{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:04:28Z","timestamp":1760144668646,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,15]],"date-time":"2024-05-15T00:00:00Z","timestamp":1715731200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at King Khalid University","award":["RGP2\/32\/44"],"award-info":[{"award-number":["RGP2\/32\/44"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study formulates a multi-objective, multi-item solid transportation issue with parameters that are neutrosophic Z-number fuzzy variables such as transportation costs, supplies, and demands. This work covers two scenarios where uncertainty in the problem can arise: the fuzzy solid transportation problem and the interval solid transportation problem. The first scenario arises when we represent data problems as intervals instead of exact values, while the second scenario arises when the information is not entirely clear. We address both models when the uncertainty alone impacts the constraint set. In order to find a solution for the interval case, we generate an additional problem. Since this auxiliary problem is typical of solid transportation, we can resolve it using the effective techniques currently in use. In the fuzzy scenario, a parametric method is used to discover a fuzzy solution to the earlier issue. Parametric analysis identifies that the best parameterized approaches to complementary problems are characterized by the application of parametric analysis. We present a suggested algorithm for determining the stability set. Finally, we provide a numerical example and sensitivity analysis for the transportation problem, which is both symmetrical and asymmetrical.<\/jats:p>","DOI":"10.3390\/sym16050615","type":"journal-article","created":{"date-parts":[[2024,5,16]],"date-time":"2024-05-16T03:24:48Z","timestamp":1715829888000},"page":"615","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Systematic Formulation into Neutrosophic Z Methodologies for Symmetrical and Asymmetrical Transportation Problem Challenges"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2358-178X","authenticated-orcid":false,"given":"Muhammad","family":"Kamran","sequence":"first","affiliation":[{"name":"Department of Mathematics, Thal University Bhakkar, Bhakkar 30000, Pakistan"},{"name":"Department of Mathematics, Uludag University, Bursa 16059, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-8951-8185","authenticated-orcid":false,"given":"Manal Elzain Mohamed","family":"Abdalla","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences and Arts (Muhyil), King Khalid University, Muhyil 61421, Saudi Arabia"}]},{"given":"Muhammad","family":"Nadeem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lahore Garrison University, Lahore 54810, Pakistan"}]},{"given":"Anns","family":"Uzair","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Thal University Bhakkar, Bhakkar 30000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7616-0500","authenticated-orcid":false,"given":"Muhammad","family":"Farman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, Near East University, Nicosia 99138, Turkey"},{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut 11022801, Lebanon"}]},{"given":"Lakhdar","family":"Ragoub","sequence":"additional","affiliation":[{"name":"Mathematics Department, Prince Mugrin University, Al Madinah 42241, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0700-5774","authenticated-orcid":false,"given":"Ismail Naci","family":"Cangul","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Uludag University, Bursa 16059, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1002\/sapm1941201224","article-title":"The distribution of a product from several sources to numerous localities","volume":"20","author":"Hitchcock","year":"1941","journal-title":"J. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1002\/nav.3800080104","article-title":"Fixed-cost transportation problems","volume":"8","author":"Balinski","year":"1961","journal-title":"Nav. Res. Logist. Q."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"11611","DOI":"10.1007\/s00500-022-07442-9","article-title":"Fixed-charge solid transportation problem with budget constraints based on carbon emission in neutrosophic environment","volume":"26","author":"Ghosh","year":"2022","journal-title":"Soft Comput."},{"key":"ref_4","first-page":"370","article-title":"An improved Vogel\u2019s approximation method for the transportation problem","volume":"16","author":"Korukoglu","year":"2011","journal-title":"Math. Comput. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1108\/FS-10-2019-0090","article-title":"A decision support system for road freight transportation route selection with new fuzzy numbers","volume":"22","author":"Meethom","year":"2020","journal-title":"Foresight"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0169-7722(96)00047-2","article-title":"Numerical solute transport simulation using fuzzy sets approach","volume":"27","author":"Dou","year":"1997","journal-title":"J. Contam. Hydrol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"103497","DOI":"10.1109\/ACCESS.2021.3098657","article-title":"Prediction of transportation costs using trapezoidal neutrosophic fuzzy analytic hierarchy process and artificial neural networks","volume":"9","author":"Singh","year":"2021","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"13373","DOI":"10.1007\/s00500-022-07482-1","article-title":"Type-2 intuitionistic fuzzy TODIM for intelligent decision-making under uncertainty and hesitancy","volume":"27","author":"Shukla","year":"2023","journal-title":"Soft Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"100239","DOI":"10.1016\/j.dajour.2023.100239","article-title":"A new definition for quartic fuzzy sets with hesitation grade applied to multi-criteria decision-making problems under uncertainty","volume":"7","author":"Arora","year":"2023","journal-title":"Decis. Anal. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","first-page":"112","article-title":"Identification of the most significant risk factor of COVID-19 in economy using cosine similarity measure under SVPNS-environment","volume":"46","author":"Majumder","year":"2021","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_13","first-page":"403","article-title":"Neutrosophic sets: An overview","volume":"2","author":"Broumi","year":"2018","journal-title":"Infin. Study"},{"key":"ref_14","first-page":"20110","article-title":"Single valued neutrosophic sets","volume":"12","author":"Wang","year":"2010","journal-title":"Infin. Study"},{"key":"ref_15","first-page":"421","article-title":"Multi-criteria group decision making model using single-valued neutrosophic set","volume":"16","author":"Das","year":"2020","journal-title":"Infin. Study"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"118968","DOI":"10.1016\/j.ins.2023.118968","article-title":"Multi-criteria decision-making model using trigonometric aggregation operators of single-valued neutrosophic credibility numbers","volume":"644","author":"Ye","year":"2023","journal-title":"Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kamran, M., Ismail, R., Al-Sabri, E.H.A., Salamat, N., Farman, M., and Ashraf, S. (2023). An optimization strategy for MADM framework with confidence level aggregation operators under probabilistic neutrosophic hesitant fuzzy rough environment. Symmetry, 15.","DOI":"10.3390\/sym15030578"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s40745-021-00363-8","article-title":"Application of neutrosophic similarity measures in COVID-19","volume":"9","author":"Das","year":"2022","journal-title":"Ann. Data Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0165-0114(78)90031-3","article-title":"Fuzzy programming and linear programming with several objective functions","volume":"1","author":"Zimmermann","year":"1978","journal-title":"Fuzzy Sets Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/0165-0114(84)90057-5","article-title":"A fuzzy approach to the transportation problem","volume":"13","author":"Chanas","year":"1984","journal-title":"Fuzzy Sets Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/0165-0114(95)00278-2","article-title":"A concept of the optimal solution of the transportation problem with fuzzy cost coefficients","volume":"82","author":"Chanas","year":"1996","journal-title":"Fuzzy Sets Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/0898-1221(96)00044-2","article-title":"An integer fuzzy transportation problem","volume":"31","author":"Tada","year":"1996","journal-title":"Comput. Math. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"12401","DOI":"10.1007\/s00500-019-03782-1","article-title":"A model for two-stage fixed charge transportation problem with multiple objectives and fuzzy linguistic preferences","volume":"23","author":"Hashmi","year":"2019","journal-title":"Soft Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/j.oceaneng.2018.01.039","article-title":"An interval type-2 fuzzy AHP and TOPSIS methods for decision-making problems in maritime transportation engineering: The case of ship loader","volume":"155","author":"Celik","year":"2018","journal-title":"Ocean Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s10479-014-1724-1","article-title":"A new approach for solving intuitionistic fuzzy transportation problem of type-2","volume":"243","author":"Singh","year":"2016","journal-title":"Ann. Oper. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3089","DOI":"10.1007\/s13042-022-01582-y","article-title":"Neutrosophic multi-objective green four-dimensional fixed-charge transportation problem","volume":"13","author":"Giri","year":"2022","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_27","first-page":"71","article-title":"An algorithm for interval-valued fuzzy fractional transportation problem","volume":"8","author":"Arora","year":"2018","journal-title":"Ski. Res. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s13748-020-00228-w","article-title":"Transportation problem with interval-valued intuitionistic fuzzy sets: Impact of a new ranking","volume":"10","author":"Bharati","year":"2021","journal-title":"Prog. Artif. Intell."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s40747-020-00153-4","article-title":"Modified Vogels approximation method for transportation problem under uncertain environment","volume":"7","author":"Pratihar","year":"2021","journal-title":"Complex Intell. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6975","DOI":"10.1007\/s12652-021-03554-6","article-title":"Intuitionistic fuzzy sustainable multi-objective multi-item multi-choice step fixed-charge solid transportation problem","volume":"14","author":"Mondal","year":"2023","journal-title":"J. Ambient. Intell. Humaniz. Comput."},{"key":"ref_31","first-page":"1479","article-title":"Fuzzy arithmetic DEA approach for fuzzy multi-objective transportation problem","volume":"22","author":"Bagheri","year":"2022","journal-title":"Oper. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"924","DOI":"10.3934\/math.2023045","article-title":"Extended DEA method for solving multi-objective transportation problem with Fermatean fuzzy sets","volume":"8","author":"Akram","year":"2023","journal-title":"Aims Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"105748","DOI":"10.1016\/j.engappai.2022.105748","article-title":"Aggregation operators of quadripartitioned single-valued neutrosophic Z-numbers with applications to diverse COVID-19 scenarios","volume":"119","author":"Borah","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_34","first-page":"18","article-title":"Sine Trigonometric Aggregation Operators with Single-Valued Neutrosophic Z-Numbers: Application in Business Site Selection","volume":"63","author":"Kamran","year":"2024","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2923","DOI":"10.1016\/j.ins.2011.02.022","article-title":"A note on Z-numbers","volume":"181","author":"Zadeh","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106517","DOI":"10.1016\/j.cie.2020.106517","article-title":"Generalized Z-numbers with hesitant fuzzy linguistic information and its application to medicine selection for the patients with mild symptoms of the COVID-19","volume":"145","author":"Ren","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.ins.2020.10.038","article-title":"A ranking model of Z-mixture-numbers based on the ideal degree and its application in multi-attribute decision making","volume":"550","author":"Xian","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"4639","DOI":"10.1007\/s00500-017-2646-z","article-title":"Neutrosophic number linear programming method and its application under neutrosophic number environments","volume":"22","author":"Ye","year":"2018","journal-title":"Soft Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/s40747-020-00204-w","article-title":"Some aggregation operators of neutrosophic Z-numbers and their multicriteria decision making method","volume":"7","author":"Du","year":"2021","journal-title":"Complex Intell. Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/615\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:42:51Z","timestamp":1760107371000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/615"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,15]]},"references-count":39,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["sym16050615"],"URL":"https:\/\/doi.org\/10.3390\/sym16050615","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,5,15]]}}}