{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T23:36:53Z","timestamp":1764977813174,"version":"3.46.0"},"reference-count":32,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2021,1,1]],"date-time":"2021-01-01T00:00:00Z","timestamp":1609459200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,4,21]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This article attempts to study cost minimizing multi-objective fractional solid transportation problem with fuzzy cost coefficients\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jisys-2020-0095_eq_001.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msubsup>\n                            <m:mrow>\n                              <m:mover accent=\"true\">\n                                <m:mi>c<\/m:mi>\n                                <m:mo>\u02dc<\/m:mo>\n                              <\/m:mover>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mi>i<\/m:mi>\n                              <m:mi>j<\/m:mi>\n                              <m:mi>k<\/m:mi>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mi>r<\/m:mi>\n                            <\/m:mrow>\n                          <\/m:msubsup>\n                        <\/m:math>\n                        <jats:tex-math>{\\tilde{c}}_{ijk}^{r}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , fuzzy supply quantities\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jisys-2020-0095_eq_002.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msub>\n                            <m:mrow>\n                              <m:mover accent=\"true\">\n                                <m:mi>a<\/m:mi>\n                                <m:mo>\u02dc<\/m:mo>\n                              <\/m:mover>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mi>i<\/m:mi>\n                            <\/m:mrow>\n                          <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>{\\tilde{a}}_{i}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , fuzzy demands\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jisys-2020-0095_eq_003.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msub>\n                            <m:mrow>\n                              <m:mover accent=\"true\">\n                                <m:mi>b<\/m:mi>\n                                <m:mo>\u02dc<\/m:mo>\n                              <\/m:mover>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mi>j<\/m:mi>\n                            <\/m:mrow>\n                          <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>{\\tilde{b}}_{j}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\/or fuzzy conveyances\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jisys-2020-0095_eq_004.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msub>\n                            <m:mrow>\n                              <m:mover accent=\"true\">\n                                <m:mi>e<\/m:mi>\n                                <m:mo>\u02dc<\/m:mo>\n                              <\/m:mover>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mi>k<\/m:mi>\n                            <\/m:mrow>\n                          <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>{\\tilde{e}}_{k}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . The fuzzy efficient concept is introduced in which the crisp efficient solution is extended. A necessary and sufficient condition for the solution is established. Fuzzy geometric programming approach is applied to solve the crisp problem by defining membership function so as to obtain the optimal compromise solution of a multi-objective two-stage problem. A linear membership function for the objective function is defined. The stability set of the first kind is defined and determined. A numerical example is given for illustration and to check the validity of the proposed approach.\n                  <\/jats:p>","DOI":"10.1515\/jisys-2020-0095","type":"journal-article","created":{"date-parts":[[2021,4,23]],"date-time":"2021-04-23T11:35:12Z","timestamp":1619177712000},"page":"620-635","source":"Crossref","is-referenced-by-count":7,"title":["On characterizing solution for multi-objective fractional two-stage solid transportation problem under fuzzy environment"],"prefix":"10.1515","volume":"30","author":[{"given":"Hamiden Abd El-Wahed","family":"Khalifa","sequence":"first","affiliation":[{"name":"Operations Research Department, Faculty of Graduate Studies for Statistical Research, Cairo University , Giza , Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pavan","family":"Kumar","sequence":"additional","affiliation":[{"name":"Mathematics Division, School of Advanced Science and Languages, VIT Bhopal University , Sehore , MP 466114 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Majed. G.","family":"Alharbi","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science and Arts, El-Methnab, Qassim University , Methnab , Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2021,4,21]]},"reference":[{"key":"2025120523322262682_j_jisys-2020-0095_ref_001","unstructured":"Shell E\n. Distribution of a product by several properties, directorate of management analysis. Proceedings of the Second symposium on linear programming. Vol. 2, Washington, DC: DCS\/Comptroller H.Q.U.S.A.F.; 1955. p. 615\u201342."},{"key":"2025120523322262682_j_jisys-2020-0095_ref_002","doi-asserted-by":"crossref","unstructured":"Haley KB\n. The solid transportation problem. Oper Res. 1962;11:446\u20138.","DOI":"10.1287\/opre.10.4.448"},{"key":"2025120523322262682_j_jisys-2020-0095_ref_003","unstructured":"Patel G\n, \nTripathy J\n. The solid transportation problem and its variants. 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