Multi-objective four dimensional imprecise TSP solved with a hybrid multi-objective ant colony optimization-genetic algorithm with diversity

In real world, most of the combinatorial optimization problems are multi-objective and it is difficult to optimize them simultaneously. In the literature, some individual algorithms (ACO, GA, etc.) are available to solve such discrete multi-objective optimization problems (MOOPs), particularly travelling salesman problems (TSPs) . Here a hybrid algorithm combining ACO and GA with diversity is developed to solve discrete multi-objective TSPs and named MOACOGAD. Generally in TSP, routes for travel are not considered as lengths of routes remain unaltered. In real life, there may be several routes for travel from one destination to another and conditions of those routes may also be different such as good, rough, bad, etc. In practical, travel costs and travel times are not defined precisely and represented by fuzzy data. When fuzzy travel costs and fuzzy travel times per unit length are involved, the lengths and conditions of the routes along-with the types of conveyances for travel become important. In some cases, risk of travel is also involved. In this paper a four dimensional imprecise TSP including source, destination, conveyances and routes under some risk factors are formulated and solved by the developed MOACOGAD. The model is illustrated numerically. As particular cases three and two dimensional multi-objective imprecise TSPs are derived and solved.