We extend our previous work on complexity indices for the travelling salesman problem (TSP) using graph spectral techniques of data mining. A complexity index is an invariant of an instance I by which we can predict the execution time of an exact algorithm for TSP for I. We consider the symMetric travelling salesman problem with instances I represented by complete weighted graphs G. Intuitively, the hardness of an instance G depends on the distribution of short edges within G.Therefore we consider some short edge subgraphs of G (minimal spanning tree and several others) as non-weighted graphs and several their invariants as potential complexity indices. Here spectral invariants (e.g. spectral radius of the adjacency matrix) play an important role. Spectral clustering algorithms are used including information obtained from the spectral gap in Laplacian spectra of short edge subgraphs.

Due to the many applications of the travelling salesman problem, solving this problem has been considered by many researchers. One of the subsets of the travelling salesman problem is the Metric travelling salesman problem in which a triangular inequality is observed. This is a crucial problem in combinatorial optimization as it is used as a standard problem as a basis for proving complexity or providing solutions to other problems in this class. The solution is used usually in logistics, manufacturing and other areas for cost minimization. Since this is an NP-hard problem, heuristic and meta-heuristic algorithms seek near-optimal solutions in polynomial time as numerical solutions. For this purpose, in this paper, a heuristic algorithm based on the minimum spanning tree is presented to solve this problem. Then, by generating 20 instances, the efficiency of the proposed algorithm was compared with one of the most famous algorithms for solving the travelling salesman problem, namely the nearest neighbour algorithm and the ant colony optimization algorithm. The results show that the proposed algorithm has good convergence to the optimal solution. In general, the proposed algorithm has a balance between runtime and the solution found compared to the other two algorithms. So the nearest neighbour algorithm has a very good runtime to reach the solution but did not have the necessary convergence to the optimal solution, and vice versa, the ant colony algorithm converges very well to the optimal solution, but, its runtime solution is very longer than the proposed algorithm.

In today’s competitive transportation systems, passengers search to find traveling agencies that are able to serve them efficiently considering both traveling time and transportation costs. In this paper, we present a new model for the traveling salesman problem with multiple transporters (TSPMT). In the proposed model, which is more applicable than the traditional versions, each city has different transporting vehicles and the cost of travel through each city is dependent on the transporting vehicles type. The aim is to determine an optimal sequence of visited cities with minimum traveling times by available transporting vehicles within a limited budget. First, the mathematical model of TSPMT is presented. Next, since the problem is NP-hard, a new hybrid parallel simulated annealing algorithm with a new coding scheme is proposed. To analyze the performance of the proposed algorithm, 50 numerical examples with different budget types are examined and solved using the algorithm. The computational results of these comparisons show that the algorithm is an excellent approach in speed and solution quality.

The travelling salesman problem is one of the fundamental operational research problems where the objective is to generate the cheapest route for a salesman starting from a given city, visiting all the other cities only once and finally returning to the starting city. In this paper, we study the travelling salesman problem in uncertain environment. Particularly, the single valued triangular neutrosophic environment is considered viewing that it is more realistic and general in real-world industrial problems. Each element in the distance matrix of the travelling salesman problem is presented as a single valued triangular neutrosophic number. To solve this problem, we enhance our novel column-row heuristic Dhouib-Matrix-TSP1 by the means of the center of gravity ranking function and the standard deviation Metric. In fact, the center of gravity ranking function is applied for defuzzification in order to convert the single valued triangular neutrosophic number to crisp number.A stepwise application of several numerical travelling salesman problems on the single valued triangular neutrosophic environment shows that the optimal or a near optimal solution can be easily reached thanks to the Dhouib-Matrix-TSP1 heuristic enriched with the center of gravity ranking function and the standard deviation Metric.