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The Shortest path problem is highly relevant in our daily lives, addressing uncertainties like traffic conditions and weather variations. To handle such uncertainties, we utilize Fuzzy Numbers. This paper focuses on Bipolar Neutrosophic Fuzzy Numbers, which have dual positive and negative aspects. They provide a robust framework for representing arc (node/edge) weights, signifying uncertain travel times between nodes. Importantly, these weights can change over time in bipolar neutrosophic fuzzy graphs. Our study introduces an extended Bellman-Ford Algorithm for identifying optimal paths and minimum times with time-dependent Bipolar Neutrosophic Fuzzy arc weights. We demonstrate its effectiveness through a step-by-step numerical example and conduct a comparative analysis to evaluate its efficiency. [ABSTRACT FROM AUTHOR]

The shortest path problem is a classic optimization problem in graph theory and computer technology. It involves identifying the shortest path between two nodes in a graph, where each edge has a numerical weight. In this paper, we put our effort into examining the use of the dynamic programming method to evaluate the shortest path (SP) between the two specified nodes in a multistage network where the parameter is a multi-value neutrosophic number (MVNN). Firstly, we propose an algorithm based on the forward and backward approach in an uncertain environment and also implement our approach in the Python-3 programming language. Furthermore, a numerical illustration has been provided to showcase the effectiveness and robustness of the novel model. [ABSTRACT FROM AUTHOR]

The most well-known subject in graph theory is the shortest path problem (SPP), which has real-world applications in several different fields of study, including transportation, emergency services, network communications, fire station services, etc. The arc weights of the applicable SP problems are typically represented by fuzzy numbers in real-world applications. In this paper, we discussed the process of finding the shortest distance in a connected graph network in which the arc weights are multi-valued neutrosophic numbers (MNNs). Moreover, here we compare our method with some of the existing results and illustrate one implementation of our method with the help of one numerical example. [ABSTRACT FROM AUTHOR]

The shortest path (SP) problem (SPP) has several applications in graph theory. It can be used to calculate the distance between the provided initial and final vertex in a network. In this paper, we employed the Fermatean neutrosophic number as the appropriate edge weight of the network to estimate the SP connecting the start and end vertex. This technique is highly useful in establishing the shortest path for the decision-maker under uncertainty. We also investigated its effectiveness in comparison to several existing methods. Finally, a few numerical tests were performed to demonstrate the validity and stability of this new technique, as well as to compare different types of shortest paths with different networks. [ABSTRACT FROM AUTHOR]