## 2 State of the Art
### Fuzzy Inference System (FIS) and Fuzzy Petri Nets  
Fuzzy Petri Nets is a combination of two different sciences--the set of fuzzy logic and Petri nets theory--which are held to provide answers to vague or unclear problems in a system that is about to be examined. Therefore, we use fuzzy Petri nets to see and provide solutions to problems that are not clear, such as an asset or system that does not have historical data but wants to get a definite answer regarding the reliability and reliability of maintenance to improve the performance of these assets. Also, several previous researchers put forward their ideas in articles they wrote as follows. [PERSON] et al. (2017) \"propose a method based on the Fuzzy Expectation-Maximization (FEM) algorithm, which integrates the evidence of the field inspection outcomes with information taken from the maintenance operators about the transition times from one state to another. Possibility distributions are used to describe the imprecision in the expert statements\" [40]. [PERSON] et al. (2017) proposed \"a new interpretation of PHM outputs to define machine degradations that are corresponding to each job. Moreover, to consider several sources of uncertainty in the prognosis process, the authors choose to model PHM outputs using fuzzy logic. Motivated by the computational complexity of the problem, Variable Neighbourhood Search (VNS) methods are developed, including well-designed local search procedures\" [41]. [PERSON] et al. (2017) mentioned that \"to solve the problem, we developed two fuzzy genetic algorithms that are based on respectively the sequential and total scheduling strategies. The one  
Figure 2: Example of a Petri net on standby (red dot = token = energy of electrical power) [39]. respecting the sequential approach consists of two phases. In the first phase, the integrated production and maintenance schedules are generated. In the second one, the human resources are assigned to maintenance activities. The second algorithm respecting a total strategy consists of developing the integrated production and maintenance schedules that explicitly satisfy the human resource constraints\" [42]. [PERSON] et al. (2019) mentioned that \"Based on the results obtained in the case study, it can conclude that the fuzzy set for calculation is more rigorous than the qualitative results. The calculated unified qualitative and fuzzy risk number shows that the plant was classified as semi-critical. It obtained the highest fuzzy risk number of 99.1452 for both blowers (BW 20 21 and BW 20 23 A) assets failure\" [43]. [PERSON] and [PERSON] (2017) mentioned that, \"also, a fuzzy logic-based risk rank calculation approach has been presented. The suggested RBM approach, together with the fuzzy inferencing process, enables us to minimize suboptimal calculations when the input values are at the boundaries of the particular ranges. Fuzzy membership functioned together with the rule base. It enabled to insert numbers with the least uncertainty\" [44]. [PERSON] et al. (2017) mentioned that, \"for this purpose, a model based on Fuzzy Axiomatic Design (FAD) is presented, wherein each evaluation has both optimistic and pessimistic fuzzy scores, as the fuzzy evaluations themselves have risks. To improve the accuracy of the presented method, a new concept called \"acceptable risk\" has been suggested\" [45]. [PERSON] et al. (2016) stated that \"to determine the weights of the indices, the fuzzy AHP is used. Subsequently, the alternatives' priorities are ranked according to the indices weighted with the VIKOR model\" [46]. According to [PERSON] (2011), \"The current contribution constitutes a review on the most representative genetic fuzzy systems relying on Mamdani-type fuzzy rule-based systems to obtain interpretable linguistic fuzzy models with a good accuracy\" [47].