# EPO Extension of Dispatching Rules to Minimize Effects of Time Uncertainty in Production Scheduling
## 1 Introduction
Such an approach to scheduling may, however, result in larger inventories of finished products in the case of make-to-order production, capital being frozen in the produced but not yet delivered production, and the impossibility of planning new tasks that have appeared and have tight deadlines, without a significant reconstruction of the schedule. Backward scheduling works on a just-in-time basis; resources are assigned as late as possible to meet the due dates required by the customers. The main advantage of this approach is the reduction in stocks of finished products and work in progress, which results in the reduction in storage costs. It also results in a more rational use of capacity--machines are used when necessary. Consequently, it is easier to react to unforeseen events such as the emergence of new orders. The main disadvantage of this approach is the lack of any time buffers, which, in the event of unexpected situations  
Figure 1: General scheme of Forward (**top**) and Backward (**bottom**) scheduling. in the production process, such as machine failure or prolonged production, may result in the failure to meet the promised shipping dates. The problem of forward or backward production scheduling can be solved in many different ways. The most frequently used methods in APS (Advanced Planning and Scheduling) systems are heuristics using static or dynamic dispatching rules. The reflection of changes in the state of production can be realized in two essential ways: through reactive or proactive scheduling. Reactive scheduling requires dynamic rescheduling of jobs each time production parameters change. It is necessary to develop algorithms that try to ensure the minimum number of changes in the schedule; but in cases of complex dependencies between jobs and continuous machine occupancy, this is not always possible. Proactive scheduling is about developing schedules that try to resist changes in parameters by modeling the applicable uncertainty. [PERSON] [12] categorizes the uncertainty that affects the production process into two groups: environmental uncertainty (related to demand or supply) and system uncertainty (related to operation yield, production lead time, quality, or failure of production systems and changes to product structures). A more detailed classification of uncertainties in production systems can be found, for example, in [13]. Uncertainty in production scheduling can be taken into account in several ways--the most frequently used is the approach based on fuzzy numbers, stochastic optimization, or computer simulation. The first applications of fuzzy logic in production planning and scheduling concerned its use in expert decision support systems in the field of aggregate production planning. Such systems were introduced, among others, by [PERSON] [14], [PERSON], and [PERSON] [15]. Later, detailed scheduling systems with the use of fuzzy logic appeared. One of the first such systems, named OPAL, was proposed by [PERSON] et al. [16] and used 5 different selection criteria and 15 different static and dynamic dispatching rules. [PERSON] and [PERSON] [17] used fuzzy logic to create aggregated rules for job shop scheduling, achieving a compromise between several dispatching rules. A similar approach was also described by [PERSON] and [PERSON] [18]. Subsequently, fuzzy logic began to be applied not only to select priority rules but also to describe uncertain production parameters. [PERSON] and [PERSON] [19] proposed a sequencing model with processing times described as trapezoidal fuzzy numbers. [PERSON] et al. [20] presented a fuzzy Longest Processing Time (LTP) algorithm for scheduling, in which the processing time of each job was described by the discrete membership function. Execution times were similarly described in [21], but this time for the constraint satisfaction approach. Later, [PERSON] et al. [22] also pointed out that uncertainty may also apply to constraints to make them more flexible.