{"id":"https://openalex.org/W2038380610","doi":"https://doi.org/10.1080/00207729808929523","title":"Probabilistic analysis of 2-opt for travelling salesman problems","display_name":"Probabilistic analysis of 2-opt for travelling salesman problems","publication_year":1998,"publication_date":"1998-03-01","ids":{"openalex":"https://openalex.org/W2038380610","doi":"https://doi.org/10.1080/00207729808929523","mag":"2038380610"},"language":"en","primary_location":{"id":"doi:10.1080/00207729808929523","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207729808929523","pdf_url":null,"source":{"id":"https://openalex.org/S129640837","display_name":"International Journal of Systems Science","issn_l":"0020-7721","issn":["0020-7721","1464-5319"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Systems Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5108637083","display_name":"Masahiro Okada","orcid":null},"institutions":[{"id":"https://openalex.org/I75917431","display_name":"Nara Institute of Science and Technology","ror":"https://ror.org/05bhada84","country_code":"JP","type":"education","lineage":["https://openalex.org/I75917431"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"MASAHIRO OKADA","raw_affiliation_strings":["Graduate School of Information Science, Nara Institute of Science and Technology , 8916-5 Takayama, Ikoma, Nara, 630-01, Japan","Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama, Ikoma, Nara 630-01, Japan#TAB#"],"affiliations":[{"raw_affiliation_string":"Graduate School of Information Science, Nara Institute of Science and Technology , 8916-5 Takayama, Ikoma, Nara, 630-01, Japan","institution_ids":["https://openalex.org/I75917431"]},{"raw_affiliation_string":"Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama, Ikoma, Nara 630-01, Japan#TAB#","institution_ids":["https://openalex.org/I75917431"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110509526","display_name":"Kouichi Taji","orcid":null},"institutions":[{"id":"https://openalex.org/I75917431","display_name":"Nara Institute of Science and Technology","ror":"https://ror.org/05bhada84","country_code":"JP","type":"education","lineage":["https://openalex.org/I75917431"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"KOUICHI TAJI","raw_affiliation_strings":["Graduate School of Information Science, Nara Institute of Science and Technology , 8916-5 Takayama, Ikoma, Nara, 630-01, Japan","Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama, Ikoma, Nara 630-01, Japan#TAB#"],"affiliations":[{"raw_affiliation_string":"Graduate School of Information Science, Nara Institute of Science and Technology , 8916-5 Takayama, Ikoma, Nara, 630-01, Japan","institution_ids":["https://openalex.org/I75917431"]},{"raw_affiliation_string":"Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama, Ikoma, Nara 630-01, Japan#TAB#","institution_ids":["https://openalex.org/I75917431"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5058813836","display_name":"Masao Fukushima","orcid":"https://orcid.org/0000-0002-4372-6376"},"institutions":[{"id":"https://openalex.org/I22299242","display_name":"Kyoto University","ror":"https://ror.org/02kpeqv85","country_code":"JP","type":"education","lineage":["https://openalex.org/I22299242"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"MASAO FUKUSHIMA","raw_affiliation_strings":["Department of Applied Mathematics and Physics , Graduate School of Engineering, Kyoto University , Kyoto, 606-01, Japan","Department of Applied Mathematics and Physics, Graduate School of Engineering, Kyoto University, Kyoto, 606-01. Japan"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Physics , Graduate School of Engineering, Kyoto University , Kyoto, 606-01, Japan","institution_ids":["https://openalex.org/I22299242"]},{"raw_affiliation_string":"Department of Applied Mathematics and Physics, Graduate School of Engineering, Kyoto University, Kyoto, 606-01. Japan","institution_ids":["https://openalex.org/I22299242"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5108637083"],"corresponding_institution_ids":["https://openalex.org/I75917431"],"apc_list":null,"apc_paid":null,"fwci":0.5353,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.74644595,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"29","issue":"3","first_page":"297","last_page":"310"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10100","display_name":"Metaheuristic Optimization Algorithms Research","score":0.9995999932289124,"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"}},"topics":[{"id":"https://openalex.org/T10100","display_name":"Metaheuristic Optimization Algorithms Research","score":0.9995999932289124,"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"}},{"id":"https://openalex.org/T10567","display_name":"Vehicle Routing Optimization Methods","score":0.9995999932289124,"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/T10848","display_name":"Advanced Multi-Objective Optimization Algorithms","score":0.9958999752998352,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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":0.7976644039154053},{"id":"https://openalex.org/keywords/heuristics","display_name":"Heuristics","score":0.7402181625366211},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6498961448669434},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.6447643637657166},{"id":"https://openalex.org/keywords/tabu-search","display_name":"Tabu search","score":0.6048079133033752},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.5579946637153625},{"id":"https://openalex.org/keywords/heuristic","display_name":"Heuristic","score":0.5508468747138977},{"id":"https://openalex.org/keywords/simulated-annealing","display_name":"Simulated annealing","score":0.540871262550354},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.4841499924659729},{"id":"https://openalex.org/keywords/probabilistic-analysis-of-algorithms","display_name":"Probabilistic analysis of algorithms","score":0.43247905373573303},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.428857684135437},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.4266446828842163},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3201403021812439},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1573415994644165}],"concepts":[{"id":"https://openalex.org/C175859090","wikidata":"https://www.wikidata.org/wiki/Q322212","display_name":"Travelling salesman problem","level":2,"score":0.7976644039154053},{"id":"https://openalex.org/C127705205","wikidata":"https://www.wikidata.org/wiki/Q5748245","display_name":"Heuristics","level":2,"score":0.7402181625366211},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6498961448669434},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6447643637657166},{"id":"https://openalex.org/C123370116","wikidata":"https://www.wikidata.org/wiki/Q1424540","display_name":"Tabu search","level":2,"score":0.6048079133033752},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.5579946637153625},{"id":"https://openalex.org/C173801870","wikidata":"https://www.wikidata.org/wiki/Q201413","display_name":"Heuristic","level":2,"score":0.5508468747138977},{"id":"https://openalex.org/C126980161","wikidata":"https://www.wikidata.org/wiki/Q863783","display_name":"Simulated annealing","level":2,"score":0.540871262550354},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.4841499924659729},{"id":"https://openalex.org/C24404364","wikidata":"https://www.wikidata.org/wiki/Q7246846","display_name":"Probabilistic analysis of algorithms","level":3,"score":0.43247905373573303},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.428857684135437},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.4266446828842163},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3201403021812439},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1573415994644165},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207729808929523","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207729808929523","pdf_url":null,"source":{"id":"https://openalex.org/S129640837","display_name":"International Journal of Systems Science","issn_l":"0020-7721","issn":["0020-7721","1464-5319"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Systems Science","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1546457065","https://openalex.org/W2024060531","https://openalex.org/W2042986967","https://openalex.org/W2045173407","https://openalex.org/W2047820376","https://openalex.org/W2053913299","https://openalex.org/W2064975723","https://openalex.org/W2069323107","https://openalex.org/W2081153368","https://openalex.org/W2103981148","https://openalex.org/W2106378689","https://openalex.org/W2141355868","https://openalex.org/W2148673189","https://openalex.org/W2163428398","https://openalex.org/W2751862591","https://openalex.org/W3107939723","https://openalex.org/W3148722492","https://openalex.org/W4246598646","https://openalex.org/W4294358747","https://openalex.org/W4301441647"],"related_works":["https://openalex.org/W2991484371","https://openalex.org/W2079721241","https://openalex.org/W2141083629","https://openalex.org/W3140135604","https://openalex.org/W173587566","https://openalex.org/W2900012203","https://openalex.org/W2392101695","https://openalex.org/W2509914111","https://openalex.org/W2388574609","https://openalex.org/W2148102208"],"abstract_inverted_index":{"In":[0,99],"this":[1,100],"paper,":[2,101],"we":[3,102,143],"propose":[4],"a":[5],"probabilistic":[6],"model":[7,184],"of":[8,18,36,60,65,69,72,106,121,148,170,196],"2-opt":[9,15],"heuristics":[10],"and":[11,29,43,79,88,113,167,192],"study":[12],"its":[13],"behaviour.":[14],"is":[16,92],"one":[17,74,91],"the":[19,34,58,67,76,80,84,89,104,118,128,133,146,155,164,168,182,189,193],"most":[20],"popular":[21],"heuristic":[22],"methods":[23],"for":[24,117],"symmetric":[25],"travelling":[26],"salesman":[27],"problems,":[28],"has":[30],"been":[31,53],"used":[32,93,161],"as":[33,40],"basis":[35],"various":[37],"meta-heuristics":[38],"such":[39],"simulated":[41],"annealing":[42],"tabu":[44],"search.":[45],"But":[46],"until":[47],"now,":[48],"few":[49],"theoretical":[50],"results":[51],"have":[52],"obtained":[54,151],"with":[55],"regard":[56],"to":[57,94,109,137,162,187],"performance":[59,191],"2-opt.":[61,153,197],"At":[62],"each":[63,174],"iteration":[64],"2-opt,":[66],"lengths":[68,105],"two":[70],"pairs":[71,108],"edges,":[73],"from":[75,82,141],"current":[77],"tour":[78,165],"other":[81],"outside":[83],"tour,":[85],"are":[86],"evaluated,":[87],"shorter":[90],"construct":[95],"an":[96],"improved":[97],"tour.":[98],"consider":[103],"these":[107,122],"be":[110,160],"random":[111,123],"variables,":[112],"derive":[114],"recurrence":[115,134],"formulae":[116,135],"distribution":[119,129,139,157],"functions":[120,130,158],"variables.":[124],"We":[125],"show":[126],"that":[127,181],"generated":[131,156],"by":[132,152],"converge":[136],"certain":[138],"functions,":[140],"which":[142],"can":[144,159],"estimate":[145,188],"length":[147,166],"local":[149],"optima":[150],"Moreover,":[154],"evaluate":[163],"probability":[169],"edge":[171],"exchange":[172],"at":[173],"iteration.":[175],"The":[176],"reported":[177],"computational":[178],"experience":[179],"shows":[180],"proposed":[183],"enables":[185],"us":[186],"average":[190],"dynamic":[194],"behaviour":[195]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}