## 2 Literature Review  
Several papers from the literature addressed the injection molding planning and scheduling problems. [PERSON] and [PERSON] [2] studied a parallel machine scheduling problem in an injection molding facility, where different products require different types of resources. The authors provide a mathematical formulation and a decomposition based approach to minimize the setup and the inventory holding costs. A decomposition strategy is also used in [3,4] where the compatibility constraints between injection molds and machines are taken into account. In both articles, the problem is decomposed into sub-problems by grouping the machines by sets according to their compatible molds. Each sub-problem is then dealt with as a Capacitated Lot-Sizing Problem (CLSP). [PERSON] and [PERSON] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Decision Level** & Strategic & Tactical & Operational \\ \hline
**Planning Horizon** & Years & Months & Weeks/Days \\ \hline
**Typical Problems** & \begin{tabular}{c} Determine the \\ number of resources \\ \end{tabular} & \
\begin{tabular}{c} Plan activities \\ Assign resources \\ \end{tabular} &
\begin{tabular}{c} Schedule activities \\ Consider emergencies \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Levels of decision. [PERSON] [3] used two heuristic procedures while [4] provides a goal programming formulation and a heuristic to minimize the production's costs. The obtained results are shown to outperform the planning and scheduling procedures currently used in their case-study injection plants. The problem we study in this paper also shares some features with the CLSP. Indeed, the lot-sizing issue reflects the fact that companies usually have to manufacture several types of products with limited production capacity. It is a very popular subject in the field of combinatorial optimization and has been defined since the early 1900s with the EOQ Economic Order Quantity [5]. Lot-sizing problems have been widely studied since then; a recent literature review can be found in [6]. Different lot-sizing models have been developed, depending on the characteristics of the production process. Although most of the lot-sizing studies focus on \"product lots\" (i.e., determining the right period and the quantity to produce for each type of product), some of them can also be related to \"process lots\"; in other words, to the decision of how many times or how long each process should be used and how the processes should be scheduled, given that each process can lead to a set of products. Such a feature appears in several industrial problems [7; 8; 9], including injection molding [10], for which a mixed integer programming model is proposed and tested on real data from a Brazilian molded pulp plant. CLSP has many extensions, including the consideration of parallel production machines per period and setup times between products of different types. However, the scheduling is not part of this issue and is traditionally made as a second step. Therefore there is a significant difference between CLSP and the problem we study. Indeed, separating assignment and scheduling or decomposing a problem can immediately eliminate several good solutions from the searching space. In this work, we take the problem as a whole and this issue may thus be referred to as a batching problem. Batching is indeed the decision of whether or not to schedule similar jobs contiguously, to avoid setup times or setup costs [11]. [PERSON] and [PERSON] [12] draw an analogy between the batching and lot-sizing problems. In the continuous time lot sizing and scheduling problem, each demand is characterized by its deadline and its size. Demands are interpreted as jobs and the demand size determines the processing time of a job. An important assumption is that the capacity (e.g., the speed of the machine) is constant over time and thus the processing time of a job does not depend on the schedule.