## 2 Literature Review
[11] presented a lot-sizing problem with a single machine that incorporated raw material perishability and analyzed how these considerations enforced specific constraints on a set of fundamental decisions, particularly for multi-level structures. Three variants of the two-level lot-sizing problem incorporating different types of raw-material perishability--fixed shelf life, functionality deterioration, and functionality-volume deterioration--were studied; the authors proposed mixed-integer programming formulations for each variant and performed computational experiments with sensitivity analyses. [PERSON] et al. [12] considered a capacitated dynamic lot-sizing problem with parallel machines for the food industry, in which a given product produced during a specified time period is used to satisfy the related demand. [PERSON] et al. [13] addressed the integrated lot-sizing and scheduling problem of food production in batch manufacturing systems with multiple shared common resources and proposed a new mixed-integer linear programming formulation with multiple objective functions. [PERSON] et al. [14] proposed a classical multi-level lot-sizing and flow-shop scheduling problem formulation to incorporate perishability issues. Regarding the shelf life of products, [PERSON] and [PERSON] [15] reviewed models for the agri-food business, where products may be perishable or not, but their focus was on procurement and harvesting planning. The only goods they were interested in were crops. [PERSON] et al. [16] introduced the planning and scheduling of decisions considering the shelf-life restrictions, product-dependent machine speeds, demand due dates, and regular and overtime working hours in the perishable supply chain. [PERSON] et al. [17] considered a three-dimensional matching model for perishable production scheduling, which was studied under two independent aspects: the relative perishability of products and the feasibility of launching/completion time. [PERSON] et al. [18] described an exact algorithm to solve a scheduling and sequencing problem in the same industry. [PERSON] et al. [19] provided a review of literature on the integration of scheduling and lot sizing for perishable food products, and they categorized the papers by the characteristics of the lot sizing and scheduling that were included in their models, as well as the strategies used to model perishability. In the present work, the studied production system is considered as a flexible job shop system. Since 1990, the flexible job shop scheduling problem (FJSP) has been extensively investigated by researchers. [PERSON] and [PERSON] [20] discussed the problem in a flexible manufacturing system with transportation times and limited buffers. They developed a mixed-integer linear programming model and heuristics to minimize the makespan, the mean completion time, and the maximum tardiness. [PERSON] and [PERSON] [21] proposed a genetic algorithm for the FJSP, where the objective function is used to minimize the makespan and the mean tardiness. [PERSON] and [PERSON] [22] took into account a special case of FJSP, where each operation could be assigned to one of two parallel machines. A tabu search algorithm for solving the sequencing and assignment problems was developed. The algorithm was compared to a branch and bound algorithm on several random test instances. [PERSON] and [PERSON] [23] studied the FJSP with attached and detached setup times, machine release dates, and technological time lag constraints. For this problem, a mixed-integer linear programming model was proposed, and a parallel genetic algorithm using multiple threads was introduced. [PERSON] et al. [24] proposed a genetic algorithm for an FJSP in which blocking constraints were taken into consideration. [PERSON] and [PERSON] [25] developed a variable neighborhood search algorithm. [PERSON] [26] presented a mixed-integer linear programming model and an iterated local search algorithm that minimize the total tardiness.