## 3 The Mixing Problem and Notation  
In systems science, by definition, a system is a set of interacting units/components with interactions among themselves and other outside entities [34]. A mathematical model \begin{table}
\begin{tabular}{c c c} \hline \hline
**Manuscript** & **Contribution** & **Control Approach** \\ \hline [PERSON] et al., 2016, [24] & This paper addresses the problem of supply chain coordination with a robust schedule coordination approach, applying optimal control theory. & Optimal control \\ \hline [PERSON] et al., 2018, [25] & This research work extends a previous survey by identifying two new directions for control theory applications and digital technology use in control-theoretic models. & Optimal control \\ \hline [PERSON] et al., 2018, [26] & A survey on optimal control applications regarding scheduling in production, supply chain, and Industry 4.0 systems via deterministic maximum principle. & Optimal control \\ \hline [PERSON] et al., 2019, [27] & This paper presents a method to study the Stackelberg differential game between a manufacturer and a supplier. & Differential game \\ \hline [PERSON], 2019, [28] & This paper presents a differential game for a single manufacturer and a single retailer in a supply chain under a consignment contract. & Optimal control \\ \hline [PERSON] and [PERSON], 2019, [29] & The cooperative advertising problem is addressed, applying optimal control theory for a supply chain under a consignment contract in a competitive environment. & Optimal control \\ \hline [PERSON] et al., 2020, [30] & An optimal control model is presented for a class of low-carbon supply chain system. A linear differential equation describes the dynamics of emission reduction level. & Optimal control \\ \hline [PERSON] et al., 2021, [31] & This paper considers supply chains modeled by partial and ordinary differential equations. & Optimal control \\ \hline [PERSON] et al., 2021, [32] & In this paper, a fractional-order digital manufacturing supply chain system is proposed with feedback controllers to control the chaotic supply chain system. & Fractional order control \\ \hline [PERSON] and [PERSON], 2021, [33] & This paper shows how a deep reinforcement learning-based optimization algorithm can synchronize inbound and outbound flows to support a business operating in the stochastic environment. & Deep reinforcement learning \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relevant optimal control for supply chains; literature review from 2016 to 2021.  
is a mathematical description of the behavior of some real-life systems, such as physical, sociological, economics, etc. [35]. Production and supply chain dynamic modeling present diverse mathematical approaches at distinct scales [36]. In this paper, we are interested in modeling the SC as a mixing problem (MP) with the intent to present it as an analogy for capacity-inventory management, taking into account decision variables (the inventory level), with production rate and capacity levels as constant parameters (which are provided), and considering the conservation of mass from the mixing problem. The mixing problem technique is also known as compartmental analysis, which has been used and applied in several branches of engineering and economy. It is mandatory to consider the supply chain as a socioeconomical problem with roots in globalization, and as a synchronization between supply and demand. HVM has the characteristic of low variability and the use of compartmental analysis, and it is suitable considering the balance rate between influx and outflux. In the context of economics, the application of this technique has been used in macroeconomic problems. MPs, also known as \"compartment analysis\" in chemistry, consist of a mixture of two or more substances in order to determine some concentration of the resulting mixture [37]. MP mathematical modeling has been used in science and engineering applications such as biological systems [38] and biomedical engineering [39].