{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T10:29:56Z","timestamp":1742984996263,"version":"3.40.3"},"publisher-location":"Boston, MA","reference-count":68,"publisher":"Springer US","isbn-type":[{"type":"print","value":"9780387747583"},{"type":"electronic","value":"9780387747590"}],"license":[{"start":{"date-parts":[[2008,1,1]],"date-time":"2008-01-01T00:00:00Z","timestamp":1199145600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2008,1,1]],"date-time":"2008-01-01T00:00:00Z","timestamp":1199145600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2008]]},"DOI":"10.1007\/978-0-387-74759-0_189","type":"book-chapter","created":{"date-parts":[[2008,8,25]],"date-time":"2008-08-25T11:03:26Z","timestamp":1219662206000},"page":"1080-1091","source":"Crossref","is-referenced-by-count":8,"title":["Fractional Programming"],"prefix":"10.1007","author":[{"given":"Hans","family":"Frenk","sequence":"first","affiliation":[]},{"given":"Siegfried","family":"Schaible","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"189_CR1_189","first-page":"359","volume-title":"Operational Research \u201869","author":"Y. Almogy","year":"1970","unstructured":"Almogy Y, Levin O (1970) Parametric analysis of a\u00a0multi-stage stochastic shipping problem. In: Lawrence J (ed) Operational Research \u201869. Tavistock Publ., London, pp 359\u2013370"},{"issue":"4","key":"189_CR2_189","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1007\/BF01840458","volume":"1","author":"K.M. Anstreicher","year":"1986","unstructured":"Anstreicher KM (1986) A\u00a0monotonic projective algorithm for fractional linear programming. Algorithmica, 1(4):483\u2013498","journal-title":"Algorithmica"},{"key":"189_CR3_189","volume-title":"Applied probability and queues","author":"S. Asmussen","year":"1987","unstructured":"Asmussen S (1987) Applied probability and queues. Wiley, New York"},{"key":"189_CR4_189","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-7600-2","volume-title":"Generalized concavity","author":"M. Avriel","year":"1988","unstructured":"Avriel M, Diewert WE, Schaible S, Zang I (1988) Generalized concavity. Plenum, New York"},{"key":"189_CR5_189","volume-title":"Discrete and fractional programming","author":"A.I. Barros","year":"1998","unstructured":"Barros AI (1998) Discrete and fractional programming. Kluwer, Dordrecht"},{"key":"189_CR6_189","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1023\/A:1008225317183","volume":"10","author":"A.I. Barros","year":"1997","unstructured":"Barros AI, Dekker R, Frenk JBG, Weeren S van (1997) Optimizing a\u00a0general optimal replacement model by fractional programming\n\t    techniques. J\u00a0Global Optim 10:405\u2013423","journal-title":"J. Global Optim."},{"issue":"1","key":"189_CR7_189","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/BF02192043","volume":"87","author":"A.I. Barros","year":"1995","unstructured":"Barros AI, Frenk JBG (1995) Generalized fractional programming and cutting plane algorithms. J\u00a0Optim Th Appl 87(1):103\u2013120","journal-title":"J. Optim. Th. Appl."},{"issue":"1","key":"189_CR8_189","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/S0966-8349(97)00011-9","volume":"5","author":"A.I. Barros","year":"1997","unstructured":"Barros AI, Frenk JBG, Gromicho J (1997) Fractional location problems. Location Sci 5(1):47\u201358","journal-title":"Location Sci."},{"issue":"2","key":"189_CR9_189","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/BF02592087","volume":"72","author":"A.I. Barros","year":"1996","unstructured":"Barros AI, Frenk JBG, Schaible S, Zhang S (1996) A\u00a0new algorithm for generalized fractional programs. Math Program 72(2):147\u2013175","journal-title":"Math. Program."},{"key":"189_CR10_189","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/BF00138690","volume":"8","author":"A.I. Barros","year":"1996","unstructured":"Barros AI, Frenk JBG, Schaible S, Zhang S (1996) Using duality to solve generalized fractional programming problems. J\u00a0Global Optim 8:139\u2013170","journal-title":"J. Global Optim."},{"key":"189_CR11_189","series-title":"Lecturer Notes Economics and Math. Systems","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1007\/978-3-642-46709-7_16","volume-title":"Generalized Convexity and Fractional Programming with Economic Applications","author":"C.R. Bector","year":"1990","unstructured":"Bector CR, Chandra S, Singh C (1990) Duality in multiobjective fractional programming. In: Cambini A, et al (eds) Generalized Convexity and Fractional Programming with Economic Applications. Lecturer Notes Economics and Math Systems, vol\u00a0345. Springer, Berlin, pp 232\u2013241"},{"key":"189_CR12_189","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1023\/A:1022616612527","volume":"96","author":"J. Benoist","year":"1998","unstructured":"Benoist J (1998) Connectedness of the efficient set for strictly quasiconcave sets for strictly quasiconcave sets. J\u00a0Optim Th Appl, 96:627\u2013654","journal-title":"J. Optim. Th. Appl."},{"key":"189_CR13_189","first-page":"37","volume-title":"Coll. Applications of Mathematics to Economics","author":"B. Bereanu","year":"1965","unstructured":"Bereanu B (1965) Decision regions and minimum risk solutions in linear programming. In: Prekopa A (ed) Coll. Applications of Mathematics to Economics. Hungarian Acad. Sci., Budapest, pp 37\u201342"},{"issue":"1","key":"189_CR14_189","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/BF00939937","volume":"68","author":"M. Boncompte","year":"1991","unstructured":"Boncompte M, Martinez-Legaz JE (1991) Fractional programming by lower subdifferentiability techniques. J\u00a0Optim Th Appl 68(1):95\u2013116","journal-title":"J. Optim. Th. Appl."},{"key":"189_CR15_189","series-title":"Lecture Notes Economics and Math. Systems","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/978-3-642-46802-5_22","volume-title":"Generalized Convexity, Proc. P\u00e9cs\/Hungary, 1992","author":"I.A. Bykadorov","year":"1994","unstructured":"Bykadorov IA (1994) On quasiconvexity in fractional programming. In: Komlosi S, Rapcsak T, Schaible S (eds) Generalized Convexity, Proc. P\u00e9cs\/Hungary, 1992. Lecture Notes Economics and Math Systems, vol\u00a0405. Springer, Berlin, pp 281\u2013293"},{"key":"189_CR16_189","series-title":"Lecture Notes Economics and Math. Systems}","volume-title":"Generalized convexity and fractional programming with economic applications, Proc. 3rd Internat. Workshop at Pisa, Italy, 1988","year":"1990","unstructured":"Cambini A, Castagnoli E, Martein L, Mazzoleni P, Schaible S (eds) (1990) Generalized convexity and fractional programming with economic applications, Proc. 3rd Internat. Workshop at Pisa, Italy, 1988. Lecture Notes Economics and Math Systems, vol\u00a0345. Springer, Berlin"},{"key":"189_CR17_189","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/978-1-4613-3341-8_23","volume-title":"Generalized Convexity, Generalized Monotonicity","author":"A. Cambini","year":"1998","unstructured":"Cambini A, Martein L (1998) Generalized concavity in multiobjective programming. In: Crouzeix JP, Martinez-Legaz JE, Volle M (eds) Generalized Convexity, Generalized Monotonicity. Kluwer, Dordrecht, pp 453\u2013467"},{"key":"189_CR18_189","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/BF01096736","volume":"3","author":"A. Cambini","year":"1993","unstructured":"Cambini A, Schaible S, Sodini C (1993) Parametric linear fractional programming for an unbounded feasible region. J\u00a0Global Optim 3:157\u2013169","journal-title":"J. Global Optim."},{"key":"189_CR19_189","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1002\/nav.3800090303","volume":"9","author":"A. Charnes","year":"1962","unstructured":"Charnes A, Cooper WW (1962) Programming with linear fractional functionals. Naval Res Logist Quart, 9:181\u2013186","journal-title":"Naval Res. Logist. Quart."},{"key":"189_CR20_189","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1287\/opre.11.1.18","volume":"11","author":"A. Charnes","year":"1963","unstructured":"Charnes A, Cooper WW (1963) Deterministic equivalents for optimizing and satisficing under chance constraints. Oper Res, 11:18\u201339","journal-title":"Oper. Res."},{"volume-title":"Data envelopment analysis: Theory, methodology and applications","year":"1994","key":"189_CR21_189","unstructured":"Charnes A, Cooper WW, Lewin AY, Seiford LM (eds) (1994) Data envelopment analysis: Theory, methodology and applications. Kluwer, Dordrecht"},{"key":"189_CR22_189","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1016\/0377-2217(78)90138-8","volume":"2","author":"A. Charnes","year":"1978","unstructured":"Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Europ J Oper Res 2:429\u2013444","journal-title":"Europ. J. Oper. Res."},{"key":"189_CR23_189","series-title":"Sigma Ser. Appl. Math.","volume-title":"Fractional programming","author":"B.D. Craven","year":"1988","unstructured":"Craven BD (1988) Fractional programming. Sigma Ser Appl Math, vol\u00a04. Heldermann, Berlin"},{"key":"189_CR24_189","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1007\/BF01582887","volume":"52","author":"J.P. Crouzeix","year":"1991","unstructured":"Crouzeix JP, Ferland JA (1991) Algorithms for generalized fractional programming. Math Program 52:191\u2013207","journal-title":"Math. Program."},{"issue":"3","key":"189_CR25_189","doi-asserted-by":"publisher","first-page":"342","DOI":"10.1007\/BF02591908","volume":"27","author":"J.P. Crouzeix","year":"1983","unstructured":"Crouzeix JP, Ferland JA, Schaible S (1983) Duality in generalized linear fractional programming. Math Program 27(3):342\u2013354","journal-title":"Math. Program."},{"issue":"1","key":"189_CR26_189","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/BF00941314","volume":"47","author":"J.P. Crouzeix","year":"1985","unstructured":"Crouzeix JP, Ferland JA, Schaible S (1985) An algorithm for generalized fractional programs. J\u00a0Optim Th Appl 47(1):35\u201349","journal-title":"J. Optim. Th. Appl."},{"volume-title":"Generalized convexity, generalized monotonicity: Recent results, Proc. 5th Internat. Symp.","year":"1998","key":"189_CR27_189","unstructured":"Crouzeix JP, Martinez-Legaz JE, Volle M (eds) (1998) Generalized convexity, generalized monotonicity: Recent results, Proc. 5th Internat. Symp. Kluwer, Dordrecht"},{"key":"189_CR28_189","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1287\/mnsc.13.7.492","volume":"13","author":"W. Dinkelbach","year":"1967","unstructured":"Dinkelbach W (1967) On nonlinear fractional programming. Managem Sci 13:492\u2013498","journal-title":"Managem. Sci."},{"key":"189_CR29_189","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1137\/0117055","volume":"7","author":"J.E. Falk","year":"1969","unstructured":"Falk JE (1969) Maximization of signal-to-noise ratio in an optical filter. SIAM J Appl Math 7:582\u2013592","journal-title":"SIAM J. Appl. Math."},{"issue":"3","key":"189_CR30_189","doi-asserted-by":"publisher","first-page":"771","DOI":"10.1023\/A:1022621621633","volume":"94","author":"J.B.G. Frenk","year":"1997","unstructured":"Frenk JBG, Dekker R, Kleijn M (1997) On the marginal cost approach in maintenance. J\u00a0Optim Th Appl 94(3):771\u2013781","journal-title":"J. Optim. Th. Appl."},{"key":"189_CR31_189","first-page":"61","volume":"9","author":"J.B.G. Frenk","year":"1998","unstructured":"Frenk JBG, Kleijn MJ (1998) On regenerative processes and inventory control. Pure Appl Math 9:61\u201394","journal-title":"Pure Appl. Math."},{"issue":"3","key":"189_CR32_189","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/BF01582229","volume":"67","author":"R.W. Freund","year":"1994","unstructured":"Freund RW, Jarre F (1994) An interior-point method for fractional programs with convex constraints. Math Program 67(3):407\u2013440","journal-title":"Math. Program."},{"key":"189_CR33_189","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/BF02192302","volume":"85","author":"R.W. Freund","year":"1995","unstructured":"Freund RW, Jarre F (1995) An interior-point method for multifractional programs with convex constraints. J\u00a0Optim Th Appl 85:125\u2013161","journal-title":"J. Optim. Th. Appl."},{"key":"189_CR34_189","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1137\/0329003","volume":"29","author":"M. Gaudioso","year":"1991","unstructured":"Gaudioso M, Monaco MF (1991) Quadratic approximations in convex non-differentiable optimization. SIAM J Control Optim 29:58\u201370","journal-title":"SIAM J. Control Optim."},{"key":"189_CR35_189","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1287\/opre.11.6.863","volume":"11","author":"P.C. Gilmore","year":"1963","unstructured":"Gilmore PC, Gomory RE (1963) A\u00a0linear programming approach to the stock cutting problem\u00a0\u2013 Part II. Oper Res 11:863\u2013888","journal-title":"Oper. Res."},{"issue":"10","key":"189_CR36_189","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1287\/mnsc.42.10.1493","volume":"42","author":"M. Gugat","year":"1996","unstructured":"Gugat M (1996) A\u00a0fast algorithm for a\u00a0class of generalized fractional programs. Managem Sci 42(10):1493\u20131499","journal-title":"Managem. Sci."},{"key":"189_CR37_189","volume-title":"Grundlagen betriebswirtschaftlicher Entscheidungen, Das Zielsystem der Unternehmung","author":"E. Heinen","year":"1971","unstructured":"Heinen E (1971) Grundlagen betriebswirtschaftlicher Entscheidungen, Das Zielsystem der Unternehmung, 2nd edn. Gabler, Wiesbaden"},{"issue":"2","key":"189_CR38_189","first-page":"13","volume":"3","author":"J.A. Hoskins","year":"1984","unstructured":"Hoskins JA, Blom R (1984) Optimal allocation of warehouse personnel: A\u00a0case study using fractional programming. Focus (UK) 3(2):13\u201321","journal-title":"Focus (UK)"},{"issue":"3","key":"189_CR39_189","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/BF02591871","volume":"26","author":"T. Ibaraki","year":"1983","unstructured":"Ibaraki T (1983) Parametric approaches in fractional programs. Math Program 26(3):345\u2013362","journal-title":"Math. Program."},{"issue":"3","key":"189_CR40_189","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1007\/BF00935361","volume":"41","author":"R. Jagannathan","year":"1983","unstructured":"Jagannathan R, Schaible S (1983) Duality in generalised fractional programming via Farkas' lemma. J\u00a0Optim Th Appl 41(3):417\u2013424","journal-title":"J. Optim. Th. Appl."},{"key":"189_CR41_189","series-title":"Lecture Notes Economics and Math. Systems","volume-title":"Generalized convexity, Proc. 4th Internat. Workshop at P\u00e9cs\/Hungary, 1992","year":"1994","unstructured":"Komlosi S, Rapcsak T, Schaible S (eds) (1994) Generalized convexity, Proc. 4th Internat. Workshop at P\u00e9cs\/Hungary, 1992. Lecture Notes Economics and Math Systems, vol\u00a0405. Springer, Berlin"},{"issue":"2","key":"189_CR42_189","first-page":"143","volume":"32","author":"H. Konno","year":"1989","unstructured":"Konno H, Inori M (1989) Bond porfolio optimiza tion by bilinear fractional programming. J\u00a0Oper Res Soc Japan 32(2):143\u2013158","journal-title":"J. Oper. Res. Soc. Japan"},{"key":"189_CR43_189","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/BF01096534","volume":"4","author":"H. Konno","year":"1994","unstructured":"Konno H, Kuno T, Yajima Y (1994) Global minimization of a\u00a0generalized convex multiplicative function. J\u00a0Global Optim 4:47\u201362","journal-title":"J. Global Optim."},{"key":"189_CR44_189","doi-asserted-by":"crossref","first-page":"1024","DOI":"10.1287\/mnsc.27.9.1024","volume":"27","author":"J.S.H. Kornbluth","year":"1981","unstructured":"Kornbluth JSH, Steuer RE (1981) Multiple objective linear fractional programming. Managem Sci 27:1024\u20131039","journal-title":"Managem. Sci."},{"key":"189_CR45_189","volume-title":"Nonlinear programming, theory and methods","author":"B. Martos","year":"1975","unstructured":"Martos B (1975) Nonlinear programming, theory and methods. North-Holland, Amsterdam"},{"key":"189_CR46_189","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1287\/moor.4.4.414","volume":"4","author":"N. Megiddo","year":"1979","unstructured":"Megiddo N (1979) Combinatorial optimization with rational objective functions. Math Oper Res 4:414\u2013424","journal-title":"Math. Oper. Res."},{"key":"189_CR47_189","volume-title":"Methods of the allocation of limited resources","author":"K.M. Mjelde","year":"1983","unstructured":"Mjelde KM (1983) Methods of the allocation of limited resources. Wiley, New York"},{"key":"189_CR48_189","first-page":"191","volume":"77","author":"A.S. Nemirovski","year":"1997","unstructured":"Nemirovski AS (1997) A\u00a0long-step method of analytical centers for fractional programs. Math Program 77:191\u2013224","journal-title":"Math. Program."},{"key":"189_CR49_189","first-page":"117","volume":"69","author":"Yu.E. Nesterov","year":"1995","unstructured":"Nesterov YuE, Nemirovski AS (1995) An interior point method for generalized linear-fractional problems. Math Program 69:117\u2013204","journal-title":"Math. Program."},{"key":"189_CR50_189","unstructured":"Von Neumann J (1937) \"\u00dcber ein \u00f6konomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. In: Menger K (ed) Ergebn. eines math. Kolloqiums, vol\u00a08, pp 73\u201383"},{"key":"189_CR51_189","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/BF00119990","volume":"1","author":"P.M. Pardalos","year":"1991","unstructured":"Pardalos PM, Phillips AT (1991) Global optimization of fractional programs. J\u00a0Global Optim 1:173\u2013182","journal-title":"J. Global Optim."},{"key":"189_CR52_189","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/BF00938758","volume":"46","author":"F. Plastria","year":"1985","unstructured":"Plastria F (1985) Lower subdifferentiable functions and their minimization by cutting planes. J\u00a0Optim Th Appl Appl. 46:37\u201353","journal-title":"J. Optim. Th. Appl."},{"key":"189_CR53_189","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/978-1-4613-0303-9_6","volume-title":"Handbook Combinatorial Optim., vol\u00a01","author":"T. Radzik","year":"1998","unstructured":"Radzik T (1998) Fractional combinatorial optimization. In: Du D-Z, Pardalos PM (eds) Handbook Combinatorial Optim., vol\u00a01. Kluwer, Dordrecht, pp 429\u2013478"},{"key":"189_CR54_189","doi-asserted-by":"publisher","first-page":"622","DOI":"10.2307\/2283542","volume":"66","author":"M.R. Rao","year":"1971","unstructured":"Rao MR (1971) Cluster analysis and mathematical programming. J\u00a0Amer Statist Assoc 66:622\u2013626","journal-title":"J. Amer. Statist. Assoc."},{"issue":"8","key":"189_CR55_189","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1287\/mnsc.22.8.858","volume":"22","author":"S. Schaible","year":"1976","unstructured":"Schaible S (1976) Fractional programming: I, Duality. Managem Sci 22(8):858\u2013867","journal-title":"Managem. Sci."},{"issue":"8","key":"189_CR56_189","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1287\/mnsc.22.8.868","volume":"22","author":"S. Schaible","year":"1976","unstructured":"Schaible S (1976) Fractional programming: II, On Dinkelbach's algorithm. Managem Sci 22(8):868\u2013873","journal-title":"Managem. Sci."},{"key":"189_CR57_189","series-title":"Mathematical Systems in Economics","volume-title":"Analyse und Anwendungen von Quotientenprogrammen, Ein Beitrag zur Planung mit Hilfe der nichtlinearen Programmierung","author":"S. Schaible","year":"1978","unstructured":"Schaible S (1978) Analyse und Anwendungen von Quotientenprogrammen, Ein Beitrag zur Planung mit Hilfe der nichtlinearen Programmierung, Mathematical Systems in Economics, vol\u00a042. Anton Hain Verlag, Meisenheim"},{"key":"189_CR58_189","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1016\/0377-2217(81)90272-1","volume":"7","author":"S. Schaible","year":"1981","unstructured":"Schaible S (1981) Fractional programming: Applications and algorithms. Europ J Oper Res 7:111\u2013120","journal-title":"Europ. J. Oper. Res."},{"issue":"1","key":"189_CR59_189","first-page":"39","volume":"27","author":"S. Schaible","year":"1983","unstructured":"Schaible S (1983) Fractional programming. Z Oper Res, Ser A\u00a027(1):39\u201354","journal-title":"Z. Oper. Res., Ser. A"},{"key":"189_CR60_189","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/978-1-4615-2025-2_10","volume-title":"Handbook Global Optim.","author":"S. Schaible","year":"1995","unstructured":"Schaible S (1995) Fractional programming. In: Horst R, Pardalos PM (eds) Handbook Global Optim. Kluwer, Dordrecht, 495\u2013608"},{"key":"189_CR61_189","first-page":"163","volume-title":"Scalar and Vector Optimization in Economic and Financial Problems, Proc. Workshop Milan, 1995","author":"S. Schaible","year":"1996","unstructured":"Schaible S (1996) Fractional programming with sums of ratios. In: Castagnoli E, Giorgi G (eds) Scalar and Vector Optimization in Economic and Financial Problems, Proc. Workshop Milan, 1995. Elioprint, Perugia, 163\u2013175"},{"issue":"4","key":"189_CR62_189","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/0377-2217(83)90153-4","volume":"12","author":"S. Schaible","year":"1983","unstructured":"Schaible S, Ibaraki T (1983) Fractional programming (invited review). Europ J Oper Res 12(4):325\u2013338","journal-title":"Europ. J. Oper. Res."},{"key":"189_CR63_189","series-title":"Proc. NATO Advanced Study Inst. Vancouver, Canada 1980","volume-title":"Generalized concavity in optimization and economics","year":"1981","unstructured":"Schaible S, Ziemba WT (eds) (1981) Generalized concavity in optimization and economics. Proc NATO Advanced Study Inst Vancouver, Canada 1980. Acad. Press, New York"},{"key":"189_CR64_189","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1023\/A:1022693032053","volume":"98","author":"C.H. Scott","year":"1998","unstructured":"Scott CH, Jefferson TR (1998) Duality for a\u00a0nonconvex sum of ratios. J\u00a0Optim Th Appl 98:151\u2013159","journal-title":"J. Optim. Th. Appl."},{"key":"189_CR65_189","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1023\/A:1008274708071","volume":"12","author":"C.H. Scott","year":"1998","unstructured":"Scott CH, Jefferson TR, Frenk JBG (1998) A\u00a0duality theory for a\u00a0class of generalized fractional programs. J\u00a0Global Optim 12:239\u2013245","journal-title":"J. Global Optim."},{"volume-title":"Continuous-time, fractional and multiobjective programming","year":"1989","key":"189_CR66_189","unstructured":"Singh C, Dass BK (eds) (1989) Continuous-time, fractional and multiobjective programming. Proc. Internat. Conf. St. Lawrence Univ., Canton, NY, 1986, Analytic Publ., Delhi, also: J. Inform. Optim. Sci. 10(1)"},{"issue":"3","key":"189_CR67_189","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1016\/0377-2217(88)90177-4","volume":"33","author":"M. Sniedovich","year":"1988","unstructured":"Sniedovich M (1988) Fractional programming revisited. Europ J Oper Res 33(3):334\u2013341","journal-title":"Europ. J. Oper. Res."},{"key":"189_CR68_189","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-0035-6","volume-title":"Fractional programming: Theory, methods and applications","author":"I.M. Stancu-Minasian","year":"1997","unstructured":"Stancu-Minasian IM (1997) Fractional programming: Theory, methods and applications. Kluwer, Dordrecht"}],"container-title":["Encyclopedia of Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-0-387-74759-0_189","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T09:50:05Z","timestamp":1720691405000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-0-387-74759-0_189"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008]]},"ISBN":["9780387747583","9780387747590"],"references-count":68,"URL":"https:\/\/doi.org\/10.1007\/978-0-387-74759-0_189","relation":{},"subject":[],"published":{"date-parts":[[2008]]}}}