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Fractional programming. (English) Zbl 0529.90088


MSC:

90C32 Fractional programming
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming

Citations:

Zbl 0494.90076
Full Text: DOI

References:

[1] Abrham, J.; Luthra, S., Comparison of duality models in fractional linear programming, Z. Operations Res., 21, 125-130 (1977) · Zbl 0358.90063
[2] Ashton, D. J.; Atkins, D. R., Multi-criteria programming for financial planning, J. Operational Res. Soc., 30, 259-270 (1979) · Zbl 0393.90048
[3] Barrodale, I., Best rational approximation and strict quasi-convexity, SIAM J. Numer. Anal., 10, 8-12 (1973) · Zbl 0262.41034
[4] Bereanu, B., Decision regions and minimum risk solutions in linear programming, (Prékopa, A., Colloquium on Applications of Mathematics to Economics (1965), Hungarian Academy of Sciences: Hungarian Academy of Sciences Budapest), 37-42 · Zbl 0142.16701
[5] Bitran, G. R.; Magnanti, T. L., Duality and sensitivity analysis for fractional programs, Operations Res., 24, 675-699 (1976) · Zbl 0361.90073
[6] Chandra, S.; Chandramohan, M., A branch and bound method for integer non-linear fractional programs, Z. Angew. Math. Mech., 60, 735-737 (1980) · Zbl 0462.90088
[7] Charnes, A.; Cooper, W. W., Programming with linear fractional functionals, Naval Res. Logist. Quart., 9, 181-186 (1962) · Zbl 0127.36901
[8] Charnes, A.; Cooper, W. W., Deterministic equivalents for optimizing and satisficing under chance constraints, Operations Res., 11, 18-39 (1963) · Zbl 0117.15403
[9] Charnes, A.; Cooper, W. W., Goal programming and multi-objective optimization: Part I, European J. Operational Res., 1, 39-54 (1977) · Zbl 0375.90079
[10] Choo, E. U., Multicriteria linear fractional programming, (PhD-Thesis (1980), University of British Columbia) · Zbl 0452.90076
[11] Craven, B. D., Mathematical Programming and Control Theory (1978), Chapman and Hall: Chapman and Hall London · Zbl 0431.90039
[12] Crouzeix, J. P.; Ferland, J. A.; Schaible, S., Duality in generalized linear fractional programming, (Technical Report No. 399 (1981), Université de Montréal) · Zbl 0548.90083
[13] Dantzig, G. B.; Blattner, W.; Rao, M. R., Finding a cycle in a graph with minimum cost to time ratio with applications to a ship routing problem, (Theory of Graphs, International Symposium (1966), Dunod: Dunod Paris), 77-83, Gordon and Breach, New York · Zbl 0189.24102
[14] Derman, C., On sequential decisions and Markov chains, Management Sci., 9, 16-24 (1962) · Zbl 0995.90621
[15] Dinkelbach, W., On nonlinear fractional programming, Management Sci., 13, 492-498 (1967) · Zbl 0152.18402
[16] Frank, M.; Wolfe, P., An algorithm for quadratic programming, Naval Res. Logist. Quart., 3, 95-110 (1956)
[17] Gilmore, P. C.; Gomory, R. E., A linear programming approach to the cutting stock problem: Part II, Operations Res., 11, 863-888 (1963) · Zbl 0124.36307
[18] Gol’stein, E. G., Theory of Convex Programming (1972), American Mathematical Society: American Mathematical Society Providence, RI, Translations of Mathematical Monographs · Zbl 0271.90035
[19] Granot, D.; Granot, F., On integer and mixed integer fractional programming problems, Ann. Discrete Math., 1, 221-231 (1977) · Zbl 0358.90043
[20] Gutenberg, E., Einführung in die Betriebswirtschaftslehre (1975), Gabler: Gabler Wiesbaden
[21] Heinen, E., Grundlagen betriebswirtschaftlicher Entscheidungen. Das Zielsystem der Unternehmung, ((1971), Gabler: Gabler Wiesbaden)
[22] T.J. Hodgson and T.J. Lowe, Production lot sizing with material handling cost considerations, AIIE Trans.; T.J. Hodgson and T.J. Lowe, Production lot sizing with material handling cost considerations, AIIE Trans.
[23] Ibaraki, T., Solving mathematical programming problems with fractional objective functions, (Schaible, S.; Ziemba, W. T., Generalized Concavity in Optimization and Economics (1981), Academic Press: Academic Press New York), 441-472 · Zbl 0534.90088
[24] Ibaraki, T., Parametric approaches to fractional programs (1982), Kyoto University, Technical report
[25] Ibaraki, T.; Ishii, H.; Iwase, J.; Hasegawa, T.; Mine, H., Algorithms for quadratic fractional programming problems, J. Operations Res. Soc. Japan, 19, 174-191 (1976) · Zbl 0349.90073
[26] Isbell, J. R.; Marlow, W. H., Attrition games, Naval Res. Logist. Quart., 3, 71-94 (1956) · Zbl 0122.15405
[27] Ishii, H.; Ibaraki, T.; Mine, H., A primal cutting plane algorithm for integer fractional programming problems, J. Operations Res. Soc. Japan, 19, 228-244 (1976) · Zbl 0405.90074
[28] Ishii, H.; Ibaraki, T.; Mine, H., Fractional knapsack problems, Math. Programming, 13, 255-271 (1977) · Zbl 0378.90071
[29] Jagannathan, R., On some properties of programming problems in parametric form pertaining to fractional programming, Management Sci., 12, 609-615 (1966) · Zbl 0143.21602
[30] Jagannathan, R.; Schaible, S., Duality in generalized fractional programming via Farka’s lemma, (Working Paper 82-8 (1982), University of Iowa) · Zbl 0502.90079
[31] Kallberg, T. G.; Ziemba, W. T., Generalized concave functions in stochastic programming and portfolio theory, (Schaible, S.; Ziemba, W. T., Generalized Concavity in Optimization and Economics (1981), Academic Press: Academic Press New York), 719-767 · Zbl 0534.90006
[32] Klein, M., Inspection-maintenance-replacement schedule under Markovian deterioration, Management Sci., 9, 25-32 (1962)
[33] Kornbluth, J. S.H.; Steuer, R. E., Goal programming with linear fractional criteria, European J. Operational Res., 8, 58-65 (1981) · Zbl 0486.90077
[34] Kornbluth, J. S.H.; Steurer, R. S., Multiple objective linear fractional programming, Management Sci., 27, 1024-1039 (1981) · Zbl 0467.90064
[35] Lasdon, L. S., Optimization Theory for Large Systems (1970), MacMillan: MacMillan London · Zbl 0224.90038
[36] Lawler, E., Combinatorial Optimization: Networks and Matroids (1976), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York · Zbl 0413.90040
[37] Mangasarian, O. L., Nonlinear fractional programming, J. Operations Res. Soc. Japan, 12, 1-10 (1969) · Zbl 0257.90043
[38] Martos, B., Math. Institute Hungarian Acad. Sci., 5, 383-406 (1960), originally published in · Zbl 0099.15101
[39] Martos, B., The direct power of adjacent vertex programming methods, Management Sci., 12, 241-252 (1965) · Zbl 0142.17002
[40] Martos, B., Nonlinear Programming: Theory and Methods (1975), North-Holland: North-Holland Amsterdam · Zbl 0357.90027
[41] Megiddo, N., Combinatorial optimization with rational objective functions, Math. Operations Res., 4, 414-424 (1979) · Zbl 0425.90076
[42] Meggido, N., Applying parallel computation algorithms in the design of serial algorithms, (Proc. 22nd IEEE Symposium on Foundation of Computer Science (1981)), 399-408
[43] Meister, B.; Oettli, W., On the capacity of a discrete, constant channel, Information and Control, 11, 341-351 (1967) · Zbl 0157.48903
[44] Mjelde, K. M., Allocation of resources according to a fractional objective, European J. Operational Res., 2, 116-124 (1978) · Zbl 0371.90084
[45] Mjelde, K. M., Methods of the Allocation of Limited Resources (1983), Wiley: Wiley Chichester · Zbl 0511.90076
[46] Ohlson, J. A.; Ziemba, W. T., Portfolio selection in a lognormal market when the investor has a power utility function, J. Financial Quantitative Anal., 11, 57-71 (1976)
[47] Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (1970), Academic Press: Academic Press New York · Zbl 0241.65046
[48] Ostrowski, A. M., Solution of Equations in Euclidean and Banach Spaces (1973), Academic Press: Academic Press New York · Zbl 0304.65002
[49] Pack, L., Maximierung der Rentabilität als preispolitisches Ziel, (Koch, H., Zur Theorie der Unternehmung Festschrift für E. Gutenberg (1962), Gabler: Gabler Wiesbaden), 73-135
[50] Pack, L., Rationalprinzip, Gewinnprinzip und Retabilität, Z. Betriebswirtschaft, 35, 525-551 (1965)
[51] Passy, U.; Keslassy, A., Pseudo duality and duality for explicity quasiconvex functions, Mimeograph Series No. 249 (1979), Technion · Zbl 0496.90075
[52] Schaible, S., Fractional programming: Transformations, duality and algorithmic aspects, (Technical Report 73-9 (1973), Stanford University)
[53] Schaible, S., Fractional programming I. Duality, Management Sci., 22, 858-867 (1976) · Zbl 0338.90050
[54] Schaible, S., Fractional programming II. On Dinkelbach’s algorithm, Management Sci., 22, 868-873 (1976) · Zbl 0346.90052
[55] Schaible, S., Duality in fractional programming: a unified approach, Operations Res., 24, 452-461 (1976) · Zbl 0348.90120
[56] Schaible, S., Minimization of ratios, J. Optimization Theory Appl., 19, 347-352 (1976) · Zbl 0308.90041
[57] Schaible, S., Analyse und Anwendungen von Quotientenprogrammen, ein Beitrag zur Planung mit Hilfe der nichtlinearen Programmierung, (Mathematical Systems in Economics, 42 (1978), Hain-Verlag: Hain-Verlag Meisenheim) · Zbl 0395.90045
[58] Schaible, S., Fractional programming: applications and algorithms, European J. Operational Res., 7, 111-120 (1981) · Zbl 0452.90079
[59] Schaible, S., A survey of fractional programming, (Schaible, S.; Ziemba, W. T., Generalized Concavity in Optimization and Economics (1981), Academic Press: Academic Press New York), 417-440 · Zbl 0535.90092
[60] Schaible, S., Bibliography in fractional programming, Z. Operations Res., 26, 7 (1982) · Zbl 0494.90076
[61] Schaible, S., Bicriteria quasiconcave programs, Cahier, Centre Etudes Recherche Opérationelle, 25 (1983) · Zbl 0505.90074
[62] Schaible, S., Fractional programming, Z. Operations Res., 27, 1 (1983) · Zbl 0527.90094
[63] Schaible, S.; Lowe, T., A note on a material control problem (1982), Purdue University
[64] Schaible, S.; Ziemba, W. T., On the concavity of the sum of lognormals is lognormal approximation in portfolio theory, (Working Paper No. 317 (1982), University of California: University of California Los Angeles) · Zbl 0573.90006
[65] Stancu-Minasian, I. M., A survey of methods used for solving the problems of functional programming. The linear case, Part I, Bull. Math. Soc. Sci. Math. R.S.R., 25, 313-319 (1981) · Zbl 0479.90076
[66] Stancu-Minasian, I. M., A survey of methods used for solving the problems of fractional programming. The Linear Case. Part II, Bull. Math. Soc. Sci. Math. R.S.R., 25, 416-430 (1981) · Zbl 0487.90087
[67] Swarup, K., Linear fractional functionals programming, Operations Res., 13, 1029-1036 (1965) · Zbl 0132.13802
[68] von Neumann, J., A model of general economic equilibrium, Rev. Econom. Studies, 13, 1-9 (1945)
[69] Wagner, H. M.; Yuan, J. S.C., Algorithmic equivalence in linear fractional programming, Management Sci., 14, 301-306 (1968) · Zbl 0153.49101
[70] Warburton, A. R., Topics in multicriteria optimization, (PhD-Thesis (1981), University of British Columbia)
[71] Zangwill, W. I., Nonlinear Programming (1969), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0191.49101
[72] Ziemba, W. T., Choosing investment portfolios when the returns have a stable distribution, (Hammer, P. L.; Zoutendijk, G., Mathematical Programming in Theory and Practice (1974), North-Holland: North-Holland Amsterdam), 443-482
[73] Ziemba, W. T.; Parkan, C.; Brooks-Hill, R., Calculation of investment portfolios with risk free borrowing and lending, Management Sci., 21, 209-222 (1974) · Zbl 0294.90004
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