中文摘要：

带柔性时间窗的开放式车辆路径问题（Opening Vehicle Routing Problem with Flexible Timewin—dows，OVRPFTW）对物流配送中的延迟或者提早具有一定程度的容忍．本文首先建立了OVRPFTW的数学模型，然后分别将Sine映射，Chebyshev映射和Logistic映射引入基本蚁群算法，构建了三种混沌蚁群算法，并将其用于求解OVRPFTW．算倒测试表明：Sine映射和Chebyshev映射能够明显地改进基本蚁群算法的优化性能，基于Sine映射和Chebyshev映射的混沌蚁群算法的求解性能优于基本蚁群算法和基于Logistic映射的混沌蚁群算法．

英文摘要：

Opening Vehicle Routing Problem with Flexible Time Windows （OVRPFTW） allows vehicles to serve customers ahead of schedule or behind schedule by a given tolerance. In this paper, the mathematical model of the OVRPFTW is formulated firstly, then three chaos Ant Colony Optimization （ACO） algorithms for solving the OVRPFTW are proposed by combining ACO with Sine mapping, Chebyshev mapping and Lo gistie mapping, respectively. Numerical results show that Sine mapping and Chebyshev mapping can signifi- cantly improve the ACO algorithm, and comparing with the basic ACO algorithm and the chaos ACO algo rithm based on Logistic mapping, the chaos ACO algorithms based on Sine mapping and Chebyshev mapping have better optimization performance.