{"id":"https://openalex.org/W3130843656","doi":"https://doi.org/10.1007/s10589-025-00660-5","title":"Two-commodity opposite direction network flow formulations for the travelling salesman problem","display_name":"Two-commodity opposite direction network flow formulations for the travelling salesman problem","publication_year":2025,"publication_date":"2025-03-22","ids":{"openalex":"https://openalex.org/W3130843656","doi":"https://doi.org/10.1007/s10589-025-00660-5","mag":"3130843656"},"language":"en","primary_location":{"id":"doi:10.1007/s10589-025-00660-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-025-00660-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-025-00660-5.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10589-025-00660-5.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5030913178","display_name":"Konstantin Pavlikov","orcid":"https://orcid.org/0000-0002-3345-5058"},"institutions":[{"id":"https://openalex.org/I177969490","display_name":"University of Southern Denmark","ror":"https://ror.org/03yrrjy16","country_code":"DK","type":"education","lineage":["https://openalex.org/I177969490"]}],"countries":["DK"],"is_corresponding":true,"raw_author_name":"Konstantin Pavlikov","raw_affiliation_strings":["Department of Business and Management, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark","Department of Business and Economics"],"raw_orcid":"https://orcid.org/0000-0002-3345-5058","affiliations":[{"raw_affiliation_string":"Department of Business and Management, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark","institution_ids":["https://openalex.org/I177969490"]},{"raw_affiliation_string":"Department of Business and Economics","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5075952282","display_name":"Niels Christian Petersen","orcid":"https://orcid.org/0000-0002-0002-2264"},"institutions":[{"id":"https://openalex.org/I177969490","display_name":"University of Southern Denmark","ror":"https://ror.org/03yrrjy16","country_code":"DK","type":"education","lineage":["https://openalex.org/I177969490"]}],"countries":["DK"],"is_corresponding":false,"raw_author_name":"Niels Christian Petersen","raw_affiliation_strings":["Department of Business and Management, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark","Department of Business and Economics"],"raw_orcid":"https://orcid.org/0000-0002-0002-2264","affiliations":[{"raw_affiliation_string":"Department of Business and Management, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark","institution_ids":["https://openalex.org/I177969490"]},{"raw_affiliation_string":"Department of Business and Economics","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5030913178"],"corresponding_institution_ids":["https://openalex.org/I177969490"],"apc_list":{"value":2990,"currency":"USD","value_usd":2990},"apc_paid":{"value":2990,"currency":"USD","value_usd":2990},"fwci":0.5813,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.63245899,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"92","issue":"3","first_page":"987","last_page":"1033"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10567","display_name":"Vehicle Routing Optimization Methods","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2209","display_name":"Industrial and Manufacturing Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10567","display_name":"Vehicle Routing Optimization Methods","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2209","display_name":"Industrial and Manufacturing Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12546","display_name":"Smart Parking Systems Research","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2215","display_name":"Building and Construction"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10100","display_name":"Metaheuristic Optimization Algorithms Research","score":0.9883999824523926,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/travelling-salesman-problem","display_name":"Travelling salesman problem","score":0.8015686869621277},{"id":"https://openalex.org/keywords/flow-network","display_name":"Flow network","score":0.5860995650291443},{"id":"https://openalex.org/keywords/flow","display_name":"Flow (mathematics)","score":0.5533940196037292},{"id":"https://openalex.org/keywords/commodity","display_name":"Commodity","score":0.5387898683547974},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4590967297554016},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.434924840927124},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.394905149936676},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.3828998804092407},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.26100611686706543},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12686127424240112},{"id":"https://openalex.org/keywords/finance","display_name":"Finance","score":0.04731708765029907}],"concepts":[{"id":"https://openalex.org/C175859090","wikidata":"https://www.wikidata.org/wiki/Q322212","display_name":"Travelling salesman problem","level":2,"score":0.8015686869621277},{"id":"https://openalex.org/C114809511","wikidata":"https://www.wikidata.org/wiki/Q1412924","display_name":"Flow network","level":2,"score":0.5860995650291443},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.5533940196037292},{"id":"https://openalex.org/C2779439359","wikidata":"https://www.wikidata.org/wiki/Q317088","display_name":"Commodity","level":2,"score":0.5387898683547974},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4590967297554016},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.434924840927124},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.394905149936676},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.3828998804092407},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.26100611686706543},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12686127424240112},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.04731708765029907}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s10589-025-00660-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-025-00660-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-025-00660-5.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},{"id":"pmh:oai:sdu.dk:openaire/de2b55a8-1048-4d66-9411-5dde9444ccd7","is_oa":true,"landing_page_url":"https://portal.findresearcher.sdu.dk/da/publications/de2b55a8-1048-4d66-9411-5dde9444ccd7","pdf_url":null,"source":{"id":"https://openalex.org/S4306400423","display_name":"University of Southern Denmark Research Portal (University of Southern Denmark)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I177969490","host_organization_name":"University of Southern Denmark","host_organization_lineage":["https://openalex.org/I177969490"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Pavlikov, K & Petersen, N C 2025, 'Two-commodity opposite direction network flow formulations for the travelling salesman problem', Computational Optimization and Applications, vol. 92, no. 3, pp. 987\u20131033. https://doi.org/10.1007/s10589-025-00660-5","raw_type":"info:eu-repo/semantics/article"},{"id":"mag:3130843656","is_oa":false,"landing_page_url":"https://portal.findresearcher.sdu.dk/en/publications/two-commodity-opposite-direction-network-flow-formulations-for-th","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"doi:10.1007/s10589-025-00660-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-025-00660-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-025-00660-5.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Decent work and economic growth","score":0.5799999833106995,"id":"https://metadata.un.org/sdg/8"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320323799","display_name":"Syddansk Universitet","ror":"https://ror.org/03yrrjy16"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3130843656.pdf"},"referenced_works_count":38,"referenced_works":["https://openalex.org/W626292722","https://openalex.org/W954848844","https://openalex.org/W1525389557","https://openalex.org/W1968429519","https://openalex.org/W1969186119","https://openalex.org/W1970025200","https://openalex.org/W1973326170","https://openalex.org/W1977129671","https://openalex.org/W1984451472","https://openalex.org/W2001994662","https://openalex.org/W2004882088","https://openalex.org/W2017708378","https://openalex.org/W2018660239","https://openalex.org/W2029242918","https://openalex.org/W2031736545","https://openalex.org/W2032686693","https://openalex.org/W2033097589","https://openalex.org/W2040865164","https://openalex.org/W2040886899","https://openalex.org/W2042852020","https://openalex.org/W2042986967","https://openalex.org/W2047194747","https://openalex.org/W2049287714","https://openalex.org/W2054177169","https://openalex.org/W2054706926","https://openalex.org/W2054991304","https://openalex.org/W2072699573","https://openalex.org/W2092353499","https://openalex.org/W2092481826","https://openalex.org/W2095546721","https://openalex.org/W2111622332","https://openalex.org/W2114552889","https://openalex.org/W2115049345","https://openalex.org/W2140425706","https://openalex.org/W2159203019","https://openalex.org/W2163428398","https://openalex.org/W2620774153","https://openalex.org/W3005757297"],"related_works":["https://openalex.org/W1525389557","https://openalex.org/W2361554335","https://openalex.org/W4311255633","https://openalex.org/W2001284284","https://openalex.org/W2368007657","https://openalex.org/W2068289888","https://openalex.org/W4250257362","https://openalex.org/W2352790364","https://openalex.org/W2391191500","https://openalex.org/W2365398175"],"abstract_inverted_index":{"Abstract":[0],"This":[1,101,182,258],"paper":[2,102,141],"considers":[3],"the":[4,20,24,27,60,66,69,73,77,81,109,116,125,128,133,152,175,188,191,198,210,216,227,253,269,273,283,287,295],"well-known":[5],"Travelling":[6],"Salesman":[7],"Problem":[8],"(TSP)":[9],"in":[10,65,139,144,151,172,187,252],"its":[11],"symmetric":[12,21,110,256,274],"and":[13,111,299],"asymmetric":[14,74,112,228,284],"versions.":[15],"A":[16],"distinctive":[17],"feature":[18],"of":[19,23,62,76,84,118,130,135,174,177,197,209,220,255,264,272],"version":[22,75],"problem":[25,37,78,126,275],"is":[26,59,142,183],"ability":[28],"to":[29,107,225,294],"formulate":[30,108,226],"it":[31],"as":[32,251],"an":[33,301],"undirected":[34],"network":[35,161,194],"optimization":[36],"using":[38,300],"$$n(n":[39,85,232],"-":[40,86,233],"1)/2$$":[41,234],"<mml:math":[42,88,235],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[43,89,236],"<mml:mrow>":[44,90,237],"<mml:mi>n</mml:mi>":[45,47,91,93,238,240],"<mml:mo>(</mml:mo>":[46,92,239],"<mml:mo>-</mml:mo>":[48,94,241],"<mml:mn>1</mml:mn>":[49,95,242],"<mml:mo>)</mml:mo>":[50,96,243],"<mml:mo>/</mml:mo>":[51,244],"<mml:mn>2</mml:mn>":[52,245],"</mml:mrow>":[53,97,246],"</mml:math>":[54,98,247],"binary":[55,99,248],"variables,":[56,249],"where":[57],"n":[58,156],"number":[61],"locations":[63],"involved":[64],"problem.":[67,285],"At":[68],"same":[70],"time,":[71],"formulating":[72,124],"generally":[79],"requires":[80],"full":[82],"set":[83],"1)$$":[87],"variables.":[100],"presents":[103],"a":[104,148,184],"new":[105,185],"approach":[106],"TSPs":[113],"based":[114,267],"on":[115],"flow":[117,129,134,162,170,195,213,219],"two":[119,136,221],"commodities.":[120],"Unlike":[121],"traditional":[122],"approaches":[123,297],"with":[127,155,230],"multiple":[131],"commodities,":[132],"commodities":[137,222],"considered":[138],"this":[140],"organized":[143],"opposite":[145,217],"directions":[146],"along":[147],"Hamiltonian":[149],"cycle":[150],"complete":[153],"graph":[154],"nodes.":[157],"The":[158],"proposed":[159,288],"two-commodity":[160,193],"formulations":[163,196,291],"are":[164,204,276,292],"strictly":[165],"stronger":[166],"than":[167],"their":[168,178],"one-commodity":[169,212],"counterparts":[171],"terms":[173],"quality":[176],"linear":[179],"programming":[180,290],"relaxations.":[181],"result":[186,259],"sense":[189],"that":[190,203,261],"existing":[192,296],"TSP":[199,229],"provide":[200],"lower":[201],"bounds":[202],"no":[205],"different":[206],"from":[207],"those":[208],"corresponding":[211],"formulations.":[214],"Moreover,":[215],"direction":[218],"allows":[223],"us":[224],"only":[231],"just":[250],"case":[254],"TSP.":[257],"suggests":[260],"various":[262],"classes":[263],"valid":[265],"inequalities":[266],"upon":[268],"polyhedral":[270],"structure":[271],"sufficient":[277],"for":[278,282],"designing":[279],"branch-and-cut":[280],"algorithms":[281],"Finally,":[286],"mathematical":[289],"compared":[293],"analytically":[298],"extensive":[302],"computational":[303],"study.":[304]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2025-10-10T00:00:00"}
