A Bi-Population Co-Evolutionary Multi-Objective Optimization Algorithm for Production Scheduling Problems in a Metal Heat Treatment Process with Time Window Constraints

Abstract

Heat treatment is a critical intermediate process in copper strip manufacturing, where strips go through an air-cushion annealing furnace. The production scheduling for the air-cushion annealing furnace can contribute to cost reduction and efficiency enhancement throughout the overall copper strip production process. The production scheduling problem must account for time window constraints and gas atmosphere transition requirements among jobs, resulting in a complex combinatorial optimization problem that necessitates dual-objective optimization of the total atmosphere transition cost of annealing and the total penalties for time window violations. Most multi-objective optimization algorithms rely on the evolution of a single population, which makes them prone to premature convergence, leading to local optimal solutions and insufficient exploration of the solution space. To address the challenges above effectively, we propose a Bi-population Co-evolutionary Multi-objective Optimization Algorithm (BCMOA). Specifically, the BCMOA initially constructs two independent populations that evolve separately. When the iterative process meets predefined conditions, elite solution sets are extracted from each population for interaction, thereby generating new offspring individuals. Subsequently, these new offspring participate in elite solution selection alongside the parent populations via a non-dominated selection mechanism. The performance of the BCMOA has undergone extensive validation on benchmark datasets. The results show that the BCMOA outperforms its competitive peers in solving the relevant problem, thereby demonstrating significant application potential in industrial scenarios.

1. Introduction

Copper strips are an essential metallic material in manufacturing sectors such as electrical machinery [1], electronic information [2], and industrial production [3]. The cold-rolling process of a copper strip begins with uncoiling, followed by steps including acid–alkali cleaning, rolling, annealing, cutting, and packaging [4]. Among these steps, annealing is an important heat treatment process serving to reduce hardness, enhance ductility, eliminate internal stress, refine grain structure, and improve microstructure, as well as mechanical properties [5].

The annealing process of copper strips in an air-cushion furnace is subject to constraints related to the specific gas atmosphere and soft time windows. Here, gas atmosphere refers to artificially configured single gases or gas mixtures that are used to protect the workpiece being processed, regulate the annealing process, or facilitate specific chemical reactions [6]. Annealing furnaces must maintain different gas atmospheres for different metals, which introduces more complex optimization challenges. Soft time window constraints allow the annealing of copper strips to be completed outside the specified time window but incur a penalty for such deviations. Due to the requirements of precision manufacturing in the entire production line, copper strips undergo different processing steps, resulting in varying soft time window constraints.

Typically, the gas atmosphere transition during annealing takes several hours. In the copper production process, annealing serves as an intermediate processing step, with other production stages both preceding and following it, as illustrated in Figure 1. Before annealing, there are processes such as acid–alkali cleaning and rolling; after annealing, there are stages like cutting and packaging. This renders the production scheduling of annealing extremely critical. The production scheduling of copper strip annealing in an air-cushion furnace aims to optimize the processing sequence of copper strips. By reducing both the atmosphere transition costs during production and the penalties incurred from time window deviations, it achieves cost reduction and efficiency improvement in the production process. This problem is a combinatorial optimization problem with two objectives and a variant of the Traveling Salesman Problem with Time Windows (TSPTW), thus inheriting NP-hard characteristics. As the number of copper coils to be processed increases, the number of feasible solutions grows exponentially.

In order to satisfy the dual-criterion requirement of the problem at hand, it is necessary to design a multi-objective optimization algorithm (MOEA) to solve it, as has been done when dealing with similar production scheduling problems [7,8,9]. Currently, the main research directions in multi-objective optimization are categorized into two types: The first type, generally referred to as decomposition-based multi-objective optimization algorithms, typically utilizes reference vectors to maintain population diversity. Zhang et al. [10] first proposed a Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D). Building upon this work, Zhang et al. [11] proposed a decomposition-based evolutionary algorithm with clustering and hierarchical estimation (MOEA/DCH). They incorporated a hierarchical estimation method, a clustering-based adaptive decomposition strategy, and a heuristic-based initialization method into the MOEA/D. Li et al. [12] incorporated the chain-partition-based adaptive weight vector adjustment method into the MOEA/D and proposed an improved decomposition-based multi-objective evolutionary algorithm for temporal data scheduling problems. Wu et al. [13] proposed a multi-objective evolutionary list scheduling algorithm, incorporating classical list scheduling into the MOEA/D. They represented genomes that had evolved via specific genetic operators using scheduling sequences and preference weights and interpreted them as scheduling solutions through specifically customized list scheduling heuristics. Li et al. [14] proposed a reinforcement-learning-based MOEA/D, which employs an initial strategy with three rules, a Q-learning-based parameter adaptation strategy, and a reinforcement-learning-based variable neighborhood search to enhance the convergence and diversity of solutions.

The second type is regarded as dominance-sorting-based multi-objective optimization algorithms, which can utilize non-dominated sorting (NS) information to determine elite solutions. Among them, NSGA-II [15] and NSGA-III [16] hold important positions. The workflow of NSGA-II primarily involves population initialization, assigning fitness via fast NS, combining parent and offspring populations, selecting optimal solutions to form a new parent population, and repeating genetic operations until the termination condition is met. This algorithm ensures convergence through an elitism strategy and maintains solution set diversity using crowding distance. NSGA-III, on the other hand, introduces the concept of reference points. It constructs hyperplanes, associates population members with reference points, prioritizes solutions close to the reference points, and maintains the number of associated solutions to preserve diversity, thereby enhancing the algorithm’s ability to handle problems in high-dimensional objective spaces. Building upon this work, Shen et al. [17] proposed a hybrid non-dominated sorting algorithm based on NSGA-II. It is based on a co-evolutionary particle swarm optimization algorithm integrated with Cauchy mutation and weight mapping crossover, which is employed to generate new solutions. Li et al. [18] proposed a multi-objective topology optimization (MOTO) methodology based on the hybridization of NSGA-II and a differential evolution algorithm, incorporating a jumping mutation operator to ensure the diversity of Pareto solutions. Li et al. [19] proposed an improved NSGA II with an adaptive crossover and mutation probability strategy and a reverse-based learning strategy.

Both types of multi-objective optimization algorithms rely on the evolution within a single population. They lack multi-population co-evolution and inter-population interaction mechanisms, making them prone to premature convergence and limited exploration of the solution space. For scheduling problems in copper strip production, although the current relevant research has focused on converter scheduling [20], electricity cost reductions [21], and melting–casting scheduling [22], to the best of the authors’ knowledge, there is no existing research on the production scheduling problems arising from the process of an air-cushion annealing furnace.

Therefore, we propose a Bi-population Co-evolutionary Multi-objective Optimization Algorithm (BCMOA) to address the described multi-objective copper strip annealing scheduling problem, which aims to minimize the total annealing cost and the penalty value for time window violations. The main contributions of this paper are as follows:

• We investigate a production scheduling problem arising from an air-cushion annealing furnace during copper strips’ heat treatment processing, which is subject to constraints based on the specific gas atmosphere and soft time windows.
• We present the BCMOA to effectively solve the described problem, which can significantly enhance the population diversity and solution efficiency through its unique dual-population structure and co-evolution mechanism.

Furthermore, to comprehensively evaluate the effectiveness of the proposed algorithm, this study conducts tests on the proposed algorithm through extensive experiments, aiming to demonstrate its efficiency and stability. Section 2 presents the problem description; Section 3 introduces the algorithm, illustrating the architecture and details of the proposed algorithm; Section 4 displays and analyzes the experimental results; and Section 5 is the conclusion section.

2. Problem Description

2.1. Industrial Process

An air-cushion continuous annealing furnace is extensively used in the heat treatment process of copper strips. It has replaced traditional coil-integrated annealing and adopts single-coil uncoiling annealing. Through the ejection of high-velocity hot gas via upper and lower nozzles, an air-cushion is formed to support the strip, avoiding contact with internal furnace structures [23]. This enables high-temperature short-time uniform annealing. During the gas atmosphere transition process in an air-cushion continuous annealing furnace, the costs primarily encompass three components: gas costs, energy consumption costs, and equipment costs. Gas costs, the most significant portion, involve two stages: purging and maintenance. During the purging stage, the original gas must be evacuated and purged until the purity meets relevant standards. During the maintenance stage, gas is supplemented gradually to compensate for micro-leakage. Special gases like hydrogen–nitrogen mixtures or ammonia decomposition gas require additional tail gas treatment systems. Energy consumption costs are primarily incurred during temperature ramping. Heating the furnace demands substantial electricity or fossil fuels [24], while cooling relies on water cooling systems that continuously circulate water. Equipment costs arise from accelerated aging of components such as seals, heating/cooling systems, gas pipelines, and valves due to frequent gas changes and temperature fluctuations. This reduces the service life and necessitates regular maintenance to ensure safety during the atmosphere transition.

Continuous air-cushion annealing furnaces are primarily utilized for the annealing of brass, copper, and other alloys, with different products requiring distinct in-furnace atmospheres. For the annealing of copper strips, a reductive protective atmosphere is predominantly required, with three commonly used types: high-hydrogen, low-hydrogen, and pure nitrogen. In contrast, inert gases or low-hydrogen environments are typically employed for brass strip annealing.

The atmosphere transition process takes a long time and significantly affects production efficiency. The transition time varies between different atmospheres. Naturally, if the subsequent product is of the same type as the current one and requires the same gas atmosphere, the transition can be avoided; otherwise, it is necessary. Furthermore, the copper strips have their specific time windows, representing the expected completion time for their processing. The described scheduling problem aims to sequence the copper strips to be processed to minimize the gas atmosphere transition cost and minimize the total penalty caused by time window deviations.

2.2. Mathematical Model

The concerned problem can be formulated as a mathematical model. To construct the model, the symbol definitions, as shown in

Table 1. Symbol definitions.

Symbol	Definition
$n$	The quantity of copper strips to be processed
$c_{ij}$	Atmosphere transition cost
$a$	Penalty coefficient for early completion
$b$	Penalty coefficient for delayed completion
$P_i$	Annealing time of copper strip i
$d_{ij}$	Atmosphere transition time between copper strips i and j
$t_i$	The start time of copper strip i’s annealing process
${\check{t}}_i$	The lower bound of the completion time window of copper strip i
${\widehat{t}}_i$	The upper bound of the completion time window of copper strip i
$M$	A sufficiently large positive number
$x_{ij}$	Binary variable: 1—transition from the i-th copper strip to the j-th copper strip; 0—otherwise
$e_i$	Binary variable: 1—copper strip i’s completion time is earlier than ${\check{t}}_i$; 0—otherwise
$l_i$	Binary variable: 1—copper strip i’s completion time is later than ${\widehat{t}}_i$; 0—otherwise
$z_1$	Total atmosphere transition cost of annealing
$z_2$	Total penalties for time window violation

, are introduced. In addition, two sets, ${\Omega }_2$ and ${\Omega }_1$, are constructed. Among them, ${\Omega }_2=\left\{1,\dots ,n\right\}$ represents the set of all copper strips to be processed, and ${\Omega }_2=\left\{1,\dots ,n\right\}$ additionally includes a virtual copper strip, 0, to represent the initial gas atmosphere. Given these, the model is developed as described in the following.

The total atmosphere transition cost of the whole annealing process can be expressed as shown in Equation (1), which depends on the production sequence of copper strips and the different gas atmospheres required by them. The penalty value for time window violations is mainly divided into two parts, penalties for being ahead of the time requirement and penalties for being behind the time requirement, as shown in Equation (2). Since the failure of copper strips to enter the next process on schedule will delay the overall processing progress, the impact of delayed completion is greater than that of early completion. Therefore, the penalty coefficient for early completion a is set to 1, and the penalty coefficient for delayed completion b is set to 10.

Constraint (3) represents that there must be one copper strip that is processed immediately after each copper strip. Constraint (4) limits that there must be one copper strip that is processed immediately before each copper strip. Constraint (5) ensures that the production of a copper strip can only start after the completion of the previous one and an atmosphere transition, thereby avoiding any overlap of consecutive production tasks. Constraint condition (6) judges whether a copper strip completes earlier than the lower bound of its time window requirement. If this is true, it forces $e_i=1$. Constraint condition (7) judges whether a copper strip completes later than the upper bound of its time window requirement. If this is true, it forces $l_i=1$. Constraint conditions (8) and (9) determine the value ranges of decision variables.

$$\min z_1={\displaystyle \sum _j{\displaystyle \sum _i{c}_{ij}{x}_{ij}}}$$

(1)

$$\min z_2=a{\displaystyle \sum _i\left\{\left({\check{t}}_{\begin{array}{c}i\end{array}}-t_i-P_i\right)e_i\right\}}+b{\displaystyle \sum _i\left\{\left(t_i+P_i-{\widehat{t}}_i\right)\cdot l_i\right\},i\in {\Omega }_1}$$

(2)

$${\displaystyle \sum _j{x}_{ij}}=1,\forall i\in {\Omega }_1$$

(3)

$${\displaystyle \sum _i{x}_{ij}=1,\forall j\in {\Omega }_1}$$

(4)

$$t_j\ge t_i+P_i+d_{ij},\forall i\in {\Omega }_1,\forall j\in {\Omega }_2$$

(5)

$${\check{t}}_i-t_i-P_i\le e_i\cdot M,\forall i\in {\Omega }_1$$

(6)

$$t_i-{\widehat{t}}_i+P_i\le l_i\cdot M,\forall i\in {\Omega }_1$$

(7)

$$t_i\ge 0,\forall i\in {\Omega }_1$$

(8)

$$x_{\mathrm{ij}},e_i,l_i\in \left\{0,1\right\},i\in {\Omega }_1,\forall j\in {\Omega }_1$$

(9)

3. Algorithm Description

Although the described problem can be formulated as the aforementioned mathematical programming model, conventional mathematical programming solvers, such as CPLEX, are limited to identifying a single optimal solution, rendering them suitable solely for single-objective optimization problems. In contrast, multi-objective optimization problems yield a set of solutions, wherein each solution constitutes a non-dominated solution relative to the others. Consequently, it is imperative to research multi-objective optimization algorithms to solve the described problem. Usually, multi-objective evolutionary algorithms are widely used to solve similar scheduling problems. In their evolutionary process, balancing the diversity and convergence of solutions is of vital importance. The evolution of a single population tends to converge prematurely, lacking diversity and exhibiting low solution efficiency [25]. In the production scheduling problems of a copper strip annealing process, issues such as complex operations involving time window constraints and local optimal solutions often arise, which impose higher requirements on the algorithm’s solution efficiency and global search capability. Thus, this work proposes a BCMOA to effectively solve the described problem. Its core idea is to enhance the diversity and convergence of the Pareto Front (PF) through bi-population co-evolution and NS, while reducing the solution complexity to improve efficiency.

3.1. The Structure of the BCMOA

The procedure of the BCMOA is depicted in Figure 2: Two populations are initialized and independently evolve via their crossover and mutation operators. At the end of each generation, elite solutions are retained through NS. After reaching a certain number of iterations, interactions, including additional crossover and mutation, are performed between the elite solutions of the two populations. The bi-population co-evolution strategy provides a strong driving force for solving the described problem. This evolutionary process continues until the maximum number of fitness functions (1) and (2) evaluations is reached. Once it terminates, the final Pareto solutions are output. The details of initialization, encoding and decoding, offspring generation, and bi-population co-evolution strategies of the BCMOA are separately introduced in the following.

3.2. Initialization and Encoding/Decoding

In the algorithm initialization process, permutation encoding is adopted to randomly generate production sequences in arrays until the population size requirement is met. These sequences are then compiled into a matrix. The encoding and decoding processes are illustrated in Figure 3. Given a production sequence encoding, the atmosphere transition time $d_{ij}$ between any two copper strips is known. Then, the starting time $t_i$ of each copper strip $i$ can be calculated sequentially according to constraint (5). Afterwards, the violations of soft time window constraints can be determined by constraints (6) and (7). Ultimately, objective functions (1) and (2) can be calculated according to these determined factors, and thus, a decoding process is completed.

3.3. Offspring Generation

To further broaden the diversity of feasible solutions, the BCMOA employs the Order Crossover (OX), Partially Matched Crossover (PMX), Inversion Mutation (IM), and Swap Mutation (SM) operators for crossover and mutation. Figure 4 illustrates the crossover processes of PMX and OX using an example with six genes. In the BCMOA framework, each population employs a fixed crossover operator, while IM and SM are separately implemented on the two offspring solutions obtained by crossover.

PMX determines the crossover region by randomly selecting two crossover points. After a crossover, gene duplication typically occurs. To repair the chromosomes, a matching relationship for each chromosome within the crossover region is established. Conflicts are resolved by applying this matching relationship to duplicate genes outside the crossover region. OX randomly selects start and end positions in two parent chromosomes. Genes within this region from Parent 1 are copied directly to the same positions in Offspring 1. The remaining genes, not present in Offspring 1, are filled in sequence from Parent 2. The other offspring is generated in a similar manner. IM reverses the order of a randomly selected segment of genes on the chromosome. SM implements mutation via pairwise gene position swapping on the chromosome.

During offspring generation, the bi-population randomly selects two parent solutions for evolution using OX/PMX, producing two offspring. After crossover, the two offspring are mutated using the IM and SM operators, respectively, to generate two new offspring. This process repeats until all parent solutions have generated offspring. Elite solutions are then preserved via NS.

3.4. Bi-Population Co-Evolution

Algorithm 1 presents the pseudo-code of a bi-population co-evolution mechanism in the BCMOA. When the algorithm iterations reach the preset iteration threshold for the dual populations, a bi-population co-evolution mechanism is triggered. In it, the PF solution sets E1 and E2 are first extracted from each population. Subsequently, the sizes of the two PF solution sets are determined as N1 and N2, and the smaller of the two sizes is used as the basis for determining the evolutionary scale. Crossover and mutation operations are then performed on the extracted PF solutions to generate offspring solutions. Next, the offspring solutions are merged with the original solutions of their respective parent populations, and NS is applied to the merged solution sets. Finally, the sorted dual-population PF solution sets are obtained, E1 and E2 are updated, and the algorithm proceeds to the next iteration.

Algorithm 1 Bi-population Co-evolution mechanism
Input: Two PF solution sets E1 and E2
Output: Two updated PF solution sets E1 and E2
1:	E3 = [ ]//Initialization
	//Step 1. Extract the number of solutions and determine the number of interactions
2:	N1 = length(E1)//Extract the number of solutions for E1
3:	N2 = length(E2)//Extract the number of solutions for E2
4:	interaction_num = min(N1, N2)//Take the smaller value of N1 and N2 as the number of interactions
	//Step 2. Crossover and mutation generate the offspring elite solution E3
5:	for (i = 1; i <= interaction_num; i++) do
6:	sol_E1 = E1 [i]//Select the i-th solution of E1
7:	sol_E2 = E2 [i]//Select the i-th solution of E2
8:	child = crossover (sol_E1, sol_E2)//Crossover
9:	child = mutation (child)//Mutation
10:	E3.append(child)//Collect descendant solutions
11:	end for
	//Step 3. Update E1 and E2
12:	merged_E1 = E1 ∪ E3//Merge E1 and E3
13:	E1 = non_dominated_sorting (merged_E1)//NS
14:	merged_E2 = E2 ∪ E3//Merge E2 and E3
15:	E2 = non_dominated_sorting (merged_E2)//NS

4. Interpretation of Result

4.1. Experimental Settings

In this section, benchmark datasets retrieved from https://homepages.dcc.ufmg.br/~rfsilva/tsptw/#instances (accessed on 27 July 2025) are utilized for testing the performance of the proposed BCMOA. The case names adhere to a specific format, NaWb.00c, where a denotes the number of cities in the case, b denotes the temporal unit for time window ranges (with larger values indicating broader time window spans), and c denotes the case identifier.

Given that the two populations in the BCMOA can undergo crossover via distinct operators, BCMOA variants with different crossover operators are first tested. Subsequently, the one exhibiting the best performance is selected for comparison with other algorithms. Naming conventions for BCMOA variants with different crossover operators are formulated: taking BCMOA-P/PO as an example, “P/PO” indicates that the inter-population crossover operator is PMX, while the crossover operators for the two populations are PMX and OX, respectively. Similarly, we have six BCMOA variants, named BCMOA-P/PO, BCMOA-P/PP, BCMOA-P/OO, BCMOA-O/PO, BCMOA-O/PP, and BCMOA-O/OO, respectively.

Other algorithms included in the comparative analysis comprise NSGA-II [15] and NSGA-III [16], with varying crossover operators. Although the concerned scheduling problem arises from an industrial process with two objective functions and a set of constraints, its core is to obtain a production sequence of the jobs to be processed, which can be represented by the encoding mechanism illustrated in Section 3.2. Given this encoding mechanism, NSGA-II and NSGA-III can be used to solve the described problem, thus being used as competitors of our proposed method. All algorithms are configured with a crossover probability of 0.9 and a mutation probability of 0.8. For NSGA-III, the number of reference point partitions is set to 10, and the iterative threshold for BCMOA is specified as 5.

In the interest of fairness, the termination criterion for all algorithms is determined by the number of fitness function evaluations. The population size is set to 200, and the number of fitness function evaluations is specified as 40,000. All algorithms are independently executed 10 times to eliminate the effects of their randomness. The mean and variance of the obtained solutions are recorded and compared in this work.

The quality of non-dominated solutions obtained by the algorithm is evaluated using three performance metrics: hypervolume (HV) [26], Inverted Generational Distance (IGD) [27], and Coverage metric (C-metric) [28]. Calculating IGD requires selecting uniformly distributed points from the true Pareto solution set. However, the true Pareto solution set of the problem remains unknown. Thus, the non-dominated solutions derived from 10 independent runs of all algorithms are assumed to approximate it. The C-metric, as it requires selecting the solution set obtained by BCMOA as the target solution set, is only used for comparisons of competitors to verify their effectiveness. Before computing HV, IGD, and C-metric, the non-dominated solutions undergo normalization.

The HV metric comprehensively assesses the inherent diversity and convergence properties of non-dominated solutions, with larger HV values being preferable. In contrast, the IGD metric focuses on the convergence characteristics of solutions, making smaller IGD values more desirable. The C-metric is used to quantify the dominance superiority problem between two solution sets, with larger C-metric values being preferable as well. The HV, IGD, and C-metric are defined as follows:

$$HV\left(x,\delta \right)={\displaystyle \underset{x\in R}{\overset{R}{\cup }}v\left(x,\delta \right)},$$

(10)

where R denotes the non-dominated solutions derived from the algorithm, δ = (1, 1) represents the total reference point, and v stands for the hypervolume enclosed by x and δ.

$$IGD={\displaystyle \frac{{\displaystyle \sum _{y\in R^{*}}d\left(y,R\right)}}{\left|R^{*}\right|}},$$

(11)

where R^* denotes the true PF, and d(y, R) denotes the Euclidean distance from an ideal solution y from R^* to the nearest solution in R.

$$C\left(A,B\right)={\displaystyle \frac{\left|x\in B\left|\exists y\in A,y\succ x or y~x\right.\right|}{\left|B\right|}},$$

(12)

where A represents the solution set obtained by applying NS to the combined results of 10 runs of the BCMOA, B represents the solution set obtained by applying NS to the combined results of 10 runs of the compared algorithms, $y\succ x$ implies that x is dominated by y (indicating that y performs no worse than x across all objectives, while strictly outperforming x in at least one objective), and $y~x$ signifies that y is equivalent to x, indicating that y and x exhibit identical performance across all objectives.

4.2. Analysis and Discussion of the Results

The results of performance evaluation metrics for the BCMOA with different crossover operators are presented in

Table 2. Comparison results of HV values obtained by BCMOA with different crossover operators.

Case	BCMOA-P/PO	BCMOA-P/PP	BCMOA-P/OO	BCMOA-O/PO	BCMOA-O/PP	BCMOA-O/OO
N20W20.001	9.596 × 10−1 (5.1 × 10−4)	9.490 × 10−1 (1.4 × 10−3)	9.536 × 10−1 (1.4 × 10−3)	9.578 × 10−1 (3.0 × 10−4)	9.528 × 10−1 (1.7 × 10−3)	9.641 × 10−1 (3.2 × 10−4)
N20W20.002	7.643 × 10−1 (2.8 × 10−3)	7.534 × 10−1 (2.3 × 10−3)	7.773 × 10−1 (1.9 × 10−3)	7.965 × 10−1 (4.7 × 10−3)	7.995 × 10−1 (7.9 × 10−3)	7.650 × 10−1 (3.3 × 10−3)
N20W20.003	9.106 × 10−1 (1.5 × 10−3)	9.082 × 10−1 (1.1 × 10−3)	9.222 × 10−1 (1.2 × 10−3)	9.155 × 10−1 (6.7 × 10−4)	8.688 × 10−1 (1.6 × 10−3)	9.159 × 10−1 (1.0 × 10−3)
N20W20.004	7.997 × 10−1 (1.9 × 10−3)	7.939 × 10−1 (6.5 × 10−3)	8.328 × 10−1 (3.7 × 10−3)	8.169 × 10−1 (4.9 × 10−3)	7.814 × 10−1 (2.4 × 10−3)	8.577 × 10−1 (3.2 × 10−3)
N20W20.005	8.853 × 10−1 (3.8 × 10−4)	8.458 × 10−1 (3.0 × 10−3)	9.060 × 10−1 (1.7 × 10−4)	8.846 × 10−1 (2.0 × 10−3)	8.711 × 10−1 (1.5 × 10−3)	9.095 × 10−1 (1.1 × 10−4)
N40W20.001	6.312 × 10−1 (3.2 × 10−2)	4.579 × 10−1 (9.6 × 10−3)	5.678 × 10−1 (1.2 × 10−2)	5.803 × 10−1 (2.6 × 10−2)	4.757 × 10−1 (1.0 × 10−2)	6.013 × 10−1 (1.6 × 10−2)
N40W20.002	6.386 × 10−1 (1.1 × 10−2)	4.804 × 10−1 (1.4 × 10−2)	6.208 × 10−1 (2.0 × 10−2)	6.279 × 10−1 (1.1 × 10−2)	5.006 × 10−1 (2.8 × 10−2)	7.562 × 10−1 (2.6 × 10−2)
N40W20.003	5.255 × 10−1 (2.0 × 10−2)	5.288 × 10−1 (9.2 × 10−3)	5.758 × 10−1 (1.3 × 10−2)	5.953 × 10−1 (1.5 × 10−2)	5.359 × 10−1 (2.1 × 10−2)	6.635 × 10−1 (1.4 × 10−2)
N40W20.004	7.371 × 10−1 (1.9 × 10−2)	5.792 × 10−1 (1.1 × 10−2)	6.392 × 10−1 (1.1 × 10−2)	6.730 × 10−1 (1.5 × 10−2)	5.497 × 10−1 (5.7 × 10−3)	7.242 × 10−1 (1.9 × 10−2)
N40W20.005	6.370 × 10−1 (1.6 × 10−2)	5.878 × 10−1 (2.6 × 10−2)	6.949 × 10−1 (2.0 × 10−2)	6.164 × 10−1 (1.9 × 10−2)	6.256 × 10−1 (3.3 × 10−2)	7.448 × 10−1 (1.9 × 10−2)
N60W20.001	5.215 × 10−1 (1.5 × 10−2)	4.821 × 10−1 (2.6 × 10−2)	5.746 × 10−1 (1.5 × 10−2)	5.690 × 10−1 (2.2 × 10−2)	5.381 × 10−1 (1.7 × 10−2)	6.641 × 10−1 (1.6 × 10−2)
N60W20.002	5.293 × 10−1 (2.1 × 10−2)	4.347 × 10−1 (8.3 × 10−3)	6.214 × 10−1 (1.3 × 10−2)	5.600 × 10−1 (1.7 × 10−2)	5.115 × 10−1 (7.0 × 10−3)	6.618 × 10−1 (1.4 × 10−2)
N60W20.003	5.878 × 10−1 (1.8 × 10−2)	5.080 × 10−1 (7.6 × 10−3)	5.852 × 10−1 (6.4 × 10−3)	5.256 × 10−1 (2.6 × 10−2)	5.507 × 10−1 (2.3 × 10−2)	6.664 × 10−1 (1.2 × 10−2)
N60W20.004	6.331 × 10−1 (1.1 × 10−2)	5.519 × 10−1 (1.8 × 10−2)	6.042 × 10−1 (9.8 × 10−3)	6.249 × 10−1 (1.8 × 10−2)	5.567 × 10−1 (3.9 × 10−2)	7.003 × 10−1 (2.2 × 10−2)
N60W20.005	5.289 × 10−1 (4.5 × 10−2)	4.700 × 10−1 (1.5 × 10−2)	5.593 × 10−1 (1.2 × 10−2)	5.193 × 10−1 (1.1 × 10−2)	5.394 × 10−1 (2.8 × 10−2)	6.130 × 10−1 (2.4 × 10−2)
N80W20.001	5.462 × 10−1 (2.5 × 10−2)	4.982 × 10−1 (1.6 × 10−2)	6.091 × 10−1 (1.8 × 10−2)	5.556 × 10−1 (2.1 × 10−2)	4.789 × 10−1 (1.1 × 10−2)	6.293 × 10−1 (2.9 × 10−2)
N80W20.002	6.330 × 10−1 (1.1 × 10−2)	6.462 × 10−1 (1.6 × 10−2)	6.743 × 10−1 (1.3 × 10−2)	6.821 × 10−1 (1.2 × 10−2)	6.140 × 10−1 (1.4 × 10−2)	7.384 × 10−1 (1.1 × 10−2)
N80W20.003	5.875 × 10−1 (5.6 × 10−3)	5.883 × 10−1 (9.9 × 10−3)	6.647 × 10−1 (1.6 × 10−2)	5.908 × 10−1 (2.4 × 10−2)	5.878 × 10−1 (1.7 × 10−2)	7.250 × 10−1 (3.4 × 10−3)
N80W20.004	6.172 × 10−1 (2.2 × 10−2)	4.662 × 10−1 (1.9 × 10−2)	5.918 × 10−1 (2.8 × 10−2)	6.380 × 10−1 (1.1 × 10−2)	5.816 × 10−1 (3.4 × 10−2)	7.463 × 10−1 (8.9 × 10−3)
N80W20.005	6.731 × 10−1 (4.4 × 10−3)	5.969 × 10−1 (6.4 × 10−3)	6.902 × 10−1 (1.2 × 10−2)	7.232 × 10−1 (2.8 × 10−2)	5.582 × 10−1 (1.3 × 10−2)	7.220 × 10−1 (1.9 × 10−2)
N100W20.001	6.786 × 10−1 (1.2 × 10−2)	5.727 × 10−1 (2.5 × 10−2)	6.739 × 10−1 (5.5 × 10−3)	5.867 × 10−1 (1.3 × 10−2)	5.244 × 10−1 (1.3 × 10−2)	7.966 × 10−1 (1.2 × 10−2)
N100W20.002	5.748 × 10−1 (9.9 × 10−3)	5.492 × 10−1 (2.2 × 10−2)	6.092 × 10−1 (1.3 × 10−2)	5.350 × 10−1 (1.6 × 10−2)	4.701 × 10−1 (1.2 × 10−2)	6.423 × 10−1 (5.3 × 10−3)
N100W20.003	5.705 × 10−1 (1.5 × 10−2)	4.740 × 10−1 (8.2 × 10−3)	6.720 × 10−1 (2.0 × 10−2)	6.033 × 10−1 (1.1 × 10−2)	4.855 × 10−1 (1.4 × 10−2)	6.898 × 10−1 (2.3 × 10−2)
N100W20.004	6.469 × 10−1 (1.9 × 10−2)	6.059 × 10−1 (1.6 × 10−2)	7.293 × 10−1 (1.1 × 10−2)	6.847 × 10−1 (1.2 × 10−2)	6.249 × 10−1 (2.4 × 10−2)	7.926 × 10−1 (8.9 × 10−3)
N100W20.005	6.185 × 10−1 (5.1 × 10−3)	5.306 × 10−1 (1.9 × 10−2)	7.304 × 10−1 (1.9 × 10−3)	5.717 × 10−1 (6.0 × 10−3)	6.176 × 10−1 (2.3 × 10−2)	7.078 × 10−1 (8.7 × 10−3)
the frequency of optimal performance	2	0	2	0	1	20

The bolded numbers indicate that the algorithm corresponding to this column performs best in this case.

and

Table 3. Comparison results of IGD values obtained by BCMOA with different crossover operators.

Case	BCMOA-P/PO	BCMOA-P/PP	BCMOA-P/OO	BCMOA-O/PO	BCMOA-O/PP	BCMOA-O/OO
N20W20.001	4.195 × 10−2 (1.7 × 10−3)	3.380 × 10−2 (4.0 × 10−4)	3.956 × 10−2 (1.1 × 10−3)	3.101 × 10−2 (1.7 × 10−4)	4.198 × 10−2 (1.2 × 10−3)	2.536 × 10−2 (8.6 × 10−5)
N20W20.002	1.377 × 10−1 (2.0 × 10−3)	1.371 × 10−1 (1.2 × 10−3)	1.119 × 10−1 (1.9 × 10−3)	1.113 × 10−1 (3.1 × 10−3)	1.177 × 10−1 (4.4 × 10−3)	1.330 × 10−1 (2.9 × 10−3)
N20W20.003	4.593 × 10−2 (4.9 × 10−4)	5.711 × 10−2 (6.0 × 10−4)	4.321 × 10−2 (2.2 × 10−4)	4.539 × 10−2 (3.8 × 10−4)	8.386 × 10−2 (1.4 × 10−3)	4.413 × 10−2 (4.3 × 10−4)
N20W20.004	1.105 × 10−1 (8.4 × 10−4)	1.189 × 10−1 (3.3 × 10−3)	8.916 × 10−2 (1.1 × 10−3)	9.929 × 10−2 (1.3 × 10−3)	1.202 × 10−1 (1.4 × 10−3)	7.434 × 10−2 (9.1 × 10−4)
N20W20.005	4.510 × 10−2 (7.9 × 10−4)	8.359 × 10−2 (3.2 × 10−3)	2.071 × 10−2 (3.5 × 10−4)	3.885 × 10−2 (1.3 × 10−3)	5.782 × 10−2 (1.6 × 10−3)	1.441 × 10−2 (1.4 × 10−4)
N40W20.001	2.603 × 10−1 (2.9 × 10−2)	4.227 × 10−1 (1.3 × 10−2)	3.099 × 10−1 (1.2 × 10−2)	3.066 × 10−1 (2.5 × 10−2)	3.930 × 10−1 (1.3 × 10−2)	2.801 × 10−1 (1.6 × 10−2)
N40W20.002	2.743 × 10−1 (6.8 × 10−3)	4.043 × 10−1 (1.6 × 10−2)	3.030 × 10−1 (1.7 × 10−2	2.827 × 10−1 (8.8 × 10−3)	3.866 × 10−1 (2.5 × 10−2)	2.113 × 10−1 (1.1 × 10−2)
N40W20.003	2.926 × 10−1 (1.7 × 10−2)	3.112 × 10−1 (8.2 × 10−3)	2.577 × 10−1 (1.0 × 10−2)	2.364 × 10−1 (1.0 × 10−2)	2.946 × 10−1 (1.6 × 10−2)	1.981 × 10−1 (7.4 × 10−3)
N40W20.004	2.031 × 10−1 (1.5 × 10−2)	3.335 × 10−1 (1.0 × 10−2)	2.921 × 10−1 (1.1 × 10−2)	2.442 × 10−1 (8.3 × 10−3)	3.643 × 10−1 (6.3 × 10−3)	2.161 × 10−1 (1.1 × 10−2)
N40W20.005	2.552 × 10−1 (1.2 × 10−2)	2.975 × 10−1 (2.4 × 10−2)	2.113 × 10−1 (9.4 × 10−3)	2.713 × 10−1 (1.4 × 10−2)	2.771 × 10−1 (2.0 × 10−2)	1.719 × 10−1 (7.0 × 10−3)
N60W20.001	3.427 × 10−1 (2.0 × 10−2)	3.849 × 10−1 (4.0 × 10−2)	2.880 × 10−1 (1.4 × 10−2)	3.081 × 10−1 (2.2 × 10−2)	3.176 × 10−1 (1.6 × 10−2)	2.057 × 10−1 (1.6 × 10−2)
N60W20.002	2.626 × 10−1 (1.7 × 10−2)	3.545 × 10−1 (7.3 × 10−3)	2.069 × 10−1 (8.5 × 10−3)	2.571 × 10−1 (1.5 × 10−2)	2.888 × 10−1 (8.2 × 10−3)	1.758 × 10−1 (6.4 × 10−3)
N60W20.003	2.003 × 10−1 (1.4 × 10−2)	2.876 × 10−1 (7.8 × 10−3)	2.064 × 10−1 (4.4 × 10−3)	2.666 × 10−1 (1.9 × 10−2)	2.422 × 10−1 (1.8 × 10−2)	1.404 × 10−1 (5.1 × 10−3)
N60W20.004	2.522 × 10−1 (7.4 × 10−3)	3.398 × 10−1 (1.8 × 10−2)	2.737 × 10−1 (5.6 × 10−3)	2.541 × 10−1 (1.3 × 10−2)	3.344 × 10−1 (3.6 × 10−2)	2.061 × 10−1 (1.4 × 10−2)
N60W20.005	3.118 × 10−1 (3.8 × 10−2)	3.575 × 10−1 (2.2 × 10−2)	2.663 × 10−1 (7.8 × 10−3)	3.013 × 10−1 (1.3 × 10−2)	3.027 × 10−1 (1.7 × 10−2)	2.234 × 10−1 (2.1 × 10−2)
N80W20.001	3.672 × 10−1 (2.3 × 10−2)	4.095 × 10−1 (1.7 × 10−2)	3.116 × 10−1 (1.4 × 10−2)	3.566 × 10−1 (1.8 × 10−2)	4.210 × 10−1 (1.5 × 10−2)	2.936 × 10−1 (2.6 × 10−2)
N80W20.002	2.102 × 10−1 (3.4 × 10−3)	1.994 × 10−1 (9.9 × 10−3)	1.908 × 10−1 (5.8 × 10−3)	1.742 × 10−1 (2.2 × 10−3)	2.279 × 10−1 (1.0 × 10−2)	1.642 × 10−1 (1.8 × 10−3)
N80W20.003	2.608 × 10−1 (6.4 × 10−3)	2.551 × 10−1 (1.0 × 10−2)	1.731 × 10−1 (1.2 × 10−2)	2.528 × 10−1 (2.0 × 10−2)	2.525 × 10−1 (1.4 × 10−2)	1.211 × 10−1 (2.7 × 10−3)
N80W20.004	2.918 × 10−1 (1.8 × 10−2)	4.441 × 10−1 (2.5 × 10−2)	3.247 × 10−1 (3.1 × 10−2)	2.743 × 10−1 (6.7 × 10−3)	3.281 × 10−1 (2.6 × 10−2)	1.906 × 10−1 (5.1 × 10−3)
N80W20.005	2.407 × 10−1 (4.1 × 10−3)	2.932 × 10−1 (5.3 × 10−3)	2.225 × 10−1 (6.2 × 10−3)	1.915 × 10−1 (1.9 × 10−2)	3.380 × 10−1 (1.5 × 10−2)	1.925 × 10−1 (8.9 × 10−3)
N100W20.001	2.140 × 10−1 (9.5 × 10−3)	3.197 × 10−1 (1.9 × 10−2)	2.102 × 10−1 (5.5 × 10−3)	3.046 × 10−1 (1.2 × 10−2)	3.485 × 10−1 (1.4 × 10−2)	1.232 × 10−1 (9.0 × 10−3)
N100W20.002	2.580 × 10−1 (8.5 × 10−3)	2.883 × 10−1 (2.0 × 10−2)	2.349 × 10−1 (8.8 × 10−3)	3.057 × 10−1 (1.6 × 10−2)	3.584 × 10−1 (1.5 × 10−2)	2.095 × 10−1 (3.8 × 10−3)
N100W20.003	3.589 × 10−1 (1.6 × 10−2)	4.371 × 10−1 (1.1 × 10−2)	2.741 × 10−1 (1.7 × 10−2)	3.342 × 10−1 (9.0 × 10−3)	4.484 × 10−1 (1.2 × 10−2)	2.541 × 10−1 (1.9 × 10−2)
N100W20.004	2.428 × 10−1 (1.1 × 10−2)	2.602 × 10−1 (1.4 × 10−2)	1.681 × 10−1 (4.9 × 10−3)	2.023 × 10−1 (7.0 × 10−3)	2.546 × 10−1 (1.7 × 10−2)	1.299 × 10−1 (2.5 × 10−3)
N100W20.005	1.685 × 10−1 (3.4 × 10−3)	2.713 × 10−1 (1.8 × 10−2)	9.451 × 10−2 (1.2 × 10−3)	2.164 × 10−1 (5.6 × 10−3)	1.893 × 10−1 (1.7 × 10−2)	1.134 × 10−1 (5.9 × 10−3)
the frequency of optimal performance	2	0	2	2	0	19

The bolded numbers indicate that the algorithm corresponding to this column performs best in this case.

. The frequency of each algorithm achieving optimal performance across various cases is displayed in the last row. Among them, the BCMOA-O/OO algorithm exhibits relatively superior performance. From the HV comparison results presented in Table 2, the BCMOA-O/OO achieves the best performance, outperforming other BCMOA variants in 20 out of 25 cases. Specifically, the BCMOA-O/OO outperforms the BCMOA-P/PP and BCMOA-O/PO in all 25 cases; it outperforms the BCMOA-P/PO and BCMOA-P/OO in 23 out of 25 cases; and it outperforms the BCMOA-O/PP in 24 out of 25 cases. From the IGD metric comparison results shown in Table 3, the BCMOA-O/OO outperforms other BCMOA variants in 19 out of 25 cases. Specifically, the BCMOA-O/OO outperforms the BCMOA-P/PP and BCMOA-O/PP in all 25 cases; it outperforms the BCMOA-P/PO, BCMOA-P/OO and BCMOA-O/PO in 23 out of 25 cases. Therefore, this study employs the BCMOA-O/OO to represent the BCMOA for subsequent comparison with NSGA-II and NSGA-III using the PMX and OX operators. The results of the comparison between BCMOA and its competitive peers are illustrated in

Table 4. Comparison results of HV values obtained by BCMOA and compared algorithms.

Case	NSGAII (PMX)	NSGAII (OX)	NSGAIII (PMX)	NSGAIII (OX)	BCMOA
N20W20.001	9.440 × 10−1 (3.4 × 10−4)	9.553 × 10−1 (3.1 × 10−4)	9.084 × 10−1 (2.9 × 10−3)	9.311 × 10−1 (8.0 × 10−4)	9.668 × 10−1 (3.4 × 10−4)
N20W20.002	7.873 × 10−1 (5.2 × 10−3)	7.628 × 10−1 (6.9 × 10−3)	6.307 × 10−1 (8.7 × 10−3)	7.294 × 10−1 (9.5 × 10−3)	8.688 × 10−1 (3.2 × 10−3)
N20W20.003	8.609 × 10−1 (2.5 × 10−3)	8.707 × 10−1 (2.3 × 10−3)	7.201 × 10−1 (5.2 × 10−3)	8.080 × 10−1 (6.3 × 10−3)	8.973 × 10−1 (2.9 × 10−3)
N20W20.004	7.589 × 10−1 (9.9 × 10−4)	7.743 × 10−1 (1.5 × 10−3)	6.320 × 10−1 (5.1 × 10−3)	6.954 × 10−1 (5.6 × 10−3)	7.766 × 10−1 (1.7 × 10−3)
N20W20.005	9.356 × 10−1 (9.1 × 10−4)	9.659 × 10−1 (3.5 × 10−4)	8.646 × 10−1 (5.7 × 10−3)	8.872 × 10−1 (2.2 × 10−3)	9.556 × 10−1 (3.3 × 10−4)
N40W20.001	6.373 × 10−1 (1.7 × 10−2)	7.815 × 10−1 (1.1 × 10−2)	2.572 × 10−1 (1.2 × 10−2)	4.719 × 10−1 (1.2 × 10−2)	7.515 × 10−1 (1.5 × 10−2)
N40W20.002	5.472 × 10−1 (1.3 × 10−2)	7.402 × 10−1 (2.3 × 10−2)	2.817 × 10−1 (1.7 × 10−2)	3.851 × 10−1 (1.2 × 10−2)	7.427 × 10−1 (1.6 × 10−2)
N40W20.003	6.289 × 10−1 (1.1 × 10−2)	7.623 × 10−1 (1.6 × 10−2)	4.432 × 10−1 (9.4 × 10−3)	5.025 × 10−1 (1.2 × 10−2)	7.856 × 10−1 (3.9 × 10−3)
N40W20.004	6.018 × 10−1 (1.4 × 10−2)	6.791 × 10−1 (9.6 × 10−3)	3.529 × 10−1 (1.3 × 10−2)	3.980 × 10−1 (3.9 × 10−3)	7.664 × 10−1 (1.5 × 10−2)
N40W20.005	5.054 × 10−1 (6. × 10−3)	6.549 × 10−1 (9.9 × 10−3)	2.360 × 10−1 (1.4 × 10−2)	3.666 × 10−1 (1.4 × 10−2)	7.566 × 10−1 (2.3 × 10−2)
N60W20.001	5.214 × 10−1 (7.0 × 10−3)	5.466 × 10−1 (1.4 × 10−2)	2.714 × 10−1 (8.3 × 10−3)	3.080 × 10−1 (7.3 × 10−3)	6.497 × 10−1 (3.3 × 10−2)
N60W20.002	5.075 × 10−1 (1.1 × 10−2)	6.746 × 10−1 (1.3 × 10−2)	2.739 × 10−1 (1.3 × 10−2)	3.248 × 10−1 (8.9 × 10−3)	7.261 × 10−1 (6.6 × 10−3)
N60W20.003	5.533 × 10−1 (2.2 × 10−2)	6.000 × 10−1 (9.8 × 10−3)	2.558 × 10−1 (5.8 × 10−3)	4.239 × 10−1 (6.5 × 10−3)	7.133 × 10−1 (1.7 × 10−2)
N60W20.004	6.183 × 10−1 (2.1 × 10−2)	7.346 × 10−1 (1.7 × 10−2)	3.685 × 10−1 (1.5 × 10−2)	3.927 × 10−1 (9.1 × 10−3)	7.653 × 10−1 (1.5 × 10−2)
N60W20.005	6.206 × 10−1 (2.9 × 10−2)	6.530 × 10−1 (1.2 × 10−2)	2.742 × 10−1 (1.7 × 10−2)	3.571 × 10−1 (5.1 × 10−3)	7.570 × 10−1 (1.8 × 10−2)
N80W20.001	4.672 × 10−1 (1.1 × 10−2)	5.650 × 10−1 (4.8 × 10−3)	2.903 × 10−1 (4.1 × 10−3)	3.616 × 10−1 (3.9 × 10−3)	7.342 × 10−1 (2.2 × 10−2)
N80W20.002	5.086 × 10−1 (1.1 × 10−2)	6.829 × 10−1 (1.1 × 10−2)	3.065 × 10−1 (1.5 × 10−2)	3.695 × 10−1 (9.6 × 10−3)	7.414 × 10−1 (1.5 × 10−2)
N80W20.003	4.402 × 10−1 (1.0 × 10−2)	5.927 × 10−1 (1.1 × 10−2)	2.588 × 10−1 (4.1 × 10−3)	3.021 × 10−1 (5.0 × 10−3)	6.960 × 10−1 (2.4 × 10−2)
N80W20.004	5.672 × 10−1 (1.4 × 10−2)	7.011 × 10−1 (2.0 × 10−2)	3.342 × 10−1 (1.4 × 10−2)	4.479 × 10−1 (1.2 × 10−2)	8.248 × 10−1 (9.6 × 10−3)
N80W20.005	4.554 × 10−1 (3.7 × 10−2)	6.187 × 10−1 (9.2 × 10−3)	2.573 × 10−1 (7.2 × 10−3)	3.076 × 10−1 (1.2 × 10−2)	6.973 × 10−1 (1.7 × 10−2)
N100W20.001	4.823 × 10−1 (1. × 10−2)	5.929 × 10−1 (4.1 × 10−3)	2.307 × 10−1 (1.0 × 10−2)	3.426 × 10−1 (4.2 × 10−3)	7.091 × 10−1 (2.2 × 10−2)
N100W20.002	4.754 × 10−1 (1.1 × 10−2)	6.519 × 10−1 (5.9 × 10−3)	2.811 × 10−1 (7.5 × 10−3)	4.176 × 10−1 (9.8 × 10−3)	7.002 × 10−1 (1.8 × 10−2)
N100W20.003	4.559 × 10−1 (1.7 × 10−2)	6.042 × 10−1 (1.8 × 10−2)	2.497 × 10−1 (8.5 × 10−3)	3.335 × 10−1 (3.9 × 10−3)	7.249 × 10−1 (2.1 × 10−2)
N100W20.004	5.153 × 10−1 (8.2 × 10−3)	6.521 × 10−1 (2.0 × 10−2)	3.162 × 10−1 (8.3 × 10−3)	4.054 × 10−1 (2.3 × 10−3)	7.654 × 10−1 (2.1 × 10−2)
N100W20.005	5.835 × 10−1 (1.3 × 10−2)	7.953 × 10−1 (7.5 × 10−3)	3.398 × 10−1 (9.1 × 10−3)	4.469 × 10−1 (5.5 × 10−3)	8.225 × 10−1 (7.6 × 10−3)
the frequency of optimal performance	0	2	0	0	23

The bolded numbers indicate that the algorithm corresponding to this column performs best in this case.

and

Table 5. Comparison results of IGD values obtained by BCMOA and compared algorithms.

Case	NSGAII (PMX)	NSGAII (OX)	NSGAIII (PMX)	NSGAIII (OX)	BCMOA
N20W20.001	4.338 × 10−2 (2.1 × 10−4)	3.594 × 10−2 (1.1 × 10−4)	1.122 × 10−1 (1.5 × 10−3)	1.045 × 10−1 (1.5 × 10−3)	2.658 × 10−2 (9.5 × 10−5)
N20W20.002	1.364 × 10−1 (2.6 × 10−3)	1.661 × 10−1 (4.8 × 10−3)	2.598 × 10−1 (4.5 × 10−3)	1.817 × 10−1 (5.7 × 10−3)	7.823 × 10−2 (1.6 × 10−3)
N20W20.003	9.275 × 10−2 (1.4 × 10−3)	7.770 × 10−2 (1.2 × 10−3)	1.721 × 10−1 (2.5 × 10−3)	1.389 × 10−1 (4.3 × 10−3)	6.004 × 10−2 (2.1 × 10−3)
N20W20.004	1.385 × 10−1 (5.1 × 10−4)	1.204 × 10−1 (3.5 × 10−4)	2.050 × 10−1 (1.6 × 10−3)	1.808 × 10−1 (3.5 × 10−3)	1.315 × 10−1 (6.5 × 10−4)
N20W20.005	5.753 × 10−2 (6.4 × 10−4)	2.983 × 10−2 (7.4 × 10−4)	1.080 × 10−1 (3.1 × 10−3)	1.043 × 10−1 (1.4 × 10−3)	3.713 × 10−1 (4.5 × 10−4)
N40W20.001	2.443 × 10−1 (9.3 × 10−3)	1.511 × 10−1 (3.9 × 10−3)	5.909 × 10−1 (2.5 × 10−2)	3.620 × 10−1 (8.0 × 10−3)	1.735 × 10−1 (5.7 × 10−3)
N40W20.002	3.577 × 10−1 (1.6 × 10−2)	1.843 × 10−1 (1.6 × 10−2)	6.467 × 10−1 (3.3 × 10−2)	5.152 × 10−1 (1.7 × 10−2)	1.795 × 10−1 (1.3 × 10−2)
N40W20.003	2.832 × 10−1 (8.1 × 10−3)	2.008 × 10−1 (4.4 × 10−3)	4.200 × 10−1 (7.9 × 10−3)	3.707 × 10−1 (8.9 × 10−3)	1.802 × 10−1 (2.0 × 10−3)
N40W20.004	3.577 × 10−1 (1.6 × 10−2)	2.992 × 10−1 (9.3 × 10−3)	6.133 × 10−1 (1.7 × 10−3)	5.562 × 10−1 (6.4 × 10−3)	2.148 × 10−1 (1.2 × 10−2)
N40W20.005	3.770 × 10−1 (7.2 × 10−3)	2.397 × 10−1 (1.1 × 10−2)	7.111 × 10−1 (2.7 × 10−2)	5.243 × 10−1 (2.1 × 10−2)	1.607 × 10−1 (1.4 × 10−2)
N60W20.001	3.394 × 10−1 (7.3 × 10−3)	3.378 × 10−1 (1.1 × 10−2)	6.173 × 10−1 (1.8 × 10−2)	5.662 × 10−1 (9.7 × 10−3)	2.462 × 10−1 (2.4 × 10−2)
N60W20.002	3.515 × 10−1 (7.6 × 10−3)	2.154 × 10−1 (7.8 × 10−3)	6.235 × 10−1 (2.1 × 10−2)	5.483 × 10−1 (1.5 × 10−2)	1.687 × 10−1 (7.4 × 10−3)
N60W20.003	4.168 × 10−1 (2.1 × 10−2)	3.571 × 10−1 (1.1 × 10−2)	7.332 × 10−1 (1.1 × 10−2)	5.201 × 10−1 (1.0 × 10−2)	2.576 × 10−1 (1.4 × 10−2)
N60W20.004	2.719 × 10−1 (1.9 × 10−2)	1.819 × 10−1 (1.0 × 10−2)	5.077 × 10−1 (2.5 × 10−2)	4.749 × 10−1 (1.4 × 10−2)	1.517 × 10−1 (1.1 × 10−2)
cN60W20.005	3.189 × 10−1 (2.1 × 10−2)	2.921 × 10−1 (1.1 × 10−2)	6.803 × 10−1 (3.6 × 10−2)	5.656 × 10−1 (7.0 × 10−3)	1.865 × 10−1 (1.2 × 10−2)
N80W20.001	4.288 × 10−1 (1.3 × 10−2)	3.265 × 10−1 (4.9 × 10−3)	6.130 × 10−1 (7.4 × 10−3)	5.285 × 10−1 (5.7 × 10−3)	1.845 × 10−1 (1.3 × 10−2)
N80W20.002	3.381 × 10−1 (1.1 × 10−2)	2.066 × 10−1 (7.3 × 10−3)	5.553 × 10−1 (1.8 × 10−2)	4.802 × 10−1 (1.2 × 10−2)	1.547 × 10−1 (6.0 × 10−3)
N80W20.003	4.722 × 10−1 (1.3 × 10−2)	3.397 × 10−1 (1.1 × 10−2)	6.842 × 10−1 (8.7 × 10−3)	6.175 × 10−1 (9.7 × 10−3)	2.342 × 10−1 (1.8 × 10−2)
N80W20.004	3.215 × 10−1 (1.1 × 10−2)	2.068 × 10−1 (1.2 × 10−2)	5.610 × 10−1 (1.8 × 10−2)	4.280 × 10−1 (1.3 × 10−2)	1.189 × 10−1 (3.5 × 10−3)
N80W20.005	4.064 × 10−1 (4.1 × 10−2)	2.306 × 10−1 (7.3 × 10−3)	6.204 × 10−1 (1.5 × 10−2)	5.233 × 10−1 (1.7 × 10−2)	1.790 × 10−1 (9.2 × 10−3)
N100W20.001	3.721 × 10−1 (1.2 × 10−2)	2.708 × 10−1 (3.2 × 10−3)	6.609 × 10−1 (2.1 × 10−2)	5.078 × 10−1 (5.8 × 10−3)	1.778 × 10−1 (1.0 × 10−2)
N100W20.002	3.836 × 10−1 (1.3 × 10−2)	2.219 × 10−1 (4.5 × 10−3)	5.871 × 10−1 (1.5 × 10−2)	4.282 × 10−1 (8.4 × 10−3)	1.899 × 10−1 (9.5 × 10−3)
N100W20.003	4.310 × 10−1 (1.7 × 10−2)	2.908 × 10−1 (1.4 × 10−2)	6.483 × 10−1 (1.5 × 10−2)	5.339 × 10−1 (6.2 × 10−3)	2.012 × 10−1 (1.0 × 10−2)
N100W20.004	3.915 × 10−1 (1.0 × 10−2)	2.689 × 10−1 (1.4 × 10−2)	5.953 × 10−1 (1.4 × 10−2)	4.873 × 10−1 (2.2 × 10−3)	1.595 × 10−1 (1.1 × 10−2)
N100W20.005	3.347 × 10−1 (1.2 × 10−2)	1.461 × 10−1 (4.5 × 10−3)	5.670 × 10−1 (1.3 × 10−2)	4.458 × 10−1 (5.4 × 10−3)	1.413 × 10−1 (4.7 × 10−3)
the frequency of optimal performance	0	3	0	0	22

The bolded numbers indicate that the algorithm corresponding to this column performs best in this case.

. The frequency of each algorithm achieving optimal performance across various cases is displayed in the final row as well. From the HV comparison results shown in Table 4, the BCMOA outperforms other algorithms in 23 out of 25 cases. Specifically, the BCMOA outperforms NSGA-II (PMX), NSGA-III (OX), and NSGA-II (PMX) in all 25 cases, while it outperforms NSGA-II (OX) in 23 out of 25 cases. From the IGD comparison results presented in Table 5, the BCMOA outperforms other algorithms in 22 out of 25 cases. Specifically, the BCMOA outperforms NSGA-II (PMX), NSGA-III (OX), and NSGA-II (PMX) in all 25 cases, while it outperforms NSGA-II (OX) in 22 out of 25 cases. The results of the comparison of the C-metric values of BCMOA with those of the other algorithms are illustrated in

Table 6. Comparison results of C-metric values of BCMOA and compared algorithms.

Case	NSGAII (PMX)	NSGAII (OX)	NSGAIII (PMX)	NSGAIII (OX)
N20W20.001	5.454 × 10−1	6.078 × 10−1	6.667 × 10−1	5.454 × 10−1
N20W20.002	8.696 × 10−1	1.538 × 10−1	7.500 × 10−1	9.091 × 10−2
N20W20.003	8.461 × 10−1	6.250 × 10−1	1	6.923 × 10−1
N20W20.004	2.105 × 10−1	0	7.857 × 10−1	2.000 × 10−1
N20W20.005	4.615 × 10−1	3.030 × 10−2	3.000 × 10−1	1
N40W20.001	1	5.000 × 10−1	1	1
N40W20.002	1	3.334 × 10−1	1	1
N40W20.003	1	0	1	1
N40W20.004	1	1	1	1
N40W20.005	1	1	1	1
N60W20.001	1	1	1	1
N60W20.002	1	8.334 × 10−1	1	1
N60W20.003	1	1	1	1
N60W20.004	1	0	1	1
N60W20.005	1	1	1	1
N80W20.001	1	1	1	1
N80W20.002	1	1	1	1
N80W20.003	1	1	1	1
N80W20.004	1	5.000 × 10−1	1	1
N80W20.005	1	1	1	1
N100W20.001	1	1	1	1
N100W20.002	1	1	1	1
N100W20.003	1	1	1	1
N100W20.004	1	1	1	1
N100W20.005	1	8.000 × 10−1	1	1

. Similarly, the performance of the BCMOA on the C-metric also demonstrates its superior dominance over the compared algorithms.

4.3. Discussion on Solution Speed

To further illustrate the solution efficiencies of the BCMOA and its competitors, we compare their solution times in solving the test cases in

Table 7. The average time consumed by the BCMOA and the compared algorithms.

Case	NSGAII (PMX)	NSGAII (OX)	NSGAIII (PMX)	NSGAIII (OX)	BCMOA
N20W20.001	16.6810	12.8953	15.2789	14.2672	7.9954
N20W20.002	16.5402	13.3836	16.0526	15.6590	9.0980
N20W20.003	16.9227	13.2171	15.7973	15.4400	8.5098
N20W20.004	16.8037	13.8636	16.4655	16.0727	9.1418
N20W20.005	17.6892	13.7185	16.0927	16.5480	9.0961
N40W20.001	16.5317	15.2617	17.4575	16.5652	9.1686
N40W20.002	17.2499	14.7497	17.3017	16.1700	9.0139
N40W20.003	16.3978	14.2238	17.0425	16.2003	9.6035
N40W20.004	17.1974	14.1613	17.6413	16.7012	9.2558
N40W20.005	17.6971	14.1751	18.0147	16.9999	9.4828
N60W20.001	15.0994	13.3947	17.2663	15.5641	9.2816
N60W20.002	16.6195	13.9883	17.5012	15.6319	9.5612
N60W20.003	16.5764	13.6365	17.1949	15.8358	9.5246
N60W20.004	16.8652	13.7312	17.4184	16.0071	9.5799
N60W20.005	17.2362	14.1337	17.3412	15.9037	9.6372
N80W20.001	17.7271	13.3909	17.9721	15.7366	9.7275
N80W20.002	17.682	13.323	17.7185	15.7811	9.6546
N80W20.003	17.7045	13.4978	17.6714	15.3711	9.5493
N80W20.004	17.5148	13.6514	17.7807	15.6800	9.7969
N80W20.005	17.6093	13.5405	17.4540	15.5288	9.5819
N100W20.001	17.8642	13.2311	17.7411	15.3952	9.8466
N100W20.002	18.6974	13.0113	17.9586	15.2789	9.6628
N100W20.003	18.3657	12.9338	17.9038	15.3798	9.9802
N100W20.004	19.2612	13.9232	18.6338	15.8405	10.2744
N100W20.005	19.6435	13.8379	18.6087	15.8183	10.0879

The bolded numbers indicate that the algorithm corresponding to this column achieved the shortest single-operation runtime in this case.

. It can be seen that the time taken by the BCMOA is significantly less than that of other algorithms. The underlying reasons may be as follows: For dominance-based multi-objective optimization algorithms, the computational load primarily stems from NS, which has a complexity of O(MN²). The BCMOA splits the NS operation of the original large population into NS operations for two smaller populations. Although the number of NS operations increases, the squared term in the complexity is halved, ultimately resulting in a significant improvement in computational speed.

In summary, the BCMOA exhibits great performance and a high solution speed, which outperforms its competitive peers in solving the described problem. This implies its high potential to be used in real-world industrial processes.

5. Conclusions

The annealing process of copper strips in air-cushion furnaces serves as a critical intermediate step in copper strip processing. Its production scheduling plays a vital role in enhancing the efficiency of the overall copper strip production process. Such a production scheduling problem needs to consider time window constraints and atmosphere transition requirements between operations, leading to a complex combinatorial optimization problem. This work proposes a Bi-population Co-evolutionary Multi-objective Optimization Algorithm (BCMOA) to solve it. The BCMOA initializes two independent populations and conducts evolutionary searches in parallel. When the iterative process reaches the preset interaction conditions, it extracts the elite solution sets from the two populations for the co-evolution between populations, generating new offspring individuals through genetic operators. Subsequently, the newly generated offspring are merged with the original parent populations, and elite solutions are screened based on the NS mechanism to retain high-quality individuals and form a new generation of populations. Through this periodic interaction and selection mechanism, the algorithm continuously strengthens the distribution diversity of feasible solution sets during the iterative process, ultimately achieving improved optimization performance. Numerous experimental results show that the BCMOA has a high solution accuracy and efficiency and significantly outperforms its competing algorithms in the three evaluation indicators of HV, IGD, and C-metric for solving related problems. Meanwhile, the BCMOA also significantly outperforms other algorithms in terms of solution speed.

Although the BCMOA effectively addresses the described scheduling problem in a copper strip production system, it overlooks the dynamic factors presented in real-world industrial processes. Our future research will incorporate dynamic factors such as uncertain process times and variable time window constraints. Furthermore, integrating data-driven and adaptive mechanisms [29,30,31] into the BCMOA to improve its effectiveness and extending it to tackle other complex scheduling problems would be valuable future directions.