{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T15:59:00Z","timestamp":1779206340778,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T00:00:00Z","timestamp":1691020800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The Transportation Problem (TP) is a special type of linear programming problem, where the objective is to minimize the cost of distributing a product from a number of sources to a number of destinations. Many methods for solving the TP have been studied over time. However, exact methods do not always succeed in finding the optimal solution or a solution that effectively approximates the optimal one. This paper introduces two new variations of the well-established Particle Swarm Optimization (PSO) algorithm named the Trigonometric Acceleration Coefficients-PSO (TrigAc-PSO) and the Four Sectors Varying Acceleration Coefficients PSO (FSVAC-PSO) and applies them to solve the TP. The performances of the proposed variations are examined and validated by carrying out extensive experimental tests. In order to demonstrate the efficiency of the proposed PSO variations, thirty two problems with different sizes have been solved to evaluate and demonstrate their performance. Moreover, the proposed PSO variations were compared with exact methods such as Vogel\u2019s Approximation Method (VAM), the Total Differences Method 1 (TDM1), the Total Opportunity Cost Matrix-Minimal Total (TOCM-MT), the Juman and Hoque Method (JHM) and the Bilqis Chastine Erma method (BCE). Last but not least, the proposed variations were also compared with other PSO variations that are well known for their completeness and efficiency, such as Decreasing Weight Particle Swarm Optimization (DWPSO) and Time Varying Acceleration Coefficients (TVAC). Experimental results show that the proposed variations achieve very satisfactory results in terms of their efficiency and effectiveness compared to existing either exact or heuristic methods.<\/jats:p>","DOI":"10.3390\/a16080372","type":"journal-article","created":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T11:13:06Z","timestamp":1691061186000},"page":"372","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Applying Particle Swarm Optimization Variations to Solve the Transportation Problem Effectively"],"prefix":"10.3390","volume":"16","author":[{"given":"Chrysanthi","family":"Aroniadi","sequence":"first","affiliation":[{"name":"Department of Food Science and Technology, University of Patras, Agrinio Campus, G. Seferi 2, 30100 Agrinio, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6896-5218","authenticated-orcid":false,"given":"Grigorios N.","family":"Beligiannis","sequence":"additional","affiliation":[{"name":"Department of Food Science and Technology, University of Patras, Agrinio Campus, G. Seferi 2, 30100 Agrinio, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,3]]},"reference":[{"key":"ref_1","first-page":"211","article-title":"A novel approximation method to obtain initial basic feasible solution of transportation problem","volume":"32","author":"Karagul","year":"2019","journal-title":"J. King Saud Univ. Eng. Sci."},{"key":"ref_2","unstructured":"Deshpande, V.A. (2010, January 5\u20136). An optimal method for obtaining initial basic feasible solution of the transportation problem. 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