{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T17:27:41Z","timestamp":1762018061058,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,12,1]],"date-time":"2017-12-01T00:00:00Z","timestamp":1512086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71471172, 61703280"],"award-info":[{"award-number":["71471172, 61703280"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>A neutrosophic number (NN) presented by Smarandache can express determinate and\/or indeterminate information in real life. NN (z = a + uI) consists of the determinate part a and the indeterminate part uI for a, u \u2208 R (R is all real numbers) and indeterminacy I, and is very suitable for representing and handling problems with both determinate and indeterminate information. Based on the concept of NNs, this paper presents for first time the concepts of neutrosophic linear equations and the neutrosophic matrix, and introduces the neutrosophic matrix operations. Then, we propose some solving methods, including the substitution method, the addition method, and the inverse matrix method, for the system of neutrosophic linear equations or the neutrosophic matrix equation. Finally, an applied example about a traffic flow problem is provided to illustrate the application and effectiveness of handling the indeterminate traffic flow problem by using the system of neutrosophic linear equations.<\/jats:p>","DOI":"10.3390\/a10040133","type":"journal-article","created":{"date-parts":[[2017,12,1]],"date-time":"2017-12-01T12:30:16Z","timestamp":1512131416000},"page":"133","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Neutrosophic Linear Equations and Application in Traffic Flow Problems"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2841-6529","authenticated-orcid":false,"given":"Jun","family":"Ye","sequence":"first","affiliation":[{"name":"Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing 312000, China"}]}],"member":"1968","published-online":{"date-parts":[[2017,12,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Koukol, M., Zaj\u00edIkov\u00e1, L., Marek, L., and TuIek, P. (2015). Fuzzy logic in traffic engineering: A review on signal control. Math. Probl. Eng., 2015.","DOI":"10.1155\/2015\/979160"},{"key":"ref_2","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_3","unstructured":"Kaufmann, A., and Gupta, M.M. (1985). Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold Company."},{"key":"ref_4","unstructured":"Hanss, M. (2005). Applied Fuzzy Arithmetic: An Introduction with Engineering Applications, Springer."},{"key":"ref_5","unstructured":"Smarandache, F. (1998). Neutrosophy: Neutrosophic Probability, Set, and Logic, American Research Press."},{"key":"ref_6","unstructured":"Smarandache, F. (2013). Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability, Education Publishing."},{"key":"ref_7","unstructured":"Smarandache, F. (2014). Introduction to Neutrosophic Statistics, Education Publishing."},{"key":"ref_8","first-page":"377","article-title":"Multiple-attribute group decision-making method under a neutrosophic number environment","volume":"25","author":"Ye","year":"2016","journal-title":"J. Intell. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1007\/s00521-015-2123-5","article-title":"Bidirectional projection method for multiple attribute group decision making with neutrosophic numbers","volume":"28","author":"Ye","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_10","first-page":"43","article-title":"Misfire fault diagnosis method of gasoline engines using the cosine similarity measure of neutrosophic numbers","volume":"8","author":"Kong","year":"2015","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1927","DOI":"10.3233\/IFS-151903","article-title":"Fault diagnoses of steam turbine using the exponential similarity measure of neutrosophic numbers","volume":"30","author":"Ye","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_12","unstructured":"Smarandache, F. (2015). Neutrosophic Precalculus and Neutrosophic Calculus, Europa-Nova."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ye, J., Yong, R., Liang, Q.F., Huang, M., and Du, S.G. (2016). Neutrosophic Functions of the Joint Roughness Coefficient (JRC) and the Shear Strength: A Case Study from the Pyroclastic Rock Mass in Shaoxing City, China. Math. Probl. Eng., 2016.","DOI":"10.1155\/2016\/4825709"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Chen, J.Q., Ye, J., Du, S.G., and Yong, R. (2017). Expressions of rock joint roughness coefficient using neutrosophic interval statistical numbers. Symmetry, 9.","DOI":"10.3390\/sym9070123"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Chen, J.Q., Ye, J., and Du, S.G. (2017). Scale effect and anisotropy analyzed for neutrosophic numbers of rock joint roughness coefficient based on neutrosophic statistics. Symmetry, 9.","DOI":"10.3390\/sym9100208"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ye, J., Chen, J.Q., Yong, R., and Du, S.G. (2017). Expression and analysis of joint roughness coefficient using neutrosophic number functions. Information, 8.","DOI":"10.3390\/info8020069"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ye, J. (2017). Neutrosophic number linear programming method and its application under neutrosophic number environments. Soft Comput.","DOI":"10.1007\/s00500-017-2646-z"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/10\/4\/133\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:52:14Z","timestamp":1760208734000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/10\/4\/133"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,1]]},"references-count":17,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["a10040133"],"URL":"https:\/\/doi.org\/10.3390\/a10040133","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2017,12,1]]}}}