## 2 FJSSP Description  
Scheduling of job shop production is defined by four main research problems represented by Job Shop Scheduling Problem (JSSP), Flexible Job Shop Scheduling Problem (FJSSP), Dynamic Job Shop Scheduling Problem (DJSSP), and Flow Shop Scheduling Problem (FSSP) [19]. The classical JSSP represents one of the most difficult workshop problems, it assumes that there is no flexibility in the resources (including machines and tools) for each operation of every job. The FJSSP is an extension of the classical job shop scheduling problem allowing an operation to be processed by any machine from a given set [1]. The FJSSP consists of assigning and sequencing \(n\) jobs in \(m\) work center, each work center can have a different number of machines or resources, and each machine may process more than one type of operation [30]. In the FJSSP, each job is formed by a sequence of consecutive operations, each operation requires one machine, and each operation has to be performed to complete the job. This problem covers two difficulties namely the machine assignment problem (how to assign the operations on the machines) and the operation sequencing problem (how to sequence the operations on the machines) [31]. The general objective of the FJSSP is to improve the organization's productivity while reducing production consumption by switching over the selectable machine and taking full advantage of the underutilized capacity, adjusting the processing workload on machines [32]. The performance measures, restrictions, and characteristics the model must comply with must be established to define the FJSSP model. The model proposed in this study is based on the following assumptions:  
* All machines are available at the beginning of the scheduling horizon and can process only one job simultaneously. * No job can start an operation until the previously assigned has finished or until a machine is available to perform that operation. Therefore, only one job operation can be executed at a time. * Processing times are represented by fuzzy numbers and modeled by triangular membership functions. * Once an operation of a job has started on a machine, it will not be interrupted until it finishes the total number of units of said job. * Staff is available to perform each operation. * Machine breakdown or downtime due to maintenance or repairs in the planning horizon are not considered. * Setup times depend on the job sequence
* Setup times are represented by fuzzy numbers and modeled by triangular membership functions. * Recirculation is allowed since a machine can perform several processes (not at the same time), so a job can be processed several times on the same machine. * A time window defines the expected completion time for each job (interval to determine whether a job is completed on time). * Machines in the same work center may have different processing times. The mathematical model of the FJSSP is based on the study by [PERSON] et al. (2018) and [PERSON] (2018), where the objective functions minimize the number of late jobs and the Makes-pan, respectively. However, in these studies, the delivery dates are not represented with time intervals but with exact dates, and they do not consider the time of preparation depending on the sequence, for which we present adjustments to the mathematical model.