An Improved Elitist GA-based Solution for Integrated Batch Planning Problem in a Steelmaking Plant

Abstract

To solve the problem of preparing the integrated batch planning for a steelmaking plant, this paper considers the continuous/quasi-continuous characteristics of the steel manufacturing process operation. On the premise of analysing the constraints of steelmaking procedure and continuous casting procedure, this paper firstly develops the charge plan model and casting plan model, and then forms and solves the integrated batch planning model. According to the actual of the production site, this paper introduces an improved elitist genetic algorithm (IEGA), defines the matching chromosome coding and decoding strategies with the actual, and suggests improving the genetic operators. Finally, we verify the proposed model and algorithm on the production data of a real steelmaking plant. We clarify the applicability of developing an integrated batch planning model based on model analysis and demonstrate the effectiveness of the IEGA through algorithm analysis.

1. Introduction

The production plan of a steelmaking plant includes contract plan, batch plan, operation plan, dynamic scheduling and other optimization problems at different levels. Among them, a batch plan is located in a transition layer between a contract plan and an operation plan. It should not only undertake the production contract pool after preparing the contract plan, but also consider the slab width, production process, delivery time and other constraints to realize continuous production at a steel plant. In the steelmaking plant, the casting procedure takes casts as the basic unit, and the steelmaking procedure takes charges as the minimum unit, so the batch plan includes two aspects: charge plan and casting plan.

A charge plan is to optimize the slab of different steel types and specifications to obtain different charges according to the constraints of steelmaking production process. A casting plan is to optimize the different charges to obtain different casts according to the constraints of continuous casting production process. The two aspects can cooperate with each other to finally realize the stable production process. This paper carries out the research from these two aspects, but before modeling, will carefully analyze and discuss the existing relevant research, and then will develop and solve the integrated batch planning model.

2. Research Status of Integrated Batch Planning Problem in a Steelmaking Plant

Scholars at home and abroad have done many researches on batch plan of a steelmaking plant. Aiming at the research on the charge plan, Quelhadj et al.¹⁾ first proposed the steelmaking plant production process is an important link of steel production, and the charge plan optimization of the steelmaking procedure can save the cost and benefit of steelmaking plant. Moreover, he modelled the charge plan optimization model of the steelmaking procedure as an integer programming model and solved it by tabu search algorithm. To allow two-strand caster to cast slabs with different widths, Cheng et al.²⁾ established a charge plan model to minimize the number of casts and the difference in adjacent slab widths within the charge, and solved it using variable neighborhood combined with simulated annealing algorithm.

For the study of the casting plan, Tang et al.^3,4) established two casting plan models respectively. The difference between them lies in whether the casts are known and whether the charges are all arranged. Yadollahpour et al.⁵⁾ presented a comprehensive two-phase approach to solve the casting plan. Considering the hot rolling procedure, the integrated batch planning mathematical model⁶⁾ based on multi-objective optimization is established, and the improved genetic algorithm is used to solve it. Aiming at the production plan and scheduling problem of Baosteel’s steelmaking plant, Tang et al.⁷⁾ considered the sequence dependent constraints and casting width selection decisions, and proposed a column generation-based branch-and-price solution approach to obtain the optimal solutions.

For the study of integrated batch planning, some references have been listed above, and many common points have been found. By analyzing the existing literature, this paper summarizes the modeling method and solving method.

2.1. Modeling Method

For the study of batch planning model of the steelmaking plant, the commonly used modeling methods are: ① conventional modeling method, ② intelligent modeling method, ③ comprehensive integration method. The main idea of the conventional modeling method is the analogy, for example, the scheduling problem of the steelmaking plant is compared to the mixed flow shop problem,⁸⁾ the charge plan problem is compared to the one-dimensional packing problem,⁹⁾ the casting plan problem is compared to the traveling salesman problem.¹⁰⁾ However, to fit classical problem with practical problem, the use of conventional modelling method should pay attention to the improvement of classical problem, which will directly affect the effect of modelling and solving. Intelligent modeling method is mainly based on artificial intelligence or computational intelligence, such as multi-agent system,^11,12) eM-Plant software^13,14) and Flexsim software.^15,16) The use of intelligent modeling method has certain advantages. It breaks through the fixed thinking of conventional modeling method and can get to the core of the problem. For example, the appearance of human-computer interaction no longer separates people from computers, making the process of information exchange between people and computers more smooth, which is very favorable to prepare and adjust the plans. The comprehensive integration method is to combine the first two methods. For example, Wang et al.¹⁷⁾ combined the optimal scheduling method, expert system, case-based reasoning and other technologies to establish a mathematical model for conflict resolution of linear programming, and obtained a conflict-free and optimized plan. However, the combination of multiple methods is naturally influenced by the advantages and disadvantages of the combination, and whether they complement each other needs to be further proved.

2.2. Solving Method

For the study of batch planning model of the steelmaking plant, the commonly used solving methods are: ① heuristic method, ② operations research method, ③ intelligent solving method. The heuristic method is to use experience in solving a problem and choose the method that has already worked, rather than systematically and with certain steps to seek answers. For example, Yang et al.¹⁸⁾ made a heuristic algorithm to solve the steelmaking-continuous casting scheduling model. The operations research method is to use the method of operations research to solve the production planning problem, which is mainly used to solve the small-scale problem. For example, Tang et al.¹⁹⁾ used the branch and bound method to determine the lower bound according to the last machine in the process shop scheduling, and can obtain the results from the computer simulation experiment. Cui et al.²⁰⁾ proposed a difference of convex functions algorithm to solve the scheduling problem of steelmaking - continuous casting process. At present, the most widely used method is intelligent solving method, which uses intelligent optimization algorithm to solve the production planning and scheduling problems, for example, tabu search algorithm,²¹⁾ genetic algorithm^22,23) or hybrid algorithm^24,25) combining multiple algorithms. Lin et al.²⁶⁾ used a modified imprecision-propagating multi-objective evolutionary algorithm to solve the multi-objective integrated model of continuous cast-hot rolling section. However, defining an appropriate intelligent algorithm is necessary to apply specific changes to adapt to a developed model and to address the premature or the local optimum problem of an algorithm itself.

As for the research on batch planning of the steelmaking plant, the amount of research is too much. But, the foreign scholars have studied relatively little, which is determined by the environment of the global metallurgical industry. However, in the existing researches, the theoretical research takes the majority, and the pursuit is mostly to improve the algorithm. The establishment of the model tends to be independent, the practicability of the algorithm or model is not considered, and the coordination before and after the batch planning of the steelmaking plant is not considered. Bases on the research of the charge plan and the casting plan, considering the integration of the two, the feedback and adjustment between the two are very important and of more practical significance under the premise of sequence.

3. Modelling of the Integrated Batch Planning Model

3.1. Problem Description

When the production capacity of the steelmaking and continuous casting equipment and the requirements of the downstream production process are met, the charge plan takes the given preselected contract pool as the input condition and the steelmaking process constraint as the main constraint condition, and the contract optimization combination into the charges. The requirements of different contracts are not consistent, and there are some differences in steel grade, specification, physical characteristics, delivery time and other factors. Therefore, under the requirement of ensuring the minimum smelting furnace capacity, the charge plan should pursue the goals of the minimum delivery time difference, the maximum finished material rate, the lowest production cost and the minimum residual material.

Based on the results of the charge plan and the constraints of the continuous casting procedure, the casting plan optimizes all the charges and combines into the casts. When organizing the production of casters, the decision maker should consider many factors, such as the difference of steel grade due to the continuous casting of molten steel between the charges of the same cast, the abnormal blank due to the adjustment of the width between the charges, the equipment adjustment cost and adjustment time brought by each cast. Therefore, the casting plan should ensure the charges that are combined for the same cast should be in accordance with the minimum steel grade, width (non-increasing order) and delivery time difference, and pursue as many charges as possible for the same cast, and continuously casting on the same caster.

Similar charge plan has been studied in literature,²⁷⁾ but this paper is not limited to charge plan, but the integrated batch planning with charge and casting. The integrated batch plan needs to combine the charge plan and the casting plan, but they cannot be independent from each other, otherwise cannot embody the meaning of integration. The following practice also proves the best charge plan does not necessarily produce the best casting plan, so the mutual feedback and compensation between the two plans are very crucial to the actual production.

3.2. Charge Plan Model

3.2.1. Application Conditions

(1) All the contracts may not be arranged into production.

(2) Number of charges is known.

(3) A single contract requirement is smaller than the furnace capacity and not decomposable. This is the reason for the furnace capacity limitation.

(4) The furnace capacity is constant. This is the requirement of the actual situation, in general, the furnace capacity is fixed.

3.2.2. Symbol Definition

All symbol definitions for this model are presented as follow:

i: Contract i;

j: Contract j (j>i);

k: Charge k;

Y_k: Residual material quantity of charge k;

L: Total number of charges to be planned;

N: Total number of contracts;

W: Furnace capacity;

g_i: Weight of contract i;

H^R: Penalty cost of residual material;

H_i^U: Penalty cost of the contract i not selected;

$H_{ij}^c/H_{ij}^w/H_{ij}^d$: When contracts i and j form a charge, the penalty cost caused by the difference of steel grade, width and delivery time;

h^c/h^w/h^d: Penalty cost coefficient caused by the difference in steel grade, width and delivery time when the contract i and contract j form a charge;

c_i/w_i/d_i: Steel grade, width and delivery time of contract i;

R_c: Maximum difference of steel grade in the same charge;

R_w: Maximum width adjustment range of the contract in the same charge;

R_d: Maximum delivery time interval between contracts in the same charge;

α₁/α₂/α₃: Weight coefficients of the three objective functions, whose sum is 1, are set equal to 1/3 in this article.

3.2.3. Mathematical Model

(1) The objective function can be defined as follows:   

$$Z_1={\sum }_{k=1}^L{\sum }_{i=1}^N{\sum }_{j>i}^N\{H_{ij}^c+H_{ij}^w+H_{ij}^d\}\cdot X_{ijk}$$(1)

  

$$Z_2={\sum }_{k=1}^L{H}^R\cdot Y_k$$(2)

  

$$Z_3={\sum }_{k=1}^L(1-X_{ik})H_i{}^U$$(3)

  

$$min\hspace{0.17em}{Z}_{charge}={\alpha }_1{Z}_1+{\alpha }_2{Z}_2+{\alpha }_3{Z}_3$$(4)

Here, Z₁ is the penalty value caused by the difference in the steel grade, width, and delivery time between the contracts in the same charge; Z₂ is the penalty value of a residual material; and Z₃ is the penalty value of the contract not selected. The objective function Z_charge is to minimize the sum of the weights of Z₁, Z₂, and Z₃.

(2) Constraints can be defined as follows:   

$$X_{ik}=\{\begin{array}{c}1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }if\mathrm{\hspace{0.17em} }\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{\hspace{0.17em} }k\mathrm{\hspace{0.17em} }\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{\hspace{0.17em} }\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} }i/j.\\ 0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }otherwise\end{array}$$(5)

  

$$\begin{array}{l}{X}_{ijk}=\\ \{\begin{array}{c}1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }if\mathrm{\hspace{0.17em} }\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{\hspace{0.17em} }k\mathrm{\hspace{0.17em} }\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{\hspace{0.17em} }\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{s}\mathrm{\hspace{0.17em} }i\mathrm{\hspace{0.17em} }and\mathrm{\hspace{0.17em} }j\mathrm{\hspace{0.17em} }at\mathrm{\hspace{0.17em} }the\mathrm{\hspace{0.17em} }same\mathrm{\hspace{0.17em} }time.\\ 0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }otherwise\end{array}\end{array}$$(6)

  

$${\sum }_{k=1}^L{X}_{ik}\le 1\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(i=1,2,\cdots ,N)$$(7)

Here, X_ik and X_ijk are the discrete variable that can adopt the values 0 or 1. Equation (7) means that each contract can only be arranged into a maximum of one charge.   

$${\sum }_{i=1}^N(g_i\cdot X_{ik})+Y_k=W\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(k=1,2,\cdots ,L)$$(8)

  

$$Y_k\ge 0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(k=1,2,\cdots ,L)$$(9)

It means that the sum of the contract weight within the charge k shall not exceed the furnace capacity, and the difference between the two is the residual material quantity Y_k in this charge.   

$$H_{ij}^c=\{\begin{array}{l}{h}^c\left|c_i-c_j\right|,\mathrm{\hspace{0.17em} }\text{if}\mathrm{\hspace{0.17em} }\text{0}\le \left|c_i-c_j\right|<R_c\\ \mathrm{\hspace{0.17em} }+\infty \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{otherwise}\end{array}$$(10)

  

$$H_{ij}^w=\{\begin{array}{l}{h}^w\left|w_i-w_j\right|,\mathrm{\hspace{0.17em} }\text{if}\mathrm{\hspace{0.17em} }\text{0}\le \left|w_i-w_j\right|<R_w\\ \mathrm{\hspace{0.17em} }+\infty \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{otherwise}\end{array}$$(11)

  

$$H_{ij}^d=\{\begin{array}{l}{h}^d\left|d_i-d_j\right|,\mathrm{\hspace{0.17em} }\text{if}\mathrm{\hspace{0.17em} }\text{0}\le \left|d_i-d_j\right|<R_d\\ \mathrm{\hspace{0.17em} }+\infty \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{otherwise}\end{array}$$(12)

It means the penalty value of contract i and contract j due to differences in steel grade, width and delivery time within the same charge.

3.3. Casting Plan Model

3.3.1. Application Conditions

(1) All charges are arranged for production. The number of charges obtained by the charge plan model is known.

(2) The number of casts is unknown. According to the known number of charges, this model can obtain the number of qualified casts.

(3) The life tundish life is known. The research in this article assumes the difference of steel grade is not big, and its life is set to be known and unchanged. Further research will allow the length of life to be customized according to the steel type.

3.3.2. Symbol Definition

All symbol definitions for this model are presented as follow:

i: Charge i;

j: Charge j (j>i);

k: Cast k;

V_k: Adjustment variable for the cast k;

S: Equipment adjustment cost caused by increasing the cast;

M: Total number of charges;

LT: Service life of the tundish;

$Q_{ij}^c/Q_{ij}^w/Q_{ij}^d$: When charge i and charge j form a cast, the penalty cost caused by the difference of steel grade, width and delivery time;

q^c/q^w/q^d: The penalty cost coefficient caused by the difference in steel grade, width and delivery time when the charge i and charge j form a cast;

$c_i^{\text{*}}/w_i^{\text{*}}/d_i^{\text{*}}$: Steel grade, width and delivery time of charge i;

$R_c^{\text{*}}$: Maximum difference of steel grade in the same cast;

$R_w^{\text{*}}$: Maximum width adjustment range of the charge in the same cast;

$R_d^{\text{*}}$: Maximum delivery time interval between charges in the same cast;

β₁/β₂: The weight coefficients of the two objective functions, whose sum is 1, are set equal to 0.5 in this article.

3.3.3. Mathematical Model

(1) The objective function can be defined as follows:   

$$Z_4={\sum }_{k=1}^M{\sum }_{i=1}^M{\sum }_{j>i}^M\{Q_{ij}^c+Q_{ij}^w+Q_{ij}^d\}\cdot X_{ijk}^{*}$$(13)

  

$$Z_5={\sum }_{k=1}^M{V}_k\cdot S$$(14)

  

$$min\hspace{0.17em}{Z}_{cast}={\beta }_1{Z}_4+{\beta }_2{Z}_5$$(15)

Here, Z₄ is the penalty value caused by the difference in the steel grade, width, and delivery time between the charges in the same cast; Z₅ is the penalty value caused by the adjustment of continuous casting machine due to the increase of casts. The objective function Z_cast is to minimize the sum of the weights of Z₄, and Z₅.

(2) Constraints can be defined as follows:   

$$X_{ik}^{*}=\{\begin{array}{c}1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }if\mathrm{\hspace{0.17em} }\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{\hspace{0.17em} }k\mathrm{\hspace{0.17em} }\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{\hspace{0.17em} }\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{\hspace{0.17em} }i/j.\\ 0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }otherwise\end{array}$$(16)

  

$$\begin{array}{l}{X}_{ijk}^{*}=\\ \{\begin{array}{c}1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }if\mathrm{\hspace{0.17em} }\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{\hspace{0.17em} }k\mathrm{\hspace{0.17em} }\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{\hspace{0.17em} }\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{\hspace{0.17em} }i\mathrm{\hspace{0.17em} }and\mathrm{\hspace{0.17em} }j\mathrm{\hspace{0.17em} }at\mathrm{\hspace{0.17em} }the\mathrm{\hspace{0.17em} }same\mathrm{\hspace{0.17em} }time.\\ 0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }otherwise\end{array}\end{array}$$(17)

  

$${\sum }_{k=1}^M{X}_{ik}^{*}=1\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(i=1,2,\cdots ,M)$$(18)

Here, $X_{ik}^{\text{*}}$ and $X_{ijk}^{\text{*}}$ are the discrete variable that can adopt the values 0 or 1. Equation (18) means that a charge can only be arranged into a maximum of one cast.   

$$1\le {\sum }_{i=1}^M{X}_{ik}^{\text{*}}\le LT\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(k=1,2,\cdots ,M)$$(19)

It means that the number of charges contained in a cast shall be at least 1 and shall not exceed the tundish life.   

$$V_k=\{\begin{array}{c}1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }if\mathrm{\hspace{0.17em} }\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{\hspace{0.17em} }k\mathrm{\hspace{0.17em} }\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s}.\\ 0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }otherwise\end{array}$$(20)

The implication is that V_k is 0, 1 variable, which is 1 if the cast k exists, and 0 if it doesn’t.   

$$Q_{ij}^c=\{\begin{array}{c}{q}^c\left|c_i^{\text{*}}-c_j^{\text{*}}\right|,\mathrm{\hspace{0.17em} }\text{if}\mathrm{\hspace{0.17em} }0\le \left|c_i^{\text{*}}-c_j^{\text{*}}\right|<R_c^{\text{*}}\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }+\infty \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{otherwise}\end{array}$$(21)

  

$$Q_{ij}^w=\{\begin{array}{c}{q}^w\left|w_i^{\text{*}}-w_j^{\text{*}}\right|,\mathrm{\hspace{0.17em} }\text{if}\mathrm{\hspace{0.17em} }\text{0}\le \left|w_i^{\text{*}}-w_j^{\text{*}}\right|<R_w^{\text{*}}\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }+\infty \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{otherwise}\end{array}$$(22)

  

$$Q_{ij}^d=\{\begin{array}{c}{q}^d\left|d_i^{\text{*}}-d_j^{\text{*}}\right|,\mathrm{\hspace{0.17em} }\text{if}\mathrm{\hspace{0.17em} }\text{0}\le \left|d_i^{\text{*}}-d_j^{\text{*}}\right|<R_d^{\text{*}}\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }+\infty \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{otherwise}\end{array}$$(23)

It means the penalty value of charge i and charge j due to differences in steel grade, width and delivery time within the same cast.

3.4. Integrated Batch Planning Model

The study of this article is not only limited to the independent study of the charge and casting plan model, but also aims to develop an integrated batch planning model for the steelmaking plant. It will also be proved later the final results are not necessarily optimal for the steelmaking plant only by independent studies. In the steelmaking process, the thrust generated by the converter procedure and the tension generated by the continuous casting procedure are the main driving forces in this section.²⁸⁾ The best production plan is to ensure the caster can be internally casted to the maximum extent. Therefore, two independent models are integrated to develop the following integrated batch planning model for the steelmaking plant.   

$$min\hspace{0.17em}{Z}_{\text{batch}}={\omega }_1{Z}_{\text{charge}}+{\omega }_2{Z}_{\text{cast}}$$(24)

The implication is to measure the results of an integrated batch plan in the form of a sum of weights. The smaller the value is, the better the result of the integrated batch plan will be. α₁/α₂/α₃ in charge plan, β₁/β₂ in casting plan, and ω₁/ω₂ here are weight coefficients of sub-objective function values, and α₁ + α₂ + α₃ = 1, β₁ + β₂ = 1, ω₁ + ω₂ = 1. According to the “honest choice” principle of game theory, the target function Z_charge, Z_cast and Z_batch all pursue an optimization of collective rationality. That is, under ideal conditions, all sub-objective function values are required to be minimized, but in the actual optimization process, it is difficult to achieve such an ideal situation. Therefore, this paper uses the weight coefficient of sub-objective function value to adjust the degree of influence of sub-objective function value on the total objective function value, in order to meet the requirements of decision makers and achieve collective optimization as much as possible.

For example, the continuous casting machine must be continuous casting. It is difficult to optimize the casting plan and charge plan at the same time, so the research focuses on the casting plan and does not strictly require the charge plan, but only needs to be able to provide enough charges, so the ω₂ must be much greater than the ω₁. When the ω₁ or ω₂ is 1, it indicates the study only considers the charge plan or the casting plan, i.e. the preparation of a single plan. Another purpose of introducing ω₁, ω₂ is to consider the introduction of hot rolling batch plan in the follow-up research, and to facilitate the unification and coordination of charge plan, casting plan and hot rolling plan. In this paper, Z_charge is obtained after iteration of charge plan, and then Z_cast is obtained by solving the casting plan with the charge plan result as input, and then Z_batch can be obtained by using Eq. (24). This is an iteration of batch planning. When all iterations of batch planning are completed, the optimal integrated batch planning results (including charge plan and casting plan results) can be obtained. The author introduced the hot rolling batch plan in the subsequent research process, and took the order difference of slab in the results of charge, casting and hot rolling plan as a subobjective function to measure the integrated batch plan. Compared with Eq. (24), the integration among charge, casting and hot rolling plan can be better demonstrated. The corresponding research results will be considered for further publication. This paper temporarily sets the coefficient to α₁ = α₂ = α₃ = 1/3, β₁ = β₂ = 0.5, ω₁ = 0.3, ω₂ = 0.7.

4. The Proposed Algorithm

The charge plan and casting plan are typically solved in the form of a large-scale combinatorial optimization problem. The complexity of the problem is often related to the quantity of contracts and charges. In the production process at steel plants, the quantity of contracts and charges are large, so the general search method is difficult to meet the expected effect. Herein, we propose an improved elitist genetic algorithm (IEGA) to solve the model. Figure 1 shows the algorithm flow.

Fig. 1.

The flow of IEGA.

Step 1. Firstly read the basic data of the preselected contract pool, including the contract number, weight, steel grade, and delivery times, etc.

Step 2. Initialize algorithm parameters, including iteration times, population size, crossover probability, and conventional mutation probability.

Step 3. Generate the population of N individuals according to the permutation coding rule.

Step 4. Decode the chromosomes of the population.

Step 5. Calculate the objective function values of current population, and record the optimal individual and its chromosomes.

Step 6. Calculate the fitness values of current populations, and record the maximum fitness, average fitness, etc.

Step 7. Judge whether reach the maximum iterations. If so, end the iteration and output the results, if not, proceed to Step 8.

Step 8. According to the improvement of the algorithm in this article, judge whether need the operation of large variation probability and output the corresponding variation probability.

Step 9. During the first selection operation, independently select N parent chromosomes from the current population.

Step 10. Independently perform the partial matching and crossover operations on the N parent chromosomes.

Step 11. Independently perform chromosome fragment reversal mutation operations on the N individuals after crossover.

Step 12. Obtain a population of size 2N by merging the parent population with the offspring population. And the second selection operation, select N individuals from 2N individuals to obtain a new generation of population.

Step 13. Go back to Step 5.

4.1. Encoding Scheme

The final solution results of the charge plan and the casting plan models have such a characteristic that all the contract numbers in the charge plan or all the charge numbers in the casting plan are different from each other. Therefore, this article chooses the encoding method of permutation coding to construct chromosomes. The charge plan uses the contract number to represent the gene value in the chromosome coding string. The chromosome coding can be expressed as {a₁,…,a_i,…,a_N}. The chromosome length is the total number N of precompiled contracts. a_i represents the contract number, a_i∈{1,2,…,N}. The chromosome of the casting plan can be expressed as {b₁,…,b_i,…,b_{L × L}}, where the chromosome length is L × L, L is the total number of charges, b_i∈{1,2,…,L × L}. In the above encoding scheme, the gene positions a_i or b_i in the same chromosome are natural numbers without repetition.

4.2. Decoding Scheme

4.2.1. Charge Plan

Figure 2 shows the chromosome decoding strategy for the charge plan problem. The example represents a feasible solution of the charge plan. The numbers 1–10 represent the contract number, the total number 10 represents the total number of preselected contracts. Here is the pseudocode of chromosome decoding for charge plan.

Fig. 2.

Schematic diagram of chromosome encoding and decoding in charge plan.

Algorithm 1: Chromosome decoding for charge plan

Input: Initialize cumulative weight (weight = 0), the number of charges (k = 0), charge plan results table (chargelist)    Known: furnace capacity (W), total number of charges (L), preselected contract pool data (orda), chromosome of charge plan (chrom)

Ouput: charge plan results table (chargelist) 1: for i = 1,2,…,N do 2:  weight accumulates the weight gi of the contract ai corresponding to chrom [i] in orda 3:  if weight < = W do 4:  record the current contract ai 5:  else 6:  weight minus gi corresponding to the current contract, and all contracts whose cumulative weight does not exceed W are taken as a charge Ak, and W–weight is recorded as residual material Yk. All results record in chargelist. 7:  if k < = L do 8:   k = k + 1; weight = gi 9:  else 10:   for loop ends 11: end for 12: return chargelist

4.2.2. Casting Plan

The algorithm should construct a two-dimensional matrix to represent the structure of the solution in the chromosome decoding process. Equation (25) shows the corresponding relationship between the gene b_i on the chromosome and the position in the two-dimensional matrix. In the decoding process, the two-dimensional matrix shall follow the following conditions:

(1) The size of the two-dimensional matrix is L × L, L represents the total number of preprogrammed charges. The rows of the two-dimensional matrix represent each charge, and the columns represent the possible casts. The number of casts is unknown, but the total number will not exceed L, so it is assumed to be L first.

(2) The value of the element in the matrix is 0 or 1. There is only one 1 in each row, and the rest are 0, which means that each charge can only be in one cast at the same time.

(3) If there is a digit 1 in a column of the matrix, the charge corresponding to that digit 1 will constitute a cast. The sum of all elements of the column shall not be greater than the tundish life. If a column in the matrix is all digits 0, the columns will not be able to form the cast, which meets the requirement the total number of casts is less than or equal to the total number of charges.

When decoding the chromosome, the position of the solution in the two-dimensional matrix is determined according to the gene b_i. The corresponding relationship between any gene b_i and the row number m and column number n in the two-dimensional matrix is as follows:   

$$\begin{array}{l}\text{m}=(b_i-1)//\text{L}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\text{(round}\mathrm{\hspace{0.17em} }\text{down)}\\ \text{n}=(b_i-1)\text{\%L}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\hspace{0.17em}\text{(take}\mathrm{\hspace{0.17em} }\text{the}\mathrm{\hspace{0.17em} }\text{remainder)}\end{array}$$(25)

Figure 3 shows the steps of chromosome decoding in the casting plan. Here is the pseudocode of chromosome decoding for casting plan.

Fig. 3.

Schematic diagram of chromosome encoding and decoding in casting plan. (Online version in color.)

Algorithm 2: Chromosome decoding for casting plan

Input: Initialize the number of casts (k = 0), the two-dimensional matrix (mid), casting plan results table (castlist)    Known: the service life of the tundish (LT), chromosome of casting plan (chrom), charge plan results table (chargelist)

Ouput: casting plan results table (castlist) 1: for i = 1,2,…,L × L do 2:  get m and n from Eq. (18) 3:  if sum(mid[m,:])==0 and sum(mid[:,n])<=LT do // Judge whether the matrix meets the above conditions ② and ③ 4:   mid[m,n]=1 5:  end if 6: end for 7: for i=1,2,..,L do 8:  if sum[:,i]!=0 do 9:   In the castlist, the row number +1* (that is, the charge number) corresponding to the digit 1 in the column is recorded to form a cast k, and record the information corresponding to the charge number in the chargelist 10:  end if 11: end for 12: return castlist

*  Notice that the row and column numbers in the two-dimensional matrix both start at 0, and the charge number starts at 1, so pay attention to the corresponding relation between them.

4.3. Calculating the Fitness Function

Fitness function adopts the fitness distribution method based on the linear ordering of the objective function value²⁹⁾ and can be written as follows:   

$$Fit(Pos)=2-SP+2\times (SP-1)\times (Pos-1)/(Nind-1)$$(26)

Here, Nind is the number of individuals in the population, Pos is the ranking position of individuals in the population; SP is the selection pressure that is generally taken as [1.0, 2.0]. When using the fitness distribution function, an objective function value of each individual in the population must be calculated in advance and these values must be arranged in descending order to obtain a ranking position of the corresponding individual. Then, according to Eq. (26), the algorithm can derive the fitness of each individual by following the convention that the higher the value of the objective function, the smaller the fitness. Thus, in the iterative process, the better individuals are selected from the population.

4.4. Genetic Operators

Genetic operators include selection operators, crossover operators and mutation operators. The selection of an operator considerably influences the convergence of an algorithm. The following compares the IEGA with the elitist genetic algorithm³⁰⁾ (EGA) to explain the specific operators and the main improvement of IEGA.

4.4.1. Selection Operators

In this article, we make two selection operators. Before making the selection, they need to calculate the fitness value of each chromosome according to Eq. (26). Compared with EGA, IEGA performs two selection operations successively, which can improve the convergence speed of the algorithm and reduce the solving time.

After statistical analysis of the current population, the first selection begins to independently select N parent chromosomes from the current population. The elitism and tournament selection method³¹⁾ adopted, which uses the traditional tournament selection method to select a certain number of individuals from the population. However, the improvement in the elite individuals must be selected.

After completing the crossover and mutation operations and merging the parent and offspring populations, the second selection begins. The purpose is to select N chromosomes from the combined 2N chromosomes as the new generation population. This time, a direct copy selection method based on fitness ordering is adopted. After calculating the fitness value of 2N chromosomes, the values are sorted from the largest to the smallest. Then N chromosomes corresponding to the high fitness value are directly copied as a new generation population.

4.4.2. Crossover Operators

This article selects the partial matching crossover operator.³²⁾ Compared with EGA, IEGA integrates conflict detection (i.e. repair algorithm in EGA) into the crossover process, so as to directly obtain the reasonable chromosome after crossover operation, and avoid nesting two functions to prolong the solving time. The following Fig. 4 shows the crossover process, which includes four steps:

Fig. 4.

Schematic diagram of crossover operator.

Step 0. Pair all the chromosomes in the order in which they existed in the population. If the number of chromosomes is odd, the last chromosome is copied directly and does not participate in the crossover operation.

Step 1. Generate randomly the starting and ending positions of the genes to be swapped in a pair of parental chromosomes (A0\A1).

Step 2. Obtain the intermediate chromosomes (B0\B1) by swapping the genes of the two chromosomes and scrambling the sequences respectively

Step 3. Conflict detection. The mapping relationships are established between the two groups of genes in A0\A1. Then the conflicting gene in B0\B1 are replaced according to the mapping relationship. Finally, a new pair of progeny chromosomes (C0\C1) is formed.

4.4.3. Mutation Operators

This article uses chromosome fragment reversal operator³³⁾ to randomly select two reversal points from individual chromosome, and then reverses sequence the genes between the two points to obtain a new progeny chromosome.

In the late iteration period, the population lacks diversity of chromosomes, and sometimes falls into the local optimal solution and cannot get out. Therefore, compared with EGA, the method of large mutation probability is added when using the reversal operator. The idea of the large mutation probability operation^34,35) is that when all the individuals in a generation are relatively concentrated, a mutation operation is performed with a probability much larger than the usual mutation probability. This mutation operation can randomly and independently produce new individuals, thus making the whole population out of precocity.

When the maximum fitness Fitmax and the average fitness Fitave of a certain generation satisfy the following formula, a mutation operation is performed with a large mutation probability.   

$$\alpha Fi{t}_{max}<Fi{t}_{ave}$$(27)

In which, α∈(0.5,1) is the density factor, which represents the concentration degree of an individual. In addition, the algorithm must adopt a great variation operation, when the model maintains the optimal objective function value at least ten times continuously during iteration. Here is the pseudocode of mutation operation.

Algorithm 3: Mutation operators

Input: mutation probability set Mu = {0.1,0.5}, current population pop, density factor α = 5/6

Output: new population newpop 1: Fitness calculation, selection operation and crossover operation are carried out for current population pop 2: if The optimal objective function value of pop has a number of consecutive occurrences in the iterative process ≥10 or satisfying Eq. (20) do 3:  Pm=Mu[2] 4: else 5:  Pm=Mu[1] 6: for i=1,2,..,N do // N: the population size 7.  P=rand() // The mutation probability of random generation (0,1) of each chromosome in the population. 8:   if P ≤ Pm do 9:   The operation of chromosome fragment reversal is performed to generate a new generation of population newpop. 10:  end if 11. end for 12: return newpop

5. Computational Experiments

5.1. Implementation

In order to verify the integrated batch planning model and IEGA of the steelmaking plant, this article uses the actual production data of a certain steelmaking plant as an example to carry out experimental verification. The experimental environment is implemented using Python3.7, and the computing platform is JetBrains PyCharm Community (Edition 2019.2.1 x64).

5.2. Input Data

From the 300 contracts data that the plants plan to produce on a certain day, this article randomly selects 40 contracts as the basic data for the application of the model (Table 1). This research solves the integrated batch planning model for 10 cycles.

Table 1. Basic parameters of charge plan model.

Contract No.	Official number	Steel grade code	Steel grade	Width/mm	Delivery time/d	Weight/t	Cancellation penalty
1	6179921	BQ37701F	24	1495	30	70	10
2	6179928	BQ69200F	24	1533	30	63	10
3	6172135	BN37704F	21	1525	30	75	10
4	6172233	CG61100F	23	1505	30	75	10
5	6156607	CB61200F	24	1474	30	72	10
6	6172129	BQ37704F	24	1474	30	65	10
7	6172242	BN47701F	22	1474	30	75	10
8	6180591	BQ38701F	23	1472	30	91	10
9	6172242	CB61200F	22	1472	30	75	10
10	6180591	AC06300R	11	1494	30	91	10
11	6180572	AC06300R	12	1488	30	81	10
12	6180570	AC06300F	11	1476	30	73	10
13	6180352	AC06300R	11	1474	30	91	10
14	6179558	AC06300R	12	1476	30	74	10
15	6180330	AC06300R	11	1472	30	61	10
16	6180615	AC06300R	11	1472	30	61	10
17	6180619	AC06300F	12	1472	30	61	10
18	6180009	AC06300R	11	1471	30	74	10
19	6180193	AC06300R	11	1471	30	72	10
20	6176823	AC06300R	12	1470	30	74	10
21	6177181	AC06300R	12	1470	30	72	10
22	6177184	AC06300F	10	1464	30	76	10
23	6180189	AC13500R	11	1464	30	62	10
24	6176670	AC13500R	10	1258	15	74	10
25	6178762	AC13500R	10	1255	15	64	10
26	6180571	AC06900F	10	1243	30	81	10
27	6180578	AC10400F	10	1243	30	61	10
28	6179559	BC06800F	12	1241	15	85	10
29	6179987	BC06800F	11	1241	15	65	10
30	6180618	AC05400F	11	1241	30	81	10
31	6122828	AK43701F	23	1569	30	66	10
32	6180060	BQ37704F	24	1525	30	73	10
33	6179923	BQ69200F	24	1524	30	65	10
34	6169455	BH02900F	15	1471	20	74	10
35	6174345	BC06900F	12	1270	15	87	100
36	6154910	AC06300R	12	1270	15	93	100
37	6179629	AC06300R	12	1269	15	93	100
38	6179877	AC06300F	12	1269	15	93	100
39	6178809	AC06300F	12	1268	15	87	100
40	6178952	AC06300R	10	1268	15	77	100

Firstly, the 40 preselected contracts data are used as the input set of the charge planning model, and the number of preparation charges is 10. The experimental parameters are: population number popsize = 500, number of iterations maxgen = 1000, crossover probability Pc = 0.95, mutation probability set Mu = {0.1, 0.5}, W = 300, h^c = 1.0, h^w = 0.02, h^d = 1.0, h^R = 0.2 R^c = 2, R^w = 200, R^d = 5.

Secondly, the integrated batch planning for the steelmaking plant is to take the results of the charge plan as the input set of the casting plan, and then to obtain the casting plan results. Therefore, before solving the casting plan model, the results of the charge plan model should be sorted out, mainly in the following three aspects:

(1) Selecting the maximum steel level of the contract in the same charge as the main steel level of the charge.

(2) Standardizing the contract width of the same charge as the standard width of the continuous casting slab. The standard width is generally accurate to 100 mm.

(3) Selecting the minimum delivery time of the contract in the same charge as the delivery time of the charge.

The sorted charge plan results are used as the input set of the casting plan model. The experimental parameters are: population number popsize = 100, number of iterations maxgen = 500, crossover probability Pc = 0.85, mutation probability Pm = 0.1, LT = 4, S = 100, q^c = 1.0, q^w = 0.02, q^d = 1.0, $R_c^{\text{*}}$ = 2, $R_w^{\text{*}}$ = 200, $R_d^{\text{*}}$ = 5.

5.3. Results of Experiments

The integrated batch planning model is solved on the basis of the solution results of the charge plan and the casting plan. Table 2 shows the results of the integrated batch planning model after 10 cycles. And the second result is the best. The optimal objective function value of batch planning is 263.02. The results of the corresponding charge plan and casting plan are respectively shown in Tables 3 and 4.

Table 2. The results of integrated batch plan model.

Times	The OOFV of the charge plan	The OOFV of the casting plan	The OOFV of the integrated batch plan	Casts	The time to solve/s
1	149.98	312	263.39	3	622.09
2	148.74	312	263.02	3	693.51
3	121.28	506	390.58	5	713.67
4	151.32	312	263.80	3	717.08
5	126.88	506	392.26	5	703.62
6	123.1	410	323.93	4	717.15
7	144.98	408	329.09	4	677.57
8	124.14	410	324.24	4	673.09
9	121.28	506	390.58	4	664.89
10	144.98	504	396.29	4	705.59

OOFV: Optimal objective function value

Table 3. The charge plan results of the second integrated batch plan.

Charge No.	Contract No.	Official number	Steel grade code	Steel grade	Width/mm	Delivery time/d	Weight/t	residual material/t
1	9	6172242	CB61200F	22	1472	30	75	59
	7	6172242	BN47701F	22	1474	30	75
	8	6180591	BQ38701F	23	1472	30	91
2	1	6179921	BQ37701F	24	1495	30	70	28
	6	6172129	BQ37704F	24	1474	30	65
	33	6179923	BQ69200F	24	1524	30	65
	5	6156607	CB61200F	24	1474	30	72
3	11	6180572	AC06300R	12	1488	30	81	55
	10	6180591	AC06300R	11	1494	30	91
	12	6180570	AC06300F	11	1476	30	73
4	18	6180009	AC06300R	11	1471	30	74	32
	19	6180193	AC06300R	11	1471	30	72
	16	6180615	AC06300R	11	1472	30	61
	15	6180330	AC06300R	11	1472	30	61
5	24	6176670	AC13500R	10	1258	15	74	20
	40	6178952	AC06300R	10	1268	15	77
	29	6179987	BC06800F	11	1241	15	65
	25	6178762	AC13500R	10	1255	15	64
6	2	6179928	BQ69200F	24	1533	30	63	23
	32	6180060	BQ37704F	24	1525	30	73
	4	6172233	CG61100F	23	1505	30	75
	31	6122828	AK43701F	23	1569	30	66
7	38	6179877	AC06300F	12	1269	15	93	27
	36	6154910	AC06300R	12	1270	15	93
	35	6174345	BC06900F	12	1270	15	87
8	37	6179629	AC06300R	12	1269	15	93	35
	39	6178809	AC06300F	12	1268	15	87
	28	6179559	BC06800F	12	1241	15	85
9	20	6176823	AC06300R	12	1470	30	74	19
	17	6180619	AC06300F	12	1472	30	61
	14	6179558	AC06300R	12	1476	30	74
	21	6177181	AC06300R	12	1470	30	72
10	23	6180189	AC13500R	11	1464	30	62	71
	22	6177184	AC06300F	10	1464	30	76
	13	6180352	AC06300R	11	1474	30	91

The optimal objective function value of the charge plan = 148.74

Table 4. The casting plan results of the second integrated batch plan.

Cast No.	Charge No.	Steel grade code	Steel grade	Width/mm	Delivery time/d	Weight/t
1	3	AC06300R	12	1500	30	245
	4	AC06300R	11	1500	30	268
	9	AC06300R	12	1500	30	281
	10	AC06300R	11	1500	30	229
2	1	BQ38701F	23	1500	30	241
	2	BQ37701F	24	1600	30	272
	6	BQ69200F	24	1600	30	277
3	5	BC06800F	11	1300	15	280
	7	AC06300F	12	1300	15	273
	8	AC06300R	12	1300	15	265

The optimal objective function value of the casting plan = 312

There are 40 contracts in the preselected contract pool, but only 35 contracts are arranged in the charge plan in Table 3. Contracts 3, 26, 27, 30 and 34 are not included in the charge plan. Besides the limitation of charges number, the reason is that when combining these five contracts and the other contracts into one charge, the steel grade, slab width or delivery time are quite different from other contracts. For example, contracts 3 and 34, when combining these two contracts and any other contracts into one charge, they will result in the penalty of large steel grade difference. When combining contracts 26, 27 and 30 into one charge with any other contract, the steel grade and slab width cannot be reasonably matched at the same time. In addition, due to the early delivery time of contracts 35–40 and the large cancellation penalty of these contracts, the preparation results also show that they are all in the plan. All these fully show that the setting of the penalty coefficient of the charge plan model is reasonable, the solution result is correct, and the solution algorithm is reasonable.

As can be seen from Table 4, among the three casts of casting plan result, the differences in steel grade, slab width and delivery time are all within the reasonable range of setting. Through the analysis of the above results, it is shown that the model and algorithm of integrated batch planning for the steelmaking plant are reasonable, and the calculation efficiency and automation degree are high.

6. Discussion and Analysis

6.1. Proposed Models

From the column of the optimal objective function value of the charge plan in Table 2, it can be found that the optimal plan should be the third or the ninth result, but the final optimal result of the integrated model is the second time. However, in Table 2, the optimal objective function values of the third and ninth times of casting plan are very poor. This is an important reflection of the need for feedback and adjustment between the charge plan and the casting plan to form an integrated batch plan model.

The solution of the integrated model has a sequence, solving the charge plan first, and then solving the casting plan, and finally solving the integrated batch plan model. Therefore, the results of the charge plan can affect the solution of the casting plan model. By comparing the second result and third result in Table 2, that is, Tables 3 and 5, Tables 4 and 6, it can be found that:

Table 5. The charge plan results of the third integrated batch plan.

Charge No.	Contract No.	Steel grade	Width/mm	Delivery time/d	residual material/t
1	24, 40, 25, 29	10, 10, 10, 11	1258, 1268, 1255, 1241	15, 15, 15, 15	20
2	4, 2, 33, 32	23, 24, 24, 24	1505, 1533, 1524, 1525	30, 30, 30, 30	24
3	15, 12, 16, 10	11, 11, 11, 11	1472, 1476, 1472, 1494	30, 30, 30, 30	14
4	9, 3, 7	22, 21, 22	1472, 1525, 1474	30, 30, 30	75
5	8, 5, 1, 6	23, 24, 24, 24	1472, 1474, 1495, 1474	30, 30, 30, 30	2
6	19, 18, 13, 23	11, 11, 11, 11	1471, 1471, 1474, 1464	30, 30, 30, 30	1
7	28, 39, 37	12, 12, 12	1241, 1268, 1269	15, 15, 15	35
8	26, 30, 27	10, 11, 10	1243, 1241, 1243	30, 30, 30	77
9	35, 38, 36	12, 12, 12	1270, 1269, 1270	15, 15, 15	27
10	20, 14, 17, 11	12, 12, 12, 12	1470, 1476, 1472, 1488	30, 30, 30, 30	10

Table 6. The casting plan results of the third integrated batch plan.

Cast No.	Charge No.	Steel grade	Width/mm	Delivery time/d	Weight/t
1	3, 6, 10	11, 11, 12	1500, 1500, 1500	30, 30, 30	286, 299, 290
2	2, 5	24, 24	1600, 1500	30, 30	276, 298
3	1, 7, 9	11, 12, 12	1300, 1300, 1300	15, 15, 15	280, 265, 273
4	4	22	1600	30	225
5	8	11	1300	30	223

(1) The total amount of residual material in the second and third times are 369 t and 285 t, respectively. In this respect, the result of the third time is better. However, according to section 3.2 and 3.3, it can be found that the penalty value of the residual material is not as large as the penalty value of the increased cast.

(2) The purpose of the steelmaking plant is to pursue continuous casting as much as possible, not exceeding the tundish life under the same cast as many charges as possible. The number of the second and third casts are three and five respectively. If the casts is too much, the equipment adjustment cost between casts of the continuous casting machine will be high, which is not desired in the actual situation.

(3) As shown in Table 6, No. 4 cast only includes one charge, and it is two steps different from the steel grade of No. 2 cast, so it cannot be cast continuously. No. 5 cast also only includes one charge, and the delivery time is 30 days different from that of the No. 3 cast. These are unreasonable plans.

(4) The main reason why the objective function value of the third integrated model is larger than the second value is the increase of the casts. In the case of studying the closely related practical problems of steel plants, the practical situation has to be considered as a factor. The third result is obviously not desired by the field managers, which will lead to increased costs, increased labor force and waste of resources.

6.2. Proposed Algorithm

Figure 5 is the iteration curves of the result of the second integrated batch planning model solved by the IEGA. The vertical coordinate represents the objective function value, and the horizontal coordinate represents the iteration times. In Fig. 5(a), after 500 iterations of solving the charge plan model, the optimization result is 148.74. In Fig. 5(b), after 100 iterations of solving the casting plan model, the optimization result is 312. The convergence rate of the algorithm is faster. The scale of the casting plan is smaller than that of the charge plan, so its optimization speed is faster. During the iteration process of the IEGA, the value of the objective function gradually converges to a smaller direction, indicating the algorithm in this article is effective.

Fig. 5.

The iteration curve of the charge plan and casting plan model solved by the IEGA. (Online version in color.)

In addition, this article has tried other methods to solve this case, and the corresponding results are shown in Table 7. The elitist genetic algorithm (EGA) and simple genetic algorithm (SGA) are tried, but can’t search for results with three casts. And the OOFV of the integrated model is obviously more advantageous for IEGA and more in line with the actual steel plant. From the solution of the charge plan alone, the OOFV of IEGA can reach 121.28 in Table 2, which is also better than EGA and SGA. From the solution of the casting plan alone, the result of IEGA is 312, which is also in an absolute advantage. As for the time of solving the problem, there is little difference among the three algorithms, which is mainly related to the population size and the number of iterations. There is also one of the most basic methods, manual production planning. Considering manual planning can only be realized when the number of preselected contracts is small, but the number is large in the article, the staff cannot achieve planning.

Table 7. Comparison of solution results of multiple methods.

Methods	The OOFV of the charge plan	The OOFV of the casting plan	The OOFV of the integrated batch plan	Casts	The time to solve/s
IEGA	148.74	312	263.02	3	693.51
EGA	123.18	410	323.95	4	754.84
SGA	2279.2	502	1035.16	5	758.95

OOFV: Optimal objective function value

In the process of solving the model, each iteration of IEGA needs to be selected twice. The first selection preserves the elite chromosomes in each generation, and the second selection preserves the population with high fitness in each generation. Partial matching crossover operation enhances the difference between individuals in each generation of population. The chromosome fragment reversal mutation operation increases the chromosome richness; in particular, in the late iteration, when the chromosomes in the population tend to be consistent, the combination of the great variation operation increases the chromosome richness to avoid the algorithm tending to the local optimal solution. Therefore, we concluded that the proposed IEGA was superior over EGA and SGA in solving the batch plan model.

7. Conclusion and Prospects

After analyzing the existing research on integrated batch planning in the steelmaking plant, this article develops the charge plan model and the casting plan model successively, and finally proposes the integrated batch planning model. In this article, the idea of modeling focuses more on the feedback and adjustment between the charge plan and the casting plan, rather than the independent study of the two.

According to the specific situation of the model, this article improves the traditional genetic algorithm and solves the planning model successfully. According to the characteristics of the planning model, the research adopts the permutation encoding method, and proposes the different chromosome decoding strategies for the charge plan model and the casting plan model. This article adopts the fitness distribution method based on the linear ordering of the objective function value in the fitness functions. The selection operator adopts the elitism and tournament selection method and the direct copy selection method based on fitness sorting. The crossover operator adopts the partial matching crossover operator. The mutation operator adopts the chromosome fragment reversal operator and combines the method of large mutation probability operation. Through the model analysis, the importance of integrated batch planning is clarified. The algorithm analysis proves that IEGA algorithm is more effective than EGA and SGA.

This study evidences the rationality of the proposed model and algorithm, but its practicability needs to be further verified. In this regard, future works will include the following subjects: 1) development and improvement of the steelmaking plant production planning system with increased versatility; 2) integrating the hot rolling plan, finally form the production plan of steelmaking - continuous casting - hot rolling; 3) studying the scheduling problem.

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