A Pareto optimization method and system for heterogeneous parallel machine production planning considering machine stability

Abstract

The present invention relates to the technical field of computational intelligence and production plan optimization, and discloses a Pareto optimization method for heterogeneous parallel machine production plan considering machine stability. The method uses a probabilistic learning model to accumulate excellent information of historical iterations to guide the construction of auxiliary populations, guide the algorithm to search for promising areas, and combine the main population and the auxiliary population through a population cooperation mechanism to generate the next generation of the main population, so that useful individual information can be continued, improve search efficiency, and further improve algorithm performance. After iterative search, the set of high-quality non-dominated solutions in the population will be retained, which can provide decision makers with diverse choices and decision support. The method is experimentally compared with advanced algorithms on examples of different scales. The present invention shows good algorithm performance, can provide decision makers with a set of high-quality non-dominated solutions that balance cost and machine stability, and provide important management inspiration for the formulation of actual production plans of enterprises.

Description

Technical Field

[0001] The present invention relates to the technical field of computational intelligence and production plan optimization, and in particular to a Pareto optimization method and system for heterogeneous parallel machine production planning taking into account machine stability. Background Art

[0002] Cost control is an important issue that manufacturing enterprises have always faced in their development process. Enterprises need to ensure product quality and production efficiency while reducing production costs as much as possible to improve competitiveness and profitability. Reducing costs as much as possible has always been a key strategy for major companies to pursue profits. With the continuous development of science and technology, enterprises also pay attention to stability and safety management. The stability of the machine is crucial to the smooth operation of the production process. Once a machine fails or becomes unstable, it will not only affect production efficiency, but may also lead to safety accidents. Therefore, enterprises need to continuously strengthen the maintenance and regular inspection of machinery and equipment to ensure their stability and reliability. Only by ensuring the stability of machinery and equipment can enterprises improve production efficiency, reduce production costs, and thus achieve the goal of maximizing profits.

[0003] In modern manufacturing, paying attention to the stability management of machinery and equipment has become an important means for enterprises to improve their competitiveness and sustainable development. However, in actual production, minimizing costs and maximizing machine stability are usually two conflicting goals. This is because some common measures to reduce costs and pursue profits often have an adverse effect on machine stability. Comprehensive consideration of machine stability and cost can not only reduce production costs and improve production efficiency, but also improve production safety and stability, promote the sustainable development of enterprises, and improve production flexibility. Therefore, for production planning optimization problems, it is of great significance to comprehensively consider machine stability and cost.

[0004] Although the parallel machine production planning problem considering machine stability has great application significance, it is difficult to solve this problem. The single-objective production planning problem considering cost is NP-hard, and the production planning problem considering both cost and machine stability is naturally NP-hard. Generally speaking, minimizing the total processing cost and maximizing the stability of the machine system are two contradictory goals, which means that there is no solution that can optimize both goals at the same time. Only a set of non-dominated solutions can provide valuable information for decision makers. Summary of the invention

[0005] 1. Technical issues to be resolved

[0006] In view of the deficiencies in the prior art, the present invention provides a Pareto optimization method and system for heterogeneous parallel machine production planning that takes into account machine stability. It has the advantages of effectively obtaining high-quality non-dominated solution sets to provide decision support for managers, and solves the above-mentioned technical problems.

[0007] (II) Technical solution

[0008] To achieve the above object, the present invention provides the following technical solution: a Pareto optimization method for heterogeneous parallel machine production planning considering machine stability, comprising the following steps:

[0009] S1. Construct a mathematical model for the production planning problem of heterogeneous parallel machines that considers the stability of the machine system, and set constraints on the established model. The specific expression is as follows:

[0010]

[0011] st

[0012]

[0013] Where M represents the agent set, N represents the task set, i∈N, j∈M, f1 represents the total processing cost, f2 represents the inverse of the machine system stability, and F ij represents the cumulative processing cost of task i assigned to agent j, S ij represents the processing cost per unit time of task i assigned to agent j, ρ j represents the failure rate of agent j, Indicates that N data are accumulated. Indicates that M data are accumulated. Indicates the cumulative multiplication of M data, q i represents the processing demand of task i, Q j represents the upper limit of the processing capacity of agent j, x ij = 1 means task i is assigned to agent j, x ij = 0 means that task i is not assigned to agent j, and st represents the constraint;

[0014] S2. Construct a Pareto evolutionary algorithm based on probability learning to solve the production planning problem in step S1, which specifically includes the following steps:

[0015] S2.1 Coding

[0016] S2.2. Set the key parameters in the Pareto evolutionary algorithm based on probability learning, including population size, crossover probability, mutation probability, learning rate, and maximum number of non-updates;

[0017] S2.3, initialize the main population, initialize the ideal point Z = (z1, z2), and initialize the probability matrix, where z1 represents the minimum value of the objective function f1 in the population, and z2 represents the minimum value of the objective function f2 in the population;

[0018] S2.4. Perform fast non-dominated sorting on the population and calculate the crowding distance;

[0019] S2.5. Use binary tournament selection to select superior individuals for genetic operations (including crossover, mutation, and selection operations. If an infeasible solution is generated, repair it) to generate offspring;

[0020] S2.6. Merge the parent and offspring populations, perform fast non-dominated sorting on the merged population, and then perform elitist selection: Select the parent population for the next generation according to the non-dominated rank. Select those with a smaller non-dominated rank first, and for those with the same non-dominated rank, select those with a larger crowding distance until the selected population meets the population size limit. At the same time, update the main population to the population after elitist selection;

[0021] S2.7. Calculate the ideal point z’=(z’1, z’2) of the current population. If z’1 < z1 or z’2 < z2, update the ideal point, where z’1 and z’2 represent the minimum values of the objective function f1 and the objective function f2 in the current population respectively;

[0022] S2.8. Select the individual set composed of the two extreme points and the knee point of the first-layer non-dominated solutions after non-dominated sorting in the population of the current iteration to update the probability matrix;

[0023] S2.9. Determine whether the learning condition is met. The learning condition is whether the cumulative number of times the ideal point does not update reaches the maximum number of non-updates. If it reaches, execute step S2.10. If it does not reach, jump to execute step S2.5;

[0024] S2.10. Use the probability matrix to help predict new solutions (if an infeasible solution is generated, repair it) to construct an auxiliary population;

[0025] S2.11. The auxiliary population and the main population in the original iteration generate the next-generation main population based on the population cooperation mechanism;

[0026] S2.12. Determine whether the termination condition is reached. If the algorithm termination condition is not reached, return to S2.5 to continue the iteration.

[0027] As a preferred technical solution of the present invention, the fast non-dominated sorting method in step S2.5 is based on the Pareto dominance relationship, and all solutions in the current population are divided into L k (k≥1) non-dominated levels. All solutions at the L k level are dominated by the solutions at the other k - 1 levels in the population. The crowding distance calculation formula in step S2.4 is as follows:

[0028]

[0029] Among them, m refers to the direction of the m-th objective function, i.e., fm Direction, D I is the crowding distance of individual I, f m [I-1] is the current non-dominated level according to the objective function f m The objective function value of the individual before individual I after the function value is sorted, f m [I+1] is the current non-dominated level according to the objective function f m The objective function value of the individual behind individual I after the function value is sorted, Represents the objective function f in the current non-dominated level m The maximum value of Represents the objective function f in the current non-dominated level m The minimum value of The crowding distance of individual I is calculated by the sum of the distances of the individual in the directions of the two objective functions. The distance in the direction of the mth objective function is the difference between the function values ​​of the individual and the surrounding individuals I+1, I-1 on the mth objective function divided by the difference between the maximum and minimum values ​​on the mth objective function.

[0030] As a preferred technical solution of the present invention, the dominant individual selection criteria in step S2.5 are: first evaluate based on the non-dominant level, the one with a low non-dominant level is the dominant individual, if the non-dominant levels are the same, the crowding distances are compared, and the one with a large crowding distance is the dominant individual, if the non-dominant level and the crowding distance are the same, one is randomly selected as the dominant individual, thereby generating a offspring population.

[0031] As a preferred technical solution of the present invention, the expression of initializing the probability matrix in step S2.3 is as follows:

[0032]

[0033] Among them, P rM×N Represents the initialized probability matrix, M and N represent the number of rows and columns of the probability matrix respectively. Since the probability matrix records the assignment of tasks to agents, the size of N is consistent with the size of the task set, and the size of M is consistent with the size of the agent set.

[0034] As a preferred technical solution of the present invention, the probability matrix update expression in step S2.8 is as follows:

[0035]

[0036] Among them, i represents the task, j represents the agent, α represents the learning rate, and P r(j,i) (gen) represents the probability that task i of the gen generation is assigned to agent j, P r(j,i)(gen-1) represents the probability of task i being assigned to agent j in the gen-1 generation, and Po represents the probability of task i being assigned to agent j in the current generation, which stores the two extreme points and one knee point in the first layer of non-dominated solutions in the current iteration.

[0037] As a preferred technical solution of the present invention, the two extreme points are the points with the minimum total processing cost in the population and the solution with the minimum inverse of machine stability, and the knee point is the point on the Pareto front with the maximum distance to the straight line connected by the two extreme points.

[0038] As a preferred technical solution of the present invention, the specific steps of using the probability matrix to help predict the new solution and construct the auxiliary population in step S2.10 are:

[0039] S2.10.1, generate the same number of random task assignment paths as the main population size;

[0040] S2.10.2. A binary tournament and roulette wheel method is used to predict the assignment of tasks to agents one by one according to the task assignment path through a probability matrix. The hybrid method here specifically generates a random selection factor. If the random selection factor is 0, the binary tournament method is used to predict the new solution. If the random selection factor is 1, the roulette wheel method is used to predict the new solution.

[0041] The present invention also implements the above-mentioned Pareto optimization method for heterogeneous parallel machine production plan considering machine stability through a Pareto optimization system for heterogeneous parallel machine production plan considering machine stability, comprising:

[0042] Establish a module for establishing a mathematical model for heterogeneous parallel machine production planning considering the stability of the machine system with the optimization objectives of minimizing the total processing cost and maximizing the machine stability;

[0043] The optimization module is used to optimize the two optimization objectives in the above model by using the Pareto evolutionary algorithm based on probability learning.

[0044] Compared with the prior art, the present invention provides a Pareto optimization method and system for heterogeneous parallel machine production planning considering machine stability, which has the following beneficial effects:

[0045] The present invention opens up new ideas for the production planning optimization problem in balancing cost and stability by adopting a heterogeneous parallel machine production planning model considering machine stability; a Pareto evolutionary algorithm based on probability learning is proposed to solve the problem, which mainly includes: a problem-oriented heuristic method and a method integrating probability learning, using a probability matrix to record excellent individual information in historical iterations, using this information as valuable knowledge to help predict promising individuals and thus guide the search to promising areas, injecting vitality into the entire search process, and improving the overall performance of the algorithm. The Pareto optimization method can obtain a set of non-dominated solutions to the heterogeneous parallel machine production planning problem considering machine stability within an acceptable CPU time, for decision makers to select according to their preferences, and provide decision support and inspiration for enterprise production planning. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 It is the overall algorithm flow chart of the present invention.

[0047] Figure 2 Schematic diagram of the dual-population search framework of the present invention.

[0048] Figure 3 This is a diagram of the encoding method of the present invention.

[0049] Figure 4 Schematic diagram of the crossover operation of the present invention.

[0050] Figure 5 This is a schematic diagram of the variation operation of the present invention.

[0051] Figure 6 It is a schematic diagram of the repair operation of the present invention. DETAILED DESCRIPTION

[0052] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0053] See also Figure 1 - Figure 6 The present invention provides the following technical solution: a Pareto optimization method for heterogeneous parallel machine production planning considering machine stability, comprising the following steps:

[0054] S1. Construct a mathematical model for the production planning problem of heterogeneous parallel machines that considers the stability of the machine system, and set constraints on the established model. The specific expression is as follows:

[0055]

[0056] st(constraint)

[0057]

[0058] This formula indicates that each task can only be assigned to one agent, and each task can only be assigned once;

[0059]

[0060] This formula indicates that the tasks assigned to an agent cannot exceed the upper limit of the agent’s processing capacity;

[0061]

[0062] This formula indicates that the value range of the decision variable is only 0 and 1;

[0063] Where M represents the agent set, N represents the task set, i∈N, j∈M, f1 represents the total processing cost, f2 represents the inverse of the machine system stability, and F ij represents the cumulative processing cost of task i assigned to agent j, S ij represents the processing cost per unit time of task i assigned to agent j, e represents the natural logarithm, ρ j represents the failure rate of agent j, Indicates that N data are accumulated. Indicates that M data are accumulated. Indicates the cumulative multiplication of M data, q i represents the processing demand of task i, Q j represents the upper limit of the processing capacity of agent j, x ij = 1 means task i is assigned to agent j, x ij =0 means task i is not assigned to agent j;

[0064] S2. Construct a Pareto evolutionary algorithm based on probability learning to solve the production planning problem in step S1, which specifically includes the following steps:

[0065] S2.1 Coding. According to the characteristics of the problem, an agent-based integer vector π is used to represent a solution for a task assignment agent. For example, π = [2, 2, 4, 3, 1, 1] means that tasks 1 and 2 are assigned to agent 2, task 3 is assigned to agent 4, task 4 is assigned to agent 3, and tasks 5 and 6 are assigned to agent 1. The coding diagram is shown in Figure 3 As shown;

[0066] S2.2. Set the key parameters in the Pareto evolutionary algorithm based on probability learning, including population size, crossover probability, mutation probability, learning rate, and maximum number of non-updates;

[0067] S2.3. Initialize the main population, initialize the ideal point Z = (z1, z2), and initialize the probability matrix, where z1 represents the minimum value of the objective function f1 in the population, and z2 represents the minimum value of the objective function f2 in the population;

[0068] S2.4. Perform fast non-dominated sorting on the population and calculate the crowding distance;

[0069] S2.5. Use binary tournament selection to select superior individuals for genetic operations (including crossover, mutation, and selection operations. If an infeasible solution is generated, repair it) to generate offspring. Among them, the selection operation uses the binary tournament selection method to select high-quality parent individuals for genetic operations to generate offspring individuals. The selection basis of the binary tournament is: randomly select two individuals. When the non-dominated ranks are different, the individual with a smaller non-dominated rank will be selected. When the non-dominated ranks are the same, compare the crowding distances between individuals, and the individual with a larger crowding distance will be selected. If the non-dominated ranks and crowding distances are both the same, randomly select one of them. The crossover operation uses two-point crossover, and the schematic diagram of this crossover operation is as Figure 4 shown. The mutation operation uses scramble mutation, and the schematic diagram of this mutation operation is as Figure 5 shown;

[0070] S2.6. Combine the parent and offspring. Perform fast non-dominated sorting on the combined population and perform elitist selection: Select the parent population for the next generation according to the non-dominated rank. Select those with a smaller non-dominated rank first. If the non-dominated ranks are the same, select those with a larger crowding distance until the selected population meets the population size limit, and at the same time update the main population to the population after elitist selection;

[0071] S2.7. Calculate the ideal point Z' = (z'1, z'2) of the current population. If z'1 < z1 or z'2 < z2, update the ideal point, where z'1 and z'2 represent the minimum values of the objective function f1 and the objective function f2 in the current population respectively;

[0072] S2.8. Select the individual set composed of the two extreme points and the knee point of the first layer of non-dominated solutions after non-dominated sorting in the population of the current iteration to update the probability matrix;

[0073] S2.9. Determine whether the learning condition is met. The learning condition is whether the cumulative number of times the ideal point does not update reaches the maximum number of non-updates. If it is reached, execute step S2.10. If it is not reached, jump back to execute step S2.5;

[0074] S2.10. Use the probability matrix to help predict new solutions (if an infeasible solution is generated, repair it) to construct an auxiliary population;

[0075] S2.11, the auxiliary population and the main population in the original iteration generate the next generation of the main population based on the population cooperation mechanism;

[0076] S2.12. Determine whether the termination condition is met. If the algorithm termination condition is not met, return to S2.5 to continue iteration.

[0077] Furthermore, the infeasible solutions mentioned in steps S2.5 and S2.10 are repaired. Since infeasible solutions, i.e. solutions that violate constraints, may often appear in the PEAPL iteration process, and infeasible solutions obviously do not meet the requirements, it is necessary to repair the infeasible solutions to convert them into feasible solutions. According to the characteristics of the problem, the infeasible solution repair operation includes: checking whether there is a task allocation that violates the constraints starting from the first gene of the infeasible solution (i.e., from the first task allocation) according to the length of the solution (i.e., the number of tasks). If a task allocation that violates the constraints is found, find other agents other than the original agent that can reassign the task without violating the constraints, record these agents, put them into a set, store them, sort them from small to large according to the comprehensive cost value, and reassign the task that violates the original agent's constraints to the agent with the smallest comprehensive cost. The repair operation diagram is shown as follows. Figure 6 As shown, the calculation formula of comprehensive cost is as follows:

[0078] g i =F ij

[0079]

[0080] f ij =ε1gi'+ε2ui'

[0081] Where i represents the task, j represents the agent, and g i It refers to the cumulative processing cost of task i assigned to agent j, u i It refers to the functional cost of assigning task i to agent j. i with u i The dimensions of are inconsistent, and they need to be normalized separately when calculating the comprehensive cost. The Min-Max normalization method is used to normalize the cumulative processing cost and functional cost separately. gi' refers to the normalized cumulative processing cost of task i assigned to agent j, ui' refers to the normalized functional cost of task i assigned to agent j, and fv ij That is, it is the comprehensive cost of assigning task i to agent j. ε1 and ε2 are weight factors. Without loss of generality, ε1=ε2=0.5 is usually taken.

[0082] Furthermore, the non-dominated method in step S2.4 is based on the Pareto dominance relationship, which divides the current population into L k(k≥1) non-dominated levels, all L k The solutions at the level are dominated by the solutions at other k-1 levels in the population, where the Pareto dominance relationship is expressed as follows:

[0083] For the problem of minimizing the objective function value, individual x1 dominates individual x2 if and only if all objective function values ​​of individual x1 are less than or equal to all objective function values ​​of individual x2.

[0084] The calculation formula of the crowding distance in step S2.4 is as follows:

[0085]

[0086] Among them, m refers to the direction of the mth objective function, that is, f m Direction, D I is the crowding distance of individual I, f m [I-1] is the current non-dominated level according to the objective function f m The objective function value of the individual before individual I after the function value is sorted, f m [I+1] is the current non-dominated level according to the objective function f m The objective function value of the individual behind individual I after the function value is sorted, Represents the objective function f in the current non-dominated level m The maximum value of Represents the objective function f in the current non-dominated level m The minimum value of The crowding distance of individual I is calculated by the sum of the distances of the individual in the directions of the two objective functions. The distance in the direction of the mth objective function is the difference between the function values ​​of the individual and the surrounding individuals I+1, I-1 on the mth objective function divided by the difference between the maximum and minimum values ​​on the mth objective function.

[0087] Furthermore, the expression of the initialization probability matrix in step S2.3 is as follows:

[0088]

[0089] in, Represents the initialized probability matrix, M and N represent the number of rows and columns of the probability matrix respectively. Since the probability matrix records the assignment of tasks to agents, the size of N is consistent with the size of the task set, and the size of M is consistent with the size of the agent set.

[0090] Furthermore, the probability matrix update expression in step S2.8 is as follows:

[0091]

[0092] Among them, i represents the task, j represents the agent, α represents the learning rate, (gen) represents the probability that task i of the gen generation is assigned to agent j, (gen-1) represents the probability of task i being assigned to agent j in the gen-1 generation, and Po represents the probability of task i being assigned to agent j in the current generation, which stores the two extreme points and one knee point in the first layer of non-dominated solutions in the current iteration.

[0093] Furthermore, in step S2.10, the specific steps of using the probability matrix to help predict the new solution and construct the auxiliary population are:

[0094] S2.10.1, generate the same number of random task assignment paths as the main population size;

[0095] S2.10.2. A binary tournament and roulette wheel method is used to predict the assignment of tasks to agents one by one according to the task assignment path through a probability matrix. The hybrid method here specifically generates a random selection factor. If the random selection factor is 0, the binary tournament method is used to predict the new solution. If the random selection factor is 1, the roulette wheel method is used to predict the new solution.

[0096] In order to verify the effectiveness of the Pareto Evolutionary Algorithm (PEAP) based on probability learning proposed in this paper, the algorithm is compared with two mainstream advanced algorithms NSGA-II and MOEA / D for solving examples of different scales. The example scales are (number of tasks × number of agents) 59×4, 59×6, 59×8, 66×4, 66×6, 66×8, 162×4, 162×6, 162×8, 245×4, 245×6, 245×8, 310×4, 310×6, 310×8, a total of 15 examples. To ensure the fairness of the comparison, all algorithms are independently repeated 11 times on each example. First, PEAPL uses the maximum number of iterations of 1000 as the algorithm termination condition. At the same time, the corresponding CPU time of PEAPL iteration 1000 times is recorded, and this time is used as the algorithm termination condition of the other two algorithms. All algorithms are based on an Intel I5 processor (1.8GHZ), 8GB of memory, Win10 operating system, and Python 3.9 programming environment.

[0097] In order to verify the comprehensive quality of the approximate Pareto frontier finally obtained by the above algorithm and thus evaluate the algorithm performance, the present invention adopts three evaluation indicators, namely: convergence, diversity, and dominance. k is obtained by k algorithms. Since the true Pareto front of this problem is unknown, S1,…S k The approximate Pareto front S* is obtained by merging the solution sets of the non-dominated solution sets.xy (S k) Denotes the solution x∈S k The shortest Euclidean distance to the solution on S*;

[0098] |S k | indicates S k The number of solutions. k The smaller the Pareto front, the better the convergence. The convergence index (γ) is calculated as follows:

[0099]

[0100] Among them, d ξk Indicates S k The Euclidean distance of the solution to the highest point in S* on the vertical axis, d ηk Indicates S k The Euclidean distance on the horizontal axis from the farthest point in S*, d λk Indicates S k The Euclidean distance between two consecutive solutions in , Indicates S k The average Euclidean distance between all solutions in , Δ k The smaller it is, the better the diversity of the Pareto frontier. The diversity index calculation (Δ) is as follows:

[0101]

[0102] The dominant index is the index indicating S k The number of solutions in S* that are not dominated by the solutions in S*, y<x means that solution y dominates solution x, Ω k The larger the value, the better the dominance of the Pareto frontier. The calculation formula of the dominance index (Ω) is as follows:

[0103]

[0104] The experimental results are shown in Tables 1 to 3, which are the statistical results of the convergence, diversity, and dominance indicators of the non-dominated solutions obtained by the three algorithms PEAPL, NSGA-II, and MPEA / D for solving 15 examples. Bold indicates the optimal value corresponding to each example. From Table 1, it can be seen that in terms of convergence indicators, PEAPL is better than the other two algorithms in 73.3% of the 15 examples. From Table 2, it can be seen that in terms of diversity indicators, PEAPL is better than the other two algorithms in 66.7% of the 15 examples. From Table 3, it can be seen that in terms of diversity indicators, PEAPL is better than the other two algorithms in 73.3% of the 15 examples. In general, the performance of PEAPL is better than the other two algorithms, so the algorithm PEAPL proposed in the present invention has effectiveness and superiority.

[0105] Table 1 Comparison results of convergence index (γ) of PEAPL, NSGA-II and MOEAD

[0106]

[0107]

[0108] Table 2 Comparison results of diversity index (Δ) of PEAPL, NSGA-II and MOEAD

[0109]

[0110] Table 3 Comparison results of PEAPL, NSGA-II, and MOEAD dominance index (Ω)

[0111]

[0112] The present invention also implements the above-mentioned Pareto optimization method for heterogeneous parallel machine production plan considering machine stability through a Pareto optimization system for heterogeneous parallel machine production plan considering machine stability, comprising:

[0113] Establish a module for establishing a mathematical model for heterogeneous parallel machine production planning considering the stability of the machine system with the optimization objectives of minimizing the total processing cost and maximizing the machine stability;

[0114] The optimization module is used to optimize the two optimization objectives in the above model by using the Pareto evolutionary algorithm based on probability learning.

[0115] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A Pareto optimization method for production planning of heterogeneous parallel machines considering machine stability, characterized by: It includes the following steps: S1. Construct a mathematical model for the heterogeneous parallel machine production planning problem considering the stability of the machine system, and set constraints for the established model. The specific expressions are as follows: Where M represents the agent set, N represents the task set, i∈N, j∈M, f1 represents the total processing cost, f2 represents the inverse of the machine system stability, and F ij represents the cumulative processing cost of task i assigned to agent j, S ij represents the processing cost per unit time of task i assigned to agent j, ρ j represents the failure rate of agent j, Indicates that N data are accumulated. Indicates that M data are accumulated. Indicates the cumulative multiplication of M data, q i represents the processing demand of task i, e represents the natural logarithm, e * represents the exponential function with natural logarithm e as base, Q j represents the upper limit of the processing capacity of agent j, x ij = 1 means task i is assigned to agent j, x ij =0 means task i is not assigned to agent j; S2. Construct a Pareto evolutionary algorithm based on probability learning to solve the production planning problem in step S1. The specific steps are as follows: S2.

1. Encoding; S2.

2. Set the key parameters in the Pareto evolutionary algorithm based on probability learning. The key parameters include population size, crossover probability, mutation probability, learning rate, and maximum non-update times; S2.

3. Initialize the main population, initialize the ideal point Z = (z1, z2), and initialize the probability matrix, where z1 represents the minimum value of the objective function f1 in the population, and z2 represents the minimum value of the objective function f2 in the population; S2.

4. Perform fast non-dominated sorting on the population and calculate the crowding distance; S2.

5. Use binary tournament selection to select dominant individuals for genetic operations to generate offspring. The genetic operations include crossover, mutation, and selection operations. If an infeasible solution is generated, it needs to be repaired; S2.

6. Merge the parent and offspring populations, perform fast non-dominated sorting on the merged population, and perform elitist selection: Select the parent population for the next generation according to the non-dominated rank. First, select those with a smaller non-dominated rank. If the non-dominated ranks are the same, then select those with a larger crowding distance until the selected population meets the population size limit. At the same time, update the main population to the population after elitist selection; S2.

7. Calculate the ideal point Z' = (z'1, z'2) of the current population. If z'1 < z1 or z'2 < z2, then update the ideal point, where z'1 and z'2 represent the minimum values of the objective function f1 and the objective function f2 in the current population respectively; S2.

8. Select the individual set composed of the two extreme points and the knee point of the first-layer non-dominated solutions after non-dominated sorting in the population of the current iteration to update the probability matrix; S2.

9. Determine whether the learning condition is reached. The learning condition is whether the cumulative number of times the ideal point does not update reaches the maximum non-update times. If it reaches, then execute step S2.

10. If it does not reach, then jump to execute step S2.5; S2.

10. Use the probability matrix to help predict new solutions to construct an auxiliary population. If an infeasible solution is generated, it needs to be repaired; S2.

11. The auxiliary population and the main population in the original iteration generate the next-generation main population based on the population cooperation mechanism; S2.

12. Determine whether the termination condition is reached. If the algorithm termination condition is not reached, return to S2.5 to continue the iteration.

2. The Pareto optimization method for heterogeneous parallel machine production planning considering machine stability according to claim 1, characterized in that: The fast non-dominated sorting method in step S2.5 is based on the Pareto dominance relationship, which divides all solutions in the current population into L k (k≥1) non-dominated levels, all L k The solutions of the level are dominated by the solutions of other k-1 levels in the population. The crowding distance calculation formula in step S2.4 is as follows: Among them, m refers to the direction of the mth objective function, that is, f m Direction, D I is the crowding distance of individual I, f m [I-1] is the current non-dominated level according to the objective function f m The objective function value of the individual before individual I after the function value is sorted, f m [I+1] is the current non-dominated level according to the objective function f m The objective function value of the individual behind individual I after the function value is sorted, Represents the objective function f in the current non-dominated level m The maximum value of Represents the objective function f in the current non-dominated level m The minimum value of The crowding distance of individual I is calculated by the sum of the distances of the individual in the directions of the two objective functions. The distance in the direction of the mth objective function is the difference between the function values ​​of the individual and the surrounding individuals I+1, I-1 on the mth objective function divided by the difference between the maximum and minimum values ​​on the mth objective function.

3. The Pareto optimization method for heterogeneous parallel machine production planning considering machine stability according to claim 1, characterized in that: The dominant individual selection criterion in step S2.5 is as follows: First, evaluate according to the non-dominated rank. Those with a lower non-dominated rank are dominant individuals. If the non-dominated ranks are the same, then compare the crowding distance. Those with a larger crowding distance are dominant individuals. If the non-dominated ranks and the crowding distances are the same, then randomly select one as the dominant individual, and thus generate the offspring population.

4. The Pareto optimization method for heterogeneous parallel machine production planning considering machine stability according to claim 1, characterized in that: The expression for initializing the probability matrix in step S2.3 is as follows: Among them, P rM×N Represents the initialized probability matrix, M and N represent the number of rows and columns of the probability matrix respectively. Since the probability matrix records the assignment of tasks to agents, the size of N is consistent with the size of the task set, and the size of M is consistent with the size of the agent set.

5. The Pareto optimization method for heterogeneous parallel machine production planning considering machine stability according to claim 4, characterized in that: The expression for updating the probability matrix in step S2.8 is as follows: Among them, i represents the task, j represents the agent, α represents the learning rate, and P r(j,i) (gen) represents the probability that task i of the gen generation is assigned to agent j, P r(j,i) (gen-1) represents the probability of task i being assigned to agent j in the gen-1 generation, and Po represents the probability of task i being assigned to agent j in the current generation, which stores the two extreme points and one knee point in the first layer of non-dominated solutions in the current iteration.

6. The Pareto optimization method for heterogeneous parallel machine production planning considering machine stability according to claim 5, characterized in that: The two extreme points are the point with the minimum total processing cost in the population and the solution with the minimum inverse of machine stability, and the knee point is the point on the Pareto front with the maximum distance to the straight line connected by the two extreme points.

7. The Pareto optimization method for heterogeneous parallel machine production planning considering machine stability according to claim 1, characterized in that: The specific steps of constructing an auxiliary population by using the probability matrix to help predict the new solution in step S2.10 are: S2.10.1, generate the same number of random task assignment paths as the main population size; S2.10.

2. A binary tournament and roulette wheel method is used to predict the assignment of tasks to agents one by one according to the task assignment path through a probability matrix. The hybrid method here specifically generates a random selection factor. If the random selection factor is 0, the binary tournament method is used to predict the new solution. If the random selection factor is 1, the roulette wheel method is used to predict the new solution.

8. A Pareto optimization system for heterogeneous parallel machine production planning considering machine stability, implementing a Pareto optimization method for heterogeneous parallel machine production planning considering machine stability as claimed in any one of claims 1 to 7, characterized in that: include: Establish a module for establishing a mathematical model for heterogeneous parallel machine production planning considering the stability of the machine system with the optimization objectives of minimizing the total processing cost and maximizing the machine stability; The optimization module is used to optimize the two optimization objectives in the above model by using the Pareto evolutionary algorithm based on probability learning.

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