Profit Maximization of TSP with Uncertain Parameters Through a Hybrid Algorithm

Here, a model of travelling salesman problem (TSP) with uncertain parameters is formulated and solved using a hybrid algorithm. For the TSP, there are some fixed number of cities and the costs and time durations for travelling from one city to another are known. Here, a travelling salesman (TS) visits and spends some time in each city for selling the company's product. The return and expenditure at each city are dependent on the time spent by the TS at that city and these are given in functional forms of t. The total time limit for the entire tour is fixed and known. Now, the problem for the TS is to identify tour programme and also to determine the stay time at each city so that total profit out of the system is maximum. In reality, profit and cost for spending time in a city by the salesman are finite but fuzzy in nature. So fuzzy expenditure and fuzzy return are taken to make the problem realistic. Here, the model is solved by a hybrid method combining the particle swarm optimization (PSO) and ant colony optimization (ACO). The problem is divided into two subproblems where ACO and PSO are used successively iteratively in a generation using one's result for the implicitly other. Numerical experiments are performed to illustrate the models. Some behavioural studies of the models and convergency of the proposed hybrid algorithm with respect to iteration numbers and cost matrix sizes are presented.