Multi-Objective Optimization of Rolling Schedule for Five-Stand Tandem Cold Mill

The optimization of rolling schedule is the main content of tandem cold rolling which will affect the quality of products directly. A rolling schedule with the objectives of minimum energy consumption, relative power margin and slippage preventing is established. First, in order to make the rolling schedule more accurate in the calculation process, a mathematical model combines with deep neural network is proposed to calculate the rolling force. Second, a multi-objective particle swarm optimizer with dynamic opposition-based learning is proposed to optimize the rolling schedule. It has a new particle learning strategy to update the moving position of particles. Moreover, opposition-based learning is proposed to make the particles jump out of local optima. Finally, the experiments are carried out based on the field data. Simulation results demonstrate that the accuracy of the rolling force is greatly improved. The proposed algorithm has a promising performance on both diversity and convergence. At the same time, the optimized rolling schedule can well balance the rolling power and prevent slipping between five stands comparing with the original rolling schedule.


I. INTRODUCTION
With the development of economy, the demand for coldrolled strip in the steel market is growing rapidly. The requirement of consumers for the quality of products has increased. Steel strip manufacturing is a complex and large industrial process. It covers multiple levels of control and multiple procedures [1]. The cold rolling process equipment, measuring instruments and their installation positions of a five-stand UCM tandem cold rolling mill in a factory are shown in Figure 1. The setting of rolling schedule affects the quality of products and energy consumption. Rolling schedule is an important aspect of the cold rolling process. A reasonable setting of rolling schedule can fully utilize the equipment capabilities, which not only improves the production and quality of steel but also reduces the consumption of rolling energy. Therefore, the optimization of rolling schedule is a typical multi-objective optimization problem.
The associate editor coordinating the review of this manuscript and approving it for publication was Olga Fink.
There are a lot of references that transforms the rolling schedule into a single-objective optimization problem by weighted method. Qi et al. integrated relatively equal load and good strip crown and flatness into one objective function in [2]. Chen et al. transformed the multi-objective optimization problem into a single-objective optimization problem in [3]. The optimization of rolling schedule needs to meet the constraints such as rolling force, rolling torque and rolling power. By introducing a penalty term into the objective function, the rolling schedule was transformed into an unconstrained problem. The function was solved by the Nelder-Mead simplex method. Compared with the rolling schedule optimized by empirical formula, the proposed method can well balance the rolling force and rolling power simultaneously. Poursina et al. integrated power distribution cost, tension cost function, perfect shape condition into one cost function in [4]. They used the genetic algorithm to optimize the cost function and achieved a good result. The case-based reasoning-Tabu search hybrid algorithm was proposed to optimize the rolling schedule in tandem cold rolling in [5]. The improved iterative method reduced the number of iterations by about half compared with the ordinary tabu search algorithm. However, in practice, because the objective functions of multi-objective optimization contradict each other and the number of optimization targets is large, it is difficult to determine the weight coefficients. In recent years, the multi-objective optimization algorithm provides a solution for solving the problem of rolling schedule. The multi-objective fuzzy theory and method was proposed to optimize the rolling schedule in [6]. Li et al. proposed a MODE/D algorithm to optimize the rolling schedule in [7]. The results showed that it can obtain a good Pareto-optimal front which made a good trade-off among the three objective functions. A differential evolution algorithm based on the evolutionary direction was proposed to solve the multiobjective optimization model of the rolling schedule. The results showed that the profile was improved and rolling energy consumption was reduced compared with the actual rolling schedule [8]. Although there have been many multiobjective optimization algorithms proposed to calculate the rolling schedule, the diversity and convergence of the solution set is not good enough. As a result, there will be problems such as the quality of products decline and high energy consumption.
The rest of this paper is organized as follows. In section 2, a hybrid rolling force model of deep neural network model combines with a mathematical model is introduced. In section 3, the mathematical models of the rolling schedule are introduced. A multi-objective particle swarm optimizer with dynamic opposition-based learning which is named MOPSO-DOL is proposed to optimize the rolling schedule. In section 4, the proposed model is used to predict the rolling force and the comparative experiment is carried out. The proposed MOPSO-DOL is tested on benchmark problems with other multi-objective optimization algorithms and it is used to optimize the rolling schedule. The optimal solutions are chosen by the weighted-sum approach as the set values of the rolling schedule. Finally, section 5 presents the conclusion.

II. THE ROLLING FORCE MODEL
The rolling force is a very important parameter in the cold rolling process. The accuracy of the rolling force model directly affects the accuracy of the output thickness. The presetting of the rolling force controls the setting of the rolling schedule.
The traditional mathematical model is a model based on the internal mechanism of objects and production processes. A new approach was proposed to predict the rolling force of cold rolled sheet based on the plastic mechanics in [15]. Most of the errors fallen within the range of ±10%. However, due to the complexity of the working environment and the coupling relationship between variables, the traditional mathematical model requires many assumptions in practice. Thus, it can't reflect the influence of disturbance factors on the rolling force.
In order to predict the rolling force more accurately, intelligence-based models are widely introduced into the field of rolling. AziGuLi et al. used the extreme learning machine and self-learning model to predict the rolling force in [16]. The artificial neural network model was used to predict the rolling force and acquired a high precision in [18]. Mahmoodkhani et al. developed an online rolling force prediction tool by combining the finite element model with the artificial neural network in [19]. The error of rolling force was reduced to 10%. But these models are all shallow machine learning models whose express ability to the complex function is restricted.
The past several years have seen increasingly rapid advances in the field of deep learning, deep neural network has a good nonlinear fitting ability, which can reflect the influence of disturbance factors on rolling force accurately. Wei et al. tried to apply deep learning to the prediction of rolling force in [20]. An MLP rolling force prediction model based on deep learning method was proposed. Simulation results showed that the model could reduce the relative error between the predicted results and the measured data to less than 3%. Many models have been established to improve the prediction accuracy of the rolling force. These models either consider only data-driven models or only mathematical models. A method combining artificial neural network and physical model was proposed to calculate the rolling force in [17], where deformation resistance and friction coefficient were calculated by physical models. The Bland-Ford-Ellis rolling force model was used to calculate the rolling force.
In this paper, we propose a method combining the mathematical model and the deep neural network to calculate the rolling force.

A. TRADITIONAL MATHEMATICAL MODEL
In this paper, the SIMS mathematical model is applied to calculate the rolling force. The SIMS model is expressed as: where F is the rolling force; B is the strip width, mm; l c is the roll contact length, mm; K is the deformation resistance of metals, K = 1.15σ ; K T is the influence coefficient of front and back tensile stress on the rolling force. The calculation formula of the contact arc length after flattening is as follows.
where R is the flattened roll radius, h 0 and h 1 are the entry thickness and exit thickness, mm. The flattened roll radius calculated by the Hitchcock formula is: where R is roll radius, mm. R can be calculated by an iterative method. Then Hill's simplified external friction stress state coefficient formula is: where ε is the relative reduction rate, µ is the coefficient of friction.
where T b and T f are the back tension and forward tension, MPa. For a cold-rolled strip, the magnitude of the deformation resistance depends on the chemical composition and cumulative deformation of metal materials. The deformation rate and deformation temperature have less influence on the deformation resistance. Therefore, the selected deformation resistance model is shown in (6).
where ε is cumulative deformation resistance, ; H 0 is the thickness of raw material, mm; h is the target thickness, mm; A, B, C, D are the coefficients of deformation resistance model associated with steel grades.
The traditional SIMS is the most commonly used rolling force model in the field. Using the SIMS model only will cause a large deviation. The coefficients in the model are constant and the model makes many assumptions that it doesn't consider the impact of disturbance factors on the rolling force.
The rolling force and the flattened roll radius need to be calculated by the iterative method, which also causes the deviation of the rolling force. Therefore, it is not appropriate to use the SIMS model only to calculate the rolling force. It needs to be corrected to adapt to the field application.

B. AGGREGATION MODEL OF DNN AND SIMS
In this paper, the method of training deep neural network in [20] is used to predict the rolling force. Roller radius, entrance thickness, initial thickness, objective thickness, forward tension, back tension, deformation resistance are chosen as the inputs of the deep neural network according to the SIMS model.
Three kinds of SIMS model combines with deep neural networks are proposed. The first combination method is marked as DNN combines with SIMS I. The mathematical model is used to calculate the main value of the rolling force, and the deep neural network is used to predict the deviation of the rolling force. The rolling force deviation is the actual rolling force minus the rolling force calculated by the mathematical model. The second combination method is marked as DNN combines with SIMS II. The deep neural network is used to predict the deviation of the rolling force. The rolling force deviation is the ratio of rolling force calculated by the mathematical model divided by the actual rolling force. The final rolling force is obtained by multiplying the calculated value of the mathematical model and the predicted ratio. The third combination method is marked as DNN combines with SIMS III. The rolling force calculated by the mathematical model is regarded as one of the inputs of the deep neural network. The third combination method is shown in Figure 2.

III. THE ROLLING SCHEDULE MODEL
In this part, the objective functions, constraints and decision variables are established firstly. Then a multi-objective particle swarm optimizer with dynamic opposition-based learning is proposed to optimize the rolling schedule. VOLUME 8, 2020

A. THE OBJECTIVE FUNCTIONS OF THE ROLLING SCHEDULE
The rolling schedule for five-stand tandem cold mill was established in [22]. The rolling power model is expressed as: where P is the rolling power, KW; M is the rolling torque, KN · m; ω is the rated speed of motor, rad/s; The model of rolling torque is as follow: where ψ is the coefficient of the rolling force arm, in this paper, ψ = 0.5. For minimizing the energy consumption generated during the rolling process, the minimum energy consumption objective function is established. The objective function can be defined as in (9).
where P i is the actual power of stand i; P ri is the rated power of stand i. In order to increase productivity and fully exploit equipment capabilities, the relative power margin objective function is established which is expressed as follows: For the purpose of preventing the occurrence of slipping during the process of rolling, a function to prevent slipping is established. Slipping refers to the phenomenon of relative movement between rolls and strips which easily causes scratches on the surface of strips. The objective function for preventing slipping is shown in (11).
where β is the weighted coefficient, and generally β is 0.4. i is the slip factor of stand i. And i is shown in (12).
The rolling mill can effectively prevent slipping when 0 < i < 0.5. Obviously, the smaller the slip factor, the smaller the probability of slippage occurs.

B. THE CONSTRAINTS OF THE OBJECTIVE FUNCTIONS
When optimizing the rolling schedule, the selection of the rolling schedule must meet the constraints. According to the reasons for the constraints, it can be divided into equipment constraints and process constraints.
The equipment constraints are as shown below. (13) where F i , P i , M i are the rolling force, rolling power, and rolling torque of ith stand respectively; F i max , P i max , M i max are the maximum rolling force, rated power of the motor and maximum rolling torque of the ith stand respectively. The process constraints are as follows.
where ε i and T i are the reduction rate and tension of each stand; ε i min , ε i max and T i min , T i max are the minimum and maximum of the reduction rate and tension of ith stand.

C. THE SELECTION OF DECISION VARIABLES
The rolling schedule for tandem cold mill involves multiple rolling parameters, and the reduction rate and tension of each stand are two important parameters. After the reduction rate and tension are determined, the rolling force, rolling torque, rolling power can be calculated according to the relevant models.
For a five-stand tandem cold mill, the entry thickness of the first stand and the exit thickness of the fifth stand are known; The entry tension of the first stand and the exit tension of the fifth stand are given according to the process conditions. Therefore, a total of eight variables that are thickness and tension between each stand are selected [3].
where X is the optimization vector, h 1 , h 2 , h 3 , h 4 are thickness between each stand; T 1 , T 2 , T 3 , T 4 are tension between each stand.

D. MULTI-OBJECTIVE OPTIMIZATION MODEL
The diversity and convergence of solution set of traditional multi-objective optimization algorithms have poor performance when optimizing the rolling schedule. The competitive mechanism based multi-objective particle swarm optimizer (CMOPSO) [9] has fast calculation speed and high convergence accuracy, but the diversity of the solution set is poor, and it is easy to fall into local optima. To address the above problems, a new algorithm called MOPSO-DOL is proposed. The algorithm has three main components: the random particle learning strategy, the dynamic opposition-based learning strategy, and the environmental selection. The random particle learning strategy is composed of elite particle selection and random particle learning. The elite particles are selected from the non-dominated sorting and crowding distance based sorting. In CMOPSO, γ elite particles are selected in order from the ranking results as the elite solution set L. After the elite solution set L is determined, the algorithm performs a pairwise competition strategy.
Suppose the particles a and b are two randomly selected particles from the elite solution, the particle p is the current particle to be updated, and θ 1 and θ 2 are the angles between the particle p and the elite particles a and b. The angles θ 1 and θ 2 are calculated separately. Particle with a smaller angle with the particle p will be used as the winning particle. The particle a is marked as a winning particle if θ 1 < θ 2 for guiding the update of the particle p.
When the winning particle is selected, the particle p updates its position and velocity by learning from the winning particle. Assume that the position of the particle p is P i , the velocity is V i , and the position of the winning particle is P w . Then the velocity update formula and position update formula of particle p are: where R 1 and R 2 are randomly generated vectors in the interval [0,1]. After that, all particles perform polynomial variations to enhance the ability to search for optimal regions. It's obvious that the competition-based method can make the particle flying faster to its convergence direction. When the ideal situation occurs, the flying direction of each particle is shown in Figure 3. It can be seen from the picture that all of the particles are flying towards their nearest convergence direction. It is assumed that the feasible particle solutions are evenly distributed in space. Thus, the particles gradually decrease from the middle to the sides. The number of particles that eventually converge on the real Pareto front will gradually decrease from the middle to the sides, which will undoubtedly reduce the distribution and convergence of the final solutions.
In order to make the particles converge to the real Pareto front more quickly, a new learning mechanism is proposed. The way of learning from the first γ elite particles is still used. The difference is that the competition based learning will not be used. A particle from the first γ elite particles is randomly picked and then let the rest of the particles learn from it by formula 16 and 17. The P w in formula 16 is changed to the position of the selected particle. Thus, the particles distribute more evenly on the real Pareto front and the convergence speed is improved. At the same time, the quality of the solution is higher when decision-makers select the rolling schedule setting parameters on the final Pareto optimal solution set.
When some of the first ten particles sorted by Pareto level and crowding distance fall into local optima, the other particles learning from them will also fall into the local optima. To solve this problem, the opposition-based learning (OBL) is proposed to make the particles jump out of the local optima.
The OBL was first proposed as a scheme in the field of machine intelligence in [10]. It was widely spread in the field of evolutionary computation and proved to be efficient. But its theoretical studies are still immature in multi-objective optimization [11]. Gao et al. proposed a velocity-free multiobjective particle swarm optimizer with centroid in [14]. The OBL was applied to generate the initial swarm and the better particle was chosen from the solution and its opposite solution. The OBL can increase the diversity of particles and accelerate the convergence rate of the population. Supposing x is a real number belonging to [a,b], its opposite solutionx is defined as follows:x The opposite solutionP is defined byP(x 1 ,x 2 , . . . ,x D ) wherê When there are particles in the first ten particles fall into local optima, the OBL method is applied to make them jump out of the local optima after every certain number of generations. In this paper, the OBL method is performed every Max-Gen/10 generations, where MaxGen is the maximum number of generations. After that, the non-dominated sorting and crowding distance sorting are re-performed. Then the random particle learning strategy is implemented and update the next generation. The environmental selection is still adopted from the SPEA2 [25]. The pseudo-code of MOPSO-DOL is shown in Algorithm 1.
To verify the effectiveness of the proposed algorithm, the computational complexity is calculated. It has the same computational complexity whether considering using the OBL to jump out of local optima or not. The worst computational complexity of non-dominated sorting is O (MN 2 ) where M is the number of objects and N is the population size. The computational complexity of the crowding distance based sorting is O (MN log N ). Because the proposed algorithm does not use external archive to store the best particles, the worst computational complexity of environment selection is O (N 3 ). Therefore, the worst computational complexity of MOPSO-DOL is O(N 3 ). In practice, the setting of rolling schedule varies with different incoming strips which shows VOLUME 8, 2020  Perform the non-dominated sorting and crowding distance sorting to P; 3: Select γ particles from P by ranking; 4: if mod(Gen, MaxGen/10) == 0 5: Update the positions of γ by formula 19; 6: Perform the non-dominated sorting and crowding distance sorting to P; 7: Select γ particles from P by ranking; 8: end if 9: for each particle p i ∈ P do 10: Randomly choose one elite particle from γ ; 11: Update the position and velocity of p i by formula 16 and 17; 12: P ← P + p i ; 13: end for 14: P ← PolynomialVariations (P ); 15: P ← EnvironmentalSelection(P, P ); 16: end while 17: return final population P that the proposed algorithm can meet the demands of production. Finally, the flowchart of the optimization of rolling schedule is shown in Figure 4.

IV. SIMULATION RESULTS AND ANALYSIS
The following simulation results consist of three parts. The first part is to verify the effectiveness of the proposed mathematical model combines with deep neural network. The second part is to illustrate the validity of the proposed MOPSO-DOL on benchmark problems. The third part is to verify the application of the MOPSO-DOL on the optimization of rolling schedule.

A. THE SIMULATION OF THE ROLLING FORCE MODEL
The field data are used to verify the accuracy of the proposed model. The material of the steel is MRT4 and the width is 910 mm. The parameters of the rolling mills are shown in Table 1. After many experiments, the number of hidden layers of deep neural network is selected as five. The batch size is taken as 100. The activation function of the hidden layer is the ReLU and the output layer uses the sigmoid activation function. The number of training data is 40000, and 1000 data are used for testing.
In order to further explore the prediction accuracy of the proposed model, the deep neural network and GA-BP [20], [21] are chosen for comparison. Mean absolute percentage error (MAPE) and root mean square error (RMSE)   Table 2. And the minimum value is shown in bold. It can be seen that most of the calculation accuracy of the rolling force model of DNN combines with the SIMS III is higher than that of the other rolling force models. When the SIMS rolling force model is added to the output of the DNN, the noise will be produced. Taking the calculated value of the SIMS rolling force as one of the inputs of the DNN is equivalent to increasing the input characteristics of the neural network. Therefore, the prediction accuracy of the neural network will be improved.
The rolling force calculated by the SIMS model and DNN combines with the SIMS III model are shown in Figure 5 and Figure 6. As can be seen from the figures, the rolling force error calculated by the traditional method is within 10%. But the rolling force error is within 5% by the proposed model. And the proposed rolling force model is applied to the optimization of rolling schedule.

B. SIMULATION RESULTS ON BENCHMARK PROBLEMS
The proposed MOPSO-DOL is tested on the benchmark problems to confirm its performance. Since the problem of rolling schedule to be solved in this paper is a three-objective problem, this paper only tests on three-objective benchmark problems. The three-objective test functions DTLZ1-DTLZ7 and WFG1-WFG9 are selected for testing. The number of decision variables for DTLZ2-DTLZ6 and all WFG test problems is set to 12, DTLZ1 is set to 7, and DTLZ7 is set to 22. The inverted generational distance (IGD) metric is used to evaluate the performance of the algorithm. The smaller the IGD value, the better the performance of the algorithm. The proposed algorithm is compared with several state-of-the-art multi-objective particle swarm optimization algorithm, MMOPSO [12], MPSOD [13], CMOPSO [9], and three famous multi-objective evolutionary algorithms (MOEAs), NSGA-II [24], MOEA/D [23], and IBEA [26]. All parameters are set as suggested in the original references. The population size is set to 100, and the maximum evolutionary generation of the population is used as the termination condition of the algorithm. Except that the maximum evolution generation of DTLZ3 is set to 1000, the rest of the test problems are set to 300. Each test problem runs independently 30 times. The test functions and the source code of the compared algorithms are provided by PlatEMO [27]. Table 3 and Table 4 show the mean and standard deviation of IGD metric of each algorithm on standard test problem. The Wilcoxon rank sum test at the significance level of 5% is adopted, and the symbol '+', '-' and '≈' represent the compared algorithm is significantly better than, worse than and similar to the proposed MOPSO-DOL. The best average of each test function is marked with a gray background. It can be seen from the tables that comparing with the other algorithms, the proposed MOPSO-DOL performs well on most benchmark test problems which proves the effectiveness of the algorithm. The proposed MOPSO-DOL is applied to the optimization of rolling schedule.

C. APPLICATION ON THE OPTIMIZATION OF ROLLING SCHEDULE
In this section, the field data are used to verify the optimization results of the proposed MOPSO-DOL. The specific parameters of the cold rolling mills are shown in Table 1. The population evolution number Maxgen of MOPSO-DOL is set to 500 and the population size is set to 100. The number of elite particles is adopted as 10. The PlatEMO [27] platform is used for verification experiments.
After the Pareto optimal solution set is obtained, decisionmakers need to select a set of solutions as the final rolling schedule settings. The weighted-sum approach in [28] is used to select the trade-off solution which is shown as follows: where w 1 , w 2 and w 3 are the weight coefficient of the three objectives and w 1 + w 2 + w 3 = 1; f i max and f i min (i = 1, 2, 3) are the maximum and minimum value on the Pareto optimal solution set of the ith objective. Decision-makers can determine the rolling schedule by setting the values of w 1 , w 2 and w 3 . For rolling power balance, set w 1 = 0.2, w 2 = 0.5, w 3 = 0.3; to prevent slippage, set In order to verify the results of the multi-objective optimization of rolling schedule, the rolling schedule calculated by the multi-objective optimization algorithm is compared with the original rolling schedule. The optimized rolling schedules are marked rolling schedule No.1 and rolling schedule No.2, respectively. The comparison results are shown in Table 5. In order to make a clearer comparison between the original rolling schedule and the optimized rolling schedule, Figure 7 and Figure 8 show the distribution of the slip factors and the rolling power of five stands.   It can obviously see that the optimized rolling schedule can well balance the slip factor and rolling power between the five stands. Because the proposed algorithm can improve the diversity and convergence of particles in the evolution process. And OBL can make particles jump out of local optima.

V. CONCLUSION
In this study, to solve the problem that it is difficult to choose the coefficients when optimizing the rolling schedule by single-objective optimization, a multi-objective particle swarm optimizer with dynamic opposition-based learning is proposed. A new particle learning strategy is proposed to update the moving position of particles. Opposition-based learning makes particles jump out of local optima. It not only improves the quality of cold-rolled products, but also reduces the rolling energy consumption. The method based on deep neural network and mathematical model is used to calculate the rolling force. The simulation results show that the proposed MOPSO-DOL performs better than other multiobjective optimization algorithms on benchmark problems. It can improve the diversity and convergence of particles. The accuracy of the rolling force calculated by the proposed method is higher than that of the other models. Most errors of the rolling force fall within the range of ±5%. Compared with the original rolling schedule, the optimized rolling schedule performs better on the rolling power and slippage preventing between five stands.
YUNLONG WANG received the B.S. degree from the School of Electrical Engineering, Hebei University of Science and Technology. He is currently pursuing the master's degree with the College of Information Science and Engineering, Northeastern University, China. His research interests include industrial big data analysis and intelligent control.
JINKUAN WANG (Member, IEEE) received the M.S. degree from Northeastern University, Shenyang, China, in 1985, and the Ph.D. degree from the University of Electro-Communications, Chofu, Japan, in 1993. In 1990, he joined the Institute of Space Astronautical Science, Japan, as a Special Member. In 1994, he was an Engineer with the Research Department, COSEL, Japan. Since 1998, he has been a Professor with the College of Information Science and Engineering, Northeastern University. His research interests include intelligent control, adaptive array, wireless sensor networks, and optimal operation of smart grid.
CHUNHUI YIN received the B.S. degree from the School of Electronic Information Engineering, South-Central University for Nationalities. She is currently pursuing the master's degree with the School of Computer and Communication Engineering, Northeastern University, China. Her research interests include industrial big data processing and data mining.
QIANG ZHAO received the Ph.D. degree in navigation guidance and control from the College of Information Science and Engineering, Northeastern University, Shenyang, China, in 2017. He is currently a Lecturer of measurement and control technology and instrumentation program with the School of Control Engineering, Northeastern University at Qinhuangdao. His research interests include electricity theft detection, fault diagnosis of renewable energy generation systems, and optimal operation of smart grid.