New improved hybrid genetic algorithm for optimizing facility layout design of reconfigurable manufacturing system

Abstract

The equipment layout design of a reconfigurable manufacturing system can be determined by a variety of algorithms. The complexity of the problem increases with the increase of dimension, and it is a typical NP hard problem. In this paper, a new improved hybrid genetic algorithm is proposed to solve this problem. Firstly, the chaos genetic algorithm based on improved Tent map is used to enhance the quality and diversity of the initial population. In order to reduce the complexity of the problem, this paper applies the association rule theory to mine the dominant blocks in the population and to combine the artificial chromosomes. After matched crossover and mutation operations on the layout encoding string, a small adaptive chaotic perturbation is applied to the genetically optimized optimal solution. Finally, through comparison of experimental results and algorithms, it can be concluded that the proposed method is superior to traditional methods in terms of both accuracy and efficiency.

Introduction

Layout design is an important research topic in the field of manufacturing system. In traditional manufacturing systems, the initial layout design is rarely changed. However, with the increasingly complex and unpredictable changes in market demands, enterprises need more flexible and responsive manufacturing systems to adapt to these changes. In case of addition of new equipment, the adjustment or reconfiguration of the layout in workshop is inevitable and frequent. Continual changes in the layout of a workshop involve the disassembly, installation and relocation of equipment, as well as the costs of stopping normal operations.

Layout design is a critical research area within the field of manufacturing systems. In traditional manufacturing environments, the initial facility layout is often considered a static element, rarely undergoing significant alterations. However, the modern manufacturing landscape is characterized by increasingly complex and unpredictable market demands, requiring enterprises to adopt more flexible and responsive manufacturing systems capable of rapid adaptation. This shift necessitates frequent adjustments and reconfigurations of the workshop layout, particularly when new equipment is introduced. Such changes involve the disassembly, installation, and relocation of existing equipment, incurring substantial costs associated with downtime and disruption to normal operations. Furthermore, the challenge of facility layout optimization in reconfigurable manufacturing systems (RMS) is significantly compounded by the high-dimensional nature of the problem. Unlike static layouts, RMS layouts must account for multiple variables, such as the dynamic location of machines, varying production demands, the size and shape of equipment, material flow, and safety requirements. These variables interact in complex ways, resulting in a vast search space, where the number of potential layout configurations grows exponentially with the number of machines and the complexity of the system. Moreover, the need to optimize multiple, often conflicting objectives (e.g., minimizing material handling costs, maximizing throughput, ensuring safety, and minimizing reconfiguration time) further exacerbates the difficulty of finding optimal or near-optimal solutions. The dynamic and time-dependent nature of RMS, where machine locations and production demands may vary over time, requires layout designs that can adapt to these changes effectively and efficiently, thereby adding another layer of complexity to the problem. This high-dimensional complexity makes traditional optimization techniques, often relying on exhaustive search or simplified models, unsuitable for real-world RMS layout problems. Therefore, the development of efficient and robust algorithms capable of handling the intricate and dynamic nature of RMS layout design is of paramount importance.”

Literature review

The increasing complexity of manufacturing systems poses significant challenges to traditional optimization methods. Exact algorithms, such as branch-and-bound and mixed-integer programming, often struggle to find solutions within reasonable timeframes as problem scale expands. This has led to the development of more efficient optimization algorithms. In recent years, metaheuristic algorithms like genetic algorithms (GA), simulated annealing (SA), ant colony optimization (ACO), particle swarm optimization (PSO), and tabu search (TS) have become popular for solving complex optimization problems, including facility layout problems^1-4

Early work in this area demonstrated the potential of these methods. Kia⁵ combined mixed-integer nonlinear programming with SA for cell layout design, focusing on constraints such as routing flexibility, machine availability, unit size, machine capacity, and machine location. Shafigh et al.⁶ developed an embedded linear programming SA algorithm that minimized costs related to machine relocation, material handling, inventory holding, machine installation, internal production, and outsourcing, while considering factors such as distance, inventory, workload, and machine location. Maniraj et al.⁷ used ACO to optimize machine allocation in a single-product line RMS, prioritizing actual operations to minimize costs. Kulturel-Konak et al.⁸explored a dual-objective approach for layout problems, using TS to reduce material handling and reconstruction costs, while carefully examining the aspect ratio of unit facilities. Pourvaziri, Hani⁹ employed an analytical method using open queuing network theory within a quadratic assignment problem framework, and Moslemipour, Ghorbanali et al.¹⁰ introduced a hybrid algorithm based on ACO, clonal selection, and robust layout design strategies. Alaa Al Hawarneh et al.¹¹proposed a grid layout model, incorporating safety and cost parameters, arranging the layout based on safety proximity levels. Recent work by Ghanei and AlGeddawy¹²developed an integrated multi-period layout planning and scheduling model, highlighting the importance of considering multiple factors in the optimization process, and Salimpour and Azab¹³have used dynamic programming for facility layout in RMS, showcasing a different approach for solving these problems.Additionally, Erik and Kuvvetli¹⁴introduced a novel material handling-driven approach. Genetic Algorithms (GA), a classic evolutionary algorithm, have demonstrated their effectiveness in solving combinatorial optimization problems^15,16.The use of Genetic Algorithms (GA) in facility layout has been a major focus of research due to their robustness, parallelism, and global search capabilities. Fan¹⁷ used a genetic algorithm with sex differentiation (GASD) to address layout problems in mass customization, aiming for cost-effective customized production within specific timeframes. Wang et al.¹⁸ and Hernandez et al.¹⁹ used enhanced GA to optimize layouts for facilities with unequal areas. Kheirkhah et al.²⁰combined GA with PSO, considering constraints like time, equipment numbers and location, to minimize the total cost of equipment handling and rearrangement. Recent research shows a significant increase in the complexity and sophistication of the algorithms. For instance, Liu and Li²¹have employed an improved genetic algorithm to solve the multi-row dynamic facility layout problem, indicating an ongoing trend towards adapting GA for dynamic environments. In contrast to traditional static layout optimization problems, Kulturel-Konak and Konak²²addressed flexibility by designing facilities based on zones, using embedded input/output points, and demonstrating that zone-based layouts can be a solution in cases where flexibility is a key requirement. Furthermore, there is a trend towards the integration of different metaheuristic approaches to leverage their respective strengths. Aihara K²³. et al. initially introduced chaotic neural networks, exploiting the sensitivity to initial values, which have been used in various optimization problems. Hybrid methods have gained traction, with Das et al.²⁴using a hybrid element heuristic for power grid energy storage, and Wei-chiang Hong²⁵combining chaos, niche search and evolutionary mechanisms for predicting complex motions of floating platforms. Pourhassan M R²⁶et al. proposed a hybrid PSO-GA for internal logistics optimization. Guangmei Hai²⁷et al. used chaos theory to optimize power control system designs, while Haitao Yuan²⁸et al. proposed a hybrid PSO-GA-SA algorithm for scheduling. Zhoubo Xu²⁹et al. used a combination of ant algorithm, chaos, and GA to solve assembly sequence programming problems, and Xue Wen³⁰et al. dynamically combined PSO and chaotic searches. Jianwen Gao³¹et al. used chaotic maps to initialize particles, combining a cloud genetic algorithm and PSO. Jiajun Ye³²et al. utilized chaotic sequences within GA to enhance its accuracy, and Li Wen³³et al. applied a chaos-enhanced algorithm to optimize PID parameters in hydraulic systems. Milisavljevic-Syed, Li, and Xia³⁴provide a detailed discussion on the realization of responsive and sustainable reconfigurable manufacturing systems, which underscores the need to view layout design within a broader context of system performance and adaptability. Maganha and Silva’s³⁵ literature review provides a comprehensive overview of the research on layout design in RMS, which lays a good foundation for the research on this topic. Additionally, Flores-Siguenza et al.³⁶ offer a systematic literature review that identifies resilience factors in facility layout problems, which is also an important aspect of RMS design. By integrating the inversion capabilities of genetic algorithms with the ergodic properties of chaotic search, it mitigates the problem of GA converging to local optima. This enhancement boosts the search speed and global convergence, rendering it highly effective for facility layout design problems.By integrating the inversion functions of genetic algorithms with the ergodic properties of chaotic search, the hybrid algorithm effectively mitigates the common issue of genetic algorithms converging to local optimum. This enhancement significantly boosts the search speed and global convergence of the algorithm, rendering it highly effective for solving facility layout design problems. Block, key block or dominant block is a recently-developed theory with obvious advantages in reducing problem complexity, increasing the speed of algorithm solving, and improving the quality of the solution. Chang P et al.³⁷ combined the probability matrix and roulette to mine the dominant block based on the binary variable probability model, and proposed a block-based distribution estimation algorithm to solve FSP using the binary variable probability model. Hsu C et al.³⁸ applied association rules to mine the gene combinations of superior individuals to form superior blocks, which greatly improved the algorithm’s computational efficiency. Despite these advancements, several research gaps remain in the literature concerning the application of hybrid metaheuristic algorithms for facility layout problems in RMS. Key challenges include: Balancing Exploration and Exploitation: While GA excel at global search, they often lack efficiency in local exploitation, which is necessary for fine-tuning the solutions. Hybrid algorithms, which combine GA with other techniques such as PSO and SA, have been used, but there is a need for more efficient hybrid approaches.

• Addressing Dynamic Aspects: The dynamic nature of RMS, such as changing product demands and frequent reconfigurations, are not always adequately addressed by existing solutions. Current methods often focus on static or simplified dynamic scenarios, which do not capture the full complexity of real-world RMS environments.

• Initial Population Diversity The effectiveness of genetic algorithms and other metaheuristic algorithms are reliant on diversity of initial populations, and most algorithms lack a strong method to ensure this.

• Integration of chaotic search: While chaotic search shows promise in improving genetic algorithms, there needs to be more efficient ways of combining the benefits of chaotic search and the convergence of GA.

• Lack of a block-based approach: Block-based methods have been used in production optimization, but not as commonly for layout design problems. The use of block-based method for facility layout needs to be explored further.

• Computational Efficiency: Many complex algorithms incur significant computational overhead, limiting their application in large-scale practical scenarios.

This study addresses these gaps by proposing a new improved hybrid genetic algorithm (NIHGA) for RMS layout optimization. It will incorporate an improved initial population based on a chaotic algorithm using the improved Tent map. The algorithm will use association rules to mine dominant blocks to reduce the problem complexity. The algorithm further improves the global and local search by refining the crossover and mutation process, and will include a small adaptive chaotic perturbation to the optimal solution of genetic optimization. The effectiveness and robustness of the algorithm will be tested through simulation experiments^42-44

Model Building

The dynamic layout optimization model for workshop equipment creates an optimal layout design across various production planning periods, considering future conditions. By assessing the material flows during each sub-planning period, the optimal equipment placement for each phase is determined. This strategy significantly lowers the total operational costs across the entire planning horizon.

According to reference³⁹, the workshop layout problem is simplified to dynamic multi-line equipment under specific constraints, with each period’s necessary conditions clearly defined. The mathematical model addressing the dynamic layout’s multi-objective optimization problem is presented below:

The minimum material handling costs

is the total cost of material handling, is the planned period, and is the number of equipment to be arranged in the workshop. is the handling cost per unit distance between units and in sub-period . is the material handling frequency between units and in sub-period . is the distance between devices between units and in sub-period , and is the total number of rows in the device layout. The following formula shows:

1

2

The formulas used to calculate the horizontal and vertical coordinates of adjacent equipment for each sub-plan period are outlined as follows:

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Minimum equipment replacement cost

When denoting as the cost of equipment replacement, it encompasses the costs associated with moving the device, which includes both installation and removal expenses. Additionally, represents the cost of moving the equipment per unit distance. This cost structure ensures that all financial aspects of equipment logistics—moving, installing, and removing—are accounted for in the overall replacement budget. The variable represents the distance that device is moved from one sub-period to the next,. Specifically, measures the displacement between the center points of the device across these adjacent sub-planning periods. The variable represents the number of cycles during which the device’s position is altered, while denotes the loss of devices resulting from each installation or disassembly, which can be minimized to a constant value.

5

6

Maximum utilization of workshop area

During the sub-planning stage, represents the total area of the layout plan’s enveloping rectangle. For each piece of equipment, , the floor area is determined by the product of its length and width. A smaller rectangular envelope for the layout device corresponds to greater utilization of the workshop area, thereby enabling a more compact layout configuration.

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The total floor space of pieces of equipment can be represented as a specific value. Consequently, the maximum utilization of the workshop area can be expressed as:

9

From the analysis presented, it is possible to derive the combined optimization objective function for the dynamic layout of workshop equipment, which integrates various parameters to maximize efficiency and space utilization.

10

Among them, are normalization factors, the first two are cost units, and the last one is an area unit, in order to ensure the unity of the dimensions, and make the minimum of their optimal results close to 1, so

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are weighting factors that must satisfy . To ensure that the device in the last row along the -direction is positioned within the workshop boundaries, a penalty function is defined as follows: represents the penalty for exceeding the workshop area in the -direction, and is a large positive penalty value .

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Solution methodology

Considering the advantages of the GA algorithm, this study integrates GA with the characteristics of layout design in RMS and proposes a new improved hybrid genetic algorithm (NIHGA). Traditional optimization methods such as branch-and-bound and mixed integer programming, often struggle with the computational complexity of large-scale facility layout problems and can get stuck in local optima. Traditional GA applications, while effective to some degree, frequently suffer from premature convergence, lack of diversity in the initial population, and often fail to fully explore the search space, leading to sub-optimal solutions. Moreover, existing GA applications have limitations in effectively balancing exploration with exploitation, thus hindering efficient and accurate search of solutions. To overcome these limitations, NIHGA is structured into six main parts, as depicted in Fig. 1:

Fig. 1.

Algorithm flow.

Note

BM represents the counter of dominant block mining; BM represents the critical value of dominant block mining.

The NIHGA overcomes the aforementioned limitations by: ① Enhancing the quality and diversity of the initial population by integrating a chaotic genetic algorithm. ②Reducing the complexity of the problem, while improving the algorithm’s efficiency, through the use of association rules for dominant block mining. ③ Improving the algorithm’s global search by adopting three different crossover strategies. ④ Using a mutation operator for mutation operation. ⑤ Optimizing the population by applying a small adaptive chaotic perturbation to the genetically optimized solution, and finally, ⑥ retaining the dominant population. The specific operations are as follows.

Generation of chaotic sequences

This paper introduces a chaotic genetic algorithm based on Tent mapping, which utilizes chaos search and the inversion of genetic algorithm optimization techniques. The central concept is to employ the ergodic properties of chaotic motion to select the initial population and to enhance the quality of the solution by introducing a small, adaptive chaotic perturbation to the current optimal solution^45,46.

The choice of the Tent map over other chaotic maps, such as the logistic map, is deliberate and based on several advantages it offers for this particular application. While the logistic map, defined within the space [0, 1], contains discontinuous points at 0.25, 0.5, and 0.75, leading to an uneven distribution of mapping points characterized by a “high sides and low middle” pattern, the Tent map offers a more uniform coverage of the search space, which enhances the diversity of the generated individuals. The uneven distribution of the logistic map can directly impact the convergence speed of the iterative process, thus reducing the algorithm’s efficiency.

The mapping equation is given by:

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In the formula, denotes the serial number of the chaotic variable, while represents the serial number of the population, where . And is the chaotic variable within the interval [0, 1]. A key advantage of the Tent map, particularly when compared to the logistic map, is its relatively uniform distribution of generated values across the [0, 1] interval. This helps in more effectively exploring the entire solution space, thereby increasing the likelihood of finding better solutions and reducing the risk of premature convergence. Unlike the logistic map where points tend to cluster, the Tent map’s more consistent distribution helps in maintaining a diverse population for a longer duration during optimization, reducing the chance of falling into local optima early in the process. Furthermore, the Tent map, while simple, exhibits robust chaotic behavior, offering a good balance between computational efficiency and the complexity necessary to explore a multi-dimensional search space.

While it is acknowledged that, due to the finite length of computer word representation, the Tent mapping iterations may converge to a fixed point or a small-period cycle²⁷, which can potentially reduce population diversity, the uniform distribution properties of the map, and the method with which it is used as described below, help to reduce this problem in our application. The dynamic layout problem of n equipment with p production planning periods is taken as an example.

1. At first, a set of random permutations between 0 and 1 are generated, and the operation is repeated p times to obtain an array.

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1. 2. By adopting the iterative method based on improved Tent map, the initial value of is iterated for 200 times to generate a series of initial individuals, and 200 groups of random numbers of 0 ~ 1 are obtained. These numbers are arranged in an ascending order within the layout range of a single period. Natural numbers 1 ~ n represent the serial number of the original number where the corresponding number was located, which is used as the coding serial number;

2. 3. The initial population is formed by selecting the first individuals () that have the highest fitness values. Firstly, the multi-line static device layout of a single plan period is coded in the order of devices from left to right and from bottom to top. Then, the code strings of the static layout of each period are concatenated in order, according to the total plan period. The phase 3 layout problem of 8 devices is taken as an example to represent the chromosome string of a dynamic device layout. .

To ensure the new obtained individuals are feasible solutions to avoid the generation of illegal progeny, the layout strings of each period are selected, crossed, and mutated, successively. Finally, the layout strings of each period of the generated sub-individuals are connected to form the entire chromosome.

Dominant block mining based on association rules

Building blocks

Considering the advantages of the GA algorithm, this study integrates GA with the characteristics of layout design in RMS and proposes a new improved hybrid genetic algorithm (NIHGA). Traditional optimization methods, such as branch-and-bound and mixed-integer programming, often struggle with the computational complexity of large-scale facility layout problems and can get stuck in local optima. Traditional GA applications, while effective to some degree, frequently suffer from premature convergence, lack of diversity in the initial population, and often fail to fully explore the search space, leading to sub-optimal solutions. Moreover, existing GA applications have limitations in effectively balancing exploration with exploitation, thus hindering efficient and accurate search of solutions.

The fundamental concept behind dominant block mining is that highly adaptable chromosomes (representing superior layout solutions) often exhibit recurring gene patterns. By identifying these patterns and consolidating them into “dominant blocks”, the complexity of the problem can be significantly reduced, leading to a more efficient and effective algorithm. In this study, association rule mining is applied to identify these dominant gene blocks from the fittest chromosomes, thus improving the algorithm’s solving efficiency and overall performance⁵².

To extract these useful gene fragments as blocks, the best-performing chromosomes (those with high fitness values) containing the equipment layout sequence from period 1 to period P are chosen from the initial population. To prepare the data for mining, the layout sequence is transformed into a series of itemsets where each position in the layout is considered as a transaction and the items are the different devices present in that position across multiple good solutions.

Figure 2 illustrates the process of converting a sequence of devices into a layout record. Suppose we choose the first four chromosomes of each generation. For the first generation, the three types of devices (i.e., the first, second and third device) can be recorded on the layout position 1 as . Secondly, there are two types of devices (device 1 and device 2) on location 2 to choose from, which are denoted as . Finally, we realize a layout record arranged by position.

Fig. 2.

Transform chromosome information into layout equipment.

Block mining

Association rule theory⁴⁰ is employed by our algorithm to identify the relationships between different device placements and extract dominant blocks. This step is explicitly part of our novel algorithm design. Our use of association rules to determine the relationship between the genes of the best individuals and forming blocks has allowed for effective extraction of the blocks.

The relationship between andcan be expressed as : implies that if happens, then happens. The association rule can be expressed as follows:

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Among them, the support degree represents the frequency with which the two data sets and concurrently appear in the transaction database . The confidence level denotes the correlation strength of the and data sets, which is determined by the lift. If the lift is greater than 1, it is a strong correlation. If the lift is less than 1, it means that and are weakly correlated, implying that if X occurs, and the probability of occurring is very small. The lift in the association rule is calculated using formula (17), and the confidence level is calculated using formula (18)⁴¹.

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The block mining process in the device layout record of database is illustrated in Fig. 3. Assuming that the minimum support threshold is 2 and the block length is 2, the first itemset of each device at each location is selected to calculate its support level in the candidate itemset (C1). If the itemset in C1 has more support than , it is denoted as frequent itemset L1.

Fig. 3.

Process of building blocks.

The support degree of the C2 candidate set is calculated, and those item sets whose support degree is less than the minimum support threshold are deleted. Given a block of length 2, the focus is to seek the entire device layout record, calculate the support degree of each device at each location, and select a device layout record which is greater than the minimum support degree. Finally, all blocks of the frequent item set are archived for further block extraction.

Dominant block extraction

Figure 4 presents the process of block extraction. First, the machines of the dominant block in the register are compared. If there are duplicate machines between the dominant blocks, the lift of the dominant block is calculated. The dominant block with a lift greater than 1 has a stronger correlation and will be retained, whereas those that do not meet the requirements will be deleted. The process of extracting dominant blocks from the saved frequent item sets, is specific to our algorithm.

Fig. 4.

Block extraction process and the combination of artificial chromosomes.

The combination of artificial chromosomes

Combining artificial chromosomes can generate chromosome populations with high competitive advantage. In this study, the dominant blocks are mined and selected on the basis of association rules and used to combine artificial chromosomes. The dominant gene blocks are directly inserted to the corresponding positions on the artificial chromosomes. Then, empty positions are randomly selected from the remaining devices to generate artificial solutions. Figure 4 shows the combination of artificial chromosomes.

Genetic operations

In order to avoid the local optimal problem, most crossover operators adopt the position-based crossover or two-point crossover mechanism after the generation of the artificial solution. In this paper, three different crossover mechanisms are designed according to the characteristics of the layout solution to improve the quality of the algorithm solution. They include the single segment, double segment and three segment crossover modes. Although these crossover modes may damage the stability of the evolutionary process, it is more important to expand the search space of solutions. Figure 5illustrates the three crossover processes^47-49.

Fig. 5.

Three crossover processes.

Chaotic disturbance

A chaotic perturbation is applied to each variable of the current generation’s optimal solution according to formula (19). The vector represents the current optimal solution, where the individual is mapped to the interval [0, 1]. After iterations, becomes the chaotic vector, representing the perturbed chaotic vector, which can be calculated using formula (19).

In formula (20), the value of lies within the range (0, 1) and decreases as the number of iterations increases. The parameteris an integer that varies depending on the objective function. Over the course of the iterations, the extent of chaotic perturbation decreases.This approach ensures a broad search space in the early stages of iteration, increasing the likelihood of finding the optimal solution, while the reduced perturbation in later stages aids in accelerating convergence toward the global optimum^50,51.

In this paper, the chaotic perturbation parameter is selected to be 3. The variable represents the number of chaotic iterations conducted.  

19

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The fitness of the newly generated population is evaluated. If the condition in Eq. (21) is met, the optimal solution is output; otherwise, the process returns to the next iteration.

21

To overcome the limitations of traditional methods and address the unique challenges posed by facility layout problems in RMS, the proposed NIHGA incorporates several novel features, distinguishing itself in both its algorithm design and problem-solving approach. Algorithmic novelty includes the use of a chaotic genetic algorithm with a Tent map for enhanced initial population quality and diversity, providing a more diverse starting point than traditional GA approaches; a unique process of dominant block identification using association rules, which is applied for the first time to the facility layout problem and specifically used in our algorithm to reduce the search space; the application of a small adaptive chaotic perturbation to the best solution to guide convergence; and the adoption of three different crossover strategies to improve the algorithm’s global search performance and the efficiency of population evolution. In its approach to the problem, NIHGA emphasizes leveraging an understanding of problem structure, applying the concept of dominant blocks within facility layout problems, and effectively combining chaos, block mining, and genetic operations. NIHGA is specifically designed to tackle the inherent complexities associated with RMS layouts, particularly the challenge of finding optimal layouts in dynamic environments. These innovations make NIHGA a unique solution not only in how it solves the problem (the algorithm), but also in how it understands and formulates a solution to the facility layout problems specific to RMS (the problem-solving approach), thereby rendering the search for optimal layouts more efficient and effective.

Instance validation

Experimental design

An engine factory intends to revamp the layout of a cylinder block assembling workshop. The size of the workshop is 16 m by 12 m. The layout plan includes 10 equipment and needs to be designed for 3 planning periods. The equipment size and logistics frequency are presented in Tables 1 and 2, respectively. Let denote the material cost per unit distance between equipment and . During the sub-planning period, the logistics matrix between these two pieces of equipment is represented by ).The horizontal distance between the two pieces of equipment is .

Table 1.

Equipment dimensions.

Equipment	1	2	3	4	5
Dimensions	2.4 × 2.0	1.0 × 0.8	3.4 × 1.6	1.4 × 1.4	2.0 × 1.6

Equipment	6	7	8	9	10
Dimensions	1.6 × 1.6	1.0 × 1.6	2.8 × 1.6	1.6 × 1.6	2.4 × 1.2

Table 2.

Logistics frequency.

The first phase		The second phase		The third phase
Logistics direction	Logistics frequency	Logistics direction	Logistics frequency	Logistics direction	Logistics frequency
1#− 2#	330	1#− 3#	430	1#− 2#	150
1#− 3#	200	2#− 4#	240	1#− 3#	50
2#− 4#	240	3#− 4#	250	1#− 4#	580
2#− 5#	90	3#− 5#	180	2#− 3#	100
3#− 4#	150	4#− 5#	490	2#− 4#	50
3#− 5#	50	5#− 6#	670	3#− 5#	150
4#− 5#	230	6#− 7#	200	4#− 5#	500
4#− 6#	100	6#− 8#	370	4#− 6#	130
4#− 8#	60	6#− 9#	100	5#− 6#	510
5#− 6#	370	7#− 8#	100	5#− 7#	140
6#− 7#	60	7#− 9#	100	6#− 7#	100
6#− 8#	410	8#− 9#	470	6#− 8#	540
7#− 9#	60	9#− 10#	670	7#− 9#	240
8#− 9#	470			8#− 10#	540
9#− 10#	530			9#− 10#	240

To simulate this scenario, we employed three optimization algorithms: the standard genetic algorithm, a chaotic genetic algorithm utilizing Tent mapping, and NHIGA, as detailed in Table 3.The population size and the number of iteration terminations . It can be seen from Table 2 that the NHIGA algorithm produces the best optimization effect.

Table 3.

Performance comparison of three optimization algorithms.

Optimization method	Optimal solution	The number of average iterations to produce the optimal solution	Average iterations Calculation Time/s
GA	23775,26.7%	121	8.92
CGA	21207,29.0%	87	7.98
NHIGA	20783,27.6%	50	5.31

Here, only the NHIGA algorithm is shown. At the same time, the equipment replacement cost, material transportation cost and workshop area utilization rate are also calculated, as shown in Fig. 6. The calculation of the target value is fast and efficient, and the result is clearly displayed.

Fig. 6.

NHIGA optimal solution display interface.

To understand the efficiency of our proposed algorithm (NIHGA), we analyze its computational complexity, which describes how the runtime of the algorithm scales with the size of the input. This analysis compares NIHGA to traditional Genetic Algorithms (GA) and a Chaotic Genetic Algorithm (CGA). The complexity of NIHGA is primarily influenced by the following core operations: initial population generation, achieved through a chaotic sequence using the improved Tent map, with complexity that exhibits a linear relationship with the population size and the number of iterations; fitness evaluation, which is proportional to the population size and the cost of evaluating a single layout; dominant block mining, which utilizes association rule mining for block extraction, adding a layer of complexity that, in the worst case, scales with the number of chromosomes used, the number of layout positions, and the square of the number of devices/items; chromosome combination, which has a complexity dependent on the number of chromosomes and the length of the dominant blocks; genetic algorithm operations like crossover and mutation, with a complexity that is a function of both population size and chromosome length; and finally, chaotic perturbation, which has a fixed computational cost.

The overall complexity of a single iteration of NIHGA is essentially a combination of these operations. While dominant block mining may contribute to higher complexity during the initial iterations, its complexity decreases over time as the blocks remain constant, thus making it less impactful during later stages. Fitness evaluation, crossover and mutation and chaotic perturbation, however, always add to the computational cost in each generation. The use of blocks also reduces the solution space, thereby impacting convergence time.

In contrast, a standard GA typically has a complexity that is dependent on the population size, the number of iterations, and the cost of fitness evaluation. A CGA, which incorporates Tent mapping, adds to that an initialization complexity proportional to population size and the number of iterations. This analysis suggests that NIHGA may have a higher per-iteration computational cost compared to GA and CGA, especially during the initial stages. However, we expect NIHGA to converge to a solution faster due to better initial populations, the use of dominant blocks, chaotic perturbation, and reduction in the solution space, leading to fewer iterations to achieve comparable fitness levels, and ultimately, a more efficient algorithm compared to traditional GA and CGA methods.

Performance comparison based on optimum results

The Performance comparison of the optimum results are shown in Tables 4 and 5. These tables provide the average values, standard deviations, and ANOVA test results to establish statistical significance. Table 4 shows the average iterations required for convergence as well as the computation time, while Table 5 shows the average fitness values obtained.

Table 4.

Computational time & iterations for convergence.

Algorithm	Average iterations	Std Dev iterations	Average time (seconds)	Std Dev time (seconds)	ANOVA (p-value)
Baseline GA	121	8.5	8.92	0.65	< 0.001
Chaotic GA	87	7.2	7.98	0.52	< 0.001
NIHGA-DB	65	5.8	6.3	0.46	< 0.001
NIHGA-P	78	6.9	6.7	0.49	< 0.001
Full NIHGA	50	4.5	5.31	0.42	< 0.001

Table 5.

Average fitness values.

Algorithm	Average fitness	Std Dev fitness	ANOVA (p-value)
Baseline GA	1.70	0.03	< 0.001
Chaotic GA	1.90	0.02	< 0.001
NIHGA-DB	1.85	0.02	< 0.001
NIHGA-P	1.85	0.01	< 0.001
Full NIHGA	1.88	0.01	< 0.001

As seen in the above two tables, the Full NIHGA, with all components, performs the best for convergence, with a lower number of average iterations and lower average computation time, compared to all the other algorithms. The standard deviation of all results shows that all algorithms, especially full NIHGA, are also stable in the results they produce. The Baseline GA and Chaotic GA perform considerably worse in terms of iterations and overall computation time due to the lack of block mining, combination, and chaotic perturbation components. The experiments also show that a standard genetic algorithm, although simpler, is significantly less efficient and effective than our proposed NIHGA. The addition of Tent mapping improves results, and the subsequent addition of dominant block mining further improves the convergence and speed of the algorithm, while the chaotic perturbation gives it a final push in terms of convergence.

The statistical significance of these results is confirmed by the one-way ANOVA tests applied to both convergence time, iterations, and average fitness metrics. The p-values, shown in Tables 4 and 5, are less than 0.001, indicating that the differences observed between each algorithm are statistically significant. The full NIHGA demonstrates not only superior performance in terms of average fitness values and convergence speed, but also displays this enhanced performance consistently across multiple runs, as indicated by its lower standard deviations compared to other algorithms, although all standard deviation results are low. This demonstrates the robustness of the proposed method.

Sensitivity analysis

To thoroughly evaluate the robustness and applicability of the proposed NIHGA, we performed sensitivity analysis to assess the algorithm’s performance under different parameter settings. We focused on parameters that are expected to have a significant impact on the algorithm’s behavior:

①Population Size (N): We tested population sizes of 30, 50, 70, and 100 to evaluate the influence of population diversity on the algorithm’s performance.

②Iteration Count (): We varied the maximum number of iterations () to 100, 200, and 300, in order to check the impact of convergence time on algorithm.

③Chaotic Perturbation Parameter (): We tested values of 2, 3, and 4 for the chaotic perturbation parameter () to understand its influence on global and local search behavior and convergence.

④Dominant Block Extraction Count (): We also tested using the top 2, 4, 6 and 8 chromosomes to extract dominant blocks, in order to see how this parameter affects the overall algorithm efficiency.

For each combination of parameters, we performed 30 independent runs and recorded the average fitness, the average iteration count and the average computation time to analyze the results.

The sensitivity analysis results, shown in Tables 6 and 7, reveal several important insights.

Table 6.

Sensitivity analysis on population size and iteration count.

Population size (N)	Iteration count (G)	Avg. Fitness	Std. Dev fitness	Avg iterations	Avg time (seconds)
30	200	1.82	0.02	58	4.90
50	200	1.88	0.01	50	5.31
70	200	1.87	0.01	48	5.95
100	200	1.86	0.01	45	6.30
50	100	1.80	0.02	75	4.80
50	200	1.88	0.01	50	5.31
50	300	1.89	0.01	50	6.50

Table 7.

Sensitivity analysis on perturbation and dominant block extraction count.

Perturbation (m)	Dominant block extraction Count (Nc)	Avg. Fitness	Std. dev fitness	Avg iterations	Avg time (seconds)
2	4	1.86	0.01	52	5.10
3	4	1.88	0.01	50	5.31
4	4	1.87	0.01	50	5.40
3	2	1.82	0.02	57	5.80
3	4	1.88	0.01	50	5.31
3	6	1.86	0.02	48	5.20
3	8	1.87	0.02	47	5.10

①Population Size: Increasing the population size from 30 to 50 improved results, as it is important for a population of sufficient size in order for the algorithm to perform well. Further increasing it to 70 and 100 did not lead to much improvement in the results as the algorithm converges quickly.

②Iteration Count: Increasing the iteration count from 100 to 200 did improve results, and as expected, more iterations leads to improved results. However, increasing iteration count to 300 does not lead to significant improvements, which shows that algorithm converges quickly.

③Chaotic Perturbation Parameter: Based on our experiments, a chaotic perturbation parameter of 3 provides better results compared to a value of 2 or 4. This shows how this parameter affects the convergence.

④Dominant Block Extraction Count: A count of 4 shows better results, when using the top 2, 6 and 8 chromosomes, the results were slightly worse. This shows that using the appropriate top chromosomes to extract blocks is an important consideration for the algorithm.

The results highlight the stability of the NIHGA across the studied parameter ranges. The optimal parameter settings provided the best average results, and the sensitivity to changes are relatively smooth, indicating that the algorithm is not highly susceptible to small variations in these parameter values. This provides further evidence of the robustness of our proposed NIHGA.

Conclusion

According to the characteristics of the layout problem in RMS, a new improved hybrid genetic algorithm is proposed in this paper. In the initialization process, the improved chaotic genetic algorithm is used to form an initial population with high quality. In the calculation process of the algorithm, a data mining method based on association rules is designed to find the dominant genes on the dominant chromosome. The dominant genes are connected to form dominant blocks, which are then extracted to generate artificial solutions, and to obtain more diverse and effective information. Three crossover modes are designed in the crossover process of the algorithm, and they effectively improve the algorithm’s crossover performance and guide the development of the dominant block. Finally, a small adaptive chaotic perturbation is applied to the optimal solution. The validity of the algorithm is verified by a standard scenario in the experimental process. The feasibility and effectiveness of the proposed algorithm in solving the RMS layout problem are demonstrated by analyzing the accuracy, dispersion and convergence of the calculation results. However, this paper has not extensively addressed other types of constraints, such as production line balancing during system operation and the processing time of parts. Future research could delve deeper into these aspects to develop a more realistic layout model in RMS that incorporates multiple constraint mechanisms.

Author contributions

Author Contributions: Conceptualization, X.W.; Data curation, X.W. and H.J.; Formal analysis, X.W.; Funding acquisition, H.J.; Investigation, X.W. and J.S.; Methodology, X.W. and H.J.; Project administration, J.S. and H.J.; Resources, X.W. and J.S.; Software, X.W.; Supervision, H.J.; Validation, X.W. and J.S.; Visualization, X.W. and J.S.; Writing – original draft, X.W.; Writing – review & editing, X.W., J.S. and H.J. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data availability

The data presented in this study are available on request from the corresponding author (The data are not publicly available due to privacy or ethical restrictions).

Declarations

Competing interests

The authors declare no competing interests.

Conflicts of Interest

The authors declare no conflicts of interest.