# Regression Approach to a Novel Lateral Flatness Leveling System for Smart Manufacturing
## 1 Introduction
Although the larger plastic deformation caused by the intermeshing of the working rolls at the entry is beneficial for the complete elimination of the sheet metal coil's initial residual stress, the rolling residual stress that is induced in the sheet metal during the multi-roll process can inevitably increase at the same time. Therefore, the total combined residual stress of the sheet metal after leveling depends on the appropriate leveling parameters. [PERSON] and [PERSON] [9] linked both the target values of sheet flatness and residual stress distribution and developed a numerical model of a seven-roll leveler that is used to determine the roll positions, resulting in a flat sheet and a defined residual stress distribution. [PERSON]'s [10] simulated results presented here are for 3/8-inch-gauge material with a yield strength of 344 N/mm\({}^{2}\). The model predicts similar results for both different gauges and different yield strengths. [PERSON] and [PERSON] [11] further investigated the correlations between the incoming strip condition, the roll intermeshes, and the residual stresses, which are evaluated to achieve both a constant flatness as well as a distinct residual stress distribution after leveling. [PERSON] et al. [12] developed the application of the well-known laser ranging techniques in flatness inspection and construction and the testing of a flatness measurement and inspection system capable of acquiring the true shape of a steel strip in real-time, leading to calculating its flatness indexes so they can be used in the quality inspection of the steel products. [PERSON] et al. [13] found an inverse technique that could be used to obtain the residual stress profile and material constants. This technique provided the best fit in a finite element analysis of bend  
Figure 1: Common symptoms of sheet metal coils. ing with the experimentally derived moment curvature relation. [PERSON] [14] and [PERSON] [15] developed a numerical algorithm for analyzing the effect of the leveling conditions on the residual stress. With the reported methods, one was able to implement the effect of plastic deformation in leveling, tension, work roll bending, and the initial state of the sheet (residual stress and curl distribution). In recent years, machine learning for smart manufacturing has been a widely discussed topic in many different applications, such as image processing, medical application, energy management, and planning data-driven digital twins [16]. Here are several examples reported in the literature that the machine learning method can improve the model-based approach. [PERSON] et al. [17] identified that the machine learning method can be used for reliability analyses and decision support systems through the use of regression analysis to predict the interval generation framework. [PERSON] and [PERSON] [18] applied logistic regression to predict construction site accidents, which can be reduced through proactive steps using prediction models. [PERSON] et al. [19] used ordinal logistic regression to the automatic scoring system for the functional movement screen (FMS). [PERSON] et al. [20] proposed machine learning approaches to develop an energy prediction algorithm for accurately forecasting the day-ahead hourly energy consumption profile of a residential building while considering the occupancy rate. [PERSON] et al. [21] identified that automated spectral analysis of surface wave inversion can be enhanced with the use of machine learning's linear regression (LR) algorithms. [PERSON] et al. [22] utilized multiple regression models to estimate the power generation of a solar power plant with changing weather conditions. [PERSON] et al. [23] developed an automated optical inspection (AOI) system for O-ring inspections using regression analysis. [PERSON] et al. [24] proposed a novel multi-objective particle swarm optimization (MOPSO) algorithm employing the Gaussian process regression (GPR)-based machine learning (ML) method for multi-variable, multi-level optimization problems with multiple constraints.