{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T18:38:48Z","timestamp":1769884728442,"version":"3.49.0"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T00:00:00Z","timestamp":1736121600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T00:00:00Z","timestamp":1736121600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["24K07950"],"award-info":[{"award-number":["24K07950"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Oper. Res. Forum"],"DOI":"10.1007\/s43069-024-00412-x","type":"journal-article","created":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T15:19:48Z","timestamp":1736176788000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Calculating Maximum Eigenvalues in Pairwise Comparison Matrices for the Analytic Hierarchy Process"],"prefix":"10.1007","volume":"6","author":[{"given":"Shunsuke","family":"Shiraishi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tsuneshi","family":"Obata","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,6]]},"reference":[{"key":"412_CR1","doi-asserted-by":"publisher","unstructured":"Brunelli M (2014) Introduction to the analytic hierarchy process. Springer, New York . https:\/\/doi.org\/10.1007\/978-3-319-12502-2","DOI":"10.1007\/978-3-319-12502-2"},{"key":"412_CR2","doi-asserted-by":"publisher","unstructured":"Tekile HA (2023) Incomplete pairwise comparison matrices and optimization techniques. Doctoral Thesis, University of Trento https:\/\/doi.org\/10.15168\/11572_376807","DOI":"10.15168\/11572_376807"},{"key":"412_CR3","volume-title":"The analytic hierarchy process","author":"TL Saaty","year":"1980","unstructured":"Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New York"},{"key":"412_CR4","doi-asserted-by":"publisher","unstructured":"Ku\u0142akowski K (2021) Understanding the analytic hierarchy process. CRC Press, Boca Raton . https:\/\/doi.org\/10.1201\/9781315392226","DOI":"10.1201\/9781315392226"},{"key":"412_CR5","volume-title":"The analytic hierarchy process: decision making (in Japanese)","author":"K Tone","year":"1986","unstructured":"Tone K (1986) The analytic hierarchy process: decision making (in Japanese). Japanese Science and Technology Press, Tokyo"},{"key":"412_CR6","doi-asserted-by":"publisher","unstructured":"Brunelli M (2018) A survey of inconsistency indices for pairwise comparisons. Int J General Syst 47(8):751\u2013771 . https:\/\/doi.org\/10.1080\/03081079.2018.1523156","DOI":"10.1080\/03081079.2018.1523156"},{"issue":"8","key":"412_CR7","doi-asserted-by":"publisher","first-page":"1206","DOI":"10.3390\/math10081206","volume":"10","author":"S Pant","year":"2022","unstructured":"Pant S, Kumar A, Ram M, Klochkov Y, Sharma HK (2022) Consistency indices in analytic hierarchy process: a review. Math 10(8):1206. https:\/\/doi.org\/10.3390\/math10081206","journal-title":"Math"},{"key":"412_CR8","doi-asserted-by":"publisher","unstructured":"Sato Y, Tan KH (2023) Inconsistency indices in pairwise comparisons: an improvement of the consistency index. Annal Operate Res 326:809\u2013830 . https:\/\/doi.org\/10.1007\/s10479-021-04431-3","DOI":"10.1007\/s10479-021-04431-3"},{"issue":"3","key":"412_CR9","doi-asserted-by":"publisher","first-page":"1078","DOI":"10.1016\/j.ejor.2023.10.035","volume":"314","author":"KC \u00c1goston","year":"2024","unstructured":"\u00c1goston KC, Csat\u00f3 L (2024) A lexicographically optimal completion for pairwise comparison matrices with missing entries. European J Operate Res 314(3):1078\u20131086. https:\/\/doi.org\/10.1016\/j.ejor.2023.10.035","journal-title":"European J Operate Res"},{"key":"412_CR10","doi-asserted-by":"publisher","unstructured":"Tekile HA, Brunelli M, Fedrizzi M (2023) A numerical comparative study of completion methods for pairwise comparison matrices. Operate Res Perspect 10:100272 . https:\/\/doi.org\/10.1016\/j.orp.2023.100272","DOI":"10.1016\/j.orp.2023.100272"},{"key":"412_CR11","doi-asserted-by":"publisher","unstructured":"Sekitani K, Yamaki N (1999) A logical interpretation for the eigenvalue method in AHP. J Operate Res Soc Japan 42(2):219\u2013232. https:\/\/doi.org\/10.15807\/jorsj.42.219","DOI":"10.15807\/jorsj.42.219"},{"issue":"2","key":"412_CR12","doi-asserted-by":"publisher","first-page":"708","DOI":"10.1016\/j.ejor.2023.09.022","volume":"313","author":"L Csat\u00f3","year":"2024","unstructured":"Csat\u00f3 L (2024) Right-left asymmetry of the eigenvector method: a simulation study. European J Operation Res 313(2):708\u2013717. https:\/\/doi.org\/10.1016\/j.ejor.2023.09.022","journal-title":"European J Operation Res"},{"issue":"1","key":"412_CR13","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1016\/j.ejor.2020.10.020","volume":"292","author":"L Csat\u00f3","year":"2021","unstructured":"Csat\u00f3 L, Petr\u00f3czy DG (2021) On the monotonicity of the eigenvector method. European J Operation Res 292(1):230\u2013237. https:\/\/doi.org\/10.1016\/j.ejor.2020.10.020","journal-title":"European J Operation Res"},{"issue":"2","key":"412_CR14","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1016\/j.ejor.2017.06.033","volume":"264","author":"S Boz\u00f3ki","year":"2018","unstructured":"Boz\u00f3ki S, F\u00fcl\u00f6p J (2018) Efficient weight vectors from pairwise comparison matrices. European J Operation Res 264(2):419\u2013427. https:\/\/doi.org\/10.1016\/j.ejor.2017.06.033","journal-title":"European J Operation Res"},{"key":"412_CR15","doi-asserted-by":"publisher","unstructured":"G\u00f3recki J, Barti D, Ram\u00edk J (2024) Robustness of priority deriving methods for pairwise comparison matrices against rank reversal: a probabilistic approach. Annal Operation Res 333:249\u2013273. https:\/\/doi.org\/10.1007\/s10479-023-05753-0","DOI":"10.1007\/s10479-023-05753-0"},{"key":"412_CR16","doi-asserted-by":"publisher","unstructured":"Ishizaka A, Lusti M (2006) How to derive priorities in AHP: a comparative study. Central European J Operation Res 14:387\u2013400 . https:\/\/doi.org\/10.1007\/s10100-006-0012-9","DOI":"10.1007\/s10100-006-0012-9"},{"key":"412_CR17","doi-asserted-by":"publisher","unstructured":"Mu E, Pereyra-Rojas M (2016) Practical decision making. An introduction to the analytic hierarchy process (AHP) using super decisions V2. Springer, New York. https:\/\/doi.org\/10.1007\/978-3-319-33861-3","DOI":"10.1007\/978-3-319-33861-3"},{"key":"412_CR18","doi-asserted-by":"publisher","unstructured":"Goepel KD (2018) Implementation of an online software tool for the analytic hierarchy process (AHP-OS). Int J Analytic Hierarchy Process 10(3):469\u2013487. https:\/\/doi.org\/10.13033\/ijahp.v10i3.590","DOI":"10.13033\/ijahp.v10i3.590"},{"issue":"102277","key":"412_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.mex.2023.102277","volume":"11","author":"RV Fran\u00e7ozo","year":"2023","unstructured":"Fran\u00e7ozo RV, Junior LSVU, Carrapateira ES, Pacheco BCS, Oliveira MT, Torsoni GB, Yari J (2023) A web-based software for group decision with analytic hierarchy process. MethodsX 11(102277):1\u201312. https:\/\/doi.org\/10.1016\/j.mex.2023.102277","journal-title":"MethodsX"},{"key":"412_CR20","doi-asserted-by":"publisher","unstructured":"Shiraishi S, Obata T (2021) On a maximum eigenvalue of third-order pairwise comparison matrix in analytic hierarchy process and convergence of Newton\u2019s method. Operation Res Forum 2(3). https:\/\/doi.org\/10.1007\/s43069-021-00077-w","DOI":"10.1007\/s43069-021-00077-w"},{"issue":"3","key":"412_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5109\/7178616","volume":"56","author":"T Obata","year":"2024","unstructured":"Obata T, Shiraishi S (2024) An initial value of Newton\u2019s method for the characteristic polynomial of 4th order PCM in AHP. Bulletin of Inf Cybern 56(3):1\u20139. https:\/\/doi.org\/10.5109\/7178616","journal-title":"Bulletin of Inf Cybern"},{"key":"412_CR22","doi-asserted-by":"publisher","unstructured":"Shiraishi S, Obata T, Daigo M (1998) Properties of positive reciprocal matrix and their application to AHP. J Operation Res Soc Japan 41(3):404\u2013414. https:\/\/doi.org\/10.15807\/jorsj.41.404","DOI":"10.15807\/jorsj.41.404"},{"issue":"2","key":"412_CR23","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1016\/0377-2217(93)90044-N","volume":"70","author":"B Aupetit","year":"1993","unstructured":"Aupetit B, Genest C (1993) On some useful properties of the perron eigenvalue of a positive reciprocal matrix in the context of the analytic hierarchy process. European J Operation Res 70(2):263\u2013268. https:\/\/doi.org\/10.1016\/0377-2217(93)90044-N","journal-title":"European J Operation Res"},{"key":"412_CR24","doi-asserted-by":"publisher","unstructured":"Borwein P, Erd\u00e9lyi T (1995) Polynomials and polynomial inequalities. Springer, New York. https:\/\/doi.org\/10.1007\/978-1-4612-0793-1","DOI":"10.1007\/978-1-4612-0793-1"},{"key":"412_CR25","doi-asserted-by":"publisher","unstructured":"Horn RA, Johnson CR (2012) Matrix analysis (2nd Ed.) Cambridge University Press, Cambridge. https:\/\/doi.org\/10.1017\/CBO9781139020411","DOI":"10.1017\/CBO9781139020411"},{"key":"412_CR26","doi-asserted-by":"publisher","unstructured":"Haier E, Wanner G (1996) Analysis by its history. Springer, New York. https:\/\/doi.org\/10.1007\/978-0-387-77036-9","DOI":"10.1007\/978-0-387-77036-9"},{"key":"412_CR27","doi-asserted-by":"publisher","unstructured":"Lang S (1986) A first course in calculus (5th Ed.). Springer, New York. https:\/\/doi.org\/10.1007\/978-1-4419-8532-3","DOI":"10.1007\/978-1-4419-8532-3"},{"key":"412_CR28","doi-asserted-by":"publisher","unstructured":"Allen MB, Isaacson EL (2019) Numerical analysis for applied science (2nd Ed.) John Wiley and Sons Inc., New Jersey. https:\/\/doi.org\/10.1002\/9781119245476","DOI":"10.1002\/9781119245476"},{"key":"412_CR29","doi-asserted-by":"publisher","unstructured":"Helmberg G, Wagner P, Veltkamp GW (1993) On Faddeev-Leverrier\u2019s method for the computation of the characteristic polynomial of a matrix and of eigenvectors. Linear Algeb Appl 185:219\u2013233. https:\/\/doi.org\/10.1016\/0024-3795(93)90214-9","DOI":"10.1016\/0024-3795(93)90214-9"}],"container-title":["Operations Research Forum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s43069-024-00412-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s43069-024-00412-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s43069-024-00412-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,30]],"date-time":"2025-03-30T11:16:47Z","timestamp":1743333407000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s43069-024-00412-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,6]]},"references-count":29,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["412"],"URL":"https:\/\/doi.org\/10.1007\/s43069-024-00412-x","relation":{},"ISSN":["2662-2556"],"issn-type":[{"value":"2662-2556","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,6]]},"assertion":[{"value":"6 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 January 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors used ChatGPT and DeepL to translate the abstract and conclusion from Japanese to English and improve its readability. Following these tools, the authors reviewed and edited the content as necessary, taking full responsibility for the publication\u2019s content.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for Publication"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing Interests"}}],"article-number":"10"}}