Statistical Modeling of Quartiles, Standard Deviation, and Buffer Time Index of Optimal Tour in Traveling Salesman Problem and Implications for Travel Time Reliability

Abstract

The traveling salesman problem (TSP) plays an important role in the field of transportation and logistics. While most studies focus on developing algorithms to find the shortest path and explore the average length of the shortest paths, the degree to which the shortest path deviates from its mean has not been studied. The study of deviation is important because it has implications for travel time reliability. Previous studies have used various indicators to measure this deviation, mainly including standard deviation, quantiles, and buffer time index (BTI). Therefore, this study aims to develop an empirical model to estimate the standard deviation, quantiles, and BTI for the optimal TSP tour. Experiments are performed to find the shortest path connecting N customers, which are generated randomly in a specified service area, using a genetic algorithm. The service area is a rectangle with ratio of length to width ranging from 1:1 to 8:1. Two types of lengths are considered: Euclidean and Manhattan. The number of customers considered ranges from 10 to 100 with intervals of 10. In the experimental design, the customers are generated randomly 500 times. The quartiles and standard deviations of the 500 shortest paths are recorded. The BTI is also calculated. Regression models are developed to estimate quartiles, standard deviation, and BTI using number of customers and parameters of service area as predictive variables. The models perform well on the testing data set. The constructed models can be used to estimate the standard deviation and reliability of travel time.