A new mathematical model and solution method for the asymmetric traveling salesman problem with replenishment arcs

This paper presents a new mathematical model for the Asymmetric Traveling Salesman Problem with Replenishment Arcs, an extension of the Asymmetric Traveling Salesman Problem, incorporating constraints on subpaths within the tour. Many existing modeling approaches to this problem require the generation of replenishment feasible or replenishment violation paths as a parameter set, which may lead to computational difficulties. Our formulation addresses these difficulties and provides direct computation of an optimal tour without relying on the set of paths as a parameter set. In this paper, we also propose a Lagrangian relaxation-based solution method. Given that ordinary Lagrangian functions can encounter duality gap in nonconvex problems, we employ a special augmented Lagrangian function, which is proven to overcome the issue of duality gap for many classes of nonconvex problems, including ours. In this paper, we utilize a hybrid solution method by combining the F-MSG method with an ant colony optimization algorithm. A similar solution method was previously used in Bulbul and Kasimbeyli (2021) [13]. In this paper, the method used in the aforementioned paper is enhanced in terms of computational complexity and solution efficiency. We assess the proposed method on 180 randomly generated instances, demonstrating that it achieves optimal solutions for almost all cases. Additionally, we apply our methodology to the aircraft maintenance routing problem, testing it on 11 instances from the aforementioned study. The results highlight the effectiveness of our approach, with an average improvement of 48.6% in solution time and a 0.93% enhancement in solution quality.