Operation Optimization in the Hot-Rolling Production Process

Article pubs.acs.org/IECR

Operation Optimization in the Hot-Rolling Production Process Li Chen,†,‡ Xianpeng Wang,‡ and Lixin Tang*,‡ †

State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110819, People’s Republic of China ‡ The Logistics Institute, Northeastern University, Shenyang, 110819, People’s Republic of China ABSTRACT: The operation optimization problem (OOP) is an important issue in production process control and optimization in process industries, because the desired solution of OOP is the optimal setting for control variables, and this setting affects the product quality to a great extent. In this paper, the OOP in the hot-rolling production process of iron and steel industry is investigated. The OOP lies between the production scheduling layer and the process control layer in the integrated automation system in iron and steel industry, and its main task is to set the optimal values for control variables (i.e., rolling force, exit thickness, etc.) based on the production process constraints. These values are then set as the targets for the process control layer. Different from previous operation optimization of hot rolling process considered in the literature, the mathematical model constructed in this paper considers more practical process constraints such as the ramping constraints, to get high-quality products. To efficiently solve this problem, a hybrid self-adaptive genetic algorithm (HSaGA) is presented. The main feature of the HSaGA is that it utilizes four different crossover operators and can self-adaptively select the operators that are most appropriate for the current problem. Such a strategy can help to improve the convergence speed and robustness of the genetic algorithm. Besides this strategy, a solution selection strategy is also proposed to select parent solutions to perform the crossover operation. This strategy divides the entire population into two partsone with good quality in the objective function value and the other one with good quality in diversityand then the parent solutions are selected from them, to improve the search diversity during evolution and, thus, avoid being trapped in a local optimum. Experiments are carried out based on benchmark test problems and practical OOP of the hot-rolling production process. The computational results show that the proposed HSaGA is superior to many state-of-the-art evolutionary algorithms in the literature for benchmark problems and can obtain better results than the empirical method for the OOP of the hot-rolling process.

1. INTRODUCTION The iron and steel industry belongs to the generalized chemical industry. In the production process of the iron and steel industry, hot rolling is an important process in which almost half of the finished products are processed. With the development of science and technology, the focus of plate and strips production in the hot-rolling process is changing from pursuing large scale, high speed, and continuity to savings, energy consumption, and improvement of product quality. The energy consumption and final product quality are heavily dependent on the setting of operation parameters in the hot-rolling production processes. The optimal parameter setting can help to improve the product quality, such as the accuracy of shape and thickness of strips, and make full use of the equipment’s capability. The main production process of the hot rolling line is illustrated in Figure 1, which is obtained from a major iron and steel enterprise in China. The hot-rolling production line mainly consists of a reheating furnace, a roughing mill, a finishing mill, laminar cooling, and a coiler. In the reheating furnace, the slabs are heated to the required rolling temperature. After a slab comes out of the reheating furnace, it is transported to the roughing mill through the conveyor, and then rolled through the roughing mill, which is generally designed to transform the thickness of slabs from 70−90 mm to 30−50 mm. After the rough rolling, the thin slab is transmitted to the finishing mill along the roller conveyor. At the entry into the finishing mill, the temperature and the thickness of this slab are first measured using a thermometer and a thickness gauge and then sent to the operation optimization system. © 2014 American Chemical Society

According to this information, the operation optimization system determines appropriate values for control variables in the finishing mill quickly, and then the slab is rolled to a thinner strip whose thickness is ∼1.2−25.4 mm through seven consecutive stands. Finally, the strip is cooled and then coiled in the coil box. Since the processing time in the roughing mill is very short (∼1 or 2 min), it is required that the operation optimization system must obtain the best control variables as soon as possible. The operation optimization system plays an important role in hot-rolling production and it lies between the production scheduling layer and the process control layer in the integrated automation system in the hot-rolling line, as shown in Figure 2. In practical production, the production scheduling layer first determines the sequence of slabs to be rolled, and then the slab’s information on width, thickness, temperature, rated power, rated rolling force, rated rolling torque, target thickness, and so on is sent to the operation optimization module. In the operation optimization module, the main task is to set the optimal values for control variables (i.e., rolling force, exit thickness of each stand, etc.) within process constraints through optimization algorithms, to guarantee a good quality of rolled products and reduce energy consumption. Finally, these values are sent to the process control layer to be set as the control targets, which, in Received: Revised: Accepted: Published: 11393

December 28, 2013 June 12, 2014 June 16, 2014 June 16, 2014 dx.doi.org/10.1021/ie404409r | Ind. Eng. Chem. Res. 2014, 53, 11393−11410

Industrial & Engineering Chemistry Research

Article

Figure 1. Main process of the hot-rolling production line.

Figure 2. Automation architecture of the hot-rolling line.

turn, guarantee the smooth running of the production and the good quality of final products. Therefore, how to get the optimal setting of control variables has become an important problem, which is called the operation optimization problem (OOP) in the

operation optimization module. OOP is essentially a real-time optimization problem in the chemical industry. The target of OOP is to determine the optimum setting parameters to get maximum economic benefits. Like the OOP in chemical industry, the 11394

dx.doi.org/10.1021/ie404409r | Ind. Eng. Chem. Res. 2014, 53, 11393−11410


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algorithms (EAs) for such types of problems has achieved great development in recent years and many types of self-adaptive strategies for evolution have been presented in the literature. Brest et al.13 presented a new DE algorithm (denoted as jDE) in which the parameter settings can be obtained by a self-adaptive control strategy, and the computational results showed that the jDE algorithm had good performance on numerical benchmark problems. Zhan et al.14 presented an adaptive particle swarm optimization (APSO) to improve the search efficiency and convergence speed. In the above two algorithms, the self-adaptive strategies are mainly focused on the control of the parameters. Some researchers also proposed the self-adaptive strategies whose focus is on the selection of both parameters and evolution strategies for EAs. The self-adaptive DE designed by Pan et al.15 constructed a list of parameter setting and evolution strategies for each trial individual, and this solution can adaptively select the appropriate parameter setting and evolution strategy according to their previous successful experiences. Zhang et al.16 proposed a so-called JADE by implementing a new mutation strategy and updating control parameters in an adaptive manner. Qin et al.17 proposed another famous self-adaptive DE algorithm called SaDE, in which both trial vector generation strategies and their control parameter values adopt a self-adaptive strategy. The selfadaptive strategy is also used in the genetic algorithm (GA). Smith et al.18 presented a genetic algorithm called APES that incorporates the self-adaptation of genetic material into recombination and mutation. Hinterding et al.19 proposed a self-adaptive genetic algorithm (SaGA) which uses population-level and individual-level adaptive parameters to self-adaptively change the population size and the mutation strength. Zhang et al.20 presented a self-adaptive strategy for mutation to improve the performance of GA. Yang et al.21 proposed a self-adaptive mutation genetic algorithm based on family competition principles and adaptive rules. Yang et al.22 proposed genetic algorithm with adaptive mutations. The proposed algorithm successfully integrated the decreasing-based Gaussian mutation and the selfadaptive Cauchy mutation for the balance of the exploitation and exploration. Korejo et al.23 presented a comparative study between a population-level adaptive mutation operator and a gen-level adaptive mutation operator on multidimensional benchmark function optimization. The results showed that the adaptive mutation operators with the gene level are usually more efficient than that with the population level for GAs. Motivated by the successful self-adaptive strategies in previous EAs, in this paper, we propose a hybrid self-adaptive genetic algorithm (HSaGA) for the real-valued single objective optimization. Different from the previous self-adaptive genetic algorithms, the HSaGA in this paper has the following features. (1) Four different types of crossover operators are adopted in the HSaGA, to improve the robustness of the HSaGA for different types of problems. (2) The self-adaptive mechanism is not focused on the mutation parameters, but on the selection of crossover operators. The selection probability of each crossover operator is based on its successful survival rate during evolution. (3) Instead of the traditional random selection method used in traditional GAs, a solution selection strategy is designed to select parent solutions for a crossover operator. This selection strategy divides the population into two parts: one with solutions that have good quality in the objective function and the other one with solutions that have inferior quality in the objective function but good quality in

main task of the OOP in hot rolling is also focused on economic benefits, that is, to achieve the optimal exit thickness of each stand through optimization algorithms, based on which the control variables (i.e., rolling force, rolling torque, temperature of each stand, etc.) can be calculated accordingly, to maximize the product quality and economic benefits. Besides the iron and steel industry, the OOP can be found in many production processes in the petrochemical industry. Tarafder et al.1 proposed a multiobjective optimization model to deal with the operation optimization of an ethane cracking furnace. Li et al.2 investigated operation optimization of the naphtha pyrolysis process and proposed a multiobjective particle swarm optimization (MOPSO) and artificial neural network (ANN) hybrid model. Currently, research on the OOP of hot rolling in the iron and steel industry mainly concentrates on the optimization of the computing models, but few papers focus on its operation optimization problem. Wang and Chen3 proposed a rolling-force calculation model and used data mining techniques to adjust the value of key coefficients. Yamada et al.4 developed a mill setup calculation system and a method to adapt the algorithms designed for the load distribution of seven-stand mill to the four-stand rolling mill. Lee and Lee5 improved the accuracy of rolling-force prediction by a long-term learning method. Wang et al.6 developed a rolling force model with temper rolling strip mill, which considers the inhomogeneous stress distribution. Besides the optimization of the computing models, some papers also focus on the optimization of the load distribution (including rolling force, rolling torque, exit thickness, reduction, crown, and so on) for the OOP. Wang et al.7 presented an algorithm called MSCGA to solve the load distribution optimization problem of a hot strip mill to achieve a balance of rolling forces. Li et al.8 introduced a Newton descendent numeric algorithm to obtain a better distribution of strip thickness along the stands, and the computational results showed that the Newton descendent numeric algorithm is suitable for online application. Yao and Yang9 presented an improved differential evolution (DE) algorithm with adaptive weight for solving the load distribution of hot rolling. Li et al.10 proposed a genetic algorithm with variable metric to optimize the rolling regulation of the hot rolling mill. In present practical production of most iron and steel enterprises in China, the OOP is solved based an empirical method in which the parameters are set by operators. The empirical method can provide the setting of exit thickness for each stand and, subsequently, the other variables can be determined by these thickness values. Although this method is feasible, the optimality of these settings cannot be guaranteed, which, in turn, may decrease the quality of strips. Therefore, it is important to distribute a reasonable exit thickness for each stand, because these exit thicknesses directly decide the stability of the rolling process and affect the output and the properties of products.11,12 Thus, the OOP is very important for the iron and steel enterprises. To achieve the reasonable exit thicknesses, we need to consider many practical factors, including the equipment limitations and the manufacturing technology limitations. Many variables in the OOP interact with each other and their calculation models are very complex. In this paper, we also investigate the setting calculation of load distribution for the OOP of the hot-rolling process and, for this problem, develop a nonlinear model that is different from the models constructed in refs 7−10. This model can be viewed as a high nonlinear single objective problem in continuous space, and it is very difficult to be solved by an exact method. So we develop a hybrid evolutionary algorithm to solve this problem. This motivation is based on the fact that the research of evolutionary 11395



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0 ≤ Ni ≤ Nimax

diversity. Such a strategy can help to improve the convergence speed and, at the same time, maintain search diversity. The rest of this paper is organized as follows. Section 2 presents the mathematical model for the operation optimization problem of the hot-rolling process. The details of the proposed HSaGA are presented in section 3. Section 4 reports and analyzes the computational results on benchmark problems and practical OOP of the hot-rolling process, and compares the HSaGA with other state-of-the-art EAs in the literature. Finally, the paper is concluded in section 5.

0 ≤ Mi ≤ Mimax hi ≤ hi − 1

i = 3 , ..., n

(4) (5)

(7) (8)

Mi ≤ β2Mi − 1

i = 3 , ..., n

N1 ≤ γ1N2

(9) (10)

Ni ≤ γ2Ni − 1

|tn − t target| tn

(3)

(6)

M1 ≤ β1M 2

i = 3 , ..., n

≤a

(11)

(12)

n

∑ Δhi = ho − hn

(13)

i=1

In the above constraints, Pi, Ni, and Mi are the rolling force, the rolling power, and the rolling torque of the ith stand, respectively, and Pimax, Nimax, and Mimax are the maximum rolling force, the maximum rolling power, and the maximum torque of the ith stand, respectively. h0 is the entrance thickness of the first stand, and hi is the exit thickness of the ith stand. α1, α2, β1, β2, γ1, γ2, a are the coefficient parameters, and Δhi = hi−1 − hi is the reduction of the ith stand. tn is the temperature of the last stand and ttarget is the target temperature of the slab at the last stand. Constraints (2)−(4) are the capacity constraints of the equipment, which require that the rolling force, the rolling power, and the rolling torque of each stand should not exceed their maximum limitations. Constraints (5) ensure that the strip’s thickness of each stand should be changed in a nonascending order. Constraints (6)−(11) are the ramping constraints. Constraints (6), (8), and (10) ensure that the rolling force, the rolling torque, and the rolling power of the first stand should be less than those of the second stand, because these values of the first stand should not be too large, so that the slab can be successfully bit in the first stand. Constraints (7), (9), and (11) require that the rolling force, the rolling torque, and the rolling power from the third stand to the last stand should be in a nonascending order, to guarantee the production stability. Constraint (12) is the temperature constraint requiring that the deviation between the exit temperature and the target temperature in the last stand should be lower than a given value a. Constraint (13) ensures that the sum of the whole reduction at all stands equals to the difference between the entrance thickness of the first stand and the exit thickness of the last stand. 2.2. Main Mechanism Models of the Hot-Rolling Process. The main mechanism models include the rolling force model, the temperature model, the rolling torque model, the velocity model and the crown model. These models have been validated in laboratories and factories. According to Sun11 and Zhao,12 these models can be briefly described as follows. (1). The Rolling Force Model. The rolling force is one of key parameters of the setup calculation of OOP. The rolling force is calculated according to the following equation:

(1)

In the objective (1), CRi, hi, and CRi/hi are the crown, exit thickness, and relative crown of the strip in the ith stand, respectively. CRr, hr, and CRr/hr are, respectively, the target crown, target thickness, and target relative crown in the last stand, which is given in advance. (2). The Constraints. In order to get reasonable exit thickness, we must consider the electric motor capacity, the technical limitations, the shape requirements, and so on, as follows: i = 1, 2 , ..., n

i = 1, 2 , ..., n

Pi ≤ α2Pi − 1

n

0 ≤ Pi ≤ Pimax

i = 1, 2 , ..., n

P1 ≤ α1P2

2. MODEL DESCRIPTION 2.1. Model of the Operation Optimization Problem (OOP). As mentioned in the previous section, the other control variables, such as the rolling force, the rolling power, and the torque, can be obtained from the exit thickness of each stand, based on the mechanism models. So, in the OOP of the hotrolling process, the control variables are the exit thickness of the first six stands, because the exit thickness at the last stand has been fixed as the target thickness of the rolled strip. In this paper, we propose a nonlinear mathematical model in which the optimization of the strip shape is taken as the objective function. This objective function is different from that considered in the work of Li et al.10 In practical hot-rolling production, we find that the decision makers usually pay more attention to the quality of the strip than the energy consumption of the hot-rolling mills. In addition, in the practical process of hot rolling, the rolling reduction is somewhat smaller for the first stand than the maximum value of the equipment capacity, in order to allow the slabs to be bitten easier. The second and third stands need to make full use of the equipment’s capacity, and thus we can give a larger impossible reduction to the two stands. From the fourth stand to the last stand, the reduction is gradually reduced in order to get the requirement shape, thickness, and other performance indicators. The constraints we consider in the model are also different from the model of Li et al.,10 in which only the technical constraints such as the capability of the stands were considered, while, in our model, we further consider the ramping requirement of the exit thickness of each stand. (1). The Objective Function. This paper takes the strip’s shape as the optimization target. In the hot-rolling process, the first three stands are mainly used to roll the slab into a thin strip, and the remaining stands are usually used to adjust the strip’s shape. Since it is recognized that a good shape can be obtained if the relative crowns of the entry stand and the target relative crown at the last stand are equal, we prefer to take the minimization of the total deviations of relative crowns between each stand and the given target as the objective function. 2 ⎛ CR i CR r ⎞ − minimize ∑ ⎜ ⎟ h hr ⎠ i=4 ⎝ i

i = 1, 2 , ..., n

Pi = Bc Q ipKlic′

(2) 11396

i = 1, 2 , ..., n



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coefficient, Lj the distance from the (j − 1)th stand to the jth stand, hn the thickness of the nth stand, and vn the velocity of the nth stand. (3). The Rolling Torque Model. The rolling torque model is given as follows:

where Bc is the average width of the slab, l′ic the distortion length of the deformed work roll, Qip the stress state coefficient, and K the deformation-resistance coefficient. The definitions of lic′ , Qip, and K are given as follows. • l′ic can be defined as lic′ =

Mi = 2Piφlic′

⎡ ⎛ P ⎞⎤ R i⎢1 + 2.2 × 10−5⎜ i ⎟⎥Δhi ⎣ ⎝ Bc Δhi ⎠⎦

where φ is the arm coefficient. (4). The Velocity Model. According to the relationship of the mills, the velocity can be set by the flow rate, based on the exit thickness of each stand; that is,

i = 1, 2 , ..., n

where Ri is the radius of the work roller. • Qip is defined as

Bc vh i i = Bc vnhn

Q ip = 0.849 − 0.3393εi + (0.2488 + 0.3393εi l + 0.0732εi 2) ic him

i = 1, 2 , ..., n

where vi is the velocity of the ith stand. (5). The Crown Model. Many factors affect the crown, and the equation can be descripted as follows:

i = 1, 2 , ..., n

where lic is the length of the deformed work roll, him the average thickness of the entry thickness and exit thickness in the ith stand, and εi the relative reduction (εi = (hi−1 − hi)/hi−1). • The parameter K is given as

CR i =

Pi F − − Eω(ωH + ωW + ω0) + Ecωc + CR 0 Kp KF

i = 1, 2 , ..., n

where Kp is the lateral stiffness coefficient of the mill, F the the bending force, KF the lateral stiffness coefficient of the bend roller, Eω the comprehensive influence coefficient of the work roller, ωH the thermal profile of the work roll crown, ωW the wear profile of the roll crown, ω0 the initial crown, Ec the influence coefficient of the controlled roll crown, ωc the controlled roll crown, and CR0 a constant. As done in the works of Sun11 and Zhao,12 only the first item (Pi/Kp) is considered in our model.

⎛u ⎞ K = 1.15σ0 exp(a1T + a 2)⎜ m ⎟ ⎝ 10 ⎠ ⎡ ⎛ e ⎞a5 ⎛ e ⎞⎤ × ⎢a6⎜ ⎟ − (a6 − 1)⎜ ⎟⎥ ⎝ 0.4 ⎠⎦ ⎣ ⎝ 0.4 ⎠

(a3T + a4)

i = 1, 2 , ..., n where σ0, a1, a2, a3, a4, a5, and a6 are the regression coefficients, T is the temperature, um is the average strain rate, and e is the degree of deformation. (2). The Temperature Model. Temperature is one of the highlights of setting the calculations for hot rolling, and it is important to calculate the temperature change accurately, because the setting of rolling force and deformation resistance is closely related to the temperature. In this paper, the radiation temperature drop model and the rolling temperature calculation model are used. (a). The Radiation Temperature Drop Model. The temperature of the slab would fall when the high-temperature slab is transported to the roller, because the slab radiates energy. The radiation temperature drop model is described as follows:

3. PROPOSED HSaGA As mentioned in the Introduction section, we prefer to develop a HSaGA for the OOP of the hot-rolling process, and the detailed procedure of the proposed HSaGA is described in this section. Please note that the proposed HSaGA can also be used to solve the other single-objective optimization problems over continuous space. In the following, we first give the overall framework of this algorithm and then elaborate each component of it in the following sections. Let P denote the population with n solutions, Xi = (xi1, xi2, ..., xiD) denote the ith solution in the population at generation g in which D is the dimension of variables, f(Xi) denotes the objective function value of solution Xi, and Cr denotes the mutation probability. In addition, we introduce a new population Pnew to store new solutions generated in each iteration and correspondingly denote Xi,new as the ith new solution in Pnew. With these notations, the procedure of the proposed HSaGA is presented in Figure 3. 3.1. Initial Population Generation Method. In general, the initial population can be generated by a random method, i.e., each dimension xij (j = 1, 2, ..., D) of a solution Xi is determined as xij = LBj + rand(0,1) × UBj, where LBj and UBj are the lower bound and upper bound of dimension xij, respectively, and rand(0,1) is a random number uniformly generated in [0,1]. This random method may be effective for many problems; however, this method cannot guarantee that a good diversity of population can always be obtained. If the initial population is not diverse enough, then the search results for complex and unimodal problems may be unsatisfactory. Therefore, in this paper, we prefer to adopt a diversification method to generate the initial population P, following the main idea of ref 24. Step 1. Divide the range of each dimension [LBj, UPj] (j = 1, ..., D) into R equal subranges.

⎡⎛ T ⎞−3 ⎤−1/3 6δσ RC ⎟ + τ⎥ TF 0 = 100⎢⎜ 100γCH ⎦ ⎣⎝ 100 ⎠

where TF0 is the radiation temperature, TRC the outlet temperature of the roughing mill, δ the emissivity, σ the Stefan− Boltzmann constant, τ the run time from the exit of the roughing mill to the entry of the finishing mill, γ the density, C the specific heat, and H the entry thickness. (b). The Rolling Temperature Calculation Model of Each Stand. The rolling temperature of each stand is calculated by follows: i ⎛ ∑ j = 1 Lj ⎞ Ti − TW ⎟ = exp⎜ −Ka ⎜ ⎟ TF 0 − TW h v n n ⎠ ⎝

i = 1, 2 , ..., n

i = 1, 2 , ..., n

where Ti is the rolling temperature of each stand, TW the cooling water temperature, Ka the comprehensive convective cooling 11397



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Figure 3. Main procedure of HSaGA.

Step 2. From the first dimension j = 1 to j = D, perform the following steps:

Step 2.4. Set i = i + 1. If i > n, j = j + 1 and then go to step 2.1; otherwise, go to step 2.2. 3.2. Update of Solutions in the Population. 3.2.1. CrossOver Operators Used in the HSaGA. Just as the differential evolution algorithm that has several types of mutation strategies21 to generate new solutions, the GA also has some different crossover operators designed for continuous optimization. The search directions of these operators are different and thus they have different search characteristics for different types of problems. Motivated by the adoption of multiple mutation strategies in

Step 2.1. Set i = 1 and the selection probability of each subrange r to be pr = 1/R. Step 2.2. Randomly select a subrange, namely, r, and then, within this range, generate a random value for xij. Step 2.3. For the selected subrange r, set pr = pr − 1/n; for the other unselected subranges, set pk = pk + 1/[n × (R − 1)]. 11398



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Table 1. Definitions of Benchmark Problems Used in the Experiment test function D

f1 (x) = ∑i = 1 xi 2 f2 (x) =

D ∑i = 1 |xi|

D

+

D ∏i = 1 |xi|

i

f3 (x) = ∑i = 1 (∑ j = 1 xj)2 f4 (x) = max|xi|

⎧ k(xi − a)m xi > a ⎪ ⎪ x +1 π D−1 f5 (x) = D {10 sin 2(πy1) + ∑i = 1 (yi − 1)2 [1 + 10 sin 2(πyi + 1)]} + (yD − 1)2u(xi , a , k , m) = ⎨ 0 − a ≤ xi ≤ a , yi = 1 + i 4 ⎪ D ⎪ m + ∑i = 1 u(xi , 10, 100, 4) ⎩ k(xi − a) xi < − a

f7 (x) =

f8 (x) =

D ∑i = 1 − xi

f9 (x) =

D ∑i = 1 [xi 2

optimal value

[−100,100]D

0

[−10,10]D

0

[−100,100]

D

0

[−100,100]

D

0

[−50,50]D

0

[−100,100]D

0

+ rand[0, 1)

[−1.28,1.28]

D

0

sin( |xi| ) + D × 418.98288727243369

[−500,500]D

0

D

0

D

f6 (x) = ∑i = 1 (⌊xi + 0.5⌋)2 D ∑i = 1 ixi 4

solution space

[−5.12,5.12]

− 10 cos(2πxi) + 10]

(

f10 (x) = − 20 exp −0.2

1 D

D

∑i = 1 xi 2

) − exp(

1 D

D

[−32,32]

)

∑i = 1 cos 2πxi + 20 + e

D

0

• First, calculate the gravity center of the three solutions as G = (X1 + X2 + X3)/3; • Second, generate D random numbers {r1, r2, ..., rD} by rj = u1/(k+1), where u ∈ [0, 1] is a random number; • Third, generate D + 1 intermediate variables Yk, according to Yk = G + ε × (Xk − G) for k = 0, ..., D, and then obtain D + 1 variables (Ck), using the expression

SaDE22 and JADE,21 in this paper, we also adopt multiple crossover strategies and develop a self-adaptive selection mechanism to adaptively select the most appropriate cross-over operator for a given optimization problem. Such a strategy will help to improve the robustness of GA for different types of problems. In the proposed HSaGA, four types of cross-over operators that are often used in GAs are adopted, i.e., the blend crossover (BLX-α),23 the simulated binary crossover (SBX),26 the simplex crossover (SPX),27 and the parent centric crossover (PCX).28 (1). BLX-α. The BLX-α operator performs cross-over operation on two solutionsX = (x1, ..., xD) and Y = (y1, ..., yD)and results in a new solution, X′ = (x1′ , ..., xD′ ), where xj′ is randomly generated in the range [ci,min − Iα, ci,max + Iα], ci,max and ci,min are, respectively, the maximum and minimum value of xi and yi, I = ci,max − ci,min, and α is a constant. (2). SBX. Similar to the BLX-α, SBX also performs on two solutionsX = (x1, ..., xD) and Y = (y1, ..., yD)and can result in two new solutions, Z1 = (z1,1, ..., z1,D) and Z2 = (z2,1, ..., z2,D) as follows. First, a random uniform number u ∈ [0, 1] is generated, and then a new random number β is generated based on u and a distribution index η, according to

⎧0 k=0 Ck = ⎨ · ⎩ rk − 1(Yk − 1 − Yk + Ck − 1) k = 1, 2 , ..., D

• Lastly, the new solution Z is obtained by Z = YD + CD. (4). PCX. Similar to the SPX operator, in our HSaGA, the PCX operator is also applied to three solutions (X1, X2, X3) to generate a new solution Z. The procedure of implementing this operator is given as follows, according to ref 28: • First, determine the mean vector G of the three solutions X1, X2, and X3. • Second, randomly select a parent solution Xp from the three solutions and the direction vector d(p) = Xp − G is then calculated. • Third, calculate the perpendicular distance Di to the line d(p) for the remaining two solutions, and then the average (D̅ ) is calculated. • Last, the new solution Z can be obtained by the expression

⎧(2u)1/(1 + η) if u ≤ 0.5 ⎪ ⎪ β = ⎨⎡ ⎤1/(1 + η) 1 ⎪ (1 ) otherwise u − ⎪ ⎣⎢ ⎦⎥ ⎩ 2

Lastly, each dimension of the two new solutions are generated using the following equations:

Z = X p + wξ ·d ̅

(p)

3

+

∑

(wη·D̅ ·ei)

i = 1, i ≠ p

z1, i = 0.5⎡⎣(1 + β)xi + (1 − β)yi ⎤⎦

where parameters wξ = N(0,σξ2), wη = N(0,ση2), and ei represents the two orthogonal bases that span the subspace perpendicular to d(p). 3.2.2. Self-Adaptive Selection Mechanism for Cross-Over Operators. To select the cross-over operator, we present a selfadaptive mechanism by introducing the concept of solution survival used in the SaDE and JADE. The success and failure memories in SaDE are also used in the HSaGA. In our HSaGA, the application of a selected cross-over operator to Xi and the other parent solutions is viewed as successful only if the new solution resulted from this operator (Xtemp in Figure 3) or the new

i = 1, 2 , ..., n

and z 2, i = 0.5⎡⎣(1 − β)xi + (1 + β)yi ⎤⎦

i = 1, 2 , ..., n

(3). SPX. Unlike BLX-α and SBX, the implementation of SPX in our HSaGA requires three solutions (X1, X2, X3) to generate a new solution (note that the SPX operator can also be performed on more than three solutions). The procedure of this cross-over operator generally includes four steps: 11399



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solution resulted from the union of this operator and a mutation process (X′i,new in Figure 3) outperforms Xi. A list L with a fixed length l (also called the learning period) is established to store the successful count skG and unsuccessful count f kG of each crossover operator Ok. During the first l generations, the selection probability of each cross-over operator is equal to be 1/K (where K is the sum of cross-over operators). This warm-up period is used to train the algorithm to analyze the performance and suitability of each crossover operator for the current optimization problem. At the end of each generation, the number of skG and f kG for each cross-over operator is counted and then added to the end of the list L. Then, from generation G = l + 1, the head node of list L will be removed so that the new record of skG and f kG can be added to the end of L. The selection probability of each crossover operator k at generation G (denoted as pkG) is calculated as follows, according to SaDE:16 pkG =

Table 4. Comparison Results for GAs with and without the Solution Selection Strategy Mean (Std Dev)

SkG K ∑k = 1 SkG G−1

Skg =

G−1

(

∑g = G − 1 skg + fkg

)

+ 0.01

k = 1 , ..., K ; G > l

Based on this calculation method, it is clear that the crossover operator with a larger number of successful applications will have Table 2. Comparison Results for the Generation Methods of Initial Population Mean (Std Dev) test function

Gen

HSaGArand

f1 f2 f3 f4 f5

1500 2000 5000 5000 1500

f6

1500

f7 f8 f9 f10

3000 9000 5000 2000

1.6 × 10−211 (0) 0 (0) 0 (0) 0 (0) 1.6 × 10−32 (5.5 × 10−48) 4.9 × 10−34 (2.4 × 10−33) 0 (0) 0 (0) 0 (0) 4.4 × 10−16 (0)

HSaGA

Sig

0 (0) 0 (0) 0 (0) 0 (0) 1.6 × 10−32 (5.5 × 10−48) 0 (0)

+ − − − −

0 (0) 0 (0) 0 (0) 4.4 × 10−16 (0)

− − − −

Gen

GArandom

GAselect

Sig

f1 f2 f3 f4 f5

1500 2000 5000 5000 1500

7.9 × 10−42 (6.4 × 10−42) 4.1 × 10−35 (2.4 × 10−35) 1.6e × 10−2 (1.0 × 10−2) 5.3 × 10−20 (5.0 × 10−20) 1.8 × 10−6 (5.5 × 10−8)

+ + + + +

f6

1500

f7

3000

0 (0)

−

f8 f9 f10

9000 5000 2000

4.6 × 10−32 (3.3 × 10−33) 5.0 × 10−126 (1.2 × 10−125) 2.7 × 103 (5.9 × 102) 8.6 × 100 (2.5 × 100) 4.0 × 10−15 (0)

0 (0) 0 (0) 0 (0) 0 (0) 3.4 × 10−1 (5.9 × 10−1) 5.0 × 10−1 (3.5 × 100)

1.3 × 103 (2.4 × 102) 0 (0) 4.4 × 10−16 (0)

+ + +

+

a larger selection probability of applying it to generate the offspring solutions. 3.2.3. Selection of Solutions for Performing Cross-Over Operator. As shown in Figure 3, different from traditional GAs, in which the parent solutions are all randomly selected, each solution of the population in our HSaGA will be taken as one of the parents and only the other parent solutions needed by the selected cross-over operator are randomly selected, according to a selection mechanism. The main idea of this selection mechanism is that we want to further improve the selection efficiency. The motivation of this selection mechanism is based on the following analysis on the random selection method used in traditional GAs. When traditional random selection method is used, all of the selected parents may be very close to each other, because the better solutions always have a larger probability to be selected based on the roulette-wheel method and, generally, better solutions have a tendency to be located in very close regions in the search space. Therefore, the traditional random selection method based on the roulette wheel may have a disadvantage of losing search diversity, which, in turn, may cause the premature phenomenon. Therefore, a selection strategy for the parent solutions is designed in this paper, to improve the convergence speed and, at the same time, maintain search diversity. During evolution, it is clear that a higher convergence speed can be obtained if only solutions with good qualities in the objective function are selected as parents; however, a higher risk of prematurity (i.e., getting trapped in local optimal regions) may occur at the same time. In contrast, if only solutions with good

where ∑g = G − 1 skg

test function

+

Table 3. Comparison Results for HSaGA and the Other GAs without the Self-Adaptive Mechanism Mean (Std Dev) test function

Gen

HSaGA

GABLX‑α

GASBX

GASPX

GAPCX

Sig

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

1500 2000 5000 5000 1500 1500 3000 9000 5000 2000

0 (0) 0 (0) 0 (0) 0 (0) 1.6 × 10−32 (5.5 × 10−48) 0 (0) 0 (0) 0 (0) 0 (0) 4.4 × 10−16(0)

0 (0) 0 (0) 0 (0) 0 (0) 3.4 × 10−1 (5.9 × 10−1) 5.0 × 10−1 (3.5 × 100) 0 (0) 1.3 × 103 (2.4 × 102) 0 (0) 4.4 × 10−16 (0)

0 (0) 0 (0) 2.9 × 100 (1.2 × 101) 0 (0) 4.3 × 10−6 (8.8 × 10−6) 1.3 × 10−4 (2.8 × 10−4) 0 (0) 2.2 × 10−5 (4.4 × 10−5) 0 (0) 4.4 × 10−16 (0)

2.6 × 10−310 (3.5 × 10−311) 6.3 × 10−208 (0) 0 (0) 0 (0) 7.3 × 10−1 (8.9 × 10−2) 6.3 × 100 (4.8 × 10−1) 0 (0) 5.7 × 103 (7.3 × 102) 0 (0) 4.4 × 10−16 (0)

2.5 × 10−3 (6.1 × 10−4) 3.9 × 101 (4.3 × 101) 9.3 × 10−1 (1.2 × 10−1) 1.1 × 100 (1.7 × 10−1) 1.8 × 100 (5.7 × 10−1) 2.4 × 10−3 (5.4 × 10−4) 1.3 × 10−22 (5.7 × 10−23) 3.4 × 103 (2.7 × 102) 1.7 × 102 (1.5 × 101) 1.9 × 101 (3.6 × 100)

− − − − − + − + − −

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Table 5. Comparison Results for HSaGA and the Other EAs for Problems with D = 30 Mean (Std Dev) test function f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

Gen 1500 2000 5000 5000 1500 1500 3000 9000 5000 2000

HSaGA50 0 (0) 0 (0) 1.4 × 10−24 (6.6 × 10−24) 0 (0) 1.6 × 10−32 (5.5 × 10−48) 7.3 × 10−6 (4.7 × 10−5) 0 (0) 4.5 × 10−6 (1.6 × 10−5) 0 (0) 4.4 × 10−16 (0)

jDE −28

SaDE −28

2.5 × 10 (3.5 × 10 ) 1.5 × 10−23 (1.0 × 10−23) 5.2 × 10−14 (1.1 × 10−13) 1.4 × 10−15 (1.0 × 10−15) 2.6 × 10−29 (7.5 × 10−29) 0 (0) 3.3 × 10−3 (8.5 × 10−4) 0 (0) 0 (0) 4.7 × 10−15 (9.6 × 10−16)

−20

JADE1 −20

4.5 × 10 (6.9 × 10 ) 1.9 × 10−14 (1.05 × 10−14) 9.0 × 10−37 (5.43 × 10−36) 7.4 × 10−11 (1.82 × 10−10) 1.2 × 10−19 (2.0 × 10−19) 0 (0) 4.8 × 10−3 (1.2 × 10−3) 4.7 × 100 (3.3 × 101) 0 (0) 4.3 × 10−14 (2.6 × 10−14)

−54

JADE2 −54

1.3 × 10 (9.2 × 10 ) 3.9 × 10−22 (2.7 × 10−21) 6.0 × 10−87 (1.9 × 10−86) 4.3 × 10−66 (1.2 × 10−65) 1.6 × 10−32 (5.5 × 10−48) 0 (0) 6.8 × 10−4 (2.5 × 10−4) 7.1 × 100 (2.8 × 101) 0 (0) 4.4 × 10−15 (0)

−60

1.8 × 10 (8.4 × 10−60) 1.8 × 10−25 (8.8 × 10−25) 5.7 × 10−61 (2.7 × 10−60) 8.2 × 10−24 (4.0 × 10−23) 1.6 × 10−32 (5.5 × 10−48) 0 (0) 6.4 × 10−4 (2.5 × 10−4) 0 (0) 0 (0) 4.4 × 10−15 (0)

Table 6. Comparison Results for HSaGA and the Other EAs for Problems with D = 100 Mean (Std Dev) test function

Gen

HSaGA100

jDE

SaDE

JADE1

JADE2

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

1500 2000 5000 5000 1500 1500 3000 9000 5000 2000

0 (0) 0 (0) 0 (0) 0 (0) 1.6 × 10−32 (5.5 × 10−48) 0 (0) 0 (0) 0 (0) 0 (0) 4.4 × 10−16 (0)

5.0 × 10−15 (1.7 × 10−15) 4.1 × 10−15 (1.1 × 10−15) 5.4 × 10−2 (2.7 × 10−2) 3.1 × 10−9 (5.9 × 10−10) 1.7 × 10−25 (7.7 × 10−26) 0 (0) 8.1 × 10−3 (9.0 × 10−4) 1.1 × 10−10 (0) 0 (0) 9.9 × 10−14 (2.0 × 10−14)

2.9 × 10−8 (3.2 × 10−8) 1.7 × 10−5 (3.8 × 10−6) 2.4 × 10−13 (5.2 × 10−13) 1.1 × 100 (4.0 × 10−1) 1.4 × 10−11 (9.1 × 10−12) 0 (0) 1.0 × 10−2 (4.9 × 10−3) 1.1 × 10−10 (0) 0 (0) 2.1 × 10−7 (1.0 × 10−7)

5.4 × 10−67 (1.6 × 10−66) 9.2 × 10−51 (2.2 × 10−50) 2.2 × 10−37 (2.5 × 10−37) 3.2 × 10−71 (8.3 × 10−71) 4.7 × 10−33 (6.8 × 10−49) 0 (0) 7.8 × 10−4 (1.4 × 10−4) 1.1 × 10−10 (0) 0 (0) 8.0 × 10−15 (0.0 × 100)

1.2 × 10−48 (1.5 × 10−48) 1.1 × 10−41 (5.1 × 10−41) 1.2 × 10−26 (2.0 × 10−26) 1.9 × 10−2 (1.5 × 10−2) 4.7 × 10−33 (2.2 × 10−34) 1.6 × 10−1 (3.7 × 10−1) 1.1 × 10−3 (2.1 × 10−4) 1.1 × 10−10 (0) 0 (0) 8.9 × 10−15 (2.1 × 10−15)

Figure 4. Comparison results of the evolution process of HSaGA and SaDE for f1−f6.

function and the another with solutions that have inferior quality in the objective function but good quality in diversity. During the implementation at each generation, this strategy first sorted the solutions in P according to the nondescending order of their objective function values, and then divided P into two parts, namely, P1 (with the first 0.5n good-quality solutions) and P2

diversity (which means that the objective function values are not good) are selected, the search diversity can be guaranteed but with the loss of a high speed of convergence. Therefore, to reach a balance of convergence speed and search diversity, the selection strategy used in our HSaGA divides the population P into two parts: one with solutions that have good quality in the objective 11401



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Figure 5. Comparison results of the evolution process of HSaGA and SaDE for f 7−f10.

Table 7. Value of the Energy Coefficient Value at Various Thicknesses coefficient

8.50 mm

K1 K2

6.035 6.702

6.014 4.558

4.369 6.517

4.467 5.215

3.045 7.992

3.001 7.950

Table 8. Value of Distribution Coefficient of Cumulative Energy Consumption φi φ range of hn

range of slab width

stand F1

stand F2

stand F3

stand F4

stand F5

stand F6

stand F7

(0, 2.2) (0, 2.2) (0, 2.2)

(0, 900) [900, 1200) [1200, 2000]

0.14 0.14 0.14

0.28 0.28 0.28

0.44 0.46 0.47

0.62 0.64 0.65

0.77 0.78 0.79

0.91 0.91 0.91

1.00 1.00 1.00

[2.2, 3.9) [2.2, 3.9) [2.2, 3.9)

(0, 900) [900, 1200) [1200, 2000]

0.13 0.14 0.14

0.27 0.28 0.28

0.44 0.46 0.47

0.61 0.63 0.64

0.76 0.77 0.77

0.90 0.90 0.90

1.00 1.00 1.00

[3.9, 8.9) [3.9, 8.9) [3.9, 8.9)

(0, 900) [900, 1200) [1200, 2000]

0.12 0.13 0.13

0.26 0.27 0.27

0.45 0.46 0.47

0.61 0.63 0.64

0.75 0.77 0.77

0.89 0.90 0.90

1.00 1.00 1.00

(with the second 0.5n solutions). Then, for a given solution Xi in the population and its corresponding cross-over operator Ok, the selection strategy used in HSaGA can be presented as follows: (1). Xi Belonging to P1. In this scenario, for the BLX-α and SBX operators, we just randomly select another solution from P1; for the SPX and PCX operators, we first select a random solution from P1 and then select another solution from P2. We use this

selection strategy because it can guarantee a good convergence speed when adopting the BLX-α and SBX operators, and a balance of convergence speed and search diversity when adopting the SPX and PCX operators. (2). Xi Belonging to P2. In this scenario, the quality of Xi in the objective function value needs to be improved, so we guide this solution to promising regions by selecting random solutions with 11402



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x6′ = x4

Table 9. Slab Information Used in the Experiments

No.

width (mm)

initial

finished thickness (mm)

1 2 3 4 5 6 7 8 9 10

1535 1285 1455 1356 1452 1428 1357 1516 1366 1348

36.7 34.7 35.2 34.6 35.6 32.3 33.8 35.9 34.3 33.5

5.7 3.92 2.9 3.7 3.6 3.35 3.83 5.1 3.46 3.5

rough rolling export temperature (°C)

finishing rolling export temperature (°C)

target crown (mm)

1067 1101 1086 1055 1087 1013 1023 1076 1085 1077

891 881 890 901 904 889 892 895 903 897

0.016 0.015 0.014 0.016 0.014 0.016 0.015 0.014 0.015 0.015

Using this repair method, the requirement of nonascending order for the rolled thickness at each stand can be satisfied and, consequently, most constraint violations caused by this requirement can be avoided.

4. COMPUTATIONAL RESULTS To test the performance of our HSaGA, computational experiments are carried out and the results are analyzed in this section. First, section 4.1 is devoted to describing the benchmark test problems and parameter settings of HSaGA used in the experiments. Then, the efficiency of the main components in HSaGA, such as the self-adaptive mechanism on multiple crossover operators, the selection strategy of parent solutions, and the initial solution generation method, are analyzed in section 4.2. In section 4.3, the proposed HSaGA is compared with other stateof-the-art evolutionary algorithms in the literature, based on the benchmark test problems. Finally, in section 4.4, the proposed HSaGA is used to solve the operation optimization problem of the hot-rolling process to show its efficiency in practical optimization problems. 4.1. Test Problem and Parameter Settings. In the experiments, 10 benchmark test problems are selected from the literature, and they are all nonlinear functions, especially ref 21. The definitions of these problems are given in Table 1, in which the first four problems are unimodal and the others are multimodal. In the experiments, the value of D is set to be 30 and 100, according to refs 21 and 17. The parameters used in the HSaGA are used as follows. The population size is set to 100, the learning period l is equal to 40, according to ref 17, and the number of subranges for each dimension used in the initial population generation method is set to be 5, according to ref 24. The parameters used in the four cross-over operators are set as their suggested values, according to refs 25−28. In the experiment, a total of 50 independent runs is performed for each problem to collect the statistical performance of each algorithm. The independent t-test is used to show the statistical difference between different algorithms and the result for each problem is given in the last column of the tables in the following sections of the paper. In the last column denoted as “Sig”, the signal “+” denotes that the performance difference between our HSaGA and the best one among the other algorithms in the tables is significant with a confidence level of 95%, while the signal “−” denotes that their performance difference is not significant. 4.2. Efficiency Analysis of the Components of HSaGA. In this section, the 10 benchmark problems with D = 30 are used to test the efficiency of the main components in our HSaGA. The stopping criterion is set as the maximum number of available generation (denoted as Gen in the following tables). 4.2.1. Efficiency of Initial Population Method. To show the efficiency of the generation method for initial population, the HSaGA with a random generation method (denoted as HSaGArand) for initial population is also tested. The comparison results of mean objective function value and the corresponding standard deviation (Std Dev) for each problem are shown in Table 2, in which the second column is the maximum available generations and the best results are shown in bold font. From the results, it can be seen that the diversification method can help the HSaGA to achieve the optimal solution in every run for f1 and f6. Taking into consideration that the multiple crossover operators with self-adaptive selection mechanism can help

good quality from P1 for each cross-over operator. That is, we just randomly select another solution from P1 for the BLX-α and SBX operators while select two random solutions from P1 for the SPX and PCX operators. 3.2.4. Solution Mutation. The mutation operator used in our HSaGA is based on the polynomial mutation operator proposed by Eshelman and Schaffer.24 That is, for a given solution X′i = (x′1, ..., xD′ ) generated by the cross-over operation, each dimension of Xi′ will be mutated by the polynomial mutation operator if rnd < pm where rnd is a random number generated in [0,1] and pm is the mutation probability. In our HSaGA, the value of pm is set to be 1/D, according to ref 29. 3.3. Constraint Handling. As shown in section 2, many constraints exist in the operation optimization of the hot-rolling process. The same thing also occurs for many optimization problems in practical industries. Therefore, a constraint handling method is adopted in our HSaGA, to compare the solutions. For two solutions Xi and Xj, Xi is said to be better than Xj whenever one of the following three conditions is satisfied: (1) Solutions Xi and Xj are both feasible, and, at the same time, solution Xi has a better objective function value than Xj; (2) Solution Xi is feasible but Xj is infeasible; and (3) Both solutions Xi and Xj are infeasible, and, at the same time, solution Xi has a smaller violation of all constraints. 3.4. Feasibility Repair. The above constraint handling method can be applied to any single-objective optimization problem. Since the violence of constraints for the OOP of the hotrolling process is often caused by the requirement that the rolled thickness at each stand must be in a nonascending order, we use a repair method for the solutions, to avoid infeasibility. For the OOP of the hot-rolling process, the solution is coded by Xi = (x1, ..., x6), where each dimension is the exit thickness of each rolling stand. Therefore, we assume that the nonascending order of each dimension after the cross-over and mutation operations is x2, x3, x1, x5, x6, and x4. Then, the repair method is applied and we obtain a new solution X′i = (x′1, ..., x′6), whose dimensions are set as follows:

x1′ = x 2 x 2′ = x3 x3′ = x1 x4′ = x5

x5′ = x6 11403



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Table 10. Comparison Results between the HSaGA and the Empirical Method Results No. 1

parameter exit thickness (mm) rolling force (MN) objective (constraint violation)

2

exit thickness (mm) rolling force (MN) objective (constraint violation)

3


4


5


6


7


8


9

exit thickness (mm) rolling force (MN)

method

F1

F2

F6

F7

empirical method HSaGA empirical method HSaGA empirical method HSaGA

25.51 27.14 20.17 17.83

18.55 17.19 18.27 23.89

12.68 9.55 7.84 11.65 9.38 7.51 22.67 16.65 11.90 21.06 15.19 12.93 6.67 × 10−8 (6.5474) 0.00 (0.0000)

6.52 6.54 11.47 9.26

5.70 5.70 8.42 8.57


19.98 24.24 20.99 15.38

13.60 14.55 15.83 20.27

8.91 6.63 5.41 9.64 7.16 5.48 17.43 14.94 9.44 17.26 15.26 11.80 1.05 × 10−6 (8.7098) 0.00 (0.0000)

4.49 4.61 9.12 8.60

3.92 3.92 6.77 7.77


18.35 23.57 27.47 19.48

11.91 13.62 19.51 24.47

7.52 5.34 4.22 8.02 5.74 4.39 20.83 18.55 11.52 23.84 18.39 12.91 6.37 × 10−7 (17.3669) 0.00 (0.0000)

3.39 3.44 11.22 12.24

2.90 2.90 8.09 8.70


19.61 22.95 24.33 19.37

13.32 12.88 17.65 24.60

8.80 6.44 5.20 8.13 6.32 5.08 18.71 16.72 10.29 20.22 13.97 10.44 1.65 × 10−6 (12.7067) 0.00 (0.0000)

4.27 4.32 9.80 8.37

3.7 3.70 7.12 7.60

empirical HSaGA empirical HSaGA empirical HSaGA empirical HSaGA empirical HSaGA empirical HSaGA

19.82 26.41 25.07 15.79

13.33 16.66 18.19 21.54

4.17 4.36 10.35 10.64

3.6 3.60 7.58 9.33

19.52 24.80 25.43 17.13

13.38 15.82 18.53 22.10

8.72 6.34 5.10 10.67 7.33 5.36 19.40 17.40 10.75 20.90 20.45 14.62 1.56 × 10−6 (13.2410) 0.00 (0.0000) 8.93 6.58 5.34 10.41 7.53 5.53 19.52 17.41 10.66 20.65 18.70 14.75 1.80 × 10−6 (13.6115) 0.00 (0.0000)

4.41 4.58 10.06 9.96

3.83 3.83 7.35 8.84


20.02 28.01 21.17 15.11

14.56 18.37 16.75 22.00

10.23 7.96 6.70 12.93 9.67 7.46 18.40 16.03 10.11 19.17 18.66 14.18 1.24 × 10−6 (11.7778) 0.00 (0.0000)

5.72 5.98 9.69 12.67

5.1 5.10 7.15 9.14


19.09 25.54 18.37 14.66

12.83 16.00 13.34 20.44

8.56 6.21 4.90 10.30 7.18 5.33 13.69 12.86 8.43 19.48 18.59 13.27 9.93 × 10−7 (12.4245) 0.00 (0.0000)

4.00 4.14 7.63 11.75

3.46 3.46 5.50 8.32

empirical method HSaGA empirical method HSaGA

18.85 24.76 19.84 14.94

12.75 15.29 14.46 20.93

8.55 10.10 14.89 18.57

4.04 4.14 9.34 11.21

3.5 3.50 6.00 8.01

method method method method method method

11404

F3

F4

6.23 7.20 13.98 17.74

F5

4.94 5.25 9.12 13.98



Article

Table 10. continued Results No.

10

parameter

method

objective (constraint violation)

empirical method HSaGA

exit thickness (mm)


rolling force (MN) objective (constraint violation)

F1

F2

F3

F4

F5

F6

F7

4.41 4.58 10.06 9.96

3.83 3.83 7.35 8.84

1.20 × 10−6 (12.2446) 0.00 (0.0000) 19.52 24.80 25.43 17.13

13.38 15.82 18.53 22.10

8.93 6.58 5.34 10.41 7.53 5.53 19.52 17.41 10.66 20.65 18.70 14.75 1.80 × 10−6 (13.6115) 0.00 (0.0000)

in ref 20, the population size of the other four algorithms is set to 100 for D = 30 and 400 for D = 100 in jDE, SaDE, and JADE. Since, in our HSaGA, the number of objective function evaluations at each generation is twice of that used in the other four algorithms, for problems with D = 30, we set the population size of our HSaGA to be 50 (denoted as HSaGA50), so that the number of total function evaluations is the same. However, for problems with D = 100, the population size of our HSaGA is still set to be 100 (denoted as HSaGA100), which means that we only use a half of the number of total function evaluations used in the other four algorithms. The comparison results of these five algorithms for problems with D = 30 and D = 100 are given in Tables 5 and 6, respectively. Since we cannot obtain the detailed results of all 50 runs for the other four algorithms, the significance column is not provided. From Table 5 with D = 30, it can be seen that our HSaGA obtains the best results for 7 out of the 10 problems and 5 of them ( f1, f 2, f4, f 7, and f10) are better than the other four algorithms. Among the other four algorithms, JADE1 and JADE2 obtain the best results for 4 out of the 10 problems and, on average, their performances are better than the jDE and SaDE. From Table 6 with D = 100, it can be seen that our HSaGA can obtain the best results for all the problems except f5, and that it obtains the better results than the other four algorithm for 7 out of the 10 problems, i.e., f1−f4, f 7, f 8, and f10. It should be noted that the number of total function evaluations of our HSaGA is only the half of that used in the other four algorithms. Therefore, it can be concluded that the proposed HSaGA is superior to the other four algorithms and that the HSaGA also has a better robustness for different algorithms. The main reasons for the better performance of the proposed HSaGA mainly lies in the self-adaptive selection mechanism with the multiple cross-over operators and the solution selection strategy for parent solutions. These strategies, as a whole, help to accelerate the convergence speed of the HSaGA to the global optimal regions and, at the same time, maintain good search diversity, to avoid being trapped in a local optimum. To give a graphical illustration of this analysis, we re-implemented the SaDE algorithm and compared the evolution processes between our HSaGA and the SaDE for these 10 problems. The comparison results are shown in Figures 4 and 5, in which the ordinate is log10(f(X)), where f(X) is the objective function value of the best solution found in each iteration. Please note that whenever the algorithm reaches the optimal solution, the evolution process line will terminate in the figures, because the optimal objective function value of each problem is zero. From this figure, it can be seen that the convergence speed of our HSaGA is much faster than that of the SaDE for all the test problems except f 8.

the algorithm to reach better solutions, the performance difference between the two methods of initial population will be larger if the multiple cross-over operators with self-adaptive selection mechanism are not adopted. Thus, they reached the conclusion that the diversification method to generate the initial solution is quite efficient. 4.2.2. Efficiency of the Self-Adaptive Selection Mechanism. To show the positive effects of the self-adaptive selection mechanism on multiple cross-over operators, the HSaGA is compared to the other four types of GAs in which only one cross-over operator is used and, thus, the self-adaptive selection mechanism is not adopted. These four types of GAs with each operator are denoted as GABLX‑α, GASBX, GASPX, and GAPCX, respectively. The comparison results between these algorithms are presented in Table 3. From the comparison results, it can be seen that the HSaGA obtains the best results for every test problem, which shows that the self-adaptive mechanism is effective and it makes the hybrid algorithm more robust, because it integrates the advantages of each cross-over operator and can self-adaptively select appropriate cross-over operators for a given problem. Among the other GAs, the GABLX‑α, GASBX, and GASPX show superior performance relative to that of GAPCX, but none of them can achieve a dominant performance with the left GAs. 4.2.3. Efficiency of Solution Selection Strategy for Performing Cross-Over Operators. To amplify the positive effect of the solution selection strategy for performing the cross-over operators, we only use the BLX-α operator and do not adopt the selfadaptive mechanism in this experiment. The GA with the solution selection strategy is denoted as GAselect, while the GA without this strategy is denoted as GArandom. The computational results are given in Table 4, from which it is clear that the solution selection strategy can obtain significantly better results for 7 out of the 10 test problems (although the result of GAselect for f 7 is better but the performance difference is not significant). So we can reach the conclusion that the solution selection strategy is very effective. 4.3. Comparison Results against Other Algorithms for Benchmark Problems. In this section, the proposed HSaGA is compared with some state-of-the-art evolutionary algorithms in the literature, such as jDE in Zhao,12 SaDE in Zhang and Sanderson,16 JADE with archive (JADE1) and JADE without archive (JADE2) in Zhang et al.20 All four of these evolutionary algorithms have adaptive strategies in parameter control or mutation strategy selection during the evolution process. The 10 best problems with D = 30 and D = 100 are used in this experiment and the results of the four algorithms (i.e., jDE, SaDE, JADE with archive, and JADE without archive) are taken from ref 20. The stopping criterion is taken as the maximum generations. As said 11405



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Figure 6. Rolling forces between the HSaGA and the empirical method for each slab instance.

4.4. Comparison Results for the Operation Optimization of the Hot-Rolling Process. 4.4.1. The Empirical Method. In order to test the validity of the algorithm for the

practical operation optimization, we compare our HSaGA with the empirical method used in the current iron and steel enterprise. In practical operations, the workers usually use an empirical 11406



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Table 11. Comparison Results between the HSaGA and the Empirical Method with Fluctuations Results No. 1

parameter exit thickness (mm) rolling force (MN) objective (constraint violation)

2


3


4


5


6


7


8


9

exit thickness (mm) rolling force (MN)

method

F1

F2

F6

F7


25.16 28.02 19.37 16.60

18.39 18.74 18.68 21.97

12.86 9.88 8.01 13.38 10.73 8.53 20.71 18.41 12.87 19.24 15.91 13.57 8.11 × 10−7 (6.5983) 5.73 × 10−9 (0.0000)

6.59 6.98 12.31 12.42

5.70 5.70 9.12 11.51


20.28 23.84 20.20 15.85

13.75 13.90 16.27 21.09

9.18 6.92 5.56 8.66 6.62 5.42 17.10 14.71 10.22 19.12 13.79 9.37 6.35 × 10−7 (8.7964) 5.01 × 10−9 (0.0000)

4.54 4.59 9.84 8.36

3.92 3.92 7.32 7.58


19.50 24.17 26.29 18.65

12.47 14.08 19.87 24.30

8.14 5.70 4.31 8.75 6.12 4.32 19.40 19.10 13.09 21.94 19.68 16.05 4.73 × 10−6 (12.9727) 1.01 × 10−8 (0.0000)

3.40 3.43 11.83 11.81

2.90 2.90 8.10 8.52


20.75 25.28 23.46 16.17

13.90 15.74 18.14 21.58

9.46 6.84 5.31 10.64 7.64 5.55 17.65 17.30 11.77 19.43 18.05 14.46 1.84 × 10−6 (10.0125) 4.42 × 10−9 (0.0000)

4.28 4.38 10.52 11.40

3.70 3.70 7.16 8.11

empirical HSaGA empirical HSaGA empirical HSaGA empirical HSaGA empirical HSaGA empirical HSaGA

20.93 25.41 23.75 17.10

13.89 15.43 18.36 22.67

4.17 4.25 10.90 11.67

3.60 3.60 7.46 8.37

18.43 23.72 27.40 18.20

12.40 15.09 20.92 23.67

9.37 6.74 5.20 9.90 7.16 5.37 17.96 17.70 12.12 20.58 17.97 13.63 1.01 × 10−6 (10.1390) 1.90 × 10−9 (0.0000) 8.48 6.15 4.79 9.62 6.84 5.14 20.21 19.65 13.28 22.23 20.57 14.61 1.21 × 10−6 (14.3617) 9.58 × 10−9 (0.0000)

3.87 4.01 11.79 13.13

3.35 3.35 7.95 9.33


19.90 24.81 24.20 17.13

13.59 15.45 18.76 22.95

9.41 6.90 5.41 10.39 7.60 5.73 18.19 17.76 12.04 19.72 18.21 13.76 1.73 × 10−6 (10.3918) 2.71 × 10−8 (0.0000)

4.40 4.53 10.71 11.87

3.83 3.83 7.26 8.44


23.04 26.92 22.52 16.62

16.17 17.46 18.53 22.22

11.17 8.62 7.04 12.27 9.43 7.24 19.37 16.68 11.53 19.03 17.20 14.29 3.50 × 10−6 (9.3211) 1.20 × 10−9 (0.0000)

5.84 5.91 10.98 11.82

5.10 5.10 8.12 8.62

empirical method HSaGA empirical method HSaGA

19.91 25.04 22.80 15.30

13.25 15.64 17.63 20.47

8.96 10.12 17.25 19.21

4.01 4.12 10.48 11.54

3.46 3.46 7.17 8.18

method method method method method method

11407

F3

F4

6.45 7.12 17.01 18.20

F5

4.99 5.28 11.65 14.24



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Table 11. continued Results No.

10

parameter

method

objective (constraint violation)

empirical method HSaGA

exit thickness (mm)


rolling force (MN) objective (constraint violation)

F1

F2

F3

F4

F5

F6

F7

4.04 4.13 10.41 8.74

3.50 3.50 7.12 7.85

1.97 × 10−6 (9.7335) 6.48 × 10−9 (0.0000) 19.31 23.48 22.56 16.64

12.99 13.82 17.49 22.25

8.87 6.43 5.01 8.61 6.42 4.91 17.12 16.88 11.57 20.24 15.51 12.06 1.58 × 10−6 (9.5845) 1.90 × 10−9 (0.0000)

From the computational results shown in Table 10, it is clear that our HSaGA obtains the optimal solutions for each slab instance. Although the empirical method also achieves solutions with good quality in the objective function, these solutions are, in fact, infeasible, because the rolling forces do not satisfy the ramping constraint. Based in Figure 6, it can be seen that the rolling force obtained by the empirical method has a large value in the first stand and then decreases at the second stand. This setting is not reasonable, because, in practical production, the rolling force of the first stand should not be very large in order to make the slab bit successfully and must be smaller than those of the second and the third stands. This requirement is used to ensure that the reduction of the first stand is not very large so that the transition of the strip through the seven stands is stable. As mentioned in section 2, the reduction (and, correspondingly, the rolling force) should be large in the second and third stands, to make full use of the ability of the equipment, but the setting obtained by the empirical method cannot meet this requirement. In contrast, the results obtained by the proposed HSaGA are very reasonable. The rolling force first increases in the first two stands and then gradually decreases in the last five stands. With this setting, the slab obtains a relatively smaller reduction in the thickness in the first stand and then a larger reduction in the second stand and the third stand. Therefore, it can be concluded that the proposed HSaGA is much superior to the empirical method in practical operation optimization of the hot-rolling production process. 4.4.2.2. Experimental Results with Fluctuations. Because of the fact that the hot-rolling production is essentially a nondeterministic and dynamic process, we further consider the fluctuations of initial thickness, initial width, and rough rolling export temperature caused by inaccurate measurements. In the experiment, the fluctuation of each parameter is assumed to be within 5% and, correspondingly, the Monte Carlo method is used to evaluate the performance of a solution. That is, for a given solution (i.e., a set of control variables), we first randomly generate 10 samples within the fluctuations, then calculate the objective function value of the solution according to each sample, and finally take the average of these 10 objective function values as the evaluation of this solution. The comparison results for the empirical method and the proposed HSaGA are given in Table 11, and the graphical illustrations of rolling force on each stand are presented in Figure 7. From the comparison results shown in Table 11, it appears that the proposed HSaGA still achieves much better solutions than the empirical method, even when fluctuations occur in the production process. In addition, these solutions are feasible while those obtained by the empirical method are still infeasible. As illustrated by Figure 7, the rolling forces obtained by our HSaGA comply with the ramping constraints and, consequently, can

formula to get the approximate load distribution. The empirical formula is described as follows:24,25 ⎧ K K 2 − 4K φ a ⎫ ⎪ 2 2 1 i n ⎪ ⎬ hi = H0 exp⎨ 2K1 ⎪ ⎪ ⎩ ⎭

where K1 and K2 are the energy consumption coefficients obtained from the field statistics, φi is the rate of the distribution coefficient of cumulative energy consumption of the ith stand, and an is the total energy consumption. The value of an can be calculated according to the following equation: ⎛ H ⎞2 ⎛H ⎞ an = K1⎜ln 0 ⎟ + K 2 ln⎜ 0 ⎟ ⎝ hn ⎠ ⎝ hn ⎠

where H0 is the entry thickness of the first stand and hn is the exit thickness of the last stand. The values of K1, K2, and φi are given in Tables 7 and 8. According to the empirical model, we can obtain the exit thickness of every stand, and then the rolling force, rolling power, and rolling torque can be obtained according to the model that is introduced in section 2. 4.4.2. Computational Results on Practical Production Data. In this section, the proposed HSaGA is used to solve the proposed operation optimization model of the hot-rolling process based on practical production data. Since there may exist fluctuations during production due to the inaccurate measurement, we carry out two types of experiments: (1) experiment without fluctuations and (2) experiment with fluctuations. In the two experiments, a set of 10 slabs is collected from the practical production and their rolling information is given in Table 9. For each of these 10 slabs, we use the empirical method and the HSaGA to obtain the values of control variables, and the computational results of the exit thickness and the rolling force of each stand. In addition, the values of α1, β1, and γ1 are set to be 1.0, while α2, β2, and γ2 are set to be 1.5, respectively. For each instance, we perform our algorithm for 10 replications and collect the average performance values. 4.4.2.1. Experimental Results without Fluctuations. When fluctuations are not considered, the rolling production process is in a deterministic and steady state. The comparison results for the empirical method and the proposed HSaGA are given in Table 10, and the graphical illustrations of rolling force on each stand are presented in Figure 6. In Table 10, the item “Objective (constraint violation)” is used to give the objective function value and the corresponding violation of constraints. If the value of the constraint violation is positive, then it shows that the obtained solution is infeasible. 11408



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Figure 7. Rolling forces between the HSaGA and the empirical method for each slab instance.

guarantee the good quality of the final products and full use of the rolling equipment. Based on these results, the proposed HSaGA seems to be very robust since the objective function values are

very close to zero. Therefore, we can reach a conclusion that the proposed HSaGA is still very efficient and effective under the production environment with fluctuations. 11409



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(9) Yao, F.; Yang, W. D. Improved differential evolution algorithm used for load distribution of hot strip rolling. in Proc. IEEE Bio-Inspired Computing: Theories and Appl.(BIC-TA 2010) 2010, 526−530. (10) Li, H. J.; Wang, G. D.; Xu, J. Z.; Liu, X. H. Optimization of draft schedules with variable metric hybrid genetic algorithm during hot strip rolling. J. Iron Steel Res. 2007, 19, 33−36. (11) Sun, Y. K. Model and control of hot strip mill; Press of Metallurgy Industry, Beijing, 2002. (12) Zhao S. Y. Research on the optimization of load distribution of HSM based on improved GA. Ph.D. Dissertation, Northeastern University, Shenyang, PRC, 2005. (13) Brest, J.; Greiner, S.; Boŝković, B.; Mernik, M. Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans. Evolut. Comput. 2006, 10, 646−657. (14) Zhan, Z. H.; Zhang, J.; Li, Y.; Chung, H. S. Adaptive particle swarm optimization. IEEE Trans. Syst., Man., Cyber. B 2009, 39, 1362− 1381. (15) Pan, Q. K.; Suganthan, P. N.; Wang, L.; Gao, L.; Mallipeddi, R. A differential evolution algorithm with self-adapting strategy and control parameters. Comput. Oper. Res. 2011, 38, 394−408. (16) Zhang, J. Q.; Sanderson, A. C. JADE: adaptive differential evolution with optional external archive. IEEE Trans. Evolut. Comput. 2009, 13, 945−958. (17) Qin, A. K.; Huang, V. L.; Suganthan, P. N. Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans. Evolut. Comput. 2009, 13, 398−417. (18) Smith, J. E.; Fogarty, T. C. Adaptively parameterised evolutionary systems: self adaptive recombination and mutation in a genetic algorithm. Presented at the 4th International Conference on Parallel Problem Solving from Nature, Berlin, Germany, September 22−26, 1996; Springer: Berlin, 1996. (19) Hinterding, R.; Michalewicz, Z.; Peachey, T. C. Self-adaptive genetic algorithm for numeric functions. The 4th International Conference on Parallel Problem Solving from Nature, Berlin, Germany, September 22−26, 1996; Springer: Berlin, 1996. (20) Zhang, G. L.; Wang, S. Y.; Li, Y. A self-adaptive genetic algorithm based on the principle of searching for things. Soft. Comput. 2010, 5, 646−653. (21) Yang, J. M.; Horngy, J. T.; Kao, C. Y. A genetic algorithm with adaptive mutations and family competition for training neural networks. Int. J. Neural Syst. 2000, 10, 333−352. (22) Yang, J. M.; Kao, C. Y. Integrating adaptive mutations and family competition into genetic algorithms as function optimizer. Soft Comput. 2000, 4, 89−102. (23) Korejo I.; Yang S. X.; Li C. H. A comparative study of adaptive mutation operators for Genetic Algorithms. Presented at the VIII Metaheuristics International Conference, Hamburg, Germany, July 13−16, 2009. (24) Nebro, A. J.; Luna, F.; Alba, E.; Dorronsoro, B.; Durillo, J. J.; Beham, A. AbYSS - adapting scatter search to multiobjective optimization. IEEE Trans. Evol. Comput. 2008, 12, 439−457. (25) Eshelman, L. J.; Schaffer, J. D. Real-coded genetic algorithms and interval-schemata. In Foundations of Genetic Algorithms 2; Whitley, L. D., Ed.; Morgan Kaufmann Publishers, CA, 1992; pp 187−202. (26) Deb, K.; Agrawal, R. B. Simulated binary crossover for continuous search space. Complex Syst. 1995, 9, 115−148. (27) Tsutsui, S.; Yamamura, M.; Higuchi, T. Multi-parent recombination with simplex crossover in real coded genetic algorithms. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 1999), July 13−17, 1999; Morgan Kaufmann: Orlando, FL, 1999; pp 657−664. (28) Deb, K.; Anand, A.; Joshi, D. A computationally efficient evolutionary algorithm for real-parameter evolution. Evol. Comput. 2002, 10, 371−395. (29) Qu, B. Y.; Suganthan, P. N. Constrained multi-objective optimization algorithm with diversity enhanced differential evolution. In Proceedings of IEEE Congress on Evolutionary Computation (CEC2010), Barcelona, Spain, July 18−23, 2010; IEEE Press: Piscataway, NJ, 2010; pp 1−5.

5. CONCLUSIONS In this paper, we investigated the operation optimization problem (OOP) of the hot-rolling process in the iron and steel industry and proposed a hybrid self-adaptive genetic algorithm for this it. The main features of the HSaGA are the self-adaptive selection mechanism performed on four cross-over operators and the solution selection strategy for parent solutions. This algorithm is first tested using benchmark problems and compared with the other four state-of-the-art evolutionary algorithms in the literature, and then applied to solve the operation optimization problem of the hot-rolling process. The computational results on benchmark problems illustrate that the proposed HSaGA is very efficient and its performance is superior to the other four powerful evolutionary algorithms in the literature. For the practical OOP of the hot-rolling process, the results show that the application of HSaGA can obtain much better results than the empirical method currently used in practical production and that the setting of control variables obtained by the HSaGA can satisfy the process constraints in the practical hot-rolling production and, thus, can help to improve the product quality.

■

AUTHOR INFORMATION

Corresponding Author

*Tel./Fax: 86-24-83680169. E-mail address: [email protected]. edu.cn. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research is partly supported by the National Natural Science Foundation of China (No. 61374203), the Fund for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 71321001), the National Natural Science Foundation of China (Grant No. 70902065) and State Key Laboratory of Synthetical Automation for Process Industries Fundamental Research Funds (Grant No. 2013ZCX04-01).

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REFERENCES

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Operation Optimization in the Hot-Rolling Production Process

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