## 3 Problem Description  
The problem of food production process scheduling considered in this study aims to schedule the operations from the pre-treatment of raw materials to the stock of finished products of a meal manufacturing process in hospital catering or, more generally, in collective catering. Figure 2 represents the typical production areas and the different material resources available for realizing the operations of the meal-making process. This problem is considered as a flexible job shop problem with sequence-dependent setup time and job splitting by integrating specific industrial constraints, such as the days of pre-treatment and production of products, the product delivery times, and the amounts of human and material resources available, in addition to the different constraints described below. The addressed problem is considered as a flexible job shop scheduling problem with sequence-dependent setup time, since each job has its own order of operations and each operation has to be assigned to one among a set of possible machines. This problem can be described by a set of \(N\) jobs, where each job \(i\) corresponds to the preparation of a dish characterized by a number of portions \(Q_{i}\) (quantity) and a set of operations \(J_{i}\) for the preparation of the dish (from raw material to finished product). It is worth highlighting that the dishes to be prepared do not have the same order of operating ranges (set of operations necessary for the preparation of the dish). For each job, there is a due date \(D_{i}\) to respect. For each operation of an operating range, there is a set of material resources that are used to realize it, such as ovens, cooking pots, cookers, induction hobs, packaging machines, cooling cells (Figure 2). These material resources can be classified into three categories: \(M_{1}\), \(M_{2}\), and \(M_{3}\). \(M_{1}\subset M\) is composed of material resources with a capacity of one portion  
Figure 1: Bibliographical summary of the resolution methods used for flexible job shop problems and the objective functions studied in the literature: 191 papers between 1990 and 2014 ([PERSON] and [PERSON] [27]). and that can not process several jobs at the same time (material resources that can perform preprocessing and cold production operations). The set \(M_{2}\subset M\) represents the material resources with a capacity greater than one portion and that cannot process several jobs at the same time (ovens, etc.). \(M_{3}\subset M\) correspond to the set of material resources with a capacity greater than one portion and that can process several jobs at the same time (cooling cells). For each material resource, there is a setup time to take into account, which corresponds to the preparation time of the machine before carrying out an operation and the cleaning time of the machine between two consecutive operations. A time window of availability is known for each material resource. Note that the corresponding machines may not be identical, involving different processing times according to the chosen machine. The setup times of machines are sequence dependent because they depends on the preceding operation on the same machine. The food production process scheduling involves two steps: (i) assignment of operations to machines and (ii) sequencing of operations on machines. As mentioned previously, in order to respect the production capacities of material resources, a job can be split into smaller sub-lots in such a way that the operations of sub-lots of a job can be performed simultaneously on different machines. This strategy, which is useful when machine capacity does not allow the treatment of the whole job, also enables a more efficient processing scheme. The criterion to minimize in the present study is the total flow time of jobs in the production system. The choice of this criterion is based on the fact that the respect of the cold chain at each stage of the product life cycle must be ensured.