An optimization method for the number of tardy jobs in single - machine scheduling considering release and due - date constraints

Abstract

The present invention discloses an optimization method for the number of tardy jobs in single - machine scheduling considering release and due - date constraints. The method particularly considers a dedicated pseudo - symmetry breaking method based on permutation group theory to reduce the computational complexity of the proposed algorithm. Based on the parity of permutations of the symmetric group of the original search space, a problem - specific pseudo - symmetry property is introduced, which halves the search space of the algorithm. The algorithm explores promising regions in the reduced search space, uses position - based crossover to generate offspring solution sequences, relies on simulated annealing based on pseudo - symmetry breaking to deeply examine neighborhood solutions, and relies on a population update mechanism based on distance and quality to ensure a healthy population. Through experimental comparisons with the current state - of - the - art algorithms on instances of multiple scales, the present invention performs excellently in various metrics, can obtain better solutions in a shorter time, and can provide higher - quality solutions for decision - makers in a short time, providing important management guidance for the actual production scheduling of enterprises.

Description

Technical Field

[0001] The present invention relates to the technical fields of intelligent optimization and production scheduling optimization, and particularly relates to an optimization method for the number of tardy jobs in single-machine scheduling considering release and due-date constraints. Background Art

[0002] In modern manufacturing, the production scheduling problem is an important type of optimization problem, and its core goal is to reasonably allocate production resources to complete production tasks in an optimal manner. As one of the basic problems of production scheduling, the single-machine scheduling problem has received extensive attention. Especially in actual production, the release time and due-date of jobs are important constraints that cannot be ignored. The technical solution described in this article aims to solve the single-machine scheduling problem with release and due-date constraints and optimize the number of tardy jobs, which has important theoretical and practical value. In the single-machine scheduling problem with release and due-date constraints, each job not only has a processing time, but also a release time and a due-date. The release time stipulates when the job can start processing, while the due-date stipulates when the job should be completed before. If the job fails to be completed before the due-date, tardiness will occur, which may lead to penalties, customer complaints and other adverse consequences in actual production.

[0003] In the existing scheduling methods, many studies focus on using exact algorithms to process simplified small-scale problem instances. The solution methods for such problems are relatively mature, mainly based on branch and bound, and can only handle problems with at most 300 jobs. However, these methods have difficulties such as long solution time, high requirements for computer performance, and inability to handle larger-scale problems, making it difficult to be applied in actual industry. In addition, when considering release time and due-date constraints, the complexity of the problem increases significantly, making it more difficult to directly apply the existing scheduling algorithms. The existing scheduling methods mainly focus on minimizing the total tardiness or the maximum tardiness, while ignoring the problem of optimizing the number of tardy jobs. In actual production, reducing the number of tardy jobs often has more direct economic and management significance than minimizing tardiness. Therefore, the present invention proposes improvements for the problems of long solution time, high computer performance requirements, and small problem scale that can be handled by the existing algorithms. Summary of the Invention

[0004] Aiming at the deficiencies of the existing technology, the present invention provides an optimization method and system for the number of tardy jobs in single-machine scheduling considering release and due-date constraints, which has the advantages of reducing the search space, high solution efficiency and good solution quality, and solves the above technical problems.

[0005] To achieve the above object, the present invention provides the following technical solution: A method for the number of tardy jobs in single-machine scheduling considering release and due-date constraints, comprising the following steps:

[0006] S1. Construct a mathematical model for the single-machine scheduling problem with release and deadline constraints, and set constraints on the model. The specific expression is as follows:

[0007]

[0008] st

[0009]

[0010] In the formula, N represents the number of workpieces to be processed, π represents the sequence of workpieces to be processed, * represents the optimal solution, Π N represents the set of all possible workpiece sequences, i∈N, Respectively represent the workpiece π i Release time, processing time and cut-off time, Indicates workpiece π i The completion time, Indicates workpiece π i-1 The completion time, Indicates workpiece π i Whether it is delayed, delayed is 1, normal delivery is 0;

[0011] S2, constructing a memetic permutation group algorithm based on permutation group theory to solve the scheduling problem in step S1;

[0012] S2.1, read the optimal sequence of the model in step S1 as the descendant solution sequence;

[0013] S2.2. Initialize the algorithm search area mark S flag And make S flag = 0 or 1, which means that the algorithm determines whether to search on the odd side or the even side in the odd-even search space. flag = 0, the algorithm searches on the even side, S flag =1, the algorithm searches on the odd side;

[0014] The permutation group theory is specifically given a symmetric group, which consists of two sets. All possible permuted elements in the symmetric group can be divided into an odd part and an even part, and each set is composed of the factorials of each element on one side of the symmetric group. The one-to-one mapping from one set to another is called the permutation group theory.

[0015] Based on the above theory, the algorithm’s search space can be divided into two sequences with the same number;

[0016] S2.3, according to the populations located on the same side obtained in S2.2, use a local search algorithm to optimize the data generated by S1 in each population, wherein the local search algorithm is an algorithm for solving optimization problems;

[0017] S2.4. Use the position-based crossover method for the data optimized in S2.3 to generate a new offspring solution sequence;

[0018] S2.5. Repair the offspring solution sequences located on the odd side of the search space in the new offspring solution sequence generated in S2.4 to the even side of the search space. If there are no offspring solution sequences on the odd side, skip this step. The specific steps are as follows:

[0019] S2.5.1. Set β N = interchange(π new , k, j) (k, j ∈ N, k ≠ j), where β N is the operation of interchanging the k-th workpiece and the j-th workpiece of π new , π new is the new offspring solution sequence, and N represents the number of workpieces to be processed;

[0020] S2.5.2. According to β N set in S2.5.1 and the permutation parity, obtain the relationship between π new and β N satisfies the following formula:

[0021] Then

[0022] In the formula, is the set of all even permutations in Π N , is the set of all odd permutations, that is, if the offspring solution sequence falls into the search space marked on the odd side, only by interchanging the positions of any two workpieces in π new , the parity of π new can be repaired to the even side marked by the search area;

[0023] S2.6. Use the simulated annealing program based on the pseudo-symmetry breaking method to check the neighborhood solutions of the individuals obtained by substituting the offspring solution sequences in S2.5 into the S1 model. The specific expression of the pseudo-symmetry breaking method is as follows:

[0024] η N = Insert(π new , k, j) (k, j ∈ N, k ≠ j)

[0025] In the formula, η N is obtained by inserting the k-th workpiece into the position of the j-th workpiece of π new . The relationship between k and j only needs to satisfy the following formula to ensure that the insertion operation does not affect the parity of the solution. The specific expression is as follows:

[0026] ηN = Insert(π new , k, j), (k, j ∈ N, k ≠ j) ∧ (|k - j| mod 2 ≠ 0)

[0027] In the formula, mod represents taking the remainder.

[0028] S2.7. Update the population according to the population update mechanism based on distance and quality, such that the number of tardy jobs in the solution generated in S2.6 is less than that in the solution with the most tardy jobs in S1. Then add the generated new solution to the population; otherwise, discard the generated new solution and skip this step. The specific steps are as follows:

[0029] S2.7.1. Determine whether the objective value of the offspring solution sequence to be inserted into the population is better than the worst solution in S1, that is, whether there is a tardy situation in S1. If it is better, execute step S2.7.2; otherwise, jump to step S2.7.4;

[0030] S2.7.2. Determine whether the sequence of the offspring solution sequence to be inserted into the population is the same as any sequence in the population. The calculation method is as follows:

[0031]

[0032] In the formula, is the relative position of the i-th job in π 1 in π 2 . If simi(π 1 , π 2 ) ≠ 0, then execute step S2.7.3; otherwise, jump to step S2.7.4. π 1 represents the sequence of the offspring solution sequence after being checked in S2.6, and π 2 represents any sequence;

[0033] S2.7.3. Replace the offspring solution sequence with the worst objective value in the population with the offspring solution sequence in S2.7.2;

[0034] S2.7.4. Discard the worst offspring solution sequence;

[0035] S2.8. Determine whether the termination condition of the algorithm is reached. The termination condition is that the running time of the algorithm is one hour. If it is reached, execute step S2.9; if not, jump to execute step S2.4;

[0036] S2.9. Output the solution found during the search process of the algorithm. Substitute the solution into the S1 model, and the obtained solution has the best minimum effect;

[0037] S3. Verify the effectiveness of the method. The specific steps are as follows:

[0038] To verify the effectiveness of the memetic permutation group algorithm based on the permutation group theory proposed in the present invention, this algorithm is compared with three currently mainstream advanced algorithms, namely VBIH, HDHHO, and IGwS, as well as the general mixed-integer programming solver CPLEX for solving examples of different scales;

[0039] The scales of the examples are 50, 300, and 700 respectively. There are 16 different examples for each scale, with a total of 48 examples. Due to memory and time limitations, the solver CPLEX can only find feasible solutions (two of which can obtain the optimal solution) for 16 examples with a scale of 50. To ensure the fairness of the comparison, all algorithms are independently and repeatedly run 20 times on each example, and the running time of 3600 seconds is used as the termination condition for all algorithms including the solver;

[0040] All algorithms use an Intel E3-1220 v5 processor (3.0GHz), 8GB of memory, a Linux operating system, and a C++ programming environment, and different random number seeds are used for each iteration;

[0041] To verify the comprehensive quality of the approximate optimal solutions finally obtained by the above algorithms, and thus evaluate the algorithm performance, the present invention adopts three evaluation indicators, namely:

[0042] 1. The historical best solution (BEST) found by each algorithm in 20 independent and repeated runs

[0043] 2. The average value (AVG) of the optimal solutions found by each algorithm in 20 independent and repeated runs

[0044] 3. The standard deviation (SD) of the optimal solutions found by each algorithm in 20 independent and repeated runs.

[0045] Advantages of the present invention

[0046] 1. The present invention discloses a method for optimizing the number of tardy jobs in single-machine scheduling with release and due-date constraints by using a memetic permutation group algorithm. This method particularly considers a dedicated pseudo-symmetry breaking method based on the permutation group theory to reduce the search space and computational workload of the proposed algorithm. Based on the symmetry group permutation parity of the original search space, the problem-specific pseudo-symmetry property in terms of the objective value is introduced. Then, the algorithm explores the promising search regions within the reduced search space, relying on position-based crossover to generate meaningful offspring solutions, relying on simulated annealing based on pseudo-symmetry breaking to deeply examine neighboring solutions, and relying on a population update mechanism based on distance and quality to ensure a healthy population. Through experimental comparison, the method of the present invention performs excellently in examples of different scales, and can provide higher-quality solutions for decision-makers in a shorter time, providing important management guidance for the actual production scheduling of enterprises.

[0047] 2. Compared with the prior art, the present invention can obtain a better solution within a reasonable time for actual industrial applications, and the performance of the algorithm is relatively stable. At the same time, it can also handle large-scale problems, which is of great help to actual applications. Description of the Drawings

[0048] Figure 1 is the overall flowchart of the algorithm of the present invention;

[0049] Figure 2 is the schematic diagram of position crossover based on the present invention;

[0050] Figure 3 is the schematic diagram of the parity exchange method based on the present invention;

[0051] Figure 4 is the schematic diagram of the insertion operation to ensure the parity remains unchanged based on the present invention. Detailed Embodiment

[0052] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative efforts belong to the scope of protection of the present invention.

[0053] The present invention provides the following technical solutions: A method for considering the number of tardy jobs in a single-machine scheduling with release and due-date constraints, as Figure 1 shown, includes the following steps:

[0054] S1. Construct a mathematical model for the single-machine scheduling problem considering release and due-date constraints, and set constraints for the model. The specific expressions are as follows:

[0055]

[0056] s.t.

[0057]

[0058]

[0059] In the formula, N represents the number of jobs to be processed, π represents the sequence of jobs to be processed, * represents the optimal solution, and Π N represents the set of all possible job sequences. i ∈ N, respectively represent the release time, processing time, and due date of job π i , represents the completion time of job π i , represents job π i-1Completion time, Indicates workpiece π i Whether it is overdue, 1 for overdue and 0 for normal delivery;

[0060] S2. Construct a memetic permutation group algorithm based on permutation group theory to solve the scheduling problem in step S1;

[0061] S2.1. Read the optimal sequence of the model in step S1 as the offspring solution sequence;

[0062] S2.2. Initialize the search area marker S of the algorithm flag And make S flag = 0 or 1, that is, determine that the algorithm searches on either the odd side or the even side of the parity search space. When S flag = 0, the algorithm searches on the even side, and when S flag = 1, the algorithm searches on the odd side;

[0063] The so-called permutation group theory specifically means that given a symmetric group, the symmetric group consists of two sets. All possible permuted elements in the symmetric group can be divided into an odd part and an even part, and each set is composed of the factorials of the elements on one side of the symmetric group. A one-to-one mapping from one set to another set is called permutation group theory;

[0064] Based on the above theory, the search space of the algorithm can be divided into two sequences with the same number;

[0065] S2.3. According to the populations on the same side obtained in S2.2, use the local search algorithm to optimize the data generated by S1 in each population. The local search algorithm is an algorithm used to solve optimization problems;

[0066] S2.4. Use the method of position-based crossover to generate new offspring solutions. As Figure 2 shown, randomly select several genes with independent probabilities in one of the parent solutions. The positions can be discontinuous. Copy the genes at these positions in parent 1 to the same positions in offspring 1, and then fill in the missing genes in offspring 1 in the order they appear in parent 2 to the vacant positions in offspring 1 to obtain offspring 1. The other offspring is obtained in the same way after exchanging the parent identities;

[0067] S2.5. Repair the offspring solution sequences on the odd side of the search space in the new offspring solution sequences generated in S2.4 to the search space on the even side. As Figure 3 shown, randomly select two different genes in the offspring solution, and exchange their positions to change the permutation parity of the solution. If there is no offspring solution sequence on the odd side, skip this step. The specific steps are as follows:

[0068] S2.5.1. Set βN = interchange(π new , k, j) (k, j ∈ N, k ≠ j), where β N is the operation of interchanging the k-th workpiece and the j-th workpiece of π new , and π new is the new offspring solution sequence, and N represents the number of workpieces to be processed;

[0069] S2.5.2. According to the β N set in S2.5.1 and the parity of the permutation, the relationship between π new and β N satisfies the following formula:

[0070] Then

[0071] In the formula, is the set of all even permutations in Π N , and is the set of all odd permutations, that is, if the offspring solution sequence falls into the search space on the odd side of the search marker, only by interchanging the positions of any two workpieces in π new , the parity of π new can be repaired to the even side marked by the search area;

[0072] S2.6. Use the simulated annealing program based on the pseudo-symmetry breaking method to check the neighborhood solutions of the individuals obtained by substituting the offspring solution sequence in S2.5 into the S1 model. Its insertion operation is as Figure 4 shown. Select two genes in the offspring solution that are separated by an odd distance, and insert one gene into the other position, which will not change the permutation parity of the solution. The specific expression of the pseudo-symmetry breaking method is as follows:

[0073] η N = Insert(π new , k, j) (k, j ∈ N, k ≠ j)

[0074] In the formula, η N is obtained by inserting the k-th workpiece into the position of the j-th workpiece of π new . Then, the relationship between k and j only needs to satisfy the following formula to ensure that the insertion operation does not affect the parity of the solution. The specific expression is as follows:

[0075] η N = Insert(π new , k, j), (k, j ∈ N, k ≠ j) ∧ (|k - j| mod 2 ≠ 0)

[0076] In the formula, mod represents taking the remainder.

[0077] S2.7. Update the population according to the population update mechanism based on distance and quality, such that the number of tardy jobs in the solution generated in S2.6 is less than that in the solution with the most tardy jobs in S1. Then add the generated new solution to the population; otherwise, discard the generated new solution and skip this step. The specific steps are as follows:

[0078] S2.7.1. Determine whether the objective value of the offspring solution sequence to be inserted into the population is better than the worst solution in S1, that is, whether there is a tardy situation in S1. If it is better, execute step S2.7.2; otherwise, jump to step S2.7.4;

[0079] S2.7.2. Determine whether the sequence of the offspring solution sequence to be inserted into the population is the same as any sequence in the population. The calculation method is as follows:

[0080]

[0081] In the formula, is the relative position of the i-th job in π 1 in π 2 . If simi(π 1 , π 2 ) ≠ 0, execute step S2.7.3; otherwise, jump to step S2.7.4. π 1 represents the sequence of the offspring solution sequence after being checked in S2.6, and π 2 represents any sequence;

[0082] S2.7.3. Replace the offspring solution sequence with the worst objective value in the population with the offspring solution sequence in S2.7.2;

[0083] S2.7.4. Discard the worst offspring solution sequence.

[0084] S2.8. Determine whether the termination condition of the algorithm is reached. The termination condition is that the running time of the algorithm is one hour. If it is reached, execute step S2.9; if not, jump to execute step S2.4;

[0085] S2.9. Output the solution found during the search process of the algorithm, substitute the solution into the S1 model, and the obtained solution has the best minimum effect;

[0086] S3. Verify the effectiveness of the method. The specific steps are as follows:

[0087] To verify the effectiveness of the memetic permutation group algorithm based on the permutation group theory proposed in the present invention, this algorithm is compared with three currently mainstream advanced algorithms, VBIH, HDHHO, and IGwS, and the general mixed-integer programming solver CPLEX for solving different-scale examples;

[0088] The scales of the numerical examples are 50, 300, and 700 respectively. There are 16 different numerical examples for each scale, with a total of 48 numerical examples. Due to memory and time limitations, the solver CPLEX can only find feasible solutions for 16 numerical examples with a scale of 50 (two of which can obtain the optimal solution). To ensure the fairness of comparison, all algorithms are independently and repeatedly run 20 times on each numerical example, and the running time of 3600 seconds is used as the termination condition for all algorithms including the solver.

[0089] All algorithms use an Intel E3-1220 v5 processor (3.0 GHz), 8 GB of memory, a Linux operating system, and a C++ programming environment. Different random number seeds are used for each iteration.

[0090] To verify the comprehensive quality of the approximate optimal solutions finally obtained by the above algorithms and thus evaluate the algorithm performance, the present invention adopts three evaluation indicators, namely:

[0091] 1. The historical optimal solution (BEST) found by each algorithm in 20 independent and repeated runs

[0092] 2. The average value (AVG) of the optimal solutions found by each algorithm in 20 independent and repeated runs

[0093] 3. The standard deviation (SD) of the optimal solutions found by each algorithm in 20 independent and repeated runs;

[0094] Table 1 shows the comparison results of CPLEX, VBIH, HDHHO, IGwS, and MPGA (the algorithm proposed in the present invention) for the case of 50 jobs. The bold numbers represent the optimal values corresponding to each numerical example. It can be seen from Table 1 that for all instances of this scale, the algorithm of the present invention is superior to the comparison algorithms in terms of BEST, AVG, and SD. For the two instances where CPLEX obtains the optimal solution, the algorithm of the present invention can also obtain this optimal solution. For the instances where CPLEX only obtains feasible solutions, the algorithm of the present invention can reach or even be better than the feasible solutions obtained by CPLEX.

[0095] Table 1 Comparison Results of CPLEX, VBIH, HDHHO, IGwS, and MPGA for 50-Scale Instances

[0096]

[0097] In the table, * represents the optimal solution, and the rest are all feasible solutions.

[0098] Table 2 shows the comparison results of the algorithm of the present invention and the comparative algorithm in instances of scale 300. It can be seen from the results that among the 16 instances at this scale, the best solutions of 15 instances are obtained by the algorithm of the present invention. For the only instance whose best solution is not obtained by the algorithm of the present invention, the result obtained by the algorithm of the present invention also has a very small gap with the best solution obtained by the comparative algorithm. In addition, in terms of the AVG and SD metrics, obviously the algorithm of the present invention has better stability than the comparative algorithm.

[0099] Table 2 Comparison Results of VBIH, HDHHO, IGwS and MPGA in Instances of Scale 300

[0100]

[0101]

[0102] Table 3 shows the comparison results of the algorithm of the present invention and the comparative algorithm in instances of scale 700. It can be seen from the results that all the best solutions of the 16 instances at this scale are obtained by the algorithm of the present invention. And in terms of the gap between the best solutions obtained by each algorithm, the best solution obtained by the present invention is significantly better than the best solution obtained by the comparative algorithm. In addition, in terms of the AVG and SD metrics, obviously the algorithm of the present invention has very good stability compared with the comparative algorithm. This also proves that the algorithm proposed by the present invention has significant advantages in dealing with large-scale problems, which can significantly increase the efficiency of dispatch decision-makers in practical applications and can provide better solutions in a shorter time.

[0103] Table 3 Comparison Results of VBIH, HDHHO, IGwS and MPGA in Instances of Scale 700

[0104]

[0105] Although the embodiments of the present invention have been shown and described, it will be understood by those of ordinary skill in the art that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. An optimization method for the number of delayed jobs in single-machine scheduling with release and deadline constraints, characterized by: The following steps are involved: S1. Construct a mathematical model for the single-machine scheduling problem with release and deadline constraints, and set constraints on the model; The specific expressions of the mathematical model and the constraints set on the model described in step S1 are as follows: st In the formula, N represents the number of workpieces to be processed, π represents the sequence of workpieces to be processed, * represents the optimal solution, Π N represents the set of all possible workpiece sequences, i∈N, Respectively represent the workpiece π i Release time, processing time and cut-off time, Indicates workpiece π i The completion time, Indicates workpiece π i-1 The completion time, Indicates workpiece π i Whether it is delayed, delayed is 1, normal delivery is 0; S2, constructing a memetic permutation group algorithm based on permutation group theory to solve the scheduling problem in step S1; Step S2 specifically includes the following steps: S2.1, read the optimal sequence of the model in step S1 as the descendant solution sequence; S2.

2. Initialize the algorithm search area mark S flag And make S flag = 0 or 1, which means that the algorithm determines whether to search on the odd or even side of the odd-even search space. flag = 0, the algorithm searches on the even side, S flag =1, the algorithm searches on the odd side; S2.3, based on the populations on the same side obtained in S2.2, use the local search algorithm to optimize the data generated by S1 in each population; S2.4, using the position-based crossover method on the data optimized in S2.3 to generate a new offspring solution sequence; S2.

5. Repair the sub-generation solution sequence on the odd side of the search space in the new sub-generation solution sequence generated in S2.4 to the search space on the even side. If there is no sub-generation solution sequence on the odd side, skip this step. S2.6, using a simulated annealing procedure based on the pseudo-symmetry breaking method, check the neighborhood solutions of the individuals obtained by substituting the offspring solution sequence of S2.5 into the S1 model; S2.7, update the population according to the population update mechanism based on distance and quality, so that the number of delayed workpieces of the solution generated by S2.6 is less than the solution with the largest number of delayed workpieces in S1, then add the generated new solution to the population, otherwise discard the generated new solution and skip this step; S2.8, determine whether the termination condition of the algorithm is met, the termination condition is that the algorithm runs for one hour, if it is met, execute step S2.9, if not, jump to step S2.4; S2.9, output the solution found during the algorithm search process, substitute the solution into the S1 model, and the solution obtained is the smallest and has the best effect; S3. Verify the effectiveness of the method.

2. The method for optimizing the number of delayed jobs in single-machine scheduling with release and deadline constraints according to claim 1, characterized in that: The specific steps of step S2.5 are as follows: S2.5.

1. Setting β N =interchange(π new ,k,j)(k,j∈N,k≠j), the β N is to convert π new The kth workpiece and the jth workpiece are exchanged, π new is the new sub-generation solution sequence, N represents the number of workpieces to be processed; S2.5.2, β set according to S2.5.1 N and permuting the parity gives π new and β N The relationship between them satisfies the following formula: In the formula, Yes N The set of all even permutations of , is the set of all odd permutations, that is, if the offspring solution sequence falls into the search space on the odd side of the search mark, just replace π new By exchanging the positions of any two workpieces, π new The parity of is fixed to the even side of the search area mark.

3. The method for optimizing the number of delayed jobs in single-machine scheduling with release and deadline constraints according to claim 1, characterized in that: In step S2.6, the specific expression based on the pseudo-symmetry breaking method is as follows: or N =Insert(π new ,k,j)(k,j∈N,k≠j) Where η N is π new Insert the kth workpiece into the position of the jth workpiece to obtain, then the relationship between k and j only needs to satisfy the following formula to ensure that the insertion operation does not affect the parity of the solution. The specific expression is as follows: η N =Insert(π new ,k,j),(k,j∈N,k≠j)∧(|kj|mod2≠0) In the formula, mod means the remainder.

4. The method for optimizing the number of delayed jobs in single-machine scheduling with release and deadline constraints according to claim 1, characterized in that: The rules of the population update mechanism based on distance and quality described in step S2.7 are as follows: S2.7.

1. Determine whether the target value of the offspring solution sequence to be inserted into the population is better than the worst solution in S1, that is, whether there is a delay in S1. If so, execute step S2.7.2; otherwise, jump to step S2.7.

4. S2.7.

2. Determine whether the sequence of the offspring solution sequence to be inserted into the population is the same as any sequence in the population. The calculation method is: In the formula, is π 1 The i-th workpiece is at π 2 The relative position in 1 ,π 2 )≠0, then execute step S2.7.3, otherwise jump to step S2.7.4, π 1 represents the sequence of subsequence solutions of S2.6 after checking, π 2 represents any sequence; S2.7.3, replace the offspring solution sequence with the worst target value in the population with the offspring solution sequence of S2.7.2; S2.7.

4. Discard the worst descendant solution sequence.

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