# Predictive Scheduling with Markov Chains and ARIMA Models
## 1 Introduction  
While conducting their activities, manufacturing enterprises establish a range of various goals. Certainly, one of the common strategic business objectives is to strengthen the market position. An enterprise that aims to broaden the group of clients, as well as foster the already existing business relations, must first and foremost be reliable and deliver quality goods within contractual deadlines [1; 2]. Therefore, proper planning of works becomes central to sound execution of production processes. Production scheduling is the solution that can boost the capacity of manufacturers, hence there are numerous scientific publications in the field [3]. Researchers are still taking active efforts to optimise the effectiveness of production jobs scheduling in order to streamline the production planning process [2; 4].  
Unfortunately, most of the proposed solutions display numerous limitations [5]. It is common practice that the job scheduling algorithms build schedules for idealized production environments, i.e., assuming a static and stable production flow [5; 6]. Thereby, a number of disruptive factors are excluded, which would bring the production to a halt in case they occur [7; 8]. As a consequence, contractual deadlines would be missed, penalties would be imposed and the manufacturer's credibility would diminish. Among the various uncertainty factors, we can highlight the following [5; 9]:  
* disruptions of resource availability (machine or robot failure)* disruptions of orders (placement of new orders)
* disruptions of processes (material shortage, poor product quality)
* disruptions associated with misestimation of the ongoing process parameters (incorrect estimation of operation times)
* disruptions related to the change in the duration of the operation (employee absence or malaise, shorter or extended operation times)  
Scheduling production in real manufacturing systems cannot afford to pretend to be disruption-free. It is, therefore, of the essence that scheduling endeavors should consider production problems under uncertainty, which is capable of having a colossal effect on the timeliness of production [10,11]. Given that the more the process changes, the greater its disorganization, the scientific literature in the field of scheduling has recently turned towards robust scheduling [5,9].  
The predictive scheduling method proposed in this work employs Markov chains and ARIMA (auto-regressive integrated moving average) models whose combination enables determining the values of the machine failure parameters (time to failure and repair time of the machine). In the next step time buffers are directly integrated into the scheduling process and determine the completion time of the production, which corresponds to the delivery date agreed with the customer.  
Section 2 summarizes the essential information regarding robust production scheduling and reviews existing literature. The new methodology for scheduling under machine failure and failure prediction is described in Section 3, and the proposed solutions and results are discussed in the subsequent section. Conclusions and plans for further research work are presented in the last section of the work.