# EPO Extension of Dispatching Rules to Minimize Effects of Time Uncertainty in Production Scheduling
## 1 Introduction
In addition to theoretical works, examples of using fuzzy logic to describe uncertainty in real production systems are presented. [PERSON] et al. [23] applied fuzzy numbers to plan and schedule under uncertain operation times in paints manufacturing, while [PERSON] et al. [24] applied fuzzy numbers to schedule the PCB (Printed Circuit Board) assembly facility. However, using fuzzy numbers to handle uncertainty in the production process requires the definition of the membership function, which is not always possible or practical; moreover, this function may change depending on the current state of the production system. For the purposes of scheduling in Industry 4.0, alternatives are proposed [25], for example, by using tolerances, but this does not make the schedule robust for larger variations in production process parameters. Therefore, the stochastic approach is becoming more and more popular for the actual planning of production systems. In the review of different methods for the flow-shop scheduling problem under uncertainties [26], the stochastic approach was used in 60% of the articles (fuzzy logic in 21%). Basically, stochastic methods can be divided into a non-simulation approach and a simulation-optimization approach. Non-simulation methods are usually based on Markov chain processes and the use of, e.g., dynamic programming. This approach was already proposed in 1968 by [PERSON] et al. [27] and evolved over the following decades [28; 29; 30]. This approach can also be used successfully in the context of scheduling in Industry 4.0, but requires the definition of some superior objective function, such as inventory optimization [31] or revenue optimization [32], which is not always the case. The simulation-optimization approach is also increasingly used in production scheduling [33; 34; 35]. However, the use of such an approach requires a lot of computing power and may be ineffective if frequent rescheduling is required. In this paper, we present a new approach to take into account the uncertainty of the job processing times. The proposed approach does not require the determination of the membership functions and guarantees that there is no need to recalculate the schedule as in the time-consuming simulation approach. The proposed approach is therefore a relatively simple but effective way to achieve the expected increase in the robustness of production schedules in the framework of Industry 4.0 and Digital Twins technology. Another advantage of this approach is that it does not increase the computational complexity of the scheduling algorithms. The remainder of this paper is organized as follows. The concept of the proposed EPO extension for dispatching rules and the principle of its operation are described in Section 3. We show how it can be used in dispatching rules for both forward and backward scheduling. Section 4 presents an example of the use of the EPO extension in a real production system, a large offset printing house characterized by a diverse range of products. Section 5 summarizes the obtained results and provides final conclusions.