A cell layout method and system based on hybrid algorithms

Abstract

This invention provides a cell layout method and system based on a hybrid algorithm, belonging to the field of cell manufacturing. The method includes: constructing a cell layout model with minimizing total transportation cost as the objective function; constructing and repairing N initial individuals using a three-segment encoding method; selecting a preset number of individuals for crossover, mutation, and re-insertion operations using a roulette wheel algorithm; merging the repaired first intermediate population with the initial population, and selecting the N individuals with the best fitness values ​​in the merged population as the second intermediate population; performing gene segment exchange, reversal, or insertion operations on individuals in the second intermediate population using a roulette wheel algorithm; comparing the fitness values ​​of individuals in the third intermediate population with those in the corresponding individuals in the second intermediate population, and updating the second intermediate population based on the acceptance probability of the simulated annealing algorithm; and selecting the individual corresponding to the final best fitness value as the optimal cell layout scheme. This invention achieves integrated optimization of cell construction and layout.

Description

Technical Field

[0001] This invention belongs to the field of cell manufacturing, and more specifically, relates to a cell layout method and system based on a hybrid algorithm. Background Technology

[0002] Currently, customer demands for products are gradually shifting from low-variety, high-volume, and rigid production to high-variety, low-volume, and flexible production, posing a significant challenge to traditional production models in the manufacturing industry. Therefore, cellular manufacturing, capable of rapidly responding to the demands of a diverse, low-volume, and personalized market, has emerged. Implementing cellular manufacturing requires addressing issues such as cell construction, cell layout, and cell scheduling; these three aspects are both independent and interdependent.

[0003] Cellular manufacturing can reduce costs and improve efficiency. Currently, most scholars focus on studying three different problems separately. However, the cellular construction scheme with the minimum number of inter-cell movements is not always consistent with the cellular construction scheme with the minimum part handling cost. First, the essence of cellular manufacturing is cost reduction; simply reducing the number of inter-cell movements does not reasonably reflect its essence. Second, due to the lack of layout data in cellular construction, solving the construction and layout problems sequentially leads to the cellular layout result depending on the quality of the cellular construction scheme, which cannot guarantee the effectiveness of the final scheme. Third, actual production factors are usually not considered, or only some factors are considered, such as the part processing sequence, the actual size of the machine, and the machine placement direction. The absence of these production factors in the problem limits the application of cellular manufacturing in most practical cases. Fourth, it assumes that machines are of equal size and uses pre-specified locations for machine placement; the distances between machines and between units are not considered in the inter-cell and intra-cell layouts. These assumptions directly lead to inaccurate calculations of part handling costs, resulting in unreasonable layout designs. Therefore, studying the integrated optimization problem of cellular construction and layout, and considering many actual production factors, highlights its necessity and has significant theoretical and practical implications.

[0004] Genetic algorithms are a simple and efficient global search method that automatically acquires and accumulates knowledge about the search space during the search process and adaptively controls the search process to find the optimal solution. They are widely used to solve various optimization problems. However, genetic algorithms have poor local search capabilities, making them relatively time-consuming and inefficient in the later stages of evolution. In practical applications, genetic algorithms are prone to premature convergence. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the present invention aims to provide a cell layout method and system based on a hybrid algorithm. This addresses the problem that existing technologies conduct separate research on cell construction and cell layout, resulting in cell construction schemes with the minimum number of cross-cell movements not always aligning with cell construction schemes with the minimum part handling costs, thus rendering cell layout designs impractical.

[0006] To achieve the above objectives, on the one hand, the present invention provides a cell layout method based on a hybrid algorithm, comprising the following steps:

[0007] S1: Assuming that all machines are projected onto the plane as rectangles, with the objective function of minimizing the total transportation cost, and with the constraints that each manufacturing unit contains at least one machine, that machines within the same manufacturing unit do not overlap in the horizontal and vertical directions, and that the horizontal coordinate of the centroid of each machine is within the width range of the corresponding manufacturing unit, a unit layout model is constructed; where the total transportation cost is the sum of the intra-unit part transportation cost and the inter-unit part transportation cost for completing all part processing.

[0008] S2: Use a three-segment coding method to construct and repair N initial individuals. Each individual contains three gene segments. The element values ​​of the three gene segments, arranged in gene order, represent the manufacturing unit number, machine sorting priority value, and machine placement direction of each machine, respectively.

[0009] S3: Based on the fact that the fitness value of the initial population is proportional to the fitness probability, a preset number of individuals are selected using the roulette wheel method to perform crossover, mutation, and re-insertion operations to obtain the first intermediate population.

[0010] S4: After repairing the individuals in the first intermediate population, merge the repaired first intermediate population with the initial population, and select the N individuals with the best fitness values ​​in the merged population as the second intermediate population.

[0011] S5: Use a roulette wheel to exchange, reverse, or insert gene segments of individuals in the second intermediate population to generate a third intermediate population;

[0012] S6: Compare the fitness values ​​of individuals in the third intermediate population with those of individuals in the corresponding second intermediate population. Combine the acceptance probability of the simulated annealing algorithm to update the second population and use it as the initial population for the next iteration. Proceed to S3. Continue until the number of iterations reaches the preset number of iterations. Decode the individual corresponding to the best fitness value in the second intermediate population of the last iteration to obtain the final cell layout method.

[0013] More preferably, the gene segment whose element value represents the manufacturing unit code corresponding to each machine is taken as the first gene segment of the individual; the gene segment whose element value represents the priority value of the machine sorting is taken as the second gene segment of the individual; and the gene segment whose element value represents the machine placement direction is taken as the third gene segment of the individual.

[0014] Individual repair methods include the following steps:

[0015] Calculate the number of machines allocated to each manufacturing unit within an individual;

[0016] Determine if the number of machines in each manufacturing unit is greater than or equal to 1. If so, retain the current individual; otherwise, re-initialize the first gene segment of the individual randomly.

[0017] More preferably, S3 includes the following steps:

[0018] Based on the principle that the initial population fitness value is proportional to the fitness probability, a roulette wheel is constructed.

[0019] Randomly generate real numbers between 0 and 1, and spin a roulette wheel N*P. s Each time, the individual corresponding to the real number falling into the sector region is selected as the selected individual; where P s The selection probability of the selection operation;

[0020] Grouping two adjacent individuals together, a crossover point is randomly generated in each group corresponding to the first and second gene segments. Genes in the gene segments before the crossover point are crossed; and several genes are randomly selected in the second gene segment for crossover.

[0021] After the crossover, two mutation points are generated in each of the three gene segments corresponding to the individual, and the gene segments between the two mutation points are reversed.

[0022] The crossover mutation generates N*P s Insert the NN*P with the best fitness value from the initial population into a population of individuals. s Individuals generate the first intermediate population.

[0023] More preferably, S5 specifically includes the following steps:

[0024] Construct a roulette wheel by assigning different probability values ​​to the swap, reverse, and insertion methods;

[0025] The exchange method involves randomly generating two intersection points in an individual and exchanging the element values ​​of the gene loci at the two intersection points; the reverse order method involves randomly generating two points in an individual and exchanging all element values ​​of the gene loci between the two points in reverse order; the insertion method involves randomly generating two points a and b in an individual. If a < b, the element values ​​corresponding to gene loci from a+1 to b are inserted before gene loci a; otherwise, the element values ​​of gene loci from b+1 to a-1 are inserted after gene loci a.

[0026] By rotating the roulette wheel, the genes in the first, second, or third gene segments of the second intermediate population can be manipulated through exchange, reversal, or insertion to generate the third intermediate population.

[0027] More preferably, the acceptance probability of the simulated annealing algorithm is:

[0028]

[0029] Where p is the acceptance probability of the simulated annealing algorithm; F(l') is the fitness value of an individual in the third intermediate population; F(l) is the fitness value of an individual in the second intermediate population; K is the annealing rate; and T is the annealing temperature.

[0030] If the fitness value of an individual in the third intermediate population is greater than that of an individual in the second intermediate population, then the individual in the third intermediate population is retained; otherwise, if the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, the individual in the third intermediate population is accepted; if the random number in the simulated annealing algorithm is greater than the acceptance probability, the individual in the second population is retained.

[0031] On the other hand, the present invention provides a cell layout system based on a hybrid algorithm, comprising:

[0032] The module for building the cell layout model is used to construct the cell layout model, assuming that all machines are projected onto the plane as rectangles. The objective function is to minimize the total transportation cost. The constraints are that each manufacturing cell contains at least one machine, machines within the same manufacturing cell do not overlap in the horizontal and vertical directions, and the horizontal coordinate of the centroid of each machine is within the width range of the corresponding manufacturing cell. The total transportation cost is the sum of the intra-cell part transportation cost and the inter-cell part transportation cost for completing all part processing.

[0033] The initial individual construction module is used to construct and repair N initial individuals using a three-segment coding method. Each individual contains three gene segments. The element values ​​of the three gene segments, arranged in gene order, represent the manufacturing unit number corresponding to each machine, the priority value of machine sorting, and the machine placement direction, respectively.

[0034] The module for constructing the first intermediate population is used to select a preset number of individuals for crossover, mutation, and re-insertion operations based on the fitness value of the initial population being proportional to the fitness probability, using the roulette wheel method to obtain the first intermediate population.

[0035] The second intermediate population construction module is used to repair individuals in the first intermediate population, merge the repaired first intermediate population with the initial population, and select the N individuals with the best fitness values ​​in the merged population as the second intermediate population.

[0036] The module for generating the third intermediate population is used to exchange, reverse, or insert gene segments of individuals in the second intermediate population using a roulette wheel approach to generate the third intermediate population.

[0037] The cell layout acquisition module compares the fitness values ​​of individuals in the third intermediate population with those of individuals in the corresponding second intermediate population. It then updates the second intermediate population by combining the acceptance probability of the simulated annealing algorithm and decodes the individual with the best fitness value in the second intermediate population from the last iteration as the final cell layout method.

[0038] More preferably, the gene segment whose element value represents the manufacturing unit code corresponding to each machine is taken as the first gene segment of the individual; the gene segment whose element value represents the priority value of the machine sorting is taken as the second gene segment of the individual; and the gene segment whose element value represents the machine placement direction is taken as the third gene segment of the individual.

[0039] Individual repair methods include the following steps:

[0040] Calculate the number of machines allocated to each manufacturing unit within an individual;

[0041] Determine if the number of machines in each manufacturing unit is greater than or equal to 1. If so, retain the current individual; otherwise, re-initialize the first gene segment of the individual randomly.

[0042] More preferably, the building block of the first intermediate population includes:

[0043] The first roulette wheel construction unit is used to construct a roulette wheel based on the fact that the initial population fitness value is proportional to the fitness probability.

[0044] Individual selection unit, used to randomly generate real numbers from 0 to 1, rotating a roulette wheel N*P s Each time, the individual corresponding to the real number falling into the sector region is selected as the selected individual; where P s The selection probability of the selection operation;

[0045] The gene crossover processing unit is used to randomly generate a crossover point in the first and second gene segments corresponding to each group of two selected adjacent individuals, crossover the genes in the gene segments corresponding to the crossover point, and randomly select several genes in the second gene segment for crossover.

[0046] The gene mutation processing unit is used to generate two mutation points in each of the three gene segments corresponding to the crossover individual, and to reverse the gene fragment between the two mutation points.

[0047] Gene insertion processing unit, used to process N*P generated by crossover mutation. s Insert the NN*P with the best fitness value from the initial population into a population of individuals. s Individuals generate the first intermediate population.

[0048] More preferably, the module for generating the third intermediate population includes:

[0049] The second roulette wheel construction unit is used to construct a roulette wheel by assigning different probability values ​​to the exchange, reverse order, and insertion methods. The exchange method involves randomly generating two intersection points in the individual and exchanging the element values ​​of the gene positions at these two intersection points. The reverse order method involves randomly generating two points in the individual and exchanging all element values ​​of the gene positions between these two points in reverse order. The insertion method involves randomly generating two points a and b in the individual; if a < b, the element values ​​corresponding to gene positions a+1 to b are inserted before gene position a; otherwise, the element values ​​of gene positions b+1 to a-1 are inserted after gene position a.

[0050] The third intermediate population generation unit is used to manipulate genes in the first, second, or third gene segments of the second intermediate population by rotating a roulette wheel and selecting exchange, reversal, or insertion methods to generate the third intermediate population.

[0051] More preferably, the acceptance probability of the simulated annealing algorithm is:

[0052]

[0053] Where p is the acceptance probability of the simulated annealing algorithm; F(l') is the fitness value of an individual in the third intermediate population; F(l) is the fitness value of an individual in the second intermediate population; K is the annealing rate; and T is the annealing temperature.

[0054] If the fitness value of an individual in the third intermediate population is greater than that of an individual in the second intermediate population, then the individual in the third intermediate population is retained; otherwise, if the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, the individual in the third intermediate population is accepted; if the random number in the simulated annealing algorithm is greater than the acceptance probability, the individual in the second population is retained.

[0055] In summary, the technical solutions conceived by this invention have the following beneficial effects compared with the prior art:

[0056] This invention assumes that all machines projected onto the plane are rectangles. Using minimum total handling cost as the objective function, and constrained by the following conditions: each manufacturing unit must contain at least one machine; machines within the same manufacturing unit must not overlap in the horizontal and vertical directions; and the x-coordinate of the centroid of each machine must be within the width range of the corresponding manufacturing unit, a unit layout model is constructed, combining the minimum handling cost of parts with the unit layout. In terms of unit layout integration optimization, a three-segment coding method is used to construct individuals. Each individual contains three gene segments, and the elements of the three gene segments, arranged in gene order, represent the manufacturing unit number and machine sequence corresponding to each machine, respectively. The priority value and machine placement direction indicate that each individual represents a unit layout information. A genetic algorithm is used to update the population through crossover, mutation, and re-insertion of individuals, and the fitness value in the population is calculated. The fitness value is the reciprocal of the minimum transportation cost. It can be seen that by calculating the fitness value, the unit layout is combined with the transportation cost. On this basis, in order to obtain the optimal fitness value and improve the global search capability and local search capability, this invention adopts a genetic algorithm that integrates simulated annealing algorithm to select the optimal unit layout method from multiple unit layout methods, thereby realizing the integrated optimization of unit construction and unit layout.

[0057] In order to improve the accuracy of layout design between and within units, this invention calculates the part handling cost based on the actual position of the machine within the unit and design factors including the actual size of the machine and the distance of the passageway. That is, the total handling cost is the sum of the part handling cost within the unit and the part handling cost between units to complete all part processing. The total handling cost is calculated based on the total distance moved by the part to complete all processing steps and the unit part handling cost within and between units.

[0058] This invention performs repair after the initial individual is constructed, and also applies a repair method to individuals in the first intermediate population. This is because some individuals may be decoded in cases where there is no machine in a manufacturing unit. Therefore, the repair method avoids this situation and can fully guarantee the effectiveness of the hybrid algorithm iteration. Attached Figure Description

[0059] Figure 1 This is a schematic diagram of the unit construction and layout integration optimization results provided in the embodiments of the present invention;

[0060] Figure 2 This is a flowchart of the hybrid algorithm provided in an embodiment of the present invention;

[0061] Figure 3 This is a schematic diagram of the three-segment encoding method provided in an embodiment of the present invention;

[0062] Figure 4 This is a schematic diagram of cross-operation provided in an embodiment of the present invention;

[0063] Figure 5 This is a schematic diagram of the mutation operation provided in an embodiment of the present invention;

[0064] Figure 6 The various algorithms provided in the embodiments of this invention are ARDI in 100 cases. k value;

[0065] Figure 7 This is the convergence graph of the genetic algorithm in Example 320251 provided in this embodiment of the invention;

[0066] Figure 8 This is the convergence graph of the particle swarm optimization algorithm in Example 320251 provided in this embodiment of the invention;

[0067] Figure 9 This is the convergence graph of the Seagull optimization algorithm in Case 320251 provided in this embodiment of the invention;

[0068] Figure 10 This is the convergence diagram of the hybrid algorithm in Case 320251 provided in the embodiment of the present invention. Detailed Implementation

[0069] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0070] This invention provides a product manufacturing unit layout method and system based on a hybrid algorithm. The overall concept is as follows: The hybrid algorithm is a genetic algorithm that integrates simulated annealing and establishes a mathematical programming model for the integrated optimization problem of unit construction and layout, encompassing all problem characteristics mentioned in the background technology. The problem characteristics are analyzed, and a three-segment encoding and decoding method is designed to address the priority order of machine allocation, machine sorting, and machine placement orientation. To improve the diversity of solutions, a segmented crossover and mutation operation is designed. To ensure the effectiveness of algorithm iteration, an initialization screening method and a repair method that satisfy problem constraints are designed. Furthermore, to improve the accuracy of layout design between and within units, the part handling cost is calculated based on the actual location of machines within the unit and design factors including actual machine dimensions and channel distances.

[0071] The problem description and mathematical description of the product manufacturing unit layout method and system based on a hybrid algorithm provided by this invention are as follows:

[0072] This invention addresses the problem of cell construction and layout integration optimization for a part with process constraints and machines within a manufacturing cell that can be arranged in both horizontal and vertical orientations. The cell layout refers to the order and orientation of the machines, taking into account both the actual size of the machines and the spacing between them. Machines within each manufacturing cell are arranged horizontally or vertically from the left side of the cell according to priority values. Furthermore, considering the distance between manufacturing cells, the cells are arranged from bottom to top. This linear cell layout method is as follows: Figure 1 As shown; Figure 1 The diagram shows the integrated optimization results of the unit construction and layout of 2 manufacturing units and 5 machines. The transportation of parts within and between units is completed by AGV carts, that is, the AGV carts transport the parts to the required machines for processing in sequence until all processing steps of the part are completed.

[0073] On one hand, the cell layout method based on a hybrid algorithm provided by the present invention includes the following steps:

[0074] S1: In order to establish a mathematical model, this invention makes the following assumptions:

[0075] A workshop has M machines (j = 1, 2, ..., M) and C manufacturing cells, processing P types of parts (i = 1, 2, ..., P), with significant differences between the different parts. The cell construction and layout integration optimization problem involves dividing the M machines into C manufacturing cells while simultaneously forming the machine layout within each cell. To facilitate the establishment of the cell layout model, the following assumptions are made:

[0076] (1) Once the processing path of a part is determined, the process flow of the part, i.e. the processing machine selected for each process and the processing sequence of all processes of the part, will also be determined.

[0077] (2) The unit distance part handling cost within and between units, the machine spacing within and between units, the aisle distance between units, the actual size of the machine, the number of manufacturing units, and the part process flow information are all known.

[0078] (3) The dimensions of each machine are not necessarily equal, and all of them are rectangles when projected onto the plane;

[0079] (4) The initial position of the part is the center position of the processing machine required for the first process of the selected processing path.

[0080] S2: Establishment of the cell layout model:

[0081] This invention establishes a unit layout model with the goal of minimizing total transportation cost; wherein, the total transportation cost is the sum of the intra-unit part transportation cost and the inter-unit part transportation cost for completing all part processing, and the total transportation cost is calculated based on the total distance moved by the part to complete all processing steps and the unit part transportation cost within and between units;

[0082] The symbols are described in Table 1;

[0083] Table 1

[0084]

[0085] The objective function is:

[0086] PC represents the total cost of moving parts within and between units for all processes of all parts. The smaller the PC value, the more effective the established unit construction and layout integration optimization scheme is.

[0087] PC = PC A +PC E (1)

[0088] Among them, PC A Total cost of handling within all process units to complete the processing of all parts; PC E The total cost of inter-unit transportation for all processes involved in processing all parts;

[0089]

[0090] Where, x j For the j-th machine; x j′ For the j'th machine;

[0091]

[0092] Among them, y k y is the ordinate of the center of unit k; k′ The ordinate of the center of unit k';

[0093] Therefore, the objective function of this invention is as follows:

[0094]

[0095] The constraints are:

[0096]

[0097]

[0098] A j =L j +(W j -L j)d j (7)

[0099] B j =W j +(L j -W j )d j (8)

[0100]

[0101]

[0102]

[0103]

[0104]

[0105] Formula (5) states that any machine must be assigned to a manufacturing cell and can only be assigned once; Formula (6) states that each manufacturing cell contains at least one machine; in Formulas (7) and (8), the width A of each machine is measured based on the machine's orientation. j and length B j In formulas (9) and (10), the width W of manufacturing cell k is calculated based on the machine dimensions allocated to manufacturing cell k. k and length L k Formula (11) indicates that machines belonging to the same manufacturing unit will not overlap in the horizontal axis (x-axis) direction; Formula (12) ensures that each manufacturing unit will not overlap in the vertical axis (y-axis) direction; Formula (13) ensures that the horizontal coordinate of the centroid of each machine belonging to the same manufacturing unit is in the interval [0~W]. k ]Inside;

[0106] S3: Cell layout method based on hybrid algorithm:

[0107] Genetic algorithms are a simple and efficient global search method that automatically acquires and accumulates knowledge about the search space during the search process and adaptively controls the search process to find the optimal solution. They are widely used to solve various optimization problems. However, genetic algorithms have poor local search capabilities, making them relatively time-consuming and inefficient in the later stages of evolution. In practical applications, genetic algorithms are prone to premature convergence.

[0108] The cell construction and layout integration optimization problem in this invention is a typical multi-constraint discrete optimization problem. Since it is an NP-hard problem, a genetic algorithm incorporating simulated annealing is proposed for convenient and rapid solution; this is called the hybrid algorithm. The characteristics of the cell construction and layout integration optimization problem are analyzed, and a three-segment encoding method is designed, considering the priority order of machine allocation, machine sorting, and machine placement orientation. To improve solution diversity, a segmented crossover and mutation operation is designed. To ensure the effectiveness of algorithm iteration, initialization screening and repair methods that satisfy problem constraints are designed. To enhance the local exploration capability of the method, a local search method incorporating simulated annealing is designed.

[0109] Hybrid algorithms are used to solve the cell construction and layout integration optimization problem:

[0110] When using a hybrid algorithm to solve the cell construction and layout integration optimization problem, as long as the encoding is appropriate, the problem parameters can directly participate in the genetic operation, thus establishing a certain correspondence between the actual description of the problem and the representation of the layout scheme, i.e., encoding and decoding. The flowchart of the designed hybrid algorithm based on simulated annealing is as follows: Figure 2 As shown;

[0111] S3.1 Three-segment coding method

[0112] When using genetic algorithms to solve the unit construction and layout integration optimization problem, the possible solutions must first be encoded as chromosomes. This is crucial for solving the problem and forms the basis of the genetic algorithm's operation. As the unit layout model shows, the machine allocation sequence, machine sorting priority value, and machine placement direction are key to solving the unit construction and layout integration optimization problem. Based on this, a three-segment encoding method based on the machine allocation sequence, machine sorting priority value, and machine placement direction is designed. Each individual contains M*3 genes. The first M gene positions represent machine allocation, with each gene position containing a positive integer from 1 to C, representing the manufacturing unit number corresponding to the machine. The first M+1: M*2 gene positions represent the machine sorting priority value, with each gene position containing a positive integer from 1 to M, and each value representing the sorting priority value corresponding to the machine. The last M gene positions represent the machine placement direction, with each gene position containing an integer from 0 to 1, representing the placement direction corresponding to the machine.

[0113] like Figure 3 As shown, the element value of the first gene of an individual is 2, indicating that machine 1 is assigned to manufacturing unit 2; the element value of the (M+1)th gene is 2, indicating that the priority value of machine 1 in the sorting is 2; the element value of the (M*2+1)th gene is 0, indicating that machine 1 is placed horizontally (i.e., the longer side of the machine is parallel to the horizontal axis), and so on.

[0114] S3.2 Population Initialization

[0115] The population is initialized randomly, and each individual is encoded using a three-segment encoding method. Since each gene position in the machine-assigned encoding segment is a randomly generated integer [1 to C], this initialization method can lead to one or more integers not appearing in the first M gene positions of the machine-assigned encoding segment, meaning that one or more manufacturing units have not received machine assignment. Generally, individuals in this situation are considered invalid solutions, which are detrimental to guiding the evolutionary direction. Therefore, an initialization screening method (repair method) that guarantees feasible solutions is needed. The specific steps are as follows:

[0116] Step 1: Individual x l , l = 1;

[0117] Step 2: Calculate individual x l The number of machines allocated to each manufacturing unit 1, 2, ..., C;

[0118] Step 3: Determine if the number of machines in each manufacturing unit is greater than or equal to 1, i.e., constraint condition (6) must be met; if the condition is met, proceed to step 4; otherwise, re-initialize individual x randomly. l The values ​​of the first M gene loci are then used to proceed to step 2;

[0119] Step 4: Let l = l + 1. If l is less than or equal to the population size N, go to step 2; otherwise, terminate the screening.

[0120] S3.3 Decoding Design

[0121] The decoding process is responsible for transforming individuals in the encoding space into feature solutions in the solution space. First, the manufacturing unit number of each machine can be determined from the first segment of the encoding result, which means that the number of machines and machine numbers contained in each unit can be known. Second, the arrangement order of the machines in each manufacturing unit can be determined from the second segment of the encoding result. Finally, the placement direction of each machine can be determined from the third segment of the encoding result. This completes the decoding of unit construction, machine layout within the unit, and layout between units.

[0122] S3.4 Calculate fitness value

[0123] The fitness value is calculated using the fitness function. The higher the fitness value of an individual, the closer it is to the target solution. The objective function for solving the cell construction and layout integration optimization problem is to minimize the total transportation cost. The total transportation cost objective is transformed into a fitness value for evaluation, which means that the higher the fitness value of the chromosome, the lower the total transportation cost.

[0124] Individual x l The fitness function is calculated using the following formula:

[0125]

[0126] Where the denominator is individual x l The total transportation cost, F(l), is the cost per individual x. l fitness value;

[0127] Operation of S3.5 Genetic Algorithm

[0128] S3.5.1 Selection Operation

[0129] The selection operation is performed using the roulette wheel method, with a selection probability of P. s Assuming N individuals are generated, first calculate the fitness value F(l) of each individual, and then calculate the probability of each fitness value based on the fitness value.

[0130] The formula for calculating the probability of a good value is as follows:

[0131]

[0132] The roulette wheel method is inspired by the random stopping of a spinning roulette wheel on a sector. The size of the sector is proportional to the probability of finding a suitable match. A real number P ∈ (0~1) is randomly generated. If P falls on the position pointed to by a sector, the individual corresponding to that sector is selected to be passed on to the next generation. The roulette wheel rotates N*P. s Next, N*P will be chosen. s Each individual is used as the population for subsequent crossover and mutation operations;

[0133] S3.5.2 Cross Operation

[0134] A crossover point is randomly generated between the first and third gene segments, and the crossover occurs in the gene segment preceding the crossover point in the parent individual; for example... Figure 4 As shown, the intersection points are 3 and 15, that is, the machine-assigned coding segment swaps the genes in the first 3 positions, and the machine-placed direction coding segment swaps the genes in the first 3 positions; the priority value encoding of the second segment of machine sorting is to randomly select three numbers from [1-M], for example, 5, 4, 3, then the order of 5, 4, 3 in the parent generation individual 1 is swapped with 3, 4, 5 in the parent generation individual 2 to produce offspring; the first segment of encoding may cause the offspring individuals to have illegal solutions, that is, there may be cases that do not satisfy constraint (6), such as Figure 4 There is no machine allocation in manufacturing unit 1 of the offspring individual 1 shown;

[0135] S3.5.3 Mutation Operation

[0136] Mutation operations expand the search space of the algorithm by adding perturbations. For an individual's three-segment encoding, a reverse mutation operation is used. This involves generating two mutation points from three segments: one randomly assigned to the machine, one based on the machine's priority order, and one representing the machine's orientation. The gene segments between these two mutation points are then reversed. This type of mutation operation will not produce illegal solutions. Figure 5 As shown, machine-assigned mutation points 2 and 3 are randomly generated, and machine-sorted priority value mutation points 8 and 10 are generated. Genes between 8 and 10 are reversed and mutated. Machine placement direction mutation points 15 and 16 are generated, and gene segments between 2 and 3, 8 and 10, and 15 and 16 are reversed.

[0137] S3.5.3 Reinsertion Operation

[0138] Since the population after the selection operation contains only N*P s There are 10 individuals, therefore the NN*P with the best fitness value in the current population will be selected. s One individual is inserted into the new population after crossover mutation, so that the population has N individuals;

[0139] The method of combining roulette wheel selection with optimal individual preservation inherits the advantages of roulette wheel selection, which ensures the selection of the best individuals without sacrificing population diversity, allowing the population to evolve in the optimal direction. On the other hand, it inherits the advantages of individual preservation, which ensures that excellent individuals can be inherited and that excellent genes are not destroyed or lost. Therefore, combining the two selection methods is of great significance to the computational accuracy and global convergence of genetic algorithms.

[0140] S3.6 Repair Method:

[0141] Crossover operation may result in the first segment of the new individual's code failing to meet constraint (6), that is, there may be a situation where no machine is assigned in a certain manufacturing unit; therefore, after the crossover operation is completed, a repair method that is consistent with the initial screening method needs to be adopted to complete the repair of illegal solutions;

[0142] For the mutated individuals, it is not difficult to find that the mutation process does not produce illegal solutions, so no repair operation is required;

[0143] S3.7 Population Update

[0144] First, calculate the fitness value of each individual in the new population. Then, merge the current population with the new population generated after selection, crossover, mutation, re-insertion, and repair. Finally, select the N individuals with the best fitness values ​​as the population for the simulated annealing local search method.

[0145] S3.8 Simulated Annealing Local Search Method

[0146] Simulated annealing introduces randomness into its search process. When iteratively updating feasible solutions, it accepts a solution that is worse than the current one with a certain probability. Therefore, it may escape the local optimum and reach the global optimum. The specific operation is as follows: taking an individual x in the current population... l As the research object, firstly, one of the following segments is randomly selected: machine-assigned coding segment, machine sorting priority value coding segment, or machine placement direction coding segment. Secondly, one of the following three neighborhood functions is randomly selected using a roulette wheel method to generate a new solution x'.

[0147] Exchange: Randomly generate two crossover points and exchange the element values ​​of the gene positions at the two crossover points;

[0148] Reverse order: Randomly generate two points and swap all element values ​​of the gene loci between the two points in reverse order;

[0149] Insertion: Randomly generate two points a and b. If a < b, insert the elements at positions a+1 to b before position a; otherwise, insert the elements at positions b+1 to a-1 after position a.

[0150] Finally, the fitness of the new solution x' is evaluated, and the solution is updated using the acceptance probability p of the Metropolis criterion of the simulated annealing algorithm.

[0151]

[0152] In the algorithm iteration, T is first set to equal the initial temperature, and then the temperature is updated according to T = KT during the iteration, where K is the annealing rate. If the fitness value of the new solution is large, the new solution is retained; otherwise, if the random number in the simulated annealing algorithm is less than or equal to the acceptance probability p, the poor new solution is accepted; if the random number in the simulated annealing algorithm is greater than the acceptance probability p, the corresponding individual in the population in S3.7 is retained; proceed to S3.5.

[0153] In the hybrid algorithm, the number of iterations is set as the algorithm termination condition. When the current number of iterations exceeds the maximum number of iterations in the parameter settings, the iteration ends, the calculation terminates, and the optimal solution is output.

[0154] On the other hand, the present invention provides a cell layout system based on a hybrid algorithm, comprising:

[0155] The module for building the cell layout model is used to construct the cell layout model, assuming that all machines are projected onto the plane as rectangles. The objective function is to minimize the total transportation cost. The constraints are that each manufacturing cell contains at least one machine, machines within the same manufacturing cell do not overlap in the horizontal and vertical directions, and the horizontal coordinate of the centroid of each machine is within the width range of the corresponding manufacturing cell. The total transportation cost is the sum of the intra-cell part transportation cost and the inter-cell part transportation cost for completing all part processing.

[0156] The initial individual construction module is used to construct and repair N initial individuals using a three-segment coding method. Each individual contains three gene segments. The element values ​​of the three gene segments, arranged in gene order, represent the manufacturing unit number corresponding to each machine, the priority value of machine sorting, and the machine placement direction, respectively.

[0157] The module for constructing the first intermediate population is used to select a preset number of individuals for crossover, mutation, and re-insertion operations based on the fitness value of the initial population being proportional to the fitness probability, using the roulette wheel method to obtain the first intermediate population.

[0158] The second intermediate population construction module is used to repair individuals in the first intermediate population, merge the repaired first intermediate population with the initial population, and select the N individuals with the best fitness values ​​in the merged population as the second intermediate population.

[0159] The module for generating the third intermediate population is used to exchange, reverse, or insert gene segments of individuals in the second intermediate population using a roulette wheel approach to generate the third intermediate population.

[0160] The cell layout acquisition module compares the fitness values ​​of individuals in the third intermediate population with those of individuals in the corresponding second intermediate population. It then updates the second intermediate population by combining the acceptance probability of the simulated annealing algorithm and decodes the individual with the best fitness value in the second intermediate population from the last iteration as the final cell layout method.

[0161] More preferably, the gene segment whose element value represents the manufacturing unit code corresponding to each machine is taken as the first gene segment of the individual; the gene segment whose element value represents the priority value of the machine sorting is taken as the second gene segment of the individual; and the gene segment whose element value represents the machine placement direction is taken as the third gene segment of the individual.

[0162] Individual repair methods include the following steps:

[0163] Calculate the number of machines allocated to each manufacturing unit within an individual;

[0164] Determine if the number of machines in each manufacturing unit is greater than or equal to 1. If so, retain the current individual; otherwise, re-initialize the first gene segment of the individual randomly.

[0165] More preferably, the building block of the first intermediate population includes:

[0166] The first roulette wheel construction unit is used to construct a roulette wheel based on the fact that the initial population fitness value is proportional to the fitness probability.

[0167] Individual selection unit, used to randomly generate real numbers from 0 to 1, rotating a roulette wheel N*Ps Each time, the individual corresponding to the real number falling into the sector region is selected as the selected individual; where P s The selection probability of the selection operation;

[0168] The gene crossover processing unit is used to randomly generate a crossover point in the first and second gene segments corresponding to each group of two selected adjacent individuals, crossover the genes in the gene segments corresponding to the crossover point, and randomly select several genes in the second gene segment for crossover.

[0169] The gene mutation processing unit is used to generate two mutation points in each of the three gene segments corresponding to the crossover individual, and to reverse the gene fragment between the two mutation points.

[0170] Gene insertion processing unit, used to process N*P generated by crossover mutation. s Insert the NN*P with the best fitness value from the initial population into a population of individuals. s Individuals generate the first intermediate population.

[0171] More preferably, the module for generating the third intermediate population includes:

[0172] The second roulette wheel construction unit is used to construct a roulette wheel by assigning different probability values ​​to the exchange, reverse order, and insertion methods. The exchange method involves randomly generating two intersection points in the individual and exchanging the element values ​​of the gene positions at these two intersection points. The reverse order method involves randomly generating two points in the individual and exchanging all element values ​​of the gene positions between these two points in reverse order. The insertion method involves randomly generating two points a and b in the individual; if a < b, the element values ​​corresponding to gene positions a+1 to b are inserted before gene position a; otherwise, the element values ​​of gene positions b+1 to a-1 are inserted after gene position a.

[0173] The third intermediate population generation unit is used to manipulate genes in the first, second, or third gene segments of the second intermediate population by rotating a roulette wheel and selecting exchange, reversal, or insertion methods to generate the third intermediate population.

[0174] More preferably, the acceptance probability of the simulated annealing algorithm is:

[0175]

[0176] Where p is the acceptance probability of the simulated annealing algorithm; F(l') is the fitness value of an individual in the third intermediate population; F(l) is the fitness value of an individual in the second intermediate population; K is the annealing rate; and T is the annealing temperature.

[0177] If the fitness value of an individual in the third intermediate population is greater than that of an individual in the second intermediate population, then the individual in the third intermediate population is retained; otherwise, if the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, the individual in the third intermediate population is accepted; if the random number in the simulated annealing algorithm is greater than the acceptance probability, the individual in the second population is retained.

[0178] Example

[0179] With the goal of minimizing the total transportation cost, four algorithms were selected to solve the example: Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Seagull Optimization (SOA), and the hybrid algorithm (GA+SA) provided by this invention. All algorithms were programmed using MATLAB 2019a, and the program ran in an environment of Intel(R) Core(TM) i5-9400 CPU@2.90GHz and 16.0GB of memory.

[0180] To fairly compare the performance of different algorithms, the population size and number of iterations for all four algorithms were 100. The material handling cost per unit distance within and between manufacturing units was 1 and 3, respectively. The distance between machines within a manufacturing unit was 1, and the passage distance between manufacturing units was 2. The number of manufacturing units C was selected as [2,3], the number of machines M as [5,8,10,15,20], and the number of parts P as [10,15,20,25,30]. These were permuted and combined. Two cases were randomly generated with the same number of machines and parts. The size of the machines was a randomly generated positive integer. 100 randomly generated cases were tested. Each case was run five times, and the average value was taken as the optimal total handling cost. The experimental results are shown in Table 2.

[0181] Table 2

[0182]

[0183]

[0184] In Table 2, the first digit represents the number of manufacturing units, the second and third digits represent the number of machines, the fourth and fifth digits represent the number of parts, and the last digit represents the case number for that scale. To verify the performance of the proposed hybrid algorithm, in addition to the optimal total handling cost, the performance of different algorithms was compared based on the Average Relative Deviation Index (ARDI), which is calculated as follows:

[0185]

[0186] Where n is the total number of cases; d is the case number; PC dkThe minimum total transport cost obtained by algorithm k for solving case d; MIN(PC) and MAX(PC) are the minimum and worst total transport costs obtained by different algorithms for solving case d; ARDI k Let k be the average relative deviation index; calculate the ARDI for each algorithm in 100 cases. k Value, result as Figure 6 As shown;

[0187] Depend on Figure 6 It can be seen that the hybrid algorithm (GA+SA) performs best in 100 cases, while the Seagull Optimization Algorithm (SOA) performs worst. The hybrid algorithm only adds simulated annealing local search to the genetic algorithm, which also proves the effectiveness of simulated annealing local search. Taking case 320251 as an example, the above four algorithms were used to encode and solve it. The optimal total transport cost obtained by GA is 6193.5, the optimal transport cost obtained by PSO is 8939.5, the optimal transport cost obtained by SOA is 9450, and the optimal transport cost obtained by the hybrid algorithm is 5666. The convergence graph is shown below. Figures 7-10 As shown.

[0188] In summary, compared with the prior art, the present invention has the following advantages:

[0189] This invention assumes that all machines projected onto the plane are rectangles. Using minimum total handling cost as the objective function, and constrained by the following conditions: each manufacturing unit must contain at least one machine; machines within the same manufacturing unit must not overlap in the horizontal and vertical directions; and the x-coordinate of the centroid of each machine must be within the width range of the corresponding manufacturing unit, a unit layout model is constructed, combining the minimum handling cost of parts with the unit layout. In terms of unit layout integration optimization, a three-segment coding method is used to construct individuals. Each individual contains three gene segments, and the elements of the three gene segments, arranged in gene order, represent the manufacturing unit code corresponding to each machine. The number, machine sorting priority value, and machine placement direction indicate that each individual represents a unit layout information. A genetic algorithm is used to update the population through crossover, mutation, and re-insertion of individuals, and the fitness value in the population is calculated. The fitness value is the reciprocal of the minimum transportation cost. It can be seen that by calculating the fitness value, the unit layout is combined with the transportation cost. Based on this, in order to obtain the optimal fitness value and improve the global search capability and local search capability, this invention adopts a genetic algorithm that integrates simulated annealing algorithm to select the optimal unit layout from multiple unit layouts, thereby achieving integrated optimization of the unit layout.

[0190] In order to improve the accuracy of layout design between and within units, this invention calculates the part handling cost based on the actual position of the machine within the unit and design factors including the actual size of the machine and the distance of the passageway. That is, the total handling cost is the sum of the part handling cost within the unit and the part handling cost between units to complete all part processing. The total handling cost is calculated based on the total distance moved by the part to complete all processing steps and the unit part handling cost within and between units.

[0191] This invention performs repair after the initial individual is constructed, and also applies a repair method to individuals in the first intermediate population. This is because some individuals may be decoded in cases where there is no machine in a manufacturing unit. Therefore, the repair method avoids this situation and can fully guarantee the effectiveness of the hybrid algorithm iteration.

[0192] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A cell layout method based on a hybrid algorithm, characterized in that, Includes the following steps: S1: Assuming that all machines are projected onto the plane as rectangles, with the objective function of minimizing the total transportation cost, and with the constraints that each manufacturing unit contains at least one machine, that machines within the same manufacturing unit do not overlap in the horizontal and vertical directions, and that the horizontal coordinate of the centroid of each machine is within the width range of the corresponding manufacturing unit, a unit layout model is constructed; where the total transportation cost is the sum of the intra-unit part transportation cost and the inter-unit part transportation cost for completing all part processing. S2: Use a three-segment coding method to construct and repair N initial individuals. Each individual contains three gene segments. The element values ​​of the three gene segments, arranged in gene order, represent the manufacturing unit number, machine sorting priority value, and machine placement direction of each machine, respectively. S3: Based on the fact that the fitness value of the initial population is proportional to the fitness probability, a preset number of individuals are selected using the roulette wheel method to perform crossover, mutation, and re-insertion operations to obtain the first intermediate population. S4: After repairing the individuals in the first intermediate population, merge the repaired first intermediate population with the initial population, and select the N individuals with the best fitness values ​​in the merged population as the second intermediate population. S5: Use a roulette wheel to exchange, reverse, or insert gene segments of individuals in the second intermediate population to generate a third intermediate population; S6: Compare the fitness values ​​of individuals in the third intermediate population with those of individuals in the corresponding second intermediate population, combine the acceptance probability of the simulated annealing algorithm, update the second population and use it as the initial population for the next iteration, and go to S3; until the number of iterations reaches the preset number of iterations, decode the individual corresponding to the best fitness value in the second intermediate population of the last iteration to obtain the final cell layout method. S3 includes the following steps: Based on the principle that the initial population fitness value is proportional to the fitness probability, a roulette wheel is constructed. Randomly generate real numbers between 0 and 1, and spin the roulette wheel N. P s Each time, the individual corresponding to the real number falling into the sector region is selected as the selected individual; where P s The selection probability of the selection operation; Grouping two adjacent individuals together, a crossover point is randomly generated for each group of individuals corresponding to the first and second gene segments. Genes in the gene segments before the crossover point are crossed; and several genes are randomly selected in the second gene segment for crossover. After the crossover, two mutation points are generated in each of the three gene segments corresponding to the individual, and the gene segments between the two mutation points are reversed. The crossover mutation generates N P s Inserting an individual into a population with the optimal fitness value from the initial population. Individuals generate the first intermediate population.

2. The unit layout method according to claim 1, characterized in that, The gene segment whose element value represents the manufacturing unit code corresponding to each machine is taken as the first gene segment of the individual; the gene segment whose element value represents the priority value of the machine sorting is taken as the second gene segment of the individual; and the gene segment whose element value represents the machine placement direction is taken as the third gene segment of the individual. Individual repair methods include the following steps: Calculate the number of machines allocated to each manufacturing unit within an individual; Determine if the number of machines in each manufacturing unit is greater than or equal to 1. If so, retain the current individual; otherwise, re-initialize the first gene segment of the individual randomly.

3. The unit layout method according to claim 1, characterized in that, S5 specifically includes the following steps: Construct a roulette wheel by assigning different probability values ​​to the swap, reverse, and insertion methods; The exchange method involves randomly generating two intersection points in an individual and exchanging the element values ​​of the gene loci at the two intersection points; the reverse order method involves randomly generating two points in an individual and exchanging all element values ​​of the gene loci between the two points in reverse order; the insertion method involves randomly generating two points a and b in an individual, and if a < b, then inserting the element values ​​corresponding to gene loci a+1 to b before gene loci a; otherwise, inserting the element values ​​of gene loci b+1 to a-1 after gene loci a. By rotating the roulette wheel, the genes in the first, second, or third gene segments of the second intermediate population can be manipulated through exchange, reversal, or insertion to generate the third intermediate population.

4. The unit layout method according to claim 3, characterized in that, The acceptance probability of the simulated annealing algorithm is: Where p is the acceptance probability of the simulated annealing algorithm; This represents the fitness value of an individual in the third intermediate population; is the fitness value of an individual in the second intermediate population; K is the annealing rate; T is the annealing temperature; If the fitness value of an individual in the third intermediate population is greater than that of an individual in the second intermediate population, then the individual in the third intermediate population is retained; otherwise, if the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, the individual in the third intermediate population is accepted; if the random number in the simulated annealing algorithm is greater than the acceptance probability, the individual in the second population is retained.

5. A cell layout system based on a hybrid algorithm, characterized in that, include: The module for building the cell layout model is used to construct the cell layout model, assuming that all machines are projected onto the plane as rectangles. The objective function is to minimize the total transportation cost. The constraints are that each manufacturing cell contains at least one machine, machines within the same manufacturing cell do not overlap in the horizontal and vertical directions, and the horizontal coordinate of the centroid of each machine is within the width range of the corresponding manufacturing cell. The total transportation cost is the sum of the intra-cell part transportation cost and the inter-cell part transportation cost for completing all part processing. The initial individual construction module is used to construct and repair N initial individuals using a three-segment coding method. Each individual contains three gene segments. The element values ​​of the three gene segments, arranged in gene order, represent the manufacturing unit number corresponding to each machine, the priority value of machine sorting, and the machine placement direction, respectively. The module for constructing the first intermediate population is used to select a preset number of individuals for crossover, mutation, and re-insertion operations based on the fitness value of the initial population being proportional to the fitness probability, using the roulette wheel method to obtain the first intermediate population. The second intermediate population construction module is used to repair individuals in the first intermediate population, merge the repaired first intermediate population with the initial population, and select the N individuals with the best fitness values ​​in the merged population as the second intermediate population. The module for generating the third intermediate population is used to exchange, reverse, or insert gene segments of individuals in the second intermediate population using a roulette wheel approach to generate the third intermediate population. The cell layout acquisition module is used to compare the fitness values ​​of individuals in the third intermediate population with those of individuals in the corresponding second intermediate population, update the second intermediate population by combining the acceptance probability of the simulated annealing algorithm, and decode the individual corresponding to the best fitness value in the second intermediate population in the last iteration as the final cell layout method. The construction module for the first intermediate population includes: The first roulette wheel construction unit is used to construct a roulette wheel based on the fact that the initial population fitness value is proportional to the fitness probability. Individual selection unit, used to randomly generate real numbers between 0 and 1, rotating roulette wheel N. P s Each time, the individual corresponding to the real number falling into the sector region is selected as the selected individual; where P s The selection probability of the selection operation; The gene crossover processing unit is used to randomly generate a crossover point in the first and second gene segments corresponding to each group of two selected adjacent individuals, crossover the genes in the gene segments corresponding to the crossover point, and randomly select several genes in the second gene segment for crossover. The gene mutation processing unit is used to generate two mutation points in each of the three gene segments corresponding to the crossover individual, and to reverse the gene fragment between the two mutation points. Gene insertion processing unit, used to process N generated by crossover mutation P s Inserting an individual into a population with the optimal fitness value from the initial population. Individuals generate the first intermediate population.

6. The unit layout system according to claim 5, characterized in that, The gene segment whose element value represents the manufacturing unit code corresponding to each machine is taken as the first gene segment of the individual; the gene segment whose element value represents the priority value of the machine sorting is taken as the second gene segment of the individual; and the gene segment whose element value represents the machine placement direction is taken as the third gene segment of the individual. Individual repair methods include the following steps: Calculate the number of machines allocated to each manufacturing unit within an individual; Determine if the number of machines in each manufacturing unit is greater than or equal to 1. If so, retain the current individual; otherwise, re-initialize the first gene segment of the individual randomly.

7. The unit layout system according to claim 5, characterized in that, The module for generating the third intermediate population includes: The second roulette wheel construction unit is used to construct a roulette wheel by assigning different probability values ​​to the exchange, reverse order, and insertion methods. The exchange method involves randomly generating two intersection points in the individual and exchanging the element values ​​of the gene positions at these two intersection points. The reverse order method involves randomly generating two points in the individual and exchanging all element values ​​of the gene positions between these two points in reverse order. The insertion method involves randomly generating two points a and b in the individual; if a < b, the element values ​​corresponding to gene positions a+1 to b are inserted before gene position a; otherwise, the element values ​​of gene positions b+1 to a-1 are inserted after gene position a. The third intermediate population generation unit is used to manipulate genes in the first, second, or third gene segments of the second intermediate population by rotating a roulette wheel and selecting exchange, reversal, or insertion methods to generate the third intermediate population.

8. The unit layout system according to claim 7, characterized in that, The acceptance probability of the simulated annealing algorithm is: Where p is the acceptance probability of the simulated annealing algorithm; This represents the fitness value of an individual in the third intermediate population; is the fitness value of an individual in the second intermediate population; K is the annealing rate; T is the annealing temperature; If the fitness value of an individual in the third intermediate population is greater than that of an individual in the second intermediate population, then the individual in the third intermediate population is retained; otherwise, if the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, the individual in the third intermediate population is accepted; if the random number in the simulated annealing algorithm is greater than the acceptance probability, the individual in the second population is retained.

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