Multi-Objective Constrained Optimization Model and Molten Iron Allocation Application Based on Hybrid Archimedes Optimization Algorithm

Abstract

The challenge of distributing molten iron involves the optimal allocation of blast furnace output to various steelmaking furnaces, considering the blast furnace’s production capacity and the steelmaking converter’s consumption capacity. The primary objective is to prioritize the distribution from the blast furnace to achieve a balance between iron and steel production while ensuring that the volume of hot metal within the system remains within a safe range. To address this, a constrained multi-objective nonlinear programming model is abstracted. A linear weighting method combines multiple objectives into a single objective function, while the Lagrange multiplier method addresses constraints. The proposed hybrid Archimedes optimization algorithm effectively solves this problem, demonstrating significant improvements in time efficiency and precision compared to existing methods.

1. Introduction

Iron and steel production plays a vital role in the Chinese economy. It encompasses various processes such as ironmaking, steelmaking, continuous casting, and hot rolling. Steel production involves multiple dynamic processes with solid coupling and uncertainties, making production scheduling a significant challenge [1]. Peng et al. [2] addressed the steelmaking–refining–continuous casting (SCC) process as a hybrid flowshop rescheduling problem. They proposed an enhanced empire competition algorithm to tackle the SCC schedule in their research on steel production scheduling. The algorithm’s design includes a multi-swap-based local search and imperialist competition to improve exploitation and restart strategies to enhance exploration ability. Yu et al. [3] studied a novel hybrid flowshop scheduling problem with operation skipping. The objective functions considered were the average sojourn time, total earliness, and total tardiness, with an added constraint for operation skipping. They developed an improved memetic algorithm and proposed enhanced local search to improve exploitability. For the steelmaking–continuous–casting dynamic scheduling problem under equipment failure, Long et al. [4] proposed a two-phase robust dynamic production scheduling method based on release time-series prediction for the scenario of uncertain tasks. The authors of [5] based on the problem specifications and constraints, a novel generic multi-objective optimization model with objectives including the makespan, the average flow time, and the total workload imbalance is formulated. Use non-dominated sorting genetic algorithm-II (NSGA-II) to solve this problem. Lv et al. [6] proposed a branch-and-bound algorithm for a single-machine scheduling problem with proportional job degradation. Kong et al. [7] developed a robust optimization-based co-optimization approach for scheduling issues in steel production processes. They considered uncertainties in processing time and the impact of deterioration. Moreover, they created a co-optimization model based on variable neighbor principles and devised a heuristic solution algorithm. The authors of [8] regarded the primary process of steel production as a hybrid flowshop scheduling problem containing multiple constraints. They established an integer planning model using specific real production constraints and proposed a batch-decoupling-based Lagrangian relaxation algorithm. After relaxing the capacity constraints using Lagrange multipliers in this algorithm, the relaxed problem decomposes the subproblems and solves them using dynamic programming. The authors of [9] propose a multi-objective optimization model that considers multiple performance measures to minimize the effect of different real-time events used in this paper. Moreover, a biased random iterative greedy method based on predictive response should be adopted to address this issue. The authors of [10] incorporated controllable processing time to decompose the whole scheduling problem into two subproblems and proposed a hybrid differential evolutionary algorithm combining the variable neighborhood decomposition search and iterative inverse list scheduling algorithms to solve the scheduling problem. Ma et al. [11] aimed to address the pressing need to reduce transportation costs and improve the turnover rate of shunting locomotives. They achieved this by creating a dictionary-order multi-objective function that combines the characteristics of the model. This function prioritizes the objective function and solves the entire model using a hybrid iterative algorithm. The authors of [12] propose a mixed-integer linear mathematical model. And propose Multi-objective Ant Lion Optimizer (MOALO), Multi-objective Keshtel Algorithm (MOKA), and Multi-objective Keshtel and Social Engineering Optimizer (MOKSEA) to solve this issue.

As an essential part of production scheduling in iron and steel enterprises, hot metal scheduling is the control center of the hot metal logistics process. The real-time rationality of hot metal scheduling directly determines the stability and efficiency of the hot metal logistics process. The scheduling of molten iron is generally divided into the following three steps: Distributing hot metal from the blast furnace to the steelmaking plant, allocating molten iron tanks to each blast furnace for transporting and determining the transportation routes for locomotives and iron tanks. The conventional hot metal scheduling management approach relies on manual tracking and dispatching of vehicles, posing significant challenges in ensuring the real-time accuracy of vehicle locations. It is difficult to accurately dispatch hot metal on time [13]. Therefore, relying on manual mode has yet to meet the dynamic scheduling needs of iron and steel enterprises. Choosing a suitable hot metal distribution scheme can effectively improve the turnover rate of the hot metal tank, increase the temperature of hot metal to the converter, improve production efficiency, and reduce costs. To address this issue, Chen et al. [14] developed an integer programming model to solve the hot metal allocation problem. The model aims to minimize the path length while ensuring that the quantity of hot metal issued at the supply point matches the hot metal received at the demand point. However, the model does not consider the stock of hot metal in the hot metal tank as a buffer. The model may have no solution in theory, but it can be solved using the hot metal in the hot metal tank as a buffer. Tang et al. [15] analyzed the actual production characteristics of hot metal distribution from the blast furnace to the converter in iron and steel enterprises. The hot metal distribution process must comply with the constraints of the arrival time of hot metal, as determined by the blast furnace production plan and the transportation time between the ironworks and the steelworks. They proposed a dynamic programming algorithm based on state-space relaxation to solve the subproblem model, but the algorithm does not consider the factor of molten iron circulation. Huang [16] established a multi-objective linear programming model to solve the distribution scheme, with the production capacity of blast furnaces and steelmaking plants constrained by the safe value of circulating molten iron. However, the capacity of blast furnaces to produce molten iron was affected by too many factors, and it took much work to increase the planned initial molten iron production within one day. Therefore, one of its objective functions, the maximization of the blast furnace’s production capacity, cannot be applied in practice. Li et al. [17] developed an intelligent control system for hot metal transportation, dynamically guiding hot metal distribution according to the demand for hot metal in steelmaking and the actual production of blast furnaces, combined with parameters such as the priority of iron consumption in steelmaking and the priority of iron distribution in blast furnaces. Li et al. [18] developed a transportation dispatching system. They evaluated the iron melt processing capacity, optimal vehicle configuration, and road planning for the production line. Lu et al. [19] simulated and evaluated the feasibility and performance indicators of the multi-scenario logistics scheme by considering various dynamic factors in the process of transporting molten iron using “crane + Cross-Train AGVs”. Lian et al. [20] designed an iterative candidate solution set and a new neighborhood structure for the Jaya algorithm, taking the maximum completion time as an optimization index, and finally proved the effectiveness of the improved Jaya algorithm with a benchmark instance set.

In summary, in recent years, various solutions have been proposed for the scheduling problem of iron and steel production, including various metaheuristic and hybrid optimization algorithms. These methods have significantly improved scheduling efficiency and solved uncertainty problems. However, different methods still face challenges in practical applications. This article improves the objective function based on the literature [16] through discussions with experts in steel enterprises. The objective function is to minimize the difference between the circulating amount of molten iron after the allocation plan and its optimal value and to establish quadratic linear programming. The multi-objective function in this article has a clear preference, and the objective function $J_1$ is the most important. Compared with $J_1$, $J_2$ can compromise with $J_3$, so we can convert the multi-objective function into a weighted sum form. We will use a hybrid AOA combining the Archimedes optimization algorithm (AOA) [21] and the limited-memory Broyden–Fletcher–Goldfarb–Shanno with bounds (L-BFGS-B) [22] algorithm to solve this problem. AOA has been tested on the CEC’17 test suite and four engineering design problems, outperforming state-of-the-art algorithms. The results indicate AOA’s high convergence speed and its ability to balance exploration and exploitation. L-BFGS-B is well suited for large-scale optimization due to its efficient Hessian matrix approximation, support for box constraints, and fast convergence rates. Therefore, we have chosen these algorithms for our solution. The rest of this article is organized as follows. Section 2 presents the problem formulation. Section 3 details the modeling process. Section 4 presents the hybrid AOA. Section 5 validates the model and algorithm through experiments. Section 6 we conclude this article.

2. Problem Formulation

The macro-scheduling of hot metal distribution and the primary factor to consider in hot metal scheduling involves balancing iron and steel production. It addresses the problem of how much hot metal produced by each blast furnace should be allocated to each steelmaking plant for smelting over a long planning period. It is necessary to coordinate the demand for molten iron in the steel mill and the production capacity of each blast furnace in the iron mill. The production capacity of each blast furnace and converter in the steel mill should be fully utilized as soon as possible. The amount of molten iron in the system should be controlled to achieve a balance between ironmaking and steelmaking, ensuring that molten iron logistics meet the weight and timing requirements of the steel mill. An iron and steel balance plan should be determined for each blast furnace and each steel plant to ensure the stable operation of both the iron and steel plants.

Take a steel plant as an example. It has two iron and three steel plants. Iron Plant A operates three blast furnaces (BFs), while Iron Plant B operates two. Steel Plant C runs three converters (CLs), while Steel Plant D runs three and Steel Plant E runs one. Steel plant dispatchers allocate iron in the following way: iron produced by Iron Plant A is prioritized for delivery to Steel Plants C and E. The iron produced by Iron Plant B is prioritized for delivery to Steel Plant D. If the iron supplied by Iron Plant B cannot meet the demand of Steel Plant D, Iron Plant A will also provide molten iron to Steel Plant D. In rare cases, Iron Plant B will also provide molten iron to Steel Plants C and E. At the same time, we need to control the amount of iron in the system at the end of each day or shift to ensure it remains within the safety stock. Based on its current iron supply, the main characteristics of iron allocation are multi-point supply and demand matching, weight considerations, and time coordination, i.e., multiple iron supply points correspond to multiple demand points, and there is a cross-correspondence between them. The correspondence between iron and steel mills is shown in Figure 1.

3. Constrained Optimization Model for Iron Allocation

Based on the detailed analysis of factors affecting the iron–steel balance and the communication with on-site scheduling experts, the characteristics of the hot metal distribution problem are summarized. We comprehensively consider the maximum production capacity of each blast furnace in the ironmaking plant, the iron consumption of each steelmaking plant, the number of furnaces, the furnace production, and the maximum production capacity of each steelmaking plant within the planned period. We take the system’s iron content at the beginning of the planning cycle, the maximum/minimum safe stock of the system’s iron content during the planning cycle, the optimal value of the system’s iron content, and the production priority of each steel plant as constraint conditions. The objective function is to minimize the difference between the system’s circulating iron content after distribution and the optimal circulating iron content, as well as the transportation time. We establish a multi-objective function nonlinear programming model with constraints for the iron–steel balancing problem to determine the weight of hot metal transported from the blast furnace to the converter in each steel plant during the planned period. In this problem, we can determine the priorities of different objective functions through communication with field scheduling experts. Therefore, this paper uses the linear weighting method to convert multiple objectives into a single objective, and equality constraints are incorporated into the objective functions by introducing Lagrange multipliers, thus transforming the original problem into an unconstrained single-objective problem.

3.1. Premise Assumptions

In order to effectively address the iron–steel balancing problem, the following key assumptions are made:

(1)

Iron and steel balancing involves adjusting the amount of iron produced by blast furnaces to meet the iron demand of the steel plant. This balancing is done based on the overall weight of the iron rather than on a tank-by-tank basis.

(2)

Intermediate iron pre-treatment processes are disregarded since the steel balancing problem concerns the iron weight balance between blast furnaces, iron supply points, and iron demand points (i.e., steel mills). By focusing on the overall flow of iron, we can ignore the intermediate iron pre-treatment processes.

3.2. Parameter and Variable Definitions

We define the parameters and variables based on the input table of steel balancing and the known conditions to provide a clear mathematical description of the steel balancing problem. The parameters and variables are defined in

Table 1. Parameters and variable definitions.

Symbol	Description
p	Index of the blast furnace, where p = {1, 2, 3, 4, 5}
q	Index of the converter, where q = {1, 2, 3, 4, 5, 6, 7}
Bp	Volume of iron in blast furnace p
Zq	Number of planned production units for converter q
BFp	Production capacity of blast furnace p
LSq	Planned production volume for converter q
LCq	Volume of output per furnace for converter q
THq	Planned iron consumption for converter q
LGq	Capacity of converter q to consume iron water in one day
Hq	Optimal iron water content in the system for converter q
C$pq$	Priority of transfer from blast furnace p to converter q
Mq	Maximum circulation of molten iron in converter q
Nq	Minimum circulation of molten iron in converter q
W$pq$	Amount of molten iron distributed from blast furnace p to converter q

.

3.3. Objective Functions

Iron–steel balancing is a multi-objective optimization problem with three objectives—$J_1$, $J_2$, and $J_3$—where $J_1$ aims to minimize the total transport priority from each blast furnace to each converter, $J_2$ aims to reduce the difference between the amount of iron in the converter system and the optimal amount of iron in the converter system after allocation, and $J_3$ aims to minimize the difference between the amount of iron in the blast furnace system and the optimal amount of iron in the blast furnace system after allocation. The objective functions are defined as follows:

(1)

During the planning cycle, the objective is to minimize the cumulative transport priority from each blast furnace to each converter. That is,

$$Min{J}_1=\sum _{p=1}^5\sum _{q=1}^7{C}_{pq}{W}_{pq}$$

(1)

(2)

Minimize the difference between the amount of iron in the converter system and the optimal amount of iron in the converter system: During the planning cycle, the sum of the amount of iron received by each converter and the amount of iron in the converter system at the beginning of the plan, minus the amount of iron consumed by the converter, should be as close as possible to the optimal amount of iron. That is,

$$Min{J}_2=\sum _{q=1}^7{(H_q-(Z_q+\sum _{p=1}^5{W}_{pq}-L{G}_q))}^2$$

(2)

(3)

Minimize the difference between the amount of iron in the blast furnace system and the optimal amount of iron in the blast furnace system: During the planning cycle, the sum of the capacity of each blast furnace and the amount of iron in the blast furnace system at the beginning of the plan, minus the amount of iron transported to the converters, should be as close as possible to the optimal amount of iron. That is,

$$Min{J}_3=\sum _{p=1}^5{(F_p-(B{F}_P+B_p-\sum _{q=1}^7{W}_{pq}))}^2$$

(3)

3.4. Constraints

The constraints are defined as follows:

(1)

Capacity constraint of the blast furnace: During iron and steel balancing, for each sub-cycle of the planning cycle, the amount of iron produced and delivered from each blast furnace cannot exceed the sum of the maximum production capacity of the blast furnace in the sub-cycle and the amount of iron initially in the furnace system. That is,

$$\sum _{q=1}^7{W}_{pq}\le B_p+B{F}_p,\forall p\in \{1,2,3,4,5\}$$

(4)

(2)

Maximum/minimum safety stock constraints for the system’s circulating iron quantity: Since blast furnace production is continuous, to ensure production safety and stability in the steelmaking area and to prevent unexpected events such as equipment failure, a certain safety stock quantity should be maintained. The inventory of the system’s circulating iron has maximum and minimum safety ranges. By controlling the iron weight to keep it within these ranges, we can prevent situations where the blast furnace cannot supply enough iron, thereby avoiding disruptions in subsequent steelmaking processes. Additionally, this control prevents excessive iron supply, which could result in an overabundance of heavy tanks and a reduction in empty tanks, ultimately impacting the safe iron output of the blast furnace. This approach mitigates the risk of exceeding the allowable weight of the system’s iron. The specified maximum/minimum safety stocks are as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill Z_q+\sum _{p=1}^5{W}_{pq}-L{G}_q\le M_q,\forall q\in \{1,2,3,4,5,6,7\}\\ \hfill Z_q+\sum _{p=1}^5{W}_{pq}-L{G}_q\ge N_q,\forall q\in \{1,2,3,4,5,6,7\}\end{array}\end{array}$$

(5)

(3)

Constraints on the amount of iron delivered: The weight of iron delivered from each blast furnace to each steelmaking plant during the planning cycle of steel balancing must be non-negative:

$$W_{pq}\ge 0,\forall p\in \{1,2,3,4,5\},\forall q\in \{1,2,3,4,5,6,7\}$$

(6)

3.5. Model Description

In order to gain a better understanding and analyze the established multi-objective constrained optimization model, we have provided a detailed description of the model’s various components and parameter settings in

Table 2. Model description.

Category	Symbol	Description	Mathematical Expression
Objective Functions	$J_1$, $J_2$, $J_3$	Refer to Section 3.3	Formulas (1)–(3)
Constraints	s.t.	Refer to Section 3.4	Formulas (4) and (5)
Decision Variable	$W_{pq}$	Refer to Table 1	$W_{pq}\ge 0$
Parameters	LS, LC, TH	Refer to Table 1	LS, LC, TH $\ge 0$

.

4. Hybrid AOA

4.1. AOA

The AOA is a new meta-heuristic optimization algorithm inspired by an interesting physical law called Archimedes’ principle. It mimics the principle of buoyancy applied to an object wholly or partially immersed in a fluid, with buoyancy proportional to the fluid’s displacement weight. In the AOA, a population is an object immersed in a liquid and is divided into global exploration and local search phases depending on whether the object immersed in the liquid has collided. If a collision occurs, the algorithm enters the global exploration stage. Otherwise, it enters the partial development stage. The migration operator ($TF$) for the two-stage switch is shown in Equation (7), where t and $t_{max}$ represent the current number of generations and the maximum number of iterations, respectively.

$$TF=exp\left({\displaystyle \frac{t-t_{max}}{t_{max}}}\right)$$

(7)

If $TF\le 0.5$, the AOA enters the global exploration stage. Otherwise, enter the partial development stage. In the initialization phase, the AOA randomly initializes the volume (vol), density (den), and acceleration (acc) of each object. In this process, the AOA evaluates the initial population and selects the current optimal individual ($x_{best}$), density ($de{n}_{best}$), volume ($vo{l}_{best}$), and acceleration ($ac{c}_{best}$) for updating the density, volume, and acceleration of other individuals. The updated density and volume are shown in Equation (8), where $de{n}_i^t$ and $de{n}_i^{t+1}$ are the densities of the $i_{th}$ and ${i+1}_{th}$ individuals in generation t, and $vo{l}_i^t$ and $vo{l}_i^{t+1}$ are the volumes of the $i_{th}$ and ${i+1}_{th}$ individuals in generation t.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & de{n}_i^{t+1}=de{n}_i^t+rand(de{n}_{best}-de{n}_i^t)\hfill \\ \hfill & vo{l}_i^{t+1}=vo{l}_i^t+rand(vo{l}_{best}-vo{l}_i^t)\hfill \end{array}\end{array}$$

(8)

When $TF<0.5$, there is a collision between the objects, and the algorithm enters the global search phase to update the acceleration of the individuals, as given by Equation (9):

$$ac{c}_i^{t+1}={\displaystyle \frac{de{n}_{mr}+vo{l}_{mr}\ast ac{c}_{mr}}{de{n}_i^{t+1}\ast vo{l}_i^{t+1}}}$$

(9)

When $TF>0.5$ there is no collision between objects and the algorithm proceeds to the local development phase. The acceleration of the individuals is updated according to Equation (10), where ${acc}_{best}$ is the acceleration of the current optimal individual.

$$ac{c}_i^{t+1}={\displaystyle \frac{de{n}_{best}+vo{l}_{best}\ast ac{c}_{best}}{de{n}_i^{t+1}\ast vo{l}_i^{t+1}}}$$

(10)

According to Equation (11), the AOA performs a normalization operation on the acceleration of the individual to update the position of the individual, where ${acc}_{i_{norm}}^{t+1}$ is the acceleration after normalization for the ith individual in generation $t+1$; u and l are used to adjust the range of normalization.

$$ac{c}_{i_{norm}}^{t+1}=u\ast \left({\displaystyle \frac{ac{c}_{best}-mi{n}_{best}}{ma{x}_{acc}-mi{n}_{acc}}}\right)+l$$

(11)

During the global search phase, the individual locations are updated according to Equation (12), where $x_i^{t+1}$ and $x_i^t$ are the positions of the $i_{th}$ individual in generation $t+1$ and generation t, $x_{rand}$ is the position of a random individual in generation t, $rand$ is a random number between 0 and 1, and $C_1$ is the stationary constant. d is the density factor, and its update equation is given by (13):

$$x_i^{t+1}=x_i^t+C_1\ast rand\ast ac{c}_{i_{norm}}^{t+1}\ast d\ast (x_{r}and-x_i^t)$$

(12)

$$d^{t+1}=exp\left({\displaystyle \frac{t_{max}-t}{t_{max}}}\right)-{\displaystyle \frac{t}{t_{max}}}$$

(13)

During the local development phase, individual locations are updated according to Equation (14), where $C_2$ is a fixed constant; $T=TF\ast C_3$, with $T\in [C_3\ast 0.3,1]$; and F is the direction factor, which is used to determine the direction of position update for the iteration and is defined in Equation (15).

$$x_i^{t+1}=x_{best}^t+F\ast C_2\ast rand\ast ac{c}_{i_{norm}}^{t+1}\ast d\ast (t\ast x_{best}-x_i^t)$$

(14)

$$F=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}P\le 0.5,\hfill \\ -1,\hfill & \mathrm{if}P>0.5.\hfill \end{array}\right.$$

(15)

where $P=2\ast rand-C_4,$ and $C_4$ is a fixed constant.

4.2. L-BFGS-B Algorithm

L-BFGS-B is an iterative mathematical optimization algorithm for solving unconstrained or constrained nonlinear optimization problems. The core idea of the algorithm is to update the parameters by approximating the inverse of the Hessian matrix of the objective function while considering bounded constraints to ensure that the values of the parameters during the optimization process are within a predetermined range. The main steps of the L-BFGS-B algorithm are as follows:

(1)

Select the initial point: Determine the initial point of the optimization problem and the initial values of the parameters;

(2)

Calculate the gradient of the objective function: Calculate the gradient of the objective function (first-order derivative) at the current point;

(3)

Constraints handling: Ensure that the values of the current parameters satisfy the constraints;

(4)

Estimate the inverse of the Hessian matrix: The L-BFGS-B algorithm uses finite memory to approximate the inverse of the Hessian matrix. During the iteration process, the algorithm maintains a finite size of historical information, through which it approximates the inverse of the Hessian matrix;

(5)

Line search: Determine the step size for the next step by performing a line search to determine the distance to move in the direction of the current gradient;

(6)

Update the parameters: Update the parameters using the step size determined by the line search to obtain the new parameter values;

(7)

Convergence test: Check whether the stopping condition is satisfied. If the stop condition is satisfied, the algorithm terminates; otherwise, return to Step 2 for the next iteration;

(8)

Iteration: Repeat Steps 2 to 7 until the stop condition is satisfied.

4.3. Hybrid AOA Process

The hybrid AOA keeps iterating the elite solution and divides the algorithm into exploration and development phases by using the migration operator TF. The exploration phase starts with the algorithm itself, which has an extensive searching range and strong global searching ability. The development phase begins with the current optimum and develops it locally using the L-BFGS-B algorithm, which has a smaller searching range and slower convergence speed. The hybrid AOA flowchart is shown in Figure 2, and the algorithm steps are detailed below.

Step 1: Initialize the algorithm parameters: The number of populations, the maximum number of iterations, and $C_1$ to $C_4$;

Step 2: Calculate the fitness of each individual in the population and record the optimal individual position $\left(x_{best}\right)$, the optimal density $\left(de{n}_{best}\right)$, the optimal volume $\left(vo{l}_{best}\right)$, and the optimal acceleration $\left(ac{c}_{best}\right)$;

Step 3: Update the density and volume according to Equation (8);

Step 4: Update the transfer operator and density factor according to Equations (7) and (13);

Step 5: If the transfer factor $TF\le 0.4$, use (9) to update the acceleration and (12) to update the individual position;

Step 6: If the transfer factor $TF>0.4$, first use the current optimal individual position as the initial point for the L-BFGS-B algorithm. If solving is successful, output the optimal result; otherwise, use (10) to update the acceleration and (14) to update the individual position;

Step 7: Check whether the iteration stop condition is satisfied. If it is, exit and output the optimal result; otherwise, repeat Steps 2–6.

5. Experiments and Analysis

Parameters such as the 9-day ironmaking plan, the under-furnace system circulation, the steelmaking plan, and the under-converter system hot metal, as well as the maximum/minimum safety amounts randomly selected from a steel mill in mid-August, are shown in

Table 3. Planned production capacity of blast furnace and circulation of molten iron in the system.

Symbol	1	2	3	4	5	6	7	8	9
BF1/t	5900	5900	5900	5900	5900	5800	6000	6100	6100
BF2/t	7500	7500	7500	7500	7500	7400	7400	7400	7400
BF3/t	5900	5900	5900	5900	5900	5600	6000	6100	6100
BF4/t	5600	5600	5600	5600	5600	5600	5600	5600	5600
BF5/t	5600	5600	5600	5600	5600	5500	5500	5700	5700
Bf1 molten iron circulation /t	300	300	200	200	350	200	300	300	300
Bf2 molten iron circulation /t	300	300	200	200	350	200	300	300	300
Bf3 molten iron circulation /t	350	200	200	200	350	200	300	300	300
Bf4 molten iron circulation /t	300	200	100	150	350	200	200	200	200
Bf5 molten iron circulation /t	300	200	100	150	300	200	200	200	200

and

Table 4. Ironmaking plan and molten iron from blast furnace hearth system.

Symbol	1	2	3	4	5	6	7	8	9
Converter 1 molten iron circulation/t	400	360	300	300	450	300	400	400	400
Converter 2 molten iron circulation/t	400	360	300	300	450	300	400	400	400
Converter 3 molten iron circulation/t	400	360	300	300	450	300	400	400	400
Converter 4 molten iron circulation/t	300	240	200	200	350	200	300	300	300
Converter 5 molten iron circulation/t	300	240	200	200	350	200	300	300	250
Converter 6 molten iron circulation/t	300	240	200	200	350	200	300	300	250
Converter 7 molten iron circulation/t	150	200	100	100	200	100	100	200	100
LS1	34	34	35	35	36	36	35	25	35
LS2	34	34	34	35	36	36	35	33	35
LS3	27	34	34	36	36	36	35	33	34
LS4	44	43	42	42	42	43	38	43	43
LS5	44	42	42	42	42	43	38	43	43
LS6	44	43	43	42	42	43	39	44	43
LS7	30	24	30	30	30	23	30	30	10
TH1-TH3 (kg/t)	880	890	870	860	860	880	880	895	895
TH4-TH6 (kg/t)	880	875	870	850	850	860	870	890	890
TH7 (kg/t)	660	650	650	650	650	650	650	680	680
M1-M3/t	600	600	600	600	600	600	600	600	600
M4-M6/t	300	300	300	300	300	300	300	300	300
M7/t	300	300	300	300	300	300	300	300	300
N1-N3/t	300	300	300	300	300	300	300	300	300
N4-N6/t	200	200	200	200	200	200	200	200	200
N7/t	100	100	100	100	100	100	100	100	100

.

5.1. Optimization Results of Molten Iron Allocation

A PC with an AMDRyzen5PRO, 2.10 GHz, 16.0 GB of RAM, and Windows 10, using the PyCharm development environment and Python 3.10 were used to establish the multi-objective optimization model for hot metal allocation constraints, which includes a total of 35 variables and 24 constraints. The process of using the model to solve the optimal value and the number of iterations for the data numbered 9 in Table 3 and Table 4 is shown in Figure 3. The final calculation results are shown in

Table 5. Allocation plan.

	Converter 1	Converter 2	Converter 3	Converter 4	Converter 5	Converter 6	Converter 7
Blast furnace 1	645	1838	1107	1268	827	407	0
Blast furnace 2	2753	2185	2405	0	0	0	0
Blast furnace 3	2096	1470	1825	0	0	0	702
Blast furnace 4	0	0	0	1647	2190	1659	0
Blast furnace 5	0	0	0	1583	1531	2482	0

,

Table 6. Circulating amount of molten iron in the blast furnace system at the end of the plan.

Blast Furnace	1	2	3	4	5
Molten iron circulation/t	306	357	302	302	303

and

Table 7. Circulating amount of molten iron in the converter system at the end of the plan.

Converter	1	2	3	4	5	6	7
Molten iron circulation/t	473	476	475	266	265	265	141

, with the accuracy accurate to a single decimal place, reflecting the actual conditions of the steel mill.

As can be seen in Figure 3, the hybrid AOA demonstrates very good optimization ability during the initial iterations, but the step size decreases in later iterations, resulting in an increased number of iterations. However, due to the small step size, the search is more refined, making it easier to find the global optimal solution and reducing the likelihood of falling into a local optimum. Therefore, the accuracy and stability of the hybrid AOA are improved.

A data value of 1123 in Table 4 indicates that on the planned day, Blast Furnace 1 of Iron Plant A should allocate 1123 tons of hot metal to Converter 1 of Steel Plant C. A data value of 0 means that there is no need for distribution on the planned day. For example, the plan for Blast Furnace 2 does not require allocation to Converter 4 on the same day. According to the data in Table 6, at the end of the plan, there remains a corresponding amount of system iron water under each blast furnace. For example, Blast Furnace 1 still holds 151 tons of system water iron and can participate in the distribution plan in the next planning cycle. The data in Table 7 are similar to that in Table 6. For example, Converter 1 holds 445 tons of system iron water, which can be used to supply the converter when the blast furnace capacity is insufficient or transportation is disrupted, thus preventing the converter from shutting down when there is no hot metal distribution.

5.2. Algorithm Comparative Analysis

Since the model is a non-convex quadratic programming problem, there are multiple local minima, some of which may be global, while others may only be local minima. So, the same algorithm may yield different optimal values after multiple experiments. Therefore, the difference between numerous optimal values is an important metric for measuring an algorithm. This paper assesses the algorithm’s stability by evaluating the maximum difference among the optimal values observed in multiple experiments.

In order to demonstrate the high efficiency of the hybrid AOA, we utilize the Powell and SLSQP algorithms to solve the optimization model for hot metal allocation. The iterative solution process is shown in Figure 4.

When comparing Figure 3 and Figure 4, it can be seen that the optimal value in Figure 4 results in a significantly reduced number of iterations. However, the algorithm may fall into a local optimum due to the considerable iteration step length. We run the Powell algorithm, SLSQP algorithm, and hybrid AOA independently ten times each to obtain the running time, number of iterations, and stability metrics for each algorithm, as shown in

Table 8. Performance Comparison of Powell, SLSQP, and Hybrid AOA Algorithms.

	Powell	SLSQP	Hybrid AOA
Average running time	3.448	0.49	0.425
Optimal value stability	32,696.113	0.07	0.003
Average number of iterations	26	26.8	57.2

.

The Powell algorithm is a numerical optimization algorithm for unconstrained optimization problems. It iteratively finds the optimal solution by conducting a one-dimensional line search at each step. The SLSQP algorithm is a sequential quadratic programming method designed for solving nonlinear optimization problems with equality and inequality constraints. This algorithm gradually approaches the optimal solution in each iteration by solving a quadratic programming subproblem. The experiment does not use Lagrange multipliers in the SLSQP algorithm; instead, constraints are integrated into the objective function to test the performance of the three algorithms from multiple perspectives. Considering the three aspects of algorithm performance, in terms of the average number of iterations, the hybrid AOA needs several iterations to finally reach convergence due to its smaller iteration steps in the later stages. Therefore, the hybrid AOA has a noticeable gap in the average number of iterations compared to the Powell and SLSQP algorithms. The running time of the SLSQP algorithm also shows good performance. However, the algorithm may not always obtain the optimal solution due to the randomly generated initial points, making it seem like it failed. The results obtained using the hybrid AOA method are superior regarding both the average run time and algorithm stability. Especially concerning stability, if applied to the actual production of a steel mill, its accuracy only needs to be accurate to a few decimal places When the algorithm is assessed based on the accuracy of its convergence conditions, the hybrid AOA significantly reduces the number of iterations and greatly improves the running time. Table 8 shows the experimental results of the hybrid AOA and the Powell algorithm on nine data groups under the same operating environment.

It can be seen in

Table 9. Comparative Effectiveness of Powell and Hybrid AOA Algorithms.

Number	Algorithm	Optimal Value	Optimal Solution	Time/s	Iterations
1	Hybrid AOA	2,193,382.03	[998.61, 1172.10, 1183.89, 693.10, 601.64, 1209.22, 0, 2769.75, 2420.78, 2218.03, 0, 0, 0, 0, 1489.43, 1664.92, 779.53, 0, 0, 0, 1974.68, 0, 0, 0, 1403.06, 2221.62, 1936.38, 0, 0, 0, 0, 2445.87, 1718.77, 1396.44, 0]	0.433	53
	Powell	2,228,256.66	[1565.38, 431.31, 1360.86, 1171.20, 1027.30, 304.11, 0, 2706.48, 2655.12, 1353.16, 0, 0, 0, 692.15, 989.52, 2173.91, 1469.73, 0, 0, 0, 1273.12, 0, 0, 0, 3348.80, 1545.37, 666.93, 0, 0, 0, 0, 21.22, 1969.23, 3571.25, 0]	4.525	17
2	Hybrid AOA	1,402,531.66	[1220.94, 1558.75, 1221.14, 747.81, 713.43, 538.94, 0, 2742.46, 2066.55, 2742.01, 0, 0, 0, 0, 1362.94, 1701.04, 1363.18, 0, 0, 0, 1473.85, 0, 0, 0, 1952.75, 1917.82, 1732.96, 0, 0, 0, 0, 1736.61, 1701.64, 2165.27, 0]	0.469	52
	Powell	1,436,098.48	[395.62, 1281.67, 2348.72, 715.94, 620.78, 641.78, 0, 3333.32, 2122.88, 1430.58, 0, 0, 0, 658.70, 1604.83, 1929.24, 1552.75, 0, 0, 0, 809, 0, 0, 0, 2479.74, 1042.34, 2084.00, 0, 0, 0, 0, 1234.08, 2665.06, 1707.35, 0]	3.095	19
3	Hybrid AOA	1,433,502.40	[1180.66, 1623.73, 1413.28, 559.74, 470.89, 767.57, 0, 3117.51, 2126.78, 2321.59, 0, 0, 0, 0, 1102.65, 1498.29, 1513.94, 0, 0, 0, 1900.99, 0, 0, 0, 1832.46, 1747.18, 2038.75, 0, 0, 0, 0, 1918.22, 2092.36, 1607.79, 0]	0.418	51
	Powell	1,495,993.32	[0, 1527.72, 2117.69, 978.27, 547.36, 282.55, 558.09, 3513.50, 1836.87, 1552.55, 0, 0, 0, 660.10, 1886.12, 1885.26, 1580.11, 0, 0, 0, 666.89, 0, 0, 0, 3333.57, 1684.25, 600.40, 0, 0, 0, 0, 1.50, 2081.44, 3535.42, 0]	4.016	25
4	Hybrid AOA	1,428,132.33	[1099.62, 2046.11, 1438.85, 527.35, 869.25, 0, 0, 3235.37, 1469.22, 2826.60, 0, 0, 0, 0, 1010.16, 1829.83, 1229.98, 0, 0, 0, 1911.21, 0, 0, 0, 2237.74, 1492.59, 1903.36, 0, 0, 0, 0, 1456.22, 1859.50, 2317.96, 0]	0.481	54
	Powell	1,749,079.47	[103.44, 1442.77, 2612.03, 283.64, 927.16, 378.31, 229.52, 4183.72, 2081.31, 1269.81, 0, 0, 0, 0, 1052.06, 1816.00, 1455.77, 0, 0, 0, 1663.18, 0, 0, 154.21, 3010.04, 2283.50, 183.01, 0, 0, 0, 0,941.36, 1019.71, 3668.92, 0]	3.833	25
5	Hybrid AOA	2,511,079.53	[1544.84, 1512.48, 1362.18, 83.61, 519.51, 813.67, 0, 2695.69, 2480.15, 2210.48, 0, 0, 0, 0, 1165, 1412.91, 1832.89, 0, 0, 0, 1425.54, 0, 0, 0, 1684.53, 1802.83, 2051.47, 0, 0, 0, 0, 2380.02, 1825.81, 1283.01, 0]	0.455	56
	Powell	2,541,979.23	[1548.78, 1207.82, 1635.80, 688.00, 426.00, 327.15, 0, 2618.72, 2233.42, 1931.22, 0, 0, 0, 605.21, 1236.62, 1962.12, 1835.71, 0, 0, 0, 811.79, 0, 0, 0, 1875.55, 635.62, 3022.38, 0, 0, 0, 0, 1589.51, 3091.06, 803.57, 0]	2.528	14
6	Hybrid AOA	1,602,163.94	[1248.21, 1720.70, 1781.37, 269.71, 900.59, 151.22, 0, 2586.00, 2630.29, 2369.64, 0, 0, 0, 0, 1762.63, 1245.86, 1445.84, 0, 0, 0, 1417.46, 0, 0, 0, 1684.79, 1850.45, 2341.65, 0, 0, 0, 0, 2370.58, 1574.06, 1832.20, 0]	0.475	50
	Powell	1,638,592.76	[0, 1719.82, 2299.22, 286.48, 657.17, 398.84, 709.89, 3592.87, 2131.57, 1864.50, 0, 0, 0, 0, 2002.82, 1740.23, 1433.52, 0, 0, 0, 702.57, 0, 0, 0, 3685.57, 1694.96, 489.13, 0, 0, 0, 0, 353.10, 1977.35, 3442.23, 0]	4.632	28
7	Hybrid AOA	2,264,160.77	[1443.18, 1762.27, 1762.42, 24.66, 141.86, 809.19, 0, 2295.30, 2498.97, 2499.33, 0, 0, 0, 0, 1676.85, 1154.09, 1153.59, 0, 0, 0, 1959.08, 0, 0, 0, 1667.26, 2148.91, 1629.93, 0, 0, 0, 0, 2196.03, 1597.16, 1552.92, 0]	0.496	58
	Powell	2,266,539.20	[355.45, 1631.85, 2972.11, 306.31, 311.22, 366.97, 0, 4076.19, 2021.04, 1158.82, 0, 0, 0, 39.36, 985.01, 1767.34, 1285.10, 0, 0, 0, 1907.23, 0, 0, 0, 2945.42, 766.18, 1733.73, 0, 0, 0, 0, 639.06, 2812.54, 1892.78, 0]	4.164	24
8	Hybrid AOA	2,804,022.05	[951.96, 1094.19, 1095.24, 1048.15, 871.67, 913.23,0, 2039.15, 2593.57, 2591.77, 0, 0, 0, 0, 967.45, 1521.87, 1522.59, 0, 0, 0, 1962.54, 0, 0, 0, 1567.74, 1887.73, 1921.44, 0, 0, 0, 0, 1910.89, 1767.40, 1798.65, 0]	0.468	55
	Powell	2,854,985.02	[79.11, 1347.05, 1687.19, 1424.73, 883.20, 549.41, 0, 2465.62, 1993.82, 1761.16, 0, 0, 0, 1005.57, 1411.70, 1866.79, 1758.23, 0, 0, 0, 948.32, 0, 0, 0, 262.53, 2948.04, 2162.25, 0, 0, 0, 0, 2845.28, 700.51, 1926.50, 0]	2.444	15
9	Hybrid AOA	1,556,698.75	[645.39, 1838.57, 1107.49, 1268.78, 827.52, 407.11, 0, 2753.07, 2185.92, 2405.93, 0, 0, 0, 0, 2096.50, 1470.49, 1825.16, 0, 0, 0, 702.74, 0, 0, 0, 1647.47, 2190.32, 1659.60, 0, 0, 0, 0, 1583.13, 1531.56, 2482.69, 0]	0.481	56
	Powell	1,556,779.20	[2058.11, 171.75, 1352.98, 1471.95, 795.29, 244.45, 0, 2378.26, 2938.68, 2031.68, 0, 0, 0, 0, 1059.42, 2383.53, 1952.33, 0, 0, 0, 701.73, 0, 0, 0, 3010.73, 1057.44, 1427.38, 0, 0, 0, 0, 18.67, 2697.98, 2878.76, 0]	4.432	23

that both the hybrid AOA proposed in this paper and the Powell algorithm can solve the problem of the hot metal intelligent allocation model. However, the algorithm proposed in this paper has clear advantages in terms of computing time. Specifically, the optimal values, running times, and number of iterations of the two algorithms are compared in Figure 5 and Figure 6.

According to the analysis in Figure 5 and Figure 6, the results from nine groups of numerical experiments show that the algorithm proposed in this paper is superior to the existing Powell algorithm in terms of optimal value and running time, Specifically, the average optimal value is reduced by 0.03%, while the average running time is increased by nearly eight times. The algorithm proposed in this paper is stable regarding both optimal value and running time.

We verified the effectiveness of the experimental results through the enterprise’s actual allocation plan. As the allocation plan of the enterprise cannot be precise to the converter level, it can only be precise to the steel plant to which the converter belongs. Therefore, we adjusted the experimental results by aligning the allocation plan with the steel plant level. The comparative data are shown in

Table 10. Comparison of allocation accuracy.

Number	Algorithm	NO. 1	NO. 2	NO. 3	Accuracy of NO. 1	Accuracy of NO. 2	Accuracy of NO. 3
1	Hybrid AOA	14,697.04	13,626.10	1974.68	98.38%	99.41%	99.26%
	Reality	14,462.80	13,706.88	1960.20
2	Hybrid AOA	15,979.01	13,207.23	1473.85	98.25%	99.93%	95.43%
	Reality	15,704.94	13,216.00	1544.40
3	Hybrid AOA	15,898.43	13,034.96	1900.99	97.45%	99.98%	99.50%
	Reality	15,502.53	13,037.82	1891.50
4	Hybrid AOA	16,185.74	12,663.97	1911.21	97.37%	99.79%	98.96%
	Reality	15,770.68	12,637.80	1891.50
5	Hybrid AOA	16,216.62	12,444.46	1425.54	99.08%	98.47%	75.37%
	Reality	16,068.24	12,637.80	1891.50
6	Hybrid AOA	16,790.54	12,975.25	1417.46	97.88%	99.12%	97.75%
	Reality	16,441.92	13,090.92	1450.15
7	Hybrid AOA	16,246.00	11,767.92	1959.08	98.37%	99.26%	96.43%
	Reality	15,985.20	11,855.93	1891.50
8	Hybrid AOA	14,377.79	13,686.90	1962.54	97.96%	99.83%	99.18%
	Reality	14,089.99	13,710.45	1978.80
9	Hybrid AOA	16,328.52	13,598.18	702.74	98.60%	99.95%	93.46%
	Reality	16,102.84	13,604.99	659.60

.

In Table 10, NO. 1, NO. 2, and NO. 3 represent Steel Plant 1, Steel Plant 2, and Steel Plant 3, respectively. We can see from the table that, except for two groups, the similarity between the iron water allocated by the algorithm and the actual allocation exceeds 96%, with some data reaching 99.9%. We analyzed the two sets of abnormal data and found that the actual allocation of iron water was relatively large, resulting in the circulation of molten iron approaching the maximum safe value. Therefore, the similarity between the algorithm and the actual results is low.

6. Conclusions

This article fully considers the resource constraints and objectives in the allocation of molten iron and establishes a multi-objective constrained non-convex optimization model for molten iron allocation. The established non-convex programming model can also be used for neural network training in deep learning, optimization of vehicle routing problems, and delivery routes. A hybrid AOA was proposed to solve the iron distribution scheme and determine the daily steel balance plan of the steel plant. Experimental studies have shown that the hybrid AOA can effectively and accurately solve the distribution of molten iron. Compared with constrained multi-objective methods, the proposed model has better stability and efficiency, making it more suitable for practical applications.