## 2 Theoretical Background and Problem Statement  
As mentioned in the previous Section, the upcoming interest in the hybrid MPC architecture, where the WIP level of a supervised production line is a control lever to be dynamically sized, requires the development of a model capable of dynamically evaluating performances in terms of throughput and cycle time. Hence, as outlined in the Introduction section, we will focus on a flow-shop line controlled by a CONWIP control mechanism whose stations are formed by single machines with a single routing through which all processed components flow. In addition, the components (i.e., jobs or WIP) can be considered units representative of processing batches that can be reasonably measured (i.e., number of jobs or parts being processed in the line). We will therefore focus on a single-product, single-routing production line in which the only variability entering in the system is related to job processing times (Figure 2). In this context, our research aims to answer the following research questions (RQs):  
* How could a model be developed to accurately predict the performance of a CONWIP-controlled flow-shop system with generalized processing time variability? * How can the performance of flow-shop balanced lines be effectively normalized to facilitate the training of machine learning algorithms in a single instance? * How will the proposed model be expanded to accommodate unbalanced production line scenarios? For the purpose of generalizing the generation of processing times, a gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\) was chosen. The choice of the gamma distribution is not accidental, as it is a distribution in which the level of variability can be controlled by modulating \(\alpha\). It is, in fact, a continuous probability distribution that includes the exponential distribution as a special case (when \(\alpha=1\)), allowing to model a variability greater than the exponential one when \(\alpha<1\) and less than the exponential one when \(\alpha>1\). Thanks to the flexibility of the gamma distribution, high variability scenarios (i.e., the ones related to customized production) as well as low variability ones (i.e., the ones related to standardized production) will be easily modeled. From a modeling point of view, the CONWIP system can be conceptually reduced to a closed-loop system in which jobs never leave the production system itself, but circulate in it indefinitely in time. While the job entering the production line and the job leaving it are obviously distinct, what remains cyclical is the WIP control card that, once released, returns at the beginning of the line allowing a new job to enter [33]. Nevertheless, for this control to be consistent, mathematical models able to relate throughput (TH) and cycle time (CT) (i.e., the dependent variables) to the WIP (i.e., the independent one) are needed. In this regard, [PERSON] and [PERSON] propose three models that are valid under well-defined production conditions: best case (i.e., the best possible operating condition), worst case (i.e., the worst possible operating condition), and practical worst case (i.e., where the operating conditions admit a wide variability). These three scenarios, whose formulas are summarized in Table 1, are built based on the following parameters:  
* \(T_{0}\), as \"the raw processing time of the line (the sum of the average processing times of each workstation at the steady state)\";
* \(r_{b}\), as \"the bottleneck rate of the line at the steady state\";
* \(W_{0}\), as \"the critical WIP of the line (i.e., the WIP level for which the line with a defined \(T_{0}\) and \(r_{b}\) achieves maximum throughput with a minimum cycle time if no variability exists in the system)\". While the best case represents the maximum performance (i.e., absence of variability and balanced line) that can be obtained with a particular WIP value in the production system, the worst case represents its minimum possible performance.