## 1 Introduction  
In many production processes, a fraction of the manufactured products can be defective due to an unreliable production system. Often, defective items incorporate substantial value and it is possible to rework them into an \"as-good-as-new\" state. Incorporating this feature results in a more ecological friendly production process as disposal reduces. Furthermore, reworking of defective items can only take place within a specific time span. For instance, in the pharmaceutical industry rework often has to take place in a short amount of time due to likely increasing contamination. In this context, we formulate a related problem and address a model formulation of the general lot sizing and scheduling problem with rework termed as GLSP-RP, which was first presented by [1]. This problem also considers lifetime constraints for defective items. As the authors report, the GLSP-RP model is hardly solvable by a general-purpose solver in an adequate amount of time. Even for small problem instances of non-practical size, it requires more than 1800 s. This leads to the necessity of faster approaches providing solutions within reasonable execution times. Matheuristics can be defined as those solution approaches obtained from the interoperation between metaheuristics and mathematical programming techniques [2]. Within matheuristics, decomposition approaches belong to one of the most relevant algorithms (some recent examples can be consulted in, for example, [3; 4; 5]). For instance, the principal idea behind proposing the matheuristic version of POPMUSIC is to decompose the problems into smaller (and likely more tractable) ones and solve them by means of an exact technique [6]. As remarked by [7] to successfully apply decomposition matheuristic approaches, one may develop an iterated procedure where the information is interchanged along the process. Adapting those ideas to well-known single point metaheuristics such as late acceptance hill-climbing (LAHC, [8]) permits, within a hill-climbing framework, to substitute the inherent method for generating candidate solutions by an exact method solving a given sub-problem. Furthermore, recently, [9] showed by means of the unrelated parallel machine scheduling problem the importance of including advanced local search methods to improve the solution quality in LAHC. This, thus, led to our research question about how LAHC would perform if we use an exact approach instead of a local search solver. Based on the previous discussion, the contribution of this paper is twofold:  
* We investigate the relationship between the original formulation of the general lot sizing and scheduling problem (GLSP) and the GLSP-RP by providing a comprehensive discussion of both problems. * For solving the GLSP-RP, we propose the Late Acceptance Hill-climbing Mathemistic (LAHCM) as a general solution framework, which is inspired by and integrates the LAHC strategy with the exact approaches, i.e., the solution of optimization models for reduced problems configured in the spirit of the LAHC. To assess its performance, we evaluate it over a set of well-defined instances. The computational results indicate that by means of our approach we are able to improve the objective function values for several instances where a general-purpose solver such as CPLEX ([[https://www.ibm.com/nl-en/analytics/cplex-optimizer](https://www.ibm.com/nl-en/analytics/cplex-optimizer)]([https://www.ibm.com/nl-en/analytics/cplex-optimizer](https://www.ibm.com/nl-en/analytics/cplex-optimizer))) is not able to find optimal solutions in the given time limit of 1800 s.  
The outline of this paper is as follows. In Section 2, those works related to the GLSP as well as its respective formulation are discussed. The general lot sizing and scheduling problem considering rework and lifetime constraints for defective items is explained in Section 3. Later, our matheuristic solution approach based on late acceptance hill-climbing is described in Section 4. The extensive computational experience is reported in Section 5.