Adjustment Factors and Applications for Analytic Approximations of Tour Lengths

Abstract

The shortest tour distance for visiting all points exactly once and returning to the origin is computed by solving the well-known Traveling Salesman Problem (TSP). Due to the large computational effort needed for optimizing TSP tours, researchers have developed approximations that relate the average length of TSP tours to the number of points n visited per tour. The most widely used approximation formula has a square root form: √n multiplied by a coefficient β. Although the existing models can effectively approximate the distance for conventional vehicles with large capacities (e.g., delivery trucks) where n is large, approximations that seek to cover large ranges of n, possibly to infinity, tend to yield poorer results for the small n values. Thus, this dissertation focuses on approximation models for the small n values, which are needed for many practical applications, such as for some recent delivery alternatives (e.g., drones). The proposed models show promise in analyzing the real-world problems in which actual tours serve few customers due to limited vehicle capacity and incorporate realistic constraints, such as the effects of a starting point location, geographical restrictions on movements, demand patterns, and service area shapes. The dissertation may open new research avenues for analyzing the new transportation alternatives and provide guidelines to planners for choosing appropriate models in designing or evaluating transportation problems. Approximation models are estimated from the following experiments: 1) a total of 60 cases are developed by considering various factors, such as point distributions and shapes of service areas. 2) Solution methods for TSP instances are compared and chosen. 3) After the TSPs are optimized for each n, the TSP tour lengths are averaged. 4) Lastly, models for the averaged TSP tour lengths are fitted with ordinary least squares (OLS) regression. After the approximations are developed, some possible extensions are explored. First, adjustment factors are designed to integrate the 60 cases within one equation. With those factors, it can be understood how approximation varies with each classification. Next, the approximations considering stochastic customer presence (i.e., probabilistic TSP) are proposed. Third, the approximated tour lengths are compared with the optimal solutions of vehicle routing problem (VRP) in actual rural and urban delivery networks. Here, some additional factors, such as a circuity factor and service zone shape, are discussed. Lastly, the proposed methodology is applied to formulate and explore various types of existing and hypothetical delivery alternatives.