CN115169643A - Unit layout method and system based on hybrid algorithm - Google Patents
Unit layout method and system based on hybrid algorithm Download PDFInfo
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Abstract
The invention provides a cell layout method and a cell layout system based on a hybrid algorithm, which belong to the field of cell manufacturing, and comprise the following steps: constructing a unit layout model by taking the minimized total carrying cost as an objective function; constructing and repairing N initial individuals by adopting a three-section coding method; selecting a preset number of individuals by adopting a roulette method to carry out crossing, variation and reinsertion operations; combining the repaired first intermediate population with the initial population, and selecting N individuals with the optimal population fitness values after combination as a second intermediate population; adopting a roulette mode to perform exchange, reverse or insertion operation on individual gene segments in the second intermediate population; comparing the fitness values of the individuals in the third intermediate population and the individuals in the corresponding second intermediate population, and updating the second intermediate population by combining the acceptance probability of the simulated annealing algorithm; and taking the individual corresponding to the final optimal fitness value as an optimal unit layout scheme. The invention realizes the integration optimization of unit construction and layout.
Description
Technical Field
The invention belongs to the field of unit manufacturing, and particularly relates to a unit layout method and system based on a hybrid algorithm.
Background
Currently, the requirement of customers for products gradually changes from the past few-variety, large-batch and rigid production to multi-variety, small-batch and flexible production, and the traditional production mode of the manufacturing enterprise faces huge challenges. Therefore, a unit production mode capable of rapidly responding to the demands of various, small-batch and individual markets is produced. The problems of unit construction, unit layout, unit scheduling and the like which need to be solved by realizing the unit production mode are mutually independent and supplement each other.
Unit production can reduce costs, improve efficiency, and currently most scholars are focused on separately investigating three different problems, and the unit construction scheme with the minimum number of movements across the unit is not always consistent with the unit construction scheme with the minimum parts handling cost. Firstly, the essence of unit production is cost reduction, i.e. the essence of unit production cannot be reasonably reflected only by reducing the number of times of cross-unit handling; secondly, due to the lack of layout data in the cell construction, the sequential solution of the construction and layout problems results in the result of cell layout depending on the quality of the cell construction scheme, and the validity of the final scheme cannot be guaranteed. Thirdly, the actual production factors are usually not considered or only some of the factors are considered, such as the processing sequence of the part process, the actual size of the machine, the placing direction of the machine and the like, and the problem that the production factors are not included limits the application of the unit production in most practical cases; fourth, assuming that the machines are equal in size, and arranging the machine positions using the pre-specified positions; inter-machine and inter-cell distances are not considered in the inter-cell and intra-cell layouts, and these assumptions directly lead to inaccuracies in the calculation of part handling costs, and thus lead to unreasonable layout designs. Therefore, the research unit construction and layout integration optimization problem and the consideration of a plurality of practical production elements highlight the necessity, and the method has important theoretical and practical significance.
The genetic algorithm is a simple and efficient global search method, can automatically acquire and accumulate knowledge about a search space in a search process, adaptively controls the search process to obtain an optimal solution, and is widely used for solving various optimization problems. However, the local search capability of the genetic algorithm is poor, so that the simple genetic algorithm is time-consuming, and the search efficiency is low in the later stage of evolution. In practical applications, genetic algorithms are prone to premature convergence.
Disclosure of Invention
In view of the defects of the prior art, the present invention aims to provide a cell layout method and system based on a hybrid algorithm, which aims to solve the problem that the prior art is separately studied in terms of cell construction and cell layout, so that a cell construction scheme with the minimum number of cross-cell movement is not always consistent with a cell construction scheme with the minimum part handling cost, and thus the cell layout design is not practical.
To achieve the above object, in one aspect, the present invention provides a cell layout method based on a hybrid algorithm, including the following steps:
s1: assuming that all the machines are rectangular in projection plane, taking the minimized total carrying cost as an objective function, and taking the constraint conditions that each manufacturing unit at least comprises one machine, the machines in the same manufacturing unit cannot be overlapped in the directions of a horizontal axis and a vertical axis, and the horizontal coordinate of the mass center of each machine is in the width interval of the corresponding manufacturing unit, constructing a unit layout model; wherein, the total carrying cost is the sum of the carrying cost of the parts in the unit and the carrying cost of the parts between the units for finishing all parts processing;
s2: constructing and repairing N initial individuals by adopting a three-section coding method, wherein each individual comprises three gene sections, and the element values of the three gene sections respectively represent the manufacturing unit number, the machine sequencing priority value and the machine placing direction corresponding to each machine according to the gene arrangement sequence;
s3: selecting a preset number of individuals by adopting a roulette method to carry out crossing, variation and reinsertion operations according to the proportion of the fitness value of the initial population and the fitness probability to obtain a first intermediate population;
s4: after individuals in the first intermediate population are repaired, the repaired first intermediate population is combined with the initial population, and N individuals with the optimal population fitness value after combination are selected as a second intermediate population;
s5: carrying out exchange, reverse or insertion operation on individual gene segments in the second intermediate population in a roulette mode to generate a third intermediate population;
s6: comparing the fitness values of the individuals in the third intermediate population and the individuals in the corresponding second intermediate population, updating the second population by combining the acceptance probability of the simulated annealing algorithm, taking the second population as the initial population of the next iteration, and turning to S3; and until the iteration times reach the preset iteration times, decoding the individuals corresponding to the optimal fitness value in the second intermediate population of the last iteration to obtain the final unit layout method.
Further preferably, the manufacturing unit code corresponding to the element value in the individual represented by each machine is used as the first gene segment of the individual; taking the gene segment of which the element value in the individual represents the priority value of machine sequencing as a second gene segment of the individual; taking the gene segment with the element value representing the machine placing direction in the individual as a third gene segment of the individual;
a method of repairing an individual comprising the steps of:
calculating the number of machines in an individual assigned to each manufacturing unit;
and judging whether the number of machines in each manufacturing unit is more than or equal to 1, if so, keeping the current individual, and otherwise, randomly initializing the first gene segment of the individual again.
Further preferably, S3 comprises the steps of:
constructing a wheel disc according to the fact that the fitness value of the initial population is in direct proportion to the fitness probability;
randomly generating 0-1 real number, and rotating the rotary disk N P s Selecting an individual corresponding to the real number falling into the sector area each time as a selected individual; wherein, P s A selection probability for a selection operation;
randomly generating a cross point on each group corresponding to the first gene segment and the second gene segment by taking the two selected adjacent individuals as a group, and crossing genes of the corresponding gene segments before the cross point; randomly selecting a plurality of gene intersections in the second gene segment;
generating two variation points on the crossed individuals corresponding to the three gene segments respectively, and reversing the sequence of the gene segments between the two variation points;
including NxP generated by cross mutation s Inserting N-N P with optimal fitness value in initial population into population of individuals s And generating a first intermediate population by each individual.
Further preferably, S5 specifically includes the following steps:
assigning probability values with different sizes to the exchange, reverse order and insertion modes to construct a wheel disc;
wherein, the exchange mode is to randomly generate two cross points in an individual and exchange the element values of the gene positions of the two cross points; the reverse order mode is that two points are randomly generated in an individual, and all element values of the gene positions between the two points are exchanged in the reverse order; the insertion mode is that two points a and b are randomly generated in an individual, if a is less than b, the element values corresponding to the a + 1-b gene positions are inserted in front of the a gene position; otherwise, inserting the element value of b + 1-a-1 gene position behind the a gene position;
and operating the genes in the first gene segment, the second gene segment or the third gene segment in the second intermediate population by rotating the wheel disc in a mode of selecting exchange, reverse order or insertion to generate a third intermediate population.
Further preferably, the acceptance probability of the simulated annealing algorithm is:
wherein p is the acceptance probability of the simulated annealing algorithm; f (l') is the fitness value of the individuals in the third intermediate population; f (l) is the fitness value of the individuals in the second intermediate population; k is the annealing rate; t is the annealing temperature;
when the fitness value of the third intermediate population individuals is larger than that of the second intermediate population individuals, the third intermediate population individuals are reserved; otherwise, when the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, accepting the third intermediate population individuals; when the random number in the simulated annealing algorithm is greater than the acceptance probability, a second population of individuals is retained.
In another aspect, the present invention provides a cell layout system based on a hybrid algorithm, including:
the unit layout model building module is used for building the unit layout model by assuming that the projection of the machine to the plane is rectangular, taking the minimum total carrying cost as an objective function and taking the constraint conditions that each manufacturing unit at least comprises one machine, the machines in the same manufacturing unit cannot be overlapped in the directions of a transverse axis and a longitudinal axis and the transverse coordinate of the mass center of each machine is in the width interval of the corresponding manufacturing unit; wherein, the total carrying cost is the sum of the carrying cost of the parts in the unit and the carrying cost of the parts between the units for finishing all parts processing;
the initial individual building module is used for building and repairing N initial individuals by adopting a three-section coding method, wherein each individual comprises three gene sections, and the element values of the three gene sections respectively represent the manufacturing unit number, the machine sequencing priority value and the machine placing direction corresponding to each machine according to the gene arrangement sequence;
the construction module of the first intermediate population is used for selecting a preset number of individuals to carry out crossing, variation and reinsertion operations by adopting a roulette method according to the proportion of the fitness value of the initial population and the fitness probability to obtain a first intermediate population;
the second intermediate population building module is used for combining the repaired first intermediate population with the initial population after repairing the individuals in the first intermediate population, and selecting N individuals with the optimal population fitness values after combination as a second intermediate population;
the generating module of the third intermediate population is used for exchanging, reversing or inserting individual gene segments in the second intermediate population in a roulette mode to generate the third intermediate population;
and the unit layout acquisition module is used for comparing the fitness values of the individuals in the third intermediate population and the individuals in the corresponding second intermediate population, updating the second intermediate population by combining the acceptance probability of the simulated annealing algorithm, and decoding the individuals corresponding to the optimal fitness values in the second intermediate population of the last iteration to serve as a final unit layout method.
Further preferably, the manufacturing unit code corresponding to the element value in the individual is used as the first gene segment of the individual; taking the gene segment of which the element value in the individual represents the priority value of machine sequencing as a second gene segment of the individual; taking the gene segment with the element value in the individual representing the machine placing direction as a third gene segment of the individual;
a method of individual repair comprising the steps of:
calculating the number of machines in an individual assigned to each manufacturing unit;
and judging whether the number of machines in each manufacturing unit is more than or equal to 1, if so, keeping the current individual, and otherwise, randomly initializing the first gene segment of the individual again.
Further preferably, the building module of the first intermediate population includes:
the first wheel disc construction unit is used for constructing a wheel disc according to the fact that the initial population fitness value is in direct proportion to the fitness probability;
an individual screening unit for randomly generating 0-1 real number and a rotary wheel disk N P s Selecting an individual corresponding to the real number falling into the sector area each time as a selected individual; wherein, P s A selection probability for a selection operation;
the gene crossing processing unit is used for randomly generating a crossing point on the first gene segment and the second gene segment corresponding to each group of individuals by taking two selected adjacent individuals as a group, and crossing the genes of the corresponding gene segments before the crossing point; randomly selecting a plurality of genes in the second gene segment for crossing;
the gene variation processing unit is used for respectively generating two variation points on the crossed individuals corresponding to the three gene segments and reversing the gene segments between the two variation points;
a gene insertion processing unit for generating cross mutation containing NxP s Inserting N-N P with optimal fitness value in initial population into population of individuals s And generating a first intermediate population by each individual.
Further preferably, the generation module of the third intermediate population includes:
the second wheel disc construction unit is used for assigning probability values with different sizes to the exchange, reverse order and insertion modes to construct a wheel disc; wherein, the exchange mode is to randomly generate two cross points in an individual and exchange the element values of the gene positions of the two cross points; the reverse order mode is that two points are randomly generated in an individual, and all element values of the gene positions between the two points are exchanged in the reverse order; the insertion mode is that two points a and b are randomly generated in an individual, if a is less than b, the element value corresponding to a + 1-b gene position is inserted before a gene position a; otherwise, inserting the element value of b + 1-a-1 gene position behind the a gene position;
and the third intermediate population generating unit is used for selecting an exchange, reverse order or insertion mode to operate the genes in the first gene segment, the second gene segment or the third gene segment in the second intermediate population through rotating the wheel disc to generate a third intermediate population.
Further preferably, the acceptance probability of the simulated annealing algorithm is:
wherein p is the acceptance probability of the simulated annealing algorithm; f (l') is the fitness value of the individuals in the third intermediate population; f (l) is the fitness value of the individuals in the second intermediate population; k is the annealing rate; t is the annealing temperature;
when the fitness value of the third intermediate population individuals is larger than that of the second intermediate population individuals, the third intermediate population individuals are reserved; otherwise, when the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, accepting the third intermediate population individuals; when the random number in the simulated annealing algorithm is greater than the acceptance probability, a second population of individuals is retained.
Generally, compared with the prior art, the above technical solution conceived by the present invention has the following beneficial effects:
the invention assumes that the projection of the machine to the plane is rectangular, takes the minimum total carrying cost as an objective function, takes the constraint conditions that each manufacturing unit at least comprises one machine, the machines in the same manufacturing unit cannot be overlapped in the directions of a transverse axis and a longitudinal axis and the transverse coordinate of the mass center of each machine is in the width interval of the corresponding manufacturing unit, constructs a unit layout model, and combines the minimum carrying cost of the part with the unit layout; in the aspect of unit layout integration optimization, individuals are constructed by adopting a three-section coding method, wherein each individual comprises three gene sections, elements of the three gene sections respectively represent the manufacturing unit number, the priority sequence value of machine sequencing and the machine placing direction corresponding to each machine according to the gene arrangement sequence, each individual can be seen to represent unit layout information, the population is updated by adopting a genetic algorithm to carry out crossing, variation and reinsertion of the individuals, and the fitness value in the population is calculated and is the reciprocal of the minimum handling cost; on the basis, in order to obtain the optimal fitness value and improve the global search capability and the local search capability, the invention adopts the genetic algorithm of the fusion simulated annealing algorithm and selects the optimal unit layout method from a plurality of unit layout methods to realize the integrated optimization of unit construction and unit layout.
In order to improve the accuracy of layout design between units and between units, the invention calculates the parts carrying cost according to the actual position of the machine in the unit and the design factors including the actual size and the channel distance of the machine, namely the total carrying cost is the sum of the parts carrying cost in the unit and the parts carrying cost between the units for completing all parts processing, and the total carrying cost is calculated according to the total distance moved by the parts for completing all processing procedures and the parts carrying cost between the units and the units.
The method repairs the initial individuals after the initial individuals are constructed, and repairs the individuals in the first intermediate population, because the condition that machines do not exist in one manufacturing unit exists after a part of the individuals are possibly decoded, the condition is avoided by adopting the repair method, and the iteration effectiveness of the hybrid algorithm can be fully ensured.
Drawings
FIG. 1 is a schematic diagram of a result of cell building and layout integration optimization provided in an embodiment of the present invention;
FIG. 2 is a flow chart of a hybrid algorithm provided by an embodiment of the present invention;
FIG. 3 is a schematic diagram of a three-stage encoding method according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of a crossover operation provided by an embodiment of the present invention;
FIG. 5 is a schematic diagram illustrating a variant operation provided by an embodiment of the present invention;
FIG. 6 is an ARDI with 100 cases of various algorithms provided by embodiments of the present invention k A value;
FIG. 7 is a convergence diagram of the case 320251 genetic algorithm provided by an embodiment of the present invention;
FIG. 8 is a convergence diagram of case 320251 particle swarm algorithm provided by an embodiment of the present invention;
FIG. 9 is a convergence diagram of case 320251 gull optimization algorithm provided by an embodiment of the present invention;
FIG. 10 is a convergence diagram of the case 320251 hybrid algorithm provided by the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention provides a product production unit layout method and system based on a hybrid algorithm, and the overall idea is as follows: the hybrid algorithm is a genetic algorithm of a fusion simulated annealing algorithm, and a mathematical programming model of a unit construction and layout integration optimization problem containing all problem characteristics in the background technology is established; analyzing problem characteristics, and designing a three-stage coding method and a three-stage decoding method for the priority sequence of machine allocation and machine sequencing and the machine placing direction; in order to improve the diversity of the solution, sectional type cross variation operation is designed, and in order to ensure the effectiveness of algorithm iteration, an initialization screening method and a repairing method which meet problem constraints are designed; in addition, to improve the accuracy of the layout design between and within cells, part handling costs are calculated based on the actual location of the machines within the cells and design factors including the actual size of the machines and the aisle distance.
The problem description and mathematical description of the product production unit layout method and system based on the hybrid algorithm provided by the invention are as follows:
the invention aims to solve the problems that a part has process constraint and machines in a manufacturing unit have unit construction and layout integration optimization in horizontal and vertical placing directions; the layout in the unit refers to the placing sequence and the placing direction of the machines, the actual size of the machines and the space between the machines in the manufacturing units are considered, and the machines in each manufacturing unit are horizontally or vertically arranged along a linear straight line from the left side of the unit according to the priority value. Further, the manufacturing units are arranged from bottom to top in consideration of the used distance between the manufacturing units, and this line type cell layout method is shown in fig. 1; FIG. 1 is a schematic diagram showing the result of the integrated optimization of unit construction and layout of 2 manufacturing units and 5 machines, wherein the parts are transported in and among the units by an AGV (automatic guided vehicle), that is, the AGV transports the parts to the required machines in sequence for processing until all the processing procedures of the parts are completed;
in one aspect, the cell layout method based on the hybrid algorithm provided by the invention comprises the following steps:
s1: in order to establish a mathematical model, the invention makes the following assumptions:
a plant with M machines (j =1,2, …, M) and C manufacturing units processes P types of parts (i =1,2, …, P) with significant variability in different parts; the problem of unit construction and layout integration optimization is that the layout of the machines in the unit is formed while M machines are divided into C manufacturing units; to facilitate building the cell layout model, the relevant assumptions are as follows:
(1) After the processing path of the part is determined, the process flow of the part, namely the processing machine selected by each procedure and the processing sequence of all procedures of the part are also determined;
(2) The cost of parts handling per unit distance within and between units, machine spacing within a unit and channel distance between units, the actual size of the machine, the number of units manufactured, and part process flow information are known;
(3) The sizes of all the machines are not necessarily equal, and the projection to the plane is a rectangle;
(4) The initial position of the part is the central position of the machining machine required for the first pass of the selected machining path.
S2: establishing a unit layout model:
the invention establishes a unit layout model with the minimized total carrying cost as the expected target; wherein, the total carrying cost is the sum of the carrying cost of the parts in the unit and the carrying cost of the parts between the units for completing all parts processing, and the total carrying cost is calculated according to the total distance moved by the parts for completing all processing procedures and the carrying cost of the parts in the unit and between the units;
the symbol description is shown in table 1;
TABLE 1
The objective function is:
PC represents the total cost of the parts transportation between units and in units required by all procedures of all parts, and the smaller the cost is, the more effective the established unit construction and layout integration optimization scheme is;
PC＝PC A +PC E (1)
wherein, PC A The total cost of handling in all process units for all parts processed; PC (personal computer) E The total cost of handling between manufacturing units for all processes for which all parts have been machined;
wherein x is j Is a jth machine; x is the number of j′ Is the j' th machine;
wherein, y k Is the unit k center ordinate; y is k′ Is the unit k' center ordinate;
thus, the objective function of the present invention is as follows:
the constraint conditions are as follows:
A j ＝L j +(W j -L j )d j (7)
B j ＝W j +(L j -W j )d j (8)
equation (5) indicates that any machine must be assigned to one manufacturing unit and can only be assigned once; formula (6) indicates that each manufacturing unit contains at least one machine; in equations (7) and (8), the width A of each machine is measured based on the machine lay direction j And length B j (ii) a In equations (9) and (10), the width W of the manufacturing unit k is calculated based on the machine size allocated to the manufacturing unit k, respectively k And a length L k (ii) a Equation (11) indicates that machines belonging to the same manufacturing cell do not overlap in the direction of the horizontal axis (x-axis); equation (12) ensures that the manufacturing units do not overlap in the direction of the longitudinal axis (y-axis); equation (13) ensures that the abscissa of the centroid of each machine belonging to the same manufacturing cell is in the interval [ 0-W ] k ]Internal;
s3: the cell layout method based on the hybrid algorithm comprises the following steps:
the genetic algorithm is a simple and efficient global search method, can automatically acquire and accumulate knowledge about a search space in a search process, adaptively controls the search process to obtain an optimal solution, and is widely used for solving various optimization problems. However, the local search capability of the genetic algorithm is poor, so that the simple genetic algorithm is time-consuming, and the search efficiency is low at the later stage of evolution. In practical applications, genetic algorithms are prone to premature convergence.
The unit construction and layout integrated optimization problem is a typical multi-constraint discrete optimization problem, and as the unit construction and layout integrated optimization problem belongs to an NP-hard problem, a genetic algorithm fused with a simulated annealing algorithm is provided for solving conveniently and rapidly, and is called a hybrid algorithm; the analysis unit construction and layout integration optimization problem characteristic is provided, and a three-section type coding method facing to machine allocation, machine sequencing priority and machine placing direction is designed. In order to improve the diversity of the solution, a sectional type cross mutation operation is designed. In order to ensure the effectiveness of algorithm iteration, an initialization screening method and a repairing method which meet problem constraints are designed. In order to enhance the local exploration capability of the method, a local search method fused with a simulated annealing algorithm is designed.
Solving a unit construction and layout integrated optimization problem by a hybrid algorithm:
when a mixed algorithm solving unit is used for building and layout integration optimization problems, as long as the coding is proper, the problem parameters can directly participate in genetic operation, and a certain corresponding relation, namely coding and decoding, can be established between the actual description of the problems and the layout scheme representation, and the designed mixed algorithm flow chart based on the simulated annealing algorithm is shown in figure 2;
s3.1 three-segment coding method
When a genetic algorithm solving unit is used for constructing and laying out an integrated optimization problem, a possible solution of the problem is firstly coded into a chromosome, which is a key for solving the problem and is also a basis for genetic algorithm operation; according to the unit distribution model, the machine distribution sequence, the size of the machine sequencing priority value and the machine placing direction are the keys for solving the problem of unit construction and layout integration optimization; based on the method, a three-section coding method based on a machine distribution sequence, the size of a machine sequencing priority value and the machine placing direction is designed; each individual comprises M x 3 genes, the first M gene positions represent machine allocation, the element value of each gene position is a positive integer of [ 1-C ], and the element value represents a manufacturing unit number corresponding to the corresponding machine; former M +1: m is 2 gene positions to represent the priority value of machine sequencing, the element value of each gene position is a positive integer of [1-M ] and is not repeated, and the element value represents the sequencing priority value corresponding to the corresponding machine; the last M gene positions represent the placing directions of the machines, the elements of each gene position are integers of [0,1], and the element values represent the placing directions corresponding to the corresponding machines;
as shown in fig. 3, the element value of the individual 1 st gene is 2, indicating that machine 1 is assigned to manufacturing unit 2; the element value of the M +1 th gene is 2, and the priority value of the sequencing of the machine 1 is 2; the element value of the Mth 2+1 gene is 0, which indicates that the machine 1 is transversely arranged (i.e. one side of the machine length is parallel to the horizontal axis), and the rest is analogized;
s3.2 population initialization
Adopting a random initialization population, and completing the coding of each individual according to a three-section coding method; since each locus of the machine-assigned code segment is a randomly generated [ 1-C ] random integer, this initialization will result in one or more integers that have not been assigned to the first M loci of the machine-assigned code segment, i.e., no machine assignment occurs within one or more manufacturing units; in general, the individuals of this kind described above belong to illegal solutions, which are disadvantageous for the guidance of the evolution direction. Therefore, it is necessary to provide an initial screening method (repairing method) that ensures a feasible solution, and the specific steps are as follows:
step 1: individual x l ，l＝1；
Step 2: calculating an individual x l The number of machines assigned to each manufacturing unit 1,2, …, C;
and step 3: judging whether the number of machines in each manufacturing unit is greater than or equal to 1, namely, a constraint condition (6) must be met; if the condition is met, turning to the step 4; otherwise re-randomly initializing individual x l The first M gene locus element values and go to step 2;
and 4, step 4: making l = l +1, and if l is less than or equal to the population number N, turning to the step 2; otherwise, the screening is terminated.
S3.3 decoding design
Firstly, the number of the manufacturing unit divided by each machine can be known from the first segment of coding result, namely the number of the machines and the number of the machines contained in each unit can be known; secondly, the arrangement sequence of the machines in each manufacturing unit can be known according to the second section of coding result; finally, the placing direction of each machine can be known according to the third section of coding results; so far, the decoding of unit construction, in-unit machine layout and inter-unit layout is completed;
s3.4 calculating the fitness value
The fitness value is calculated through a fitness function, and the larger the fitness value of an individual is, the closer the solution is to the target is; solving a unit construction and layout integrated optimization problem, wherein an objective function is to minimize the total handling cost, and the total handling cost objective is converted into a fitness value to evaluate, namely the larger the fitness value of the required chromosome is, the smaller the total handling cost is;
individual x l The fitness function calculation formula is as follows:
wherein, the denominator is the individual x l Total cost of handling, F (l) is the individual x l A fitness value of;
s3.5 operation of the genetic Algorithm
S3.5.1 selection operation
Selecting a roulette method with a selection probability of P s (ii) a Supposing to generate N individuals, firstly calculating a fitness value F (l) of each individual, and then calculating the probability of each fitness value according to the fitness value;
the fitness probability calculation formula is as follows:
the roulette method is inspired by the fact that a rotating roulette randomly stops in a certain sector, and the size of the sector area is in direct proportion to the size of the fitness probability; randomly generating a real number P epsilon (0-1), and if the P value falls into the position pointed by a certain sector area, selecting an individual corresponding to the area to be inherited to the next generation; disk rotation N P s Then, N x P will be selected s An individual makes thisThe population is subjected to subsequent operations such as cross mutation and the like;
s3.5.2 crossover operation
Randomly generating a cross point on the first gene segment and the third gene segment, and crossing the gene segments before the cross point of the parent individuals; as shown in FIG. 4, the cross points are 3 and 15, i.e., the machine allocation code segment exchanges genes at the first 3 positions, and the machine placement direction code segment exchanges genes at the first 3 positions; the priority value code of the second section of machine sequencing is to randomly select the number in three [1-M ], such as 5,4,3, and then the order of 5,4,3 in the individual 1 in the parent is changed with 3,4,5 of the individual 2 in the parent to generate the child; the first segment of encoding may cause the child individual to have illegal solution, that is, the condition that the constraint (6) is not satisfied may occur, as shown in fig. 4, there is no machine allocation in the manufacturing unit 1 of the child individual 1;
s3.5.3 mutation operation
The mutation operation is to increase the search space of the disturbance lifting algorithm, the three-section coding of an individual adopts reverse sequence mutation operation, namely two mutation points are respectively generated at the machine distribution coding section, the machine sequencing priority value coding section and the machine placing direction coding section at random, and the gene segments between the two mutation points are inverted, so that the mutation operation can not generate illegal solutions; as shown in fig. 5, machine-assigned variation points 2 and 3 are randomly generated, machine-ordered variation points 8 and 10 with priority values are randomly generated, and the genes between 8 and 10 are subjected to reverse order mutation operation; placing the gene segments between the direction change points 15 and 16, the reverse order 2 and 3, 8 and 10, and 15 and 16 in the machine;
s3.5.3 reinsertion operation
Only N × P in the population after the selection operation s Individual, so as to optimize the fitness value of the current population s Inserting the individual into the new population after the cross variation, so that the individual in the population is N;
by utilizing the method of combining roulette and optimal individual reservation, on one hand, the method inherits the advantages of the roulette method, namely, the preferred selection is ensured without losing the diversity of the population, and the population is enabled to evolve towards the optimal direction; on the other hand, the method inherits the advantages of an individual retention method, namely ensuring excellent individuals to be inherited and ensuring that excellent genes are not destroyed and lost; therefore, the combination of the two selection methods has great significance on the calculation precision and the global convergence of the genetic algorithm;
s3.6 repair method:
the crossover operation may result in a situation where the constraint (6) cannot be met for the first segment of code for the new individual, i.e. there may be no machine allocation within a certain manufacturing unit; therefore, after the cross operation is finished, a repairing method consistent with the initial screening method needs to be adopted to complete the repairing of the illegal solution;
for the individuals after mutation, the mutation process can not generate illegal solutions easily, so that the repair operation is not needed;
s3.7 updating populations
Firstly, calculating the fitness value of each individual of a new population, then combining the current population with the new population generated after selection, crossing, mutation, reinsertion and repair, and finally selecting N individuals with the optimal fitness values as the population of the simulated annealing local search method;
s3.8 simulated annealing local search method
The simulated annealing algorithm introduces random factors in the searching process, and receives a solution worse than the current solution with a certain probability when iteratively updating the feasible solution, so that the local optimal solution can possibly jump out to reach the global optimal solution; the specific operation is as follows: with individual x in the current population l For a research object, one of a machine distribution coding section, a machine sequencing priority value coding section or a machine placing direction coding section is selected at random, and one of the following three neighborhood functions is selected at random to generate a new solution x' in combination with a roulette mode;
exchanging: randomly generating two cross points, and exchanging element values of gene positions of the two cross points;
reversing the sequence: randomly generating two points, and exchanging all element values of the gene positions between the two points in a reverse order;
inserting: randomly generating two points a and b, and if a is less than b, inserting elements at positions a + 1-b in front of the position a; otherwise, inserting the element values of the b + 1-a-1 positions into the position a;
finally, carrying out fitness evaluation on the new solution x', and updating the solution by utilizing the acceptance probability p of the simulated annealing algorithm Metropolis criterion;
in algorithm iteration, firstly setting T equal to an initial temperature, and then updating the temperature according to T = KT in iteration, wherein K is an annealing rate; if the fitness value of the new solution is larger, the new solution is reserved; otherwise, when the random number in the simulated annealing algorithm is less than or equal to the acceptance probability p, accepting the poor new solution, and when the random number in the simulated annealing algorithm is greater than the acceptance probability p, reserving the corresponding individuals in the S3.7 population; turning to S3.5;
in the hybrid algorithm, setting the iteration times as an algorithm termination condition, finishing the iteration when the current iteration times of the algorithm exceeds the maximum iteration times in the parameter setting, and terminating the calculation and outputting an optimal solution;
in another aspect, the present invention provides a cell layout system based on a hybrid algorithm, including:
the unit layout model building module is used for building the unit layout model by assuming that the projection of the machine to the plane is rectangular, taking the minimum total carrying cost as an objective function and taking the constraint conditions that each manufacturing unit at least comprises one machine, the machines in the same manufacturing unit cannot be overlapped in the directions of a transverse axis and a longitudinal axis and the transverse coordinate of the mass center of each machine is in the width interval of the corresponding manufacturing unit; wherein, the total carrying cost is the sum of the carrying cost of the parts in the unit and the carrying cost of the parts between the units for completing the processing of all the parts;
the initial individual building module is used for building and repairing N initial individuals by adopting a three-section coding method, wherein each individual comprises three gene sections, and the element values of the three gene sections respectively represent the manufacturing unit number, the machine sequencing priority value and the machine placing direction corresponding to each machine according to the gene arrangement sequence;
the construction module of the first intermediate population is used for selecting a preset number of individuals to carry out crossing, variation and reinsertion operations by adopting a roulette method according to the proportion of the fitness value of the initial population and the fitness probability to obtain a first intermediate population;
the second intermediate population building module is used for merging the restored first intermediate population and the initial population after restoring the individuals in the first intermediate population, and selecting N individuals with the optimal population fitness values after merging as a second intermediate population;
the generating module of the third intermediate population is used for exchanging, reversing or inserting the individual gene segments in the second intermediate population in a roulette mode to generate the third intermediate population;
and the unit layout acquisition module is used for comparing the fitness values of the individuals in the third intermediate population and the individuals in the corresponding second intermediate population, updating the second intermediate population by combining the acceptance probability of the simulated annealing algorithm, and decoding the individuals corresponding to the optimal fitness values in the second intermediate population of the last iteration to serve as a final unit layout method.
Further preferably, the manufacturing unit code corresponding to the element value in the individual is used as the first gene segment of the individual; taking the gene segment of which the element value in the individual represents the machine-ordered priority value as a second gene segment of the individual; taking the gene segment with the element value representing the machine placing direction in the individual as a third gene segment of the individual;
a method of repairing an individual comprising the steps of:
calculating the number of machines in an individual assigned to each manufacturing unit;
and judging whether the number of machines in each manufacturing unit is more than or equal to 1, if so, keeping the current individual, and otherwise, randomly initializing the first gene segment of the individual again.
Further preferably, the building module of the first intermediate population includes:
the first wheel disc construction unit is used for constructing a wheel disc according to the fact that the initial population fitness value is in direct proportion to the fitness probability;
individual screening unit for randomizationGenerating a real number of 0-1, rotating the wheel disk N x P s Selecting an individual corresponding to the real number falling into the sector area each time as a selected individual; wherein, P s A selection probability for a selection operation;
the gene crossing processing unit is used for randomly generating a crossing point on the first gene segment and the second gene segment corresponding to each group of individuals by taking two selected adjacent individuals as a group, and crossing the genes of the corresponding gene segments before the crossing point; randomly selecting a plurality of gene intersections in the second gene segment;
the gene mutation processing unit is used for respectively generating two mutation points and gene segments between the two mutation points in a reverse order in the crossed individuals corresponding to the three gene segments;
gene insertion processing unit for generating cross mutation containing NxP s Inserting N-N P with optimal fitness value in initial population into population of individuals s And generating a first intermediate population by each individual.
Further preferably, the generation module of the third intermediate population includes:
the second wheel disc construction unit is used for assigning probability values with different sizes to the exchange, reverse order and insertion modes to construct a wheel disc; wherein, the exchange mode is to randomly generate two cross points in an individual and exchange the element values of the gene positions of the two cross points; the reverse order mode is that two points are randomly generated in an individual, and all element values of the gene positions between the two points are exchanged in the reverse order; the insertion mode is that two points a and b are randomly generated in an individual, if a is less than b, the element values corresponding to the a + 1-b gene positions are inserted in front of the a gene position; otherwise, inserting the element value of b + 1-a-1 gene position behind the a gene position;
and the third intermediate population generating unit is used for selecting an exchange, reverse order or insertion mode to operate the genes in the first gene segment, the second gene segment or the third gene segment in the second intermediate population through rotating the wheel disc so as to generate a third intermediate population.
Further preferably, the acceptance probability of the simulated annealing algorithm is:
wherein p is the acceptance probability of the simulated annealing algorithm; f (l') is the fitness value of the individuals in the third intermediate population; f (l) is the fitness value of the individuals in the second intermediate population; k is the annealing rate; t is the annealing temperature;
when the fitness value of the third intermediate population individuals is larger than that of the second intermediate population individuals, the third intermediate population individuals are reserved; otherwise, when the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, accepting the third intermediate population individuals; when the random number in the simulated annealing algorithm is greater than the acceptance probability, a second population of individuals is retained.
Examples
Aiming at minimizing the total carrying cost, four algorithms, namely a Genetic Algorithm (GA), a particle swarm algorithm (PSO), a gull optimization algorithm (SOA) and a hybrid algorithm (GA + SA) provided by the invention are selected to solve the embodiment, all the algorithms are programmed by MATLB 2019a, the program running environment is Intel (R) Core (TM) i5-9400 CPU@2.90GHz, and the memory is 16.0GB;
in order to fairly compare the performances of different algorithms, the population quantity and the iteration times of the four algorithms are both 100, the material handling cost of unit distance between a manufacturing unit and a manufacturing unit is respectively 1 and 3, the distance between machines in the manufacturing unit is 1, and the channel distance between the manufacturing units is 2; selecting a manufacturing unit number C of [2,3], a machine number M of [5,8,10,15,20], a part number P of [10,15,20,25,30], respectively arranging and combining the manufacturing unit number C, the part number P of [5,8,10,15,20], randomly generating two cases under the scale of the same machine number and part number, wherein the size of the machine is a positive integer generated randomly, testing 100 randomly generated cases, respectively operating five times for each case, and taking an average value as an optimal total carrying cost, wherein the experimental result is shown in table 2;
TABLE 2
The first digit of the case in table 2 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 number of cases of that scale; to verify the performance of the proposed hybrid algorithm, in addition to the optimal total handling cost, the performance of the different algorithms is compared according to an Average Relative Deviation Index (ARDI), which is calculated as follows:
wherein n is the total number of cases; d is case number; PC (personal computer) dk Solving the case d for the algorithm k to obtain the minimum total carrying cost; MIN (PC) and MAX (PC) are minimum total carrying cost and worst total carrying cost obtained by solving the case d by different algorithms; ARDI k Is the average relative deviation index value of algorithm k; respectively calculating the ARDI of each algorithm under 100 cases k Values, results are shown in fig. 6;
as can be seen from fig. 6, the performance of the hybrid algorithm (GA + SA) is the best in 100 cases, and the performance of the gull optimization algorithm (SOA) is the worst; the hybrid algorithm is only added with a simulated annealing local search method on the basis of the genetic algorithm, and the effectiveness of the simulated annealing local search is also proved. Selecting one case 320251, and using the four algorithms to carry out coding solution on the case 320251, wherein the optimal total carrying cost obtained by GA is 6193.5, the optimal carrying cost obtained by PSO is 8939.5, the optimal carrying cost obtained by SOA is 9450, the optimal carrying cost obtained by a mixed algorithm is 5666, and a convergence graph of the cases is shown in figures 7-10.
In summary, compared with the prior art, the invention has the following advantages:
the invention assumes that the projection of the machine to the plane is all rectangular, takes the minimum total carrying cost as an objective function, takes the constraint conditions that each manufacturing unit at least comprises one machine, the machines in the same manufacturing unit cannot be overlapped in the directions of a horizontal axis and a vertical axis, and the horizontal coordinate of the mass center of each machine is in the width interval of the corresponding manufacturing unit, constructs a unit layout model, and combines the minimum carrying cost of the part with the unit layout; in the aspect of unit layout integration optimization, individuals are constructed by adopting a three-section coding method, wherein each individual comprises three gene sections, elements of the three gene sections respectively represent a manufacturing unit number, a machine sequencing priority value and a machine placing direction corresponding to each machine according to a gene arrangement sequence, each individual can be seen to represent unit layout information, a genetic algorithm is adopted to carry out individual intersection, variation and reinsertion to update a population, and a fitness value in the population is calculated, wherein the fitness value is the reciprocal of the minimum handling cost, and the unit layout and the handling cost are combined by calculating the fitness value; on the basis, in order to obtain the optimal fitness value and improve the global searching capability and the local searching capability, the invention adopts the genetic algorithm of the fusion simulated annealing algorithm to realize the integrated optimization of the unit layout from the optimal unit layout in various unit layouts.
In order to improve the accuracy of layout design between units and between units, the invention calculates the parts carrying cost according to the actual position of the machine in the unit and the design factors including the actual size and the channel distance of the machine, namely the total carrying cost is the sum of the parts carrying cost in the unit and the parts carrying cost between the units for completing all parts processing, and the total carrying cost is calculated according to the total distance moved by the parts for completing all processing procedures and the parts carrying cost between the units and the units.
The method repairs the initial individuals after the initial individuals are constructed, and the individuals in the first intermediate population are also repaired, because the condition that no machine exists in one manufacturing unit exists after a part of the possible individuals are decoded, the repairing method is adopted to avoid the condition, and the iterative effectiveness of the hybrid algorithm can be fully ensured.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (10)
1. A cell layout method based on a hybrid algorithm is characterized by comprising the following steps:
s1: assuming that the projection of the machines to the plane is all rectangular, taking the minimized total carrying cost as an objective function, and taking the constraint conditions that each manufacturing unit at least comprises one machine, the machines in the same manufacturing unit cannot be overlapped in the directions of a horizontal axis and a vertical axis, and the horizontal coordinate of the mass center of each machine is in the width interval of the corresponding manufacturing unit, constructing a unit layout model; wherein, the total carrying cost is the sum of the carrying cost of the parts in the unit and the carrying cost of the parts between the units for completing the processing of all the parts;
s2: constructing and repairing N initial individuals by adopting a three-section coding method, wherein each individual comprises three gene sections, and the element values of the three gene sections respectively represent the manufacturing unit number, the machine sequencing priority value and the machine placing direction corresponding to each machine according to the gene arrangement sequence;
s3: selecting a preset number of individuals by adopting a roulette method to carry out crossing, variation and reinsertion operations according to the proportion of the fitness value of the initial population and the fitness probability to obtain a first intermediate population;
s4: after individuals in the first intermediate population are repaired, the repaired first intermediate population is combined with the initial population, and N individuals with the optimal population fitness value after combination are selected as a second intermediate population;
s5: carrying out exchange, reverse or insertion operation on individual gene segments in the second intermediate population in a roulette mode to generate a third intermediate population;
s6: comparing the fitness values of the individuals in the third intermediate population and the individuals in the corresponding second intermediate population, updating the second population by combining the acceptance probability of the simulated annealing algorithm, taking the second population as the initial population of the next iteration, and turning to S3; and until the iteration times reach the preset iteration times, decoding the individuals corresponding to the optimal fitness value in the second intermediate population of the last iteration to obtain the final unit layout method.
2. The cell layout method according to claim 1, wherein the gene segment corresponding to the manufacturing cell code corresponding to the element value representation in the individual is taken as the first gene segment of the individual; taking the gene segment of which the element value in the individual represents the machine-ordered priority value as a second gene segment of the individual; taking the gene segment with the element value representing the machine placing direction in the individual as a third gene segment of the individual;
a method of repairing an individual comprising the steps of:
calculating the number of machines in an individual assigned to each manufacturing unit;
and judging whether the number of machines in each manufacturing unit is more than or equal to 1, if so, keeping the current individual, and otherwise, randomly initializing the first gene segment of the individual again.
3. The cell layout method according to claim 2, wherein S3 comprises the steps of:
constructing a wheel disc according to the fact that the fitness value of the initial population is in direct proportion to the fitness probability;
randomly generating 0-1 real number, and rotating the rotary disk N P s Selecting an individual corresponding to the real number falling into the sector area each time as a selected individual; wherein, P s A selection probability for a selection operation;
randomly generating a cross point on the first gene segment and the second gene segment corresponding to each group of individuals by taking two selected adjacent individuals as a group, and crossing genes of the corresponding gene segments before the cross point; randomly selecting a plurality of gene intersections in the second gene segment;
generating two variation points on the crossed individuals corresponding to the three gene segments respectively, and reversing the sequence of the gene segments between the two variation points;
including NxP generated by cross mutation s Inserting N-N P with optimal fitness value in initial population into population of individuals s And generating a first intermediate population by each individual.
4. The cell layout method according to claim 3, wherein S5 specifically comprises the steps of:
assigning probability values with different sizes to the exchange, reverse order and insertion modes to construct a wheel disc;
wherein, the exchange mode is to randomly generate two cross points in an individual and exchange the element values of the gene positions of the two cross points; the reverse order mode is that two points are randomly generated in an individual, and all element values of the gene positions between the two points are exchanged in the reverse order; the insertion mode is that two points a and b are randomly generated in an individual, if a is less than b, the element values corresponding to the a + 1-b gene positions are inserted in front of the a gene position; otherwise, inserting the element value of b + 1-a-1 gene position behind the a gene position;
and operating the genes in the first gene segment, the second gene segment or the third gene segment in the second intermediate population by rotating the wheel disc in a mode of selecting exchange, reverse order or insertion to generate a third intermediate population.
5. The cell layout method of claim 4, wherein the acceptance probability of the simulated annealing algorithm is:
wherein p is the acceptance probability of the simulated annealing algorithm; f (l') is the fitness value of the individuals in the third intermediate population; f (l) is the fitness value of the individuals in the second intermediate population; k is the annealing rate; t is the annealing temperature;
when the fitness value of the third intermediate population individuals is larger than that of the second intermediate population individuals, the third intermediate population individuals are reserved; otherwise, when the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, accepting the third intermediate population individuals; when the random number in the simulated annealing algorithm is greater than the acceptance probability, a second population of individuals is retained.
6. A cell placement system based on a hybrid algorithm, comprising:
the unit layout model building module is used for building the unit layout model by assuming that the projection of the machine to the plane is rectangular, taking the minimum total carrying cost as an objective function and taking the constraint conditions that each manufacturing unit at least comprises one machine, the machines in the same manufacturing unit cannot be overlapped in the directions of a transverse axis and a longitudinal axis and the transverse coordinate of the mass center of each machine is in the width interval of the corresponding manufacturing unit; wherein, the total carrying cost is the sum of the carrying cost of the parts in the unit and the carrying cost of the parts between the units for finishing all parts processing;
the initial individual building module is used for building and repairing N initial individuals by adopting a three-section coding method, wherein each individual comprises three gene sections, and the element values of the three gene sections respectively represent the manufacturing unit number, the machine sequencing priority value and the machine placing direction corresponding to each machine according to the gene arrangement sequence;
the construction module of the first intermediate population is used for selecting a preset number of individuals to carry out crossing, variation and reinsertion operations by adopting a roulette method according to the proportion of the fitness value of the initial population and the fitness probability to obtain a first intermediate population;
the second intermediate population building module is used for combining the repaired first intermediate population with the initial population after repairing the individuals in the first intermediate population, and selecting N individuals with the optimal population fitness values after combination as a second intermediate population;
the generating module of the third intermediate population is used for exchanging, reversing or inserting the individual gene segments in the second intermediate population in a roulette mode to generate the third intermediate population;
and the unit layout acquisition module is used for comparing the fitness values of the individuals in the third intermediate population and the individuals in the corresponding second intermediate population, updating the second intermediate population by combining the acceptance probability of the simulated annealing algorithm, and decoding the individuals corresponding to the optimal fitness values in the second intermediate population of the last iteration to serve as a final unit layout method.
7. The cell layout system according to claim 6, wherein the manufacturing cell code corresponding to the element value in the individual is represented by the corresponding manufacturing cell code of each machine as the first gene segment of the individual; taking the gene segment of which the element value in the individual represents the machine-ordered priority value as a second gene segment of the individual; taking the gene segment with the element value representing the machine placing direction in the individual as a third gene segment of the individual;
a method of repairing an individual comprising the steps of:
calculating the number of machines in an individual assigned to each manufacturing unit;
and judging whether the number of machines in each manufacturing unit is more than or equal to 1, if so, keeping the current individual, and otherwise, randomly initializing the first gene segment of the individual again.
8. The cell layout method of claim 7, wherein the building blocks of the first intermediate population comprise:
the first wheel disc construction unit is used for constructing a wheel disc according to the fact that the initial population fitness value is in direct proportion to the fitness probability;
an individual screening unit for randomly generating 0-1 real number and a rotary wheel disk N P s Selecting an individual corresponding to the real number falling into the sector area each time as a selected individual; wherein, P s A selection probability for a selection operation;
the gene crossing processing unit is used for randomly generating a crossing point on the first gene segment and the second gene segment corresponding to each group of individuals by taking two selected adjacent individuals as a group, and crossing the genes of the corresponding gene segments before the crossing point; randomly selecting a plurality of gene intersections in the second gene segment;
the gene variation processing unit is used for respectively generating two variation points on the crossed individuals corresponding to the three gene segments and reversing the gene segments between the two variation points;
gene insertion processing unit for generating cross mutation containing NxP s Inserting N-N P with optimal fitness value in initial population into population of individuals s And generating a first intermediate population by each individual.
9. The cell layout method according to claim 8, wherein the generating module of the third middle population comprises:
the second wheel disc construction unit is used for assigning probability values with different sizes to the exchange, reverse order and insertion modes to construct a wheel disc; wherein, the exchange mode is to randomly generate two cross points in an individual and exchange the element values of the gene positions of the two cross points; the reverse order mode is that two points are randomly generated in an individual, and all element values of the gene positions between the two points are exchanged in the reverse order; the insertion mode is that two points a and b are randomly generated in an individual, if a is less than b, the element values corresponding to the a + 1-b gene positions are inserted in front of the a gene position; otherwise, inserting the element value of b + 1-a-1 gene position behind the a gene position;
and the third intermediate population generating unit is used for selecting an exchange, reverse order or insertion mode to operate the genes in the first gene segment, the second gene segment or the third gene segment in the second intermediate population through rotating the wheel disc so as to generate a third intermediate population.
10. The cell layout method of claim 9, wherein the acceptance probability of the simulated annealing algorithm is:
wherein p is the acceptance probability of the simulated annealing algorithm; f (l') is the fitness value of the individuals in the third intermediate population; f (l) is the fitness value of the individuals in the second intermediate population; k is the annealing rate; t is the annealing temperature;
when the fitness value of the third intermediate population individuals is larger than that of the second intermediate population individuals, the third intermediate population individuals are reserved; otherwise, when the random number in the simulated annealing algorithm is less than or equal to the acceptance probability, accepting the third intermediate population individuals; when the random number in the simulated annealing algorithm is greater than the acceptance probability, a second population of individuals is retained.
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