The travelling salesman problem (TSP) is a problem whereby a finite number of nodes are supposed to be visited exactly once, one after the other, in such a way that the total weight of connecting arcs used to visit these nodes is minimized. We propose a labelling method to solve the TSP problem. The algorithm terminates after K−1 iterations, where K is the total number of nodes in the network. The algorithm’s design allows it to determine alternative tours if there are any in the TSP network. The computational complexity of the algorithm reduces as iterations increase, thereby making it a powerful and efficient algorithm. Numerical illustrations are used to prove the efficiency and validity of the proposed algorithm.
In this article, the authors propose a maximum flow algorithm based on flow matrix. The algorithm only requires the effort to reduce the capacity of the underutilized arcs to that of the respective flow. The optimality of the algorithm is proved by the max-flow min-cut theorem. The algorithm is table-based, thus avoiding augmenting path and residual network concepts. The authors used numerical examples and computational comparisons to demonstrate the efficiency of the algorithm. These examples and comparisons revealed that the proposed algorithm is capable of computing exact solutions while using few iterations as compared to some existing algorithms.
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