Estimating the Tour Length for the Close Enough Traveling Salesman Problem

Debdatta Sinha Roy,Xingyin Wang,Edward Wasil,Bruce Golden

doi:10.3390/a14040123

Abstract

We construct empirically based regression models for estimating the tour length in the Close Enough Traveling Salesman Problem (CETSP). In the CETSP, a customer is considered visited when the salesman visits any point in the customer’s service region. We build our models using as many as 14 independent variables on a set of 780 benchmark instances of the CETSP and compare the estimated tour lengths to the results from a Steiner zone heuristic. We validate our results on a new set of 234 instances that are similar to the 780 benchmark instances. We also generate results for a new set of 72 larger instances. Overall, our models fit the data well and do a very good job of estimating the tour length. In addition, we show that our modeling approach can be used to accurately estimate the optimal tour lengths for the CETSP.

Highlights

Operations researchers have long been interested in estimating the length of tours and routes in the traveling salesman problem (TSP) and the vehicle routing problem (VRP)
AvgR and SZ are used to capture the geometric features unique to the Close Enough Traveling Salesman Problem (CETSP). These two independent variables would not be used in a regression model that estimates TSP tour lengths
We recommend the model with the least number of independent variables (Best Bayesian information criterion (BIC) model with eight variables) to estimate the Steiner zone variable neighborhood search (SZVNS) tour lengths for CETSP instances with random node locations

Summary

Introduction

Operations researchers have long been interested in estimating the length of tours and routes in the traveling salesman problem (TSP) and the vehicle routing problem (VRP). Nicola et al [9] developed empirically based regression models for estimating the travel distance in the TSP, in the capacitated VRP with time windows, and in the multi-region, multi-depot pickup and delivery problem. Approaches that generate fast, accurate estimates for the travel distance are highly desirable in practice for a wide range of real-world routing problems. We construct empirically based regression models for estimating the tour length in the Close Enough Traveling Salesman Problem (CETSP). In order to generate high-quality solutions quickly, heuristics have been developed and tested for the CETSP These include the use of supernodes by Gulczynski et al [16].

Regression Models and Fitness Measures

Regression Results

Best Subset Model Selection

Model Validation

Conclusions and Future Directions