# The continuous pollution routing problem

## Abstract

In this paper, we presented an ε-accurate approach to conduct a continuous optimization on the pollution routing problem (PRP). First, we developed an ε-accurate inner polyhedral approximation method for the nonlinear relation between the travel time and travel speed. The approximation error was controlled within the limit of a given parameter ε, which could be as low as 0.01% in our experiments. Second, we developed two ε-accurate methods for the nonlinear fuel consumption rate (FCR) function of a fossil fuel-powered vehicle while ensuring the approximation error to be within the same parameter ε. Based on these linearization methods, we proposed an ε-accurate mathematical linear programming model for the continuous PRP (ε-CPRP for short), in which decision variables such as driving speeds, travel times, arrival/departure/waiting times, vehicle loads, and FCRs were all optimized concurrently on their continuous domains. A theoretical analysis is provided to confirm that the solutions of ε-CPRP are feasible and controlled within the predefined limit. The proposed ε-CPRP model is rigorously tested on well-known benchmark PRP instances in the literature, and has solved PRP instances optimally with up to 25 customers within reasonable CPU times. New optimal solutions of many PRP instances were reported for the first time in the experiments.

Publication DOI: https://doi.org/10.1016/j.amc.2020.125072 College of Engineering & Physical Sciences > School of Engineering and Technology > Mechanical, Biomedical & Design This work is partly supported by the National Natural Science Foundation of China under Grant Nos. 71871003 , 71871003 , 71971009 , and 71971013 . © 2020, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Continuous optimization,Convex programming,Emission reduction,Vehicle routing problem,Computational Mathematics,Applied Mathematics 1873-5649 19 Feb 2024 08:30 17 Feb 2020 16:14 http://www.scop ... tnerID=8YFLogxK (Scopus URL) https://www.sci ... 0412?via%3Dihub (Publisher URL) Article 2020-12-15 2020-02-05 2020-01-19 Xiao, Yiyong Zuo, Xiaorong Huang, Jiaoying Konak, Abdullah Xu, Yuchun ( 0000-0001-6388-813X)