# Predictive Scheduling with Markov Chains and ARIMA Models
## 2 Existing Work on Robust Production Scheduling
### Machine Failure as the Major Uncertainty Factor  
Although many uncertainty factors can be named, the failure of technological machinery is still considered to be the central problem in manufacturing. This disturbance is regarded to have the greatest impact on performed processes. Failure will not only halt the production but its consequences will linger throughout the remaining production process [5,7]. From the analyzed scientific papers dealing with the topic of machine failure in scheduling, it can be seen that the search for methods that will enable approaching the problem of machine failure and predicting its occurrence are very much in place. Developing effective prediction methods is extremely important from the perspective of robust scheduling [18,19,26]. To this end, a probability distribution is among the most widely used approaches in the field of failure analysis. Researchers employ typical distributions and their combinations. The failure description proposed by [PERSON] [27] applies a uniform distribution. A similar solution is proposed by [PERSON] and [PERSON] [28], who, however, assume that the probability of failure is constant. In contrast, in their description of production process disturbances, [PERSON] et al. [29] implement a normal distribution, while [PERSON] and [PERSON] [21] utilize an exponential distribution. The authors propose the use of interesting approaches, such as the methods based on combinations of various distributions. The latter is used by [PERSON] et al. [4], who combine normal, triangular and exponential distributions. Recently, researchers have also investigated the application of typical key performance indicators (KPIs) used in maintenance, e.g., MTTF (mean time to failure), MTBF (mean time between failures) and MTTR (mean time to repair). In their direct application of the indicators, [PERSON] [9] and [PERSON] [5] analyze specially prepared scenarios that assume a certain frequency of machine failure, i.e., high, medium or low, to study the consequences of machine downtime and propose solutions to absorb the emerging disruptions to the schedule. With respect to the indirect use of the indicators, [PERSON] et al. [30,31] propose the use of the aforementioned reliability indicators indirectly for the purpose of estimating Weibull distribution parameters, while [PERSON] and [PERSON] [32] in their study, couple the said indicators with statistical methods. These studies, however, suffer from the major drawback--the acquisition and use of the respective quantities is treated quite theoretically and lacks practical verification on real data of machine failure rates [9,31]. In addition to the methods referenced in the preceding paragraphs, a range of alternative failure prediction methods can be found in the literature. [PERSON] et al. [20] propose accumulating failures to a single occurrence, describing it by means of the MTTR parameter and their original indicator, MBL (machine breakdown level). In turn, [PERSON] and [PERSON] [33] determine failure rates from the analysis of machine load time distributions, and [PERSON] et al. in [34] use artificial neural networks for failure analysis. Although constituting an interesting and important voice in the robust production scheduling studies, these models are associated with certain limitations. Their verification is often carried out on test data, which may not be the most accurate representation of actual problems in manufacturing systems. Secondly, the questions arise as to insufficient argumentation regarding the selection of the solutions. Consequently, the key aspect of implementing historical data in studies of the failure rate of machines is omitted. The need to use real data on uncertainty factors is also emphasized by [PERSON] et al. [29] and [PERSON] et al. [35]. Only real knowledge on process disruptions can actually solve actual the problems that result from their occurrence.