Expected Length of the Shortest Path of the Traveling Salesman Problem in 3D Space

Abstract

Finding the shortest path of the traveling salesman problem (TSP) is a typical NP-hard problem and one of the basic optimization problems. TSP in three-dimensional space (3D-TSP) is an extension of TSP. It plays an important role in the fields of 3D path planning and UAV inspection, such as forest fire patrol path planning. Many existing studies have focused on the expected length of the shortest path of TSP in 2D space. The expected length of the shortest path in 3D space has not yet been studied. To fill this gap, this research focuses on developing models to estimate the expected length of the shortest path of 3D-TSP. First, different experimental scenarios are designed by combining different service areas and the number of demand points. Under each scenario, the specified number of demand points is randomly generated, and an improved genetic algorithm and Gurobi are used to find the shortest path. A total of 500 experiments are performed for each scenario, and the average length of the shortest path is calculated. The models to estimate the expected length of the shortest path are proposed. Model parameters are estimated and k-fold cross-validation is used to evaluate the goodness of fit. Results show that all the models fit the data well and the best model is selected. The developed models can be used to estimate the expected length of the shortest path of 3D-TSP and provide important references for many applications.