## 2 Theoretical Fundamentals and State of the Art
As a digitalized supply chain in the metal processing industry usually consists of multiple smart factories, often owned by different companies and in general delocalized, smart factories include several CPPSs. A CPPS serves as an extension of a CPS, referring to a system capable of acquiring, storing, analyzing data in real-time using Internet technologies, and reintegrating information from the virtual to the physical world [48], partly dismantling the classical automation pyramid [49]. The reintegration of information involves human-machine interaction, which can be realized with Human Machine Interfaces (HMIs) as shown in, e.g., [2; 35]. Building upon the concept of the CPS, a CPPS is using automation technology to a greater extend [2; 48]. According to [48; 50; 51], the following characteristics must be fulfilled: a CPPS: (i) consists of superordinate systems within systems; (ii) consists of connected and cooperative elements acting situationally appropriate between all layers of a production environment; (iii) enhances real-time decision making. Figure 1: RAMI 4.0 serving as a structured approach for implementing I 4.0 by visualizing the horizontal and vertical integration along the Life Cycle & Value Stream [46]. A further extension of these criteria is proposed by [52], adding two additional conditions to be met, especially considering SMEs: (iv) a user-centered CPPS consists of HMIs that are tailored to the application and the respective end-user; (v) a user-centered CPPS for SMEs is resilient and has a short amortization time. Apparent assistance for the establishment of a CPPS of this kind can be the implementation of DTs, serving as a decision-making support system for the further optimization of the respective production process. According to [53], a common definition of a DT is a virtual representation of a physical product. This definition can be further refined according to [2; 53], depending on the field of application. In the metal industry, a DT can be seen as a partial or complete representation of a production chain, shifting the focus more onto production planning and optimization, or as a representation of one or more process steps in the production chain for the manufacturing of a semi-finished or finished product, whereby the focus shifts on the simulation [2]. To generate advantage of a DT, data from physical space and the information derived from it must be reintegrated from the digital back into physical space. This data transfer between a physical and virtual entity leads to three differentiations depending on its degree of automation [53]. The Digital Model (DM) includes data transfer between the physical and digital space without automated data transfer from the digital to the physical domain or vice versa [53]. A Digital Shadow (DS) consists of a unilateral automated data transfer, in the sense that data is automatically transferred from one domain to the other, while the reverse has to be executed manually [53]. The DT, on the other hand, includes an automated bilateral data transfer, resulting in an algorithm-based process adaption and can adapt the digital domain by utilizing near real-time process data [53]. The fundament of a DM, DS or DT can be a White Box Model (WBM), Black Box Model (BBM) or a mixture of both, defined as Grey Box Model (GBM) [2]. WBMs are based on real physical relationships build up upon known parameters and mathematical correlations. Therefore, the output and how output-related results are obtained are comprehensible [2; 54; 55; 56]. BBM are often used where a mathematical description based on real physical relationships is too complex [54; 55; 57] or not available in the required depth and time. As a result, its logic and working methods are not transparent in comparison to a WBM [54]. The output of a BBM has no real physical mechanisms compared to the WBM and is based on stochastic approaches and the correlation of data [2]. The GBM represents the combination of a WBM and a BBM, aiming to merge the benefits of both [2; 54].