## 4 Late Acceptance Hill-Climbing Matheuristic Template  
The late acceptance hill-climbing (LAHC) algorithm is a single-point iterative search algorithm proposed by [8] that accepts non-improving movements when the objective function value of a candidate has a better value than the one number of iterations before. Namely, while in the standard hill-climbing algorithm a candidate solution is compared to the direct predecessor, in LAHC the candidate is compared with the current solution several iterations before. LAHC has been successfully applied to several optimization problems. In the following, some of them are briefly reviewed. The authors of [8] apply the LAHC to exam timetabling problems reporting a competitive performance. This problem is later addressed by [50] where they present a hybridization of an adaptive artificial bee colony (ABC) and LAHC. The latter is applied as a local search procedure during the exploitation of the ABC algorithm. The results indicate that this hybridization performs better than other modifications of the ABC algorithm. In [51], the authors apply the LAHC to the traveling purchaser problem showing a suitable performance for solving that problem. The authors of [52] use this algorithm for balancing two-sided assembly lines with multiple constraints. The LAHC, in this case, is able to provide the same solutions as those provided by the optimization model implemented using LINGO for the small-sized instances. For large-sized instances, the model is not able to provide an optimal solution in reasonable computational time, thus leaving LAHC the best option for those instances. The author of [53] applies LAHC to the liner shipping fleet reposition problem. The author compares it to simulated annealing (SA) reporting that LAHC performs well on small instances, but on larger instances, SA exhibits better performance. The authors of [54] experimentally examine their proposed method and compare it to search techniques that employ a cooling schedule, i.e., simulated annealing, threshold accepting and the great deluge algorithm for the traveling salesman problem. In [55], the authors present a cutoff time strategy based on the coupon collector's problem and apply it to the LAHC algorithm. The authors tested that approach on the Travelling Salesman Problem, the Quadratic Assignment Problem, and the Permutation Flow-shop Scheduling Problem. Their results show that the proposed strategy is a valid stopping method for a one-point stochastic local search algorithm that accepts worsening moves. The authors of [56] apply the LAHC for finding train composition causing greatest fatigue damage in railway bridges. In [9], the authors propose an LAHC approach to solve the unrelated parallel machine scheduling problem with sequence and machine-dependent setup times, the reported results indicated the relevancy of including advanced local search methods to improve LAHC performance. This raises the research question related to using an exact solver instead of a local search within LAHC. Based on the previous discussion, there are two reasons for developing and proposing the Late Acceptance Hill-Climbing Matheuristic (LAHCM) in this paper. First, the LAHC as a metaheuristic reportedly performs well for the traveling purchaser problem and the traveling salesman problem. The traveling purchaser problem incorporates a choice option regarding the inclusion of nodes. As including rework into dynamic lot sizing shows a high similarity to this property and lot sizing models are frequently formulated as traveling salesman based formulations including the GLSP (see [26]), it is worth examining the application of LAHC for solving our GLSP model formulation. Second, considering the relevant performance of this strategy in the related literature and the benefits of exact approaches in terms of solution quality, the interplay between the two types of approaches is considered in LAHCM. In this sense, exact approaches are exploited to solve subproblems optimally and generate candidate solutions within LAHC.