# Model Predictive Control for Flexible Job Shop Scheduling in Industry 4.0 +
The solution of the proposed scheduling problem is done on the basis of a Petri Net (PN) model, which is generated automatically from the initial description of the scheduling problem. The generation algorithms exploit the modular formulation of the initial scheduling problem, where the manufacturing units and the production orders can be considered as separate modules, and transforms it into a well-structured PN. The PN offers an algebraic representation that is used to determine the schedule for the FMS. The result of the scheduling scheme are manufacturing decisions that are intended to be directly applied to the system in real-time. In order to exploit the flexibility of the FMS, it is important to keep the scheduling scheme flexible as well. Therefore, it is formulated in the form of a feedback control law, more precisely in the form of Model Predictive Control (MPC) [14]. MPC has the advantage that an optimization problem, in which the economic objective of the FMS can be considered, is solved while computing the scheduling inputs. The feedback mechanism of the MPC allows to immediately react to changes in the manufacturing system and to adjust its behavior accordingly. The remainder of the paper is structured as follows. The description that we use for the scheduling problem will be further described in Section 2. It is kept in a rather general way such that it can be used for various specific cases. It is a generalization of the widely discussed job shop problem (JS) [15] and allows more general production plans with respect to the JS. This general form of a scheduling problem has, for example, practical applications in printing shops [16], and it can also be used to describe several special cases and practical applications thereof. A detailed description of the presented scheduling problem together with a classification with respect to a common classification scheme and a comparison with more common scheduling problems in literature can be found in Section 2. In this regard, we especially set the focus on the flexibility of the employed problem description. In Section 3, we describe the automatic generation of a PN for the considered class of FMSs, which is used as mathematical basis to solve the scheduling problem. In this context, we slightly extend the classical notion of a PN in order to distinguish between autonomously running production processes and conscious production decisions. The generation process is discussed and the resulting PN is analyzed with respect to its relevant properties for the scheduling of the FMS. Finally, two MPC schemes are introduced in Section 4, which determine the optimal scheduling decisions at the current time step with respect to a given cost function. We relate the presented approach to common control strategies for manufacturing and for PNs from literature. Most importantly, we provide proven guarantees of the MPC schemes that assure the completion of the scheduling problem based on a few natural assumptions. In Section 5, we apply one of the presented MPC schemes in the simulation of two scheduling problems from literature. The results show that decent closed loop performance is achieved. Section 6 summarizes and discusses the results with respect to the initially formulated objectives. It is explained how the results from Section 4 can be used in order to implement a suitable MPC formulation for a given manufacturing problem. The flexibility of the proposed solution is laid out and possible changes in the problem description are analyzed with respect to their effects on the MPC schemes. Further research opportunities are identified that enhance the presented methods. In Section 7, we draw a brief conclusion of our work. The paper at hand is an extension of our conference papers [17; 18] and repeats previous results in some indicated it paragraphs. Besides many additional explanations and bringing those two related works in their mutual context, our previous work is extended in various ways.