{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T17:35:09Z","timestamp":1776101709815,"version":"3.50.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2016,3,7]],"date-time":"2016-03-07T00:00:00Z","timestamp":1457308800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput & Applic"],"published-print":{"date-parts":[[2017,11]]},"DOI":"10.1007\/s00521-016-2243-6","type":"journal-article","created":{"date-parts":[[2016,3,7]],"date-time":"2016-03-07T01:53:30Z","timestamp":1457315610000},"page":"3341-3351","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Branch and bound computational method for multi-objective linear fractional optimization problem"],"prefix":"10.1007","volume":"28","author":[{"given":"Deepak","family":"Bhati","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pitam","family":"Singh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,3,7]]},"reference":[{"issue":"2","key":"2243_CR1","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1080\/1055678031000105242","volume":"18","author":"S Schaible","year":"2003","unstructured":"Schaible S, Shi J (2003) Fractional programming: the sum-of-ratio case. Optim Method Softw 18(2):219\u2013229","journal-title":"Optim Method Softw"},{"key":"2243_CR2","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/0022-247X(81)90123-2","volume":"80","author":"MA Henson","year":"1981","unstructured":"Henson MA (1981) On sufficiency of the Kuhn\u2013Tucker conditions. J Math Anal Appl 80:545\u2013550","journal-title":"J Math Anal Appl"},{"key":"2243_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03199-5","volume-title":"Global optimization: deterministic approaches","author":"R Horst","year":"1996","unstructured":"Horst R, Tuy H (1996) Global optimization: deterministic approaches. Springer, Berlin"},{"key":"2243_CR4","volume-title":"Proceedings of the second Berkeley symposium on mathematical statistics and probability","author":"HW Kuhn","year":"1951","unstructured":"Kuhn HW, Tucker AW (1951) Nonlinear programming. In: Neyman J (ed) Proceedings of the second Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley"},{"key":"2243_CR5","volume-title":"Generalized convexity and vector optimization. Nonconvex optimization and its applications","author":"SK Mishra","year":"2009","unstructured":"Mishra SK et al (2009) Generalized convexity and vector optimization. Nonconvex optimization and its applications. Springer, Berlin"},{"issue":"2","key":"2243_CR6","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1080\/09720510.2006.10701219","volume":"9","author":"OM Saad","year":"2006","unstructured":"Saad OM (2006) Finding propoer efficient solutions in fuzzy multi-objectve programming. J Stat Manag Syst 9(2):485\u2013496","journal-title":"J Stat Manag Syst"},{"key":"2243_CR7","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1002\/nav.3800380610","volume":"38","author":"C Singh","year":"1991","unstructured":"Singh C, Hanson MA (1991) Multi-objectve fractional programming duality theory. Navel Res Logist 38:925\u2013933","journal-title":"Navel Res Logist"},{"issue":"10","key":"2243_CR8","doi-asserted-by":"crossref","first-page":"2375","DOI":"10.1093\/bioinformatics\/bti379","volume":"21","author":"M Leber","year":"2005","unstructured":"Leber M, Kaderali L, Schnhuth A, Schrader R (2005) A fractional programming approach to efficient DNA melting temperature calculation. Bioinformatics 21(10):2375\u20132382","journal-title":"Bioinformatics"},{"issue":"1","key":"2243_CR9","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0377-2217(94)00034-A","volume":"82","author":"MH Goedhart","year":"1995","unstructured":"Goedhart MH, Spronk J (1995) Financial planning with fractional goals. Eur J Oper Res 82(1):111\u2013124","journal-title":"Eur J Oper Res"},{"key":"2243_CR10","doi-asserted-by":"crossref","first-page":"4639","DOI":"10.1007\/s11269-010-9683-z","volume":"24","author":"AA Fasakhodi","year":"2010","unstructured":"Fasakhodi AA, Nouri SH, Amini M (2010) Water resources sustainability and optimal cropping pattern in farming systems: a multi-objective fractional goal programming approach. Water Res Manag 24:4639\u20134657","journal-title":"Water Res Manag"},{"key":"2243_CR11","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1007\/s00291-004-0191-5","volume":"27","author":"JP Costa","year":"2005","unstructured":"Costa JP (2005) An interactive method for multi objective linear fractional programming problem. OR Spectrum 27:633\u2013652","journal-title":"OR Spectrum"},{"issue":"6","key":"2243_CR12","doi-asserted-by":"crossref","first-page":"820","DOI":"10.1007\/s10958-009-9603-z","volume":"161","author":"JP Costa","year":"2009","unstructured":"Costa JP, Alves MJ (2009) A reference point technique to compute non-dominated solutions in MOLFP. J Math Sci 161(6):820\u2013831","journal-title":"J Math Sci"},{"key":"2243_CR13","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.apm.2013.05.046","volume":"38","author":"A Valipour","year":"2014","unstructured":"Valipour A, Yaghoobi MA, Mashinchi M (2014) An intertive approach to solve multi objective linear fractional programming problems. Appl Math Model 38:38\u201349","journal-title":"Appl Math Model"},{"issue":"3","key":"2243_CR14","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/0165-0114(89)90258-3","volume":"32","author":"H Yano","year":"1989","unstructured":"Yano H, Sakawa M (1989) Interactive fuzzy decision making for generalized multi objective linear fractional programming problems with fuzzy parameters. Fuzzy Sets Syst 32(3):245\u2013261","journal-title":"Fuzzy Sets Syst"},{"key":"2243_CR15","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/S0165-0114(01)00060-4","volume":"125","author":"M Chakraborty","year":"2002","unstructured":"Chakraborty M, Gupta S (2002) Fuzzy mathematical programming for multi objective linear fractional programming problem. Fuzzy Sets Syst 125:335\u2013342","journal-title":"Fuzzy Sets Syst"},{"issue":"2","key":"2243_CR16","first-page":"80","volume":"6","author":"N Guzel","year":"2005","unstructured":"Guzel N, Sivri M (2005) Taylor series solution of multi-objective linear fractional programming problem. Trakya Univ J Sci 6(2):80\u201387","journal-title":"Trakya Univ J Sci"},{"key":"2243_CR17","unstructured":"Guzel N (2013) A proposal to the solution of multi-objective linear fractional programming problem. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013. Article ID 435030:1\u20134"},{"key":"2243_CR18","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/s10107-005-0582-7","volume":"103","author":"J Linderoth","year":"2005","unstructured":"Linderoth J (2005) A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math Program Ser 103:251\u2013282","journal-title":"Math Program Ser"},{"issue":"2","key":"2243_CR19","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s12597-012-0067-4","volume":"49","author":"V Sharma","year":"2012","unstructured":"Sharma V (2012) Multi-objective integer nonlinear fractional programming problem: a cutting plane approach. OPSEARCH 49(2):133\u2013153","journal-title":"OPSEARCH"},{"key":"2243_CR20","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1023\/A:1008314922240","volume":"18","author":"H Konno","year":"2000","unstructured":"Konno H, Fukaishi K (2000) Branch and bound algorithm for solving low rank linear multiplicative and fractional programming problems. J Glob Optim 18:283\u2013299","journal-title":"J Glob Optim"},{"key":"2243_CR21","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10957-007-9323-9","volume":"137","author":"HP Benson","year":"2008","unstructured":"Benson HP (2008) Global maximization of a generalized concave multiplicative function. J Optim Theory Appl 137:105\u2013120","journal-title":"J Optim Theory Appl"},{"key":"2243_CR22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-010-9647-8","volume":"146","author":"HP Benson","year":"2010","unstructured":"Benson HP (2010) Branch-and-bound outer approximation algorithm for sum-of-ratios fractional programs. J Optim Theory Appl 146:1\u201318","journal-title":"J Optim Theory Appl"},{"key":"2243_CR23","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.cam.2008.01.003","volume":"223","author":"PP Shen","year":"2009","unstructured":"Shen PP, Duan YP, Pei YG (2009) A simplicial branch and duality bound algorithm for the sum of convex-convex ratios problem. J Comput Appl Math 223:145\u2013158","journal-title":"J Comput Appl Math"},{"key":"2243_CR24","unstructured":"Zhou XG, Cao BY (2013) A Simplicial branch and bound duality bounds algorithm to linear multiplicative programming. J Appl Math Volume 2013: 1\u201310, Article ID: 984168"},{"key":"2243_CR25","doi-asserted-by":"crossref","first-page":"837","DOI":"10.4134\/CKMS.2005.20.4.837","volume":"20","author":"DS Kim","year":"2005","unstructured":"Kim DS (2005) Multiobjective fractional programming with a modified objective function. Commun Korean Math Soc 20:837\u2013847","journal-title":"Commun Korean Math Soc"},{"key":"2243_CR26","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1002\/nav.3800240416","volume":"24","author":"S Schaible","year":"1977","unstructured":"Schaible S (1977) A note on the sum of a linear and linear fractional functions. Naval Res Logist Q 24:691\u2013693","journal-title":"Naval Res Logist Q"},{"key":"2243_CR27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-007-9199-8","volume":"135","author":"HP Benson","year":"2007","unstructured":"Benson HP (2007) Solving sum of ratios fractional programs via concave minimization. J Optim Theory Appl 135:1\u201317","journal-title":"J Optim Theory Appl"},{"key":"2243_CR28","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/s10957-007-9284-z","volume":"135","author":"H Kanno","year":"2007","unstructured":"Kanno H, Tsuchiya K, Yamamoto R (2007) Minimization of ratio of function defined as sum of the absolute values. J Optim Theory Appl 135:399\u2013410","journal-title":"J Optim Theory Appl"},{"key":"2243_CR29","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s10957-008-9499-7","volume":"141","author":"HJ Chen","year":"2009","unstructured":"Chen HJ, Schaible S, Sheu RL (2009) Generic algorithm for generalized fractional programming. J Optim Theory Appl 141:93\u2013105","journal-title":"J Optim Theory Appl"},{"issue":"1","key":"2243_CR30","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1023\/A:1014857230393","volume":"113","author":"ST Hackman","year":"2002","unstructured":"Hackman ST, Passy U (2002) Maximizing linear fractional function on a pareto efficient frontier. J Optim Theory Appl 113(1):83\u2013103","journal-title":"J Optim Theory Appl"},{"issue":"3","key":"2243_CR31","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1023\/B:JOTA.0000006959.01650.59","volume":"119","author":"HP Benson","year":"2003","unstructured":"Benson HP (2003) Generating sum of ratios test problems in global optimization. J Optim Theory Appl 119(3):615\u2013621","journal-title":"J Optim Theory Appl"},{"issue":"1","key":"2243_CR32","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1023\/B:JOTA.0000026129.07165.5a","volume":"121","author":"HP Benson","year":"2004","unstructured":"Benson HP (2004) On the global optimization of sums of linear fractional functions over a convex set. J Optim Theory Appl 121(1):19\u201339","journal-title":"J Optim Theory Appl"},{"issue":"1","key":"2243_CR33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1013072027218","volume":"112","author":"HP Benson","year":"2002","unstructured":"Benson HP (2002) Global optimization algorithm for the nonlinear sum of ratios problem. J Optim Theory Appl 112(1):1\u201329","journal-title":"J Optim Theory Appl"},{"key":"2243_CR34","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1006\/jmaa.2001.7650","volume":"263","author":"HP Benson","year":"2001","unstructured":"Benson HP (2001) Global optimization algorithm for the non-linear sum of ratios problem. J Math Anal Appl 263:301\u2013315","journal-title":"J Math Anal Appl"},{"key":"2243_CR35","doi-asserted-by":"crossref","first-page":"2396","DOI":"10.1016\/j.apm.2009.11.006","volume":"34","author":"P Shen","year":"2010","unstructured":"Shen P, Jin L (2010) Using canonical partition to globally maximizing the non-linear sum of ratios. Appl Math Model 34:2396\u20132413","journal-title":"Appl Math Model"},{"key":"2243_CR36","first-page":"319","volume":"158","author":"YJ Wang","year":"2004","unstructured":"Wang YJ, Zhang KC (2004) Global optimization of non-linear sum of ratios problem. Appl Math Appl 158:319\u2013330","journal-title":"Appl Math Appl"},{"key":"2243_CR37","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.amc.2005.09.047","volume":"176","author":"PP Shen","year":"2006","unstructured":"Shen PP, Wang CF (2006) Global optimization for sum of ratios problem with coefficient. Appl Math Comput 176:219\u2013229","journal-title":"Appl Math Comput"},{"key":"2243_CR38","doi-asserted-by":"crossref","first-page":"1812","DOI":"10.1016\/j.amc.2006.11.047","volume":"188","author":"H Jiao","year":"2007","unstructured":"Jiao H, Shen P (2007) A note on the paper global optimization of non-linear sum of ratios. Appl Math Comput 188:1812\u20131815","journal-title":"Appl Math Comput"},{"key":"2243_CR39","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1016\/j.amc.2009.02.024","volume":"212","author":"P Shen","year":"2009","unstructured":"Shen P, Chen Y, Yuan M (2009) Solving sum of quadratic ratios fractional programs via monotonic function. Appl Math Comput 212:234\u2013244","journal-title":"Appl Math Comput"},{"key":"2243_CR40","doi-asserted-by":"crossref","first-page":"2301","DOI":"10.1016\/j.cor.2013.03.012","volume":"40","author":"P Shen","year":"2013","unstructured":"Shen P, Li W, Bai X (2013) Maximizing for the sum of ratios of two convex functions over a convex set. Comput Oper Res 40:2301\u20132307","journal-title":"Comput Oper Res"},{"key":"2243_CR41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2007.01.022","volume":"214","author":"PP Shen","year":"2008","unstructured":"Shen PP, Wang CF (2008) Global optimization for sum of generalization fractional functions. J Comput Appl Math 214:1\u201312","journal-title":"J Comput Appl Math"},{"key":"2243_CR42","doi-asserted-by":"crossref","unstructured":"Jin L, Hou XP (2014) Global optimization for a class non-linear sum of ratios problems Problems in Engineering Volume 2014: Article ID: 103569","DOI":"10.1155\/2014\/103569"},{"key":"2243_CR43","doi-asserted-by":"crossref","unstructured":"Gao Y, Jin S (2013) A global optimization algorithm for sum of linear ratios problem. J Appl Math Volume 2013: 1\u201310, Article ID: 276245","DOI":"10.1155\/2013\/276245"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00521-016-2243-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-016-2243-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-016-2243-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-016-2243-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T20:53:31Z","timestamp":1718398411000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00521-016-2243-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,7]]},"references-count":43,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2017,11]]}},"alternative-id":["2243"],"URL":"https:\/\/doi.org\/10.1007\/s00521-016-2243-6","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,3,7]]}}}