M. KOVA^I^ et al.: ROLL WEAR MODELING USING GENETIC PROGRAMMING – INDUSTRY CASE STUDY 319–325 ROLL WEAR MODELING USING GENETIC PROGRAMMING – INDUSTRY CASE STUDY MODELIRANJE OBRABE VALJEV Z GENETSKIM PROGRAMIRANJEM – PRIMER IZ INDUSTRIJE Miha Kova~i~ 1* , Andrej Mihevc 2 , Milan Ter~elj 3 1 [tore Steel d.o.o., @elezarska cesta 3, 3220 [tore; Institute of Metals and Technology, Lepi pot 11, 1000 Ljubljana; College of Industrial Engineering, Mariborska cesta 2, 3000 Celje, Slovenia 2 [tore Steel d.o.o., @elezarska cesta 3, 3220 [tore, Slovenia 3 Faculty of Natural Sciences and Engineering, Department of Materials and Metallurgy, A{ker~eva cesta 12, 1000 Ljubljana, Slovenia Prejem rokopisa – received: 2018-05-18; sprejem za objavo – accepted for publication: 2018-12-06 doi:10.17222/mit.2018.104 [tore Steel Ltd. is one of the largest flat spring steel producers in Europe. Using the continuous rolling line (10 stands – 6 horizontal, 4 vertical), all the rolled dimensions, including round (more than 80 nominal diameters), flat (more than 650 shapes and dimensions) and square bars (13 different sizes), can be rolled each month. The purpose of the research was to identify the parameters affecting the working roll wear in the hot-rolling process. For this purpose, we collected data during the 2013 annual production on the first stand of the continuous roll mill for rolling of diameters from 20 mm to 58 mm for which data of the groove shape and surface, roll diameter, contact time, carbon equivalent, rolling temperature and quantity of the rolled material are available. After roll wear-out they are machined using a turning operation. The root cause “why the rolls were machined” was not collected. To evaluate the roll wear-out, the quantity of rolled material before the machining of rolls was used. Prediction of the quantity of rolled material before the machining of rolls was conducted using linear regression and genetic programming. The developed models were validated using the data from 2014. The validation showed that in the case of excluding the fatigue cracks from collected data the prediction could be improved drastically. The results of the research can be used in practice for predicting roll wear and consequently roll maintenance on the basis of rolling schedule quantities. Keywords: roll wear, hot rolling, prediction, linear regression, genetic programming [tore Steel d.o.o. je ena izmed najve~jih proizvajalcev plo{~atega, vzmetnega jekla v Evropi. S pomo~jo kontinuirane valjarske proge (10 ogrodij – 6 horizontalnih, 4 vertikalne) lahko mese~no izvaljajo vse dimenzije, vklju~ujo~ okrogle (80 premerov), kvadratne (13 stranic) in plo{~ate (ve~ kot 650 oblik in dimenzij) palice. Namen raziskave je bil prepoznati parametre, ki vplivajo na obrabo valjev med vro~im valjanjem. V te namene so se v letu 2013 zbirali podatki pri prvem horizontalnem ogrodju kontinuirane valjarske proge pri valjanju okroglih palic premerov od 20 mm do 58 mm: oblika in povr{ina kalibra, premer valjev, kontaktni ~as, ogljikov ekvivalent, temperatura valjanja in koli~ina izvaljanega materiala. Po obrabi valjev se le-ti mehansko obdelajo – postru`ijo. Pravi razlog za mehansko obdelavo valjev se ni navajal. Za ovrednotenje obrabe valjev smo uporabili koli~ino materiala, ki se je valjal pred mehansko obdelavo valjev. Za napovedovanje koli~ine materiala, ki se je valjal pred mehansko obdelavo valjev, smo uporabili linearno regresijo in genetsko programiranje. Oba razvita modela smo ovrednotili z uporabo podatkov iz leta 2014. Analiza je pokazala, da se zmo`nost napovedovanja pri neupo{tevanju utrujenostnih napak drasti~no izbolj{a. Rezultati so prakti~no uporabni pri napovedovanju obrabe valjev ter posledi~no vzdr`evanju valjev v skladu s plani valjanja. Klju~ne besede: obraba valjev, vro~e valjanje, napovedovanje, linearna regresija, genetsko programiranje 1 INTRODUCTION During the hot rolling of long bars the roll with the grooves allows for dimensional changes to the rolled bars. Due to the constant contact between the hot-rolled material and the cooled roll, the surface of the latter gradually wears out. Also, surface cracks can occur due to temperature gradients influenced by thermo-mecha- nical-tribological rolling conditions (e.g., roll and rolled material, coolant temperature, rolling speed). Conse- quently, one must determine influences that the rolls have on the surface defects of rolled material. 1–5 Accord- ingly, knowing about roll wear is essential. Practical approaches for roll wear reduction during hot rolling can be classified as: • changing existing roll material, 6 • using roll surface coatings, 1,3 • using lubricants, 2,4,7,8 • changing roll geometry (i.e. grooves) 9–12 • using different rolling regimes. 2,13–16 Though there are several well-known mathematical models for roll wear (i.e., Archard, Yasada, Lim and Ashby, Sibakin, Oike, Somers, Tong and Chakko), 1,3 but none of them can be used practically in an industrial environment, where the specifics of several different steel grades and where different rolling regimes are pro- duced and bound to delivery dates, come into play. In our extremely flexible industrial environment, the following were analyzed during a one-year period: the influences of groove geometry, its area, roll diameter, contact time, carbon equivalent of rolled material (more than 200 serially produced different steel grades), rolling Materiali in tehnologije / Materials and technology 53 (2019) 3, 319–325 319 UDK 004.414.23:620.193.95:62-222 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 53(3)319(2019) *Corresponding author e-mail: miha.kovacic@store-steel.si temperature and quantity of the rolled material on the wear of roll of the first stand of the continuous roll mill for long round bars (diameters from 20 mm to 58 mm). In the paper the experimental setup, including the industrial environment and the monitored parameter descriptions, is presented after the introduction. After- wards, we present the roll wear prediction, represented as a quantity of the rolled material before the machining of rolls, using genetic programming and linear regres- sion. Also, the results and the validation thereof are commented on. Finally, conclusions are drawn and directions for future work are outlined. 2 MATERIALS AND METHODS [tore Steel Ltd. is a small, flexible steel mill where more than 200 steel grades with varying chemical com- positions are produced. The scrap is melted in the elec- tric arc furnace. After tapping, the melt is ladle-treated and finally cast using a two-strand continuous caster. The cooled cast billets (180 mm x 180 mm) are reheated up to 1250 °C and rolled. After exiting the reheating fur- nace, the material goes through the descaling device and duo reversible rolling stand with 800-mm-diameter rolls. The rolled material makes 7 passes. The final rolling diameters achieved using the same rolling stand range from 95 mm to 110 mm. Before it enters the continuous rolling line with 460 mm diameter rolls (700 mm length), the material is rolled using a duo reversible rolling stand with 650-mm-diameter rolls. After exiting the duo reversible stand, after 5 passes (the last is a by-pass), the material cools down while the rolling temperature is achieved. The temperature is measured using infrared pyrometer. Note that the rolling is conducted without lubrication. The number of passes using the continuous rolling line: • 20–36 mm: 9 passes, • 37–48 mm: 7 passes, • 50–58 mm: 5 passes. The continuous rolling line (Figure 1) itself consists of a descaling device, 6 horizontal and 4 vertical stands, three hot shears – of which two are used for cutting the first and the last end of the rolled bar, while the third is used for cutting the final dimensions before the material enters the cooling bed. The continuous rolling line rolls are double-layered rolls made of steel (outer working layer) and nodular cast iron (the core). The working layer chemical composition is presented in Table 1. The same layer consists of pear- lite and bainite, depending on the required hardness. The working layer thickness depends on the groove dimen- sions. In general, 30 mm is added, on the basis of the grooves’ geometry. The core consists basically of pear- lite, but within this free cementite and spherical pearlite could be found. After rolls of the continuous roll mill wear-out, or fatigue cracks occur, they are additionally machined using a turning operation. The machining is conducted in [tore Steel Ltd. After several machining operations, reaching the core layer, the rolls should be replaced by new ones. M. KOVA^I^ et al.: ROLL WEAR MODELING USING GENETIC PROGRAMMING – INDUSTRY CASE STUDY 320 Materiali in tehnologije / Materials and technology 53 (2019) 3, 319–325 Table 1: Rolls working layer chemical composition CS iM nPSC rN iM o 3.0 %–3.8 % 1.0 %–1.8 % 0.3 %–0.8 % <0.060 % <0.020 % 0.5 %–1.2 % 1.0 %–3.0 % 0.1 %–0.5 % Figure 1: Continuous rolling line During 2013 the life of rolls of the first stand (Fig- ure 2) of the continuous roll mill for long round bars (diameters from 20 mm to 58 mm) was analyzed. The following parameters were monitored: • area of the individual groove after machining oper- ation in mm 2 (AG), • diameter of rolls after machining operation in mm (DR), • average contact time in s (TC), • average carbon equivalent in % (CE), • average rolling temperature before entering the first rolling stand in °C (TR), • quantity of rolled material before using machining operation in kg (Q’). After the rolls wear out, the removal of the affected area is conducted using a machining process. The remo- val depth depends on the severity of the affected area. It must be emphasized that detailed information about the root cause for the machining of rolls was not available. Table 2 shows the reduction of roll diameter (DR) and the area of the grooves on the first stand of the conti- nuous rolling line (AG) during the roll life cycle (before replacement). Table 2: The area of the individual groove after the machining oper- ation (practical example of roll life cycle) Roll diameter OV60/1 (mm 2 ) OV70/1 (mm 2 ) OV70/2 (mm 2 ) OV70/3 (mm 2 ) OV85/1 (mm 2 ) 460 122859 142267 148647 157561 166376 419 111327 128786 134473 142394 150459 418 111046 128457 134127 142024 150071 416 110484 127800 133436 141284 149295 413.8 109865 127076 132675 140470 148441 412.3 109443 126583 132156 139916 147859 410 108796 125827 131361 139065 146966 407.3 108037 124939 130428 138066 145918 405.2 107446 124249 129702 137289 145102 403 106827 123525 128941 136475 144248 400.5 106124 122703 128077 135550 143278 The contact time is the time between the workpiece entering and exiting the deformation zone. This time is calculated using the following equation: t l v c = , where l is contact length (1) lRh =⋅ Δ , where R is roll radius, (2) h = h 0 – h 1 , where h 0 and h 1 are the effective heights of the ingoing and outgoing workpiece, (3) v v Rf r i i n = ∏ _ , where (4) v is the roll circumferential speed, v r is the rolling speed and R_f i are the reduction factors for a calculation of the rolling speed at the i-1 stand. For instance, the reduction factors for rolling of round bar 21 mm using conti- nuous rolling line in [tore for stands H1, H2, V3, H4, V5, H6, V7, H8 and V9 are 1.189, 1.144, 1.286, 1.191, 1.148, 1, 1, 1 and 1, respectively. The following equation for carbon equivalent (CE), a single-number value which covers several influential alloying chemical elements, was used: 17 CE C =+ + + + + + + % %Mn %Si %Cr %Mo+%V %Cu %Ni 6 51 5 (5) Selected parameters, collected during the 2013 annual production, are presented in Table 3. Table 3: Selected parameters, collected during the 2013 annual pro- duction Area of the individual groove after machining operation in mm 2 (AG) Diameter of rolls after machining operation in mm (DR) Average contact time in s (TC) Average carbon equivalent in %( CE) Average rolling temperature before entering the first rolling stand in °C (TR) Quantity of rolled material before using machining operation in kg (Q’) 111327 419.0 0.0506 0.699 930 776725 134473 419.0 0.0596 0.739 930 1199800 128786 419.0 0.0591 0.825 930 2325295 142394 419.0 0.0699 0.763 930 738240 134473 419.0 0.0561 0.789 930 356043 150459 419.0 0.0626 0.715 930 535983 111046 418.0 0.0482 0.681 930 1153125 134127 418.0 0.0600 0.698 930 2145300 128457 418.0 0.0577 0.830 930 1514790 142024 418.0 0.0664 0.853 930 678110 150071 418.0 0.0605 0.705 930 456370 109865 413.8 0.0505 0.666 930 1369770 132675 413.8 0.0588 0.752 930 2210725 148441 413.8 0.0564 0.709 930 143935 139989 412.5 0.0699 0.813 930 745308 126649 412.5 0.0582 0.818 930 2497160 139989 412.5 0.0579 0.672 930 232620 132225 412.5 0.0590 0.825 930 744396 147936 412.5 0.0643 0.762 930 619500 109499 412.5 0.0497 0.674 930 1209050 132225 412.5 0.0602 0.738 930 1217867 108796 410.0 0.0580 0.778 930 1366531 139065 410.0 0.0671 0.797 930 716430 131361 410.0 0.0608 0.838 930 645393 Materiali in tehnologije / Materials and technology 53 (2019) 3, 319–325 321 M. KOVA^I^ et al.: ROLL WEAR MODELING USING GENETIC PROGRAMMING – INDUSTRY CASE STUDY Figure 2: Grooves on the first stand of the continuous rolling line 146966 410.0 0.0557 0.632 930 159770 108796 410.0 0.0492 0.683 930 1136585 131361 410.0 0.0597 0.735 930 3848941 124939 407.3 0.0580 0.781 930 1134305 138066 407.3 0.0697 0.822 930 594478 130428 407.3 0.0610 0.771 930 363901 145918 407.3 0.0685 0.638 930 103334 108037 407.3 0.0494 0.634 930 1420125 130428 407.3 0.0587 0.724 930 2005551 124249 405.2 0.0568 0.798 890 1333585 137289 405.2 0.0675 0.843 890 804196 129702 405.2 0.0582 0.765 890 544345 145102 405.2 0.0616 0.693 890 603295 107446 405.2 0.0511 0.654 890 1127800 129702 405.2 0.0591 0.728 890 1956695 123525 403.0 0.0575 0.858 930 1637455 136475 403.0 0.0640 0.856 930 370090 128941 403.0 0.0546 0.798 930 253489 144248 403.0 0.0597 0.741 930 179320 3 RESULTS On the basis of the collected data (Table 4), the pre- diction of the quantity of rolled material before the machining of the rolls was conducted using linear regression and genetic programming. For the fitness function average relative deviation between predicted and experimental data was selected. It is defined as: Δ= − = ∑ () ’ ’ QQ Q n ii i i n 1 (6) where n is the size of monitored data and Q’ i and Q i are the actual and the predicted quantities of the rolled material before the machining of rolls, respectively. 3.1 Linear regression prediction On the basis of the linear regression results, it is possible to conclude that the model does not predict in a significant manner the quantity of rolled material before the machining of the rolls (p>0.05, ANOVA) and that only 19.43 % of the total variances can be explained by the independent variables’ variances (R-square). The only significantly influential parameter (p<0.05) is the area of the individual groove after machining (AG). The linear-regression model is: Q = –31.45·AG + 27377·DR + 18891118·TC + + 1140389·CE – 3227·TR + 5072217 (7) Its relative deviation from the experimental data is 71.10 %. The calculated influences of the individual parameters (individual variables) on the quantity of rolled material before the machining of the rolls are presented in Figure 3. It is possible, on the basis of the same figure, to conclude that the area of the individual groove after machining is most influential. 3.2 Genetic programming prediction Genetic programming is probably the most general evolutionary optimization method. 18–20 Evolutionary opti- mization methods mimic the natural evolution of living things and can be used for solving different problems (see, for example 21–24 ). The organisms in genetic pro- gramming that undergo adaptation are in fact mathe- matical expressions (models) for predicting the quantity of the rolled material before the machining of rolls. The models – that is, computer programs – consist of the selected function (i.e., basic arithmetical functions) and terminal genes (i.e., independent input parameters, and random floating-point constants). Typical function genes are: addition (+), subtraction (–), multiplication (*) and division (/), and terminal genes (e.g. x, y, z). Random computer programs for calculating the various forms and lengths are generated by means of the selected genes at the beginning of the simulated evolu- tion. The varying of the computer programs is conducted M. KOVA^I^ et al.: ROLL WEAR MODELING USING GENETIC PROGRAMMING – INDUSTRY CASE STUDY 322 Materiali in tehnologije / Materials and technology 53 (2019) 3, 319–325 Figure 4: Crossover operation (out of two parental organisms the off springs with randomly distributed genetic material are evolved) Figure 3: Calculated influences of individual parameters on the quan- tity of rolled material before the machining of rolls using a linear-re- gression model [kg] with genetic operations (e.g., crossover, mutation) during several iterations, called generations. The crossover operation is presented in Figure 4. After the completion of the variation of the computer programs a new gene- ration is obtained. Each result obtained from an indivi- dual program from a generation is compared with the ex- perimental data. The process of changing and evaluating the organisms is repeated until the termination criterion of the process is fulfilled. In-house genetic programming system, programmed using AutoLISP, which is integrated in AutoCAD – com- mercial computer-aided design software, was used. 25–27 Its settings were: • size of the population of organisms: 1000, • maximum number of generations: 100, • reproduction probability: 0.4, • crossover probability: 0.6, • maximum permissible depth in the creation of the population: 6, • maximum permissible depth after the operation of crossover of two organisms: 10, • smallest permissible depth of organisms in generating new organisms: 2. The genetic operations of reproduction and crossover were used. For the selection of organisms the tournament method with tournament size 7 was used. The AutoLISP based in-house genetic programming system was run 100 times in order to develop 100 inde- pendent civilizations. Each run lasted approximately 1 min on a I7 Intel processor and 8 GB of RAM. The best mathematical model obtained from 100 runs of the genetic programming system is: M. KOVA^I^ et al.: ROLL WEAR MODELING USING GENETIC PROGRAMMING – INDUSTRY CASE STUDY Materiali in tehnologije / Materials and technology 53 (2019) 3, 319–325 323 F i g u r e5 :C alculated influences of individual parameters on the quantity of rolled material before the machining of the rolls using the genetic programming model [kg] Its relative deviation from the experimental data is 43.93 %, which is 1.62-times better than a linear regres- sion model. The calculated influences of the individual parameters (individual variables) on the quantity of rolled material before the machining of the rolls are pre- sented in Figure 5. Based on the same figure it is possi- ble to conclude that the roll diameter after machining is the most influential parameter. 4 DISCUSSION Based on linear regression and genetic programming modeling we can conclude: • both models’ performances are relatively low, • the most influential parameters (area of the individual groove after machining operation, diameter of the rolls after machining operation) depend on the ma- chining of the rolls and, consequently, additional data and a precise description of the root cause for ma- chining of rolls are needed. Accordingly, additional data was gathered during the period of an additional year, 2014 (Table 4). In all 16 cases the rolls were machined only due to their having worn out. No fatigue cracks were noted during the year 2014. Table 4: Selected parameters, collected during the 2014 annual pro- duction Area of the individual groove after machining operation in mm 2 (AG) Diameter of rolls after machining operation in mm (DR) Average contact time in s (TC) Average carbon equivalent in %( CE) Average rolling temperature before entering the first rolling stand in °C (TR) Quantity of rolled material before using machining operation in kg (Q’) 142268 460.0 0.0612 0.812 930 946000 122859 460.0 0.0538 0.687 930 1196360 148647 460.0 0.0632 0.741 930 2146800 141347 457.2 0.0617 0.853 930 33195370 165290 457.2 0.0613 0.651 930 6043100 147679 457.2 0.0633 0.745 930 23035450 121341 454.6 0.0516 0.682 930 14102600 146781 454.6 0.0628 0.725 930 17868800 140492 454.6 0.0621 0.767 930 30325250 146781 454.6 0.0619 0.755 930 9038860 164280 454.6 0.0677 0.628 930 6610820 139637 452.0 0.0613 0.812 930 16725950 120609 452.0 0.0517 0.630 930 12177050 145882 452.0 0.0623 0.767 930 29562300 138749 449.3 0.0609 0.839 930 11691900 144948 449.3 0.0624 0.787 930 20959950 The relative deviations from the experimental data of the linear regression (Equation (7)) and genetically (Equation (8)) developed model are 55.72 % and 31.56 %, respectively. The improvement in performance of both models for the quantity of rolled material before the machining of rolls suggests that the previously gathered data (from the year 2013) should be filtered out from instantaneous root causes (e.g., fatigue cracks) and only roll wear out should be taken into account. 5 CONCLUSIONS [tore Steel Ltd. is a small, flexible steel mill where more than 200 steel grades with varying chemical com- positions are produced. The material is rolled using a continuous rolling line, which rolls are double layered and made of steel (outer working layer) and nodular cast iron (the core). After roll wear-out – that is, after the rolls became worn out – they are machined using a turning operation. In the research the life of rolls of the first stand (Fig- ure 2) of the continuous roll mill for long round bars (diameters from 20 mm to 58 mm) was analyzed for 2013. The following parameters were monitored: • area of the individual groove after machining oper- ation in mm 2 , • diameter of rolls after machining operation in mm, • average contact time in s, • average carbon equivalent in %, • average rolling temperature before entering the first rolling stand in °C, • quantity of rolled material before using machining operation in kg. On the basis of the collected data (Table 3), the pre- diction of the quantity of rolled material before the machining of rolls was conducted using linear regression and genetic programming. For genetic programming, an in-house system, programmed using AutoLISP, was used. For the fitness function average relative deviation between predicted and experimental data was selected. The relative deviations from experimental data of linear regression and genetically developed model are 71.10 % and 43.39 %, respectively. The genetically developed model outperformed the linear regression model by 1.62-times. Nevertheless, both models have relatively poor performance. Additionally, both models were validated using additionally gathered data from 2014, but in this case the data without fatigue cracks were used. The relative deviations from the experimental data of linear regression and genetically developed model are 55.72 % and 31.56 %, respectively. The drastically im- proved performance of both models can be attributed to filtering the data – the data regarding machining of the rolls due to fatigue cracks was removed, which also indicates that both models can be accordingly developed again. In the future only data related to roll wear out will be gathered. Consequently, the relatively precise prediction of roll wear and roll maintenance, based on the rolling schedule quantities, will be possible. It must also be emphasized that the adjusted methodology can be used in various rolling mill environments. M. KOVA^I^ et al.: ROLL WEAR MODELING USING GENETIC PROGRAMMING – INDUSTRY CASE STUDY 324 Materiali in tehnologije / Materials and technology 53 (2019) 3, 319–325 Acknowledgment The authors would like to thank Professor Jason Blake for proofreading, which greatly helped the read- ability of the paper. 6 REFERENCES 1 C. Bataille, E. Luc, M. Bigerelle, R. Deltombe, M. Dubar, Rolls wear characterization in hot rolling process, Tribol. Int., 100 (2016), doi:10.1016/j.triboint.2016.03.012 2 D. Strasser, M. Bergmann, B. Smeulders, D. Paesold, K. Krimpel- stätter, P. Schellingerhout, A. Kainz, K. Zeman, A novel model- based approach for the prediction of wear in cold rolling, Wear, 376–377 (2017), doi:10.1016/j.wear.2016.12.056 3 S. Spuzic, K. N. Strafford, C. Subramanian, G. Savage, Wear of hot rolling mill rolls: an overview, Wear, 176 (1994) 2, doi:10.1016/ 0043-1648(94)90155-4 4 X. Yu, Z. Jiangad, J. Zhao, D. Wei, J. Zhou, C. Zhoud, Q. Huang, The role of oxide-scale microtexture on tribological behaviour in the nanoparticle lubrication of hot rolling, Tribol. Int., 93 (2016), doi:10.1016/j.triboint.2015.08.049 5 M. Nilsson and M. Olsson, Microstructural, mechanical and tribo- logical characterisation of roll materials for the finishing stands of the hot strip mill for steel rolling, Wear, 307 (2013) 1–2, doi:10.1016/j.wear.2013.09.002 6 P. Andersson, J. Levén, and B. Hemming, Hot rolling tests with steel bars and silicon nitride rolls, J. Mater. Process. Technol., 209 (2009) 2, doi:10.1016/j.jmatprotec.2008.02.069 7 Y. Bao, J. Sun, and L. Kong, Effects of nano-SiO 2 as water-based lubricant additive on surface qualities of strips after hot rolling, Tribol. Int., 114 (2017), doi:10.1016/j.triboint.2017.04.026 8 X. Wenzhen, Z. Jingwei, W. Hui, Z. Xianming, Z. Xiaoming, X. Jianzhong, J. Sihai, J. Zhengyi, Effects of oil-in-water based nano- lubricant containing TiO 2 nanoparticles in hot rolling of 304 stainless steel, Procedia Eng., 207 (2017), doi:10.1016/j.proeng. 2017.10.901 9 R. Servin-Castañeda, A. M. Garcia-Lara, R. D. Mercado-Solís, C. A. Vega-Lebrun, Development of mathematical model for control wear in backup roll for hot strip mill, J. Iron Steel Res. Int., 21 (2014)1, doi:10.1016/S1006-706X(14)60008-X 10 R. Wang et al., Strip shape control capability of hot wide strip rolling mills, J. Univ. Sci. Technol. Beijing, Miner. Metall. Mater., 15 (2008) 1, doi:10.1016/S1005-8850(08)60018-3 11 J. G. Lenard, Primer on Flat Rolling, 2 nd ed., Department of Mechanical and Mechatronics Engineering, University of Waterloo, Ontario 2014, 31–37 (Roll design), doi:10.1016/B978-0-08-099418- 5.00003-2 12 X. B. Ma, D. C. Wang, H. M. Liu, C. C. Wen, Y. Zhou, Large con- cave roll technology for hot rolled silicon steel, Ironmak. Steelmak. 45 (2018) 1, doi:10.1080/03019233.2016.1240841 13 J. Mian, L. Xuejun, W. Jigang, W. Guangbin, A precision on-line model for the prediction of thermal crown in hot rolling processes, Int. J. Heat Mass Transf., 78 (2014), doi:10.1016/j.ijheatmasstransfer. 2014.07.061 14 M. Stürmer, J. Dagner, P. Manstetten, H. Köstler, Real-time simu- lation of temperature in hot rolling rolls, J. Comput. Sci., 5 (2014)5, doi:10.1016/j.jocs.2014.04.003 15 X. Cheng, Z. Jiang, J. Zhao, D. Wei, L. Hao, Investigation of oxide scale on ferritic stainless steel B445J1M and its tribological effect in hot rolling, Wear, 338–339 (2015), doi:10.1016/j.wear.2015.06.014 16 Y. Fu, H. Yu, Application of mathematical modeling in two-stage rolling of hot rolled wire rods, J. Mater. Process. Technol., 214 (2014) 9, doi:10.1016/j.jmatprotec.2014.04.017 17 M. Bruneau, C.-M. Uang, R. Sabelli, Ductile Design of Steel Struc- tures, 2 nd ed., McGraw-Hill Professional, 2011 18 J. R. Koza, Genetic Programming: On the Programming of Com- puters by Means of Natural Selection, MIT Press, Cambridge, MA, USA, 1992 19 J. R. Koza, Genetic Programming II: Automatic Discovery of Reus- able Programs, MIT Press, Cambridge, MA, USA, 1994 20 K. John, F. H. Bennett III, D. Andre, M. A. Keane, Genetic Pro- gramming III: Darwinian Invention and Problem Solving, Morgan Kaufmann Publishers Inc., San Francisco 1999 21 M. Sekuli}, V. Peji}, M. Brezo~nik, M. Gostimirovi}, M. Had`istevi}, Prediction of surface roughness in the ball-end milling process using response surface methodology, genetic algorithms, and grey wolf optimizer algorithm, Adv. Prod. Eng. Manag., 13 (2018)1, doi:10.14743/apem2018.1.270 22 L. Wang, X. Zhu, Z. Xie, Container assignment optimization con- sidering overlapping amount and operation distance in rail-road transshipment terminal, Adv. Prod. Eng. Manag., 12 (2017)4 , doi:10.14743/apem2017.4.264 23 P. D. Dubrovski, M. Brezocnik, Porosity and nonwoven fabric verti- cal wicking rate, Fibers Polym., 17 (2016) 5, doi:10.1007/s12221- 016-6347-5 24 J. Gotlih, M. Brezo~nik, J. Bali~, T. Karner, B. Razbor{ek, K. Gotlih, Determination of accuracy contour and optimization of workpiece positioning for robot milling, Adv. Prod. Eng. Manag., 12 (2017)3, doi:10.14743/apem2017.3.254 25 M. Kova~i~, F. Dolenc, Prediction of the natural gas consumption in chemical processing facilities with genetic programming, Genet. Program. Evolvable Mach., (2016), doi:10.1007/s10710-016-9264-x 26 M. Kova~i~, Modeling of total decarburization of spring steel with genetic programming, Mater. Manuf. Process., 30 (2014)4 , doi:10.1080/10426914.2014.961477 27 M. Kova~i~, B. [arler, Genetic programming prediction of the natural gas consumption in a steel plant, Energy, 66 (2014), doi:10.1016/j.energy.2014.02.001 M. KOVA^I^ et al.: ROLL WEAR MODELING USING GENETIC PROGRAMMING – INDUSTRY CASE STUDY Materiali in tehnologije / Materials and technology 53 (2019) 3, 319–325 325