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Author's personal copy Simulation Modelling Practice and Theory 22 (2012) 61–73
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Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
Application of genetic algorithms to optimization of rolling schedules based on damage mechanics Mehrdad Poursina a,⇑, Noushin Torabian Dehkordi b, Amin Fattahi c, Hadi Mirmohammadi d a
Faculty of Engineering, University of Isfahan, Isfahan, Iran Mechanical Engineering Department, Isfahan University of Technology, Isfahan, Iran c Faculty of Engineering, Islamic Azad University, Khomeini Shahr Branch, Iran d Mobarakeh Steel Company, Isfahan, Iran b
a r t i c l e
i n f o
Article history: Received 27 June 2011 Received in revised form 29 November 2011 Accepted 30 November 2011
Keywords: Rolling schedules Process optimization Genetic algorithms Damage evolution
a b s t r a c t It is well known that tandem cold rolling is one of the most widely used processes in the manufacture of various sheet products with high accuracy and production rate. This paper deals with an optimization problem for tandem cold rolling. A genetic algorithm is developed to optimize the reduction schedules from the power consumption and damage evolution points of view. Damage-coupled finite element simulations are employed to determine the damage objective function. The dominant parameters of the rolling process are calculated using an experimental–analytical model, obtained from an industrial tandem rolling mill. Generally, in rolling process damage and power have conflicting natures and none of them can be improved without degrading the other. In this paper, in the first step, power and damage are optimized independently and some reduction schedules are introduced to minimize power consumption or damage evolution during the process and the results are compared with the experimental observations. Afterwards power and damage are optimized simultaneously by defining a multi-objective function and employing the Pareto optimality; a set of optimized reduction schedules are provided to optimize the power and damage based on the preference ordering of the decision makers in tandem mill. This multi-objective optimization enables the mill operators to select the most appropriate optimized schedule according to the mill necessities. Finally the optimal schedules are numerically simulated to investigate the efficiency of the damage optimized schedule. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In tandem cold rolling, strip is rolled on a continuous production line, passing from one mill stand to another at high speed without stopping between the stands. Tandem rolling plays an important role in various industries as tandem mills can operate at very high speeds in which large reductions can be obtained with relatively close tolerance on flatness and thickness. In order to meet today’s market requirements, it is necessary to efficiently calculate and control the various process variables of a tandem cold rolling mill, to maintain high levels of product quality as well as productivity and to minimize the overall rolling cost. Some of these objectives can be achieved by proper selection of the reduction schedules [1,2]. Therefore, many attempts have been made to propose rolling schedules to optimize the operation of tandem mills with respect to the mill throughput, power consumption or other dependent parameters. Venkata Reddy and Suryanarayana [2] calculated the power consumption for various reduction schedules and presented a setup model for maximizing the throughput of ⇑ Corresponding author. Tel.: +98 311 7934516; fax: +98 311 7932746. E-mail address:
[email protected] (M. Poursina). 1569-190X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2011.11.005
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tandem rolling mills. Dixit and Dixit [3] applied fuzzy set theory to obtain an optimum reduction schedule in tandem rolling to minimize the energy consumption. Wang et al. [4] utilized a genetic algorithm method to develop an optimum design of rolling schedules for tandem cold rolling mills based on the power distribution, tensions, strip flatness and rolling constraints. Pires et al. [5] applied a non-linear simplex optimization method to setup an industrial cold mill, taking the quality and productivity aspects of the rolling mill into consideration. Murakami et al. [6] optimized the pass schedules considering the distributions and variations in strips properties and rolling conditions. Pires et al. [7] proposed an adaptation procedure for the setup generation of a four-stand tandem rolling mill, in order to optimize the friction and yield stress coefficients. Yang et al. [8] used ANN and Fuzzy method to optimize a multi-objective function in order to facilitate the design of new systems aiming at equating the relative load, preventing slip and obtaining the best shape. Overall, the optimization procedures conducted in the previous researches, were mostly aimed at maximizing the mill throughput and minimizing the operating cost, however, optimization of tandem cold rolling mills with respect to damage evolution through the strip is a missing point in the literature. Strip tearing during tandem cold rolling is one of the manufacturing issues in the rolling industry that can significantly increase production costs. In the previous paper of authors [9], an explicit finite element code coupled with the improved Lemaitre damage model was developed to predict strip tearing in a five-stand tandem rolling mill. Although previous researches have been carried out on using genetic algorithm in the rolling process, there has not been any attempt in using damage mechanics as a tool for optimization. In the present work, in addition to the power consumption optimization, the rolling schedules are investigated and optimized from the continuum damage mechanics point of view. The Genetic Algorithm (GA) method along with damage-coupled finite element simulations are employed to optimize the rolling schedules at an industrial tandem mill in order to minimize the probability of strip tearing and the mechanical defects of the final products which are induced by damage evolution during the process. In the first step, an empirical–analytical based code is developed to simulate the rolling process and calculate the rolling parameters. The simulation results are validated using the experimental observations obtained from the five-stand tandem mill at Isfahan Mobarakeh Steel Company (IMSC). Furthermore, a GA based optimization procedure is conducted for the purpose of minimizing the power consumption and the results are compared to the power optimization results reported in the literature. In the next step, a genetic algorithm is used to optimize the process with respect to damage evolution through the strip, independently of the power optimization procedure. In order to prevent unrealistic results, the optimal schedules obtained from the damage optimization are checked against the industrial rolling constraints, such as the maximum power capacity of drivers and the maximum obtainable reduction at each stand. Afterwards, an optimization procedure is carried out based on the Pareto optimality, by considering both damage evolution and power consumption, simultaneously. Finally, all of the obtained optimized schedules are simulated using a damage-coupled finite element model, in order to investigate damage evolution through the strip for each schedule.
2. Industrial tandem cold rolling mill Fig. 1 illustrates a schematic diagram of a five-stand tandem cold rolling mill at IMSC; each rolling stand consists of a pair of work rolls beside a pair of backup rolls. The distance between two adjacent stands is 4.5 m. The mechanical properties of the rolling mill are listed in Table 1. Based on industrial reports, strip tearing, which accounts for a large portion of the amount of productivity that is lost, mostly occurs at Stands 4 and 5.
Strip
Work roll Rolling direction
Backup roll
Fig. 1. Schematic diagram of an industrial five-stand tandem cold rolling mill.
Table 1 Mechanical properties of the industrial rolling mill [10]. Work rolls diameter (mm) Maximum roll force (ton) Initial strip thickness (mm) Final strip thickness (mm)
510–585 3000 1.5–3.5 0.18–3
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3. Modeling of the rolling process One-dimensional mathematical models for analyzing cold rolling process have been developed by many researchers [11–13]. These theories have been widely used in practice because of their ability in prediction of the roll force, roll torque and pressure distributions with reasonable accuracy. In the present study, an empirical–analytical approach, which is currently used at IMSC, is employed to develop a setup model for tandem cold rolling and evaluate the process parameters such as the roll forces, roll torques and the powers. In this model, the material is considered to be isotropic, rolls are assumed to be rigid and the coefficient of friction is constant at each rolling stand. The flattened roll radius, RB, can be obtained from the following relation [12]:
RB ¼ C 3 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 23 R2AW
ð1Þ
where RAW is the radius of the work roll and C3 is defined as follows:
C 3 ¼ RAW þ
1 ðA RAW K M Þ2 2ðh1 h2 Þ
ð2Þ
where h1 and h2 refer to the strip thicknesses at the entry and delivery sides of the roll stand. Moreover, A and KM are defined as:
A¼
16ð1 m2roll Þ ; pEroll
KM ¼
K 1 þ 2K 2 3
ð3Þ
In which, mroll and Eroll are the Poison’s ratio and the Young’s modulus of the work roll, respectively. For steel work rolls A = 2.2 105 mm2/N. Moreover, K1 and K2 represent the material yield stresses at the entry and the exit of the rolling stand, respectively. In the presence of inter-stand tensions, K1 and K2 are obtained from the following relations [12]:
h1 rb K 1 ¼ 1:1547 KF 0 þ KF T 1 h0 h2 rf K 2 ¼ 1:1547 KF 0 þ KF T 1 h0
ð4Þ ð5Þ
In these equations, KF0 represents the initial material yield stress, KFT is the increase in the material stress in the case of 100% reduction and h0 is the initial thickness of the strip. In addition, rb and rf are the back and front tension stresses, respectively. The roll torque, T, can be calculated as follows [12]:
T ¼ B R B Dh K 1 K2 RB 1 4 1 l RB 3 4 D2 þ D3 BG C D D2 þ D5 1þ 1 D2 þ BG l ð2C þ 1Þ BG 3 3 2 5 5 K1 h2 6h2 h1 þ rb rf h2 cdotB RAW h2 ð6Þ In which, B is the width of the strip, Dh =(h1 h2) is the thickness difference, l represents the friction coefficient and BG is the bite angle. Moreover, C and D are defined as:
1 l2 RB KF T h2 1þ 2 h2 K 1 h0 BF D¼ BG
C¼
ð7Þ ð8Þ
where BF is the neutral point angle. For the simplification and linearization purpose, an adaptive factor, obtained from experimental data, is applied to the calculated roll torque, T. The roll power, P, can be obtained from the following relation:
P ¼T x
ð9Þ
where x is the angular velocity of the work rolls. Adaptation factors are multiplied to roll force, roll torque and forward slips. These factors are obtained from Ref. [10]. 4. Effect of reduction schedule on damage evolution In the previous paper of authors [9] a damage-coupled finite element analysis was conducted to investigate strip tearing for an industrial five-stand tandem cold rolling mill; the improved Lemaitre damage model was utilized to predict damage evolution through the strip for various reduction schedules by distributing the strip thickness in arithmetic, geometric and
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harmonic series and for the reduction schedule proposed by Roberts [14]. The simulations were carried out for a special case in which the initial and final thicknesses of the strip were 2 mm and 0.6 mm, respectively, which corresponds to a total reduction of 70%. Table 2 shows the pure reduction percentage values at each rolling stand, ri, for these rolling schedules; ri is obtained from the following relation:
ri ð%Þ ¼
h1 h2 100 h1
ð10Þ
where i represents the stand number and h1 and h2 refer to the entry and exit strip thicknesses for that rolling stand, respectively. Moreover, the numerical predictions of the damage values at each stand, obtained from Ref. [9], are listed in Table 3. As it was shown in Ref. [9], the reduction schedule is an effective factor in damage evolution through the strip and it is necessary to select a proper reduction schedule to reduce damage in the process. Moreover it was mentioned that in addition to the reduction schedule, there are other effective factors for strip tearing during tandem rolling such as the defects in the input coils, problems related to the mechanical and controlling equipment and the variations of the friction coefficient, the effects of which were addressed in Ref. [9]. However the reduction schedule is the only manageable factor which can be controlled by the mill operator, therefore in the present work, the reduction schedule is considered as the dominant factor that govern the damage evolution and in order to develop a damage optimized schedule, the effect of other factors are not considered. In the present study, in order to investigate the effect of reduction distribution on damage evolution, the total reduction percentage at each rolling stand, Ri, which is defined as Eq. (11), is considered.
Ri ð%Þ ¼
h0 h2 100 h0
ð11Þ
where h2 represents the exit strip thickness for that rolling stand and h0 is the initial strip thickness (the entry strip thickness for Stand 1). By this definition the reduction percentage at Stand 5 would be equal to the total strip thickness reduction and therefore has the same value for all rolling setups (R5 ¼ Rtotal ¼ 20:6 ¼ 70%). However, the setups differ in the values of total 2 reduction percentage at Stands 1-4. Fig. 2 illustrates the total reduction distribution among the rolling stands, calculated from Eq. (11), for the rolling setups introduced in Table 2. The values of damage parameter and total reduction percentage are compared in Table 4 for the rolling setups. From this table it is clear that for Stands 1–4, damage parameter at each stand, Di, increases along with an increase in the reduction percentage, Ri; for the arithmetic setup, for which the total reduction has the minimum value at Stands 1–4, the damage parameter attains a minimum value among the rolling setups, and for this case Dmax = 0.15 at Stand 5. These numerical results are confirmed by the physical nature of damage evolution during the process and the analytical relations; as indicated in Ref. [9], the large plastic strains induced at the rolling stands are the primary reason for damage evolution through the strip and the influence of inter-stand tensions is negligible. Moreover, the equivalent plastic strain at each rolling stand equals to [14]: p eq
e
2 h1 ¼ pffiffiffi ln h2 3
ð12Þ
In addition, the Lemaitre damage model defines the damage evolution rate as a function of plastic flow as follows [15]:
Dep Y s DD ¼ 1D S
ð13Þ
Table 2 Percentage reduction schedules for a five-stand mill [9]. Stand No.
Roberts
Harmonic
Geometric
Arithmetic
1 2 3 4 5
32.50 29.62 23.15 15.06 1.61
31.35 23.81 19.31 16.11 13.84
21.15 21.17 21.15 21.12 21.08
13.90 16.14 19.25 23.84 31.30
Table 3 Damage parameter at each rolling stand for different reduction schedules [9]. Stand No.
Roberts
Harmonic
Geometric
Arithmetic
1 2 3 4 5
0.04 0.1 0.17 0.22 0.24
0.035 0.078 0.13 0.19 0.22
0.016 0.045 0.08 0.12 0.175
0.007 0.025 0.05 0.09 0.15
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80
Total Reduction (%)
70 60 Roberts
50
Harmonic
40
G eometric
30
Arithmetic
20 10 0 1
2
3 Stand No.
4
5
Fig. 2. Total reduction values at each rolling stand for different Setups.
Table 4 Total reduction and damage parameter for different rolling setups. Stand 1
Stand 2
Stand 3
Stand 4
Stand 5
Roberts
R (%) D
32.5 0.04
52.5 0.1
63.5 0.17
69 0.22
70 0.24
Harmonic
R (%) D
31.35 0.035
47.70 0.078
57.8 0.13
64.5 0.19
70 0.22
Geometric
R (%) D
21.15 0.016
38 0.045
51 0.08
61.5 0.12
70 0.175
Arithmetic
R (%) D
13.9 0.007
28 0.025
42 0.05
56 0.09
70 0.15
where S and s are material- and temperature-dependent properties and Y is the damage strain energy release rate. According to Eqs. (11) and (12), increasing the total reduction at each stand leads to decreasing the exit strip thickness for that stand and consequently increasing the plastic strains. Moreover, from Eqs. (12) and (13) it is clear that the damage parameter at each stand increases along with an increase in the plastic strains. Fig. 3 illustrates the variations of the maximum damage parameter versus the value of the function F(D), defined as:
FðDÞ ¼ R1 þ R2 þ R3 þ R4
ð14Þ
According to Fig. 3, the maximum damage value increases by increasing the value of the F(D) function. Therefore, it can be concluded that a way to minimize damage evolution through the strip is to develop a reduction schedule in which the sum of the total reduction percentage values at the first four stands is minimum. The optimization process is given in the following sections.
0.27
Damage Paramete
0.24 0.21 0.18 0.15 0.12 120
140
160
180 F(D) (%)
200
220
Fig. 3. Variations of the maximum damage value versus F(D).
240
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5. Soft computing techniques in metal forming optimization Soft computing, SC, is a collection of methodologies including Evolutionary Computation, Fuzzy Logic, Neuro-computing, and Probabilistic Computing [16]. SC aims at exploiting tolerance for imprecision, uncertainty and partial truth to achieve tractability, robustness and low cost solutions. It differs from conventional techniques as it incorporates human knowledge into the solution methodology. This paper focuses on genetic algorithm as a SC technology for the optimization problem. 5.1. Genetic algorithm The genetic algorithm is a search technique combining the Darwin theory of ‘‘survival of the fittest’’ together with stochastic structured data. This method has been successfully used in optimizing various metal forming processes [17–19]. This model is formulated through projection of individuals in a population, referred to as chromosomes. Each individual is evaluated in a related fitness function. New individuals are produced through the application of crossover and mutation operators while the old and redundant outliers are eliminated. One repetition of this cycle results in a new generation. The first generation of this process is produced randomly but in the following generations, fitness value is calculated based on effectiveness of each individual’s result. 5.2. Design variables and constraints Since the initial and final thicknesses of strip are known before rolling, the magnitudes of reduction for the five stands are not independent. Therefore the reductions for the first four stands are considered as the design variables and the last stand reduction percentage is considered as a dependent variable. It should be mentioned that in the developed code the interstand tensions are calculated using a modified system of nonlinear differential equations according to Ref. [20]. Two categories of constraints are considered in this paper. The first category consists of the industrial constraints and the second one includes the constraints obtained from the literature. 5.2.1. Industrial constraints In the industrial rolling mill, the maximum power and the maximum obtainable reduction at each stand are limited to the corresponding values due to the mechanical design of the mill and the electrical drive motors. The fifth stand of the tandem cold rolling mill has two vital tasks at IMSC. As the strip is annealed after rolling, in order to prevent strip adhesion during the annealing process, the surface average roughness, Ra, should be increased to 1–2 lm at the last stand. Moreover, the final shape of the strip is determined at this stand. Therefore, the amount of reduction percentage for the last stand, r5, is minimum among all rolling stands; the lower and upper limits of r5 are 5% and 10% respectively [10]. Additionally, the friction coefficient for this stand is the highest amongst all stands and reaches 0.1 [10]. The yield strength of the strip increases during tandem cold rolling process, therefore, the strip has the minimum magnitude of yield strength at the first stand. Experience shows that if the reduction percentage for Stand 1, r1, is greater than 35% the strip tearing occurrence is probable [10]. 5.2.2. Other constraints Too high inter-stand tensions may lead to strip tearing and too low tensions may cause looping of the strip. Therefore, as it has been reported in the literature, the upper and the lower limits of the inter-stand tension should be respectively equal to 2% and 33% of the yield stress of the strip, ry [2]. The next constraint is considered on forward slip. The magnitudes of forward slip are considered between 2% and 10% for 5–60% reduction [14]. The above mentioned industrial constraints in addition to the constraints reported in the literature are listed in Table 5. Moreover, the upper and lower bounds of the reduction as well as the maximum power at each stand at IMSC are presented in Table 6.
Table 5 Industrial and literature constraints for the five-stand rolling mill at IMSC. Upper bound of reduction at Stand 1 Lower bound of reduction at Stand 5 Upper bound of reduction at Stand 5 Lower limit of the inter-stand tension Upper limit of the inter-stand tension Minimum forward slip Maximum forward slip
35% 5% 10% 0.02ry 0.33ry 2% 10%
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M. Poursina et al. / Simulation Modelling Practice and Theory 22 (2012) 61–73 Table 6 Power and reduction constraints at IMSC [10]. Stand No.
Upper bound of reduction (%)
Lower bound of reduction (%)
Maximum power (KW)
1 2 3 4 5
35 40 40 40 10
15 15 15 15 5
3900 4800 4800 4800 5850
5.3. Objective functions Rolling process can be optimized from different aspects. For instance, some criteria can be considered for forming energy or power, slip, number of stands and wear of tools. In this paper, in addition to power optimization, the rolling process is optimized from continuum damage mechanics point of view, which has not been carried out in the previous researches, on the authors’ knowledge. 5.3.1. Power objective function An important goal which the rolling setups should meet is minimizing the power consumption and consequently maximizing the throughput of the rolling mill. Hence, the power consumption objective function can be defined as the sum of the powers of the rolling stands, described by the following relation:
FðpowerÞ ¼ FðPÞ ¼
N X ðPi Þ
ð15Þ
i¼1
where N is the total number of the stands which is equal to 5, i represents the stand number and Pi is the power consumption at each rolling stand. 5.3.2. Damage objective function In order to reduce the probability of strip tearing occurrence, the damage evolution during the process should be minimized. As it was mentioned in the previous sections, a way to meet this aim is to minimize the sum of the total reduction percentage values at Stands 1–4. Therefore, the damage objective function can be defined as follows:
FðDamageÞ ¼ FðDÞ ¼
N1 X ðRi Þ
ð16Þ
i¼1
where Ri represents the total reduction percentage at each rolling stand, which is obtained from Eq. (11). 5.3.3. Multi-objective function In order to optimize the power and damage simultaneously in a process, the total objective function is defined as:
FðTotalÞ ¼ ½FðDamageÞ; FðPowerÞ
ð17Þ
In a multi-objective optimization problem (MOP), a number of conflicting objective functions are to be optimized simultaneously. An ideal solution, at which each objective function gets its optimal value, usually does not exist due to the conflicting nature of objective functions. The solution of a MOP is associated with the definition of Pareto-optimal solutions [21]. The concept of Pareto optimality was introduced at the turn of the previous century by the Swiss economist Pareto [22]. A solution is said to be Pareto optimal if the value of any objective function, F(D) cannot be improved without degrading at least one of the other objective functions [22]. Two Pareto-optimal solutions are not comparable without some preference ordering, provided by the decision maker. A genetic algorithm can be used to find a representative set of Pareto-optimal solutions. 5.4. Genetic operators and optimization scheme Basically, the developed GA consists of four main operators to create a new generation: selection, cross-over, mutation and replacement. The general steps of the GA are schematically illustrated in Fig. 4. In the present study, the stochastic uniform method is applied as the selection approach. Stochastic uniform method is similar to the roulette-wheel selection method but individuals are selected simultaneously in this approach. In this method, individuals are mapped to a segmented line. The fitness value of each individual determines the size of the related segment on the line, exactly as in the roulette-wheel selection method. In a case that N individuals are needed to be selected, equally spaced pointers are placed over the line as many as the number of individuals to be selected. The distance between the pointers is equal to 1/N. A random number is generated in the range of [0, 1/N] as the position of the first pointer. In comparison
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Initial population
Generating new population
Fitness function evaluation
Selection (Stochastic uniform method)
Ranking
Cross-over (Scattered method) Replacement the old generation with the new one
Mutation
Fitness function evaluation
Is the stopping criterion satisfied?
Yes Stop
No Loop Fig. 4. General steps of the genetic algorithm.
Table 7 The industrial rolling setup properties.
Reduction (ri%) Angular velocity (rad/s) Coefficient of friction Roll diameter (mm)
Stand 1
Stand 2
Stand 3
Stand 4
Stand 5
28.8 260.3 0.045 527.4
28.8 358.9 0.044 550
25.9 464.6 0.045 573.3
21.6 589.4 0.06 576.1
5 627.4 0.085 583.4
with the roulette-wheel method, the stochastic uniform method ensures the selection of an offspring which is closer to what is deserved. The scattered method and the adaptive feasible approach are used for the cross-over and mutation functions, respectively. In scattered method, a random binary vector is created. When the element of the vector is 1, the gene is selected from the first parent and if the element of the vector is 0, the gene is selected from the second parent. At the end, selected genes are combined to form a child. Adaptive feasible method randomly generates directions that are adaptive with respect to the last successful or unsuccessful generation. The general genetic model was constructed in MATLAB Software 7.9. The initial population of individuals was assigned to 40. The crossover rate was set to 0.8. The evolutionary process is stopped if the mean fitness of the best solution group does not change during a specified number of generations, 10 generations, otherwise the new population will evolve into the next generation. In the numerical calculations, an industrial rolling program, belonging to IMSC, with the properties listed in Table 7 was considered. For this rolling setup, the initial and final thicknesses of the strip were 2.5 mm and 0.7 mm respectively. The strip width was equal to 1170 mm. The power consumption for this case was calculated using the empirical–analytical model introduced in Section 3 and was used to investigate the efficiency of the developed GA for rolling optimization. 6. Results and discussions 6.1. Power optimization In this section, the optimization process aims at minimizing the power consumption, hence, Eq. (15) is considered as the objective function. In the first step, the optimization process was carried out without consideration of the reduction and
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power constraints, listed in Table 6, in order to investigate the optimum schedule in the absence of the industrial constraints. In this case, which is referred to as Case 1P hereafter, the reduction value at all stands is considered in the range of 5–40%. The GA optimization procedure converged after 52 generations. Fig. 5 illustrates the objective function evolution. The obtained optimal solution for the reduction values corresponds to the vector r = [40, 29.6, 26.5, 5, 5], where the first component refers to the reduction percentage at Stand 1 and the last one corresponds to the reduction percentage at Stand 5. In the next step, the optimization process was conducted by applying the industrial constraints, introduced in Table 6. The optimum reduction values for this case, referred to as Case 2P hereafter, were obtained as r = [35, 33.6, 28, 5, 5]. The optimal rolling schedules and their corresponding power consumption values for these two cases are compared to the empirical rolling schedule, currently used at IMSC, in Table 8. 6.2. Damage optimization In this section, the damage objective function, defined by Eq. (16), is minimized, independently of the power objective function. The optimization procedure was carried out for both Cases 1D and 2D: in the absence of the industrial reduction constraints and with consideration of them. The obtained optimized reduction schedules as well as the power consumption for each case are presented in Table 9. In the following section, the power and damage will be simultaneously optimized using the Pareto search method.
Best
Fitness
Mean
Generation Fig. 5. Evolution of the best and mean fitness.
Table 8 Rolling parameters for the industrial and power optimizes setups. Stand No.
1 2 3 4 5 Total power
Reduction (r%)
Power (kW)
Industrial setup
Case 1P
Case 2P
Industrial setup
Case 1P
Case 2P
28.8 28.8 25.9 21.6 5
40 29.6 26.5 5 5
35 33.6 28 5 5
2651 3785 3800 4560 2593 17,389
3782 4133 3442 579 3890 15,826
3050 4800 3683 579 3890 16,002
Table 9 Rolling parameters for industrial and damage optimized setups. Stand No.
Reduction (r%) Industrial setup
1 28.8 2 28.8 3 25.9 4 21.6 5 5 Damage function value, F(D) Total power
Total reduction (R%)
Power (kW)
Case 1D
Case 2D
Industrial setup
Case 1D
Case 2D
5 5 13.8 40 40
18.3 29.1 30.1 23.2 10
28.8 49.3 62.4 70.6 72 211.1
5 9.75 22.2 53.3 72 90.25
18.3 42.1 59.5 68.9 72 188.8
Industrial setup
Case 1D
Case 2D
2651 3785 3800 4560 2593
289 504 2182 9988 9259
1284 4178 4787 4800 4606
17,389
22,222
19,655
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6.3. Multi-objective function optimization In contrast to single-objective optimization, for multi-objective problems, there is no single global solution and it is often necessary to determine a set of points that all fit a predetermined definition for an optimum [22]. The predominant concept in defining an optimal point in this research is that of Pareto optimality. In this approach, the aim is to find as many Pareto-optimal solutions as possible. The decision maker is going to select among these solutions according to his preferences, i.e. according to the significance he gives to each one of the objective functions fi(x). P To this end the weighting approach is employed. A cost function JðxÞ ¼ ni¼1 wi fi ðxÞ is minimized with respect to x [22]. The weights, wi, represent the importance that is attributed to each objective fi(x). The minimum x⁄ of J(x) is found through a genetic algorithm. In this work f1(x) is the power function and f2(x) is the damage function and the reduction percentage is the optimization variable. As a matter of fact, employing this method provides the operator of the rolling mill with the opportunity to choose an optimal reduction schedule from a variety of available optimum points, based on the importance level of the power consumption and damage evolution during the process. The multi-objective function, defined by Eq. (17), was optimized using the above mentioned approach and the optimization results as well as their corresponding numerical values are presented in Fig. 6 and Table 10, respectively. 6.4. Discussions It can be concluded from the optimization results listed in Table 8 that the power consumption would be minimum for Case 1P, however, in this case the practical rolling constraints are violated and it is not applicable in practice. Therefore, the rolling schedule obtained for Case 2P is proposed as the optimal schedule, in order to minimize power consumption in the process. This optimal schedule predicts a descending trend for the reduction distribution among the rolling stands; the maximum reductions should be applied at the initial stands and it corresponds to the results reported in the literature [2–4]. In this case the power consumption equals to 16,002 kW which indicates a reduction of 8% comparing to the industrial setup. Moreover, according to Table 9, the magnitude of damage function equals to 211.1, 90.25 and 188.8 for the industrial setup, Case 1D and Case 2D, respectively. Therefore, the best results are obtained for Case 1D, however, for this case the industrial reduction constraints are violated at Stands 4 and 5 and the power constraints are violated at all stands excluding
Damage Function
230
1 2
220
3 4
210
5 6
200
7 8
9
190
10
180 1.72
1.74
1.76
1.78
1.8
1.82
1.84
1.86
Power Function
1.88 x 104
Fig. 6. Pareto search for power and damage optimization.
Table 10 Pareto optimal solutions. Optimal point No.
Power function (kW)
Damage function
r1 (%)
r2 (%)
r3 (%)
r4 (%)
r5 (%)
1 2 3 4 5 6 7 8 9 10
17,217 17,357 17,516 17,746 17,972 18,155 18,269 18,346 18,633 18,773
225.6 220.6 217.7 212.6 204.5 197.4 195.6 191.7 187.2 182.0
34.9 32.2 32.2 29.2 26.0 21.6 21.4 19.9 19.3 17.0
30.7 30.6 28.9 30.4 28.8 29.3 29.1 28.7 26.3 26.1
23.0 25.8 24.8 24.1 27.3 30.3 29.0 29.9 29.9 30.8
15.1 15.2 17.3 20.3 20.3 22.0 23.4 22.8 26.1 26.9
5.0 5.4 6.6 6.2 8.3 7.1 7.8 9.5 9.2 9.9
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Stand 4. Consequently, the optimum results obtained for Case 2D can be proposed as the optimum schedule for minimizing damage evolution during the process. The damage evolution during the process for this optimized schedule as well as the other optimized setups will be investigated in the following section, by using finite element simulations. It should be noticed that in the absence of the reduction constraints (Case 1D) the damage optimization results predict an ascending trend for the reduction distribution among the rolling stands; the maximum reduction values should be applied at the last two stands and it is in contrast to the power optimization results for Case 1P, for which the reduction distribution had a descending trend. From the mechanical point of view, power consumption and damage evolution are conflicting with each other, i.e. improvement of one objective causes the degradation of the other and it is due to the contrasting effects which the work hardening phenomenon has on these two competing objectives; the yield strength of the strip is minimum at the first stand while it is maximum at the last stand due to the work hardening. As the power at each stand increases by increasing the yield strength, the total power consumption would be minimum if large amounts of reduction are applied at the initial stands, while, applying large reductions at the last stands, in which the strip yield strength is high, would reduce damage evolution during the process. Finally, from the multi-objective optimization results illustrated in Fig. 6, there are various optimum points available to choose based on the rolling mill conditions and necessities. In this figure, Points 1 and 10 represent extreme cases. In the cases that the power optimization is of the utmost importance, the optimum point should be selected from the left half of the diagram (Points 1–4) and when the power is of the secondary importance and damage optimization outweigh the power optimization, the point should be selected from the right half of the diagram (Points 6–10). Finally, if the power and damage are of the same level of importance, i.e. for the cases in which the prevention of strip tearing is as important as saving energy, the reduction schedule corresponding to Point 5 would be the best option to be employed in the industrial rolling mill. The magnitudes of reduction for the five stands corresponding to point 5 are listed by bold values in Table 10. 7. Finite element simulation In order to investigate the damage evolution in each optimal schedule obtained in the previous sections, in this section, the power and damage optimized schedules as well as the Pareto optimal solution, corresponding to Point 5 in Fig. 6, are simulated using the damage-coupled finite element model developed in Ref. [9]. The optimal reduction schedules, previously obtained for these cases, are rephrased in Table 11. The value of the damage parameter at each stand, which was obtained from the numerical simulations, is shown in Fig. 7 for each optimal rolling schedule. Moreover, the numerical results for the damage optimized schedule are shown in Fig. 8. It is clear from Fig. 7 that the damage parameter attains its minimum value for the damage optimized schedule; in this case the maximum damage value is Dmax = 0.18 at Stand 5, which indicates a reduction of 18% in damage evolution in comparison with the industrial setup, for which Dmax = 0.22. Therefore, the numer-
Table 11 Industrial and optimal percentage reduction schedules for a five-stand mill. Stand No.
Damage optimized setup
Power optimized setup
Pareto optimal setup
Industrial setup
1 2 3 4 5
18.3 29.1 30.1 23.2 10.0
35.0 33.6 28.0 5.0 5.0
26.0 28.8 27.3 20.3 8.3
28.8 28.8 25.9 21.6 5.0
Damage Parameter
0.25 0.2
Power optimized setup
0.15
Pareto optimal setup Damage optimized setup
0.1
Industrial setup
0.05
0
1
2
3
4
5
Stand No. Fig. 7. Damage parameter at each stand for different reduction schedules.
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Damage
Rolling Direction
Symmetry
(a)
(b)
(c) (d)
(e) Fig. 8. Damage evolution through the strip for the damage optimized setup at: (a) Stand 1, (b) Stand 2, (c) Stand 3, (d) Stand 4, (e) Stand 5.
ical results confirm the efficiency of the developed optimal schedule for minimizing damage evolution during the process. On the other hand, for the power optimized reduction schedule, the damage value at each stand attains its maximum value among the reduction schedules; Dmax = 0.24 at Stand 5 for this case and it is due to the conflicting nature of the damage evolution and power consumption during the process. 8. Conclusion This work was an attempt towards a genetic algorithm based optimization of the reduction schedules in tandem cold rolling process. Power consumption and damage evolution were independently optimized in order to minimize the production cost and the strip tearing probability in an industrial five-stand rolling mill. An empirical–analytical model was employed to calculate the governing parameters of the process. A genetic algorithm was developed for the optimization procedure with consideration of two categories of constraints: the industrial constraints and the constraints obtained from the literature. The power optimization results predicted a descending trend for the reduction distribution among the rolling stands and it was consistent with the optimal schedules reported in the literature. The optimization results indicated a reduction of 8% in power consumption. In the presence of the industrial constraint, the damage optimization results predicted an upward trend for the reduction values from Stand 1 to Stand 3 and a downward trend from Stand 3 to Stand 5, while in the absence of the industrial constraints it predicted an ascending trend for the reduction values which is in contrast to the obtained power optimized schedule. This inconsistency stems from this fact that the work hardening phenomenon reduces the damage evolution while increases the power consumption during the rolling process. In order to solve this inconsistency, the Pareto optimality was used to optimize the damage-power multi-objective function and a variety of optimal points was offered, so that the decision maker can select the best one according to the mill conditions and necessities; for a case study with the total reduction of 72% in which damage and power are of the same level of importance, the reduction distribution of 26%, 28.8% 27.3%, 20.3% and 8.3% was proposed for Stands 1–5, respectively. Finally the obtained optimal schedules were simulated using a damage-coupled finite element model to investigate damage evolution during the process for each schedule. The numerical results confirmed the efficiency of the proposed damage optimized schedule as the optimization resulted in a reduction of 18% in damage evolution. References [1] W.L. Roberts, Flat Processing of Steel, Marcel Dekker, New York, 1995. [2] N. Venkata Reddy, G. Suryanarayana, A set-up model for tandem cold rolling mills, J. Mater. Process Technol. 116 (2001) 269–277.
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