Flow shop scheduling optimization method based on preventive maintenance and degradation

Abstract

The invention discloses a flow shop scheduling optimization method based on preventive maintenance and degradation, and the method comprises the steps: considering the interaction among scheduling, production, maintenance and degradation for a flow shop scheduling problem, and building a mathematical model with the minimum completion time and the minimum equipment average idle time as an optimization target; the influence of equipment degradation on actual machining time and preventive maintenance time is considered, and a workpiece machining time degradation model and an equipment preventive maintenance time degradation model are established; and a preventive maintenance strategy is researched and established by considering the relationship among faults, maintenance, preventive maintenance and progress. And finally, obtaining a reasonable scheduling scheme according to the characteristics of the problem. According to the scheduling optimization method provided by the invention, the relationship among production, equipment maintenance and equipment degradation can be effectively coordinated and balanced, so that the efficiency of a manufacturing system is improved, the production cost is saved, and optimal utilization of resources is realized.

Description

Technical Field

[0001] The invention relates to the technical field of flow shop scheduling, and in particular to a flow shop scheduling optimization method based on preventive maintenance and degradation. Background Art

[0002] In the actual production system, as the manufacturing system continues to run, equipment degradation and equipment failure are particularly prone to occur during the production process. Therefore, it is necessary to perform preventive maintenance on the equipment to reduce the failure rate and ensure stable operation of the equipment.

[0003] Preventive maintenance (PM) is to inspect, maintain, and service the equipment before it fails, so as to reduce the probability of accidents and ensure that the equipment is operating in an efficient and stable state. Preventive maintenance is required several times during the life cycle, and if an occasional failure occurs, post-fault maintenance is performed immediately after the accidental failure.

[0004] However, preventive maintenance activities will take up production time, which may lead to additional consumption of production resources. For the production scheduling system, when to perform maintenance is an important issue to ensure the stable operation of equipment and reduce production resources. If there are too many maintenance steps in the processing of the equipment, the processing time of the workpiece will be occupied during the maintenance period, resulting in delays in the completion of the task. However, if there are too few maintenance steps, the equipment will seriously degrade due to the accumulation of usage time, and failures will frequently occur during the processing process, making it impossible to complete the project as planned. It can be seen that production scheduling, equipment degradation and equipment maintenance are interrelated, and the correlation diagram between the various factors is shown in the figure below. Figure 1 Therefore, coordinating and balancing the relationship between production scheduling, equipment degradation and equipment maintenance, and a scientific and reasonable maintenance strategy are essential in production scheduling, which can improve the efficiency of the manufacturing system, effectively protect equipment, thereby reducing production costs and achieving optimal utilization of resources and energy.

[0005] In current research, when studying the joint optimization of production and preventive maintenance, most people think that the processing time is fixed, or only consider the impact of equipment degradation on the processing time. However, there are few studies on the impact of equipment degradation on maintenance time and maintenance cycle. To this end, in view of the above problems and shortcomings, the present invention considers the impact of equipment degradation on processing time and preventive maintenance time, establishes linear degradation models for processing time and preventive maintenance time, adopts failure functions that obey Weibull distribution to describe equipment degradation and failure, adopts reliability thresholds as constraints for performing preventive maintenance, and takes the minimization of equipment completion time and average idle time as the joint optimization goal.

[0006] In recent years, intelligent algorithms have developed rapidly, especially the advantages of intelligent algorithms in dealing with multi-objective optimization problems are very obvious. Existing research shows that despite the increasing complexity of scheduling problems, the non-dominated sorting genetic algorithm (NSGA-II) has become an important solution method for workshop scheduling with multi-objective optimization as the core. However, in the process of solving the job-shop scheduling problem, the crowding distance sorting mechanism adopted by the NSGA-II algorithm to maintain the diversity of the population has the defect of uneven distribution of the Pareto optimal solution set. In addition, the crowding distance sorting mechanism has a high computational complexity. To this end, the present invention proposes an NSGA-II algorithm (NSGA-II-ALS) based on adaptive local search. The adaptive local search strategy, through the setting of adaptive parameters, enables the algorithm to have a larger search range in the early stage and a smaller search range when close to the Pareto frontier, thereby improving the search efficiency of the local search, changing the search and sorting rules of the traditional NSGA-II algorithm, effectively maintaining the diversity of the population, and simplifying the computational complexity. Summary of the invention

[0007] The purpose of the present invention is to provide a flow shop scheduling optimization method based on preventive maintenance and degradation. In order to achieve the above purpose, the present invention provides the following technical solution, which includes the following steps:

[0008] S1. Establishing a linear degradation model of processing time and a linear degradation model of preventive maintenance time respectively;

[0009] The linear degradation model function expression of the processing time is:

[0010]

[0011] In the formula, p [k],i represents the actual processing time of equipment i in processing the kth workpiece; is the standard processing time; t [k-1],i is the service life of equipment i after processing the k-1th workpiece; δ is the degradation coefficient of equipment processing time;

[0012] The linear degradation model function expression of the preventive maintenance time is:

[0013] tp [k],i =tp 0 +γt [k-1],i (5)

[0014] Where, tp [k],i is the actual maintenance time of equipment i after processing the kth workpiece; tp 0 is the standard maintenance time; γ is the maintenance time degradation factor;

[0015] S2. Use the failure function that obeys the Weibull distribution to describe the degradation and failure of the equipment, use the reliability threshold as the constraint condition for performing preventive maintenance, and take the minimum completion time and average idle time of the equipment as the joint optimization goal;

[0016] S3, construct an adaptive local search strategy, use the adaptive local search strategy to obtain the intersection points and sparse points according to the non-dominated sorting results, and obtain the optimal solution;

[0017] The adaptive local search strategy is as follows: first, non-dominated sorting is performed on all current populations, and intersection points and sparse points are obtained according to the sorting results, which are defined as the intersection area and the sparse area center, and the solution with the smallest sparsity among the current non-dominated solutions is taken as the sparse solution; secondly, local search is performed around the intersection points and sparse points. During the local search process, the extreme optimization strategy and the random search strategy are used to simultaneously generate local solutions, and adaptive parameters are set to dynamically adjust the local search range;

[0018] The process of using extreme optimization strategy and random search strategy to generate local solutions at the same time is as follows: first, the extreme optimization strategy is used to generate a new population around the sparse solution. When the extreme optimization mutation method generates local solutions, each local solution only changes one decision variable of the sparse solution; at the same time, the random search strategy is used to increase the convergence speed of the algorithm.

[0019] Furthermore, the joint optimization objective in step S2 is expressed as:

[0020] minF1=E(C [n],m ) (6)

[0021]

[0022] Where F1 is the maximum completion time; F2 is the average equipment idle time; i is the equipment number, m is the number of equipment, k is the position number of the workpiece on the equipment, n is the number of workpieces, and C [k],i The completion time of equipment i to process the kth workpiece, C [n],m is the maximum completion time p [k],i represents the actual processing time of equipment i in processing the kth workpiece, S [1],i is the start processing time of the first workpiece on equipment i, S [1],1 =0; Formula (6) indicates that the optimization goal is to minimize the maximum completion time, that is, the expected completion time of the last workpiece processed on the last equipment; Formula (7) indicates that the optimization goal is to minimize the average equipment idle time, and the equipment idle time is the time from the equipment start time to the completion time when the workpiece is not processed;

[0023] The constraints are expressed as:

[0024] S [1],i =E(C[1],i-1 ),i=2,...,m (8)

[0025] E(C [1],i )=S [1],i +p [1],i +tr[u(t [1],i )-u(0)],i=1,…,m (9)

[0026] S [k],1 =E(C [k-1],1 )+y [k-1],1 city [k-1],1 ,k=2,K,n (10)

[0027] E(C [k],1 )=S [k],1 +p [k],1 +tr[u(t [k],1 )-u(t [k-1],1 )],k=2,K,n (11)

[0028] S [k],i =max[E(C [k-1],i )+y [k-1],i city [k-1],i ,E(C [k],i-1 )],k=2,...,m;i=2,...,n(12)

[0029] E(C [k],i )=S [k],i +p [k],i +tr[u(t [k],i )-u(t [k-1],i )],k=2,...,m;i=2,...,n (13)

[0030] t [k],i =t [k-1],i (1-y [k-1],i )+p [k],i ,k=1,...,m;i=1,...,n (14)

[0031]

[0032] Wherein, equation (8) represents the start time of the first position of the workpiece on any equipment; constraint equation (9) represents the expected completion time of the first position of the workpiece on any equipment; constraint equation (10) represents the start time of the workpiece at any position on the first equipment; constraint equation (11) represents the expected completion time of the workpiece at any position on the first equipment; constraint equation (12) represents the start time of processing at any position on any equipment; constraint equation (13) represents the completion time of processing at any position on any equipment; constraint equation (14) represents that the service life of the machine is the accumulation of processing time, but after preventive maintenance, the service life returns to zero, t [k],i is the service life of the equipment when equipment i completes processing the kth workpiece; t [0],i = 0, that is, the service life of all equipment before operation is 0; constraint (15) represents the preventive maintenance strategy, which considers that whether equipment i performs preventive maintenance after processing the k-1th workpiece depends on the service life of the equipment when the workpiece at the kth position is completed. If the service life of the equipment at the kth position is greater than or equal to the reliability critical value R L The corresponding equipment age is [k-1],i =1, perform preventive maintenance; otherwise y [k-1],i = 0, do not perform preventive maintenance, and directly process the next workpiece, y [k],i is a 0 / 1 preventive maintenance decision variable; Formula (16) indicates that each position in each device can only process one workpiece, x i,j,k is a 0 / 1 processing order decision variable, which is 1 if workpiece j is processed at the kth position of any device i, otherwise it is 0; Formula (17) indicates that the workpiece can only be processed at one position on the same device; Formula (18) represents the processing time of the device i to process the kth workpiece.

[0033] Furthermore, in the step S3 adaptive local search strategy, the method for obtaining the center of the boundary area and the sparse area is as follows:

[0034] Suppose the t-th generation population is P, the number of objective functions is m, and the population P is sorted by non-dominated sorting and crowding distance sorting. The first-level non-dominated solution P1 is the non-dominated solution of the current population. The first m individuals in the non-dominated solution set P1 are intersection points, and the m+1th individual is a sparse point, corresponding to the center of the m intersection areas and the center of the sparse areas.

[0035] Furthermore, in the step S3 adaptive local search strategy, the method for obtaining the sparsity is as follows:

[0036] Assume that the total number of populations is N, that is, there are N solutions. First, normalize the objective function value and assume that the i-th solution X i The target vector is (f i (X i ),…,f m (X i), the normalized formula is:

[0037]

[0038] where f jmin and f jmax are the minimum and maximum values ​​of the jth objective function corresponding to all current solutions respectively; the i-th solution X i The sparsity calculation formula is as follows:

[0039]

[0040] where n i is the target function space with the target vector F(X i ) The number of other target vectors whose Euclidean distance is less than r, where r ranges from 0 <r<1。

[0041] Furthermore, in the step S3 adaptive local search strategy, the extreme optimization strategy method adopted is as follows:

[0042] Assume that the center solution of the current searched area is X = (x1, x2, ..., x n ), n is the number of decision variables, the total population is N, then the number of local solutions generated is n, and the mutation formula for generating local solutions is:

[0043] X i =(x1,…,x′ i ,…,x n ),0<i≤n (21)

[0044] x′ i =x i +α·β max (x i ),0<i≤n (22)

[0045]

[0046] β max (x i )=max[x i -l i ,u i -x i ],0<i≤n (24)

[0047] Where: x i is the decision variable; h is a random number between 0 and 1; q is a positive real number, called the shape parameter; l i is the lower bound of the ith decision variable; u i is the upper bound of the i-th decision variable; β max (x i ) is the current decision variable xi Maximum value that can be changed.

[0048] Furthermore, in step S3, the random search strategy method adopted in the adaptive local search strategy is as follows:

[0049] Assume that the central solution of the current search area is X = (x1, x2, ..., x n ), n is the number of decision variables, the total population is N, then the number of local solutions generated is 20% of the total population, and the mutation formula is:

[0050]

[0051] x′ k =rand(γ)x k ,k=1,2,…,[0.2N]; j=1,2,…,[0.2N / n] (26)

[0052] Among them, γ is the search range parameter, which is used to determine the size of the search range; rand(γ) represents a random number between (0,γ); the above mutation strategy generates a total of (m+1)(n+[0.2N]) local solutions, and m is the number of objective functions.

[0053] Furthermore, in step S3, in the adaptive local search strategy, according to the maximum optimization time T max Set adaptive parameters and adjust parameters dynamically as follows:

[0054] The set preventive maintenance time degradation factor γ has a value range of (0.05, 1.2), and the search range adjustment formula is:

[0055]

[0056] Among them, t is the time from the start of optimization to the current moment, and the maximum optimization time T max The value of is greater than 5.

[0057] Compared with the prior art, the present invention has the following beneficial effects:

[0058] 1. The flow shop scheduling optimization method based on preventive maintenance and degradation proposed in the present invention takes into account the impact of equipment degradation on processing time and preventive maintenance time, and establishes a linear degradation model of processing time and preventive maintenance time. Based on this model, the scheduling method proposed in the present invention can effectively coordinate and balance the relationship between production, equipment degradation and equipment maintenance, improve the efficiency of the manufacturing system, and save production costs.

[0059] 2. The flow shop scheduling optimization method based on preventive maintenance and degradation proposed in the present invention constructs an NSGA-II algorithm (NSGA-II-ALS) based on adaptive local search. The adaptive search strategy in NSGA-II-ALS has a larger search range in the early stage and a smaller search range when approaching the Pareto frontier through the setting of adaptive parameters, which improves the search efficiency and search ability of local search. At the same time, the adaptive local search strategy of NSGA-II-ALS more effectively approaches the Pareto frontier, improving the uniformity of solution distribution and the range of solution distribution. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments recorded in the present invention, and for those of ordinary skill in the art, other drawings can also be obtained based on these drawings.

[0061] Figure 1 This is a relationship diagram between influencing factors and scheduling.

[0062] Figure 2 This is an illustration of the preventive maintenance strategy provided by the present invention.

[0063] Figure 3 This is a flow chart of the NSGA-II-ALS algorithm provided by the present invention.

[0064] Figure 4 This is the scheduling Gantt chart obtained by the NSGA-II-LS provided by the present invention. DETAILED DESCRIPTION

[0065] The main purpose of the present invention is to establish and solve a degraded flow shop scheduling model based on completion time and average equipment idle time. The interaction between scheduling, production, maintenance and degradation is considered, and a mathematical model including completion time and average equipment idle time is constructed. At the same time, based on the NSGA-II algorithm, the present invention proposes an improved algorithm-adaptive local search strategy (NSGA-II-ALS), which obtains intersection points and sparse points according to the non-dominated sorting results. Subsequently, an example experiment was carried out using the NSGA-II-ALS algorithm to obtain the optimization solution set and Pareto chart, and the Gantt chart of the optimal solution was drawn. It is proved through experiments that the scheduling model proposed by the present invention can effectively coordinate and balance the relationship between production, equipment maintenance and equipment degradation, thereby improving the efficiency of the manufacturing system, saving production costs, and achieving optimal utilization of resources.

[0066] In order to better understand the technical solution, the method of the present invention is described in detail below with reference to the accompanying drawings.

[0067] The flow shop scheduling problem studied in this invention is described as follows: There are n workpieces J = {1, 2, 3, ..., n} in the scheduling period, which pass through m equipment M1, M2, ..., M m The workpieces are processed in the same order on each device, can only be processed once on each device, and each device can only process one workpiece at the same time, that is, the replacement flow workshop, and the buffer space between each device is unlimited. The problem studied by this invention is based on the following assumptions: all workpieces are independent and can be processed at zero time; processing preparation time, minor repair preparation time and maintenance preparation time are not counted.

[0068] Given the processing constraints of n workpieces on m equipment in a flow shop production mode:

[0069] C [1],1 =p [1],1 (1)

[0070] C [k],1 =C [k-1],1 +p [k],1 ,k=1,2,…,n (2)

[0071] C [k],i =max[C [k-1],i ,C [k],i-1 ]+p [k],i ,i=1,2,…,m;k=1,2,…,n (3)

[0072] In the formula, i is the equipment number, m is the number of equipment, k is the position number of the workpiece on the equipment, n is the number of workpieces, and p is the number of workpieces. [k],i represents the actual processing time of equipment i in processing the kth workpiece, C [k],i represents the completion time of processing the kth workpiece on equipment i, then the maximum completion time C max It can be expressed as C [n],m .

[0073] In view of the above problems, the present invention provides a flow shop scheduling optimization method based on preventive maintenance and degradation, comprising the following steps:

[0074] S1. Establish the linear degradation model of processing time and the linear degradation model of preventive maintenance time respectively.

[0075] Traditional flow shop scheduling assumes that the equipment is in a stable state without degradation. In fact, as the equipment is used for a longer time, the equipment itself will degrade. This invention focuses on the impact of equipment degradation on processing time, preventive maintenance cycle and preventive maintenance duration, and establishes a linear degradation model of processing time and preventive maintenance time.

[0076] 1. Linear degradation model of processing time

[0077] Equipment will degrade during operation during the scheduling period, and the relative degradation of component functions, such as the passivation of machining tools, will lead to an extension of the actual workpiece processing time of the equipment. The older the equipment is when the workpiece begins to be processed, the greater the degradation effect on the processing time. The literature uses the linear deterioration of the processing start time to represent the actual processing time. The present invention comprehensively considers the standard processing time and gives the linear degradation model function expression of the processing time as follows:

[0078]

[0079] In the formula, p [k],i represents the actual processing time of equipment i in processing the kth workpiece; is the standard processing time; t [k-1],i is the service life of equipment i after processing the k-1th workpiece; δ is the degradation coefficient of equipment processing time,

[0080] 2. Preventive maintenance time linear degradation model

[0081] During the processing, the increase in the service life of the equipment leads to an increase in the equipment failure rate and a gradual decrease in reliability. In order to ensure that the equipment operates in a stable state, maintenance operations need to be performed on the equipment. Therefore, preventive maintenance activities should be performed before the equipment reliability falls below the set reliability threshold. Since the service life of the equipment is different when performing preventive maintenance, the degree of damage to the equipment is also different, and the maintenance level required is also different. In order to represent the execution of different types of maintenance operations according to different equipment states, with reference to the linear degradation of the processing time, the maintenance activity time is designed to be an increasing function of the equipment service life. The linear degradation model function expression of the preventive maintenance time is:

[0082] tp [k],i =tp 0 +γt [k-1],i (5)

[0083] Where, tp [k],i is the actual maintenance time of equipment i after processing the kth workpiece; tp 0 is the standard maintenance time; γ is the maintenance time degradation factor.

[0084] The present invention takes the reliability of the equipment as the limit condition for preventive maintenance. When the equipment completes the current processing, the current reliability of the equipment is compared with the reliability threshold value as the basis for whether to perform preventive maintenance steps. In order to demonstrate the maintenance decision-making method, a schematic diagram of the preventive maintenance strategy is drawn, as shown in FIG. Figure 2 shown.

[0085] S2. A failure function that obeys the Weibull distribution is used to describe the degradation and failure of equipment. A reliability threshold is used as a constraint for performing preventive maintenance. The minimum completion time and average idle time of the equipment are taken as the joint optimization goal.

[0086] The joint optimization objective is expressed as:

[0087] minF1=E(C [n],m ) (6)

[0088]

[0089] Where F1 is the maximum completion time; F2 is the average equipment idle time; i is the equipment number, m is the number of equipment, k is the position number of the workpiece on the equipment, n is the number of workpieces, and C [k],i The completion time of equipment i to process the kth workpiece, C [n],m is the maximum completion time p [k],i represents the actual processing time of equipment i in processing the kth workpiece, S [1],i is the start processing time of the first workpiece on equipment i, S [1],1 =0; Formula (6) indicates that the optimization goal is to minimize the maximum completion time, that is, the expected completion time of the last workpiece processed on the last equipment; Formula (7) indicates that the optimization goal is to minimize the average equipment idle time, and the equipment idle time is the time from the equipment start time to the completion time when the workpiece is not processed;

[0090] The constraints are expressed as:

[0091] S [1],i =E(C [1],i-1 ),i=2,...,m (8)

[0092] E(C [1],i )=S [1],i +p [1],i +tr[u(t [1],i )-u(0)],i=1,…,m (9)

[0093] S [k],1 =E(C [k-1],1 )+y [k-1],1 tp [k-1],1 ,k=2,K,n (10)

[0094] E(C [k],1) =S [k],1 +p [k],1 +tr[u(t [k],1 )-u(t [k-1],1 )],k=2,K,n (11)

[0095] S [k],i =max[E(C [k-1],i )+y [k-1],i tp [k-1],i, E(C [k],i-1 )],k=2,…,m; i=2,…,n (12)

[0096] E(C [k],i )=S [k],i +p [k],i +tr[u(t [k],i )-u(t [k-1],i )],k=2,...,m; i=2,...,n (13)

[0097] t [k],i =t [k-1],i (1-y [k-1],i )+p [k],i ,k=1,...,m; i=1,...,n (14)

[0098]

[0099] Wherein, equation (8) represents the start time of the first position of the workpiece on any equipment; constraint equation (9) represents the expected completion time of the first position of the workpiece on any equipment; constraint equation (10) represents the start time of the workpiece at any position on the first equipment; constraint equation (11) represents the expected completion time of the workpiece at any position on the first equipment; constraint equation (12) represents the start time of processing at any position on any equipment; constraint equation (13) represents the completion time of processing at any position on any equipment; constraint equation (14) represents that the service life of the machine is the accumulation of processing time, but after preventive maintenance, the service life returns to zero, t [k],i is the service life of the equipment when equipment i completes processing the kth workpiece; t [0],i = 0, that is, the service life of all equipment before operation is 0; constraint (15) represents the preventive maintenance strategy, which considers that whether equipment i performs preventive maintenance after processing the k-1th workpiece depends on the service life of the equipment when the workpiece at the kth position is completed. If the service life of the equipment at the kth position is greater than or equal to the reliability critical value R L The corresponding equipment age is [k-1],i =1, perform preventive maintenance; otherwise y [k-1],i = 0, do not perform preventive maintenance, and directly process the next workpiece, y [k],i is a 0 / 1 preventive maintenance decision variable; Formula (16) indicates that each position in each device can only process one workpiece, x i,j,kis a 0 / 1 processing order decision variable, which is 1 if workpiece j is processed at the kth position of any device i, otherwise it is 0; Formula (17) indicates that the workpiece can only be processed at one position on the same device; Formula (18) represents the processing time of the device i to process the kth workpiece.

[0100] The symbols are explained as follows.

[0101] 1. Parameters:

[0102] n is the number of workpieces;

[0103] m is the number of devices;

[0104] j Workpiece number j = {1, 2, ..., n}

[0105] i Device serial number i={1,2,3,...,m}

[0106] k The position number of the workpiece on the equipment k = {1, 2, 3, ... n}

[0107] p j,i The processing time of workpiece j on equipment i;

[0108] The standard processing time of the kth workpiece processed by equipment i;

[0109] tr Minor repair time after failure;

[0110] tp 0 Standard preventive maintenance schedule;

[0111] δ [k ] ,i The processing time degradation factor of equipment i when processing the kth workpiece;

[0112] δ basic processing time degradation factor;

[0113] γ preventive maintenance time degradation factor;

[0114] R L Reliability critical value;

[0115] t L The equipment service life corresponding to the reliability critical value.

[0116] 2. Decision variables:

[0117] C [k],i The completion time of equipment i to process the kth workpiece;

[0118] S [k],i The start time of equipment i processing the kth workpiece;

[0119] S[1],i The start time of the first workpiece on equipment i, S [1],1 =0;

[0120] p [k],i The actual processing time of the kth workpiece processed by equipment i;

[0121] tp [k],i The actual maintenance time of equipment i after processing the kth workpiece;

[0122] t [k],i The service life of equipment i when it finishes processing the kth workpiece; t [0],i =0, that is, the service life of all equipment before the start of operation is 0;

[0123] x i,j,k 0 / 1 processing order decision variable, which is 1 if workpiece j is processed at the kth position of any equipment i, otherwise it is 0;

[0124] y [k],i 0 / 1 preventive maintenance decision variable, which is 1 if equipment i is repaired after processing the kth workpiece, and 0 otherwise.

[0125] S3. Construct an adaptive local search strategy (NSGA-II-ALS). Use the adaptive local search strategy to obtain the intersection points and sparse points according to the non-dominated sorting results to obtain the optimal solution.

[0126] In view of the problem of large computational complexity of the non-dominated sorting genetic algorithm (NSGA-II), the present invention proposes an NSGA-II algorithm based on adaptive local search (NSGA-II-ALS). The adaptive local search strategy is: first, non-dominated sorting is performed on all current populations, and intersection points and sparse points are obtained according to the sorting results, which are defined as the intersection area and the center of the sparse area, and the solution with the smallest sparsity among the current non-dominated solutions is taken as the sparse solution; secondly, local search is performed around the intersection points and sparse points. During the local search process, extreme optimization strategy and random search strategy are used to simultaneously generate local solutions to improve the quality and convergence speed of the solution, and adaptive parameters are set to dynamically adjust the local search range. The NSGA-II-ALS algorithm has outstanding advantages in terms of convergence speed and solution quality in a relatively short time.

[0127] 1. Obtain a regional center

[0128] In the present invention, the method for obtaining the center of the boundary area and the sparse area is as follows:

[0129] Let the t-th generation population be P and the number of objective functions be m. According to the non-dominated sorting and crowding distance sorting of population P, the first-level non-dominated solutions P1 are the non-dominated solutions of the current population. It can be known from the mechanism of crowding distance sorting that the number of intersection points is equal to the number of objective functions. In population P1, the crowding distances of the first m individuals are infinite, and the crowding distance of the (m + 1)-th individual is the largest point except for the boundary points, which means that the density of solutions around the (m + 1)-th individual is the lowest. Thus, it can be seen that the first m individuals in the non-dominated solution set P1 are intersection points, and the (m + 1)-th individual is a sparse point, corresponding to the centers of m intersection regions and the sparse region.

[0130] The present invention performs local search centered on intersection points and sparse points. An intersection point is a maximum or minimum value in at least one objective vector direction. Searching centered on intersection points is a measure to ensure the distribution range of the population. The area around the point with the lowest crowding distance is the region where the solutions are the sparest. Conducting local search around sparse points can increase the uniformity of the population distribution. Therefore, it is meaningful to conduct local search centered on intersection points and the point with the minimum crowding distance.

[0131] 2. Calculate the sparsity

[0132] The crowding distance method used in the traditional NSGA-II algorithm cannot fully guarantee the uniformity of the population. Therefore, NSGA-II-ALS uses the density of solutions to measure the sparsity (SP) of a solution. The method for calculating sparsity is as follows:

[0133] Let the total number of the population be N, that is, there are N solutions. First, normalize the objective function values. Let the objective vector of the i-th solution X i be (f i (X i ), …, f m (X i )) and the normalization formula is:

[0134]

[0135] where f jmin and f jmax are respectively the minimum and maximum values of the j-th objective function values corresponding to all current solutions; the formula for calculating the sparsity of the i-th solution X i after normalization is as follows:

[0136]

[0137] where n i is the number of other objective vectors in the objective function space whose Euclidean distance from the objective vector F(X i ) is less than r. The value range of r is 0 < r < 1, and in all experiments of the present invention, 0.1 is taken.

[0138] 3 Adaptive Local Search

[0139] NSGA-II-ALS takes the solution with the smallest sparsity among the current non-dominated solutions as the sparse solution. The setting of the sparse solution gives NSGA-II-ALS the following advantages: when the solution distribution is uneven, this method effectively improves the exploration of the space around the sparse solution, thereby making the solution distribution uniform; when the non-dominated solution distribution is uniform, the solution sparsity at both ends of the solution set is the smallest, and the algorithm begins to explore beyond the edge, thereby increasing the extensiveness of the solution.

[0140] After the sparse solution is set, the present invention uses two mutation strategies to generate local solutions at the same time. First, the extreme optimization strategy is used to generate a new population around the sparse solution. When the extreme optimization mutation method generates a local solution, each local solution only changes one decision variable of the sparse solution. The extreme optimization strategy improves the local search ability when the population is close to the Pareto optimal solution, effectively improving the accuracy of the solution.

[0141] The process of the present invention using the extreme optimization strategy and the random search strategy to simultaneously generate local solutions is as follows: first, the extreme optimization strategy is used to generate a new population around the sparse solution. When the extreme optimization mutation method generates local solutions, each local solution only changes one decision variable of the sparse solution; at the same time, the random search strategy is used to increase the convergence speed of the algorithm.

[0142] Among them, the limit optimization strategy method adopted by the present invention is as follows:

[0143] Assume that the center solution of the current searched area is X = (x1, x2, ..., x n ), n is the number of decision variables, the total population is N, then the number of local solutions generated is n, and the mutation formula for generating local solutions is:

[0144] X i =(x1,…,x′ i ,…,x n ),0<i≤n (21)

[0145] x′ i =x i +α·β max (x i ),0<i≤n (22)

[0146]

[0147] β max (x i )=max[x i -l i ,u i -x i ],0<i≤n (24)

[0148] Where: x i is the decision variable; h is a random number between 0 and 1; q is a positive real number, called the shape parameter, and in the experiment of the present invention, q is 11; l i is the lower bound of the ith decision variable; u i is the upper bound of the i-th decision variable; β max (x i ) is the current decision variable x i Maximum value that can be changed.

[0149] The extreme optimization mutation method only changes one decision variable at a time, and has a strong fine-tuning ability, but the search range is small. In order to increase the convergence speed of the algorithm, the second mutation strategy, the random search strategy, is also adopted.

[0150] The random search strategy method adopted by the present invention is as follows:

[0151] Assume that the central solution of the current search area is X = (x1, x2, ..., x n ), n is the number of decision variables, the total population is N, then the number of local solutions is 20% of the total population (if it cannot be divided evenly, round it up), and the mutation formula is:

[0152]

[0153] x′ k =rand(γ)xk,k=1,2,…,[0.2N]; j=1,2,…,[0.2N / n] (26)

[0154] Among them, γ is the search range parameter, which is used to determine the size of the search range; rand(γ) represents a random number between (0,γ); the above mutation strategy generates a total of (m+1)(n+[0.2N]) local solutions, and m is the number of objective functions.

[0155] 4. Adaptive parameter setting

[0156] For the random local search strategy, a larger search range in the early stage of the algorithm can improve the global search capability and enable the population to quickly approach the Pareto frontier. After approaching the Pareto frontier, a smaller search range can improve the population's ability to approach the Pareto frontier. In the adaptive local search strategy of the present invention, according to the maximum optimization time T max Set adaptive parameters and adjust parameters dynamically as follows:

[0157] According to many years of experience and experimental research, the preventive maintenance time degradation factor γ set in the present invention has a value range of (0.05, 1.2), and the search range adjustment formula is:

[0158]

[0159] Among them, t is the time from the start of optimization to the current moment, and the maximum optimization time T max The value of is greater than 5. The proposed adaptive formula has good adaptability to different setting values.

[0160] The Pareto frontier of most practical optimization problems is unknown. The search range adjustment method proposed in the present invention does not require knowing the Pareto frontier and requires very few parameters to be set, which is suitable for solving practical problems.

[0161] 5. Algorithm Process

[0162] The specific algorithm flow chart of NSGA-II-ALS proposed in the present invention is as follows Figure 3 As shown. According to the actual MOP problem, the algorithm parameters are set: the initial population size is N; the maximum optimization time is T max ; Decision vector dimension n; Decision variable value upper bound u=(u1,u2,L,u n ) and the lower bound l=(l1,l2,L,l n );Number of objective functions m;Shape parameter q;Crossover probability p c and mutation probability p m ; Crossover parameter η c and the mutation parameter η m The specific algorithm flow of NSGA II-ALS is as follows:

[0163] Step 1: Randomly initialize the population P within the value range I ={X 1 ,X 2 ,L,X N} Among them, X i =(x′1,x′2,L,x′ n ), i=1,2,...,n.

[0164] Step 2: P I Perform non-dominated and crowded distance sorting, and all non-dominated solutions in the current population are denoted as P C , determine the center of the boundary area and the center of the sparse area according to the sorting results.

[0165] Step 3: Use the standard NSGA-II algorithm to calculate P I The population in the crossover mutation, forming the offspring P M .

[0166] Step 4: Perform local search around the intersection center and the sparse area center to generate a total of (m+1)(n+[0.2N]) local solutions, which are set as population P N .

[0167] Step 5: Place P I , P M and P N Merge and sort all solutions by non-dominated and crowded distance, and select the best N solutions to form the next generation population P O , and assume P I =P O .

[0168] Step 6: Repeat steps 2-5 until the maximum optimization time T is reached. max or target accuracy, proceed to step 7.

[0169] Step 7: Current P I The non-dominated solution in is the optimal solution.

[0170] Example

[0171] 1. Methods

[0172] In order to test the feasibility and effectiveness of the method of the present invention, NSGA-II-ALS was compared with several other genetic algorithms to analyze the performance of the algorithms under multiple groups of different case scales. The specific experimental parameters are as follows.

[0173] The initial population size is 50, the maximum number of iterations is 100 generations, and the crossover probability P c =0.9, mutation probability P m =0.1, minor repair time tr=5min, standard preventive maintenance time tp 0 =15min, reliability critical value R L =0.8, Weibull shape parameter β=2, size parameter α=150, processing time degradation factor δ=0.01, preventive maintenance time degradation factor, γ=0, 0.05, 0.1, 0.2, the scheduling problem scale n×m is 10×6, 10 random experiments are conducted for each scale, and there are 30 cases in total. The standard processing time of the workpiece on each equipment is shown in Table 1.

[0174] Table 1 Standard processing schedule

[0175]

[0176] 2. Results and Analysis

[0177] In order to specifically demonstrate the experimental results of this scheduling problem and reflect the effectiveness of the improved algorithm studied in this invention, NSGA-II-ALS, traditional NSGA-II and particle swarm optimization algorithm (PSO) are used to simulate and optimize this scheduling problem.

[0178] When γ = 0, the degradation of preventive maintenance is not considered. According to NSGA-II-ALS, the scheduling order of the optimal solution in the solution set, except for the boundary points, is: 9-10-1-4-5-8-2-3-7-6. The preventive maintenance decision matrix is ​​shown in the following formula. The resulting Gantt chart is as follows: Figure 4 As shown in (a).

[0179]

[0180] When γ = 0.05, NSGA-II-ALS finds that the optimal scheduling order of the solution set, except for the boundary points, is: 9-10-1-4-5-2-8-3-6-7, and the preventive maintenance decision matrix is ​​shown in the following formula. The resulting Gantt chart is as follows Figure 4 As shown in (b).

[0181]

[0182] when γ = 0.1, NSGA-II-ALS finds that the optimal scheduling sequence of the solution set, except for the boundary points, is: 9-4-10-1-5-8-2-3-7-6, and the preventive maintenance decision matrix is ​​shown in the following formula. The resulting Gantt chart is as follows Figure 4 As shown in (c).

[0183]

[0184] When γ = 0.2, NSGA-II-ALS finds that the optimal scheduling order of the solution set, except for the boundary points, is: 9-10-1-4-8-6-3-5-2-7, and the preventive maintenance decision matrix is ​​shown in the following formula. The resulting Gantt chart is as follows Figure 4 As shown in (d).

[0185]

[0186] 3. Conclusion

[0187] In actual production systems, production scheduling, equipment degradation and equipment maintenance are interrelated. Based on this, the present invention takes the scheduling of degraded flow workshops as the research object, considers the interaction between scheduling, production, maintenance and degradation, and establishes a mathematical model of completion time and average equipment idle time. Finally, a reasonable scheduling scheme is obtained according to the characteristics of the problem. Aiming at the flow workshop scheduling problem, the present invention establishes a mathematical model with the minimum completion time and average equipment idle time as the optimization goal; considering the impact of equipment degradation on the actual processing time and preventive maintenance time, a workpiece processing time degradation model and an equipment preventive maintenance time degradation model are established; considering the relationship between failure, maintenance, preventive maintenance and progress, a preventive maintenance strategy is studied and established.

[0188] According to the NSGA-II algorithm, encoding, crossover and mutation operations are performed. On this basis, a local search strategy is proposed to obtain boundary points and sparse points according to the non-dominated sorting results. Local search is performed near boundary points and sparse points through extreme optimization strategy and random search strategy. This method not only ensures the diversity and uniformity of the population, but also improves the quality of the solution and the convergence speed.

[0189] Through statistical analysis, the following conclusions were drawn:

[0190] (1) The scheduling method proposed in the present invention can effectively coordinate and balance the relationship between production, equipment degradation and equipment maintenance, improve the efficiency of the manufacturing system, effectively protect the equipment, thereby saving production costs and achieving optimal utilization of resources and energy.

[0191] (2) The local search strategy of NSGA-II-ALS proposed in this invention more effectively approaches the Pareto frontier while improving the uniformity and range of solution distribution.

[0192] (3) In the NSGA-II-ALS proposed in the present invention, boundary points and sparse points can be directly obtained from the non-dominant sorting results without calculating density and gradient, which has small calculation amount, high solution efficiency and strong search ability.

[0193] Based on the above analysis, management and practice insights are proposed. For the degenerate flow shop manufacturing industry, the model proposed by the present invention can be used as the basic model of such systems, providing managers with an effective management method and new methods and means.

[0194] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, a person skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features thereof may be replaced by equivalents. However, such modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A flow shop scheduling optimization method based on preventive maintenance and degradation, characterized in that: The following steps are involved: S1. Establishing a linear degradation model of processing time and a linear degradation model of preventive maintenance time respectively; The linear degradation model function expression of the processing time is: In the formula, p [k],i represents the actual processing time of equipment i in processing the kth workpiece; is the standard processing time; t [k-1],i is the service life of equipment i after processing the k-1th workpiece; δ is the degradation coefficient of equipment processing time; The linear degradation model function expression of the preventive maintenance time is: tp [k],i =tp 0 +γt [k-1],i (5) In the formula, tp [k],i is the actual maintenance time of equipment i after processing the kth workpiece; tp 0 is the standard maintenance time; γ is the maintenance time degradation factor; S2. Use the failure function that obeys the Weibull distribution to describe the degradation and failure of the equipment, use the reliability threshold as the constraint condition for performing preventive maintenance, and take the minimum completion time and average idle time of the equipment as the joint optimization goal; S3, construct an adaptive local search strategy, use the adaptive local search strategy to obtain the intersection points and sparse points according to the non-dominated sorting results, and obtain the optimal solution; The adaptive local search strategy is: firstly, non-dominated sorting is performed on all current populations, and the junction points and sparse points are obtained according to the sorting results, which are defined as the junction area and the sparse area center, and the solution with the smallest sparsity among the current non-dominated solutions is taken as the sparse solution; Secondly, local search is performed around the intersection points and sparse points. During the local search process, the extreme optimization strategy and random search strategy are used to simultaneously generate local solutions, and adaptive parameters are set to dynamically adjust the local search range. The process of using extreme optimization strategy and random search strategy to generate local solutions at the same time is as follows: first, the extreme optimization strategy is used to generate a new population around the sparse solution. When the extreme optimization mutation method generates local solutions, each local solution only changes one decision variable of the sparse solution; at the same time, the random search strategy is used to increase the convergence speed of the algorithm.

2. The flow shop scheduling optimization method based on preventive maintenance and degradation according to claim 1 is characterized in that: The joint optimization objective in step S2 is expressed as: minF1=E(C [n],m )(6) Where, F1 is the maximum completion time; F2 is the average equipment idle time; i is the equipment number, m is the number of equipment, k is the position number of the workpiece on the equipment, n is the number of workpieces, C [k],i The completion time of equipment i to process the kth workpiece, C [n],m is the maximum completion time p [k],i represents the actual processing time of equipment i in processing the kth workpiece, S [1],i is the start processing time of the first workpiece on equipment i, S [1],1 =0; Formula (6) indicates that the optimization goal is to minimize the maximum completion time, that is, the expected completion time of the last workpiece processed on the last equipment; Formula (7) indicates that the optimization goal is to minimize the average equipment idle time, and the equipment idle time is the time from the equipment start time to the completion time when the workpiece is not processed; The constraints are expressed as: S [1],i =E(C [1],i-1 ),i=2,...,m (8) E(C [1],i )=S [1],i +p [1],i +tr[u(t [1],i )-u(0)],i=1,...,m (9) S [k],1 =E(C [k-1],1 )+y [k-1],1 tp [k-1],1, k=2,K,n (10) E(C [k],1 )=S [k],1 +p [k],1 +tr[u(t [k],1 )-u(t [k-1],1 )],k=2,K,n (11) S [k],i =max[E(C [k-1],i )+y [k-1],i tp [k-1],i ,E(C [k],i-1 )],k=2,…,m;i=2,…,n (12) E(C [k],i )=S [k],i +p [k],i +tr[u(t [k],i )-u(t [k-1],i )],k=2,…,m;i=2,…,n (13) t [k],i =t [k-1],i (1-y [k-1],i )+p [k],i ,k=1,…,m;i=1,…,n (14) Wherein, equation (8) represents the start time of the first position of the workpiece on any equipment; constraint equation (9) represents the expected completion time of the first position of the workpiece on any equipment; constraint equation (10) represents the start time of the workpiece at any position on the first equipment; constraint equation (11) represents the expected completion time of the workpiece at any position on the first equipment; constraint equation (12) represents the start time of processing at any position on any equipment; constraint equation (13) represents the completion time of processing at any position on any equipment; constraint equation (14) represents that the service life of the machine is the accumulation of processing time, but after preventive maintenance, the service life returns to zero, t [k],i is the service life of the equipment when equipment i completes processing the kth workpiece; t [0],i = 0, that is, the service life of all equipment before operation is 0; constraint (15) represents the preventive maintenance strategy, which considers that whether equipment i performs preventive maintenance after processing the k-1th workpiece depends on the service life of the equipment when the workpiece at the kth position is completed. If the service life of the equipment at the kth position is greater than or equal to the reliability critical value R L The corresponding equipment age is [k-1],i =1, perform preventive maintenance; otherwise y [k-1],i = 0, do not perform preventive maintenance, and directly process the next workpiece, y [k],i is a 0 / 1 preventive maintenance decision variable; Formula (16) indicates that each position in each device can only process one workpiece, x i,j,k is a 0 / 1 processing order decision variable, which is 1 if workpiece j is processed at the kth position of any device i, otherwise it is 0; Formula (17) indicates that the workpiece can only be processed at one position on the same device; Formula (18) represents the processing time of the device i to process the kth workpiece.

3. The flow shop scheduling optimization method based on preventive maintenance and degradation according to claim 1 is characterized in that: In the adaptive local search strategy of step S3, the method for obtaining the center of the boundary area and the sparse area is as follows: Suppose the t-th generation population is P, the number of objective functions is m, and the population P is sorted by non-dominated sorting and crowding distance sorting. The first-level non-dominated solution P1 is the non-dominated solution of the current population. The first m individuals in the non-dominated solution set P1 are intersection points, and the m+1th individual is a sparse point, corresponding to the center of the m intersection areas and the center of the sparse areas.

4. The flow shop scheduling optimization method based on preventive maintenance and degradation according to claim 1 is characterized in that: In the adaptive local search strategy of step S3, the method for obtaining the sparsity is as follows: Assume that the total number of populations is N, that is, there are N solutions. First, normalize the objective function value and assume that the i-th solution X i The target vector is (f i (X i ),…,f m (X i ), the normalized formula is: where f jmin and f jmax are the minimum and maximum values ​​of the jth objective function corresponding to all current solutions respectively; the i-th solution X i The sparsity calculation formula is as follows: Where n i is the target function space with the target vector F(X i ) The number of other target vectors whose Euclidean distance is less than r, where r ranges from 0 <r<1。 5. The flow shop scheduling optimization method based on preventive maintenance and degradation according to claim 1 is characterized in that: In step S3, the adaptive local search strategy adopts the following extreme optimization strategy: Assume that the center solution of the current searched area is X = (x1, x2, ..., x n ), n is the number of decision variables, the total population is N, then the number of local solutions generated is n, and the mutation formula for generating local solutions is: X i =(x 1, …,x′ i ,…,x n ),0<i≤n (21) x′ i =x i +a·b max (x i ),0<i≤n (22) β max (x i )=max[x i -l i ,u i -x i ],0<i≤n (24) Where: x i is the decision variable; h is a random number between 0 and 1; q is a positive real number, called the shape parameter; l i is the lower bound of the ith decision variable; u i is the upper bound of the i-th decision variable; β max (xi ) is the current decision variable x i Maximum value that can be changed.

6. The flow shop scheduling optimization method based on preventive maintenance and degradation according to claim 1 is characterized in that: In step S3, the random search strategy method used in the adaptive local search strategy is as follows: Assume that the central solution of the current search area is X = (x1, x2, ..., x n ), n is the number of decision variables, the total population is N, then the number of local solutions generated is 20% of the total population, and the mutation formula is: x′ k =rand(γ)x k ,k=1,2,…,[0.2N];j=1,2,…,[0.2N / n] (26) Among them, γ is the search range parameter, which is used to determine the size of the search range; rand(γ) represents a random number between (0,γ); the above mutation strategy generates a total of (m+1)(n+[0.2N]) local solutions, and m is the number of objective functions.

7. The flow shop scheduling optimization method based on preventive maintenance and degradation according to claim 1 is characterized in that: In step S3, in the adaptive local search strategy, according to the maximum optimization time T max Set adaptive parameters and adjust parameters dynamically as follows: The set preventive maintenance time degradation factor γ has a value range of (0.05, 1.2), and the search range adjustment formula is: Among them, t is the time from the start of optimization to the current moment, and the maximum optimization time T max The value of is greater than 5.

Images