Improved genetic algorithm based on rule optimization strategy for fibre allocation

ABSTRACT

In modern manufacturing industry, in order to adapt to changes in the general environment, the manufacturing industry must improve production efficiency. To this end, this article introduces an improved genetic algorithm based on rule selection to tackle the nondeterministic polynomial hard problem stemming from inventory fibre resources and fibre selection principles in optical cable production. The algorithm aims to maximize inventory score and minimize fibre segmentation rate. It employs a permutation encoding approach to link the genetic algorithm with fibre allocation solutions and applies a self-attention mechanism to determine subset solution weight within each solution. To boost the recombination of favourable gene segments from different chromosomes, a rule optimization strategy is integrated into the crossover operation based on the weights. This operations enhance the algorithm's global search capability and convergence speed. A feasibility repair strategy is then used to inspect and rectify chromosomes, preventing the generation of illegal solutions. The legitimate mutation operation, founded on weight optimization rules, effectively reduces the algorithm's running time by avoiding illegal solutions. By leveraging actual production data from an optical cable manufacturer for simulation, the experimental results confirm the effectiveness of the improved genetic algorithm in addressing the fibre allocation problem. Comparative simulations with the unimproved genetic algorithm and a stepwise greedy algorithm underscore the superiority of the improved genetic algorithm in resolving the fibre allocation problem.

KEYWORDS:

• Fibre segmentation rate
• self-attention mechanism
• permutation encoding
• feasibility repair
• genetic algorithm

1. Introduction

There are many combinatorial optimization problems in various fields such as logistics planning, production scheduling, network design (Luo et al., 2021), financial investment, and more. It is effective to enhance resource utilization efficiency, reduce production costs, optimize decision-making, and yield various other benefits when such problems are solved. In the era of industrial intelligence, with the increasing personalization of customer requirements, manufacturing enterprises are undergoing transformation towards leveraging industrial big data and industrial intelligence to adapt to intense market competition. In this context, it is imperative to solve these optimization problems.

Due to its significant practical value, this problem has attracted considerable attention from researchers worldwide since the 1950s. Most combinatorial optimization problems have been confirmed to be NP(Nondeterministic Polynomial)-hard. In general, it is difficult to find a polynomial time algorithm to solve them. This implies that, for large-scale problems, finding the exact optimal solution may be impractical, requiring the use of heuristic algorithms or approximation algorithms for resolution. Nikas et al. introduced a robust variant of the augmented ε-constraint algorithm (Nikas et al., 2022), AUGMECONR, for obtaining exact solutions to multi-objective linear programming problems. Although precise algorithms, such as the robust variant of the augmented ε-constraint algorithm, can find optimal solutions for combinatorial optimization problems, they often rely on expert knowledge to formulate mathematical programming models for each problem, and the solving time exponentially increases with problem complexity. To solve the MaxCut problem in different types of undirected graphs (Yi et al., 2024), Yi X and Huo J C proposed an iterative quantum algorithm based on quantum gradient descent. They analysed the algorithm's robustness to random errors and Gaussian errors through simulation. While iterative quantum algorithms and other approximation algorithms can find solutions to combinatorial optimization problems within a reasonable time, they suffer from drawbacks such as low solution accuracy, difficulty in parameter tuning, and poor generalization,etc. To address the weaknesses of traditional genetic algorithms (Xue et al., 2020), such as weak search capability and long running times, when solving the flexible job shop scheduling problem (FJSP) (Zheng et al., 2023), Luo X et al. proposed a genetic algorithm based on a new population generation mechanism and an improved single-point mutation. Although common optimization algorithms (H. Li et al., 2023; Zhao et al., 2023) can find solutions to combinatorial optimization problems (Dhaenens et al., 2010) within a reasonable time, determining the optimal hyperparameter values is challenging.

In optical cable production, there are complex combination optimization problems in the optical fibre selection process. Various factors such as fibre colour, length, storage duration and fibre segmentation rate, are crucial indicators in fibre allocation of optical cable production. High fibre segmentation rate negatively impacts production efficiency and the quality of inventory fibres. Factors, such as fibre colour, length, storage duration and the order of fibre selection, play significant roles in determining the quality of inventory fibres. Therefore, during fibre distribution, how to allocate limited inventory of optical fibre resources and provide the optimal fibre allocation production combination plan is a crucial problem, which must be solved in the transformation of optical cable manufacturing enterprises.

In this article, we propose an improved genetic algorithm based on rule optimization strategy. Specifically, the novel algorithm, by utilizing permutation encoding (Berger et al., 1972) to establish the association between algorithm and fibre combination, introduces a new mutation approach that eliminates the necessity to reconstruct chromosomes post-mutation(i.e. avoiding the generation of illegal chromosomes (X. Wu & Cao, 2022)). Additionally, to enhance the probability of recombining favourable gene segments from different chromosomes, a crossover method is employed based on correlation weights. In scenarios which is characterized by a small population size, limited iteration counts and a vast search space, simulation results demonstrate that the proposed algorithm exhibits superior search capabilities and faster population convergence speed compared to other algorithms. Specifically, the main contributions of this article are as follows:

• A novel encoding scheme is proposed to address combinatorial optimization problems in the new scenario.

• A new mutation approach is introduced that eliminates the necessity to reconstruct chromosomes post-mutation.

• To enhance the probability of recombining favourable gene segments from different chromosomes, a crossover method is employed based on correlation weights.

We organize the remainder of this article as follows. In Section 2, the problem description and modelling of fibre allocation is introduced. In Section 3, the novel algorithm proposed in this article is described in detail. In Section 4, the algorithm performance is tested on common test cases, comparative experiments are conducted, and the experimental results is analysed. Finally, the conclusions and future directions are presented in Section 5.

2. Problem description and modelling

2.1. Problem description

The problem of fibre allocation in optical cable production can be described as follows: in the scheduling task, there are bundle number set $N$, the length set $L$, the number of tubes $K$, the number of fibres $P$, $L_i$ is the length of product i, $N_i$ is the number of bundles of product i, each length l corresponds to a bundle number $n$, that is, the elements of the bundle number set $N$ and the length set $L$ have a one-to-one correspondence. Subject to meeting the production scheduling requirements, manufacturers devise a reasonable fibre allocation plan, also known as a fibre allocation plan, based on factors such as the quantity and length of stocked fibres, the total length of scheduled production $\mathrm{\Sigma }{L}_i\ast N_i$, the number of tubes ($K$), and the number of fibres ($P$). The total number of fibres required to produce the optical cable is equal to $K\ast P$. The objective is to increase the proportion of ‘high-quality’ fibres from the existing inventory and reduce the fibre segmentation rate during fibre allocation. Furthermore, during the allocation process, the principles for fibre selection are as follows:

1. Prioritize the use of coloured fibres to reduce the proportion of coloured fibres in inventory.

2. Minimize the fibre segmentation rate during the process of producing optical cable.

3. Prioritize the utilization of return-to-inventory fibres to reduce the proportion of return-to-inventory fibres in inventory.

4. Prioritize the use of fibres with length less than 8 km to reduce the proportion of short-length fibres in inventory.

5. Prioritize the use of fibres with long storage duration to reduce the proportion of fibres with extended storage times in inventory.

To provide a more visual understanding of the number of tubes and the number of fibres, an illustration is given using the cross-sectional diagram of a specific model of optical cable product, as shown in Figure 1.

Figure 1. Cross-sectional diagram of the optical cable.

In Figure 1, small coloured rings represent tubes, and the number of these rings defines the tube count. Within each tube, different coloured dots represent fibres, and the number of these dots defines the fibre count. Specifically, the optical cable in the illustration consists of 8 tubes, with each tube comprising 8 fibres.

The overall fibre allocation framework diagram is shown in Figure 2. The process of fibre allocation is composed of three parts: (1) Encoding the bundle quantity set according to the order requirements into multiple chromosomes to form the population. (2) After the population undergoes weight evaluation, crossover, inspection, repair, and legitimacy variation to generate a new population, the chromosomes are then decoded based on the length requirements. Subsequently, an adequate number of optical fibres are selected from the inventory based on the number of tubes ($K$) and fibres ($P$), and the corresponding fitness value is calculated. (3) After multiple iterations, the algorithm stops running and outputs the globally optimal chromosome along with its fitness value.

Figure 2. Framework of fibre allocation.

2.2. Integer programming model of fibre allocation

In order to provide an optimal fibre allocation solution in complex scenarios with multiple constraints, an integer programming model (Chae & Regan, 2020) is established with the objectives of maximizing inventory score and minimizing fibre segmentation rate. The symbols defined in the model are shown in Table 1. The integer programming model for the multi-constraint (Meng Ang et al., 2020) fibre allocation problem is as follows: (1) $\begin{array}{rl}& \mathrm{Maximize}F\\ & s.t.\\ & C_i=\alpha \left(c_i\right),\alpha \left(c_i\right)=\{0,c_i\in G;1,c_i\in M\},\mathrm{\forall }i\in I,\end{array}$(1) (2) $\begin{array}{rl}& U_i=J\left(k_i\right),J\left(k_i\right)=\{0,k_i\in Q;1,k_i\in Q^{\prime }\},\mathrm{\forall }i\in I,\end{array}$(2) (3) $\begin{array}{rl}& L_i=\text{β}\left(l_i\right),\text{β}\left(l_i\right)=\{0,l_i\le 8;1,l_i>8\},\mathrm{\forall }i\in I,\end{array}$(3) (4) $\begin{array}{rl}& D_i=1/(1+(t_2-t_1\left)\right),\mathrm{\forall }i\in I,\end{array}$(4) (5) $\begin{array}{rl}& X_i=0.25\ast (r_1\ast C_i+r_2\ast U_i+r_3\ast L_i+r_4\ast D_i),\mathrm{\forall }i\in I,\end{array}$(5) (6) $\begin{array}{rl}& \mathcal{U}=\left(1/I\right)\sum _{i\in I}{X}_i,\mathrm{\forall }i\in I,\end{array}$(6) (7) $\begin{array}{rl}& r=a/\varrho ,\end{array}$(7) (8) $\begin{array}{rl}& S_1=5\ast d\left(r\right),d\left(r\right)=\{0,r\ge 0.2;0.2-r,0\le r<0.2\},\end{array}$(8) (9) $\begin{array}{rl}& S_2={\mathcal{U}}_2-{\mathcal{U}}_1,\end{array}$(9) (10) $\begin{array}{rl}& F={\gamma }_1\ast S_1+{\gamma }_2\ast S_2.\end{array}$(10) The objective of the model is to maximize the inventory fibre score and minimize the fibre segmentation rate. Equation (1(1) $\begin{array}{rl}& \mathrm{Maximize}F\\ & s.t.\\ & C_i=\alpha \left(c_i\right),\alpha \left(c_i\right)=\{0,c_i\in G;1,c_i\in M\},\mathrm{\forall }i\in I,\end{array}$(1) ) ensures the priority use of coloured fibres; Equation (2(2) $\begin{array}{rl}& U_i=J\left(k_i\right),J\left(k_i\right)=\{0,k_i\in Q;1,k_i\in Q^{\prime }\},\mathrm{\forall }i\in I,\end{array}$(2) ) ensures the priority use of return-to-inventory fibres; Equation (3(3) $\begin{array}{rl}& L_i=\text{β}\left(l_i\right),\text{β}\left(l_i\right)=\{0,l_i\le 8;1,l_i>8\},\mathrm{\forall }i\in I,\end{array}$(3) ) ensures the priority use of fibres with a length of 8 km or less; Equation (4(4) $\begin{array}{rl}& D_i=1/(1+(t_2-t_1\left)\right),\mathrm{\forall }i\in I,\end{array}$(4) ) ensures the priority use of fibres with long storage time; Equation (5(5) $\begin{array}{rl}& X_i=0.25\ast (r_1\ast C_i+r_2\ast U_i+r_3\ast L_i+r_4\ast D_i),\mathrm{\forall }i\in I,\end{array}$(5) ) ensures that the above four selection principles are considered simultaneously in e selection; Equation (6(6) $\begin{array}{rl}& \mathcal{U}=\left(1/I\right)\sum _{i\in I}{X}_i,\mathrm{\forall }i\in I,\end{array}$(6) ) is defined as the inventory score; Equation (7(7) $\begin{array}{rl}& r=a/\varrho ,\end{array}$(7) ) is defined as the fibre segmentation rate; Equation (8(8) $\begin{array}{rl}& S_1=5\ast d\left(r\right),d\left(r\right)=\{0,r\ge 0.2;0.2-r,0\le r<0.2\},\end{array}$(8) ) ensures that the fibre segmentation rate is considered in fibre selection; Equation (9(9) $\begin{array}{rl}& S_2={\mathcal{U}}_2-{\mathcal{U}}_1,\end{array}$(9) ) ensures that the inventory score is considered in fibre selection; Equation (10(10) $\begin{array}{rl}& F={\gamma }_1\ast S_1+{\gamma }_2\ast S_2.\end{array}$(10) ) is the evaluation metric of fibre allocation results. The larger the value, the better the fibre allocation results.

Table 1. Definition of symbols in the integer programming model.

symbol	define
I	The aggregate set of optical fibre inventory quantities,i=1,2,…,$|I|$
G, M	The set of colour$\{B,\mathrm{OR},G,\mathrm{BR},\mathrm{GR},R,\mathrm{BL},Y,V,P,\mathrm{AQ}\},\left\{N\right\}$
N,B,OR,G,BR,GR,R,BL,Y,V,P,AQ	Colourless, Blue, Orange-Yellow, Green, Brown, White, Red, Black, Yellow, Blue-Purple, Purple, Light Green
$C_i$	Quantitative score of colour dimension for the ith fibre
$c_i$	The colour of the ith fibre
$Q,Q^{\prime }$	The set of return-to-inventory fibre, the set of non-return-to-inventory fibre
$k_i$	Whether the ith fibre is a return-to-stock fibre
$U_i$	Quantitative score of return-to-stock dimension for the ith fibre
l	The length of fibre
$L_i$	Quantitative score of length dimension for the ith fibre
$t_1,t_2$	The time of fibre entry into inventory, the time of fibre usage
$D_i$	Quantitative score of time dimension for the ith fibre
$r_1,r_2,r_3,r_4$	Weighting coefficients for calculating the overall score of a single fibre
$X_i$	The overall score of the ith fibre
$\mathcal{U}$	The score of inventory
${\mathcal{U}}_1$	The score of inventory before fibre allocation
${\mathcal{U}}_2$	The score of inventory after fibre allocation
a	The total quantity of fibres to be segmented during fibre allocation
ϱ	The total quantity of fibres during fibre allocation
r	Fibre segmentation rate
$S_1$	Quantitative score of fibre segmentation dimension
$S_2$	Quantitative score of inventory fibre dimension
${\gamma }_1,{\gamma }_2$	Weighting coefficients for calculating evaluation metrics
F	The evaluation function of fibre allocation result

3. Improved genetic algorithm based on rule optimization strategy

Many combinatorial optimization problems are NP-hard problems (W. Li et al., 2020), which are difficult to solve accurately in polynomial time (Kononov et al., 2022). In the context of fibre allocation of optical cable production, constraints, such as limited inventory resource, selection principle and uncertain combination of optical cable length, further complicate the problem. This article proposes an improved genetic algorithm based on rule optimization strategy (IGA-ROS). The overall framework of the algorithm is shown in Figure 3. With regard to the high randomness in conventional genetic algorithm crossover and mutation operations, a self-attention mechanism (Y. Zhang et al., 2022) is employed to identify ‘unsociable’ segments within the chromosome, prioritizing their involvement in crossover and mutation operations, to improve the probability of recombination of advantageous gene segments. To ensure the ‘legality’ of chromosomes within the population during evolution, a feasibility repair strategy (J.-Q. Li et al., 2020) is introduced in the iterative optimization process. Additionally, during the iterative process, an elite retention strategy (M. Wu et al., 2022) is introduced to retain the chromosomes with the top 20% fitness values (Wang et al., 2023), entering directly into the next generation, so as to guide the population to further converge (M. Zhang et al., 2021).

Figure 3. Framework of IGA-ROS.

3.1. Chromosome encoding and decoding

According to the above, the fibre allocation problem of optical cable production can be decomposed into two stages: task decomposition and task recombination. Therefore, the chromosome construction adopts the permutation encoding approach, where the chromosome length equals the total number of optical cable production tasks, and the gene length equals the number of different types of optical cable production tasks. Genes represent the results of task decomposition and recombination for fibre allocation in optical cable production, where the different positions correspond to the quantities of different types of optical cables in the gene. Two examples of encoding a production task ‘3.14*2+4.78*3+5.98*4’ is shown in Figure 4, where 3.14, 4.78, and 5.98 represent the length of three different requirements of optical cables, and 2, 3, and 4 represent the quantity of optical cables. Because the genetic algorithm operations need to ensure that the chromosome length is consistent and the sum of the corresponding positions of all subsets is equal to the corresponding positions of [2,3,4], the insufficient subset segments are filled with [0,0,0]. The detail principle of encoding is shown in Algorithm 1.

Figure 4. Example of encoding diagram.

Because the optical cable production process requires consideration of the optical cable type, the number of tubes, and the number of fibres to select the fibre length and quantity, decoding is essentially equivalent to multiplying the numerical values by their corresponding fibre lengths and summing them in the genes. This process yields the selected fibre length. Additionally, fibre selection interval calculations are performed based on fixed rules and the selected fibre length. Furthermore, considering the multiplication of the number of tubes and the number of fibres, the corresponding quantity of fibres is selected from the fibre selection interval. Examples of gene fragment decoding and chromosome decoding are shown in Figures 5 and 6, respectively.

Figure 5. Decoding diagram of gene segment.

Figure 6. Decoding diagram of chromosome.

3.2. The elite reservation strategy

The selection operation, which involves choosing individuals from the population, guides the algorithm to evolve towards better and more advantageous directions in the early iterations. This article adopts an elite reservation strategy as the selection strategy. Firstly, Individuals, whose fitness values rank in the top 20% of the current population, are directly transferred to the next generation. Finally, the individuals with top 50% fitness values serve as the ‘breeding’ population, and the remaining 80% of individuals in the next generation are generated through particular crossover and mutation operations.

3.3. Evaluate the correlation of individual segments

In the population, each individual represents a fibre allocation solution. Within individual, different segments have various influences on the fitness value. Individuals with higher fitness values are selected to form the ‘breeding’ population. The weight of each segment is evaluated using a self-attention mechanism, where segments with lower weights are considered ‘unsociable’ and are prioritized for crossover and mutation. The structure diagram of the self-attention mechanism is shown in Figure 7, where ‘a’ represents the input (gene segment), ‘b’ represents the output (segment correlation weight). It is worth noting that the decoding length of the zero segments is equal to 0, so it does not affect the inventory fibre. Therefore, the weight of the zero segments does not participate in the crossover and repair operations.

Figure 7. Self-Attention structure diagram.

3.4. Crossover operation

Crossover operation (Koohestani, 2020) involves recombining a certain number of genes from both father individual and mother individual to create new individuals. The essence of the crossover operation is to recombine advantageous gene segments from different chromosomes to produce more excellent offspring (Deng et al., 2022). However, traditional crossover operation, which is characterized by high randomness, are marked by slow population convergence and lower optimization capabilities. To solve the above problem, this article introduces a novel crossover approach. Firstly, two individuals are randomly selected from the ‘breeding’ population, where they serve as father individual and mother individual for chromosome crossover. Next, the highest-weight segment of father individual replaces the lowest-weight segment of mother individual, where the probability of recombining advantageous gene segments from different chromosomes is increased, and vice versa. An example of the crossover operation is shown in Figure 8.

Figure 8. Example diagram of crossover operation.

3.5. Feasibility repair strategy

After the aforementioned crossover operation, the decoded chromosome may not be a feasible solution (Ahmed Memon et al., 2021). It requires feasible checking and repairing. Specific operations : firstly, the sum of each segment in the chromosome is compared with the sum of each segment in a feasible solution; then, the difference between them is calculated; finally, the crossed chromosomes are repaired based on the difference and priority is given to retaining the highest weight fragments and crossover fragments. If other fragments cannot meet the crossover chromosome feasibility correction, then the crossover fragment will be corrected. It is worth noting that, because the highest weight segment belongs to the subset solutions and is given the highest priority for retention, it is possible to complete the chromosome repair without modifying the highest weight segment. An example of the correction operation is shown in Figure 9.

Figure 9. Example diagram of correction operation.

3.6. Feasibility mutation operation

The essence of mutation is to introduce randomness and diversity to expand the search space and prevent the algorithm from getting stuck in local optimum. Mutation operation (Han & Xiao, 2022) makes the algorithm more exploratory and creative, in which more potential solutions is enabled to discover during the search process. However, traditional mutation operation may disrupt the feasibility of the chromosome, leading to infeasible solutions after decoding. Therefore, traditional mutation operation is not suitable for the fibre allocation scenario. This article introduces the concept of ‘active’ mutation position and ‘passive’ mutation position, which is based on traditional mutation operations. ‘active’ mutation position: the position where mutation occurs first; ‘passive’ mutation point: after the ‘active’ mutation occurs, other corresponding positions are forced to mutate in order to ensure the ‘legitimacy’ of the chromosome. The ‘feasible’ mutation operation allows the algorithm to be more exploratory and creative while ensuring the ‘feasibility’ of the chromosome. When ‘active’ mutation occurs at a particular position in an individual, other corresponding positions for different segments are forced to undergo ‘passive’ mutation to compensate for the difference caused by ‘active’ mutation. Similar to the crossover operation preservation strategy, the optimal and crossover segments are prioritized retainment during mutation. An example of the mutation operation is shown in Figure 10.

Figure 10. Example diagram of mutation operation.

3.7. Algorithm steps

In response to the characteristics of the optical cable of fibre allocation problem, this article proposes an improved genetic algorithm based on rule optimization, the IGA-ROS algorithm framework is shown in Figure 3. This article formulates the objective function of maximizing the inventory score and minimizing the fibre segmentation rate during fibre allocation. Therefore, the larger the value of the objective function, the better the fibre allocation results. The steps of the algorithm are as follows:

• Step 1. Initialize various hyperparameters for the IGA-ROS algorithm.

• Step 2. Generate sample data based on order requirements and place it in the solution pool.

• Step 3. Train the attention mechanism network.

• Step 4. Determine whether the termination condition is met. If satisfied, then extract the attention mechanism network layer and randomly select a specified quantity of sample from the solution pool as the initialization population. Otherwise, proceed to the next iteration of the loop.

• Step 5. Decode the population and evaluate the fitness.

• Step 6. The elite reservation strategy, which involves the direct enter of the next generation population by the top 20% of individuals based on fitness.

• Step 7. The top 50% of individuals with fitness values serve as the ‘breeding’ population, and in the next generation population, the remaining 80% of individuals are generated based on the correlation weight crossover operation.

• Step 8. Conduct ‘feasibility’ check and correction on the individual, which is generated in Step 7.

• Step 9. Determine whether to perform mutation operations for the individuals from Step 8.

• Step 10. Determine whether the termination condition is met. If satisfied, proceed to Step 11; otherwise, go back to Step 5.

• Step 11. Output the optimal value and average population fitness value for each iteration.

4. Experimental design and results analysis

4.1. Test cases

To validate the reliability and efficiency of the algorithm, test cases (Faramarzi et al., 2020) are generated by real fibre data from a optical cable production factory. The test cases cover common production combinations in the optical cable manufacturing industry, including: combination one ‘produce 20 reels of 2-kilometre optical cable, 20 reels of 2.5-kilometre optical cable, and 20 reels of 3.5-kilometre optical cable’; combination two ‘produce 20 reels of 2-kilometre optical cable, 10 reels of 2.5-kilometre optical cable, and 12 reels of 3.5-kilometre optical cable’; combination three ‘produce 20 reels of 2-kilometre optical cable, 10 reels of 2.5-kilometre optical cable, 9 reels of 3-kilometre optical cable, 12 reels of 3.5-kilometre optical cable, and 10 reels of 4-kilometre optical cable’; the number of tubes 8, the number of fibres 8, the score of initial inventory fibre is 0.6031.

The naming format for the test cases is $\mathrm{\Sigma }{L}_i\ast N_i$, where $L_i$ represents the optical cable length and $N_i$ represents the number of optical cable reels.

4.2. Parameter setting

In IGA-ROS, the solution pool size is set to 1000, the total number of iterations for training network is 500, the learning rate (Lewkowycz et al., 2020) is 1e-5, the batch size is 64, and after the training network is completed, 50 samples are randomly selected from the solution pool (Altabeeb et al., 2021) as the initialization population, that is, the population size is 50, the number of iterations is set to 50, the crossover rate is set to 0.8, and the mutation rate is set to 0.02.

The genetic algorithm is one of the most representative metaheuristic algorithms for combinatorial optimization problems. Therefore, we will consider the Genetic Algorithm with Feasibility Correction Strategy (GA-FCS) and the stepwise greedy algorithm as comparative methods for IGA-ROS, with GA-FCS parameters are set the same as the latter. Here, the Stepwise Greedy Algorithm (Qian et al., 2019) specifically refers to treating the fibre allocation problem as a multi-stage selection problem. In each stage, the algorithm selects the length interval with the lowest segmentation rate, and then chooses the fibre with the lowest score within that interval until the fibre allocation demand is satisfied.

In order to eliminate the influence of randomness of the algorithm and measure the solution effect of the algorithm, two performance metrics ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ are introduced, and their definitions are shown in Equations (11(11) $\begin{array}{rl}& {\mathrm{E}}_1=\frac{\overline{{\upsilon }_1}-{\upsilon }_3}{{\upsilon }_3}\times 100\mathrm{\%},\end{array}$(11) ) and (12(12) $\begin{array}{rl}& {\mathrm{E}}_2=\frac{\overline{{\upsilon }_1}-\overline{{\upsilon }_2}}{\overline{{\upsilon }_2}}\times 100\mathrm{\%}.\end{array}$(12) ) (Liu et al., 2024). Among them, $\overline{{\upsilon }_1}$ and $\overline{{\upsilon }_2}$ are the average optimal solutions after five repeated tests of the same test case of IGA-ROS and GA-FCS, respectively, and ${\upsilon }_3$ is the optimal solution obtained by the stepwise greedy algorithm. $\overline{{\upsilon }_1}$, $\overline{{\upsilon }_2}$ and ${\upsilon }_3$ reflect the average calculation results of IGA-ROS, GA-FCS and Greedy on the same test case, respectively, with the purpose of reducing the impact of accidental factors and enhancing the trustworthiness of the experimental results. If ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ are positive, then it proves that IGA-ROS has a better optimization effect than the stepwise greedy algorithm and the general approximation algorithm. (11) $\begin{array}{rl}& {\mathrm{E}}_1=\frac{\overline{{\upsilon }_1}-{\upsilon }_3}{{\upsilon }_3}\times 100\mathrm{\%},\end{array}$(11) (12) $\begin{array}{rl}& {\mathrm{E}}_2=\frac{\overline{{\upsilon }_1}-\overline{{\upsilon }_2}}{\overline{{\upsilon }_2}}\times 100\mathrm{\%}.\end{array}$(12) All experiments were conducted on a personal computer with an Intel(R) core(TM) i7-13700H CPU @ 2.40 GHz and 16.0 GB RAM. The experimental program was implemented in Python 3.8.

4.3. Experimental comparison

To comparing the experimental results of IGA-ROS, GA-FCS, and the stepwise greedy algorithm, a curve graph, which depicts fifty iterations for test case one, is shown in Figure 11. In test case one, IGA-ROS obtains the optimal value, and the average fitness of the population nearly converges to the optimal value. It can be observed that, with the introduction of the rule selection strategy, the optimal solution curve and the population's average fitness curve of IGA-ROS exhibit a more significant upward trend compared to GA-FCS, ultimately achieving a better optimal value.

Figure 11. Simulation illustration.

To further compare the experimental results of IGA-ROS, GA-FCS, and the greedy algorithm in different test cases, the results are presented in Table 2. In the test case one, the greedy algorithm exhibits the fastest solving speed but yields suboptimal results. The relative deviation between the optimal solutions of GA-FCS and IGA-ROS is within 0.52%. Although the solving time of IGA-ROS is slightly longer than that of GA-FCS, the optimization capability and population convergence speed have been significantly improved. As the scale of the test cases increases, the advantages of IGA-ROS become more pronounced. From test case two to test case three, the solution time of the stepwise greedy algorithm increases exponentially and the solution effect is still poor. However, the solution results of IGA-ROS are better than those of GA-FCS and stepwise greedy algorithm in all calculation examples. In test case two, compared with stepwise greedy algorithm, the solution quality improvement rate is even as high as 25.52%. Although the solution time has increased as the scale of the calculation example increases, the solution time increment is more acceptable than the stepwise greedy algorithm. In contrast, although GA-FCS achieves slightly lower inventory scores (approximately 0.01-0.03) than IGA-ROS in the three combinatorial cases, there is still a noticeable gap in inventory improvement compared to IGA-ROS, when considering the vast quantity of fibre in the entire inventory. The effectiveness of the rule optimization strategy is evident in this context.

Table 2. Statistical summary of test case results.

Test case	Algorithm name	Solution time	Inventory score	Segmentation rate	Optimal fitness	Optimal average fitness	$\overline{{\upsilon }_1}$	$\overline{{\upsilon }_2}$	${\upsilon }_3$	${\mathrm{E}}_1$	${\mathrm{E}}_2$
2*20+2.5*20+3.5*20	IGA-ROS	698.36	0.6323	0.35%	1.0117	1.0077	1.0103			10.80%	0.52%
	GA-FCS	622.87	0.6099	0.00%	1.0068	0.8559		1.0051
	Greedy	68.43	0.6364	2.43%	0.9118				0.9118
2*20+2.5*10+3.5*12+4*15	IGA-ROS	714.56	0.6108	0.00%	1.0077	1.0053	1.0052			25.52%	0.28%
	GA-FCS	632.56	0.6084	0.00%	1.0053	0.9120		1.0024
	Greedy	319.31	0.6124	4.17%	0.8008				0.8008
2*20+2.5*10+3*9+3.5*12+4*10	IGA-ROS	796.69	0.6324	0.38%	1.0103	0.9330	1.0086			10.53%	0.61%
	GA-FCS	722.22	0.6086	0.00%	1.0055	0.8825		1.0025
	Greedy	1268.29	0.6196	2.08%	0.9125				0.9125

In conclusion, under complex scenarios that consider multiple fibre selection principles and the constraint of inventory fibre quantity, as the sheath number of products and inventory expands, the stepwise greedy algorithm cannot solve a high-quality solution within an ideal time; and IGA-ROS relies on feasible repair strategy and rule optimization strategy to find solution that are better than GA-FCS and stepwise greedy in a relatively acceptable time.

5. Conclusion

This article proposes an improved genetic algorithm, IGA-ROS, for fibre allocation scenarios. The algorithm is based on a rule optimization strategy and utilizes novel permutation encoding to align closely with the fibre allocation problem. A repairable strategy is introduced to ensure individual effectiveness throughout the iteration process. Additionally, a solution pool is created to train a neural network model with a self-attention mechanism layer. This layer is trained to evaluate the correlation of chromosome segments within the population. The rule optimization strategy aims to increase the likelihood of recombining excellent gene segments from different chromosomes by assigning evaluation weights. The effectiveness of IGA-ROS is validated through solving multi-scale instances derived from practical production scenarios, showcasing its superiority over traditional methods. Unlike conventional fibre allocation approaches, IGA-ROS considers factors such as production scheduling tasks, fibre selection criteria, and production order segmentation rates, closely aligning with the requirements of optical cable production. Today's intelligent transformation of the manufacturing industry is strongly advocated and has extremely high application potential. In the future, how to make timely and reasonable dynamic responses to dynamic inventory optical fibres deserves further study.

Author Contributions

Feng Tan: Data curation, Software, Investigation. Zhipeng Yuan: Investigation, Validation, Visualization. Yong Zhang: Conceptualization, Writing – original draft, Funding acquisition. Sheng Tang: Methodology, Formal analysis, Writing – review & editing. Feng Guo and Shuai Zhang: Investigation, Writing – review & editing.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

Data availability statement

Because it is production data from a certain manufacture, the authors do not have permission to share data.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Grant No. 62273264)