{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T15:33:43Z","timestamp":1767972823305,"version":"3.49.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,2,8]],"date-time":"2020-02-08T00:00:00Z","timestamp":1581120000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,2,8]],"date-time":"2020-02-08T00:00:00Z","timestamp":1581120000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"FCT","award":["UID\/MAT\/00324\/2019"],"award-info":[{"award-number":["UID\/MAT\/00324\/2019"]}]},{"name":"FCT","award":["UID\/MAT\/04721\/2019"],"award-info":[{"award-number":["UID\/MAT\/04721\/2019"]}]},{"name":"FCT","award":["UID\/MAT\/00297\/2019"],"award-info":[{"award-number":["UID\/MAT\/00297\/2019"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2020,5]]},"DOI":"10.1007\/s40314-020-1075-2","type":"journal-article","created":{"date-parts":[[2020,2,8]],"date-time":"2020-02-08T17:02:26Z","timestamp":1581181346000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Reciprocal matrices: properties and approximation by a transitive matrix"],"prefix":"10.1007","volume":"39","author":[{"given":"Nat\u00e1lia","family":"Bebiano","sequence":"first","affiliation":[]},{"given":"Ros\u00e1rio","family":"Fernandes","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0395-5972","authenticated-orcid":false,"given":"Susana","family":"Furtado","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,2,8]]},"reference":[{"key":"1075_CR1","doi-asserted-by":"publisher","first-page":"69","DOI":"10.5614\/itbj.sci.2009.41.2.1","volume":"41","author":"P Astuti","year":"2009","unstructured":"Astuti P, Garnadi AD (2009) On eigenvalues and eigenvectors of perturbed pairwise comparison matrices. ITB J Sci 41:69\u201377","journal-title":"ITB J Sci"},{"key":"1075_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2014\/349729","volume":"2014","author":"J Ben\u00edtez","year":"2014","unstructured":"Ben\u00edtez J, Carri\u00f3n L, Izquierdo J, P\u00e9rez-Garc\u00eda R (2014) Characterization of consistent completion of reciprocal comparison matrices. Abst Appl Anal 2014:1\u201312","journal-title":"Abst Appl Anal"},{"key":"1075_CR3","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.ijar.2018.01.004","volume":"95","author":"V Cernanov\u00e1","year":"2018","unstructured":"Cernanov\u00e1 V, Koczkodaj WW, Szybowski J (2018) Inconsistency of special cases of pairwise comparisons matrices. Int J Approx Reason 95:36\u201345","journal-title":"Int J Approx Reason"},{"key":"1075_CR4","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/S0024-3795(97)00329-7","volume":"272","author":"MT Chu","year":"1998","unstructured":"Chu MT (1998) On the optimal consistent approximation to pairwise comparison matrices. Linear Algebra Appl 272:155\u2013168","journal-title":"Linear Algebra Appl"},{"key":"1075_CR5","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/j.laa.2005.02.002","volume":"403","author":"G Dahl","year":"2005","unstructured":"Dahl G (2005) A method for approximating symmetrically reciprocal matrices by transitive matrices. Linear Algebra Appl 403:207\u2013215","journal-title":"Linear Algebra Appl"},{"key":"1075_CR6","volume-title":"Essay on the application of analysis to the probability of majority decisions","author":"N de Condorcet","year":"1785","unstructured":"de Condorcet N (1785) Essay on the application of analysis to the probability of majority decisions. l\u2019Imprimerie Royale, Paris"},{"key":"1075_CR7","volume-title":"Graph theory","author":"R Diestel","year":"2005","unstructured":"Diestel R (2005) Graph theory. Springer, Berlin"},{"key":"1075_CR8","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/S0024-3795(01)00261-0","volume":"331","author":"J Ding","year":"2001","unstructured":"Ding J, Pye WC (2001) On the spectrum and pseudoinverse of a special bordered matrix. Linear Algebra Appl 331:11\u201320","journal-title":"Linear Algebra Appl"},{"key":"1075_CR9","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/S0022-460X(73)80373-6","volume":"31","author":"CJ Dodds","year":"1973","unstructured":"Dodds CJ, Robson JD (1973) The description of road surface roughness. J Sound Vib 31:175\u2013183","journal-title":"J Sound Vib"},{"key":"1075_CR10","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/S0024-3795(03)00476-2","volume":"385","author":"L Elsner","year":"2004","unstructured":"Elsner L, van den Driessche P (2004) Max-algebra and pairwise comparison matrices. Linear Algebra Appl 385:47\u201362","journal-title":"Linear Algebra Appl"},{"key":"1075_CR11","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1016\/j.laa.2009.10.005","volume":"432","author":"L Elsner","year":"2010","unstructured":"Elsner L, van den Driessche P (2010) Max-algebra and pairwise comparison matrices. II. Linear Algebra Appl 432:927\u2013935","journal-title":"Linear Algebra Appl"},{"key":"1075_CR12","first-page":"99","volume":"4","author":"A Farkas","year":"2007","unstructured":"Farkas A (2007) The analysis of the principal eigenvector of pairwise comparison matrices. Acta Polytechnica Hungarica 4:99\u2013115","journal-title":"Acta Polytechnica Hungarica"},{"key":"1075_CR13","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1016\/S0024-3795(99)00208-6","volume":"302\u2013303","author":"A Farkas","year":"1999","unstructured":"Farkas A, R\u00f3zsa P, Stunbnya E (1999) Transitive matrices and their applications. Linear Algebra Appl 302\u2013303:423\u2013433","journal-title":"Linear Algebra Appl"},{"key":"1075_CR14","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1002\/nla.318","volume":"10","author":"A Farkas","year":"2003","unstructured":"Farkas A, Lancaster P, R\u00f3zsa P (2003) Consistency adjustments for pairwise comparison matrices. Numer Linear Algebra Appl 10:689\u2013700","journal-title":"Numer Linear Algebra Appl"},{"key":"1075_CR15","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1016\/j.laa.2003.12.034","volume":"385","author":"A Farkas","year":"2004","unstructured":"Farkas A, Gyorgy A, R\u00f3zsa P (2004) On the spectrum of pairwise comparison matrices. Linear Algebra Appl 385:443\u2013462","journal-title":"Linear Algebra Appl"},{"key":"1075_CR16","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1016\/0038-0121(86)90045-5","volume":"20","author":"J Fichtner","year":"1986","unstructured":"Fichtner J (1986) On deriving priority vectors from matrices of pairwise comparisons. Soc Econ Plan Sci 20:341\u2013345","journal-title":"Soc Econ Plan Sci"},{"key":"1075_CR17","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1016\/S0377-2217(02)00755-5","volume":"154","author":"SI Gass","year":"2004","unstructured":"Gass SI, Rapcs\u00e1k T (2004) Singular value decomposition in AHP. Eur J Oper Res 154:573\u2013584","journal-title":"Eur J Oper Res"},{"key":"1075_CR18","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810817","volume-title":"Matrix analysis","author":"RA Horn","year":"1985","unstructured":"Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge"},{"key":"1075_CR19","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/0022-2496(79)90005-1","volume":"19","author":"CR Johnson","year":"1979","unstructured":"Johnson CR, Beine WB, Wang TJ (1979) Right-left asymmetry in an eigenvector ranking procedure. J Math Psychol 19:61\u201364","journal-title":"J Math Psychol"},{"issue":"4","key":"1075_CR20","doi-asserted-by":"publisher","first-page":"485","DOI":"10.3233\/FI-2014-1055","volume":"132","author":"WW Koczkodaj","year":"2014","unstructured":"Koczkodaj WW, Szwarc R (2014) On axiomatization of inconsistency indicators for pairwise comparisons. Fundamenta Informaticae 132(4):485\u2013500","journal-title":"Fundamenta Informaticae"},{"key":"1075_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3233\/FI-2016-1336","volume":"144","author":"WW Koczkodaj","year":"2016","unstructured":"Koczkodaj WW, Mikhailov L, Redlarski G, Soltys M, Szybowski J, Wajch E, Yuen KKF, Tamazian G (2016) Important facts and observations about pairwise comparisons. Fundamenta Informaticae 144:1\u201317","journal-title":"Fundamenta Informaticae"},{"key":"1075_CR22","doi-asserted-by":"publisher","first-page":"1247","DOI":"10.1017\/S0266466609990715","volume":"26","author":"A Luati","year":"2010","unstructured":"Luati A, Proietti T (2010) On the spectral properties of matrices associated with trend filters. Econ Theory 26:1247\u20131261","journal-title":"Econ Theory"},{"key":"1075_CR23","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1002\/nla.1811","volume":"20","author":"S Noschese","year":"2013","unstructured":"Noschese S, Pasquini L, Reichel L (2013) Tridiagonal Toeplitz matrices: properties and novel applications. Numer Linear Algebra Appl 20:302\u2013326","journal-title":"Numer Linear Algebra Appl"},{"key":"1075_CR24","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1016\/0022-2496(77)90033-5","volume":"32","author":"TL Saaty","year":"1977","unstructured":"Saaty TL (1977) A scaling method for priorities in hierarchical structures. J Math Psychol 32:234\u2013281","journal-title":"J Math Psychol"},{"key":"1075_CR25","volume-title":"The analytical hierarchy process","author":"TL Saaty","year":"1980","unstructured":"Saaty TL (1980) The analytical hierarchy process. McGraw-Hill, New York"},{"key":"1075_CR26","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/S0377-2217(02)00227-8","volume":"145","author":"TL Saaty","year":"2003","unstructured":"Saaty TL (2003) Decision-making with the AHP: why is the principal eigenvector necessary. Eur J Oper Res 145:85\u201391","journal-title":"Eur J Oper Res"},{"key":"1075_CR27","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/0270-0255(84)90008-3","volume":"4","author":"TL Saaty","year":"1984","unstructured":"Saaty TL, Vargas LG (1984) Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Math Model 4:309\u2013324","journal-title":"Math Model"},{"key":"1075_CR28","doi-asserted-by":"publisher","first-page":"404","DOI":"10.15807\/jorsj.41.404","volume":"41","author":"S Shiraishi","year":"1998","unstructured":"Shiraishi S, Obata T, Daigo M (1998) Properties of a positive reciprocal matrix and their application to AHP. J Oper Res Soc Jpn 41:404\u2013414","journal-title":"J Oper Res Soc Jpn"},{"key":"1075_CR29","volume-title":"Numerical solution of partial differential equations","author":"GD Smith","year":"1978","unstructured":"Smith GD (1978) Numerical solution of partial differential equations. Clarendon Press, Oxford"},{"key":"1075_CR30","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1016\/0038-0121(86)90008-X","volume":"20","author":"AE Steenge","year":"1986","unstructured":"Steenge AE (1986) Saaty\u2019s consistency analysis: an application to problems in static and in dynamic input-output models. Soc Econ Plan Sci 20:173\u2013180","journal-title":"Soc Econ Plan Sci"},{"key":"1075_CR31","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/0038-0121(84)90015-6","volume":"18","author":"D Stojanovic","year":"1984","unstructured":"Stojanovic D (1984) The model based on the matrix of economic growth. Soc Econ Plan Sci 18:167\u2013169","journal-title":"Soc Econ Plan Sci"},{"key":"1075_CR32","volume-title":"Mathematical economics","author":"A Takayama","year":"1974","unstructured":"Takayama A (1974) Mathematical economics. The Dryden Press, Hinsdale"},{"key":"1075_CR33","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1007\/s40844-015-0007-0","volume":"12","author":"NM Tran","year":"2015","unstructured":"Tran NM (2015) The PerronRank family: a brief review. Evol Inst Econ Rev 12:213\u2013221","journal-title":"Evol Inst Econ Rev"},{"key":"1075_CR34","first-page":"1","volume":"2014","author":"H Zhang","year":"2014","unstructured":"Zhang H, Sekhari A, Ouzrout Y, Bouras A (2014) Optimal inconsistency repairing of pairwise comparison matrices using integrated linear programming and eigenvector methods. Math Probl Eng 2014:1\u201316","journal-title":"Math Probl Eng"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-1075-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-020-1075-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-1075-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T02:28:54Z","timestamp":1612837734000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-020-1075-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,8]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,5]]}},"alternative-id":["1075"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-1075-2","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,8]]},"assertion":[{"value":"24 April 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 February 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"50"}}