aspg-21-2934-1791475929 20261008161209 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group International Journal of Neutrosophic Science IJNS 2692-6148 2690-6805 2025 25 1 Shubham Kumar Tripathi VIT-AP University, Inavolu, Beside AP Secretariat, Amaravati, AP, India Kottakkaran Sooppy Nisar Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia; Saveetha School of Engineering, SIMATS, Chennai, Said Broumi Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, B.P 7955, Morocco Ranjan Kumar VIT-AP University, Inavolu, Beside AP Secretariat, Amaravati, AP, India In the domain of optimization, linear programming (LP) is recognized as an exceptionally effective method for ensuring the most favorable outcomes. Within the context of LP, the minimum cost flow (MCF) problem is fundamental, with its primary objective being to reduce the transportation costs for a single item moving through a network, under the constraints related to capacity. This network is made up of supply nodes, directed arcs, and demand nodes and each arc has an associated cost and capacity constraint, these factors are certain. However, in practical scenarios, these factors are susceptible to variation due to causal uncertainty. The neutrosophic set theory has surfaced as a challenging approach to tackle the uncertainty that is often encountered in optimization processes. In this manuscript, our primary objective is to address the minimal cost flow (MCF) problem while accounting for the uncertainty inherent in the neutrosophic set. We specifically focus on the cost aspect as SVTN numbers and introduce a new approach based on a customized ranking function handmade for the MCF problem a pioneering endeavor within the field of neutrosophic sets. Additionally, we present numerical example to validate the effectiveness and robustness of our model. 2025 81 92 2934 https://creativecommons.org/licenses/by/4.0/ 10.54216/IJNS.250107 https://www.americaspg.com/journal/21/article/2934 https://www.americaspg.com/storage/31720178252.pdf