The Traveling Salesman Problem (TSP) is a well known problem in operations research with various studies and applications. In this paper, we address a variant of the TSP in which the customers are divided into several priority groups and the order of servicing groups can be flexibly changed with a rule called the d-relaxed priority rule. The problem is called the Clustered Traveling Salesman Problem with Relaxed Priority Rule (CTSP-d). We propose two new Mixed Integer Linear Programming (MILP) models for the CTSP-d and a metaheuristic based on Iterated Local Search (ILS) with operators designed for or adapted to the problem. The experimental results obtained on the benchmark instances show that two new models performs better than previous ones, and ILS also proves its performance with 13 new best known solutions found and significant stability compared to existing metaheuristics.

旅行商问题（TSP）是运筹学中众所周知的问题，有各种研究和应用。在本文中，我们提出了 TSP 的一种变体，其中将客户分为几个优先级组，并且可以使用称为d宽松优先级规则的规则灵活更改服务组的顺序。该问题称为具有宽松优先级规则的集群旅行商问题 (CTSP- d )。我们为 CTSP- d提出了两个新的混合整数线性规划 (MILP) 模型以及基于迭代局部搜索（ILS）的元启发式算法，其中的运算符是为问题设计或适应问题的。在基准实例上获得的实验结果表明，两个新模型的性能优于以前的模型，ILS 还通过发现的 13 个新的最知名解决方案证明了其性能，并且与现有的元启发法相比具有显着的稳定性。