# Model Predictive Control for Flexible Job Shop Scheduling in Industry 4.0 +
## 3 Model Generation
### Generation of the Discrete Time Petri Net Model
The product agents then solve their local optimization problem and compete with each other for the machines. This requires coordination between the agents in order to arrive at an optimal production schedule. Depending on the interaction mechanism, this leads to a distributed optimization problem that approximates the centralized optimization problems used in the MPC problems (6) and (16). The smaller individual optimization problems will be easier to solve, but the global optimal solution might only be obtained in special cases and potentially only after multiple iterations between the competing agents [63]. ## 7 Conclusions

Motivated by the changes through the implementation of Industry 4.0, we developed a novel scheduling scheme for flexible manufacturing system that is based on Model Predictive Control. The initial problem formulation of a job shop with multipurpose machines and sequence flexibility captures a wide class of relevant scheduling problems. This generality allows the direct application of the presented methods on a variety of important special cases. The initial scheduling problem is then automatically transformed into a Petri Net formulation, which we slightly adapted to better suit its purpose as basis for the feedback control mechanism that we use for scheduling. This transformation mechanism serves as a blueprint that can be adapted to similar problem descriptions in order to bring them into a form that allows the application of control theoretic tools. The production schedule is generated with MPC, where the cost of the production process is explicitly considered. The MPC formulation allows to rigorously prove the completion of the initial scheduling problem in closed loop based on verifiable conditions on the elements involved. Due to the feedback mechanism that is inherent in MPC, the scheduling scheme can react to changes in the problem setup and to exploit the flexibility of the manufacturing system. The presented approach can serve as a general guideline on how to automatically transform complex scheduling problems into a form in which MPC can be applied. The presented solution only considers a part of the properties that need to be taken into account in production planning and scheduling but it has the potential to be enhanced and adapted in order to include further criteria. Furthermore, in those cases, the possibility to provide proven guarantees based on the MPC problem formulation will be a significant advantage. Conceptualization, P.W. and F.A. ; methodology, P.W. and F.A. ; formal analysis, P.W. ; investigation, P.W. ; writing--original draft preparation, P.W. ; writing--review and editing, P.W. and F.A. ; visualization, P.W. ; supervision, F.A. ; project administration, F.A. ; funding acquisition, F.A. Both authors have read and agreed to the published version of the manuscript. This work was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant AL 316/12-2-279734922. Not applicable. The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. ## Appendix A Proof of Lemma 1

We have to proof that there is no steady state \((x^{s},u^{s})\) with \(u^{s}\
eq 0,u^{s}\in\mathbb{N}^{m}\). As no entry in \(u^{s}\) can be negative, the effect of a positive entry in \(u^{s}\) through \(B^{+}\) can only be compensated by the effect of a positive entry in \(u^{s}\) through \(B^{-}\), and vice versa. In the same sense the effect that an input arc \((P,T)\) has when the transition \(T\) fires in the firing count vector \(u^{s}\) can only be reversed by an output arc \((T^{\prime},P^{\prime})\) of a transition \(T^{\prime}\) in \(u^{s}\). Note that \(T=T^{\prime}\) would be possible.