## 1 Introduction
Within artificial intelligence techniques, genetic algorithms are usually efficient for the optimization of complex systems, represent a good solver for combinatorial problems, provide a wide range of solutions [19], and have been used to solve many problems related to the FJSSP [20], becoming a powerful and successful technique for solving of NP-Hard problems due to the logic of evolutionary principles [21; 22]. The genetic algorithm as an evolutionary algorithm has advantages as it is relatively easy to understand and apply and presents fault tolerance [23]. However, although it has been proven that this metaheuristic efficiently solves complex optimization problems, the parameter values must prevent premature convergences and promote the finding of global solutions instead of local solutions [24; 25]. [PERSON], [PERSON], [PERSON], and [PERSON] [26] presented a study considering a flexible job shop system, fuzzy parameters, and time windows, introduced a multi-objective programming problem with fuzzy time windows, and solved the problem through a multi-group genetic algorithm. Similarly, [PERSON], [PERSON], and [PERSON] [27] developed a model that incorporates time windows, capacity, and space constraints and uses mixed integer programming to limit cyclic activities. [PERSON], [PERSON], and [PERSON] [26] use a rolling window rescheduling strategy and dynamic scheduling for the FJSSP with fuzzy delivery time, considering a trapezoidal delivery window to minimize energy consumption, maximum makespan, and consumer dissatisfaction, and solving this problem with an immune genetic algorithm. Other studies have added restrictions to the FJSSP models to adapt to realistic problems and seek to fulfill several objectives considering multi-objective uncertainty environ ments [25; 28]. In this sense, [PERSON] et al. [29] propose a model where the processing time can be exact or fuzzy depending on the availability of job data, and the FJSSP is solved using a hybrid algorithm between a genetic algorithm and the particle swarm optimization algorithm. In most of these models, the fuzzy variables receive triangular membership functions due to the ease of their construction. Based on the abovementioned, this article aims to address the FJSSP considering due-window and fuzzy setup times with triangular membership functions to minimize the weighted penalties for tardiness/earliness through a genetic algorithm. The present study contributes in multiple ways: Firstly, it develops a methodology to calculate the possibility of tardiness and earliness of a job, comparing the due-date time window with the fuzzy set of the completion time and using this result to calculate the penalty of tardiness and earliness in the objective function. Secondly, the study adapts GA and deterministic methods (heuristics and rules) for the specific assumptions of the proposed FJSSP model. Thirdly, it presents a novel solution representation for each chromosome of the GA in two ways, one to calculate the total penalty (objective function) and another for the mutation and crossover operators to minimize the chance of infectable chromosomes, reducing the use of repairing operators and the algorithm computing time. Fourthly, the proposed algorithm solves a real-world case study in the textile sector, demonstrating the algorithm's applicability to industries with complex production systems.