{"id":"https://openalex.org/W3190279398","doi":"https://doi.org/10.1007/s43069-021-00077-w","title":"On a Maximum Eigenvalue of Third-Order Pairwise Comparison Matrix in Analytic Hierarchy Process and Convergence of Newton\u2019s Method","display_name":"On a Maximum Eigenvalue of Third-Order Pairwise Comparison Matrix in Analytic Hierarchy Process and Convergence of Newton\u2019s Method","publication_year":2021,"publication_date":"2021-07-08","ids":{"openalex":"https://openalex.org/W3190279398","doi":"https://doi.org/10.1007/s43069-021-00077-w","mag":"3190279398"},"language":"en","primary_location":{"id":"doi:10.1007/s43069-021-00077-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s43069-021-00077-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s43069-021-00077-w.pdf","source":{"id":"https://openalex.org/S4210215153","display_name":"Operations Research Forum","issn_l":"2662-2556","issn":["2662-2556"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["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":"Operations Research Forum","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/s43069-021-00077-w.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004186830","display_name":"Shunsuke Shiraishi","orcid":"https://orcid.org/0000-0003-2401-1938"},"institutions":[{"id":"https://openalex.org/I42766147","display_name":"University of Toyama","ror":"https://ror.org/0445phv87","country_code":"JP","type":"education","lineage":["https://openalex.org/I42766147"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Shunsuke Shiraishi","raw_affiliation_strings":["School of Economics, University of Toyama, 3190 Gofuku, Toyama, 930-8555, Japan","School of Economics, University of Toyama, Toyama, Japan"],"raw_orcid":"https://orcid.org/0000-0003-2401-1938","affiliations":[{"raw_affiliation_string":"School of Economics, University of Toyama, 3190 Gofuku, Toyama, 930-8555, Japan","institution_ids":["https://openalex.org/I42766147"]},{"raw_affiliation_string":"School of Economics, University of Toyama, Toyama, Japan","institution_ids":["https://openalex.org/I42766147"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007236158","display_name":"Tsuneshi Obata","orcid":"https://orcid.org/0000-0002-5924-112X"},"institutions":[{"id":"https://openalex.org/I188815454","display_name":"Oita University","ror":"https://ror.org/01nyv7k26","country_code":"JP","type":"education","lineage":["https://openalex.org/I188815454"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Tsuneshi Obata","raw_affiliation_strings":["Faculty of Science and Technology, Oita University, 700 Dan-noharu, Oita, 870-1192, Japan","Faculty of Science and Technology, Oita University, Oita, Japan"],"raw_orcid":"https://orcid.org/0000-0002-5924-112X","affiliations":[{"raw_affiliation_string":"Faculty of Science and Technology, Oita University, 700 Dan-noharu, Oita, 870-1192, Japan","institution_ids":["https://openalex.org/I188815454"]},{"raw_affiliation_string":"Faculty of Science and Technology, Oita University, Oita, Japan","institution_ids":["https://openalex.org/I188815454"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5007236158"],"corresponding_institution_ids":["https://openalex.org/I188815454"],"apc_list":{"value":2190,"currency":"EUR","value_usd":2780},"apc_paid":{"value":2190,"currency":"EUR","value_usd":2780},"fwci":0.42,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.69406048,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"2","issue":"3","first_page":"30","last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T14175","display_name":"Intuitionistic Fuzzy Systems Applications","score":0.19990000128746033,"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/T14175","display_name":"Intuitionistic Fuzzy Systems Applications","score":0.19990000128746033,"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/T10050","display_name":"Multi-Criteria Decision Making","score":0.16599999368190765,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12709","display_name":"Optimization and Mathematical Programming","score":0.027400000020861626,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7782211303710938},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.7008568048477173},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5922698378562927},{"id":"https://openalex.org/keywords/analytic-hierarchy-process","display_name":"Analytic hierarchy process","score":0.5748982429504395},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5571776032447815},{"id":"https://openalex.org/keywords/consistency","display_name":"Consistency (knowledge bases)","score":0.5521541833877563},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5242089033126831},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.44779443740844727},{"id":"https://openalex.org/keywords/newtons-method","display_name":"Newton's method","score":0.4238448143005371},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4236159026622772},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.37519729137420654},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.24022918939590454},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.16398954391479492},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.07266989350318909},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.07251614332199097},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.062180787324905396}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7782211303710938},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.7008568048477173},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5922698378562927},{"id":"https://openalex.org/C87345402","wikidata":"https://www.wikidata.org/wiki/Q485202","display_name":"Analytic hierarchy process","level":2,"score":0.5748982429504395},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5571776032447815},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.5521541833877563},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5242089033126831},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.44779443740844727},{"id":"https://openalex.org/C85189116","wikidata":"https://www.wikidata.org/wiki/Q374195","display_name":"Newton's method","level":3,"score":0.4238448143005371},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4236159026622772},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.37519729137420654},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.24022918939590454},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.16398954391479492},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.07266989350318909},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.07251614332199097},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.062180787324905396},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s43069-021-00077-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s43069-021-00077-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s43069-021-00077-w.pdf","source":{"id":"https://openalex.org/S4210215153","display_name":"Operations Research Forum","issn_l":"2662-2556","issn":["2662-2556"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["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":"Operations Research Forum","raw_type":"journal-article"},{"id":"pmh:oai:toyama.repo.nii.ac.jp:00018443","is_oa":true,"landing_page_url":"http://hdl.handle.net/10110/00020869","pdf_url":"https://toyama.repo.nii.ac.jp/?action=repository_uri&item_id=18443","source":{"id":"https://openalex.org/S4306401765","display_name":"University of Toyama Repository for Academic Information (University of Toyama)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I42766147","host_organization_name":"University of Toyama","host_organization_lineage":["https://openalex.org/I42766147"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal Article"},{"id":"pmh:oai:RePEc:spr:snopef:v:2:y:2021:i:3:d:10.1007_s43069-021-00077-w","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s43069-021-00077-w","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:irdb.nii.ac.jp:00844:0005010660","is_oa":true,"landing_page_url":"https://toyama.repo.nii.ac.jp/records/18443","pdf_url":null,"source":{"id":"https://openalex.org/S7407056385","display_name":"Institutional Repositories DataBase (IRDB)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I184597095","host_organization_name":"National Institute of Informatics","host_organization_lineage":["https://openalex.org/I184597095"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Operations Research Forum","raw_type":"journal article"}],"best_oa_location":{"id":"doi:10.1007/s43069-021-00077-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s43069-021-00077-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s43069-021-00077-w.pdf","source":{"id":"https://openalex.org/S4210215153","display_name":"Operations Research Forum","issn_l":"2662-2556","issn":["2662-2556"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["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":"Operations Research Forum","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.8100000023841858,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3190279398.pdf","grobid_xml":"https://content.openalex.org/works/W3190279398.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1524771255","https://openalex.org/W1856705291","https://openalex.org/W1871214897","https://openalex.org/W1995514749","https://openalex.org/W2093184664","https://openalex.org/W2119468588","https://openalex.org/W2128971439","https://openalex.org/W2168936936","https://openalex.org/W2284724473","https://openalex.org/W2292276779","https://openalex.org/W2893188218","https://openalex.org/W3096385332","https://openalex.org/W3135293561","https://openalex.org/W4214643344","https://openalex.org/W4240954528","https://openalex.org/W4248241950"],"related_works":["https://openalex.org/W2619886288","https://openalex.org/W2410864841","https://openalex.org/W380774767","https://openalex.org/W3134930408","https://openalex.org/W4287965002","https://openalex.org/W4389396049","https://openalex.org/W1977674833","https://openalex.org/W2395251459","https://openalex.org/W2028983668","https://openalex.org/W2084939379"],"abstract_inverted_index":{"Abstract":[0],"Nowadays,":[1],"the":[2,16,29,39,48,51,59,78,93,98,102,107,118,130,133,141,145,153,157,165],"analytic":[3,30],"hierarchy":[4,31],"process":[5],"is":[6,63,101,122,127],"an":[7],"established":[8],"method":[9,139],"of":[10,18,38,50,58,81,86,97,106,132,144,160],"multiple":[11],"criteria":[12],"decision":[13,52],"making":[14],"in":[15,28,117],"field":[17],"Operations":[19],"Research.":[20],"Pairwise":[21],"comparison":[22,41,84,110,148],"matrix":[23,42,85],"plays":[24],"a":[25,82],"crucial":[26],"role":[27],"process.":[32],"The":[33,55,112],"principal":[34],"(maximum":[35],"magnitude)":[36],"eigenvalue":[37,62,105],"pairwise":[40,83,109,147],"can":[43],"be":[44,66],"utilized":[45],"for":[46],"measuring":[47],"consistency":[49,68],"maker\u2019s":[53],"judgment.":[54],"simple":[56],"transformation":[57],"maximum":[60,103,166],"magnitude":[61,104,167],"known":[64],"to":[65,129,140,164],"Saaty\u2019s":[67],"index.":[69],"In":[70],"this":[71],"short":[72],"note,":[73],"we":[74,150],"shed":[75],"light":[76],"on":[77],"characteristic":[79,99,142],"polynomial":[80,143],"third":[87],"order.":[88],"We":[89],"will":[90],"show":[91],"that":[92,152],"only":[94],"real-number":[95,114],"root":[96,115],"equation":[100],"third-order":[108,146],"matrix.":[111,134],"unique":[113],"appears":[116],"area":[119],"where":[120],"it":[121],"greater":[123],"than":[124],"3,":[125],"which":[126],"equal":[128],"order":[131],"By":[135],"applying":[136],"usual":[137],"Newton\u2019s":[138],"matrix,":[149],"see":[151],"sequence":[154],"generated":[155],"from":[156],"initial":[158],"value":[159],"3":[161],"always":[162],"converges":[163],"eigenvalue.":[168]},"counts_by_year":[{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}