{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:34:03Z","timestamp":1777703643487,"version":"3.51.4"},"reference-count":18,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2015,8,31]],"date-time":"2015-08-31T00:00:00Z","timestamp":1440979200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2015,8,31]]},"abstract":"<jats:p>\n                    In this paper, the fuzzy dual matrix system as\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>x<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>+<\/mml:mo>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>B<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>=<\/mml:mo>\n                      <mml:mi>C<\/mml:mi>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>x<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>+<\/mml:mo>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>D<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                    <\/mml:math>\n                    in which\n                    <jats:italic>A<\/jats:italic>\n                    ,\n                    <jats:italic>C<\/jats:italic>\n                    are\n                    <jats:italic>n<\/jats:italic>\n                    \u00a0\u00d7\u00a0\n                    <jats:italic>n<\/jats:italic>\n                    crisp matrices and\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                      <mml:mover accent=\"true\">\n                        <mml:mi>B<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                    <\/mml:math>\n                    ,\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                      <mml:mover accent=\"true\">\n                        <mml:mi>D<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                    <\/mml:math>\n                    are\n                    <jats:italic>n<\/jats:italic>\n                    \u00a0\u00d7\u00a0\n                    <jats:italic>n<\/jats:italic>\n                    LR fuzzy matrices is studied. To obtain the solution of fuzzy dual system, we initially solve 1-cut and use minimization problem to calculate the spreads. Afterwards, we explain the relationship between the solutions of the systems of\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>x<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>+<\/mml:mo>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>B<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>=<\/mml:mo>\n                      <mml:mi>C<\/mml:mi>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>x<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>+<\/mml:mo>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>D<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                    <\/mml:math>\n                    and\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mi>C<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>x<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>=<\/mml:mo>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>D<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:munder>\n                        <mml:mo>\u2296<\/mml:mo>\n                        <mml:mi>h<\/mml:mi>\n                      <\/mml:munder>\n                      <mml:mover accent=\"true\">\n                        <mml:mi>B<\/mml:mi>\n                        <mml:mo stretchy=\"true\">\u02dc<\/mml:mo>\n                      <\/mml:mover>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:math>\n                    . Finally, some examples are solved to illustrate the proposed method.\n                  <\/jats:p>","DOI":"10.3233\/ifs-151781","type":"journal-article","created":{"date-parts":[[2016,1,15]],"date-time":"2016-01-15T12:24:38Z","timestamp":1452860678000},"page":"575-581","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":6,"title":["An efficient method for solving LR fuzzy dual matrix systems"],"prefix":"10.1177","volume":"30","author":[{"given":"F.","family":"Babakordi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T.","family":"Allahviranloo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. Adabitabar","family":"firozja","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Science and Research Branch, Islamic Azad University, Ghaemshahr, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2015,10,8]]},"reference":[{"key":"e_1_3_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(96)00270-9"},{"key":"e_1_3_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(98)00102-X"},{"key":"e_1_3_1_4_2","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1016\/j.fss.2005.09.005","article-title":"Fuzzy linear system of the form A\n                  1\n                  x\u00a0+\u00a0b\n                  1\u00a0=\u00a0A\n                  2\n                  x\u00a0+\u00a0b\n                  2","volume":"157","author":"Muzzilio S.","year":"2006","unstructured":"MuzzilioS. and ReynaertsH., Fuzzy linear system of the form A1x\u00a0+\u00a0b1\u00a0=\u00a0A2x\u00a0+\u00a0b2, Fuzzy Sets and Systems 157 (2006), 939\u2013951.","journal-title":"Fuzzy Sets and Systems"},{"key":"e_1_3_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(03)00793-8"},{"key":"e_1_3_1_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2003.12.085"},{"key":"e_1_3_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2004.02.020"},{"key":"e_1_3_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2005.08.047"},{"key":"e_1_3_1_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2005.07.048"},{"key":"e_1_3_1_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2005.02.018"},{"key":"e_1_3_1_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2005.07.036"},{"key":"e_1_3_1_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2010.02.042"},{"key":"e_1_3_1_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2010.05.009"},{"key":"e_1_3_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2012.01.002"},{"key":"e_1_3_1_15_2","first-page":"86","article-title":"Solution of LR-type fuzzy system of linear algebraic equations","volume":"35","author":"Abramovich F.","year":"1988","unstructured":"AbramovichF., WagenknechtM. and KhurginY.I., Solution of LR-type fuzzy system of linear algebraic equations, s.l.: Busefal 35 (1988), 86\u201399.","journal-title":"s.l.: Busefal"},{"key":"e_1_3_1_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(86)90026-6"},{"key":"e_1_3_1_17_2","doi-asserted-by":"crossref","unstructured":"ZimmermannH.J. 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