A neutrosophic vague Floyd–Warshall algorithm for multi-criteria shortest-path problems

Abstract

Routing decisions in computer networks frequently involve imprecise, incomplete, and uncertain information arising from variations in bandwidth, delay, throughput, load, and link reliability. The classical Floyd–Warshall algorithm requires scalar and consistently ordered edge weights; therefore, it cannot directly evaluate multiple routing criteria with conflicting optimisation directions. This study develops a direction-aware Neutrosophic Vague Floyd–Warshall framework for multi-criteria routing under uncertainty. An illustrative directed network comprising five nodes, 11 edges, and nine routing criteria was analysed. The linguistic assessments were represented using a corrected seven-level neutrosophic vague scale and converted into numerical scores through a neutrosophic vague score function. Length, delay, cost, hop count, and load were treated as cost criteria, whereas bandwidth, maximum transmission unit, throughput, and reliability were treated as benefit criteria. The criterion scores were directionally normalised and aggregated using equal weights to produce a nonnegative composite penalty for each edge. These penalties were subsequently processed using the Floyd–Warshall recurrence, while a next-node matrix was employed to reconstruct the globally optimal routes. The results identified 1→3→5 as the optimal route from node 1 to node 5, with a total composite penalty of 0.6852. Independent enumeration of all feasible simple paths confirmed this result. The proposed framework resolves inconsistencies associated with direct vector comparisons and greedy Boolean edge selection while providing a transparent approach for uncertain multi-criteria routing. Future research should validate the framework using large-scale, dynamic, and real-time network data and expert-derived criterion weights.