## 2 Literature Review
Another fundamental assumption is that jobs are not allowed to be split, which means that a certain demand must always be processed from the beginning to the end. Of course, several demands for the same item may be grouped together and processed by lots to save setup costs. Problems with these assumptions can thus be referred to as a batching and scheduling problem (BSP) rather than a lot sizing and scheduling problem. The relationship between batching and lot-sizing is also analyzed in [13], where the studied lot sizing problem is solved as a batch sequencing problem. [PERSON] and [PERSON] [14] review the literature on scheduling with batching, in the particular case of parallel machines. In this paper, parallel machines are classified as: identical, when the processing times depend only of the jobs, regardless of the machine on which they are processed; uniform, if the processing times depend on the jobs and the machines' speed; or unrelated, when the processing time is a function of both the job and the machine where it is allocated. One of the specificities of our problem is that the processing time of the jobs depends only on the jobs. But because of the incompatibility between some molds and some machines, not all the machines can process all the jobs. To the best of our knowledge, this aspect has not been considered in batching problems but [15; 16] solved some scheduling problems with eligibility restrictions using constructive heuristics. The eligibility aspect can also be found in [17] but in the flowshop case. Nevertheless, our problem can be seen as a problem with parallel unrelated machines, where the processing time will be: (1) the time required to process the job if the mold is compatible with the selected machine or (2) infinite if the mold is not compatible with the selected machine. A batch-schedubling problem with parallel unrelated machines is dealt with in [18], where the authors compare the performances of four heuristics to minimize the total weighted tardiness. Their study showed the superiority of a two-level batching heuristic and a simulated annealing approach over heuristics that use priority rules such as earliest weighted due date or shortest weighted processing time. Reference [19] also addressed a parallel unrelated batch scheduling problem with sequence dependent setup times and developed a GRASP metaheuristic to minimize the makespan. Reference [20] provide a constructive algorithm and a simulated annealing to portion and schedule batches in a multi-stages semi-conductor manufacturing plant where the parallel machines per stage are non-identical. Because the weighted tardiness minimization in an unrelated parallel machine scheduling problem, \(m\Big{|}\Big{|}\sum_{j=1}^{n}w_{j}T_{j}\), is already NP-hard [21], our problem is also NP-hard. Indeed, the \(Rm\Big{|}\Big{|}\sum_{j=1}^{n}w_{j}T_{j}\) is a particular case of our problem when the jobs have equal weights, the setup-times are null and each mold is required by only one job. As seen in this literature review, whether for solving the lot-sizing or batching problems, the developed methods are quite sophisticated constructive approaches or optimization approaches. Our aim in this paper is to provide a high performing optimization method which is easy to develop and use. Although in this article the proposed method will only be tested in a case study company, injection molding is a common industrial activity and several plants may share the same features. Our solving method will thus not be limited to a single use-case and this work may be considered as a first step toward a final objective which is to generalize the optimization tool to other scheduling problems with new characteristics. In the following section, we provide a mathematical model that will be useful to spot the limit of the exact resolution for this NP-hard problem in terms of solving times and to ensure the quality of our method for the small-sized instances.