Cliffs, N.J.
Avriel, M. and Schaible, S. (1978), Second–Order Characterizations of Pseudo–Convex Functions, Mathematical Programming 14, 170–185.
Avriel, M., Diewert, W.E., Schaible, S. and Zang, I. (1988), Generalized Concavity, Plenum Press, New York–London.
Avriel, M., Diewert, W.E., Schaible, S. and Ziemba, W.T. (1981), Introduction to Concave and Generalized Concave Functions, in Schaible, S. and Ziemba, W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 21–50.
Awerbuch, S., Ecker, J.G. and Wallace, W.A. (1976), A Note: Hidden Nonlinearities in the Application of Goal Programming, Management Science 22, 918–920.
Aylawadi, D.R. (1977), Duality for Homogeneous Fractional Programming with Nonlinear Constraints, Journal of Mathematical Sciences 12 (13), 2932.
Babayev, D.A. (1974), Methode der Lösung einer Klasse Nichtlinearer Programmierungsprobleme, Akademiya Nauk Azerbaidzan SSR. Dokl. 30 (9), 3–6.
Babayev, D.A. (1975), Mathematical Models for Optimal Timing of Drilling on Multilayer Oil and Gas Fields, Management Science 21, 1361–1369.
Babayev, D.A. (1976), A General Fractional Programming Problem, Central Ekonom.–Mat. Inst., Akad, Nauk SSSR, Moscow. [Russian]
Bakhshi, H. C. (1979/80), A Study of Sensitivity in Extreme Point Linear Fractional Functional Programming Problems, J. Math. Sci. 14–15.
Bakhshi, H.C. (1979), Sensitivity Analysis in Linear Fractional Functionals Programming Problems with Extreme Point Restriction, SCIMA Journal of Management Science and Applied Cybernetics 8, 6–15.
Bakhshi, H.C. and Puri, M.C. (1979), An Efficient Technique for Extreme Point Mathematical Programming Problems, Cahiers du Centre d’ Etudes de Recherche Operationelle 21, 257–268.
Baluta, M., Dobrescu, V. and Paun, G. (1979), Algorithm for Solving a Combinatorial Problem of Optimal Selection, Economic Computation and Economic Cybernetics Studies and Research 1, 87–95.
Banker, R.D. (1980), A Gametheoretic Approach to Measuring Efficiency, European Journal of Operational Research 5, 262–266.
Banker, R.D., Charnes A. and Cooper, W.W. (1984), Some Models for Estimating Technical and Scale Inefficiencies in Data Envelope Analysis, Management Science 30, 1078–1092.
Banker, R.D., Charnes, A., Cooper, W.W. and Schinnar, A.P. (1981), A BiExtremal Principle for Frontier Estimation and Efficiency Evaluations, Management Science 27, 1370–1382.
Bansal, S. (1981), Absolute Value Linear Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 23, 43–52.
Bansal, S. (1982), On Extreme Point Programming Problems, Journal of Information and Optimization Sciences 3, 173–184.
Barlow, R.E. and Proschan, F. (1965), Mathematical Theory of Reliability, J. Wiley, New York.
Barrodale, I. (1973), Best Rational Approximation and Strict Quasiconvexity, SIAM J. of Numerical Analysis 10, 8–12.
Barrodale, I., Powell, M.J.D. and Roberts, F.D.K. (1972), The Differential Correction Algorithm for Rational Lam–Approximation, SIAM J. Numerical Analysis 9, 493–504.
Bayalinov, E.B. (1981), On a Question of Duality in Fractional–Linear Programming. lzvestiya Akademii Nauk Kirgizskoi SSR 2, 10–20, 101. [Russian]
Bayalinov, E.B. (1982), The Effect of Changes in the Conditions of a Linear Fractional Programming Problem on the Optimum of the Target Function, Application of Mathematical Economics Methods in Improving Control of the Economy,’ihm’, Frunze, 144–155. [Russian]
Bayalinov, E.B. (1986), A Method of Successive Reduction of Residues for Solving a Problem of Fractional–Linear Programming, Application of Mathematics in Economics, ‘Ilim’, Frunze, 153–162, 194. [Russian]
Bayalinov, E.B. (1986), Second Algorithm for a Method of Successive Improvement of a Plan in Fractional–Linear Programming, Applications of Mathematics in Economics, 143–153, 194. [Russian]
Bayalinov, E.B. (1987), Economic Interpretation of Dual Estimates in Linear Fractional Programming, Izvestiya Akademii Nauk Kirgizskoi SSR 3 , 8–11, 81. [Russian]
Bayalinov, E.B. (1988), Coincidence of Optimal Plans for Problems of Linear and Linear–Fractional Programming, Izvestiya Akademii Nauk Kirgizskoi SSR 3 , 9–15, 91. [Russian]
Bayalinov, E.B. (1988), On a System of Interconnected Dual Estimates, Izvestiya Akademii Nauk Kirgizskoi SSR 2 , 8–12, 91. [Russian]
Bayalinov, E.B. (1988), The Economic Meaning of Dual Variables in Linear– Fractional Programming, Ekonomika i Matematicheskie Metody 24 (3), 558–561. [Russian]
Bazaara, M.S. and Shetty, C.M. (1979), Nonlinear Programming, Theory and Algorithms, J. Wiley, New York.
Beale, E.M.L. (1980), Fractional Programming with Zero–One Variables, in Fiacco, A.V. and Kortanek, K.O., (eds.), Extremal Methods and Systems Analysis, Springer, Berlin, 430–432.
Beato–Moreno, A. and Ruiz–Canales, P. (1991), A Comparison of the Efficiency of Two Methods for Integral Fractional Linear Programming, [Spanish, English Summary], XV Jornades Luso–Espanholas de Matematica, Vol. IV (Proceedings of the XVth Portugese–Spanish Conference on Mathematics), 355–360. [Spanish]
Becher, O. (1965), Das Problem der Optimalen Routenwahl einer Transporteinheit bei Beschäftigung mit Gelegenheitsfahrten, Zeitschrift für die gesamte Staatswissenschaft 121, 196–221.
Bector, C.R. (1968), Certain Aspects of Duality in Nonlinear Indefinite Functional Programming, Ph.D. Thesis, Dept. of Mathematics, I.I.T. Kanpur (India).
Bector, C.R. (1968), Duality in Fractional and Indefinite Programming, Zeitschrift für Angewandte Mathematik und Mechanik 48, 418–420.
Bector, C.R. (1968), Nonlinear Fractional Functional Programming with Nonlinear Constraints, Zeitschriftfür Angewandte Mathematik und Mechanik 48, 284–286.
Bector, C.R. (1968), Programming Problems with Convex Fractional Functions, Operations Research 16, 383–391.
Bector, C.R. (1970), Some Aspects of Nonlinear Indefinite Fractional Functional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 12, 22–34.
Bector, C.R. (1970), Some Aspects of Quasi–Convex Programming, Zeitschrift für Angewandte Mathematik und Mechanik 50, 495–502.
Bector, C.R. (1971), Indefinite Quadratic Fractional Functional Programming, Metrika 18, 21–30.
Bector, C.R. (1973), Duality in Linear Fractional Programming, Utilitas Mathematica 4, 155–168.
Bector, C.R. (1973), Duality in Nonlinear Fractional Programming, Utilitas Mathematica 4, 8193.
Bector, C.R. (1973), Duality in Nonlinear Fractional Programming, Zeitschrift für Operations Research 17, 183–193.
Bector, C.R. (1973), On Convexity, Pseudo–Convexity and Quasi–Convexity of Composite Functions, Cahiers du Centre d’Etudes de Recherche Operationelle 15, 411–428.
Bector, C.R. (1974), A Note on a Dual Fractional Program, Cahiers du Centre d’Etudes de Recherche Operationelle 16, 107–115.
Bector, C.R. and Bector, M.K. (1989), Fritz John Sufficient Optimality Conditions and Duality for Generalized Minimax Program, Journal of Information and Optimization Sciences 10 (1), 193–205.
Bector, C.R. and Bhatia, B.L. (1984), An Optimization Theorem with Applications in some Mathematical Programming Problems, Utilitas Mathematica 26, 249–258.
Bector, C.R. and Bhatia, B.L. (1985), Duality for a Multiple Objective Nonlinear Nonconvex Program, Utilitas Mathematica 28, 175–192.
Bector, C.R. and Bhatia, B.L. (1985), Generalized Duality for Nonlinear Programming in Complex Space, Econom Comput. Econom. Cybernet. Stud. Res., 20 (2), 75–80.
Bector, C.R. and Bhatia, B.L. (1986), Nature of RENYI’s Entropy and Associated Divergence Function, Naval Res. Logist. 34 (4), 741–746.
Bector, C.R. and Bhatt S.K. (1976), A Linerization Technique for Solving Interval Linear Fractional Programs, in: Proceedings of the Fifth Manitoba Conference on Numerical Mathematics, Congressus Numerantium XVI, Winnipeg, Utilitas Mathematica, 221–229.
Bector, C.R. and Bhatt, S.K. (1977), Duality for a Nonconvex Program in a Real Banach Space, Zeitschrift für Angewandte Mathematik und Mechanik 57 (3), 193–194.
Bector, C.R. and Bhatt, S.K. (1978), Pseudo–Monotonic Interval Programming, Naval Research Logistics Quarterly 25, 309–314.
Bector, C.R. and Bhatt, S.K. (1985), Nonlinear Programming in Complex Space: Necessary and Sufficient Conditions, Revue Roumaine de Mathematiques Pures et Appliquees 30 (7), 497–503.
Bector, C.R. and Cambini, A. (1990), Fractional Programming—Some Recent Results. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 86–98.
Bector, C.R. and Chandra, S. (1986), Duality for Pseudolinear Minmax Programs, Asia–Pacific J. of Operational Research 2, 86–94.
Bector, C.R. and Chandra, S. (1986), First and Second Order Duality for a Class of Nondifferentiable Fractional Programming Problem, Journal of Information and Optimization Sciences 7 (3), 335–348.
[ 137]Bector, C.R. and Chandra, S. (1986), Second Order Duality for GeneralizedFractional Programming, Methods of Operations Research 56, 11–28.
Bector, C.R. and Chandra, S. (1987), (Generalized) Bonvexity and Higher Order Duality for Fractional Programming, Opsearch 24 (3), 143–154.
Bector, C.R. and Chandra, S. (1992), Generalized Continuous Fractional Programming Duality: A Parametric Approach, Utilitas Mathematica 42, 39–60.
Bector, C.R. and Grover, T.R. (1973), Minimizing Certain Nonconvex Quadratic Fractional Programs by Ranking the Extreme Points, in: Proceedings of the Second Manitoba Conference on Numerical Mathematics, Congressus Numerantium, Winnipeg, Utilitas Mathematica, 95–100.
Bector, C.R. and Jolly, P.L. (1979), Pseudo Monotonic Integer Programming, Proceedings of the Manitoba Conference on Numerical Mathematics and Computing, 211–218.
Bector, C.R. and Jolly, P.L. (1984), Programming Problems with Pseudomonotonic Objectives, Math. Operationsforschung Statist. Ser. Optimization 15 (2), 217–229.
Bector, C.R. and Kumar, U. (1984), Duality for Multiple Objective Fractional Programs, Cahiers du Centre d’Etudes de Recherche Operationelle 26 (34), 201–207.
Bector, C.R. and Suneja, S.K. (1988), Duality in Generalized Fractional Programming Involving Nondifferentiable Functions, Congressus Numerantium 62, 159–164.
Bector, C.R. and Suneja, S.K. (1988), Duality in Nondifferentiable Generalized Fractional Programming, Asia Pacific Journal of Operational Research 5 (2), 134–139.
Bector, C.R., Bector, M.K. and Klassen, J.E. (1977), Duality for a Nonlinear Programming Problem, Utilitas Mathematica 11, 87–99.
Bector, C.R., Bhatia, D. and Aggarwal, S. (1992), Multiobjective Fractional Programming Duality: A Nondifferentiable Case, Congressus Numerantium 87, 77–85.
Bector, C.R., Chandra, S. and Bector, M.K. (1985), Optimality Conditions and Duality for Minmax Programs under Generalized Convexity, Research Report No. 85–20, Department of Actuarial and Management Sciences, The University of Manitoba.
Bector, C.R., Chandra, S. and Bector, M.K. (1989), Generalized Fractional Programming Duality: A Parametric Approach, Journal of Optimization Theory and Applications 60 (2), 243–260.
Bector, C.R., Chandra, S. and Bector, M.K. (1991), Duality for Minmax Problems, Congressus Numerantium 80, 33–48.
Bector, C.R., Chandra, S. and Durga–Prasad, M.V. (1988), Duality in Pseudolinear Multiobjective Programming, Asia–Pacific J. Oper. Res. 5 (2), 150–159.
Bector, C.R., Chandra, S. and Durga–Prasad, M.V. (1988), Nonlinear Fractional Programming Duality for Hanson–Mond Class of Functions with Nondifferentiable Terms, Congressus Numerantium 62, 165–170.
Bector, C.R., Chandra, S. and Gulati, T.R. (1974), Duality for Complex Nonlinear Fractional Programming Over Cones, in: Proceedings of the Third Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica, 87–103.
Bector, C.R., Chandra, S. and Gulati, T.R. (1976), Duality for Fractional Control Problems, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica, 231–241.
Bector, C.R., Chandra, S. and Gulati, T.R. (1977), A Lagrangian Approach toDuality for Complex Nonlinear Fractional Programming Over Cones, Mathematische Operationsforschung and Statistik, Series Optimization 8, 17–25.
Bector, C.R., Chandra, S. and Singh, C. (1990), Duality in Multiobjective Fractional Programming. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 232–241.
Bector, M.K., Husain, I., Chandra, S. and Bector, C.R. (1988), A Duality Model for a Generalized Minmax Program, Naval Research Logistics 35 (5), 493–501.
Bedi, M.K. (1978), Duality for a Special Class of Quasi–Convex Programming Problems, Zeitschrift für Angewandte Mathematik and Mechanik 58, 165166.
Behringer, F.A. (1977), Lexicographic Quasiconcave Multiobjective Programming, Zeitschrift für Operations Research 21, 103–116.
Belen ‘Kij, A.S. (1980), Minimax Problem with Linear Constraints, Automation and Remote Control 41, 562–568; translated from Autom. Telemekh 4, 151–158. [Russian]
Belenova, N.K. and Kim, K.V. (1984), On Algorithms for Solving a Balance Problem with a Uniform Criterion. Algorithms for Solving Network Problems, Akad. Nauk SSSR, Tsentral. Ekonom. Mat. Inst., Moskow, 2532. [Russian]
Bell, E.J. (1965), Primal–Dual Decomposition Programming, Ph.D. Thesis, Operations Research Center, University of California, Berkeley.
Bellman, R. (1957), Dynamic Programming, Princeton University Press, Princeton.
Belov, J.A. (1981), Two–Level Optimization for a Single–Step Stochastic Programming Problem with Probability Constraints, Kievskii Gosudarstvennyi Universitet. Mezhvedomstvennyi Nauchnyi Sbornik. Vychislitelnaya i Prikladnaya Matematika 43, 78–85, 158. [Russian]
[ 165]Belykh, V.M. and Gavurin, M.K. (1980), Minimization Algorithm for a Linear–Fractional Function,Vestnik Leningradskogo Universiteta, Matematika, Mehanika, Astronomija 4, 10–15. [Russian]
Belykh, V.M. and Gavurin, M.K. (1981), An Algorithm for Minimization of a Fractional–Linear Function, Mathematische Operationsforschung und Statistik, Serie Optimization 12 (1), 10–15, 115. [Russian]
Benadada, Y. (1989), Approches de Résolution du Problème de Programmation Fractionaire Généralisée, Thèse de Doctorat, Université de Montréal.
Benadada, Y. and Ferland, J.A. (1988), Partial Linearization for Generalized Fractional Programming, Zeitschrift für Operations Research 32 (2), 101–106.
Benadada, Y., Crouzeix, J.P. and Ferland, J.A. (1990), An Interval–Type Algorithm for Generalized Fractional Programming. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 106–120.
Benadada, Y., Crouzeix, J.P. and Ferland, J.A. (1993), Rate of Convergence of a Generalization of Newton’s Method, Journal of Optimization Theory and Applications 78, 599–604.
Benson, H.P. (1985), Finding Certain Weakly–Efficient Vertices in Multiple Objective Linear Fractional Programming, Management Science 31 (2), 240–245.
Beoni, C. (1986), A Generalization of Fenchel Duality Theory, Journal of Optimization Theory and Applications 49 (3), 375–387.
Bereanu, B. (1963), Distribution Problems in Stochastic Linear Programming and Minimum Risk Solutions, Dissertation Abstract, Faculty of Mathematics and Mechanics, Bucharest. [Romanian]
Bereanu, B. (1964), Programme de Risque Minimal en Programmation Lineaire Stochastique, Compte Revue Academie Science, Paris, 981–983.
Bereanu, B. (1964), Solutions of Minimum Risk in Linear Programming, Analele Universitatii Bucuresti, Seria Stiintele Naturii, MatematicaMecanica 13, 121–140. [Romanian]
Bereanu, B. (1965), Decision Regions and Minimum Risk Solutions in Linear Programming, in Prekopa, A., (ed.), Colloquium on Applications of Mathematics to Economics, Hungarian Academy of Sciences, Budapest, 37–42.
Bereanu, B. (1972), Quasi–Convexity, Strict Quasi–Convexity and Pseudo–Convexity of Composite Objective Functions, Revue Francaise d’Automatique, Informatique et Recherche Operationelle, 15–26.
Bergthaller, C.A. (1970), Quadratic Equivalent of the Minimum Risk Problem, Revue Roumaine de Mathematiques Pures et Appliquees 15, 17–23.
Bernard, J.C. (1986), Quelques Aspects Théoriques en Programmation Fractionnaire Généralisée et Algorithmes, Thèse de Doctorat, Université de Montréal.
Bernard, J.C. and Ferland, J.A. (1989), Convergence of Interval–Type Algorithms for Generalized Fractional Programming, Mathematical Programming 43 (3), 349–363.
Bessent, A. and Bessent W. (1979), Determining the Comparative Efficiency of Schools through Data Envelopment Analysis, Research Report CCS 361, Center for Cybernetic Studies, University of Texas, Austin.
Bessent, A., Bessent, W., Elam, J. and Clark, T. (1988), Efficiency Frontier Determination by Constrained Fleet Analysis, Operations Research 36 (5), 785–796.
Bessent, A., Bessent, W., Kennington, J. and Reagan, B. (1982), An Application of Mathematical Programming to Assess Productivity in the Houston Independent School District, Management Science 28, 13551367.
Bhatia, D. and Datta, N. (1985), Necessary Conditions and Subgradient Duality for Nondifferentiable and Nonconvex Multi–objective Programming Problem, Cahiers du Centre d’Etudes de Recherche Operationelle 27 (12), 131–139.
Bhatia, D. and Gupta, B. (1980), Efficiency in Certain Nonlinear Fractional Vector Maximization Problems, Indian Journal of Pure and Applied Mathematics 11, 669–672.
Bhatia, D. and Jain, P. (1991), Nondifferentiable Multiobjective Fractional Programming with Hanson–Mond Classes of Functions, Journal of Information and Optimization Sciences 12 (1), 35–47.
Bhatia, D. and Pandey, S. (1991), A Note on Multiobjective Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 33 (1–2), 3–11.
Bhatia, D. and Pandey, S. (1991), Subgradient Duality and Duality for Multiobjective Fractional Programming Involving Invex Functions, Cahiers du Centre d’Etudes de Recherche Operationelle 33 (3–4), 167181.
Bhatia, H.L. (1978), Solid Transportation Problem in Linear Fractional Programming, Revue Belge de Statistique d’ Informatique et de Recherche Operationelle 18, 35–50.
Bhatt, S.K. (1973), An Existence Theorem for a Fractional Control Problem, Journal of Optimization Theory and Applications 11, 379–385.
Bhatt, S.K. (1973), Sequential Unconstrained Minimization Technique for a Non–Convex Program, Cahiers du Centre d’Etudes de Recherche Operationelle 15, 429–435.
Bhatt, S.K. (1974), Generalized Pseudo–Convex Programming in Real Banach Space and Duality, Cahiers du Centre d’Etudes de Recherche Operationelle 16, 7–16.
Bhatt, S.K. (1981), Linearization Technique for Linear Fractional and Pseudo–Monotonic Programs Revisited, Cahiers du Centre d’Etudes de Recherche Operationelle 23 (1–2), 53–56.
Bhatt, S.K. (1989), Equivalence of Various Linearization Algorithms for Linear Fractional Programming, Zeitschrift für Operations Research 33 (1), 3943.
Bhatt, S.K. and Rosenbloom, E.S. (1985), A Dynamic Programming Approach to Generalized Linear Fractional Programs, Cahiers du Centre d’Etudes de Recherche Operationelle 27 (3–4), 207–212.
Bitran, G.R. (1979), Experiments with Linear Fractional Problems, Naval Research Logistics Quarterly 26, 689–693.
Bitran, G.R. and Magnanti, T.L. (1976), Duality and Sensitivity Analysis for Fractional Programs, Operations Research 24, 675–699.
Bitran, G.R. and Novaes, A.G. (1973), Linear Programming with a Fractional Objective Function, Operations Research 21, 22–29.
Blau, R.A. (1973), Decomposition Technique for the Chebychev Problem, Operations Research 21, 1157–1162.
Böhm, H.H. (1962), Die Maximierung der Kapitalrentabilität, Zeitschrift für Betriebswirtschaft 32, 489–512.
Boncompte, M. (1985), Programacion Fraccional Generalizada, Thesis, Universidad di Barcelona, Facultad De Matematicas.
Boncompte, M. (1990), Analisi Subdifferencial en Programacion Fraccional, Doctoral Dissertation, Departamento de Matematica Aplicada y Analisis, Universidad di Barcelona. [Spanish]
Boncompte, M. and Martinez–Legaz, J.E. (1991), Fractional Programming by Lower Subdifferentiability Techniques, Journal of Optimization Theory and Applications 68 (1), 95–116.
Borde, J. (1985), Quelques Aspects Theoriques et Algorithmiques en Quasiconvexite, Doctoral Thesis, Universite de Clermont II, Departement de Mathematiques Appliquees.
Borde, J. and Crouzeix, J.P. (1987), Convergence of a Dinkelbach–Type Algorithm in Generalized Fractional Programming, Zeitschrift für Operations Research. Serie A 31 (1), 31–54.
Borwein, J.M. (1976), Fractional Programming Without Differentiability, Mathematical Programming 11, 283–290.
Boyadzhiev, A.V. (1981), On a Nonlinear Pragramming Problem, God. Vissh. Uchebn. Zaved, Prilozhna Mat. 17 (1), 107–112. [Bulgarian]
Boyd, G. and Färe, R. (1984), Measuring the Efficiency of Decision Making Units: A Comment, European Journal of Operational Research 15 (3), 331–332.
Bradley, G.H. (1971), Transformation of Integer Programs to Knapsack Problems, Discrete Mathematics 1 (1), 29–45.
Bradley, S.P. and Frey, S.C. (1974), Fractional Programming with Homogeneous Functions, Operations Research 22, 350–357.
Brännlund, U. (1993), On Relaxation Methods for Nonsmooth Convex Optimization, Doctoral Thesis, Department of Mathematics, Royal Institute of Technology Stockholm, Sweden.
Brender, D.M. (1963), A Surveilance Model for Recurrent Events, IBM Watson Research Center Report.
Brosh, I. (1981), Optimal Cargo Allocation on Board a Plane: A Sequential Linear Programming Approach, European Journal of Operational Research 8 (1), 40–46.
Bühler, W. (1975), A Note on Fractional Interval Programming, Zeitschrift für Operations Research 19, 29–36.
Bühler, W. and Dick, R. (1972/73), Stochastische Lineare Optimierung, Teil I, Zeitschrift für Betriebswirtschaft 42, 1972, 667–692; Teil II, Zeitschrift für Betriebswirtschaft 43, 101–120.
Bühler, W. and Newinger, N. (1973), On the Convergence of Martos’ Hyperbolic Programming Algorithm, Arbeitsbericht, Lehrstuhl für Unternehmensforschung, Rheinisch–Westfälische Technische Hochschule, Aachen.
Bum–Il, L. and Dong–Wan, T. (1989), An Interactive Procedure for Fuzzy Programming Problems with Linear Fractional Objectives, Comput. Ind. Eng. (UK) 16 (2), 269–275.
Burkard, R.E., Dlaska, K. and Klinz. B. (1993), The Quickest Flow Problem, Zeitschrift für Operations Research 37, 31–58.
Burley, S.P. (1974), Dynamic Generalizations of the Von Neumann Model, Math. Mod. Econ., Amsterdam, 27–33.
Bykadorov, I.A. (1985), A Problem of Linear–Fractional Programming,Akademiva Nauk SSSR Sibirskoe Otdelenie, Institut Matematiki, Optimizatsiya 35 (52), 51–55,159. [Russian]
Bykadorov, I.A. (1985), Finite Systems of Linear–Fractional Inequalities, Akademiya Nauk SSSR Sibirskoe Otdelenie, Institut Matematiki, Optimizatsiya 35 (52), 43–50, 158. [Russian]
Bykadorov, I.A. (1986), Conditions for the Quasiconvexity of Sums of Linear–Fractional Functions, Akademiya Nauk SSSR Sibirskoe Otdelenie, Institut Matematiki, Optimizatsiya 39 (56), 25–41, 158. [Russian]
Bykadorov, I.A. (1989), Differential Quasiconvexity Criteria for Finite Sums of Linear Fractional Functions, Preprint No. 24, AN SSSR Sibirskoe Otdelenie, Institut Matematiki, Novosibirsk. [Russian]
Bykadorov, I.A. (1990), Differential Quasiconvexity Criteria for Ratios of Polynomials of Several Variables, Preprint No. 12, AN SSSR Sibirskoe Otdelenie, Institut Matematiki, Novosibirsk. [Russian]
Bykadorov, I.A. (1990), Quasiconvexity Conditions for some Classes of Fractional Functions, Preprint No. 18, AN SSSR Sibirskoe Otdelenie, Institut Matematiki, Novosibirsk. [Russian]
Bykadorov, I.A. (1990), Quasiconvexity Criteria for Objective Functions in Fractional Programming Problems, Doctoral Thesis in Philosophy (Mathematics), Institut Matematiki, Novosibirsk. [Russian]
Bykadorov, I.A. (1994), On Quasiconvexity in Fractional Programming. Komlosi, S., Rapcsak, T. and Schaible, S. (eds.), Generalized Convexity, Proceedings Pécs/Hungary, 1992; Lecture Notes in Economics and Mathematical Systems 405, Springer–Verlag, Berlin, to appear.
Bykadorov, I.A. and Pozhidaev, D.M. (1993), On an Approach to Solving Special Classes of Multiextremal Problems, Preprint No. 3, AN SSSR Sibirskoe Otdelenie, Institut Matematiki, Novosibirsk.
Cabot, A.V. (1978), Maximizing the Sum of Certain Quasiconcave Functions Using Generalized Benders Decomposition, Naval Research Logistics Quarterly 25, 473–482.
Callahan, J.R. and Bector, C.R. (1973), Optimization with General Stochastic Objective Functions, Proceedings of the Third Manitoba Conference on Numerical Mathematics, 127–137.
Cambini, A. (1977), Un Algoritmo per il Massimo del Quoziente di Due Forme Affini con Vincoli Lineari, Paper No. A–42, Department of Operations Research, University of Pisa, Italy.
Cambini, A. (1978), Sulla Programmazione Lineare Frazionaria Stocastica, Paper No. A–50, Dipartimento di Recerco Operativa e Scienze Statistiche, Universita di Pisa, Italy.
Cambini, A. (1979), Sulla Esistenza di Moltiplicatori Esponenziali per i Problem di Programmazione Lineare Frazionaria, Publication No. 67, Serie A. Dipartimento di Ricerca Operativa e Scienze Statistiche, Universita di Pisa, Italy.
Cambini, A. (1981), An Algorithm for a Special Class of Fractional Programs, in Schaible,S. and Ziemba, W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 491–508.
Cambini, A. (1987), Concavità Generalizzata e Programmazione Frazionaria: Stato dell’Arte (Relazione Invitata), Atti XI Convegno A.M.A.S.E.S., Aosta.
Cambini, A. and Martein, L. (1986), A Modified Version of Martos’s Algorithm for the Linear Fractional Problem. Proceedings of Xth Symposium on Operations Research, Methods of Operations Research 53, 33–44.
Cambini, A. and Martein, L. (1986), On the Fenchel–Like and Lagrangian Duality in Fractional Programming. Xth Symposium on Operations Research, Methods of Operations Research 53, 21–32.
Cambini, A. and Martein, L. (1989), Equivalence in Linear Fractional Programming, Dipartimento di Statistica e Matematica Applicata all’ Economia, Universita di Pisa, Report No. 28; to appear in Optimization 23 (1992).
Cambini, A. and Martein, L. (1990), Linear Fractional and Bicriteria Linear Fractional Programs. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 155–166.
Cambini, A. and Sodini, C. (1982), Un Algoritmo per un Problema di Programmazione Frazionaria non Lineare Derivante da un Problema di Selezione del Portafoglio, Publication No. 88, Serie A. Dipartimento di Ricerca Operativa e Scienze Statistiche, Universita di Pisa, Italy. Also in: Atti del Quinto Convegno A.M.A.S.E.S., Perugio 22–24 Ottobre, 1981, 4961
Cambini, A. and Sodini, C. (1983), Sulla Equivalenza di Alcuni Algorithmi di Programmazione Lineare Frazionaria, Report No. A–105, Dipartimento di Matematica, Universita di Pisa.
Cambini, A. and Sodini, C. (1987), On Parametric Linear Fractional Programs, Report No. 6, Dipartimento di Statistica e Matematica Applicata all’ Economia, Universita di Pisa.
Cambini, A., Castagnoli, E., Martein, L., Mazzoleni, P. and Schaible, S., (eds.), (1990), Generalized Convexity and Fractional Programming with Economic Applications. Proceedings of an International Workshop at Pisa, 1988, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin.
Cambini, A., Martein, L. and Pellegrini, L. ( 1978), Decomposition Methods and Algorithms for a Class of Non–linear Programming Problems, First Meeting AFCET–SMF Palaiseau, Ecole Polytechnique Palaiseau Paris, 179–189.
Cambini, A., Martein, L. and Schaible, S. (1989), On Maximizing a Sum of Ratios, Journal of Information and Optimization Sciences 10 (1), 65–79.
Cambini, A., Martein, L. and Sodini, C. (1983), An Algorithm for Two Particular Nonlinear Fractional Programs. Proceedings of VIIth Symposium on Operations Research, Methods of Operations Research 45, 61–70.
Cambini, A., Schaible, S. and Sodini, C. (1993), Parametric Linear Fractional Programming for an Unbounded Feasible Region, J. of Global Optimization 3, 157–169.
Cambini, R. (1994), A Class of Nonlinear Programs: Theoretical and Algorithmic Results. Komlosi, S., Rapcsak, T. and Schaible, S. (eds.), Generalized Convexity, Proceedings Pécs/Hungary, 1992; Lecture Notes in Economics and Mathematical Systems 405, Springer–Verlag, BerlinHeidelberg–New York, to appear.
Canestrelli, E. and Giove, S. (1993), Bidimensional Approach to Fuzzy Linear Goal Programming. Delgado, Kaeprzyk, Verdegay and Vile (eds.), Fuzzy Optimization—Recent Advances, to appear.
Carotenuto, L., Muraca, P. and Raiconi, G. (1988), Observation Strategy for a Parallel Connection of Discrete–Time Linear Systems, Mathematics and Computers in Simulation 30 (5), 389–403.
Castagnoli, E. and Mazzoleni, P. (1990), Differentiable (a,20–Concave Functions. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345 Springer, Berlin, 52–76.
Cernov, J.P. (1970), A Certain Problem of Parametric Linear–Fractional Programming, Izvestija Akademii Nauk Kirgizskoi SSR 3, 20–27. [Russian]
Cernov, J.P. (1970), Several Problems of Parametric Fractional Linear Programming, Optimal Planirovanie Vyp. 16, 98–111. [Russian]
Cernov, J.P. (1971), The Problems of Fractional Programming with Linear Separable and Quadratic Functions, Econom. i Mat. Metody 7, 721–732. [Russian]
Cernov, J.P. (1972), An Application of the 8–Method to the Solution of Fractional Programming Problems with Separable Functions, Mathematical Methods for the Solution of Economic Problems (Suppl. to Ekonom. i Mat. Metody 3), 68–73. [Russian].
Cernov, J.P. and Bayalinov, E.B. (1981), A Dual Fractional Linear Programming Problem, Mathematical Modeling of Economic Processes, ‘Ilim’, Frunze, 115–122. [Russian]
Cernov, J.P. and Bayalinov, E.B. (1981), Linear Analogue of a Fractional Linear Programming Problem, Mathematical Modeling of Economic Processes, ‘Ilim’, Frunze, 109–115, 236. [Russian]
Cernov, J.P. and Lange, E.G. (1970), A Transport Problem of Fractional Programming, Optimal. Planirovanie Vyp. 16, 112–132. [Russian]
Cernov, J.P. and Lange, E.G. (1974), An Application of the Method of Successive Computations to the Solution of a Certain Class of Fractional Concave Programming Problems, Mathematical Methods of Solution of Economic Problems (Suppl. to Ekonom. i Mat. Metody 5), 37–49. [Russian]
Cernov, J.P. and Lange, E.G. (1978), Zadachi Nelineinogo Programmirovanya s Udelnymi Ekonomicheskimi Pokazatelyami. (Nonlinear Programming Problems with Specific Economic Indices), Metody i Prilozheniya. (Methods and Applications), ‘Ilim’, Frunze, 291 pp. [Russian]
Cernov, J.P., Lange, E.G. and Zusupbaev, A. (1979), Application of the Successive Calculation Method to Solving a Production Allocation Problem with a Fractional–Convex Functional, lzvestija Akademii Nauk Kirgizskoi SSR 2, 27–34. [Russian]
Chadha, S.S. (1967), A Decomposition Principle for Fractional Programming, Opsearch 4, 123–132.
Chadha, S.S. (1969), An Extension of Upper Bounded Technique for a Linear Fractional Program, Metrika 20, 25–35.
Chadha, S.S. (1971), A Dual Fractional Program, Zeitschrift für Angewandte Mathematik und Mechanik 51, 560–561.
Chadha, S.S. (1971), A Linear Fractional Functional Program with a Two Parameter Objective Function, Zeitschrift für Angewandte Mathematik und Mechanik 51, 479–481.
Chadha, S.S. (1971), A Linear Fractional Functionals Program with Variable Coefficients, Revue de la Faculte des Sciences de l’Universite d’Istanbul, Serie A 36, 7–13.
Chadha, S.S. (1973), A Generalized Upper Bounded Technique for a Linear Fractional Program, Metrika 20, 25–35.
Chadha, S.S. (1973), Duality Theorems for A Generalized Linear and Linear Fractional Program, Cahiers du Centre d’Etudes de Recherche Operationelle 15, 167–173.
Chadha, S.S. (1977), A Decomposition Principle for a Generalized Linear and Piece–wise Linear Program, Trabajos de Estadistica e Investigacion Operativa 28, 85–92.
Chadha, S.S. (1983), Quadratic Parametric Linear Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 25 (1–2), 23–28.
Chadha, S.S. (1984), A Dual Non–Linear Program, Acta Math. Appl. Sin., Engl. Ser. 1, 163–167.
Chadha, S.S. (1987), Hyperbolic Programming—A new Criteria, Economic Computation and Economic Cybernetics Studies and Research 22 (4), 8388.
Chadha, S.S. (1988), A Parametric Linear Fractional Program, J. Comb. Inform. System Sci. 13 (3–4), 106–113.
Chadha, S.S. (1988), Duality Theorems for a Class of Non–Linear Programming Problems, Trab. Invest. Oper. 3 (1), 141–148.
Chadha, S.S. (1989), Multiparametric Linear Fractional Functionals Programming, Trabajos de Investigacion Operativa (Spain) 4 (1), 3–10.
Chadha, S.S. (1992), Optimization of a Quadratic Fractional Function, Cahiers du Centre d’Etudes de Recherche Operationelle 34 (1), 3–6.
Chadha, S.S. and Gupta, J.M. (1976), Sensitivity Analysis of the Solution of a Generalized Linear and Piece–wise Linear Program, Cahiers du Centre d’Études de Recherche Operationelle 18, 309–321.
Chadha, S.S. and Kaul, R.N. (1972), A Dual Nonlinear Program, Metrika 19,18–22.
Chadha, S.S. and Kaul, R.N. (1972), A Linear Fractional Functionals Program with Variable Coefficients, Journal of Mathematical Sciences 7, 15–20.
Chadha, S.S. and Shivpuri S. (1973), A Simple Class of Parametric Linear Fractional Functionals Programming, Zeitschrift für Angewandte Mathematik und Mechanik 53, 641 646.
Chadha, S.S. and Shivpuri, S. (1977), Parametrization of a Generalized Linear and Piece–wise Linear Programming Problem, Trabajos de Estadistica e Investigacion Operativa 28, 151–160.
Chadha, S.S. and Shivpuri, S. (1980), Enumerative Technique for an Extreme Point Fractional Program, European Journal of Operational Research 4, 54–59.
Chambers, D. (1967), Programming the Allocation of Funds Subject to Restrictions on Reported Results, Operational Research Quarterly 10, 407–431.
Chandra, S. (1967), Linear Fractional Functional Programming, Journal of the Operations Research Society of Japan 10, 1–2.
Chandra, S. (1967), The Capacitated Transportation Problem in Linear Fractional Programming, Journal of the Operations Research Society of Japan 10, 18–26.
Chandra, S. (1968), Decomposition Principle for Linear Fractional Functional Programs, Revue Francaise d’Informatique et de Recherche Operationelle 2, 65–72.
Chandra, S. (1969), Duality in Banach Spaces; Transformation, Assignment and Large–Structured Fractional Programs, Ph.D. Thesis, Dept. of Mathematics, Indian Institute of Technology, Kanpur (India).
Chandra, S. (1972), Strong Pseudo–Convex Programming, Indian Journal of Pure and Applied Mathematics 3, 278–282.
Chandra, S. and Chandramohan, M. (1976), Duality and Algorithmic Aspects of Integer Linear Fractional Programming, Research Report, Department of Mathematics, Indian Institute of Technology, New Delhi.
Chandra, S. and Chandramohan, M. (1979), An Improved Branch and Bound Method for Mixed Integer Linear Fractional Programs, Zeitschrift für Angewandte Mathematik und Mechanik 59, 575–577.
Chandra, S. and Chandramohan, M. (1979), Duality in Mixed Integer Nonconvex and Nondifferentiable Programming, Zeitschrift für Angewandte Mathematik und Mechanik 59, 205–209.
Chandra, S. and Chandramohan, M. (1980), A Branch and Bound Method for Integer Non–linear Fractional Programs, Zeitschrift für Angewandte Mathematik und Mechanik 60, 735–737.
Chandra, S. and Chandramohan, M. (1980), A Note on Integer Linear Fractional Programming, Naval Research Logistics Quarterly 27, 171174.
Chandra, S. and Gulati, T.R. (1976), A Duality Theorem for a Nondifferentiable Fractional Programming Problem, Management Science 23, 32–37.
Chandra, S. and Husain, I. (1989), Symmetric Dual Continuous Fractional Programming, Journal of Information and Optimization Sciences 10 (1), 241–255.
Chandra, S. and Kumar, V. (1991), Equivalent Lagrangians for Generalized Fractional Programming, Technical Report, Dept. of Mathematics, Indian Institute of Technology, Kanpur (India).
Chandra, S. and Saxena, P.K. (1983), Fractional Transportation Problem with Impurities, Advances in Management Studies 2, 335–349.
Chandra, S. and Saxena, P.K. (1984), Fractional Transportation Problem with Penalty Costs and Impurities, Advances in Management Studies 3, 57–65.
Chandra, S. and Saxena, P.K. (1987), Cost/Completion–Date Tradeoffs in Quadratic Fractional Transportation Problem, Economic Computation and Economic Cybernetics Studies and Research 22 (3), 67–72.
Chandra, S., Craven, B.D. and Husain, I. (1985), Continuous Programming Containing Arbitrary Norms, J. Austr. Math. Soc. Ser. A 39, 28–38.
Chandra, S., Craven, B.D. and Mond, B. (1985), Nonlinear Programming Duality and Matrix Game Equivalence, J. Austr. Math. Soc. Ser. B 26, 422–429.
Chandra, S., Craven, B.D. and Mond, B. (1985), Symmetric Dual Fractional Programming, Zeitschrift für Operations Research, Serie A 29 (1), A59–A64.
Chandra, S., Craven, B.D. and Mond, B. (1986), Generalized Fractional Programming Duality: A Ratio Game Approach, Australian Mathematical Society, Journal Series B, Applied Mathematics 28 (2), 170–180.
Chandra, S., Craven, B.D. and Mond, B. (1990), Vector–Valued Lagrangian and Multi–Objective Fractional Programming Duality, Numerical Functional Analysis and Optimization 11 (3–4), 239–254.
Chandra, S., Craven, B.D. and Mond, B. (1991), Multiobjective Fractional Programming Duality. A Lagragian Approach, Optimization 22 (4), 549556.
Chandra, S., Mond, B. and Durga–Prasad, M.V. (1988), Constrained Ratio Games and Generalized Fractional Programming, Zeitschrift für Operations Research 32 (5), 307–314.
Chandrasekaran, R. (1977), Minimum Ratio Spanning Trees, Networks 7, 335342.
Chandrasekaran, R. and Tamir, A. (1984), Optimization Problems with Algebraic Solutions: Quadratic Fractional Programs and Ratio Games, Mathematical Programming 30 (3), 326–339.
Chandrasekaran, R. and Tamir, A. (1984), Polynomial Testing of the Query “Is a b >— c°?” with Application to Finding a Minimal Cost Reliability Ratio Spanning Tree, Discrete Appl. Math. 9, 117–123.
Chandrasekaran, R., Anija, Y.P. and Nair, K.P.K. (1981), Minimal Cost Reliability Ratio Spanning Tree. Studies on Graphs and Discrete Programming, Annals of Discrete Mathematics 11, 53–60.
Charnes, A. and Cooper, W.W. (1962), Programming with Linear Fractional Functionals, Naval Research Logistics Quarterly 9, 181–186.
Charnes, A. and Cooper, W.W. (1962), Systems Evaluation and Repricing Theorems, Management Science 9, 33–49.
Charnes, A. and Cooper, W.W. (1963), Deterministic Equivalents for Optimizing and Satisficing Under Chance Constraints, Operations Research 11, 18–39.
Charnes, A. and Cooper, W.W. (1963), Programming with Linear Fractional Functionals, a Communication, Naval Research Logistics Quarterly 10, 273–274.
Charnes, A. and Cooper, W.W. (1973), An Explicit General Solution in Linear Fractional Programming, Naval Research Logistics Quarterly 20, 449467.
Charnes, A. and Cooper, W.W. (1977), Goal Programming and Multi–Objective Optimization, Part I, European Journal of Operational Research 1, 39–54.
Charnes, A. and Cooper, W.W. (1980), Auditing and Accounting for Program Efficiency and Management Efficiency in Not–for–Profit Entities, Accounting, Organizations and Society 5, 87–107.
Charries, A. and Cooper, W.W. (1980), Management Science Relations for Evaluation and Management Accountability, Journal of Enterprise Management 2, 143–162.
Charnes, A. and Cooper, W.W. (1981), The Effective Solution of the Class of Non–Convex Multi–Objective Problems with Ratio Goals, Operational Research ‘81, North–Holland, Amsterdam, 20–24.
Charnes, A. and Cooper. W.W. (1984), The Non–Archimedean CCR Ratio for Efficiency Analysis: A Rejoinder to Boyd and Färe, European Journal of Operational Research 15 (3), 333–334.
Charries, A. and Granot, D. (1976), Constrained Non–Cooperative von Neumann Ratio Games, Working Paper No. 368, Faculty of Commerce and Business Administration, University of British Columbia, Vancouver.
Charnes, A. and Neralic, L. (1989), Sensitivity Analysis in Data Envelopment Analysis, Glasnik Matematicki 24 (44), No.1, 211–226.
Charries, A., Cooper, W.W. and Rhodes, E. (1978/79), Measuring the Efficiency of Decision Making Units, European Journal of Operational Research 2, 429444; also Corrections, op. cit. 3, 339.
Charries, A., Cooper, W.W. and Rhodes, E. (1981), Evaluation Program and Managerial Efficiency, An Application of Data Envelopment Analysis for Program Follow Through, Management Science 27, 668–697.
Charnes, A., Cooper, W.W. and Thrall, R.M. (1986), Classifying and Characterizing Effinciencies and Inefficiencies in Data Development Analysis, Operations Research Letters 5 (3), 105–110.
Charnes, A., Cooper, W.W., Golany, B. and Seiford, L. (1984), Foundations of Data Envelopment Analysis for Pareto–Koopmans Efficient Emperical Production Functions, J. Econometrics 30, 91–107.
Charries, A., Cox, L. and Lane, M. (1970), A Note on the Redesigning of a Rate Structure for Allocation of State Funds to Educational Institutions, Working Paper 70–49 of Project GUM, University of Texas, Austin.
Charnes, A., Granot, D. and Granot, F. (1976), A Note on Explicit Solutions in Linear Fractional Programming, Naval Research Logistics Quarterly 23, 161–167.
Charnes, A., Granot, D. and Granot, F. (1976), An Algorithm for Solving General Fractional Interval Programming Problems, Naval Research Logistics Quarterly 23, 53–65.
Charnes, A., Granot, D. and Granot, F. (1978), On Solving Linear Fractional Interval Programming Problems, Cahiers du Centre d’Etudes de Recherche Operationelle 20, 45–57.
Chatterjee, S. and Sen, R. (1985), On a Certain Type of Bicriterion Programming Problems, Bulletin Mathematique de la Societe des Sciences Mathematiques de la Republique Socialiste de Roumanie, Nouvelle Serie 29 (4), 297–306.
Cheney, E.W. and Loeb, H.L. (1961), Two New Algorithms for Rational Approximation, Numerische Mathematik 3, 72–75.
Chernov, Y.P. and Bayalinov, E.B. (1981), A Dual Fractional–Linear Programming Problem, Mathematical Modeling of Economic Processes, ‘Ilim’, Frunze, 115–122, 236. [Russian]
Chernov, Y.P. and Bayalinov, E.B. (1981), Linear Analogue of a Fractional–Linear Programming Problem, Mathematical Modeling of Economic Processes, ‘Ilim’, Frunze, 109–115, 236. [Russian]
Chew, K.L. and Choo, E.U. (1984), Pseudolinearity and Effficiency, Mathematical Programming 28 (2), 226–239.
Chong, J.H. and Kim, J.O. (1975), A Solution of Linear and Fractional Linear Programming with Both Restrictions, Su–hak kwa Mul–li 19, 9–15. [Korean]
Choo, E.U. (1980), Multicriteria Linear Fractional Programming, Ph.D. Thesis, University of British Columbia, Vancouver.
Choo, E.U. (1984), Proper Efficiency and the Linear Fractional Vector Maximum Problem, Oper. Res. 32 (1), 216–220.
Choo, E.U. and Atkins, D.R. (1980), An Interactive Algorithm for Multicriteria Programming, Computer and Operations Research 7, 81–87.
Choo, E.U. and Atkins, D.R. (1982), Bicriteria Linear Fractional Programming, Journal of Optimization Theory and Applications 36, 204–220.
Choo, E.U. and Atkins, D.R. (1983), Connectedness in Multiple Linear Fractional Programming, Management Science 29 (2), 250–255.
Choo, E.U., Schaible, S. and Chew, K.P. (1985), Connectedness of the Efficient Set in Three Criteria Quasiconcave Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 27, 213–220.
Chowdhuri, S. and Breuer, M.A. (1988), Optimization Algorithms for a Class of Nonlinear Programming Problems, Comput. Math. Appl. 15 (3), 175184.
Christov, G. (1979), Hyperbolic Optimization Problem, Proceedings of IX. International Symposium on Mathematical Programming, Budapest, 1976, A. Prékopa (ed.), North–Holland, Amsterdam.
Christov, G. (1983), Properties and Method for Solving Fractional–Linear Optimization Problems, Doklady Bolgarskoi Akademii Nauk, Comptes Rendus de l’Academie Bulgare des Sciences 36 (1), 61–64.
Christov, G., Karamiteva, Z. and Stoyanov, T.E. (1985), A Numerical Method and Program for Solving a Linear–Fractional Optimization Problem, Mathematics and Mathematical Education (Sunny Beach), Slunchev Bryag, Bulg. Akademii Nauk, Sofia, 566–570. [Bulgarian]
Chung, K.J. (1989), A Note on Maximal Mean/Standard Deviation Ratio in an Undiscounted MDP, Oper. Res. Lett. 8 (4), 201–203.
Climaco, J.C.N. and Cardoso, D.M. (1990), Linear Fractional Programming, a new Bicriteria Approach, Belg. J. Oper. Res. Stat. Comput. Sci 29 (3), 324.
Climaco, J.C.N., Martins, E.Q.V. and Rosa, M.S. (1981), A Simplex Based Algorithm for the Minimal Average Speed Path Problem, Technical Report, Dept. of Mathematics, University of Coimbra, Coimbra, Portugal.
Colantoni, C.S., Manes, R.P. and Whinston, A. (1969), Programming, Profit Rates and Pricing Decisions, Accounting Review, 467–481.
Collatz, L. (1989), Some Advantages of Rational Approximation Compared with Polynomial Approximation. Chui, C.K., Shumaker, L.L. and Ward, J.D. (eds.) Approximation Theory 6, Academic Press, New York, 145148.
Collatz, L. and Wetterling, W. (1971), Optimierungsaufgaben, 2nd ed., Springer, Berlin.
Conde, E. and Ruiz–Canales, P. (1991), A Branch–and–Bound Algorithm for Discrete Fractional Programming, [Spanish, English Summary], XV Jornades Luso–Espanholas de Matematica, Vol. IV (Proceedings of the XVth Portugese–Spanish Conference on Mathematics), 361–366. [Spanish]
Cook, W.D., Kirby, M.J.L. and Mehndiratta, S.L. (1975), A Linear Fractional Max–Min Problem, Operations Research 23, 511–521.
Corban, A. (1973), Duality in Transportation Problems in Fractional Programming, Studii si Cercetari Matematice 25, 347–357. [Romanian]
Corban, A. (1973), Programming with Fractional Linear Objective Function, Revue Roumaine de Mathematique Pures et Appliquees 18, 633–637.
Corban, A. (1974), Duality in Non–Linear Programming, Studii si Cercetari Matematice 26, 375–399. [Romanian]
Corban, A. (1975), Non–Linear Three–Dimensional Programming, Revue Roumaine de Mathematiques Pures et Appliquees 20, 1043–1059.
Craven, B.D. (1978), Mathematical Programming and Control Theory, Chapman and Hall, London.
Craven, B.D. (1981), Duality for Generalized Convex Fractional Programs, in Schaible, S. and Ziemba, W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 473–490.
Craven, B.D. (1988), Fractional Programming, Sigma Series in Applied Mathematics 4, Heldermann Verlag, Berlin.
Craven, B.D. (1988), Fractional Programming—A Survey, Opsearch 25 (3), 165–176.
Craven, B.D. and Mond, B. (1973/76), The Dual of a Fractional Linear Program, Journal of Mathematical Analysis and Applications 42, 507512; Journal of Mathematical Analysis and Applications 55, 807.
Craven, B.D. and Mond, B. (1975), On Fractional Programming and Equivalence, Naval Research Logistics Quarterly 22, 405–410.
Craven, B.D. and Mond, B. (1976), Duality for Homogeneous Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 18, 413–417.
Craven, B.D. and Mond, B. (1979), A Note on Duality in Homogenous Fractional Programming, Naval Research Logistics Quarterly 26, 153156.
Craven, B.D. and Mond, B. (1979), On Duality for Fractional Programming, Zeitschrift für Angewandte Mathematik und Mechanik 59, 278–279.
Craven, B.D. and Mond, B. (1983), On Maximizing a Ratio of Optimization Problems, Cahiers du Centre d’Etudes de Recherche Operationelle 25 (12), 29–34.
Crouzeix, J.P. (1977), Contributions a l’Etude des Fonctions Quasiconvexes, Doctoral Thesis, Université de Clermont, France.
Crouzeix, J.P. (1981), A Duality Framework in Quasiconvex Programming, in Schaible, S. and Ziemba,W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 207–225.
Crouzeix, J.P. and Lindberg, P.O. (1986), Additively Decomposed Quasiconvex Functions, Mathematical Programming 35, 42–57.
Crouzeix, J.P., and Ferland, J.A. (1991), Algorithms for Generalized Fractional Programming, Mathematical Programming 52 (Ser. B), 191–207.
Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1981), Duality in Generalized Linear Fractional Programming, Technical Report No. 399, Departement d’Informatique et de Recherche Operationelle, Universite de Montreal.
Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1983), Duality in Generalized Linear Fractional Programming, Mathematical Programming 27 (3), 342354.
Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1984), Erratum: “Duality in Generalized Linear Fractional Programming”, Mathematical Programming 29 (2), 243.
Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1985), An Algorithm for Generalized Fractional Programs, Journal of Optimization Theory and Applications 47 (1), 35–49.
Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1986), A Note on an Algorithm for Generalized Fractional Programs, Journal of Optimization Theory and Applications 50 (1), 183–187.
Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1990), Improved Analysis of the Generalized Convexity of a Function in Portfolio Theory. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 287–294.
Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1992), Generalized Convexity of Functions on Affine Subspaces with an Application to Potential Functions, Mathematical Programming 56, 223–232.
Csebfalvi, A. and Csebfalvi, G. (1994), Post–Buckling Analysis of Frames by a Hybrid Path–Following Method. Komlosi, S., Rapcsak, T. and Schaible, S. (eds.), Generalized Convexity, Proceedings Pécs/Hungary, 1992; Lecture Notes in Economics and Mathematical Systems 405, Springer–Verlag, Berlin–Heidelberg–New York, to appear.
Dantzig, G.B., Blattner, W. and Rao, M.R. (1966), All Shortest Routes From a Fixed Origin in a Graph, in Theory of Graphs, Intern. Symp., Dunod, Paris, and Gordon and Breach, New York, 85–90.
Dantzig, G.B., Blattner, W. and Rao, M.R. (1966), Finding a Cycle in a Graph with Minimum Cost to Time Ratio with Applications to a Ship Routing Problem, in Theory of Graphs, Intern. Symp., Dunod, Paris, and Gordon and Breach, New York, 77–83.
Das, C. and Swarup, K. (1975), Complex Fractional Functionals Programming with Nonlinear Constraints, Zeitschrift für Angewandte Mathematik und Mechanik 55, 441–442.
Datta, N. (1982), Efficiency in Multi–Objective Fractional Functional Programming, Journal of Information and Optimization Sciences 3, 262268.
Datta, N. and Bhatia, D. (1984), A Note on Duality Theory for Concave Convex Fractional Programming Problem in Complex Space, Indian Journal of Pure and Applied Mathematics 15 (12), 1289–1295.
Datta, N. and Bhatia, D. (1984), Algorithm to Determine an Initial Efficient Basic Solution for a Linear Fractional Multiple Objective Transportation Problem, Cahiers du Centre d’Etudes de Recherche Operationelle 26 (12), 127–136.
De Angelis, V. (1979), Linear Programming with Uncertain Objective Function, Minimax Solution for Relative Loss, Calcolo 16, 125–141.
Delman, C. (1962), On Sequential Decisions and Marcov Chains, Management Science 9, 16–24.
Deshpande, D.V. and Zionts, S. (1980), Sensitivity Analysis in Multiple Objective Linear Programming: Changes in the Objective Function Matrix, Multiple Criteria Decision Making Theory and Application, Lecture Notes in Economics and Mathematical Systems 177, Springer, Berlin–New York, 26–39.
Deumlich, R. and Elster, K.H. (1978), Duality Theorems for Nonconvex Optimization Problems, Mathematische Operationsforschung und Statistik, Serie Optimization 9, 335–347.
Deumlich, R. and Elster, K.H. (1980), Duality Theorems and Optimality Conditions for Nonconvex Optimization Problems, Mathematische Operationsforschung und Statistik, Serie Optimization 11 (2), 181–219.
Deumlich, R. and Elster, K.H. (1981), A Contribution to Duality Theory of Nonlinear Programming, Methods of Mathematical Programming, PWNWarszawa 31–40.
Deumlich, R. and Elster, K.H. (1983), (D—Conjugation and Nonconvex Optimization, I. A Survey, Mathematische Operationsforschung und Statistik, Serie Optimization 14, 125–149.
Deumlich, R. and Elster, K.H. (1984), (D—Conjugation and Nonconvex Optimization, II. A Survey, Mathematische Operationsforschung und Statistik, Serie Optimization 15 (4), 499–515.
Deumlich, R. and Elster, K.H. (1984), On a Class of Nonconvex Optimization Problems, Proceedings of the VIIIth Symposium in Operations Research, University of Karlsruhe/Germany, August 22–25, 1983. Lecture Notes in Economics and Mathematical Systems, 226, Springer Verlag, Berlin–New York, 13–29.
Deumlich, R. and Elster, K.H. (1985), (D—Conjugation and Nonconvex Optimization, III. A Survey, Optimization 16 (6), 789–803.
Deumlich, R. and Elster, K.H. (1985), Fractional Programming in View of Generalized Conjugation, Proceedings of the IXth Symposium in Operations Research, Methods of Operations Research 49, 3–16.
Deumlich, R. and Elster, K.H. (1986), On Perturbations of Certain Nonconvex Optimization Problems, J. Optim Theory Appl. 48 (1), 81–93.
Dexter, N.S., Yu, M.W. and Ziemba, W.T. (1980), Portfolio Selection in a Lognormal Market when the Investor has a Power Utility Function, Computational Results, in Dempster, M.A.H., (ed.), Stochastic Programming, Academic Press, New York, 507–523.
Dezhurko, L.F. and Fam–Tkhe–Long (1983), On Methods for Solving the General Fractional–Linear Programming Problem, Doklady Akademii Nauk BSSR 27 (7), 595–598. [Russian]
Ding, B. and Wang, C.L. (1991), Stability Analysis for Linear and Linear Fractional Programs, Congressus Numerantium 80, 107–115.
Dinkelbach, W. (1962), Die Maximierung eines Quotienten zweier linearer Funktionen unter linearen Nebenbedingungen, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1, 141–145.
Dinkelbach, W. (1967), On Nonlinear Fractional Programming, Management Science 13, 492–498.
Dombi, P. (1985), On Extremal Points of Quasiconvex Functions, Mathematical Programming 33, 115–119.
Dormany, M. (1979), Specialis Alaku Parametrikus Linearis es Hiperbolikus Programozasi Feladatok Megoldasa Revide alt Szimplex Modszerral, Nehez. Musz. Egyetem Kozl. 4, 25 (1), 53–62. [Hungarian]
Dormany, M. (1983), On Solving a Bicriterion Program, Alkalmaz. Mat. Lapok 9 (3–4), 393–404. [Hungarian]
Dorn, W.S. (1962), Linear Fractional Programming, IBM Research Report RC–830.
Dorn, W.S. (1963), Nonlinear Programming, A Survey, Management Science 9, 171–208.
Dragomirescu, M. (1972), An Algorithm for the Minimum–Risk Problem of Stochastic Programming, Operations Research 20, 154–164.
Drezner, Z., Schaible, S. and Simchi–Levi, D. (1990), Queuing–Location Problems on the Plane, Naval Research Logistics 37, 929–935.
Duca, D. (1977), On a Fractional Programming Problem, Studii si Cercetari Matematice 29, 487–497. [Romanian]
Duca, D. (1979), On the Ordering of the Efficient Extremal Points in a Vectorial Programming Problem, Stud. Univ. Babes – Bolyai, Math. 24, 57–63. [Romanian]
Duca, D. (1980), On the Hierarchy of Extremal Solutions in a Problem of Fractional Programming, Studii si Cercetari Matematice 32, 179–185. [Romanian]
Duca, E. (1979), An Algorithm for the Calculation of Efficient Extreme Points in Problems of Vector Programming, Operations Research Proceedings of the 3rd Colloquium, Cluj–Napoca (Romania), 90–97.
Dumitru, V. and Luban, F. (1980), Mathematical Programming Real Variable Models for Integer and Mixed–Integer Problems, Econ. Comp. & Econ. Cyb. Stud. & Res. 14 (3), 69–79.
Dumitru, V. and Luban, F. (1981), Mathematical Programming Real Variable Models for Integer and Mixed–Integer Problems and some Applications to Production Scheduling, Math. Rev. Anal Numer. Theor. Approximation, Math. 23 (46), 11–23.
Dutta, D., Rao, J.R. and Tiwari, R.N. (1992), Sensitivity Analysis in Fuzzy Linear Fractional Programming Problem, Fuzzy Sets and Systems 48 (2), 211–216.
Dutta, D., Tiwari, R.N. and Rao, J.R. (1992), Multiple Objective Linear Fractional Programming—A Fuzzy Set–Theoretic Approach, Fuzzy Sets and Systems 52 (1), 39–45.
Egisapetov, E.G. (1969), Teaching Pattern Recognition with the Help of Decision Functions, lzvestja Akademii Nauk SSSR, Tekhnicheskaya Kibernetika (USSR) 5, 84–93. [Russian]
Egorova, E.N. (1986), On an Approach to the Solution of Special Problems of Fractional–Cubic Programming, Akademiya Nauk SSSR. Sibirskoe Otdelenie. Institut Matimatiki. Optimizatsiya 39 (56), 81–98. [Russian]
Egorova, E.N. (1988), On a Mistaken Generalization of Methods of Linear and Linear Fractional Programming, Metody Optimiz. v Econ.–Mat. Modelir., 129–133. [Russian]
Egudo, R.R. (1988), Multiobjective Fractional Duality, Bulletin of the Australian Mathematical Society 37 (3), 367–378.
Eichhorn, W. (1972), Effektivität von Produktionsverfahren, Operations Research Verfahren 12, 98–115.
Eichhorn, W. (1978), Functional Equations in Economics, Addison–Wesley, New York.
Eichhorn, W. (1990), Generalized Convexity in Economics: Some Examples. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 266–275.
Ellero, A. (1993), Soluzioni Ottime di Livello in Programmazione Lineare Frazionaria e in Alcune sue Generalizzazioni: Aspetti teorici ed Algortmici, Tesi di Dottorato di Recerca, Università di Triesta–Università di Venezia. [Italian]
Ellero, A. and Moretti Tomasin, E. (1992), A Computational Comparison Between Algorithms for Linear Fractional Programming, J. of Information and Optimization Sciences 13, 343–362.
Ellero, A. and Moretti Tomasin, E. (1992), Equivalence Between Two Algorithms for Linear Bicriteria Programming Problems. Mazzoleni, P. (ed.), Generalized Concavity, Proceedings, Pisa, April 1992, Tecnoprint S.N.C., Bologna, 41–52.
Elmaghraby, S.E. and Arisawa, S. (1972), On Hyperbolic Programming with a Single Constraint and Upper–Bounded Variables, Management Science 19, 42–45.
Elster, K.H. (1979), On the Theory of Nonconvex Optimization Problems. Third Symposium on Operations Research, University Mannheim, 1978, Operations Research Verfahren 31, 173–184.
Elster, K.H. (1990), On Duality Results in Nonconvex Optimization, Functional Analysis, Optimization, and Mathematical Economics, 235247.
Elster, K.H. and Deumlich, R. (1981), Lagrange Duality for Nonconvex Optimization Problems, Operations Research Verfahren 43, 53–61.
Elster, K.H. and Wolf, A. (1990), Generalized Convexity and Fractional Optimization. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 219–231.
Elton, E.J., Gruber, M.J. and Padberg, M.W. (1976), Simple Criteria for Optimal Portfolio Selection, Journal of Finance 31, 1341–1357.
Elton, E.J., Gruber, M.J. and Padberg, M.W. (1977), Simple Criteria for Optimal Portfolio Selection with Upper Bounds, Operations Research 25, 952–957.
Elzinga, J., Hearn, D. and Randolph, W.D. (1976), Minimax Multifacility Location with Euclidian Distances, Transportation Science 10, 321–336.
Emol’eva, L.G. (1972), A Problem of Fractional Nonlinear Programming, Kibernetika (Kiev) 2, 45–47. [Russian]
Eriksson, J. (1980), A Note on Solution of Large Sparse Maximum Entropy Problems with Linear Equality Constraints, Mathematical Programming 18 (2), 146–154.
Ezio, M. (1976), Equilibrium Points of National N–Person Games, Journal of Mathematical Analysis and Applications 54 (1), 1–4.
Faaland, B.H. and Jacob, N.L. (1981), The Linear Fractional Portfolio Selection Problem, Management Science 27, 1383–1389.
Faiziev, N.G., Serman, I.M. and Nuritdinov, S. (1976), Application of a Certain Class of Target Functions in the Optimization of Engineering Processes, Dokl. Akad. Nauk Uz. SSR. 6, 16–18. [Russian]
Falk, J.E. (1969), Maximization of Signal–to–Noise Ratio in an Optical Filter, SIAM Journal of Applied Mathematics 7, 582–592.
Falk, J.E. and Palocsay, S.W. (1992), Optimizing the Sum of Linear Fractional Functions. Collection: Recent Advances in Global Optimization, 221–258.
Falk, J.E., Polacsay, S.W., Sacco, W.J., Copes, W.S. and Champion, H.R. (1992), Bounds on a Trauma Outcome Function Via Optimization, Operations Research 40 (S1), S86–S95.
Färe, R. and Hunsaker, W. (1986), Notions of Efficiency and their Reference Sets, Management Science 32 (2), 237–243.
Ferland, J.A. and Potvin, J.Y. (1985), Generalized Fractional Programming: Algorithms and Numerical Experimentation, European Journal of Operational Research 20 (1), 92–101.
Filipovich, E.I. (1964), On Solving Fractional Programming Problems, lzd. Gor’skogo Isledovatel’scogo Fiziko–Matematiceskogo Instituta Gor’kii. [Russian]
Flachs, J. (1981), Global Saddle–Point Duality for Quasi–concave Programs, Mathematical Programming 20, 327–347.
Flachs, J. (1982), Global Saddle–Point Duality for Quasiconcave Programs, II., Mathematical Programming 24 (3), 326–345.
Flachs, J. (1985), Generalized Cheney–Loeb–Dinkelbach–Type Algorithms, Mathematics of Operations Research 10, 674–687.
Flachs, J. and Pollatschek, M. A. (1982), Equivalence Between a Generalized Fenchel Duality Theorem and a Saddle–Point Theorem for Fractional Programs, Journal of Optimization Theory and Applications 37 (1), 2332.
Florian, M. and Robillard, P. (1970), A Note on Hyperbolic Programming, Publication No. 31, Département d’Informatique, Université de Montréal.
Florian, M. and Robillard, P. (1971), Programmation Hyperbolique en Variables Bivalentes, Revue Francaise d’Informatique et de Recherche Operationelle 1, 3–9.
Fox, B. (1966), Markov Renewal Programming by Linear Fractional Programming, SIAM Journal of Applied Mathematics 14, 1418–1432.
Fox, B. (1969), Finding Minimum Cost–Time Ratio Circuits, Operations Research 17, 546–550.
Frair, L. (1982), A Scheduling Procedure to Minimize Community Annoyance from Airport Noise, Technical Report, Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University.
Freund, R.W. and Jarre, F. (1993), A Polynomial–Time Algorithm for Fractional Programs with Convex Constraints, Numerical Analysis Manuscript 93–08, AT&T Bell Laboratories, Murray–Hill, NJ; to appear in Zeitschrift für Angewandte Mathematik und Mechanik.
Freund, R.W. and Jarre, F. (1993), An Interior–Point Method for Multi–Fractional Programs with Convex Constraints, Numerical Analysis Manuscript 93–07, AT&T Bell Laboratories, Murray–Hill, NJ.
Gabasov, R. and Dezhurko, L.F. (1985), An Adaptive Method of Solution of the General Fractional–Linear Programming Problem, Doklady Akademii Nauk BSSR 29 (8), 685–687, 763. [Russian]
Galambos, G. and Imreh, B. (1984), Solution of One–Dimensional Cutting Stock Problems by Column–Generation, Alkamaz. Mat. Lapok 10 (1–2), 73–85. [Hungarian]
Gamidov, S.I. and Dem’yanov, V.F. (1984), Minimization of the Ratio of Maximum and Minimum Functions, Vestn. Leningr. Univ. 1984, No. 13; Mat. Mech. Astron. 3, 14–18. [Russian]
Garg, K.C. and Swarup, K. (1978), Complementary Programming with Linear Fractional Objective Function, Cahiers du Centre d’ Etudes de Recherche Operationelle 20, 83–94.
Garg, K.C. and Swarìtp, K. (1978), Linear Fractional Functional Complementary Programming with Extreme Point Optimization, Indian Journal of Pure and Applied Mathematics 9, 556–563.
Garg, K.C. and Swarup, K. (1980), The Use of Cuts in Linear Fractional Functional Complementary Programming, Zeitschrift für Angewandte Mathematik und Mechanik 60, 53–54.
Gasparotto, G. (1987), On the Charnes–Cooper Transformation in Linear Fractional Programming, Report No. 3, Dipartimento di Statistica e Matematica Applicata all’ Economia, Universita di Pisa.
Gass, S.I. (1985), Linear Programming: Methods and Applications, 5th ed., McGraw–Hill, New York.
Gaudioso, M. and Monaco, M.F. (1991), Quadratic Approximations in Convex Non–differentiable Optimization, SIAM J. Control and Optimization 29 (1), 58–70.
Gavurin, M.K. (1982), Linear Fractional Programming on an Unbounded Set, Vestnik Lenigradskogo Universiteta. Matematika, Mekhanika, Astronomiya 4, 12–16, 110. [Russian]
Gavurin, M.K. (1982), On the Minimization of a Function Expressed Through Linear–Fractional Functions, The Tohoku Mathematical Journal, Second Series 34 (2), 5–8, 124. [Russian]
Geoffrion, A.M. (1967), Solving Bi–criterion Mathematical Programs, Operations Research 15, 39–54.
Geoffrion, A.M. (1967), Stochastic Programming with Aspiration and Fractile Criteria, Management Science 13, 672–679.
Geoffrion, A.M., Dyer, J.S. and Feinberg, A. (1972), An Interactive Approach for Multicriterion Optimization with an Application to the Operation of an Academic Department, Management Science 19, 357–368.
Gilmore, P.C. and Gomory, R.E. (1963), A Linear Programming Approach to the Cutting Stock Problem—Part II, Operations Research 11, 863–888.
Glover, F. and Woosley, R.E. (1970), Aggregating Diophantine Equations, Report 70–4, University of Colorado.
Gogia, N.K. (1967), The Multiplex Method for Linear Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 9, 123–133.
Gogia, N.K. (1968), Revised Simplex Algorithm for the Linear Fractional Functionals Programming Problem, Mathematical Student 36, 55–57.
Gogia, N.K. (1969), The Non–Linear Fractional Functional Programming Problem with Separable Functions, Journal of Mathematical Sciences 4, 77–84.
Gol’stein, E.G. (1967), Dual Problems of Convex and Fractionally–Convex Programming in Functional Spaces, Soviet Mathematics Doklady 8, 212216.
Gol’stein, E.G. (1972), Theory of Convex Programming, Translation of Mathematical Monographs, American Mathematical Society.
Gol’stein, E.G. (1973), Konvexe Optimierung, Akademie–Verlag, Berlin,
Gol’stein, E.G., Borisova, E.P. and Dubson, M.S. (1984), A Dialog System of Vector Optimization with Linear–Fractional Criteria and Linear Constraints, Wissenschaftliche Zeitschrift der Hochschule Ilmenau 30 (5), 17–31. [Russian]
Goldfarb, D. and Mehrotra, S. (1989), A Self–Correcting Version of Karmarkar’s Algorithm, SIAM J. Numer. Anal. 26 (4), 1006–1015.
Goldfarb, D. and Xiao, D. (1989), A Primal Projective Interior Point Method for Linear Programming, Technical Report, Department of Industrial Engineering and Operations Research, Columbia University.
Golitschek, M.V. (1987), The Cost–to–Time Ratio Problem for Large or Infinite Graphs, Discrete Appl. Math. 16 (1), 1–9.
Golub, G.H. and Underwood, R. (1970), Stationary Values of the Ratio of Quadratic Forms Subject to Linear Constraints, Zeitschrift für Angewandte Mathematik und Physik 21, 318–326.
Gondran, M. (1980), Les Problemes de Ratio Minimum en Optimization Combinatoire, Note EDF HI 13433–02.
Gondran, M. (1982), Optimization with Rational Objective Functions, Bull. Direction Etudes Rech. Ser. C Math. Inform. 1, 49–54.
Goswami, M.K. and Sharma, J.K. (1986), Enumerative Technique for Fractional Programming Problem, Methods of Operations Research 56, 93–100.
Goswami, M.K. and Sharma, J.K. (1988), Cutting Plane Technique for Linear Fractional Complementary Programming Problem, Acta Cienc. Indica 14 (2), 75–83..
Goswami, M.K. and Sharma, J.K. (1990), Fractional Fixed Charge Complementary Programming Problem, Methods of Operations Research 61, 21–35.
Granot, D. and Granot, F. (1976), On Solving Fractional (0,1) Programs by Implicit Enumeration, Canadian Journal of Operational Research and Information Processing 14, 241–249.
Granot, D. and Granot, F. (1977), On Integer and Mixed Integer Fractional Programming Problems, Annals of Discrete Mathematics 1, 221–231.
Grinold, R.C. and Stanford, R.E. (1976), Limiting Distributions in a Linear Fractional Flow Model, SIAM Journal on Applied Mathematics 30, 402406.
Grunspan, M. (1971), Fractional Programming: A Survey, Technical Report 50, Department of Industrial and Systems Engineering, University of Florida.
Grunspan, M. and Thomas, M.E. (1973), Hyperbolic Integer Programming, Naval Research Logistics Quarterly 20, 341–356.
Guerra, F. and Verdaguer, R. (1988), An Algorithm for Two–Criterion Linear Fractional Programming, Investigacion Operacional 9 (3), 3–13. [Spanish]
Gugat, M. (1991), A Method for General Restricted Rational TchebycheffApproximation, Technical Report, Department of Mathematics, University of Trier, Germany.
Gulati, T.R. (1975), Optimality Criterion and Duality in Complex Fractional and Indefinite Programming, Ph.D. Thesis, I.I.T. New Delhi.
Gulati, T.R. (1979), Duality for a Nondifferentiable Fractional Program, Cahiers du Centre d’Etudes de Recherche Operationelle 21, 325–330.
Gulati, T.R. and Chandra, S. (1975), A Duality Theorem for Complex Fractional Programming, Zeitschrift für Angewandte Mathematik and Mechanik 55, 348–349.
Gulati, T.R. and Islam, M.A. (1988), Proper Efficiency in a Linear Fractional Vector Maximization Problem with Generalized Convex Constraints, European Journal of Operational Research 36 (3), 339–345.
Gulati, T.R and Islam, M.A. (1989), Efficiency in Linear Fractional Vector Maximization Problem with Nonlinear Constraints, Optimization 20 (4), 477–482.
Gulati, T.R. and Talaat, N. (1991), Duality in Nonconvex Multiobjective Programming, Asia–Pacific J. Oper. Res. 8 (1), 62–69.
Gulati, T.R. and Talaat, N. (1991), Duality in Nonconvex Vector Minimum Problems, Bull. Austral. Math. Soc. 44 (3), 501–509.
Gupta, A.K. and Sharma, J.K. (1981), Fractional Functional Programming, A Brief Survey, Technical Report, Saharanpur.
Gupta, B. (1981), Finding the Set of all Efficient Solutions for the Linear Fractional Multi–Objective Program with Zero–One Variables, Opsearch 18 (4), 204–214.
Gupta, B. (1982), Existence and Duality Relations for Multi–Objective Programs in Complex Space, Opsearch 19 (3), 178–182.
Gupta, B. (1983), Linear Fractional Vector Maximization Problem. Existence and Duality, Cahiers du Centre d’Etudes de Recherche Operationelle 25 (1–2), 35–40.
Gupta, B. (1983), Programming with Multi–Objective Linear Fractional Functionals, Acta Ciencia Indica Mathematics 9 (1–4), 195–201.
Gupta, B. and Swamp, K. (1981), On Extreme Point Fractional Programming, Portugaliae Mathematica 37 (1–2), 13–29.
Gupta, B., Swamp, K. and Banwarilal (1981), Stochastic Fractional Programming under Chance Constraints with Random Technology Matrix, Gujarat Statistical Review 8 (1), 23–34.
Gupta, R.K. (1971), Basic Feasible Solutions and Decomposition Principle for Linear Fractional Functional Programming Problem, Trabajos de Estradistica 22, 185–193.
Gupta, R.K. (1973), A Simple Class of Parametric Linear Fractional Functionals Programming Problem, Cahiers du Centre d’Etudes et de Recherche Operationelle 15, 185–196.
Gupta, R.K. (1974), A Simple Class of Parametric Linear Fractional Functional Programming Problem, Erratum, Cahiers du Centre d’Etudes de Recherche Operationelle 16, 179.
Gupta, R.K. (1977), Decomposition Method and Transportation Type Problems with a Fractional Objective Function, Zeitschrift für Angewandte Mathematik und Mechanik 57, 81–88.
Gupta, R.K. and Bector, C.R. (1968), Nature of Quotients, Products and Rational Powers of Convex (Concave)–Like Functions, Mathematical Student 36, 63–67.
Gupta, R.K. and Bhatt, S.K. (1972), Generalized Pseudo–Convex Functions, Cahiers du Centre d’Etudes de Recherche Operationelle 14, 213–222.
Gupta, R.K. and Swarup, K. (1969), Approximate Method of Solution for Nonlinear Fractional Programming, Zeitschrift für Angewandte Mathematik und Mechanik 49, 753–756.
Gupta, R.K. and Swarup, K. (1974), A Cutting Plane Algorithm for Extreme Point Linear Fractional Functional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 16, 161–177.
Gupta, R.K. and Swarup, K. (1978), On Extreme Point Fractional Programming, Portugaliae Mathematica 37, 13–29.
Gupta, S.N. and Jain, A.K. (1986), Optimization with the Ratio of Independent Normal Variates, Acta Ciencia Indica Mathematics 12 (3), 209–212.
Gupta, S.N. and Jain, R.K. (1986), Stochastic Fractional Programming under Chance Constraints with Random Technology Matrix, Acta Ciencia Indica Mathematics 12 (3), 191–198.
Gupta, S.N. and Swamp, K. (1979), Stochastic Fractional Functionals Programming, Ricerca Operativa 9 (10), 65–79.
Gupta, S.N. and Swarup, K. (1980), Duality in Stochastic Fractional Functionals Programming, Ricerca Operativa 10 (3), 53–63.
Gupta, S.N., Jain, A.K. and Swarup, K. (1987), Stochastic Linear Fractional Programming with the Ratio of Independent Cauchy Variates, Naval Research Logistics 34 (2), 293–305.
Gutenberg, E. (1975), Einführung in die Betriebswirtschaftslehre, Gabler, Wiesbaden.
Gwinner, J. (1987), A General Farkas Lemma and Applications in Duality. XIth Symposium on Operations Research, Methods of Operations Research 57, 25–48.
Gwinner, J. and Jeyakumar, V. (1993), A Solvability Theorem and Minimax Fractional Programming, Zeitschrift für Operations Research 37, 1–12.
Hagopian, J.D. and Frisch, I.T. (1972), Capacitance, Inductance and Resistance Minimization in RLC Networks, IEEE Trans. Circuit Theory CT–19, 383385.
Haimovici, A. and Rimer, S. (1962), On a Problem of Generalized Linear Programming, Bul. Inst. Politehn. Iasi 8 (12), 1–2, 9–14. [Romanian]
Halpern, J. (1972), Ratios in Planning, Budgeting and Bounds on Resource Requirements, Operations Research 20, 974–983.
Hammer, P.L. and Rudeanu, S. (1968), Boolean Methods in Operations Research and Related Areas, Springer, Berlin – Heidelberg – New York.
Hannan, E.I__ (1977), Effects of Substituting a Linear Goal for a Fractional Goal in the Goal Programming Problem, Management Science 24, 105107.
Hannan, E.L. (1981), On an Interpretation of Fractional Objectives in Goal Programming as Related to Papers by Awerbuch et al. and Hannan, Management Science 27, 847–848.
Hansen, P., Poggi de Aragäo, M.V. and Ribeiro, C. (1991), Hyperbolic 0–1 Programming and Query Optimization in Information Retrieval, Mathematical Programming 52 (Ser. B), 255–263.
Hanson, M.A. (1981), On Sufficiency of the Kuhn–Tucker Conditions, Journal of Mathematical Analysis and Applications 80 (2), 545–550.
Hanson, M.A. (1989), Continuous–Time Programming, Journal of Information and Optimization Sciences 10 (1), 129–140.
Hanssman, F. (1968), Probability of Survival as an Investment Criterion, Management Science, Theory 15 (1), 33–48.
Hartmann, K. (1973), Einige Aspekte der Ganzzahligen Linearen Quotientenoptimierung, Wiss. Z. Tech. Hochschule Chem. Launa–Merseburg 15, 413418.
Hartmann, K. (1975), Gemischt Ganzzahlige Lineare Quotientenoptimierung nach dem Schnittvverfahren von Gomory, Mathematische Operationsforschung und Statistik 6, 845–854.
Hartmann, K. (1975), Rein ganzzahlige lineare Quotientenoptimierung nach dem Schnittebenverfahren von Gomory, Mathematische Operationenforschung und Statistik 6, 33–53.
Hartwig, H. (1975), Ein simplexartiger Lösungsalgorithmus für pseudolineare Optimierungsprobleme, Studia Scientarum Mathematicarum Hungarica 10, 213–236.
Harvath, I. (1974), Linear Fractional Programming with Additional Constraints, Revista de Analiza Numerica si Teoria Aproximatiei 3, 71–77. [Romanian]
Harvath, I. (1981), Asupra Programarii Fractionare Lineare cu Restrictii Suplimentaro, Informatica Pentru Conducere, Orizont ‘81, Relizari si Aplicatii, Cluj–Napoca, 101–102.
Harvey, C.M. (1981), Stochastic Programming Models for Decreasing Risk Aversion, J. Oper. Res. Soc. 32 (10), 885–889.
Hashizume, S., Fukushima, M., Katoh, N. and Ibaraki, T. (1987), Approximation Algorithms for Combinatorial Fractional Programming Problems, Mathematical Programming 37 (3), 255–267.
Hearn, D. and Randolph, W.D. (1973), Dual Approaches to Quadratically Constrained Quadratic Programming, Technical Report, Department of Industrial and Systems Engineering, University of Florida, Gainesville.
Heinen, E. (1970), Betriebliche Kennzahlen, Eine organisationstheoretische und kybernetische Analyse, in Lindhardt, H., Penzkofer, P. und Scherpf, P., (eds.), Dienstleistungen in Theorie und Praxis, Stuttgart, 227–236.
Heinen, E. (1971), Grundlagen betriebswirtschaftlicher Entscheidungen, Das Zielsystem der Unternehmung, 2. Aufl., Gabler, Wiesbaden.
Helbig, S. (1990), Optimality Criteria in Disjunctive Optimization and some Applications, Methods of Operations Research 62, 67–78.
Hillier, F.S. and Lieberman, G.J. (1990), Introduction to Operations Research, 5th Edition, McGraw–Hill, New York.
Hirche, J. (1975), Zur Extremwertannahme und Dualität bei Optimierungsproblemen mit linearem und gebrochen linearem Zielfunktionsanteil, Zeitschrift für Angewandte Mathematik und Mechanik 55,184–185.
Hirche, J. (1977), Nichtkonvexe Optimierungsprobleme mit zusammengesetzten Zielfunktionen, Dissertation B, Martin–LutherUniversität, Halle.
Hirche, J. (1978), Berichtigung: Über eine Klasse Nichtkonvexer Optimierungs–probleme, Zeitschrift für Angewandte Mathematik und Mechanik 58 (8), 367.
Hirche, J. (1979), Vektorminimumprobleme mit verallgemeinert konvexen Zielfunktionen, Bericht des 24. Internationalen wissenschaftlichen Kolloquiums der Technischen Hochschule Ilmenau, Vortragsreihe Bl , 13—16.
Hirche, J. (1980), Optimierungsprobleme mit zusammengesetzten Zielfunktionen und Vektoroptimierung, Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau 26, 123–134.
Hirche, J. (1981), A Note on Minimizing a Polynomial of a Linear Fractional Function, Academie de la Republique Populaire de Roumane. Revue Roumaine de Mathematiques Pures et Appliquees 26 (9), 1193–1195.
Hirche, J. (1981), Zur Lösung von Optimierungsproblemen mit Monoton–Linear Zusammengesetzten Zielfunktionen, Beiträge zur Num. Math. 9, 87–94.
Hirche, J. (1983), Verallgemeinerte Konvexität bei Summen und Produkten Linearer und Gebrochen–Linearer Funktionen, Wiss. Z. Univ. Halle 32, 9199.
Hirche, J. (1984), On Programming Problems with a Linear plus Linear–Fractional Objective Function, Cahiers du Centre d’Etudes de Recherche Operationelle 26 (1–2), 59–64.
Hirche, J. (1985), Some Remarks on Generalized Convexity of Sums and Products, Zeitschrift für Angewandte Mathematik und Mechanik 65 (1), 62–63.
Hirche, J. and Tan, H.K. (1977), Über eine Klasse nichtkonvexer Optimierungsprobleme, Zeitschrift für Angewandte Mathematik und Mechanik 57, 247–253.
Hirche, J., Köhler, J. and Stieblitz, V. (1975), Ein effektives Lösungsverfahren für das hyperbolische Transportproblem, Beiträge zur Analysis 7, 151156.
Ho, J.K. (1986), A Parametric Subproblem for Dual Methods in Decomposition, Math. Oper. Res. 11, 611 650.
Hodgson, T.J. and Lowe, T.J. (1982), Production Lot Sizing with Material Handling Cost Considerations, AILE Transactions 14, 44–51.
Hordijk, A. (1974), Dynamic Programming and Marcov Potential Theory, Mathematical Centre Tracts 51, Amsterdam.
Horibe, Y. (1974), On the Optimum Ratio of Word Length in Communication, IEEE Trans. Inform. Theory IT–20, 756–758.
Horst, R. (1984), On the Convexification of Nonlinear Programming Problems: An Applications Oriented Survey, European Journal of Operations Research 15, 382–392.
Hoskins, J.A. and Blom, R. (1984), Optimal Allocation of Warehouse Personnel: A Case Study Using Fractional Programming, FOCUS (U.K.) 3 (2), 13–21.
Ibaraki, T. (1980), Recent Advances in Mathematical Programming. V. Fractional Programming, Systems and Control (Shisutemu to Seigyo) 24 (12), 787–797. [Japanese]
Ibaraki, T. (1981), Solving Mathematical Programming Problems with Fractional Objective Functions, in Schaible, S. and Ziemba, W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 441–472.
Ibaraki, T. (1983), Parametric Approaches to Fractional Programs, Mathematical Programming 26 (3), 345–362.
Ibaraki, T., Ischii, H., Iwase, J., Hasegawa, T. and Mine, H. (1976), Algorithms for Quadratic Fractional Programming Problems, Journal of the Operations Research Society of Japan 19, 174–191.
Ichimori, T., Ishii, H. and Nishida, T. (1981), Algorithm for One Job Machine Job Sequencing with Precedence Constraints, Journal of the Operations Research Society of Japan 24, 159–168.
Lida, K. (1989), Optimal Stopping of a Contact Investigation in Two–Stage Search, Mathematica Japonica 34 (2), 169–190.
Intriligator, M.D. (1971), Mathematical Optimization and Economic Theory, Prentice–Hall, Englewood Cliffs, N.J.
Isbell, J.R. and Marlow, W.H (1956), Attrition Games, Naval Research Logistics Oiunrterly 3, 71–94.
Ishii, H. and Nishida, T. (1984), Stochastic Linear Knapsack Problem: Probability Maximization Model, Mathematica Japonica 29 (2), 273–281.
Ishii, H., Ibaraki, T. and Mine, H. (1976), A Primal Cutting Plane Algorithm for Integer Fractional Programming Problems, Journal of the Operations Research Society of Japan 19, 228–244.
Ishii, H., Ibaraki, T. and Mine, H. (1977), Fractional Knapsack Problems, Mathematical Programming 13, 255–271.
Ishii, H., Nishida, T. and Daino, A. (1979), Fractional Set Covering Problems, Technology Reports of the Osaka University 29, 319–326.
Ivanov, E.H. and Nehse, R. (1983), Relations between Generalized Concepts of Convexity and Conjugacy, Mathematische Operationsforschung and Statistik, Serie Optimization 13 (1), 1–9.
Jacobsen, S. (1967), Comparison of the Primal, Dual and Primal–Dual Decomposition Algorithms, Notes on Operations Research 6, Report ORC 67–17, Operations Research Center, University of California, Berkeley.
Jagannathan, R. (1966), On Some Properties of Programming Problems in Parametric Form Pertaining to Fractional Programming, Management Science 12, 609–615.
Jagannathan, R. (1973), Duality for Nonlinear Fractional Programs, Zeitschrift für Operations Research 17, Series A, 1–3.
Jagannathan, R. (1985), An Algorithm for a Class of Nonconvex Programmingg Problems with Nonlinear Fractional Objectives, Management Science 31 (7), 847–851.
Jagannathan, R. and Schaible, S. (1983), Duality in Generalized Fractional Programming via Farkas’ Lemma, Journal of Optimization Theory and Applications 41 (3), 417–424.
Jagannathan, R. and Schaible, S. (1984), An Application of Farkas’ Lemma to a Nonconvex Minimization Problem, in Walter, W., (ed.), General Inequalities IV, 4th Int. Conf. on General Inequalities, Oberwolfach/Germany 1983, ISNM 71, 365–367.
Jain, O.P. (1979), Duality for Fractional Functional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 21 (1), 81–86.
Jeyakumar, V. (1985), First and Second Order Fractional Programming Duality, Opsearch 22 (1), 24–41.
Jeyakumar, V. and Mond, B. (1992), On Generalized Convex Mathematical Programming, Australian Mathematical Society Journal, Series B Applied Mathematics 34 (1), 43–53.
Joksch, H , (1964), Programming with Fractional Linear Objective Functions, Naval Research Logistics Quarterly 11, 197–204.
Jüttler, H. (1967), Die Lineare’Quotientenoptimierung als Hilfsmittel für die Entscheidungsfindung, Rechentechnik – Datenverarbeitung 11.
Jüttier, H. (1969), Untersuchungen zu Fragen der Operationsforschung und ihrer Anwendungsmöglichkeiten auf ökonomische Problemstellungen unter besonderer Berücksichtigung der Spieltheorie, Dissertation, Wirtschafts–wissenschaftliche Fakultät der Humoldt–Universität, Berlin, 144–181.
Kabe, D.G. (1980), Direct Solutions to the m–Median and Fractional Transportation Problems, Ind. Math. 30, 1–27.
Kabe, D.G. (1980), On a Certain Linear Fractional Programming Problem, Indian Journal of Pure and Applied Mathematics 11, 1411–1413.
Kabe, D.G. (1984), Direct Solutions to some Linear Programming Problems, Ind. Math. 34 (1), 1–20.
Kac’janjuk, S.A. (1969), A Certain Problem in Infinite Linear Fractional Programming, ‘Zurnal Vycislitel not Matematiki i Matematiceskoi Fiziki 9, 413–417. [Russian]
Kacnel’son, L.Z. and Neizvestnyi, M.M. (1976), Isoextremal Cebysev Fractions with a Denominator of Second Degree, Latvian Mathematical Yearbook 17, 24–29. [Russian]
Kallberg, J.G. and Ziemba, W.T. (1981), Generalized Concave Functions in Stochastic Programming and Portfolio Theory, in Schaible, S. and Ziemba, W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 719–767.
Kaltinska, R. (1981), An Algorithm for the Solution of the Hyperbolic Transportation Problem, Seminarbericht, Humboldt Universität Berlin, Sektion Mathematik 39, 104–113. [Russian]
Kanchan, P.K. (1976), Linear Fractional Functional Programming, Acta Ciencia Indica 2, 401–405.
Kanchan, P.K. (1977), Upper Bounds in Linear and Piecewise Linear Programming, Acta Ciencia Indica 3, 357–360.
Kanchan, P.K., Holland, A.S.B. and Sahney, B.N. (1981), Transportation Techniques in Linear–Plus–Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 23, 153–157.
Kang, R.S. and Chong, Y.C. (1989), On the Minimization of the Sum of Two Linear Fractional Functions, Cho son Min ju ui In min Kong hwa kuk Kwa hak won. T’ong bo. Academy of Science of the People’s Republic of Korea. Bulletin 4, 2–4. [Korean]
Karp, R.M. (1977), A Characterization of the Minimum Cycle Mean in a Digraph, Memorandum No. UCB/ERL 147, Electronic Research Laboratory, College of Engineering, University of California, Berkeley.
Kas’yanyuk, S.A. and Kucher, B.M. (1967), Monotonic Programming, Dupovidi Akademii Nauk URSR (USSR) A, 18–21. [Ukrainian]
Kaska, J. (1969), Duality in Linear Fractional Programs, EconomickoMatematicky Obzor 5, 442–453. [Czech]
Kaska, J. and Pisek, M. (1964), Linearni Lomene Programovani a Jeho uplate Neni v Planovani Stavebni vyroby, Pozemni Stavby 2.
Kaska, J. and Pisek, M. (1965), Linearni Lomene Programovani, Statistika a Demografie V, 191–207.
Kaska, J. and Pisek, M. (1966), Quadratic–Linear Fractional Programming, Economicko–Matematicky Obzor 2, 169–173. [Czech]
Kaska, J. and Pisek, M. (1967), Convex–Concave Fractional Programming, Economicko–Matematicky Obzor 3, 457–464. [Czech]
Kasyanyuk, S.A. (1969), A Certain Problem in Infinite Linear Fractional Programming, Z. Vycisl. Mat. i Mat. Fiz. 9, 413–417. [Russian]
Kataoka, S. (1963), A Stochastic Programming Model, Econometrica 11, 1839.
Kataoka, S. (1967), Stochastic Programming Maximum Probability Model, Hitotsubashi Journal of Arts and Science 8, 51–59.
Kaufmann, E.H. and Taylor, G.D. (1981), Uniform Approximation by Rational Functions Restricted Denominators, Journal of Approximation Theory 32, 9–26.
Kaul, R.N. and Bhatia, D. (1974), Generalized Linear Fractional Programming, Ekonomicko–Matematicky Obzor 10, 322–330.
Kaul, R.N. and Chadha, S.S. (1971), Duality in Nonlinear Fractional Programming Problems, Ekonomicko–Matematicky Obzor 7, 141–148.
Kaul, R.N. and Datta, N. (1981), On the Solution of Separable Programming Problem with a Fractional Objective Function, Cahiers du Centre d’ Etudes de Recherche Operationelle 23, 159–169.
Kaul, R.N. and Gupta, B. (1980), Efficiency and Linear Vector Maximum Value Problem, Zeitschrift für Angewandte Mathematik und Mechanik 60, 112–113.
Kaul, R.N. and Gupta, B. (1981), Multi–Objective Programming in Complex Space, Zeitschrift für Angewandte Mathematik und Mechanik 61 (11), 599–601.
Kaul, R.N. and Lata, M. (1974), A Method of Decomposition for Linear Fractional Programming, Opsearch 11, 183–192.
Kaul, R.N. and Lyall, V. (1989), A Note on Nonlinear Fractional Vector Maximization, Opsearch 26 (2), 108–121.
Kaul, R.N., Kaur, S. and Lyall, V. (1986), Duality in Inexact Fractional Programming with Set–Inclusive Costraints, Journal of Optimization Theory and Applications 50 (2), 279–288.
Kaul, R.N., Lyall, V. and Kaur, S. (1988), Semilocal Pseudolinearity and’ Efficiency, European J. of Operational Research 36, 402–4.10.
Kaur, S. (1981), Inexact Fractional Programming with Set Inclusive Constraints, Cahiers du Centre d’Etudes de Recherche Operationelle 23, 171–181.
Kaur, S. (1982), Subgradient Duality in Fractional Programming, Indian Journal of Pure and Applied Mathematics 13 (3), 287–298.
Kaur, S. and Bhatia, D. (1983), Duality Theory for Generalized Fractional Programs, Indian Journal of Pure and Applied Mathematics 14 (2), 257264.
Kawohl, B. (1985), Rearrangements and Convexity of Level Sets in Partial Differential Equations, Lecture Notes in Mathematics 1150, Springer Verlag, Heidelberg.
Kawohl, B. (1986), Geometrical Properties of Level Sets of Solutions to Elliptic Problems, Proc. Symp. Pure Math. 45, 25–36.
Kern, W. (1960), Rentabilitätsanalyse, Zeitschrift für handelswissenschaftliche Forschung 12, 17–40.
Kern, W. (1971), Kennzahlensysteme als Niederschlag interdependenter Unternehmungsplanung, Zeitschrift für betriebswirtschaftliche Forschung 23, 701–718.
Khalitov, N.T. (1988), A Linear–Fractional Programming Problem in Singular Cases, J. Soy. Math. 40, (6) 725–727; Translation from Issled. Pritel. Mat. 1, 1973, 38–40. [Russian].
Khan, S.U. and Bari, A. (1977), A Procedure for Integer Solutions to some Allocation Problems, Pure and Applied Mathematical Sciences 5, 25–32.
Khan, Z.A. ( 1990), Converse Duality in Nonlinear Fractional Programming, Asia Pacific Journal of Operational Research 7 (1), 9–15.
Khristov, G., Karamiteva, Z. and Stoyanov, T.E. (1985), A Numerical Method and Program for Solving a Linear–Fractional Integer Optimization Problem, Mathematics and Mathematical Education, Blgar. Akad. Nauk, Sofia, 566–570. [Bulgarian]
Kirsch, W. (1968), Gewinn und Rentabilität, Gabler, Wiesbaden.
Klafszky, E., Mayer, J. and Terlaky, T. (1989), Linearly Constrained Estimation by Mathematical Programming, European Journal of Operational Research 42 (3), 254–267.
Klein, M. (1962), Inspection – Maintenance – Replacement Schedule Under Marcovian Deterioration, Management Science 9, 25–32.
Kleinmann, P. (1978), Quantitative Sensitivitätsanalyse bei Parametrischen Optimierungsaufgaben, Seminarberichte, Humboldt Universität Berlin, 9. Sektion Mathematik, 105 pp.
Klevachev, V.I. (1968), Solving the Problem of Linear Fractional Programming, Kibernetika 4, 27–31.
Kloock, J. (1974), Kurzfristige Produktionsplanungsmodelle auf der Basis von Entscheidungsfeldern mit alternativer Fremd–und Eigenfertigung (mit variablen Produktionstiefen), Zeitschrift für betriebswirtschaftliche Forschung 26, 671–682.
Köhler, J. (1981), An Algorithm for the Solution of Integer Hyperbolic Programming Problems, Mathematische Optimierungstheorie und Anwendungen, Internationale Tagung, Eisenach 1981, 89–92.
Köhler, J. and Hirche, J. (1975), Mehrfache Zerlegung hyperbolischer Optimierungsaufgaben, Beiträge zur Analysis 7, 143–149.
Komlosi, S., Rapcsak, T. and Schaible, S. (eds.) (1994), Generalized Convexity, Proceedings Pécs/Hungary, 1992; Lecture Notes in Economics and Mathematical Systems 405, Springer–Verlag, Berlin–Heidelberg–New York, to appear.
Konno, H. and Inori, M. (1989), Bond Portfolio Optimization by Bilinear Fractional Programming, Journal of the Operations Research Society of Japan 32 (2), 143–158.
Konno, H. and Kuno, T. (1990), Generalized Linear Multiplicative and Fractional Programming. Computational Methods in Global Optimization, Annals of Operations Research 25 (1–4), 147–161.
Konno, H. and Yajima, Y. (1992), Minimizing and Maximizing the Product of Linear Fractional Functions, Collection: Recent Advances in Global Optimization, 259–273.
Konno, H., Yajima, Y. and Matsui, T. (1991), Parametric Simplex Algorithms for Solving a Special Class of Nonconvex Minimization Problems, J. of Global Optimization 1, 65–81.
Kombluth, J.S.H. (1973), A Survey of Goal Programming, Omega 1, 193–205.
Kombluth, J.S.H. (1979), Indifference Regions and Marginal Utility Weights in Multiple Objective Linear Fractional Programming, Working Paper 7902–3, Dept. of Decision Sciences, The Wharton School, University of Pennsylvania.
Kombluth, J.S.H. (1981), Multiple Objective Linear Fractional Programming Algorithms. Some Computational Experience, Lecture Notes in Economics and Mathematical Systems 190, 173–198.
Kornbluth, J.S.H. (1982), Max–Min Programming with Linear Fractional Functions; Algorithms and Examples. Essays and Surveys on Multiple Criteria Decision Making, Lecture Notes in Economics and Mathematical Systems 209, Springer, Berlin–New York, 204–213.
Kornbluth, J.S.H. (1983), Ratio Goals in Manpower Planning Models, INFOR—Canadian J. Oper. Res. Inform. Process. 21 (2), 151–154.
Kornbluth, J.S.H. (1986), On the Use of Multiple Objective Linear Programming Algorithms to Solve Problems with Fractional Objectives, European Journal of Operational Research 23 (1), 78–81.
Kornbluth, J.S.H. and Salkin, G.R. (1972), A Note on the Economic Interpretation of the Dual Variables in Linear Fractional Programming, Zeitschrift für Angewandte Mathematik und Mechanik 52, 175–178.
Kornbluth, J.S.H. and Salkin, G.R. (1974), The Optimal Dual Solution in Linear Fractional Decomposition Problems, Operations Research 22, 183189.
Kornbluth, J.S.H. and Salkin, G.R. (1975), A Note on Returns to Scale in Linear Fractional Programming, Zeitschrift für Angewandte Mathematik und Mechanik 55, 757–758.
Kornbluth, J.S.H. and Steuer, R.E. (1980), On Computing the Set of all Weakly Efficient Vertices in Multiple Objective Linear Fractional Programming, in Fandel, G., and Gal, T., (eds.), Multiple Criteria Decision Making – Theory and Application, Lecture Notes in Economics and Mathematical Systems 177, 189–202.
Kornbluth, J.S.H. and Steuer, R.E. (1981), Goal Programming with Linear Fractional Criteria, European Journal of Operational Research 8, 58–65.
Kornbluth, J.S.H. and Steuer, R.E. (1981), Multiple Objective Linear Fractional Programming, Management Science 27, 1024–1039.
Körth, H. (1967), Zur Quotientenoptimierung, Berichte des XII. Internationalen wissenschaftlichen Kolloquiums, Technische Hochschule Ilmenau, 95–99.
Körth, H. (1969), Ein Zerlegungsprinzip für die hyperbolische Optimierung, Wissenschaftliche Zeitschrift der Humboldt–Universität zu Berlin, Ges. – Sprachw. R. XVIII, 827–829.
Körth, H. (1969), Untersuchungen zur nichtlinearen Optimierung ökonomischer Erscheinungen und Prozesse unter besonderer Berücksichtigung der Quotientenprogrammierung sowie der Lösung ökonomisch–mathematischer Modelle bei Existenz mehrerer Zielfunktionen, Habilitationsschrift, Humboldt–Universität, Berlin, Sektion Wirtschaftswissenschaft.
Körth, H. (1970), Transportoptimierung mit hyperbolischer Zielfunktion, Wissenschaftliche Zeitschrift der Humboldt–Universität zu Berlin, Ges. – Sprachw. R. XIX, 737–740.
Körth, H. (1971), Hyperbolische Transportoptimierung mit Beschränkung der Variablen, Wissenschaftliche Zeitschrift der Humboldt–Universität zu Berlin, Ges. – Sprachw. R. XX, 521–524.
Körth, H. (1979), Verallgemeinerte hyperbolische Optimierungsaufgabe, Berichte des 24. Internationalen wissenschaftlichen Kolloquiums, Technische Hochschule Ilmenau, 4, 41–43.
Kovacs, A. and Stahl, J. (1973), Decomposition Procedure in Case of Maximizing the Index of Enterprise Interest, Szigma 6, 105–114. [Hungarian]
Kovacs, A. and Stahl, J. (1976), On Large Scale Linear Fractional Programs, Lecture Notes in Computer Science 41, 353–361.
Kreko, B. (1974), Optimierung, Nichtlineare Modelle, Deutscher Verlag der Wissenschaften, Berlin, and Akadémiai Kiadó, Budapest.
Kreutzberger, O. (1978), Bemerkungen zur Quotienten–und Produktoptimierung mit Anwendung beim Risikoproblem der stochastischen Optimierung, Wissenschaftliche Zeitschrift der MartinLuther–Universität Halle – Wittenberg, Mathematisch–Naturwissenschaftliche Reihe 27, 75–79.
Krupitskij, A.E. (1983), Minimization of the Sum of Two Linear–Fractional Functions on a Convex Polyhedral Set, Vestnik Lenigradskogo Universiteta Matematika, Mekhanika, Astronomiya 3, 15–20. [Russian]
Krystyna, Z. (1969/70), On the Optimum Rational Function Connected with the ADI–Method, Zastors. Mat. 11, 337–352.
Kucher, B.N. (1967), On One Algorithm for Solving a Problem of Monotone Programming, Dopovidi Akad. Nauk. Ukrain. RSR Ser. A 10, 869–872. [Russian]
Kydland, F. (1969), Simulation of Linear Operators, Institute of Shipping Research, Norwegian School of Economics and Business Administration, Bergen; translated and reprinted from Sosialoekonomen 23.
Kydland, F. (1972), Duality in Fractional Programming, Naval Research Logistics Quarterly 19, 691–697.
Lal, S.N., Mukherjee, R.N. and Singh, A.K. (1990), Sufficiency of Exact Penalty Minimization and Fractional Programming, Opsearch 27 (3), 165170.
Lange, E.G. and Artjuhin, A.V. (1974), Some Problems of Linear–Fractional Parametric Programming, in Mathematical Methods for the Solution of Mathematical–Economic Problems, Izdat, “Ilim”, Frunze, 5–19. [Russian]
Lasdon, L.S. (1970), Optimization Theory for Large Systems, MacMillan, London.
Lata, M. (1975), Strong Pseudoconvex Programming in Banach Space, Indian Journal of Pure and Applied Mathematics 6, 45–48.
Lata, M. and Mittal, B.S. (1975), An Operator Theory for a Class of Linear Fractional Programming Problems, I, Zeitschrift für Angewandte Mathematik und Mechanik 55, 133–140.
Lata, M. and Mittal, B.S. (1976), A Decomposition Method for Interval Linear Fractional Programming Problems, Zeitschrift für Angewandte Mathematik und Mechanik 56, 153–159.
Lata, M. and Mittal, B.S. (1976), An Operator Theory for a Class of Linear Fractional Programming Problems, II, Zeitschrift für Angewandte Mathematik und Mechanik 56, 75–88.
Lawler, E.M. (1972), Optimal Cycles in Graphs and the Minimal Cost–to–Time Ratio Problem, Technical Report ERL–M343, Department of Electrical Engineering, University of California, Berkeley.
Lawler, E.M. (1976), Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York.
Lazarev, I.A. (1975), Determination of the Extremum in a Nonregular Network Problem with Fixed Costs Involving Penalties, Izvestija Academii Nauk SSSR, Tekhnicheskaya Kibernetika 2, 36–46. [Russian]
Lee, S.M. and Clayton, E.R. (1972), A Goal Programming Model for Academic Resource Allocation, Management Science 18, B395–B408.
Lee, Y.H. and Shin, K.G. (1987), Optimal Reconfiguration Strategy for a Degradable Multimodule Computing System, Journal of the Association for Computing Machinery 34 (2), 326–348.
Leleno, J. (1977), Remarks on the Algorithm of B. Martos for the Solution of Hyperbolic Programming Problems, Przeglad Stalystyczny 24, 399–408. [Polish]
Lemke, C.E. and Powers, T.J. (1961), A Dual Decomposition Principal, RPI Math. Report 48, Rensselaer Polytechnic Institute, Troy, N.Y.
Lin, C.Y. (1981), Comparison of Duality Models in Nonlinear Fractional Programming, Numerical Mathematics. A Journal of Chinese Universities. Gaodeng Xuexiao Jisuan Shxue Xuebao 3 (3), 270–272. [Chinese]
Lin, C.Y. (1982), Duality Theory for Multi–Objective Fractional Programming, Numerical Mathematics. A Journal of Chinese Universities. Gaodeng Xuexiao Jisuan Shxue Xuebao 4 (4), 289–299. [Chinese]
Lin, C.Y. (1983), The Fundamental Theorem of Multiobjective Fractional Programming, Acta Mathematicae Applicatae Sinica. Yingyong Shuxue Xuebao 6 (2), 247–250. [Chinese]
Lintner, J. (1965), The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets, Review of Economics and Statistics 18, 13–37.
Liu, S.Y. (1990), Symmetric Duality of Fractional Programming, Xi’an Jiaotong Daxue Xuebao. Journal of Xi’an Jiaotong University 24 (6), 135138. [Chinese]
Lommis, L.H. (1946), On a Theorem of Von Neumann, Proc. Nat. Acad. Sci. (USA) 32, 213–215.
Lücke, W. (1969), Produktions–und Kostentheorie, Physica–Verlag, Würzburg–Wien.
Luhandjula, M.K. (1984), Fuzzy Approaches for Multiple Objective Linear Fractional Optimization, Fuzzy Sets and Systems 13 (1), 11–23.
Lupsa, L. (1978), Asupra Alurii Unor Functii Hiperbolice, Studia Univ. BabesBolyai, Math 2, 66–72.
Lupsa, L. (1978), Remarques Concernant le Rapport entre les Problemes de Programmation Quadratique Indefinie et les Problemes de Programmation Hyperbolique, Studia Universitatis Babes – Bolyai. Mathematica 23, 5054.
Lyall, V. and Kaur–Suneja, S. (1987), Decomposition in Generalized Fractional Programming and its Optimal Dual Solution, Indian Journal of Pure and Applied Mathematics 18 (11), 973–978.
Lysenko, A.I, Dabagjian, A.V. and Gorelyi, A.V. (1977), Analytic Solution of a Nonlinear Programming Problem, Avtomat. Sistemy Upravlenijayi Pribory Avtomat 42 (3), 53–62. [Russian]
Mahajan, D.G. and Vartak, M.N. (1977), Generalization of Some Duality Theorems in Nonlinear Programming, Mathematical Programming 12, 293–317.
Major, D.C. (1969), Benefit–Cost Ratios for Projects In Multiple Objective Investment Programming, Water Resources Research 6 (5), 1174–1178.
Makhmudov, R.A. (1988), On an Algorithm for Solving a Covering Problem with Linear Fractional Functional, Izv. Akad. Nauk Az. SSR, Ser. Fiz.Tekh. Mat. Nauk 3, 128–130. [Russian]
Manas, M. (1968), On Transformations of Quasi–Convex Programming Problems, Ekonomicko–Matematicky Obzor 4, 93–99.
Mangasarian, O.L. (1969), Nonlinear Fractional Programming, Journal of the Operations Research Society of Japan 12, 1–10.
Mangasarian, O.L. (1969), Nonlinear Programming, McGraw–Hill, New York.
Mangasarian, O.L. (1970), Convexity, Pseudo–Convexity and Quasi–Convexity of Composite Functions, Cahiers du Centre d’Etudes de Recherche Operationelle 12, 114–122.
Manjurov, D.M. (1968), Investigation of a Linear–Fractional Programming Problem when Certain Parameters are Varied, Izvestija Akademii Nauk Azerbaidzanskoi SSR Serija Fiziko–Techniceskin i Matematiceskih Nauk 4, 89–95. [Russian]
Manjurov, D.M. (1973), A Certain Generalized Problem of Linear–Fractional Programming, Numerical Mathematics, Izdat, “Elm”, Baku, 88–103. [Russian]
Manjurov, D.M. (1975), Block Methods for Fractional Linear Programming, Questions of Mathematical Cybernetics and Applied Mathematis 1, Izdat. “Elm”, Baku, 51–71. [Russian]
Mao, J.C.T. (1970), Essentials of Portfolio Diversification Strategy, Journal of Finance 25, 1109–1121.
Marchi, A. (1990), Sulla Relazione tra un Problema Bicriteria ed un Problema Frazionario, Report No. 33, Department of Statistics and Applied Mathematics, University of Pisa/Italy. Also Atti del XV Convego A.MA.S.E.S., Grado 1991, 381–392.
Marchi, A. (1994), On the Relationship Between Bicriteria Problems and Nonlinear Programming Problems. Komlosi, S., Rapcsak, T. and Schaible, S. (eds.), Generalized Convexity, Proceedings Pécs/Hungary, 1992; Lecture Notes in Economics and Mathematical Systems 405, Springer–Verlag, Berlin–Heidelberg–New York, to appear.
Marchi, E. (1976), Equilibrium Points of Rational n–Person Games, J. Math Anal. Appl. 54, 1–4.
Mareschal, B. (1988), Weight Stability Intervals in Multicriteria Decision Aid, European Journal of Operational Research 33 (1), 54–64.
Markowitz, H.M., Schaible, S. and Ziemba, W.T. (1992), An Algorithm for Portfolio Selection in a Lognormal Market, The International Review of Financial Analysis 1, 109–113.
Martein, L. (1985), Maximum of the Sum of a Linear Function and a Linear Fractional Function, Rivista di Matematica per le Scienze Economiche e Sociali 8 (1), 13–20. [Italian]
Martein, L. (1988), Applicazioni della Programmazione Frazionaria nel Campo Economico–Finanziario, Report No. 14, Dipartimento di Statistica e Matematica Applicata all’ Economia, Università di Pisa/Italy.
Martein, L. (1988), On Generating the Set of all Eficient Points of a Bicriteria Linear Fractional Problem, Report No. 13, Dipartimento di Statistica e Matematica Applicata all’ Economia, Università di Pisa.
Martein, L. (1990), On the Bicriteria Maximization Problem, Cambini, A. et al. (eds.), Generalized Convexity and Fractional Programming with Economic Applications, Proceedings, Pisa 1988, Springer Verlag, Berlin–New York, 75–84.
Martein, L. and Pellegrini, L. (1977), Su un’estensione di una Particolare Classe di Problemi di Programmazione Frazionaria, Paper No. A–48, Department of Operations Research, University of Pisa, Italy.
Martein, L. and Pellegrini, L. (1977), Un Algoritmo per la Determinazione del Massimo di una Particolare Funzione Razionale Fratta Soggetta a Vincoli Lineari, Paper No. A–45, Dept. of Operations Research, University of Pisa, Italy.
Martein, L. and Pellegrini, L. (1979), On an Extension of a Particular Class of Fractional Progamming Problems, Proceedings of the First AMASES Meeting, 213–228. [Italian]
Martein, L. and Schaible, S. (1989), On Solving a Linear Program with One Quadratic Constraint, Rivista di Matematica per le Scienze Economiche e Sociali 10 (1–2), 75–90.
Martein, L. and Sodini, C. (1982), Un Algoritmo per un Problema di Programmazione Frazionaria non Lineare e non Convessa, Publication No. 93, Serie A, Dipartimento di Ricerca Operativa e Scienze Statistiche, Università di Pisa/Italy.
Martin, W.R. III (1978), An Augmented Formulation of the Transportation Problem with Application to Resource Allocation and Production Systems, Command and Information Systems, General Electric Company, Sunnyvale, California.
Martinez–Legaz, J.E. (1981), Un Concepto Generalizado de Conjugacion, Applicacion a las Functiones Quasiconvexas, Doctoral Thesis, Universidad de Barcelona.
Martinez–Legaz, J.E. (1983), Exact Quasiconvex Conjugation, Zeitschrift für Operations Research 27, 257–266.
Martinez–Legaz, J.E. (1985), Some New Results on Exact Quasiconvex Duality. Proceedings of IXth Symposium on Operations Research, Methods of Operations Research 49, 47–62.
Martins, E.Q.V. (1984), An Algorithm to Determine a Path with Minimal Cost/Capacity Ratio, Discrete Appl. Math. 8, 189–194.
Martos, B. (1964), Hyperbolic Programming, Naval Research Logistics Quarterly 11, 135–155; originally published in Math. Institute of Hungarian Academy of Sciences 5, 1960, 383–406. [Hungarian]
Martos, B. (1965), The Direct Power of Adjecent Vertex Programming Methods, Management Science 12, 241–252.
Martos, B. (1968), Errata, Management Science 14, 255–256.
Martos, B. (1975), Nonlinear Programming, Theory and Method, North–Holland, Amsterdam.
Marusciac, I. (1973), Metode de Rezolvare a Problemelor de Programare Neliniara, Editura Dacia, Cluj, 90–99.
Marusciac, I. (1974), Asupra Unei Programari Hiperbolice, Stud. Cerc. Mat. 26 (3), 419–430.
Matriasin, N.P. and Makeeva, V.K. (1974), Mathematical Programming. [Russian]
Mazzoleni, P. (1973), Stationary Values of the Ratio of Quadratic Polynomials, Technical Report No. 45, The Hatfield Polytechnic.
Mazzoleni, P. (1974), The Dual for a Particular Fractional Program, Technical Report No. 59, Numerical Optimization Centre, The Hatfield Polytechnic.
Mazzoleni, P. (1975), Some Experience on a Moving–Truncation Method Applied to a Nonlinear Programming Problem with a Fractional Objective Function, in: Dixon, L.C.W. (ed.), Towards Global Optimization, North–Holland, Amsterdam, 350–360.
Mazzoleni, P. (1975), Teoria Della Dualita per Una Classe di Problemi con Funzione Obietivo Fratta, Boll. Un. Mat. Ital. IV (11), 571–577.
Mazzoleni, P. (ed.). (1992), Generalized Concavity, Proceedings, Pisa, April 1992, Tecnoprint S.N.C., Bologna.
Megiddo, N. (1979), Combinatorial Optimization with Rational Objective Functions, Mathematics of Operations Research 4, 414–424.
Megiddo, N. (1981), Applying Parallel Computation Algorithms in the Design of Serial Algorithms, Proc. of 22nd IEEE Symposium on Foundation of Computer Science, 399–408.
Meister, B. and Oettli, W. (1967), On the Capacity of a Discrete, Constant Channel, Information and Control 11, 341–351.
Meister, B. and Oettli, W. (1973), Two Classes of Algorithms for Concave Optimization and the Calculation of the Capacity of Discrete Memoryless Channels, Elektron. Informationsverarbeit. Kybernst. 9, 189–195.
Mensch, G. (1972), The (0,1) Indefinite Programming Problem, Technical Report, International Institute of Management, Berlin.
Mesko, I. (1993), Optimizations of Investments in the Multiphase Production Process, Proceedings of International Conference on Design to Manufacture in Modern Industry, Bled/Slovenia, 258–262.
Mészâros, C. and Rapcsâk, T. (1993), On Sensitivity Analysis for a Class of Decision Systems, Working Paper 93–7, Laboratory of Operations Research and Decision Systems, Hungarian Academy of Sciences, Budapest.
Meyer, C. (1976), Kennzahlen und Kennzahlensysteme, Poeschel, Stuttgart.
Mine, H., Fukushima, M. and Ryang, Y.J. (1978), Parametric Nonlinear Programming for General Cases and its Application to some Problems, Memoirs of the Faculty of Engineering, Kyoto University 40 (3), 198–211.
Miscenko, I.I. (1980), A Possible Method of Optimization on Infinite Matrices of Limiting Conditions, Kievskii Gosudarstvennyi Universitet. Issledovanie Operatsii i ASU 15, 76–80,135. [Russian]
Misra, S. and Das, C. (1981), The Sum of a Linear and Linear Fractional Function and a Three Dimensional Transportation Problem, Opsearch 18, 139–157.
Mjelde, K.M. (1978), Allocation of Resources According to a Fractional Objective, European Journal of Operational Research 2, 116–124.
Mjelde, K.M. (1978), Convex–Concave Fractional Programming with Each Variable Occuring in a Single Constraint, BIT, Nordisk Tidskrift for Informationsbehandling 18, 202–210.
Mjelde, K.M. (1978), Sufficiency of Kuhn–Tucker Optimality Conditions for a Fractional Programming Problem, BIT, Nordisk Tidskrift for Informationsbehandling 18, 454–456.
Mjelde, K.M. (1979), Allocation Problems Involving the Replenishment of Resources, Operations Research Verfahren 35, 313–314.
Mjelde, K.M. (1979), Convex–Concave Fractional Programming – Evaluation of Solutions and Optimality Conditions, BIT, Nordisk Tidskrift for Informationsbehanding 19, 270.
Mjelde, K.M. (1979), Location of a Discrete Resource and its Allocation According to a Fractional Objective, European Journal of Operational Research 4, 49–53.
Mjelde, K.M. (1981), Improvement of a Method of the Evaluation of Allocations of Resources to Activities, European Journal of Operational Research 8 (1), 86–87.
Mjelde, K.M. (1981), Properties of Optimal Allocations of Resources According to a Fractional Objective, Journal of the Operational Research Society 32, 405–408.
Mjelde, K.M. (1982), Cost–Effective Allocations of Bounded and Binary Resources in Polynomial Time, European Journal of Operational Research 11 (1), 176–180.
Mjelde, K.M. (1983), Componentwise Fractional Programming with Application to Resource Allocation, Modeling, Identification and Control. A Norwegian Research Bulletin 4 (2), 117–123.
Mjelde, K.M. (1983), Fractional Resource Allocation with S–shaped Return Functions, Journal of Operational Research Society 34, 627–632.
Mjelde, K.M. (1983), Methods of the Allocation of Limited Resources, J. Wiley and Sons, Chichester.
Mjelde, K.M. (1986), An Incremental and Parametrical Algorithm for Convex–Concave Fractional Programming with a Single Constraint, European Journal of Operational Research 23 (3), 391–395.
Mohnar, Z. and Mohnar, V.T. (1976), Solution of a Hyperbolic Zero–One Programming Problem by the Branch and Bound Method, EkonomickoMatematicky Obzor 12, 428–437. [Czech]
Mond, B. (1972), Fractional Programming, Optimization, Proc. Sem., Austral. Nat. Univ., Canberra, Univ. Queensland Press, St. Lucia, 172–181.
Mond, B. (1978), A Class of Nondifferentiable Fractional Programming Problems, Zeitschrift für Angewandte Mathematik und Mechanik 58, 337341.
Mond, B. (1981), On Algorithmic Equivalence in Linear Fractional Programming, Mathematics of Computation 37, 185–187.
Mond, B. and Craven, B.D. (1973), A Note on Mathematical Programming with Fractional Objective Functions, Naval Research Logistics Quarterly 20, 577–581.
Mond, B. and Craven, B.D. (1975), Nonlinear Fractional Programming, Bulletin of the Australian Mathematical Society 12, 391–397.
Mond, B. and Craven, B.D. (1975), On Fractional Programming and Equivalance, Naval Research Logistics Quarterly 22, 405–410.
Mond, B. and Craven, B.D. (1979), A Duality Theorem for a Nondifferentiable Nonlinear Fractional Programming Problem, Bulletin of the Australian Mathematical Society 20, 397–406.
Mond, B. and Schechter, M. (1978), Duality for a Homogeneous Fractional Programming Problem, Journal of Optimization Theory and Applications 25,349–359.
Mond, B. and Schechter, M. (1980), Duality in Homogeneous Fractional Programming, Journal of Information and Optimization Sciences 1, 271280.
Mond, B. and Weir, T. (1981), Generalized Concavity and Duality, in Schaible, S. and Ziemba, W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 263–279.
Mond, B. and Weir, T. (1982), Duality for Fractional Programming with Generalized Convexity Conditions, Journal of Information and Optimization Sciences 3 (2), 105–124.
Montanov, V. (1971), Optimization Problems with a Linear Fractional Functional, Ekonom. i Mat. Metody 7, 586–592. [Russian]
Morioka, T., Ohnishi, M. and Ibaraki, T. (1987), Optimal Inspection and Replacement Problem of Markovian Deterioration System and its Computational Algorithms, Reliability Theory and Applications, World Science Publishing, Singapore, 245–254.
Morita, H., Ishii, H. and Nishida, T. (1989), Stochastic Linear Knapsack Programming Problem and its Application to a Portfolio Selection Problem, European Journal of Operational Research 40 (3), 329–336.
Morris, A.J. (1978), Generalization of Dual Structural Optimization Problems in Terms of Fractional Programming, Quarterly of Applied Mathematics 36, 115–119.
Mukherjee, R.N. (1991), Generalized Convex Duality for Multiobjective Fractional Programs, J. Math. Anal. Appl. 162 (2), 309–316.
Munteanu, E. and Rado, F. (1960), Calcul Sarjelor Celor Mai Economice la Cuptoarele de Topit Fonta, Studii si Cercetari Matematice, Cluj, Fasciola AnexaXI, 149–158.
Nabeya, S. (1965), On Linear Fractional Programming, Hitotsubashi Journal of Arts and Sciences 5, 58–64.
Nakonechnyi, A.N. (1985), Optimization of the Checking and Replacement Times of Systems, Dokl. Akad. Nauk. Ukrain. SSR Ser. A 7, 64–67. [Russian]
Narihisa, H. (1978), An Algorithm for Solving the Stochastic Programming Problem, Memoirs of the Defense Academy 18, 151–160.
Nath, B., Rahawendra and Mukherjee, R.N. (1988), Necessary Conditions for Nondifferentiable Fractional Multi–Objective Programming, International Journal of Management and Systems 4 (3), 153–159.
Nauss, R.M. (1988), On the Use of Internal Rate of Return in Linear and Integer Programming, Operations Research Letters 7 (6), 258–289.
Nesterov, J.E. and Nemirovsky, A.S. (1991), An Interior–Point Method for Generalized Linear–Fractional Programming, Research Report, Central Economical and Mathematical Institute, USSR Acad. Sci., Moscow/Russia.
Neumann, K. and Morlock, M. (1993), Operations Research, Carl Hanser Verlag, München–Wien.
Nguyen, N.T. (1977), The Generalized Beale Method for Pseudo–Convex Functions, Bulletin Mathematique de la Societe des Sciences Mathematiques de la Republique Socialiste de Roumanie, Nouvelle Serie 21, 67–81. [Russian]
Nikitin, A.I. and Nuriev, U.G. (1982), A Heuristic Algorithm for the Solution of a Problem of Fractional–Linear Boolean Programming, Izvestiya Akademii Nauk Azerbaidzhanskoi SSR. Seriya Fiziko Tekhnicheskikh i Matimaticheskikh Nauk 3 (5), 112–117. [Russian]
Noble, B. (1969), Applied Linear Algebra, Prentice–Hall, Englewood Cliffs, NJ.
Nykowski, I. (1966), Problem Pogodzenia Kilku Kryteriow w Jednym Programie Liniowym, Preglad Statystyczny 13 (4), 367–375.
Nykowski, I. and Zolkiewski, Z. (1981), A Linear Model with a Linear Fractional Objective Function, and Multi–Objective Programming, Polska Akademia Nauk.. Komitet Statystyki i Ekonometrii. Przeglad Statystyczny 28 (3–4), 181–196. [Polish]
Nykowski, I. and Zolkiewski, Z. (1983), On Some Connections Between Bicriteria and Fractional Programming Problems. Essays and Surveys on Multiple Criteria Decision Making, Lecture Notes in Economics and Mathematical Systems 209, Springer, Berlin–New York, 300–309.
Nykowski, I. and Zolkiewski, Z. (1985), A Compromise Procedure for the Multiple Objective Linear Fractional Programming Problem, European Journal of Operational Research 19 (1), 91–97.
Ohlson, J.A. and Ziemba, W.T. (1976), Portfolio Selection in a Lognormal Market when the Investor has a Power Utility Function, Journal of Financial and Quantitative Analysis 11, 57–71.
Ohnishi, M. (1992), Policy Iteration and Newton–Raphson Methods for Markov Decision Processes under Average Cost Criterion, Computers and Mathematics with Applications 24 (1–2), 147–155.
Okabe, A. (1982), Spatial Distributions Maximizing or Minimizing Geary’s Contiguity Ratio, J. Fac. Engrg. Univ. Tokio Ser. B. 36 (3), 525–528.
Ondran, M. (1982), Optimization with Rational Objective Functions, Bull. Direction Etudes Rech. Ser. C. Math. Inform. 1, 49–54.
Ottaviani, M. (1989), Alcuni Risultati su Problemi di Ottimo sotto Vincoli Lineari, in Mazzoleni, P., (ed.), Atti del XIII Convegno A.M.A.S.E.S., Verona 1989, Pitagora Editrice, Bologna, 649–659.
Ottaviani, M. and Pacelli, G. (1993), Fractional Programming and Characterization of some Vertices of the Feasible Region, J. of Optimization Theory and Applications 79, 333–344.
Pacelli, G. (1989), Ottimo di Una Funzione Frazionaria Lineare su Una Sfera di Uno Spazio Affine, in Mazzoleni, P., (ed.), Atti del XIII Convegno A.MA.S.E.S., Verona 1989, Pitagora Editrice, Bologna, 661–676.
Pack, L. (1962), Maximierung der Rentabilität als preispolitisches Ziel, in Koch, H., (ed.), Zur Theorie der Unternehmung, Festschrift fiir E. Gutenberg, Gabler, Wiesbaden, 73–135.
Pack, L. (1965), Rationalprinzip, Gewinnprinzip and Rentabilität, Zeitschrift far Betriebswir tscha t 35, 525–551.
Pang, J.S. (1980), A Parametric Linear Complementarity Technique for Optimal Portfolio Selection with a Risk–Free Asset, Operations Research 28, 927–941.
Pardalos, P.M. (1986), An Algorithm for a Class of Nonlinear Fractional Problems Using Ranking of the Vertices, BIT (Nordisk Tidskrift for Informations–behandling) 26 (3), 392–395.
Pardalos, P.M. and Phillips, A.T. (1991), Global Optimization of Fractional Programs, J. of Global Optimization 1, 173–182.
Parkash, O., Saxena, P.C. and Patkar, V. (1979), Duality in a Class of Nonlinear Fractional Programming Problems, National Academy Science Letters 2, 267–268.
Parkash, O., Saxena, P.C. and Patkar, V. (1984), Nondifferentiable Fractional Programming in Complex Space, Zeitschrift für Angewandte Mathematik und Mechanik 64(1), 59–62.
Passy, U. (1979), Fractional Programming Using Pseudo Duality, Operations Research Verfahren 31, 481–493.
Passy, U. (1981), Pseudo Duality and Non–Convex Programming, in Schaible, S. and Ziemba, W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 239–261.
Passy, U. (1981), Pseudo Duality in Mathematical Programs with Quotients and Products, Journal of Optimization Theory and Applications 33, 349–374.
Passy, U. and Keslassy, A. (1979), Pseudo Duality and Duality for Explicitly Quasiconvex Functions, Mimeograph Series No. 249, Faculty of Industrial Engineering and Management, Technion, Haifa.
Passy, U. and Keslassy, A. (1983), Duality for a Class of Quasiconvex Programs, Journal of Optimization Theory and Applications 40 (4), 515536.
Passy, U. and Prisman, E.Z. (1985), A Convex–Like Duality Scheme for Quasi–Convex Programs, Mathematical Programming 32 (3), 278–300.
Patkar, V. and Stancu–Minasian, I.M. (1981), Aproaches for Solving a Class of Nondifferentiable Nonlinear Fractional Programming Problems, National Academy of Science Letters 4 (12), 477–480.
Patkar, V. and Stancu–Minasian, I.M. (1982), On Disjunctive Linear Fractional Programming, Economic Computation and Economic Cybernetics Studies and Research 16 (2), 87–96.
Patkar, V. and Stancu–Minasian, I.M. (1985), Duality in Disjunctive Linear Fractional Programming, European Journal of Operational Research 21 (1), 101–105.
Patkar, V. and Stancu–Minasian, I.M. (1985), Parametric Algorithm for a Class of Disjunctive Linear Fractional Programs, Bulletin Mathematique de la Societe des Sciences Mathematiques de la Republique Socialiste de Roumanie, Nouvelle Serie 29 (3), 279–284.
Patkar, V. and Stancu–Minasian, I.M. (1990), Recent Results in Disjunctive Linear Fractional Programming. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 99–105.
Patkar, V. and Stancu–Minasian, I.M. (1991), A Disjunctive Linear Fractional Max–Min Problem, Portugal. Math. 48 (1), 67–73.
Patkar, V., Saxena, P.C. and Parkash, O. (1979), Linear Fractional Functional Programming in Complex Space, Zeitschrift für Angewandte Mathematik and Mechanik 59, 276–278.
Patkar, V., Saxena, P.C. and Parkash, O. (1979), Linear Piecewise Linear Programs with Variable Coefficients, Pure Appl. Math. Sci. 10 (1–2), 51-56.
Patkar, V., Saxena, P.C. and Parkash, O. (1979), On a Class of Quadratic Fractional Programming Problems, National Academy Science Letters 2, 29–30.
Patkar, V., Saxena, P.C. and Parkash, O. (1980), On a Discrete Non–Linear Fractional Programming Problem, National Academy of Science Letters 3 (7), 204–205.
Patkar, V., Saxena, P.C. and Parkash, O. (1981), Dual Program for a Convex Fractional Function, Economic Computation and Economic Cybernetics Studies and Research 15 (1), 77–80.
Peteanu, V. and Tigan, S. (1983), Asupra unor Probleme de Max–Min in grafe, Lucrarile celui de al III–lea Seminar de Teoria grafurilor, Brasov, 46–53.
Peteanu, V. and Tigan, S. (1984), Interval Goal Programming with Linear Fractional Criteria, Preprint Babes–Bolyai University of Cluj–Napoca, 122.
Peteanu, V. and Tigan, S. (1984), On Some Discrete Fractional Max–Min Problems. Application to Max–Min Problems in Graphs, L’Analyse Numérique et la Théorie de !Approximation 13 (2), 167–173.
Peteanu, V. and Tigan, S. (1986), The Multiobjective Linear–Fractional Programming and Interval Goal Programming, Itinerant Seminar on Functional Equations, Approximation and Convexity; Iasi 26 Oct., Univ. “Al. I. Cuza”, Facultatea de Matematica, 40–45.
Picard, J.C. and Queyranne, M. (1979), A Network Flow Solution to some No near 0 1 Programming Problems, with Applications to Graph Theory, Networks 12 (2), 141–159.
Plavka, J. (1992), The 0(n3) Algorithm for a Special Case of the Maximum Cost–to–Time Ratio Cycle Problem and its Coherence with an Eigenproblem of a Matrix, Zeitschrift für Operations Reseach 36,417— 422.
Podkaminer, L. (1970), Maksymalizacja Ilorazu Dwu Form Liniowych Przy Liniowych Warunkach Ograniczjacych, Przeglad Statyst 17, 93–103. [Polish]
Podkaminer, L. (1971), Dual Prices and Other Parameters of the Optimal Solution with a Criterion Function in the Form of a Quotient, Przeglad Statyst 18, 333–338. [Polish]
Pogodin, V.P. (1974), Study of a Certain Parametric Linear–Fractional Programming Problem, Mathematical Methods in Economics Research, Izdat. “Nauka”, Moscow, 44–50. [Russian]
Pollack, E.G., Novaes, A. and Frankel, E.G. (1965), Optimization and Integration of Shipping Ventures (A Parametric Linear Programming Algorithm), Ship Building and Marine Eng. Monthly, 267–281.
Prabha, S. (1982), Parametrizing a Column Vector in a Linear Fractional Programming Problem, Journal of Information and Optimization Sciences 3 (3), 290–304.
Puri, M.C. (1973), Enumerative Technique for Extreme Point Linear Fractional Programming Problems, SCIMA Journal of Management Science and Applied Cybernetics 2, 1–8.
Puri, M.C. (1974), Extreme Point Linear Fractional Functional Programming, Zeitschrift far Operations Research 18, 131–139.
Puri, M.C. (1975), Strong Cut Cutting Plane Method for Extreme Point Linear Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 17, 65–69.
Puri, M.C. and Swamp, K. (1974), Extreme Point Linear Fractional Functional Programming, Zeitschrift für Operations Research 18, 131–139.
Puri, M.C. and Swamp, K. (1975), Strong–Cut Cutting Plane Method for Extreme Point Linear Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 17, 65–69.
Radzik, T. (1992), Algorithms for some Linear and Fractional Combinatorial Optimization Problems, PhD. Thesis, Department of Computer Science, Stanford University, Stanford.
Radzik, T. (1992), Newton’s Method for Fractional Combinatorial Optimization, Proceedings of the 33rd IEEE Annual Symposium on Foundations of Computer Science, 659–669.
Radzik, T. (1993), Parametric Flows, Weighted Means of Cuts, and Fractional Combinatorial Optimization, in Pardalos, P.M., (ed.), Complexity in Numerical Optimization, World Scientific Publishing, 351–386.
Ramos, P.C.F. (1981/82), Generalized Fractional Programming, PhD. Thesis, Florida State University.
Rang, S.Y. and Li, J.U. (1987), A Solution Method of the Generalized Parametric Fractional Linear Programming Problem, Cho son Min ju ju ui In min Kong hwa kuk Kwa hak won. T’ong bo. Academy of Science of the People’s Republic of Korea. Bulletin 2, 4–6. [Korean]
Rani, O. and Kaul, R.N. (1973), Duality Theorems for a Class of Nonconvex Programming Problems, Journal of Optimization Theory and Applications 11, 305–308.
Rani, O. and Shivpuri, S. (1977), An Algorithm for Linear Fractional Functionals Programming Problems, Zeitschrift für Angewandte Mathematik und Mechanik 57, 75–80.
Rao, M.R. (1971), Cluster Analysis and Mathematical Programming, Journal of the American Statistical Association 66 (335), 622–626.
Rapcsâk, T. (1991), On Pseudolinear Functions, European J. of Operational Research 50 (3), 353–360.
Reichmann, T. and Lachnit, L. (1976), Planung, Steuerung, und Kontrolle mit Hilfe von Kennzahlen, Zeitschrift für betriebswirtschaftliche Forschung 28, 705–723.
Reinhardt, H.E. (1966), A Maximization Problem Suggested by Baker vs. Can, American Mathematics Monthly 73, 1069–1073.
Reiss, S.P. (1979), Rational Search, In formation Processing Letters 8, 89–90.
Rhode, P. (1978), Verfahren zur ganzzahligen linearen Quotientenprogrammierung, Diplomarbeit, Köln.
Rios, S. and Giron, F.J. (1975), The Portfolio Selection Problem when the Returns Have Stable Distributions, Trabajos de Estadistica e Investigacion Operativa 26, 301–318. [Spanish]
Ritter, K. (1967), A Parametric Method for Solving Certain Nonconcave Maximization Problems, Journal of Computer and System Science 1, 4454.
Robillard, P. (1971), (0,1) Hyperbolic Programming Problems, Naval Research Logistics Quarterly 18, 47–57.
Robinson, S.M. (1971), Numerical Solution of the Irreductible Von Neumann Economic Model, Technical Summary Report No. 1142, Mathematics University of Wisconsin, Madison.
Robinson, S.M. (1972), A Linearization Technique for Solving the Irreductible Von Neumann Economic Model, Technical Summary Report No. 1290, Mathematics Research Center, University of Wisconsin, Madison.
Robinson, S.M. (1972), Computational Solution of Ratio Games by Iterative Linear Programming, Proceedings of the 17th Conference of Army Mathematicians, U.S. Army Research Office Report No. 72–1, Durham, NC.
Rothblum, U.G. (1985), Ratios of Affine Functions, Mathematical Programming 32 (3), 357–365.
Rubalskii, G.B. (1990), Integer Minimization of Fractional–Separable Functions, Akademiya Nauk SSSR. Zhurnal Vychislitelnoi Matematiki i Matimaticheskoi Fiziki 30 (10), 1454–1466. [Russian]
Rubinstein, G.S. (1970), Duality in Mathematical Programming and Some Problems of Convex Analysis, Mathematical Surveys 25, 171–200. [Russian]
Ruiz, J.A. (1975), Modelo Matematico para la Planificacion Calendaria Optima de los Talleres de Homos Martin, Investigation Operational (Cuba), 14, 619.
Rutledge, R.W. (1967), A Simplex Method for Zero–One Mixed Integer Linear Programs, Journal of Mathematical Analysis and Applications 18, 377390.
Ryang, S.Y. (1987), A Solution Method of the Linear Fractional Functional Program with a Parameter Objective Function, Academy of Science of the People’s Democratic Republic of Korea. Research Center for Physics and Mathematics. Su hak 2, 20–24. [Korean]
Rybashov, M.V. and Dudnikov, E.E. (1965), Parametric Method of Solving Fractional Programming Problems by Analogue Computers, Doklady Akademii Nauk SSSR 161, 1289–1290. [Russian]
Rybin, S.V. (1985), Analysis of a Conical Programming Problem, Vestn. Lenigr. Univ. No. 15, Mat. Mekh. Astron 3, 105–107. [Russian]
Saipe, A.L. (1975), Solving a (0,1) Hyperbolic Program by Branch and Bound, Naval Research Logistics Quarterly 22, 497–515.
Sakaguchi, M. (1975), A Maximization for Markov Chains Related to Traffic Flow Problems, Rep. Statist. Appl. Res. Un. Japan, Sci. Engrs. 22 (3), 2023.
Sakawa, M. and Yano, H. (1985), Interactive Decision Making for Multiobjective Linear Fractional Programming Problems with Fuzzy Parameters, Cybernetics and Systems 16 (4), 377–394.
Sakawa, M. and Yano, H. (1988), An Interactive Fuzzy Satisficing Method for Multiobjective Linear Fractional Programming Problems, Fuzzy Sets and Systems 28 (2), 129–144.
Sakawa, M. and Yumine, T. (1983), Interactive Fuzzy Decision–Making for Multiobjective Linear Fractional Programming Problems, Large Scale Systems. Theory and Applications 5 (2), 105–113.
Sakawa, M. and Yumine, T. (1983), Interactive Fuzzy Decision–Making for Multiobjective Linear Fractional Programming Problems, Systems and Control (Shisutemu to Seigyo) 27 (2), 138–146. [Japanese]
Sakawa, M., Yano, H. and Takahashi, J. (1989), Pareto Optimality for Multiobjective Linear Fractional Programming Problems with Fuzzy Parameters, The Transactions of the Institute of Electronics, Information and Communication Engineers (Japan) J72–A (6), 931–937; Information Sciences 63 (1–2), 33–53. (1992).
Sakawa, M., Yano, H. and Yumine, T. (1986), An Interactive Fuzzy Satisfying Method for Multiobjective Linear Fractional Programming Problems, Trans. Inst. Electron. & Commun. Eng. Jpn. Part A (Japan) J69A (1), 3241. [Japanese]
Samyrkanov, S. (1973), Solution of a Fractional Quadratic Programming Problem, Izdat. “Ilim”, Frunze, 55–65. [Russian]
Sannomiya, N., Nishikawa, Y. and Inazu, M. (1979), A Method for Solving Multi–Objective Programming Problems with Linear Fractional Functions, Syst. & Control 23 (1), 61–62. [Japanese]
Saxena, P.C. (1978), Duality Theorem for Fractional Functional Programming in Complex Space, Portugaliae Mathematica 37, 87–92.
Saxena, P.C. and Aggarwal, S.P. (1980), Parametric Linear Fractional Functional Programming, Economic Computation and Economic Cybernetics Studies and Research 14, 87–97.
Saxena, P.C. and Parkash, O. (1980), On a Discrete Non–Linear Fractional Programming Problem, National Academy Science Letters – India 3, 204205.
Saxena, P.C. and Patkar, V. (1978), Linear Fractional Programming in Complex Space, Portugaliae Mathematica 37, 73–80.
Saxena, P.C. and Patkar, V. (1978), Non–Linear Non–Differentiable Fractional Programming in Complex Space, Cahiers du Centre d’Etudes de Recherche Operationelle 20, 183–193.
Saxena, P.C., Patkar, V. and Parkash, O. (1979), A Note on an Algorithm for Integer Solution to Linear and Piecewise Linear Programs, Pure and Applied Mathematical Sciences 9, 31–36.
Saxena, P.C., Patkar, V. and Parkash, O. (1979), A Note on Duality for a Pseudoconvex Functional Programming Problem, National Academy of Science Letters 2 (6), 231–232.
Saxena, P.C., Patkar, V. and Parkash, O. (1979), Linear Fractional Functional Programming in Complex Space, Zeitschrift für Angewandte Mathematik und Mechanik 59, 276–278.
Scaruppe, L. (1967), Die Quotientenoptimierung, Diplomarbeit, Humboldt–Universität, Berlin, Mathematisches Institut.
Schaible, S. (1971), Beiträge zur quasi–konvexen Programmierung, Doctoral Dissertation, Universität Köln, Mathematisches Institut.
Schaible, S. (1972), Quasi–Convex Optimization in General Real Linear Spaces, Zeitschrift für Operations Research 16, 205–213.
Schaible, S. (1973), Fractional Programming: Transformations, Duality and Algorithmic Aspects, Technical Report 73–9, Department of Operations Research, Stanford University, Stanford.
Schaible, S. (1973), Quasiconcave, Strictly Quasiconcave and Pseudoconcave Functions, Operations Research Verfahren 17, 308–316.
Schaible, S. (1973), Transformationen nichtlinearer Quotientenprogramme in konvexe Programme, in Jacob, H. et al. (eds.), Proceedings in Operations Research 2, 351–361.
Schaible, S. (1974), Maximization of Quasiconcave Quotients and Products of Finitely Many Functionals, Cahiers du Centre d’Etudes de Recherche Operationelle 16, 45–53.
Schaible, S. (1974), Nonlinear Fractional Programming, Operations Research Verfahren 19, 109–115.
Schaible, S. (1974), Parameter–Free Convex Equivalent and Dual Programs of Fractional Programming Problems, Zeitschrift für Operations Research 18, 187–196.
Schaible, S. (1975), A Note on ‘Quadratic Fractional Functionals Programming’ by S.P. Aggarwal, Cahiers du Centre d’Etudes de Recherche Operationelle 17, 95–96.
Schaible, S. (1975), Marginalwerte in der Quotientenprogrammierung, Zeitschrift für Betriebswirtschaft 45, 649–658.
Schaible, S. (1976), Duality in Fractional Programming: A Unified Approach, Operations Research 24, 452–461.
Schaible, S. (1976), Fractional Programming: I, Duality, Management Science 22, 858–867.
Schaible, S. (1976), Fractional Programming: II, On Dinkelbach’s Algorithm, Management Science 22, 868–873.
Schaible, S. (1976), Minimization of Ratios, Journal of Optimization Theory and Applications 19, 347–352.
Schaible, S. (1977), A Note on the Sum of a Linear and Linear–Fractional Function, Naval Research Logistics Quarterly 24, 691–693.
Schaible, S. (1977), Duality in Ratio Optimization, Operations Research Verfahren 25, 131–132.
Schaible, S. (1977), Recent Results in Fractional Programming, Operations Research Verfahren 23, 271–272.
Schaible, S. (1978), Analyse und Anwendungen von Quotientenprogrammen, Ein Beitrag zur Planung mit Hilfe der nichtlinearen Programmierung, Mathematical Systems in Economics 42, Hain–Verlag, Meisenheim.
Schaible, S. (1981), A Survey of Fractional Programming, in Schaible, S. and Ziemba, W.T., (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 417–440.
Schaible, S. (1981), Fractional Programming – State of the Art, in Brans, J.P., (ed.), Operational Research ‘81, North–Holland, Amsterdam, 479–493.
Schaible, S. (1981), Fractional Programming: Applications and Algorithms, European Journal of Operational Research 7, 111–120.
Schaible, S. (1982), Bibliography in Fractional Programming, Zeitschrift für Operations Research. Serie A 26 (7), 211–241.
Schaible, S. (1983), Bicriteria Quasiconcave Programs, Cahiers du Centre d’Etudes de Recherche Operationelle 25 (1–2), 93–101.
Schaible, S. (1983), Fractional Programming, Zeitschrift für Operations Research. Serie A 27 (1), 39–54.
Schaible, S. (1984), Simultaneous Optimization of Absolute and Relative Terms, Zeitschrift für Angewandte Mathematik und Mechanik 64 (8), 363364.
Schaible, S. (1985), Fractional Programming with Several Ratios. IXth Symposium on Operations Research, Methods of Operations Research 49, 77–83.
Schaible, S. (1985), Neuere Entwicklungen in der Quotienten–Programmierung, in Kreikebaum, H., Liesegang, G., Schaible, S. and Wildemann, H., (eds.), Industriebetriebslehre in Wissenschaft und Praxis, Festschrift für Th. Ellinger, Duncker–Humblot Verlag, Berlin, 285–312.
Schaible, S. (1988), Multi–Ratio Fractional Programming—A Survey. Optimization, Parallel Processing and Applications, Lecture Notes in Economics and Mathematical Systems 304, Springer, Berlin, 57–66.
Schaible, S. (1989), Fractional Programming—Some Recent Developments, Journal of Information and Optimization Sciences 10 (1), 1–14.
Schaible, S. (1989), Multiratio Fractional Programming—Analysis and Applications, (invited lecture), in Mazzoleni, P., (ed.), Atti del XIII Convegno A.MA.S.E.S., Verona 1989, Pitagora Editrice, Bologna, 47–86.
Schaible, S. (1990), Introduction to Generalized Convexity. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 2–13.
Schaible, S. (1992), Some Recent Results in Fractional Programming, in Mazzoleni, P., (ed.), Generalized Concavity, Proceedings, Pisa, April 1992, Tecnoprint S.N.C., Bologna, 7–14.
Schaible, S. (1994), Fractional Programming, in Gass, S.I. and Harris, C.M., (eds.), Encyclopedia of Operations Research and Management Science, Kluwer Academic Publishers, to appear.
Schaible, S. and Ibaraki, T. (1983), Fractional Programming, (Invited Review), European Journal of Operational Research 12 (4), 325–338.
Schaible, S. and Lowe, T. (1983), A Note on a Material Control Problem, HE Transactions 15, 177–179.
Schaible, S. and Ziemba, W.T. (1982), On the Concavity of the Sum of Lognormals is Lognormal Approximation in Portfolio Theory, Working Paper No. 317, University of California, Los Angeles.
Schaible, S. and Ziemba, W.T. (1985), Generalized Concavity of a Function in Portfolio Theory, Zeitschrift für Operations Research 29, 161–186.
Schaible, S. and Ziemba, W.T., (eds.) (1981), Generalized Concavity in Optimization and Economics, Academic Press, New York.
Schechter, M. (1989), An Extension of the Channes–Cooper Method in Linear Fractional Programming, Journal of Information and Optimization Sciences 10 (1), 97–104.
Schroeder, R.G. (1970), Linear Programming Solutions to Ratio Games, Operations Research 18, 300–305.
Schroeder, R.G. (1974), Resource Planning in University Management by Goal Programming, Operations Research 22, 700–710.
Scott, C.H. and Jefferson, T.R. (1980), Fractional Programming Duality via Geometric Programming Duality, Journal of Australian Mathematical Society, Series B 21, 398–401.
Scott, C.H. and Jefferson, T.R. (1981), Conjugate Duality for Fractional Programs, Journal of Mathematical Analysis and Applications 84 (2), 381389.
Scott, C.H. and Jefferson, T.R. (1987), Nonstandard Posynomial Geometric Programs, International Journal of Systems Science 18 (8), 1467–1474.
Scott, C.H. and Jefferson, T.R. (1989), Conjugate Duality in Generalized Fractional Programming, Journal of Optimization Theory and Applications 60 (3), 475–483.
Seelbach, H. (1968), Rentabilitätsmaximierung bei Variablem Eigenkapital, Zeitschrift für Betriebswirtschaft 38, 237–256.
Sen, R. and Chatterjee, S. (1983), Page Cuts for Mixed Integer Interval Linear Fractional Programming, 1983 Proceedings of the International Conference on Systems, Man and Cybernetics 2, Bombay and New Delhi/India, 29 Dec. 1983–7 Jan. 1984, (New York, USA: IEEE 1983), 813–816.
Sen, R. and Chatterjee, S. (1984), On an Algorithm for Solving Absolute Value Linear Fractional Programming with Integer Interval Linear Constraints, Proceedings of the 1984 IEEE International Conference on Systems, Man and Cybernetics, Halifax/Canada, 10–12 Oct. 1984, (New York, USA: IEEE 1984), 158–162.
Sengupta, J.K. (1972), Stochastic Programming, Methods and Applications, North–Holland, Amsterdam,
Seshan, C.R. (1980), An Algorithm for Ranking the Extreme Points for a Linear Fractional Objective Function, Journal of the Indian Institute of Science, Section B – Physical and Chemical Series 62, 119–121.
Seshan, C.R. (1980), On Duality in Linear Fractional Programming, Proceedings of the Indian Academy of Sciences, Mathematical Sciences 89, 35–42.
Seshan, C.R. and Achary, K.K. (1980), A Branch and Bound Algorithm for a Transportation Type Problem with Piecewise Linear Convex Objective Function, Zeitschrift für Angewandte Mathematik und Mechanik 60, 303–307.
Seshan, C.R. and Tikekar, V.G. (1980), Algorithms for Integer Fractional Programming, Journal of the Indian Institute of Science, Section B – Physical and Chemical Series 62, 9–16.
Sgurev, V.S. (1990), A Maximum Network Flow and a Capacity in Networks with Nonlinear Constraints, Problemi na Tekhnicheskata Kibernetika i Robotikata (Problems of Engineering Cybernetics and Robotics) 31, 1823.
Shapley, L.S. (1953), Stochastic Games, Proceedings of the National Academy of Sciences 39, 1095–1100.
Sharma, I.C. (1967), Feasible Direction Approach to Fractional Programming Problems, Opsearch 4, 61–72.
Sharma, I.C. (1973), Transportation Technique in Non–Linear Fractional Prng ammina, Trnhnins de Estadistica y de Investigacion Operativa 24, 131–139.
Sharma, I.C. and Swarup, K. (1972), On Duality in Linear Fractional Functionals Programming, Zeitschrift für Operations Research 16, 91100.
Sharma, J.K. (1978), Extensions and Special Cases of Transportation Problem: A Survey, Indian Journal of Pure and Applied Mathematics 9, 928–940.
Sharma, J.K. (1978), Programming with Fractional Non–Linear Objective Function and Transportation Technique, Revue Roumaine de Mathematiques Pures et Appliquees 23, 1227–1234.
Sharma, J.K. and Swarup, K. (1977), Transportation Fractional Programming with Respect to Time, Ricerca Operativa 7, 458–459.
Sharma, J.K., Gupta, A.K. and Gupta, M.P. (1980), Extension of Simplex Technique for Solving Fractional Programming Problem, Indian Journal of Pure and Applied Mathematics 11, 961–968.
Shepilov, M.A. (1980), Methods for Solving Fractional Mathematical Programming Problems, Kibernetica 16, 93–98, [Russian], translated in Cybernetics (USA) 16, 104–111.
Shi, Y.G. (1986), A Minimization Problem in the Mean Norm Using Generalized Rational Functions, Math. Numer. Sin (China) 8 (2), 205208.
Shivpuri, S. and Chadha, S.S. (1978), Multiparametric Linear Fractional Functionals Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 20, 103–108.
Shor, N.Z. and Solomon, D.I. (1989), Dekompozitsionnye Metody v DrobnoLineinom Programmirovanii. (Decomposition Methods in Linear Fractional Programming), ‘Shitiintsa’, Kishinev. [Russian]
Shukla, D.P. and Kanchan, P.K. (1978), Sum of Linear and Linear Fractional Programming, Acta Ciencia Indica 4, 199–201.
Shvartsman, A.P. (1965), An Algorithm of Fractional Linear Programming, Ekonomicheskie i Matematicheskie Metody (USSR) 1, 558–566. [Russian]
Sideri, E.A. (1989), A Cutting Plane Algorithm for Min–Max Fractional Programming, Journal of Information and Optimization Sciences 10 (1), 177–192.
Sideri, E.A. (1990), A Modified Kelley’s Cutting Plane Algorithm for some Special Nonconvex Problems. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 121–142.
Sideri, E.A. (1992), A New Local Characterization of Pseudoconvex Functions and their Nonsmooth Extensions, Giannessi, F. (ed.), Nonsmooth Optimization Methods and Applications, Gordon and Breach Science Publishers, Amsterdam, 409–420.
Singh, C. (1981), Optimality Conditions in Fractional Programming, Journal of Optimization Theory and Applications 33, 287–294.
Singh, C. (1982), Convex Programming with Set–Inclusive Constraints and its Applications to Generalized Linear and Fractional Programming, Journal of Optimization Theory and Applications 38 (1), 33–42.
Singh, C. (1984), Optimality Conditions for Fractional Min–Max Programming, Journal of Mathematical Analysis and Applications 100 (2), 409–415.
Singh, C. (1986), A Class of Multiple–Criteria Fractional Programming Problems, Journal of Mathematical Analysis and Applications 115 (1), 202–213.
Singh, C. (1986), Nondifferentiable Fractional programming with Hanson–Mond Classes of Functions, Journal of Optimization Theory and Applications 49 (3), 431–447.
Singh, C. (1988), Generalized Fractional Programming with Hanson–Mond Classes of Functions, Journal of Information and Optimization Sciences 9 (2), 219–230.
Singh, C. and Dass, B.K., (eds.) (1989), Continuous–Time, Fractional and Multiobjective Programming. Proceedings of a Conference at St. Lawrence University, Canton, NY, 1986, Journal of Information and Optimization Sciences 10 (1).
Singh, C. and Hanson, M.A. (1986), Saddlepoint Theory for Nondifferentiable Multiobjective Fractional Programming, Journal of Information and Optimization Sciences 7 (1), 41–48.
Singh, C. and Hanson, M.A. (1991), Multiobjective Fractional Programming Duality Theory, Naval Research Logistics 38, 925–933.
Singh, C. and Rueda, N. (1990), Generalized Fractional Programming: Optimality and Duality Theory, Journal of Optimization Theory and Applications 66 (1), 149–159.
Singh, C., Suneja, S.K. and Reuda, N.G. (1992), Pre–Invexity in Multiobjective Fractional Programming, Journal of Information and Optimization Sciences 13 (2), 293–302.
Sinha, S.M. and Aylawadi, D.R. (1983), Optimality Conditions for a Class of Nondifferential Fractional Programming Problem, Indian Journal of Pure and Applied Mathematics 14 (2), 167–174.
Sinha, S.M. and Wadhwa, V. (1970), Programming with a Special Class of Nonlinear Functionals, Unternehmensforschung 14, 215–219.
Slowinski, R. (1986), A Multicriteria Fuzzy Linear Programming Method for Water Supply System Development Planning, Fuzzy Sets and Systems 19, 217–237.
Slowinski, R. (1987), An Interactive Method for Multiobjective Linear Programming with Fuzzy Parameters and its Application to Water Supply Planning, Optimization Models Using Fuzzy Sets and Possibility Theory, Theory Decis. Libr., Ser B 4, 396–414.
Slusarczyk, C. (1981), Modification of the Bitran–Novaes Method for Solving Problems with a Fractionally Linear Objective Function in the Case of an Unbounded Set of Feasible Solutions, Polska Akademia Nauk.. Komitet Statystyki i Ekonometrii. Przeglad Statystyczny 28 (1–2), 63–73. [Polish]
Slusarczyk, C. (1986), On a Certain Property of Linear–Fractional Function and its Application, Przeglad Statyst. 33 (4), 403–413. [Polish]
Smith, J.D. (1972), A Mathematical Investigation into the Optimum Ratio of Plough and Conveyor Speeds in Multi–Plough Bidirectional Cutting, Int. J. Rock Mech. and Min. Sci. (G.B.) 9 (6), 767–781.
Smyrev, V.I. (1968), A Method of Solution of the ‘von Neumann’ model, Optimal. Planirovanie. 11, 76–87. [Russian]
Smyrev, V.I. (1976), A Procedure of Newtonian Type for the Determination of Expansion Rates in the ‘von Neuman’ model, Optimizatsija. 18, 36–47. [Russian]
Smyreva, N.V. (1974), An Algorithm for a Parametric Linear–Fractional Programming Problem, Optimizacija 14, 83–102. [Russian]
Sniedovich, M. (1986), C–Programming and the Minimization of Pseudolinear and Separable Functions, Operations Research Letters 5, 185–189.
Sniedovich, M. (1987), A New Look at Fractional Programming, Journal of Optimization Theory and Applications 54 (1), 113–120.
Sniedovich, M. (1987), On Fractional Programming Problems in Engineering Optimization, Engineering Optimization (U.K.) 12 (3), 247–250.
Sniedovich, M. (1988), Fractional Programming Revisited, European Journal of Operational Research 33 (3), 334–431.
Sniedovich, M. (1989), Analysis of a Class of Fractional Programming Problems, Mathematical Progamming 43 (3), 329–347.
Sniedovich, M. and Vazirinejad, S. (1990), A Solution Strategy for a Class of Nonlinear Knapsack Problems, American Journal of Mathematical and Management Sciences 10 (1–2), 51–71.
Sobel, M.J. (1985), Maximal Mean/Standard Deviation Ratio in an Undiscounted MDP, Operations Research Letter 4, 157–159.
Sodini, C. (1990), Equivalence and Parametric Analysis in Linear Fractional Programming. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 143–154.
Solomon, D.I. (1979), Generalized Linear Fractional Programming Problems, Matematiceskie Issledovaniya 52, 206–215. [Russian]
Solomon, D.I. (1979), On a Certain Procedure to Transform Mathematical Programming Problems Having Connected Constraints and Variables, Matematiceskie Issledovaniya 52, 199–205. [Russian]
Solomon, D.I. (1979), On a Method in Linear–Fractional Programming with a Block–Diagonal Matrix, Izv. Akad, Nauk. Mold, SSR, Ser Fiz. – Tekh. Mat. Nauk 3, 68–70. [Russian]
Solomon, D.I. (1983), A Problem of Integer Fractional–Linear Programming, Akademya Nauk Moldayskoi SSR. Institut Matimaticki s Vychislitelnym Tsentrom Matematicheskie Issledovaniya 72, 122–131. [Russian]
Solomon, D.I. (1983), Application of the Method of Generalized Gradient Descent in the Solution of Problems of Fractional–Linear programming, Izvestiya Akademii Nauk Moldayskoi SSR. Seriya Fiziko Tekhnicheskikh i Matematicheskikh Nauk 1, 7–13. [Russian]
Solomon, D.I. (1984), Generalization of the Linear and Fractional–Linear Transportation Problem, Izvestiya Akademii Nauk Moldayskoi SSR. Seriya Fiziko Tekhnicheskikh i Matematicheskikh Nauk 1, 13–18. [Russian]
Solomon, D.I. (1985), The Principle of Decomposition by Variables with Application of the Method of Generalized Gradient Descent in the Solution of Fractional–Linear Programming Problems, Akademya Nauk Moldayskoi SSR. Institut Matimaticki s Vychislitelnym Tsentrom Matematicheskie Issledovaniya 82, 121–134, 156. [Russian]
Solomon, D.I. (1986), An Iterative Algorithm for the Solution of the Generalized Fractional–Linear Programming Problem, Akademya Nauk Moldayskoi SSR. Institut Matimaticki s Vychislitelnym Tsentrom Matematicheskie Issledovaniya 87, 161–164, 184. [Russian]
Solomon, D.I. (1987), A Parametric Method for Solving Problems of Fractional Linear Programming, Akademya Nauk Moldayskoi SSR. Institut Matimaticki s Vychislitelnym Tsentrom Matematicheskie Issledovaniya 96, 124–134, 172. [Russian]
Solomon, D.I. (1988), Decomposition Algorithms for Solving Generalized Linear–Fractional Programming Problems, Akademya Nauk Moldayskoi SSR. Institut Matimaticki s Vychislitelnym Tsentrom Matematicheskie Issledovaniya 100, 133–141, 153. [Russian]
Solomon, D.I. (1988), Decomposition Methods in Linear–Fractional Programming, Akademya Nauk Moldayskoi SSR. Institut Matimaticki s Vychislitelnym Tsentrom Matematicheskie Issledovaniya 100, 115–132, 152. [Russian]
Soyster, A.L. and Lev, B. (1978), An Interpretation of Fractional Objectives in Goal Programming as Related to Papers by Awerbuch et. al., and Hannan, Management Science 24, 1546–1549.
Sriram, M. and Stevens, W.F. (1973), An Example of the Application of Nonlinear Programming to Chemical–Process Optimization, Operations Research 21, 296–304.
Stahl, J. (1964), K& újebb Eljârâs Hiperbolikus Programazâsi Feladatok Megoldí sâra, Publications of the Mathematical Institute of the Hungarian Academy of Sciences 9, Series B, fasc. 4, 743–754.
Stahl, J. (1982), On the Decomposition of Fractional Programming Problem, Szigma 15, 289–292.
Stancu–Minasian, I.M. (1974), A Three–Dimensional Transportation Problem with a Special Structured Objective Function, Bulletin Mathematique 18, 385–397.
Stancu–Minasian, I.M. (1976), Asupra Problemei cu Risc Minim Multiplu I: Cazul a Doua Functii Obiectiv, Stud. Cerc. Mat. 28 (5), 617–623.
Stancu–Minasian, I.M. (1976), Asupra Problemei de Risc Minim Multiplu II: Cazul a r (r>2) Functii Obiectiv, Stud. Cerc. Mat. 28 (6), 723–734.
Stancu–Minasian, I.M. (1976), Asupra Problemei lui Kataoka, Studii si Cercetari Matematice 28, 95–111.
Stancu–Minasian, I.M. (1976), Criterii Multiple in Programarea Stochastica, Teza de Doctorate, Centrul de Statistica Matematica, Bucuresti.
Stancu–Minasian, I.M. (1977), On Stochastic Programming with Multiple Objective Functions, Proceedings of the 5th Conference on Probability Theory, Brasov, 429–436.
Stancu–Minasian, I.M. (1978), On a Class of Non–Linear Fractional Programming Problems, Revue Roumaine de Mathematiques Pures et Appliquees 23, 285–290.
Stancu–Minasian, I.M. (1978), On the Transportation Problem with Multiple Objective Functions, Bull. Math. Soc. Sci. Math., R.S.R., u. Ser. 22 (70), 315–328.
Stancu–Minasian, I.M. (1979), On the Multiple Minimum Risk Problem, Bulletin Mathematique de la Societe des Sciences Mathematiques de la Republique Socialiste de Roumanie, Nouvelle Serie 23 (4), 427–437.
Stancu–Minasian, I.M. (1980), Applications of Fractional Programming, Economic Computation and Economic Cybernetics Studies and Research 14, 69–86.
Stancu–Minasian, I.M. (1980), Programarea Stocastica cu mai multe Functii Obiectiv. (Stochastic Programming with Multiple Objective Functions), Editura Academiei Republucii Socialiste Romania, Bucharest, 259pp. [Romanian]
Stancu–Minasian, I.M. (1981), A Survey of Methods used for Solving the Linear Fractional Programming Problems with Several Objective Functions. Fifth Symposium on Operations Research, Operations Research Verfahren 40, 159–162.
Stancu–Minasian, I.M. (1981), A Survey of Methods used for Solving the Problems of Fractional Programming. The Linear Case. I, Bulletin Mathematique de la Societe des Sciences Mathematiques de la Republique Socialiste de Roumanie, Nouvelle Serie 25 (3), 313–319.
Stancu–Minasian, I.M. (1981), A Survey of Methods used for Solving the Problems of Fractional Programming. The Linear Case. II, Bulletin Mathematique de la Societe des Sciences Mathematiques de la Republique Socialiste de Roumanie, Nouvelle Serie 25 (4), 415–430.
Stancu–Minasian, I.M. (1981), Asupra Unei Probleme de Programare Fractionara, Buletin Stiintiftc, Seria Technica–Matematica IV, Institutul de Invatamint Superior Sibiu, 37–42.
Stancu–Minasian, I.M. (1981), Bibliography of Fractional Programming 1960 – 1976, Pure and Applied Mathematika Sciences 13, 35–69.
Stancu–Minasian, I.M. (1981), Fractional Programming in Complex Space: The State of the Art, Academie de la Republique Populaire de Roumaine. Revue Roumaine de Mathematiques Pures et Appliquees 26 (3), 481–491.
Stancu–Minasian, I.M. (1983), A Second Bibliography of Fractional Programming, Pure and Applied Mathematika Sciences 17 (1–2), 87–102.
Stancu–Minasian, I.M. (1985), A Third Bibliography of Fractional Programming, Pure and Applied Mathematika Sciences 22 (1–2), 109–122.
Stancu–Minasian, I.M. (1985), An Overview of Separable Fractional Programming Problem, L’Analyse Numerique et la Theorie de L’Approximation 14 (1), 91–96.
Stancu–Minasian, I.M. (1992), A Fourth Bibliography of Fractional Programming, Optimization 23, 53–71.
Stancu–Minasian, I.M. and Patkar, V. (1985), A Note on Nonlinear Fractional Max–Min Problem, National Academy of Science Letters 8 (2), 39–41.
Stancu–Minasian, I.M. and Patkar, V. (1986), A Note on Duality Theory for an Indefinite Functional Programming Problem, Universitatis Babes Bolyai. Studia. Mathematica 31 (1), 23–26.
Stancu–Minasian, I.M. and Tigan, S. (1984), The Minimum Risk Approach to Special Problems of Mathematical Programming. The Distribution Function of the Optimal Value, L’Analyse Numerique et la Theorie de L’Approximation 13 (2), 175–187.
Stancu–Minasian, I.M. and Tigan, S. (1985), The Minimum Risk Approach to Max–Min Bilinear Programming, Analele Stiintifice ale Universitatii ‘Al. I. Cuza’ din Iasi. Seria Noua. Sectiunea I a Matematica 31 (2), 205–209.
Stancu–Minasian, I.M. and Tigan, S. (1985), The Minimum Risk Approach to the Bottleneck Transportation Problem, Itinerant Seminar on Functional Equations, Approximation and Convexity, Preprint 85–6, Univ. “BabesBolyai”, Cluj–Napoca/Romania, 203–208.
Stancu–Minasian, I.M. and Tigan, S. (1985), The Vectorial Minimum–Risk Problem. Proceedings of the Colloquium on Approximation and Optimization, University Cluj–Napoca, 321–328.
Stancu–Minasian, I.M. and Tigan, S. (1987), Criteriul Riscului Minim in Programarea Stochastica, Lucrarile Sesiunii Stiintifice a Centrului de Calcul al Univertatii Bucuresti, 20–21 Februarie, 392–397.
Stancu–Minasian, I.M. and Tigan, S. (1987), The Stochastic Linear–Fractional Max–Min Problem, Itinerant Seminar on Functional Equations, Approximation and Convexity, University ‘Babes–Bolyai’, Cluj–Napoca, 275–280.
Stancu–Minasian, I.M. and Tigan, S. (1988), A Stochastic Approach to some Linear Fractional Goal Programming Problems, Kybernetika 24 (2), 139149.
Stancu–Minasian, I.M. and Tigan, S. (1988), Generalized Pseudofractional Max–Min Problems, Itinerant Seminar on Functional Equations, Approximation and Convexity, University ‘Babes–Bolyai’, Cluj–Napoca, 295–302.
Stancu–Minasian, I.M. and Tigan, S. (1988), Inexact Mathematical Programming, Seminar on Optimization Theory, University ‘BabesBolyai’, Cluj–Napoca, 99–116.
Stancu–Minasian, I.M. and Tigan, S. (1990), Multiobjective Mathematical Programming with Inexact Data, in Slowinski, R. and Teghem, J. (eds.) Stochastic Versus Fuzzy Approach to Multiobjective Mathematical Programming Under Uncertainty, Kluwer Academic Publishers, 395–418.
Stancu–Minasian, I.M. and Tigan, S. (1990), On some Fractional Programming Models Occuring in Minimum–Risk Problems. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 295–324.
Stancu–Minasian, I.M. and Tigan, S. (1994), Fractional Programming Under Uncertainty, in Komlosi, S., Rapcsak, T. and Schaible, S., (eds.), Generalized Convexity, Proceedings Pécs/Hungary, 1992; Lecture Notes in Economics and Mathematical Systems 405, Springer–Verlag, BerlinHeidelberg–New York, to appear.
Stançu–Minasian, LM., Duca, D.I. and Nishida, T. (1990), Multiple Objective Linear Fractional Optimization in Complex Space, Math. Japonica 35 (1), 195–203.
Steuer, R.E. (1986), Multiple Criteria Optimization. Theory, Computation and Application, John Wiley & Sons Inc., New York, 546 pp.
Storey, C. (1979), Optimization Using Rational Functions, Operations Research Verfahren 31, 613–617.
Storoy, S. (1983), Ranking of Vertices in the Linear Fractional Programming Problem, BIT (Nordisk Tidskrift for Informationsbehandling) 23 (3), 403405.
Subrahmanyam, M.B. (1986), A Note on Best Constants in Discrete Inequalities, Aequationes Mathematicae 30 (2–3), 208–211.
Sumanth, D.J. and Carrillo, M.A. ( 1989), Productivity Modeling of Group Machine Loading with Variable Processing Times, IIE Transactions 21 (1), 27–34.
Suneja, S.K., Bector, C.R. and Singh, C. (1991), Duality in Multiobjective Fractional Programming Involving Strongly and Weakly Invex and Related Functions, Opsearch 28 (3), 153–164.
Suneja, S.K., Singh, C. and Kaul, R.N. (1992), Optimality and Duality in Continuous–Time Nonlinear Fractional Programming, Australian Mathematical Society Journal, Series B Applied Mathematics 34 (2), 229244.
Suppe, C. (1976), Hyperbolische Optimierungsprobleme mit homogenen Funktionen, Ekonomicko–Mathematicky Obzor 12, 438–443.
Swarup, K. (1965), Linear Fractional Functionals Programming, Operations Research 13, 1029–1036.
Swarup, K. (1965), Programming with Quadratic Fractional Functionals, Opsearch 2, 23–30.
Swamp, K. (1965), Some Aspects of Linear Fractional Functionals Programming, Australian Journal of Statistics 7, 90–104.
Swarup, K. (1966), Fractional Programming with Nonlinear Constraints, Zeitschrift fir Angewandte Mathematik and Mechanik 46, 468–469.
Swamp, K. (1966), Transportation Technique in Linear Fractional Programming, Journal of Royal Naval Scientific Service, 256–260.
[ 1058] Swamp, K. (1967), Computational Technique for Linear Fractional Program, J. Math. Sci. 2 (1), 17–20.
Swamp, K. (1967), Mathematical Programming, Ph.D. Thesis, Delhi University, Delhi, 67–73.
Swamp, K. (1967), Some Aspects of Duality in Fractional Programming, Zeitschrift für Angewandte Mathematik and Mechanik 47, 204–205.
Swamp, K. (1967), Some Properties of Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 9, 82–86.
Swarup, K. (1968), Duality for Transportation Problems in Fractional Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 10, 46–54.
Swamp, K. (1968), Duality in Fractional Programming, Unternehmensforschung 12, 106–112.
Swamp, K. (1968), Fractional Programing, Mathematical Student 36, 58–62.
Swamp, K. (1968), Note on Linear Fractional Functionals Programming, Metrika 13 (1), 72–77.
Swamp, K. (1968), On Varying all the Parameters in a Linear Fractional Functionals Programming, Metrika 13, 196–205.
Swamp, K. (1970), Upper Bound Problem in Linear Fractional Functionals Programming, Metrika 15, 81–85.
Swamp, K. (1972), Some Aspects of Fractional Programming, Matematichki Vesnik n. Ser. 9 (24), 97–100.
Swamp, K. (1973), A Primal–Like Algorithm for (0,1) Integer Fractional Programming Problem, Trabajos de Estadistica y de Investigacion Operativa XXIV, 123–136.
Swamp, K. (1974), On Duality in Nonlinear Fractional Programming Problems, Zeitschrift für Angewandte Mathematik and Mechanik 54, 734.
Swamp, K. and Bedi, M.K. (1976), Linear Fractional Functionals Programming with Absolute–Value Fractional Functionals, Cahiers du Centre d’Etudes de Recherche Operationelle 18, 367–375.
Swamp, K. and Sharma, I.C. (1970), Programming with Linear Fractional Functionals in Complex Space, Cahiers du Centre d’Etudes de Recherche Operationelle 12, 103–109.
Taha, H.A. (1971), Hyperbolic Programming with Bivalent Variables, Technical Report 71–7, Dept of Ind. Eng., University of Arkansas, Fayetteville.
Taha, H.A. (1975), An Algorithm for Zero–One Fractional Programming, AIIE Transactions 7, 29–34.
Tammer, E.–C. (1974), Dualität und Stabilität in der hyperbolischen Optimierung, Dissertation A, Humboldt–Universität, Berlin, Sektion Mathematik.
Tammer, F,–C, (1974), Dualitätstheorie für hyperbolische und stückweise–lineare konvexe Optimierungsprobleme, Mathematische Operationenforschung und Statistik 5, 93–108.
Tammer, E.–C. (1977), Parametrische hyperbolische Optimierungsprobleme mit Parametern in der Zielfunktion, Mathematische Operationsforschung und Statistik, Series Optimization 8, 207–225.
Tang, H.W. and Li, G.B. (1986), Karmarkar’s Algorithm and Fractional Linear Programming, Journal of Dalian Institute of Technology. Dalian Gong Xueyuan Xuebao 25, 79–83. [Chinese]
Teterev, A.G. (1970), A Certain Generalization of Linear and Fractional–Linear Programming, Matekon 6, 246–259; originally in Ekonomika i Matematiceskie Metody 5, 1969, 440–447. [Russian].
Thanassoulis, E. (1985), An Adaptation of PASEB for the Solution of Multi–Objective Linear Fractional Programming Problems, Journal of the Operational Research Society 36 (2), 155–161.
Thang, N.N. (1977), Generalization of Beale’s Method to a Pseudoconvex
Function, Bulletin Mathematique de la Societe des Sciences Mathematiques de la Republique Socialiste de Roumanie 21 (1–2), 67–81. [Russian]
Tigan, E. (1989), Algorithms to Minimize the Cost Capacity Ratio, Econom. Comput. Econom. Cybernet Stud. Res. 24 (4), 53–58.
Tigan, S. (1971), Sur Quelques Problèmes d’Affectation. Direction Scientifique. Note de Travail No. 157, SEMA, Paris.
Tigan, S. (1971), Sur une Méthode de Decomposition pour le Problème de Program–mation Monotone, Mathematica (Cluj) 13 (36), 347–354.
Tigan, S. (1972), On A Certain Problem of Nonlinear Fractional Programming, Revista de Analiza Numerica si Teoria Approximatiei 1, 215–226. [Romanian]
Tigan, S. (1972), Sur Quelques Problèmes d’Affectation à Applications Economiques, Elektronische Datenverarbeitung in der Wissenschaft, Wirtschaft und Verwaltung, 107–123.
Tigan, S. (1973), On a Method for Fractional Optimization Problems. Application to Stochastic Optimization Problems, Proceeding of the Computer Science Conference, Szekesfehervar, Hungary, 351–355.
Tigan, S. (1975), Sur le Probleme de la Programmation Vectorielle Fractionnaire, Revue d’ Analyse Numerique et de la Theorie de l’ Approximation 4, 99–103.
Tigan, S. (1975), Sur une Methode Pour la Resolution d’un Probleme d’Optimisation Fractionaire par Segments, Revue d’ Analyse Numerique et de la Theorie de l’ Approximation 4, 87–97.
Tigan, S. (1980), On the Max–Min Nonlinear Fractional Problem, L’Analyse Numerique et la Theorie de L’Approximation 9 (2), 283–288.
Tigan, S. (1980), Remarques sur Certains Problems de Programmation PseudoLineaire par Morceaux, Rev. Anal. Numer. Theor. Approximation 9, 129132.
Tigan, S. (1981), Asupra Unor Method de Rezolvare a Unor Probleme Particulare de Programmare Fractionara, Informatica Pentru Conducere Orizont ‘81, Realizari si Aplicatii, Cluj, Napoca, 92–93.
Tigan, S. (1982), Eficienta si Propriu–Eficienta Pentru Programarea Fractionara Vectoriala, Seminarul Itinerant de Ecuatii Functionale, Aproximare si Convexitate, Cluj Napoca.
Tigan, S. (1982), Problema Fractionara a Arborelui Minim, Buletinul Român de Informatica 1, 9–15.
Tigan, S. (1982), Sur quelques Problemes de Programmation Pseudo–Fractionnaire. (Some Problems of Pseudofractional Programming), L’Analyse Numerique et la Theorie de L’Approximation 11 (1–2), 167–174.
Tigan, S. (1983), A Parametrical Method for Max–Min Nonlinear Fractional Problems, Itinerant Seminar on Functional Equations, Approximation and Convexity, University ‘Babes Bolyai’, Cluj–Napoca, 175–184.
Tigan, S. (1983), On the Linearization Technique for Quasimonotonic Optimization Problems, L’Analyse Numerique et la Theorie de L’Approximation 12 (1), 89–96.
Tigan, S. (1988), Numerical Methods for Solving some Max–Min Pseudofractional Problems, Proceedings of the Second Symposium of Mathematics and its Applications, Academia SR Romania, Timisoara, 93–97.
Tigan, S. (1988), On Some Procedures for Solving Fractional Max–Min Problems, L’Analyse Numerique et la Theorie de L’Approximation 17 (1), 73–91.
Tigan, S. and Stancu–Minasian, I.M. (1989), Fractional Goal Programming with Inexact Data, Itinerant Seminar on Functional Equations, Approximation and Convexity, University ‘Babes Bolyai’, Cluj–Napoca, 311–318.
Tigan, S. and Stancu–Minasian, I.M. (1990), An Application of Warburton Procedure to a Max–Min Fractional Programming Problem, Itinerant Seminar on Functional Equations, Approximation and Convexity, Cluj–Napoca.
Tigan, S. and Stancu–Minasian, I.M. (1991), On a Bicriterion Max–Min Fractional Problem, L’Analyse Numerique et la Theorie de L’Approximation 20 (1–2), 117–125.
Tobin, J. (1958), Liquidity Preference as Behavior Towards Risk, Review of Economic Studies 26, 65–86.
Tran–Quoc–Chien (1985), Nondifferentiable and Quasidifferentiable Duality in Vector Optimization Theory, Kybernetika 21, 298–312.
Tran–Quoc–Chien (1986), Fenchel–Lagrange Duality in Vector Fractional Programming via Abstract Duality Scheme, Kybernetika 22 (4), 299–319.
Tung, C.T., Chan, G.H. and Chew, K.L. (1987), Finding a Minimal–Ratio Elementary Path in a Network, Asia–Pac. J. Oper. Res. 4 (2), 151–157.
Tuy, H. (1991), Polyhedral Annexation, Dualization and Dimension Reduction Technique in Global Optimization, J. of Global Optimization 1, 229–244.
Uberti, M. (1986), Misurazione dell’ Onerosità dei Contratti di Leasing, Report No. 38, Istituto di Matematica Finanziaria, Università di Torino/Italy.
Uberti, M. (1988), Nota su un’ Applicazione Finanziaria della Programmazione Frazionaria, Quaderno dell’ Istituto di Matematica Finanziaria, Università di Torino/Italy.
Vajda, S. (1975), Problems in Linear and Non–Linear Programming, C. Griffin and Co., London.
Varma, G.K. (1969), Some Aspects of Parametric Linear Fractional Programming, Journal of the Indian Statistical Association 7, 162–167.
Vanna, G.K. (1972), General Parametric Linear Fractional Programming, Metrika 19, 11–17.
Varma, G.K. (1972), On Parametric Linear Fractional Functionals Programming, Trabajos Estadistica y de Investigacion Operativa 23, 149157.
Vartak, M.N. and Gupta, I. (1987), Duality Theory for Fractional Programming Problems under n–Convexity, Opsearch 24 (3), 163–174.
Verdaguer, R. and Iglesias, L. (1990), A Modified Fractional Simplex Method, Investigacion Operacional 11 (1), 3–10. [Spanish]
Verma, V. (1990), Constrained Integer Linear Fractional Programming Problem, Optimization 21 (5), 749–757.
Verma, V. and Puri, M.C. (1990), On Wolf s Method for Solving Linear
Fractional Programming Problem, Opsearch 27 (3), 176–179.
Verma, V., Bakhshi, H.C. and Puri, M.C. (1990), Ranking in Integer Linear Fractional Programming Problems, Zeitschrift für Operations Research 34 (5), 325–334.
Verma, V., Khanna, S. and Puri, M.C. (1987), ‘Bad–Points’ in Linear Fractional Program: A Comparitive Study, Cahiers du Centre d’Etudes de Recherche Operationelle 29 (1–2), 123–131.
[ 1121] Verma, V., Khanna, S. and Puri, M.C. (1990), On Martos’ and CharnesCooper’s Approach vis–a–vis ‘Singular–Points’, Optimization 20 (4), 415420.
Verma, V., Puri, M.C. and Arora, S.R. (1989), Some Special Situations in Linear Fractional Programming Problem: An Algorithmic Comparison, Opsearch 26 (2), 96–107.
Vial, J.P. (1989), A Unified Approach to Projective Algorithms for Linear Programming. Optimization, Proc. 5th French–German Conf., Varetz/France 1988, Lecture Notes Math. 1405, 191–220.
Vlach, M. (1990), Rubinstein Duality Scheme for Vector Optimization. Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems 345, Springer, Berlin, 252–264.
Vogel, W. (1977), Vektoroptimierung in Produkträumen, Hain–Verlag, Meisenheim.
Von Neumann, J. (1937), Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes, in Menger, K., (ed.), Ergebnisse eines mathematischen Kolloqiums, 8, Leipzig und Wien, 73–83.
Von Neumann, J. (1945), A Model of General Economic Equilibrium, Review of Economic Studies 13, 1–9.
Wadhwa, V. (1969), Programming with Separable Fractional Functionals, Journal of Mathematical Sciences 4, 51–60.
Wadhwa, V. (1972), Linear Fractional Programs with Variable Coefficients, Cahiers du Centre d’Etudes de Recherche Operationelle 14, 223–232.
Wadhwa, V. (1974), Parametric Linear Fractional Programming, SCIMA Journal of Management Science and Applied Cybernetics 3, 21–29.
Wagner, H.M. (1975), Principles of Operations Research with Applications to Management Decisions, 2nd ed., Prentice–Hall, Englewood–Cliffs, N.J.
Wagner, H.M. and Yuan, S.C. (1968), Algorithmic Equivalence in Linear Fractional Programming, Management Science 14, 301–306.
Wang, C.L. (1988), The Principle and Models of Dynamic Programming. II, III, Journal of Mathematical Analysis and Applications 135 (1), 268–283, 284–296.
Wang, Cl— and Wu,, Y, (1991), Optimal Control Problems with Nonstandard Cost Functions, Congr. Numer. 80, 129–137.
Warburton, A.R. (1985), Parametric Solution of Bicriterion Linear Fractional Programs, Operations Research 33 (1), 74–84.
Warren, D. and Thomas, J.B. (1985), Signal–to–Noise Ratio and Central Limit Theorem Considerations in non–Gaussian Defection, Proceedings of the 23rd Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, USA, 2–4 Oct. 1985, Urbana–Champaign, University of Illinois, 35–44.
Wdowiak, J. (1977), A Discrete Linear Fractional Programming Problem, Przeglad Statystyczny 24, 483–497. [Polish]
Wdowiak, J. (1978), A Discrete Linear Fractional Programming Problem II, Przeglad Statystyczny 25, 133–140. [Polish]
Weber, R. (1983), Pseudomonotonic Multiobjective Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 25 (1), 115–128.
Weir, T. (1985), A Note on Duality for Fractional Programming Problems, Opsearch 22 (4), 241–247.
Weir, T. (1986), A Dual for a Multiple Objective Fractional Programming Problem, Journal of Information and Optimization Sciences 7 (3), 261269.
Weir, T. (1986), A Duality Theorem for a Multiple Objective Fractional Optimization Problem, Bulletin of the Australian Mathematical Society 34 (3), 415–425.
Weir, T. (1986), A Note on Invex Functions and Duality in Generalized Fractional Programming, Report No. 4, Department of Mathematics, Australian Defense Force Academy.
Weir, T. (1989), Duality for Nondifferentiable Multiple Objective Fractional Programming Problems, Utilitas Mathematica 36, 53–64.
Weir, T. (1989), On Duality in Multiobjective Fractional Programming, Opsearch 26 (3), 151–158.
Weir, T. (1991), Symmetric Dual Multiobjective Fractional Programming, Australian Mathematical Society. Journal. Series A 50 (1), 67–74.
Weir, T. and Mond, B. (1990), Duality for Fractional Programming Without a Constraint Qualification, Utilitas Mathematica 38, 193–197.
Weir, T., Mond, B. and Egudo, R.R. (1992), Duality Without Constraint Qualification for Multiobjective Fractional Programming, Asia Pacific Journal of Operations Research 9 (2), 195–206.
Werner, J. (1988), Duality in Generalized Fractional Programming. Trends in Mathematical Optimization, Internationale Schriftenreihe Numerische Mathematik 84, 341–351.
Whinston, A. (1964), A Dual Decomposition Algorithm for Quadratic Programming, Cahiers du Centre d’Etudes de Recherche Operationelle 6, 188–201.
Wilde, F. (1968), Bestimmung eines optimalen Produktionsprogramms mit maximaler grundfondsbezogener Rentabilität mit Hilfe der hyperbolischen Optimierung, Thesis, Wirtschaftswissenschaftliche Fakultät der Humboldt–Universität, Berlin.
Williams, H.P. (1974), Experiments in the Formulation of Integer Programming Problems, Mathematical Programming Studies 2, 180–197.
[ 1154] Wingo, D.R. (1983), Maximum Likelihood Models for Fitting the Burr–Type XII Distribution to Life Test Data, Biomedical Journal 25, 77–84.
Wolf, H. (1983), A New Solution Approach to the Linear Fractional Programming Problem, Diskussionsbeiträge des Fachbereichs Wirtschaftswissenschaft der Fernuniversität Hagen, Nr. 67.
Wolf, H. (1983), Die parametrische Analyse eines linearen Quotientenprogramms mit einem Skalarparameter in der rechten Seite, Diskussionsbeiträge des Fachbereichs Wirtschaftswissenschaft der Fernuniversität Hagen, Nr. 74.
Wolf, H. (1985), A Parametric Method for Solving the Linear Fractional Programming Problem, Operations Research 33 (4), 835–841.
Wolf, H. (1986), Parametric Analysis for Fractional Programs, Proceedings of Xth Symposium on Operations Research, Methods of Operations Research 53, 215–222.
Wolf, H. (1986), Parametric Analysis in Linear Fractional Programming, Operations Research 34 (6), 930–937.
Wolf, H. (1986), Solving Special Nonlinear Fractional Programming Problems via Parametric Linear Programming, European Journal of Operational Research 23 (3), 396–400.
Wolfe, O.B., Hawaleshka, O. and Mohamed, A.M. (1986), User Friendly Micro Computer Program for Solving Fractional and Linear Programming Problems, Proceedings of the 8th Annual Conference on Computers and Industrial Engineering, Orlando, FL, USA, 19–21 March, 1986, Comput. & Ind. Eng. (G.B.) 11 (1–4), 225–231.
[ 1162] Wolkowicz, H. (1981), Bounds for the Kantorovich Ratio, Technical Report, Dept. of Mathematics, University of Alberta, Edmonton.
Xiao, D. and Goldfarb, D. (1990), A Path–Following Projective Interior Point Method for Linear Programming, Technical Report, Department of Industrial Engineering and Operations Research, Columbia University.
Xu, Z.K. (1983), A Method for a Fractional Programming Problem, Journal of Shanghai University of Technology 1, 66–72.
Xu, Z.K. (1987), The Use of Programming with p+l Parameters to Solve Problems of Generalized Fractional Programming, Chinese Academy of Science. Kexue Tongbao 32 (19), 1444–1446. [Chinese]
Xu, Z.K. (1988), Saddle–Point Type Optimality Criteria for Generalized Fractional Programming, Journal of Optimization Theory and Applications 57 (1), 189–196.
Xu, Z.K. (1990), On Inexact Fractional Programming, Journal of Systems Science and Mathematical Sciences. Xitong Kexue yu Shuxue 10 (4), 377382.[Chinese]
Yadav, S.R. and Mukherjee, R.N. (1984), A Generalized Big–M Method for Fractional Programming and some Problems on Parametric Fractional Programming, Istanbul Teknik Universitesi Bulteni. Bulletin of the Technical University of Istanbul 37 (4), 465–475.
Yadav, S.R. and Mukherjee, R.N. (1984), On Parametric Fractional Programming, Istanbul Teknik Universitesi Bulteni. Bulletin of the Technical University of Istanbul 37 (3), 273–282.
Yadav, S.R. and Mukherjee, R.N. (1990), Duality for Fractional Minimax Programming Problems, Australian Mathematical Society. Journal. Series B. Applied Mathematics 31 (4), 484–492.
Yadav, S.R., Prasad, S. and Mukherjee, R.N. (1988), A Dual Differentiable Exact Penalty Function in Fractional Programming, Indian Journal of Pure and Applied Mathematics 19 (6), 513–515.
[ 1172] Yamada, K. (1963), On Linear Fractional Programming, The Hitotsubashi Review, 49. [Japanese]
Yano, H. and Sakawa, M. (1989), Interactive Fuzzy Decision Making for Generalized Multiobjective Linear Fractional Programming Problems with Fuzzy Parameters, Fuzzy Sets and Systems 32 (3), 245–261.
Yi, Y.W. and Zhang, Y.S. (1989), An Optimal Planning Method for Water Resource Systems, Information and Control. Xinxi yu Kongzhi 18 (5), 5154. [Chinese]
Ying, M.Q. (1986), Several Theoretical Problems on Multiple Objective Programming (continued), Qufu Shifan Daxue Xuebao. Ziran Kexue Ban. Journal of Qufu Normal University. Natural Sciences Education 12 (4), 24–34. [Chinese]
Yon, R.S. (1986), A Solution Method of the Fractional n–Index Transportation Problem, SUHAK 3, 1–8. [Korean]
Yon, R.S. (1907), A Solution Method for Linear Fractional Functional Programming with a Parameter Objective Function, SUHAK 2, 20–24. [Chinese]
Yon, R.S. (1987), A Solution Method of the Fractional n–Index Transportation Problem with Pass Capacity, SUHAK 1, 21–28. [Korean]
Zalmai, G.J. (1986), Duality for a Class of Continuous–Time Homogeneous Fractional Programming Problems, Zeitschrift für Operations Research. Serie A 30 (1), 43–48.
Zalmai, G.J. (1986), Optimality Conditions for a Class of Nondifferentiable Minmax Programming Problems, Optimization 17 (4), 453–465.
Zalmai, G.J. (1987), Duality for a Class of Continuous–Time Fractional Programming Problems, Utilitas Mathematica 31209–218.
Zalmai, G.J. (1988), Optimality Conditions and Subgradient Duality for Minmax Programming Problems with Applications, Utilitas Mathematica 34, 193–222.
Zalmai, G.J. (1989), Optimality Conditions and Duality for Constrained Measurable Subset Selection Problems with Minmax Objective Functions, Optimization 20 (4), 377–395.
Zalmai, G.J. (1990), Duality for Generalized Fractional Programs Involving n–Set Functions, Journal of Mathematical Analysis and Applications 149 (2), 339–350.
Zalmai, G.J. (1990), Optimality Conditions and Duality for a Class of Continuous–Time Generalized Fractional Programming Problems, Journal of Mathematical Analysis and Applications 153 (2), 356–371.
[ 1186] Zemel, E. (1981), On Search over Rationals, Operations Research Letters 1, 34–38.
Zhang, S. (1990), Linear, Matroid and Polymatroid Fractional Optimization, Technical Report, Econometric Institute, Erasmus University, Rotterdam.
Zhang, S. (1991), Stochastic Queue Location Problems, Doctoral Dissertation, Econometric Institute, Erasmus University, Rotterdam.
Zheng, Q. (1988), Theory and Methods for Global Optimization—An Integral Approach. Advances in Optimization and Control, Proc. Conf., Optimization Days, Montreal/Canada 1986. Lecture Notes Econ. Math. Syst. 302, 15–37.
Ziemba, W.T. (1974), Choosing Investment Portfolios when the Returns have a Stable Distribution, in Hammer, P.L. and Zoutendijk, G. (eds.), Mathematical Programming in Theory and Practice, North–Holland, Amsterdam, 443–482.
Ziemba, W.T. and Vickson, R.G. (eds.), (1975), Stochastic Optimization Models in Finance, Academic Press, New York.
Ziemba, :V.T., Parkan, C. and Brooks–Hill, R. (1974), Calculation of
Investment Portfolios with Risk Free Borrowing and Lending, Management Science 21, 209–222.
Zionts, S. (1968), Programming with Linear Fractional Functionals, Naval Research Logistics Quarterly 15, 449–451.
Zólkiewski, Z. (1983), A Multicriteria Linear Programming Model with Linear Fractional Objective Functions, Ph.D. Thesis, The Central School of Planning and Statistics, Warsaw. [Polish]
Zólkiewski, Z. (1984), Multiobjective Linear Fractional Programming Problem, Przgl. Stat. 31, 359–373. [Polish]
Zólkiewski, Z. (1984), On Computing L.–Compromise Solutions of the Multiple Objective Linear Fractional Programming (MOLFP) Problem, Ceskoslovenska Akademie Ved. Ekonomicko Matematicky Obzor 20 (2), 197–202.
Zsigmond, I. (1987), Mixed Integer Linear Fractional Programming by a Branch and Bound Technique, Annales Universitatis Scientiarum Budapestinensis. Sectio Computatorica 7, 117–130.
Zusupbaev, A. (1973), The Production Allocation Problem with a Fractional Linear of Fractional Convex Functional and with Unknown Quantities of Production and Consumption, Some Mathematical Optimization Methods and their Application in the Economy of Kirghizia, ‘Ilim, Frunze, 19–29. [Russian]
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Schaible, S. (1995). Fractional Programming. In: Horst, R., Pardalos, P.M. (eds) Handbook of Global Optimization. Nonconvex Optimization and Its Applications, vol 2. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2025-2_10
Download citation
DOI: https://doi.org/10.1007/978-1-4615-2025-2_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5838-1
Online ISBN: 978-1-4615-2025-2
eBook Packages: Springer Book Archive