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In this study particle swarm optimization (PSO) is modified and hybridised with genetic algorithm (GA) using one’s output as the other's input to solve Traveling Salesman Problem(TSP). Here multiple velocity update rules are introduced to modify the PSO and at the time of the movement of a solution, one rule is selected depending on its performances using roulette wheel selection process. Each velocity update rule and the corresponding solution update rule are defined using swap sequence (SS) and swap operation (SO). K-Opt operation is applied in a regular interval of iterations for the movement of any stagnant solution. GA is applied on the final output swarm of the PSO to search the optimal path of the large size TSPs. Roulette wheel selection process, multi-point cyclic crossover and the K-opt operation for the mutation are used in the GA phase. The algorithm is tested in crisp environment using different size benchmark test problems available in the TSPLIB. In the crisp environment the algorithm gives approximately 100% success rate for the test problems up to considerably large sizes. Efficiency of the algorithm is tested with some other existing algorithms in the literature using Friedman test. Some approaches are incorporated with this algorithm for finding solutions of the TSPs in imprecise (fuzzy/rough) environment. Imprecise problems are generated from the crisp problems randomly, solved and obtained results are discussed. It is observed that the performance of the proposed algorithm is better compared to the some other algorithms in the existing literature with respect to the accuracy and the consistency for the symmetric TSPs as well as the Asymmetric TSPs.

A swap sequence-based particle swarm optimization (SSPSO) technique and genetic algorithm (GA) are used in tandem to develop a hybrid algorithm to solve generalized traveling salesman problem. Local search algorithm K-Opt is occasionally used to move any stagnant solution. Here, SSPSO is used to find the sequence of groups of a solution in which a tour to be made and cities from different groups of the sequence are selected using GA. The K-Opt algorithm (for K = 3) is used periodically for a predefined number of iterations to improve the quality of the solutions. The algorithm is capable of solving the problem in crisp as well as in imprecise environment. For this purpose, a general fitness evaluation rule for the solutions is proposed. The efficiency of the algorithm is tested in crisp environment using different size benchmark problems available in TSPLIB. In crisp environment, the algorithm gives 100% success rate for problems up to considerably large sizes. Imprecise problems are generated from crisp problems randomly using a rule and are solved using the proposed approach. The obtained results are discussed. Moreover it is observed that the proposed algorithm finds multiple optimal paths, when they exist, both for the crisp problems and their fuzzy variations.

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