Meta-model based optimization of hot rolling processes in the metal industry

CIplus Band 6/2017 Meta-model based optimization of hot rolling processes in the metal industry Christian Jung, Martin Zaefferer, Thomas Bartz-Beielstein, Günter Rudolph The final publication is available at Springer via http://dx.doi.org/10.1007/s00170-016-9386-6 Noname manuscript No. (will be inserted by the editor) Meta-model based optimization of hot rolling processes in the metal industry Christian Jung · Martin Zaefferer · Thomas BartzBeielstein · G¨unter Rudolph the date of receipt and acceptance should be inserted later Abstract To maximize the throughput of a hot rolling mill, the number of passes has to be reduced. This can be achieved by maximizing the thickness reduction in each pass. For this purpose, exact predictions of roll force and torque are required. Hence, the predictive models that describe the physical behavior of the product have to be accurate and cover a wide range of different materials. Due to market requirements a lot of new materials are tested and rolled. If these materials are chosen to be rolled more often, a suitable flow curve has to be established. It is not reasonable to determine those flow curves in laboratory, because of costs and time. A strong demand for quick parameter determination and the optimization of flow curve parameter with minimum costs is the logical consequence. Therefore parameter estimation and the optimization with real data, which were collected during previous runs, is a promising idea. Producers benefit from this data-driven approach and receive a huge gain in flexibility when rolling new materials, optimizing current production, and increasing quality. This concept would also allow to optimize flow curve parameters, which have already been treated by standard methods. In this article, a new data-driven apFaculty for Computer and Engineering Sciences Cologne University of Applied Sciences, 51643 Gummersbach, Germany [email protected] Faculty for Computational Intelligence TU Dortmund University, 44227 Dortmund, Germany [email protected] proach for predicting the physical behavior of the product and setting important parameters is presented. We demonstrate how the prediction quality of the roll force and roll torque can be optimized sustainably. This offers the opportunity to continuously increase the workload in each pass to the theoretical maximum while product quality and process stability can also be improved. Keywords flowcurve ·Kriging ·meta-model ·metal · hot rolling 1 Introduction The complex process of hot rolling requires very accurate physical models. Several different physical models are used in hot rolling mills. These models include the slab or ingot heating at the furnace, where the ther-5 mal behavior is modeled, as well as the rolling process itself. The task of the different models is to predict the thermal and physical behavior of the product and set important parameters for achieving maximum product throughput while increasing the quality of the10 final product. There are several process and product parameters which play important roles. They are independent of the plant type and applicable to steel and aluminum hot mills. Some of them, for example plant geometries or drive parameters, remain constant during15 rolling. Other parameters may vary during rolling but are not dominated by the material, e.g., the maximum possible thickness reduction depends mainly on the actual thickness and the work rolls currently installed. The maximum redcution is of course also dependent20 from the friction between the rolls and the material but this effect is not as huge as the geometrical limitations. Parameters, which depend on the material of the product, are usually hard to optimize, because (i) the number of different materials is steadily increasing due25 to market requirements and (ii) measurements of the process are only indirectly correlated to the material. The most important parameters for the prediction of the roll force are the flow curve parameters of each material. A more efficient method for optimizing these30 parameters is necessary to increase the flexibility and reduce the cost of the rolling process. A recent approach uses artificial intelligent techniques for optimization of shape rolling sequences [17]. Especially in the field of cold rolling, several methods35 for simulation and optimization were published [18, 24]. There were also studies, which were based on finite element methods [25]. In this paper we propose metamodel based optimization strategies for the task of hot rolling mill flow curve parameter optimization. Meta40 models, also referred to as surrogate models, simplify 2 Christian Jung et al. the simulation optimization as the run times are generally much shorter than the original function evaluations [1, 16] and are a proven strategy for global optimization [15]. The results are compared to classical optimization45 strategies. Hence, this paper addresses the following research questions: (R-1) Can the rolling process benefit from meta-model based optimization? (R-2) How timeand cost efficient is this approach in50 comparison to the established industry procedures? This paper is structured as follows. Section 2 introduces technical terms and fundamental principles of hot rolling and the related parameters. It describes the cur-55 rent state-of-the-art approach in industry. Problems related to the rolling process are described in Sec. 3. Our methodology is detailed and compared to existing approaches in Sec. 4. The experimental setup is described in Sec. 5. Results are presented in Sec. 6. The paper60 concludes with a discussion in Sec. 7. 2 Hot Rolling 2.1 Fundamentals and Technical Terms Before describing some of the main aspects, the fundamental technical terminology will be introduced. Fig-65 ure 1 shows a common hot strip mill for steel. The process in general is very similar to an aluminum mill. The work flow of the process is from the left side to the right side. The main components are a reheating furnace, a reversing roughing mill, a continuous finishing70 mill, a cooling line, and a downcoiler. The coilbox between roughing mill and finishing mill is more or less optional. It is used to achieve better temperature profiles and allows a more compact rolling mill. The process starts with the charging of the furnace.75 Here, slabs which are usually at room temperature are charged and reheated to temperatures around 1200 deg C for steel mills and 500 deg C for aluminum mills. Fig. 1 Hot Rolling mill for steel with coilbox. The roughing mill consists of one horizontal stand with four rolls, the socalled quarto stand and an optional vertical rolling stand with two rolls, the so-called edger (not shown in the figure). The slab geometries may vary. Usually, they have an input thickness,hinit, between 200 mm and 300 mm,80 width of 600 mm to 2500 mm and length between 3 m and 10 m for steel production. For aluminum, the thickness after discharge is commonly around 600 mm because the temperature loss of aluminum during rolling is much less than for steel. After discharging, the first ma-85 jor process is to reduce the thickness of the slab by 30 mm to 40 mm. This is done in a so-called roughing mill (RM). The roughing mill in a conventional hot strip mill consists of one horizontal stand with four rolls, which is then called quarto stand and an optional vertical rolling90 stand with two rolls, the so-called edger (not shown in the figure). The edger is optional and has the task to reduce the width of the slab and to improve the shape of the slab especially at both ends. The reduction from the initial thickness hinit down95 to the target thickness htarget is done in several deformation steps which are called passes. In each of these deformation steps, the thickness of the slab will be continuously reduced until the target thickness is attained. The reduction has to be split to several passes because100 the feasible reduction in one pass is limited. The deformation in the roughing mill can be done in both operating directions. Hence, each pass changes the direction of movement of the slab. The total number of passes has to be odd, since the slab has to be moved to105 the next process step. After rolling in the roughing mill, the slab is transferred in the direction of the finishing mill. If a coilbox is used, the material is coiled first and then directly uncoiled to start the rolling in the finishing mill. Here, the product is rolled in several stands to110 the final thickness (specified by by the customer) and is directly cooled afterwards. The additional cooling line is only used for steel mills. Finally, the product is coiled in the downcoiler. One important quality criterion for the final prod-115 uct is the deviation of the actual thickness from the target thickness htarget. The thickness deviation at the head of the product is typically a result of the roll force prediction accuracy of the physical model. The head of the product is the first part which encounters the120 deformation. In finishing mills we will most often find thickness gauges after the last stand. These are used to control the thickness once the head passes the gauge so the roll force deviation would mainly be responsible for the head thickness. Additionally, the roll force is used125 as reference for the bending and for profile control and is therefore also very important for the product profile. Inline control of the thickness is usually not installed in other mill types such as plate mills. This is because the total length of the product is much smaller than130 in mills with coil production, where the total length Meta-model based optimization of hot rolling processes in the metal industry 3 might accumulate to more than 1000m. In that case, a feedback of the measured thickness during rolling can be used to adjust the roll gap and therefore the final thickness. For mill types without inline control, it is135 crucial to improve the roll force prediction in order to minimize the thickness deviation. Another aspect is the throughput of the mill, which should always be maximized. To maximize the throughput, the number of deformation steps, i.e. the number140 of passes, should be minimized without violating other constraints. Two of the most limiting parameters when trying to increase the reduction in each pass are (i) the maximum roll force capability and (ii) the maximum roll torque capability of the stand. Therefore, it is es-145 sential to have good models for the prediction of roll force and torque. As mentioned above, deviations in roll force prediction will also affect the thickness and therefore the quality of the product. Because current market require-150 ments cover a very wide range of materials and geometries it is important to increase the model quality for the roll force prediction for all products which may be rolled on these mills. Ideally, the roll force prediction is completely independent of the geometry and other155 parameters and will only depend on the quality of the material parameters. Each material is usually classified according to its chemistry. A material database stores mechanical and thermophysical parameters for the description of the different properties of each class. These160 parameters are used for the prediction of behavior during the deformation process and are therefore of major importance to the rolling process. 2.2 Flow Curve and Roll Force Model The flow curve parameters are most relevant for the roll force model. The flow curve expresses the material resistance during plastic deformation in dependence on the chemistry, the temperature, the deformation and the deformation rate. The deformation ϕ, also called effective, logarithmic deformation or true strain is expressed by: ϕ= ln h0 h1 . Here, h0is the input thickness and h1is the output thickness of the pass. The first flow curve formulas were developed by Geleji and Ekelund around 1950. These formulas were only linearly dependent on the temperature and only valid for standard low carbon steel [10]. Afterwards several other formulas were developed with polynomial components and exponential terms which also took into account the deformation and deformation rate. While the first formulas were only valid for some low alloyed carbon steels, Hajduk developed formulas which were also valid for some medium and high alloyed carbon steels [9],[8]. A good overview and description of the different flow curves can be found in [10, 12, 26]. These formulas model the deformation resistance,kf, in dependence on the deformation ϕ, the deformation rate, ˙ϕ, and the temperature,ϑ. The deformation resistance or flow stress expresses the stress which is needed to sustain a plastical deformation. In general, the deformation rate can be formulated as: kf=AKϕKϑK˙ϕ, where A∈R+is a constant factor and the terms K(·) 165 represent functions of the corresponding variables ϕ, ˙ϕ, and ϑ, respectively. The most common model for these formulas was developed by Hensel and Spittel [10]. It was extended at the University of Freiberg. Thus, these extensions are called Freiberger Approach. The170 extended versions of this flow curve model gives a better approximation of the flow stress within high deformation grades. Some of the available flow curve models, which are typically used in process models for hot rolling, are presented in Table 1. Their corresponding175 equations read as follows. kf=kf,0A0A1em1ϑA2ϕm2em4 ϕA3˙ϕm3(1) kf=A0em1ϑϕm2em4 ϕ˙ϕm3(2) kf=A0em1ϑϕm2em4 ϕ(1 + ϕ)m5ϑem7ϕ˙ϕm8ϑ(3) kf=kf,0A1em1ϑA2ϕm2A3˙ϕm3(4) The multipliers Ai(i= 0,1,2,3) can be reduced to one parameter, A. The parameters mj(j= 1,2,...,8) are defining the exponential behavior of the materials in dependence of the temperature ϑ, the deformation180 ϕ, and the deformation rate ˙ϕ. The parameters ϑ,ϕ, and ˙ϕare defining the working point in each deformation. The value kf,0used in the equations (1) and (4) is the basic deformation and is calculated by empirical formulations based on the chemistry. Each material is185 classified according to its chemistry and gets its own parameter set Mwith parameter values m1to m8and A0to A3, respectively. Usually, there is one parameterset Mfor each material, which is then valid for a specific equation only. Besides the parameter values for190 the models also the valid region of these parameters is stored. Summarizing, the parameters kf, ϕ, ˙ϕ, and ϑ, the multiplicators Ai, mj, (i= 0,1,2,3; j= 1,...,8), and related functions kf,0, Kϕ, K ˙ϕ, Kϑ, are used. Nowadays, hundreds of different materials are known.195 The parameter kfis almost linearly correlated with the 4 Christian Jung et al. Table 1 Overview of typical equations for the regression of the material flow stress. The multipliers Aiare reduced to one parameter, A. Parameters mjare defining the exponential behavior of the materials in dependence of the temperature ϑ, the deformation ϕor the deformation rate ˙ϕwhich defines the working point. Parameter kf,0is defined by a simple equation based on the chemistry. Entries in the column ”Equation” refers to the equations defined on p. 3. Eq. Name #Params Parameter List (1) Freiberg 1 5 A, m1, m2, m3, m4 (2) Freiberg 4 5 A, m1, m2, m3, m4 (3) Freiberg 8 7 A, m1, m2, m4, m5, m7, m8 (4) Hensel Spittel 4A, m1, m2, m3 roll force and roll torque. Thus, the model prediction quality and herewith the process stability and product quality are correlated to the parameter-set M. It is therefore important to optimize those parameters in or-200 der to increase the model quality and to ensure a stable process with maximum throughput and product quality. A standard procedure for obtaining these parameters is the measurement of the deformation resistance in a laboratory. Those measurements can be used for a205 regression onto one of the formulas shown in Equations (1) to (4). Of course, other formulas exist, and might be used for regression. Especially when trying to model the deformation of micro alloyed or high alloyed steel or when complex materials with phase transformations210 should be described these other models might be more suitable. 2.3 Description of the analytical model The analytical model used for the calculation of the roll force is based on the elementary rolling theory [12, 27].215 Some of the limitations of that theory are compensated with correction functions. For example, one of the requirements for the elementary theory is that during each pass there is a complete plastic deformation of the whole material. For products with thickness above 500220 mm this is clearly not given. Therefore, a compensation curve, which is empirically determined, is applied. For the calculation of the roll force in each pass the deformation zone is divided into single stripe-like elements and the force balance for each stripe is calculated. The225 solution yields to the basic differential equation of the plastically deformation theory which was developed by Karman in 1925. When calculating the roll force for one pass, the flow resistance has to be considered. This flow resistance kfis material dependent and is influ-230 enced by the parameter values of ϕ, ˙ϕand ϑ. Furthermore, the forces induce temperature into the material so the calculation of the next pass depends on the previous passes. Optimizing the flow curve by analyzing the rolling results is not sufficient. If the parameter of235 the flow curve changes, the whole process has to be simulated again and then the calculated roll forces based on the new flow curve can be compared with the original feedback, i.e., measurements of roll force, torque, temperature, and speed. Additionally, it is also not suf-240 ficient to optimize the result of a single product because the parameters ϕ, ˙ϕand ϑmay not vary enough to achieve stable results. Therefore it is preferable to consider a campaign with a wide variation of product geometries, temperature ranges, and deformations.245 2.4 Standards in Industry Currently, the available concepts for the optimization of flow curve parameters are mostly dealing with determination of those parameters in laboratory rather than optimizing those parameters with real process data.250 Traditionally, the parameters are measured with small samples of one piece in laboratory and are then generalized for every material which is close to the sample in terms of material composition. Some companies are modifying the flow curve parameters with linear mod-255 els. That is, they are determining the prediction accuracy of their model and are varying some of the influence parameters. Most of the research in this area is on the development of suitable flow curve equations especially for high and micro alloyed steel [11, 19, 28] rather260 than using a data driven approach for the optimization of those parameters. 3 Problem Description According to our best knowledge, flow curve parameter determination in laboratory as described in Sec. 2 will265 require several weeks and costs several thousand Euro for the required materials. Sometimes, this is not affordable and therefore not a suitable way to determine those parameters. Hence, we are looking for a cheaper approach to parameter estimation.270 Due to their highly nonlinear behavior, the flow curve equations cannot be solved directly. Furthermore, a different roll force would result in a different temperature balance of the product and thus the temperature in the next pass differs from the original calculation.275 Because these aspects cannot be neglected, we have to simulate a whole scenario when testing new parameter sets for the flow curve of a specified material. The calculations of the roll forces and roll torques within this simulation are afterwards compared with the measure-280 ments to get a quality criterion for the new parameter Meta-model based optimization of hot rolling processes in the metal industry 5 set. The detailed description of the simulation scenario is presented in Chapter 4. It is important that the simulation scenario behaves in the same way as the online process. Therefore, the285 simulation uses feedback of the measured speed, the reduction, and temperature to calculate the new settings. This enables the simulation to achieve the same working point as in the online process. Another problem can be the amount of data. The simulation of a290 whole batch where only one material group was rolled consists of thousands of different deformation steps and will therefore be highly expensive in terms of simulation time. Several optimization algorithms require bound constraints of the optimized parameters. In our case,295 parameters Aand miare dependent on each other. The only limitation which can be set is a plausible region for the resulting value kfof the basic deformation. In hot mills, the maximum basic deformation value for kfis usually below 300 N mm2, but always positive. Then, for a300 given maximum working range of the deformation, deformation rate, and temperature, the feasibility of the parameter set can be tested. Due to the fact that every company has usually its own classification system it might be that materials305 which are grouped together in one company are separated in other companies. In this case, the optimization, which has been done in the first company cannot be directly used for other companies and has to be renewed every time.310 Summarizing, it is desirable to optimize the process in order to –reliably estimate valid flow curves, –reduce lab costs, –save time,315 –determine parameters in their working environments, and –make the process more flexible and adapt to new (material) changes quickly. 4 Methodology320 4.1 Simulation environment The environment of the online process is shown in Fig. 2. First of all the model gets information about the product which includes initial geometry data, discharging temperature data, and information about the chemistry325 of the product. The discharge temperature is an initial temperature field for the product. One part of the rolling model calculates the temperatures losses during the whole process. Finally, the important parameter ϑ is a result of the temperature losses from discharge to330 that point of the process. Furthermore, target data is Fig. 2 Model environment in the real-world process for each pass: The product and customer data such as material description, initial and target geometry (1) are combined with operator data (2) and are fed to the model which calculates all required settings (3) for the next rolling pass (setup for next pass). After rolling of this pass the model gets feedback (4) of the just rolled pass and combines this information for the recalculation of the previous pass and for the next calculations (2-4). Additionally, changes from the operator for the next pass are send to the model. The product and customer data are only product and not pass dependent and may only be send once. To enable a simulation of this process every inand output of the model is stored in a database. Fig. 3 Model environment for the offline simulation. The data which has been collected in the real word process is send to the model which calculates a new setup for the next pass. This setup may be different from the original one. But because of the fact that we also store the feedback from the drives and gauges the model will receive also the original setting and recalculates the pass as it really has been rolled. also coming from the customer. Both data can be seen as constant and are denoted with Nconst , i.e., the operator cannot change them as they are part of the production planning system which handles the orders of the335 mill owner. Afterwards, the Nconst is used to calculate the first settings for the mill and this result is shown to the operator. The setting consists of roll gap settings, speed settings, geometry, temperature and time calculations. With this data the first planned settings NSet,340 which also include the parameter ϕ, ϑ and ˙ϕ, are calculated and are shown to the operator. The operator can interfere and modify the way this product is rolled. This is referred to as rolling strategy. This rolling strategy defines how a product is rolled which includes rolling345 speed, number of passes, deformations, speed settings, 6 Christian Jung et al. possible rolling breaks and much more. These strategies may also be specific constraints like absolute reduction, deformation ϕ, force, torque, but also other restrictions to the process like drive limitations.350 If the operator is satisfied with the settings NSet calculated by the model, the settings are sent to the plant where the first pass is rolled. After this pass, all measurements collected during rolling which include forces, torques, speeds, temperatures, gap settings, de-355 lay times, and several more data, are sent to the model and the database but are also shown to the operator. Now, the operator and also the model can adapt the settings for the next passes and react to any unexpected behavior of the mill. Usually, no big changes are made360 by the operator and the rolling of the further passes is started directly. Again, for each pass the settings are sent to the plant and the feedback of measurements is received from the plant. For analysis and offline simulations all inputs and outputs are stored in a database.365 This database is the basis of the offline simulation shown in Figure 3. Here, we feed the same model which was used in the online process with the data stored during the real-world process. Therefore we can guarantee that the model reacts in the same way as it would react370 in the rolling process. The data, which was coming from the operator and the geometry data are taken from the database. Hence, the model will not recognize if it is used for an offline calculation or for an online scenario. Although it may calculate different settings for the pro-375 cess, it will receive the original feedback from the plant and calculate everything based on the original settings. For a product with 19 passes the model is triggered 20 times. The first trigger creates the initial setup and all other triggers are feedbacks for the 19 passes with380 which the original products were rolled. The only difference are the parameters used for the calculation of the roll force. Therefore, we have a calculated and measured value for the force and torque for each product and pass for every run. After each simulation run, the parame-385 ters may be changed and results of different parameters may be compared. This enables the optimization of the flow curve parameters. Minimization of the Root Mean Square Error (RMSE) of the predicted roll force is the optimization objective.390 4.2 Surrogate Modeling If the simulation runs or the original problem in general are very expensive in terms of evaluation it would be very time consuming to perform parameter optimization on those original scenarios. Therefore, we use a395 surrogate-model based optimization approach. Surrogate models are supposed to replace the original, expensive simulation model and are expected to be cheaper to calculate. The analytical models, as introduced in Sec. 2.2,400 are cheaper to evaluate in comparison to the finite element method models. In this paper, the term surrogate model is used to describe data driven models, which are built from an analytical model. Therefore, the simulation runs on the analytical model with a specified set of405 products is the expensive model. The products which are simulated usually belong to the same material group and have previously been rolled in a series on a real rolling mill. Each product is calculated as it would be done during rolling. Therefore, time delays which oc-410 curred during the real production are also taken into account. Our data is sourced from a reversing aluminum hot mill, which has typically around 19 passes. Hence, we have more than 20 calculations for each product be-415 cause the model is triggered after finishing each pass. The data of each pass is sent to the model, which may react on unpredicted circumstances. In general, data-driven surrogate models can be any kind of models, e.g., artificial neural networks, linear420 models, Kriging, random forest and others. A detailed overview of surrogate model based numerical optimization is presented by Jin [13] and Jones [15]. One framework for surrogate-model based optimization is sequential parameter optimization (SPO) [3].425 SPO combines methods from classical DoE and modern Design and Analysis of Computer Experiments (DACE) [2, 4] based on Kriging models. Algorithm 1 presents the pseudo code of SPO, as adapted for the application of hot mill parameter opti-430 mization. Note, that we will use the notation x(i), y(i) for the data from the i-th pass which is passed to the surrogate model. During the first stage of experimentation, SPO explores the search space of the optimization problem A, which is treated as a black box. A set of in-435 put design points xis passed to A. Usually these are created by a space filling design, e.g. Latin hypercube sampling. Each call of the objective function produces some output yregarding its performance. SPO now tries to determine a functional relation-440 ship between xand y.SPO thus uses a model Y(x) as surrogate for the hot mill simulation model A. As mentioned above, the chosen model type is Kriging Kriging is frequently used for surrogate-model based optimization, because it provides a powerful and flexible predic-445 tor. It also provides an estimate of the variance or error of each prediction. The observations are interpreted as realizations of a stochastic process. A gaussian kernel is used to model the correlation between observations [23]. Meta-model based optimization of hot rolling processes in the metal industry 7 Algorithm 1: SPO-based hot mill simulation tuning. // phase 1, collect initial knowledge about the optimized process: 1let Abe the hot mill simulation model we want to tune; 2generate an initial design DES = {x(1),...,x(n)}of n parameter vectors; 3let k=k0be the initial number of replications for determining estimated responses; 4foreach x ∈DES do 5run Awith xto determine the estimated response yof x; // phase 2, building, using and improving a surrogate model: 6while stop criteria not reached do 7build surrogate model Y(x) based on DES and {y(1),...,y(|DES|)}; 8optimize the model w.r.t some cost function and constraints, thus produce a set DES’ of dnew parameter vectors ; 9run Awith each x∈DES’ to determine the response; 10 extend the design by DES = DES ∪DES’; // phase 3, final exploitation and fine tuning: 11 use local optimizer for the best pparameter sets x1...p ∈DES, without constraints In the sequential improvement loop SPO optimizes450 the surrogate model Y(x) over the considered space of input variables by means of a cost function. Once the new set of design points DES’ has been selected, the required evaluations of DES’ are performed. Based on DES’, the surrogate model Y(x) is updated.455 In step 8 of Algorithm 1, a search on the surrogate model is performed. Here, the constraints of the mill parameterization problem have to be considered. As the constraints are not expensive to evaluate, they are evaluated together with the surrogate model itself.460 For the inequality constrained optimization, we use the popular method developed by Powell [21, 22], which does not require any derivatives of the objective function to be available. During this optimization step 8, the next point xto evaluate in the sequential loop of465 SPO is determined. For expensive, global, black-box optimization Jones [14] introduced efficient global optimization (EGO). EGO exploits the information given from a Kriging model, i.e., the predicted mean and variance, to compute the expected improvement (EI) of a470 given solution. EI can hence be used as a cost function during step 8, as an alternative to the predicted value of the Kriging model. In step 11, the well known downhill simplex algorithm introduced by Nelder and Mead [20] is used to im-475 prove the best found results by a local optimization procedure. We choose the downhill simplex implementation in the nloptr R package. During local refinement, constraints are disregarded because they no longer play a role in the region of good solutions.480 5 Experimental Setup In our case, the parameter optimization was based on Equation (1). The feasible range was 0 ≤kf≤300. That is, solutions that result into negative kfvalues or kfvalues larger than 300 are considered to be infeasible. The usual working point for our test data was in the following range: 0≤ϕ≤0.5, 0 ≤˙ϕ≤600, 500 ≤ϑ≤600. With that said, the optimization problem to be solved in this study is defined as follows: –Parameters to be changed are the flow curve parameter vector mand the consolidated parameter Aof485 the flow curve. –The deviation of simulated roll force from the measured roll force is minimized. –Computational constraints: The evaluations of the objective function is expensive. (see Section 4.2)490 The parameters which represent the search space and were subject to optimization in this study are summarized in Table 2. To evaluate the success of the optimization, the resulting parameter set is compared to a well-known, es-495 tablished parameter set used in practice so far. This old parameter set has been determined by experts according to best knowledge from literature on similar materials. We have chosen the SPO toolbox (SPOT) to con-500 duct the experiments [5]. SPOT itself has parameters as well that are set according to the authors experience: –The chosen surrogate model is Kriging, based on code by Forrester et al. [7]. Parameter: seq.predictionModel.func.505 –The initial design consists of 40 candidate solutions, which are created by Latin Hypercube Sampling (LHS). Parameter: init.design.size. Table 2 Upper and lower bounds for the parameter set M introduced in section 2.2 which was used during the optimization. All parameters are of type FLOAT. Factor Low High A0 2 m1-0.01 0 m2-0.3 0.4 m30 0.2 m4-0.1 0.1 14 Christian Jung et al. 8. Hajduk, M., Zidek, M., Elfmark, J., Kopec, S.: Derivation of mean values of inherent deformation resistance in hot rolling of tonnage steel. Hutnicke825 Listy 27(8), 567 (1972) 9. Hajduk, M., et al.: Effect of improper selection of the rpm of vertical and horizontal drives on balanced rolling force distribution in a universal rolling mill. Hutnicke Listy 27(8), 259 (1972)830 10. 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Datum der Veröffentlichung: 14.11.2017 Herausgeber / Editorship Prof. Dr. Thomas Bartz-Beielstein, Prof. Dr. Wolfgang Konen, Prof. Dr. Boris Naujoks, Prof. Dr. Horst Stenzel Institute of Computer Science, Faculty of Computer Science and Engineering Science, TH Köln, Steinmüllerallee 1, 51643 Gummersbach url: www.ciplus-research.de Schriftleitung und Ansprechpartner/ Contact editor’s office Prof. Dr. Thomas Bartz-Beielstein, Institute of Computer Science, Faculty of Computer Science and Engineering Science, TH Köln, Steinmüllerallee 1, 51643 Gummersbach phone: +49 2261 8196 6391 url: http://www.spotseven.de eMail: [email protected] ISSN (online) 2194-2870 This work has been partially supported by the MIWF NRW under grant agreement FH-STRUKTUR 2014/10 (ISAFAN).