An elongation of the single-valued neutrosophic set is an interval-valued neutrosophic set. It has been demonstrated to deal indeterminacy in a decision-making problem. Real-world problems have some kind of uncertainty in nature and among them; one of the influential problems is solving the shortest path problem (SPP) in interconnections. In this contribution, we consider SPP through Bellman’s algorithm for a network using interval-valued neutrosophic numbers (IVNNs). We proposed a novel algorithm to obtain the neutrosophic shortest path between each pair of nodes. Length of all the edges is accredited an IVNN. Moreover, for the validation of the proposed algorithm, a numerical example has been offered. Also, a comparative analysis has been done with the existing methods which exhibit the advantages of the new algorithm.

Classical inventory models (IM) serve as quantitative tools for determining the optimal order quantities, timing of orders, and safety stock levels for specific inventory items or item groups. Zadeh (1965. Fuzzy sets. Information and Control, 8, 338–353) introduced fuzzy theory and Dubois and Parade (1988. Fuzzy logic in expert systems: The role of uncertainty management. Fuzzy Sets and Systems, 28, 3–17) presented the study of fuzzy inventory model, which, however, exhibits limitations in effectively handling uncertainty, inaccuracies, and imprecise data. In 1999, Smarandache presented the idea of neutrosophic set theory to handle uncertainty. Using trapezoidal neutrosophic numbers, this study extends the idea of neutrosophic sets to inventory management, concentrating on resolving the uncertainty associated with holding costs, ordering costs, and shortage costs. First time within the literature of the neutrosophic set, our new method not only addresses existing problems but can also tackle other issues that no other authors have successfully resolved so far. Additionally, we conduct a comparative analysis of our proposed model against existing models in this article. Based on this comparative study, our findings assert the superior performance of our proposed model in relation to some of the existing models. In conclusion, we wrap up our research by presenting graphical, logical, and tabular comparisons with the existing methods.

Dijkstra’s algorithm (DA) is a very popular approach for finding the shortest route (SR) in the shortest route problem (SRP). The SRP becomes a challenging and complex problem in real life scenarios. The Fermatean neutrosophic set is a mathematical model that combines Fermatean sets with neutrosophic sets. It can handle the unclear, ambiguous, inconsistent, confusing, and uncertain information that comes from real-world problems. Decision-makers face difficulty accurately determining the precise membership (MG) and non membership levels due to the lack of appropriate data available. The FNS can handle this problem. In this study, we consider the interval FNS to describe the arc weight of a neutrosophic graph (NG). This SRP is called an interval Fermatean neutrosophic shortest route problem (IFNSRP). A modified DA is presented to solve this IFNSRP in an uncertain environment. The effectiveness of the presented method is illustrated with a numerical instance of a neutrosophic network.

American scientific publishing group

The traveling salesman problem (TSP) is an important and well known classical combinatorial network optimization problem in operation research, where the TSP finds a shortest possible route through a set of n nodes such that each and every node are visited exactly one time except for the starting node. In this problem, the arc lengths are generally considered to represent the traveling time or travelling cost rather than geographical distance. It is not possible to predict the exact arc length because the traveling time or traveling cost fluctuated with payload, weather, traffic conditions and so on. neutrosophic set theory provides a new tool to handle the uncertainties in TSP. In this paper, we concentrate on TSP on a network in which neutrosophic set, Instead of real number is assigned to edge as edge weight. We propose a mathematical model for a TSP with neutrosophic arc lengths. We present the utility of neutrosophic sets as arc length for TSP. An algorithmic method based on Genetic Algorithm (GA) is proposed for solving this problem. We have designed a new heuristic crossover and heuristic mutation our proposed GA. We have used a numerical example to illustrate the effectiveness of our proposed algorithm.

American scientific publishing group