CN113805545A - A Method for Generating Combined Scheduling Rules for Flexible Flow Shop Considering Batch Processing - Google Patents
A Method for Generating Combined Scheduling Rules for Flexible Flow Shop Considering Batch Processing Download PDFInfo
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Abstract
本发明提供一种考虑批处理的柔性流水车间组合调度规则生成方法。本发明方法，包括以下步骤：对本发明涉及的FFSP‑BPM问题进行分解并构建相应的调度规则，对调度规则进行组合以生成求解原问题的组合调度规则；考虑工件动态到达、工时不确定环境下不兼容工件族组批作业的FFSP‑BPM问题构建多目标优化模型；采用改进基因表达式编程算法生成组合调度规则求解FFSP‑BPM问题，并针对算法容易出现重复个体、陷入局部最优的不足提出重复个体剔除、变邻域搜索、自适应遗传算子的改进策略。该方法针对考虑不兼容工件族并行批处理的柔性流水组批车间调度过程，采用组合调度规则生成的方法，提高生产效率，同时能够有效指导调度过程。
The invention provides a method for generating a combined scheduling rule for a flexible flow shop considering batch processing. The method of the present invention includes the following steps: decomposing the FFSP-BPM problem involved in the present invention and constructing corresponding scheduling rules, combining the scheduling rules to generate combined scheduling rules for solving the original problem; considering the dynamic arrival of workpieces and the uncertain working hours environment A multi-objective optimization model is constructed for the FFSP-BPM problem that is not compatible with the group batch operation of the workpiece family; the improved gene expression programming algorithm is used to generate combined scheduling rules to solve the FFSP-BPM problem. Improved strategies for removing duplicate individuals, variable neighborhood search, and adaptive genetic operators. For the flexible flow group batch shop scheduling process considering the parallel batch processing of incompatible workpiece families, the method adopts the method of generating combined scheduling rules, which improves the production efficiency and can effectively guide the scheduling process.
Description
Technical Field
The invention relates to the field of dynamic workshop scheduling, in particular to a method for generating a flexible flow workshop combined scheduling rule considering batch processing.
Background
With the improvement of social and economic levels, the market demand tends to diversify and individually customize products, and the modern manufacturing mode is gradually changed from the original mass production to a flexible multi-variety small-batch production mode. The flexible flow production can flexibly adapt to market changes, realizes the mixed line operation of various products, and is a mainstream mode of discrete manufacturing enterprises for coping with the small-batch production of various products. The flexible flow production considering batch processing is a typical application scene, is widely applied to a plurality of fields such as semiconductor manufacturing and the like, and is mainly characterized in that a batch production mode capable of simultaneously processing a plurality of limited operations is combined on the basis of the original flexible flow production.
The batch production mode can achieve the aims of avoiding frequent production preparation time, simplifying material handling and improving production efficiency. However, the combination of this method and the flexible flow job scenario increases the complexity of production operation management, and the scheduling process needs to select processing equipment, determine the processing sequence of the workpieces on the equipment, and consider how to batch the workpieces. Meanwhile, the frequent change of the market environment enables the workpiece to have the characteristic of dynamic arrival, and random dynamic events such as working hour fluctuation and the like can occur due to different equipment states in the production process. Due to the dynamic factors, an enterprise cannot acquire scheduled global task information in advance and can only make a decision according to real-time production data, so that full batch operation is difficult to realize in a batch processing stage. Therefore, it is necessary to provide an efficient Scheduling method based on real-time production data for solving the Problem of Flexible Flow-sheet Scheduling with Batch Process Machines (FFSP-BPM) in a Flexible Flow shop with Batch Process Scheduling features in a dynamic environment, so as to adapt to the current production mode of multiple varieties of small batches, and respond to dynamic factors in the production Process in time while improving the production efficiency.
In the current research methods related to real-time scheduling, a scheduling rule method (DR) is widely applied. The method has lower time complexity, can be flexibly suitable for various problems, and in addition, the scheduling decision based on the scheduling rule is given before implementation, so the method can respond to dynamic factors in time and is very suitable for solving the FFSP-BPM problem with higher complexity and real-time scheduling requirement. However, the method for solving the FFSP-BPM problem has two defects: 1) the FFSP-BPM problem is a multi-decision combined optimization problem of multiple stages, multiple equipment types and flexible paths, and the traditional scheduling rule method is single in consideration dimension and cannot be directly solved. 2) The scheduling rule is a method based on local information, the performance of the scheduling rule depends on various factors such as a scheduling target and system configuration, a single scheduling rule cannot adapt to all dynamic environments, and dynamic selection of the scheduling rule needs to be realized by combining different scheduling scenes and performance indexes.
Disclosure of Invention
Aiming at the multi-target FFSP-BPM problem of incompatible workpiece family batch operation under the environment of dynamic arrival of workpieces and uncertain working hours, the method for generating the combined scheduling rule of the flexible flow shop considering batch processing is provided. The invention firstly introduces a scheduling rule method which is widely applied in the field of dynamic scheduling, secondly constructs a combined scheduling rule for solving the original problem by utilizing the thought of problem decomposition, and finally provides an FFSP-BPM combined scheduling rule generation method based on a Gene Expression Programming (GEP) algorithm. The technical means adopted by the invention are as follows:
a method for generating a combined scheduling rule of a flexible flow shop considering batch processing comprises the following steps:
s1, decomposing the FFSP-BPM problem, constructing corresponding scheduling rules aiming at the decomposed sub-problem analysis problem factors, and combining the scheduling rules corresponding to the sub-problems to generate a combined scheduling rule for solving the original problem;
s2, constructing a multi-objective optimization model aiming at FFSP-BPM problems of incompatible workpiece family batch operation under the environment of dynamic arrival and uncertain man-hour of the workpiece;
s3, generating a combined scheduling rule for solving the multi-objective optimization model by adopting a gene expression programming algorithm, and providing improved strategies of repeated individual elimination, variable neighborhood search and adaptive genetic operators aiming at the defects that repeated individuals are easy to appear in the algorithm and the algorithm is trapped in local optimization.
Further, the FFSP-BPM problem is specifically: when different types of workpieces dynamically arrive at a workshop, K processing stages are sequentially carried outThe production task can be completed, wherein the first stage is a batch processing stage, l is more than or equal to 1 and less than or equal to K, the stage comprises a plurality of batch processors with the load limiting quantity of B, the batch processing task of a limited number of workpieces can be completed, other stages are single-piece processing, and M is used for processing the workpieces(k)A parallel machine component, wherein M(k)More than or equal to 1, workpieces can only be processed by one equipment of the process in each processing stage, different types of workpieces have different processing time in the same processing stage, the processing time fluctuation caused by the new and old states of the equipment is considered, the processing time of the same type of workpieces on different equipment in the same stage is also different, batch processing equipment can only process the same type of workpieces at the same time, the operation time depends on the longest processing time in the operation contained in each batch, namely, a parallel batch processing operation mode which is incompatible with a workpiece family is adopted.
Further, based on the FFSP-BPM problem, the following assumptions are made: (1) the buffer area between adjacent stages is an infinite buffer area; (2) the batch size is only related to the number of workpieces in the batch; (3) batch phase jobs have minimum capacity requirements.
Further, in the S1, decomposing the FFSP-BPM problem specifically includes decomposing the original problem into a flexible flow shop scheduling problem and a batch processing scheduling problem;
aiming at the scheduling problem of the flexible flow shop, the job priority is judged according to the job parameters so as to realize job sequencing and the equipment priority is judged according to the equipment parameters so as to realize equipment selection;
aiming at the sub-problem of batch processing scheduling, workpieces are firstly distinguished according to workpiece families, the workpieces in each product type are subjected to job sequencing according to priorities, then feasible batches are generated according to batch requirements, and finally, for all the feasible batches of all the workpiece types, a batch forming process is guided through batch parameters and the batch priorities are determined, so that the batch sequencing of incompatible workpiece families is realized.
Further, the step S2 includes the following steps:
s2-1: and defining parameters and decision variables of the multi-objective optimization model with the maximum completion time and the average delay deviation.
(1) Parameter definition: the workpiece type index is I, where I is 1, 2 … …, I, the workpiece index is j, where j is 1, 2 … …, N, the machining stage index is K, where K is 1, 2 … …, K, the batch set is B, where B is 1, 2 … …, B, the equipment index is M, where M is 1, 2 … …, M, djIs the delivery date of the workpiece j, riTime of entry of workpiece j into production System, sjkFor the time at which the workpiece j begins to be machined at stage k, pjkmFor the machining time, p, of the workpiece j on the apparatus m of stage kjkFor the actual machining time of workpiece j in stage k, OjkFor the workpiece j, the process is completed in stage k, i.e. the k-th process, C, of the workpieceikTime TO complete processing of workpiece j at stage k, TOjkIs OijkType of operation of lijFor correspondence of workpiece j and product type i, TMmOf type m, CMmFor the processing capacity of the plant m, MBmIs the minimum batch size of the batch processing apparatus m, BmNumber of batches, P, for a batch processing apparatus mmbIs the processing time of batch b on batch processing equipment m;
(2) decision variables:
s2-2: and defining a multi-objective optimization model objective function and constraint conditions of the maximum completion time and the average delay deviation.
The maximum time-out CmaxRefers to the finish time of the last workpiece; the average delay deviation (AE/L) is used for measuring the time between the completion time and the delivery date of the workpieceDegree of difference, tending to be delivered on time;
the optimization objective is represented as:
f1＝min(maxjCjK) (4)
wherein f is1、f2Respectively representing a function aimed at minimizing the maximum completion time and the mean delay deviation, CjKRepresenting the time of completion of the workpiece j in the last processing stage K, djIndicating the delivery date of the workpiece j;
according to the characteristics of the actual production problem, the problem model should satisfy the following constraints:
constraint (6) indicates that a tool can only process one workpiece at a time, constraint (7) indicates that the workpiece can only be processed once at each stage, constraint (8) indicates that the part cannot be processed until it reaches the production system, constraint (9) ensures that discrete processing tools can only process one workpiece at a time, constraint (10) indicates that processing of the next process can only begin until the previous stage is completed, constraint (11) defines the completion time of the workpiece at stage k, constraint (12) indicates the actual processing time of the workpiece at stage k, constraint (13) indicates that the batch processing time on the batch processing tool is equal to the maximum of the workpiece processing times in the batch, constraint (14) ensures that a workpiece is allocated to only one batch, constraint (15) specifies that the batch size cannot be less than the minimum batch requirement and cannot exceed the batch processing tool capacity, constraint (16) indicates that workpieces assigned to the same batch have the same start time in the batch phase, constraint (17) indicates that workpiece j can only belong to one product type i, and constraint (18) indicates that workpieces of a batch can only come from one product type.
Further, step S3 includes the following steps:
s3-1, constructing an endpoint set and a function set based on the classic scheduling rules, adopting the classic scheduling rules as the endpoint set, and selecting the operators { +, -,/, - √ v, ()2The GEP algorithm represents different scheduling rules through the linear combination of the endpoint set and the function set;
s3-2, based on the combined dispatching rule coding, according to the three-segment real number coding mode, respectively representing the dispatching rules of the equipment selection process, the job sorting process and the batch sorting process;
s3-3, decoding the chromosome to generate a group of dispatching rule function expressions containing equipment selection rules, job sorting rules and batch sorting rules;
s3-4, defining the effective length of the gene based on the genetic operation of the effective length, designing the genetic operator based on the effective length, and carrying out the genetic operation on the elements in the effective length;
s3-5: aiming at the defects that the algorithm is early converged, easily falls into local optimum and is easy to generate repeated individuals, an algorithm improvement strategy is introduced: 1) a repeated individual eliminating algorithm for eliminating the weight of the population after genetic operation to ensure the diversity of the population; 2) two neighborhood structures for improving the local search capability of the algorithm; 3) an adaptive genetic operator for adjusting the genetic probability.
Further, in the step S3-4, the effective length-based genetic operation includes four modes, i.e., a crossover operator, a mutation operator, an RIS insertion operator, and a non-dominated sorting and selecting operator.
Further, the algorithm improvement strategy of step S3-5 includes:
(1) repeated individual elimination algorithm
Step 1: traversing the merged population, and searching for repeated individuals.
Step 2: determining an individual to be operated: if the two individuals which are mutually repeated are respectively located in the parent population and the offspring population, the parent individuals corresponding to the repeated individuals in the offspring population are used as the individuals to be operated, and if the two individuals which are mutually repeated are both located in the offspring population, one individual corresponding to the parent individual is randomly selected as the individual to be operated;
step 3: repeated individual removal: repeatedly carrying out variation and RIS string insertion operation on the individuals to be operated until the generated individuals are not repeated with other individuals in the combined population;
after repeated individuals are removed, all the individuals in the combined population after the duplication removal are subjected to non-dominated sorting, and the optimal N individuals are selected as a new generation population P according to the quality degree of the individuals in the populationi+1；
(2) Variable neighborhood search algorithm
Performing non-domination sorting on the offspring population generated after the repeated individuals are removed, and performing variable neighborhood search on the individuals in the non-domination stage so as to improve the local search capability of the algorithm;
the neighborhood structure definition includes: randomly scrambling and rearranging the neighborhood, namely scrambling and rearranging all end point set elements in the effective length of the current gene segment; forward insertion neighborhood, namely randomly selecting two positions of the end point set of the two elements in the effective length part of the current gene fragment, inserting the element corresponding to the latter position into the former position, and sequentially moving the elements of the other end point sets backwards;
(3) adaptive genetic operator
Carrying out dimensionless on the performance targets by adopting a max-min standardization method, taking the sum of the standardized targets as the fitness value of the current individual, and endowing the poorer individual with higher genetic operation probability by adopting a self-adaptive genetic operator, wherein the concrete formula is as follows:
in the formula, PmaxAnd PminThe maximum value and the minimum value of the genetic operation probability are respectively, f is the fitness value of the current individual to be operated, and for the crossover operator, f is the larger value of the fitness of the two individuals to be crossed. f. ofmaxAnd favgThe maximum value and the average value of the fitness of all individuals in the population to be operated are respectively.
The technical conception of the invention is as follows: the invention aims at the scheduling problem of a flexible flow line batch workshop which is incompatible with parallel batch processing of a workpiece family under the environment of dynamic arrival of workpieces and uncertain working hours, takes the minimized maximum completion time and average delay deviation as optimization targets, introduces a scheduling rule method widely applied to the field of real-time scheduling, constructs a combined scheduling rule for solving the original problem by using the thought of problem decomposition, and further provides a FFSP-BPM combined scheduling rule generation method based on gene expression programming.
The effective effects of the invention are mainly shown in that: in consideration of the defect that the traditional scheduling rule method has single dimension and cannot directly solve the FFSP-BPM combined decision optimization problem related by the invention, the scheduling rule generation algorithm based on gene expression programming and capable of effectively mining functional relation is provided. Aiming at the defects that the traditional gene expression programming algorithm has premature convergence, is easy to fall into local optimum, is easy to generate repeated individuals and the like, the improvement method is provided from the two aspects of problem characteristics and algorithm performance, and the algorithm performance is further improved.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a diagram of FFSP-BPM scheduling issues in view of incompatible workpiece families in accordance with an embodiment of the present invention;
FIG. 2 is a diagram illustrating a process of generating a problem resolution and combined scheduling rule according to an embodiment of the present invention;
FIG. 3 is a K-expression diagram in an embodiment of the present invention;
FIG. 4 is a diagram illustrating a decoding process according to an embodiment of the present invention;
FIG. 5 is a diagram of a single point forward cross, a two point sequential cross, and a genetic cross in accordance with an embodiment of the present invention;
FIG. 6 is a diagram of single point variations in an embodiment of the present invention;
FIG. 7 is a RIS plug string diagram in accordance with an embodiment of the present invention;
FIG. 8 is a diagram of repetitive individuals in an embodiment of the present invention;
FIG. 9 is a flow chart of a variable neighborhood search in an embodiment of the present invention;
FIG. 10 is a comparison algorithm coverage boxplot in an embodiment of the present invention;
FIG. 11 is a non-dominated calat diagram of an example of the s3b4f2n30_0.8_2 algorithm resulting from the improved GEP algorithm in an embodiment of the present invention;
FIG. 12 is a Gantt chart of an example of an s3b4f2n30_0.8_2 algorithm derived from the SP-ATC-RMSTB classic schedule rule in an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Aiming at the multi-target FFSP-BPM problem of incompatible workpiece family batch operation under the environment of workpiece dynamic arrival and uncertain working hours, the invention firstly introduces a scheduling rule method which is widely applied in the field of dynamic scheduling, secondly constructs a combined scheduling rule for solving the original problem by using the thought of problem decomposition, and finally provides a FFSP-BPM combined scheduling rule generation method based on gene expression programming. The method specifically comprises the following steps:
s1: constructing a combined scheduling rule based on a problem decomposition idea: the method comprises the steps of decomposing an FFSP-BPM original problem, respectively analyzing problem factors aiming at the decomposed sub-problems to construct corresponding scheduling rule algorithms, and then combining scheduling rules corresponding to the sub-problems to generate a combined scheduling rule for solving the original problem.
S2: and constructing a multi-objective optimization model aiming at the FFSP-BPM problem of incompatible workpiece family batch operation under the environment of dynamic arrival and uncertain working hours of the workpiece.
S3, adopting an improved gene expression programming algorithm to generate a combined scheduling rule to solve the FFSP-BPM problem, and providing improved strategies of repeated individual elimination, variable neighborhood search and adaptive genetic operators aiming at the defects that the algorithm is easy to generate repeated individuals and fall into local optimum.
Before step S1, the FFSP-BPM scheduling problem involved in the present invention is problem-defined. Specifically, as shown in fig. 1, when different types of workpieces dynamically arrive at a workshop, a production task can be completed through K processing stages in sequence, wherein the l (l is more than or equal to 1 and less than or equal to K) stage is a batch processing stage, the stage comprises a plurality of batch processors with the load limiting quantity of B, the batch processing task of a limited number of workpieces can be completed, the other stages are single-piece processing, and the M is used for processing the workpieces(k)(M(k)Not less than 1), the workpiece can only be processed by one device of the process at each processing stage, the processing time of different types of workpieces at the same processing stage is different, the processing time fluctuation caused by the new and old states of the device is considered, and the processing time of the same type of workpieces at different devices at the same stage is also different. Batch processing facilities can only process the same type of workpiece simultaneously, with the job time depending on the longest processing time in the jobs contained in each batch, i.e., in parallel batch processing jobs that are incompatible with the workpiece family.
The following assumptions are proposed based on the above: (1) the buffer area between adjacent stages is an infinite buffer area; (2) the batch size is only related to the number of workpieces in the batch; (3) batch phase jobs have minimum capacity requirements.
The problem decomposition-based combined scheduling construction process in step S1 is shown in fig. 2, and specifically includes the following steps:
s1-1: FFSP-BPM scheduling problem decomposition. The flexible flow shop batch scheduling problem can be decomposed into a flexible flow shop scheduling problem and a batch processing scheduling problem. For the former, firstly, the processing sequence of the workpiece at each stage needs to be decided, namely, the operation priority is judged through the operation parameters so as to realize operation sequencing (the workpiece is selected by the equipment); secondly, the work equipment of the workpiece needs to be determined, namely equipment selection (workpiece selection equipment) is realized by judging equipment priority through equipment parameters. For the sub-problem of batch processing scheduling, the decision process is divided into three steps, firstly, workpieces are distinguished according to workpiece families, job sorting is carried out on the workpieces in each product type according to priorities, then feasible batches are generated according to the batch requirements, and finally, for all the feasible batches of all the workpiece types, the batch forming process is guided through batch parameters and the batch priorities are determined, so that the batch sorting (equipment selection batches) of incompatible workpiece families is realized.
S1-2: the main problem factor of the combined scheduling rule structure influencing the operation sorting subproblem is that workpieces need to be determined according to the operation sorting rule based on the related attributes of the workpieces such as arrival time, delivery date, process number, processing time and the like. The main problem factor affecting the equipment selection subproblem is equipment, and the priority of the equipment needs to be determined according to the equipment selection rule based on the information of the relevant attributes of the equipment such as the working efficiency, the earliest available time, the number of waiting workpieces and the like. The problem factor affecting the batch generation subproblem is a workpiece, and firstly, an alternative workpiece range is determined according to a Time Window strategy (TW) based on the attribute of the workpiece arrival Time; and secondly, generating all feasible batches meeting the minimum capacity requirement according to the attribute of the workpiece type. The problem factor affecting the lot ordering sub-problem is the workpieces in all feasible lots, and the candidate lot priority needs to be determined according to the lot ordering rule according to the related attributes of the workpiece family, such as lot, operation time, mean delivery date, lot weight, etc.
And combining the 3 types of heuristic rules analyzed above, and forming a combined dispatching rule algorithm for solving the FFSP-BPM by matching with a time window strategy for solving batch generation subproblems. Combined scheduling rules (DR)HFSP-BPM) Expressed by equation (1), JS represents a job sort rule, MS represents a device selection rule, and BS represents a batch sort rule.
DRHFSP-BPM＝{MS,JS,BS} (1)
And designing a decision variable from the solution of the sub-problems, providing related parameter definition to construct a mathematical model, and performing joint decision on the three sub-problems to generate an FFSP-BPM real-time scheduling scheme in a dynamic environment. Step S2 includes the following steps:
s2-1: and defining parameters and decision variables of the multi-objective optimization model with the maximum completion time and the average delay deviation.
(1) Decision variables
The final objective of the general flow shop scheduling subproblem is to determine the work-on time of the workpiece at each stage by the job sequencing rule, so the corresponding decision variables are shown in formula (2).
The final purpose of the parallel machine scheduling subproblem is to determine the working equipment of the workpiece at each stage through equipment selection rules, so the corresponding decision variables are shown in equation (3).
The final purpose of the sub-problem of batch scheduling is to determine the processing batch and the operating equipment of the workpiece in the batch processing stage according to the batch sorting rule, and because the invention considers the batch processing process of incompatible workpiece families, if the operating equipment is determined according to the equipment selection rule and then the workpiece family classification, batch generation and sorting are carried out on the to-be-processed operation of the equipment, the utilization rate of the batch processing equipment is greatly reduced, so the invention does not adopt the equipment selection rule to decide the equipment to be operated by the workpiece in the batch processing stage, but synthesizes the feasible batches of all the equipment for sorting, and the corresponding decision variable is shown in the formula (4).
To sum upThe decision variables of the FFSP-BPM scheduling problem are the combined scheduling rules DR including the equipment selection rule MS, the job scheduling rule JS and the batch scheduling rule BSHFSP-BPMJS, MS, BS. The scheduling objective of the FFSP-BPM scheduling problem researched by the invention is to determine DRHFSP-BPMTherefore, the processing sequence of the workpieces in the batch processing stage, the processing batches and the processing equipment and the optimal processing sequence of the workpieces in the discrete processing stage are determined, and the start-up and completion time of each process of the workpieces is further determined so as to achieve the performance goal of the production system.
(2) Related parameter definition
Table 1 shows the relevant parameters of the model:
TABLE 1 parameter definitions
S2-2: and defining a multi-objective optimization model objective function and constraint conditions of the maximum completion time and the average delay deviation.
The present invention considers two objective functions: maximum time-out CmaxAverage delay bias (AE/L). The maximum completion time refers to the completion time of the last workpiece; the average delay deviation measures the difference degree between the completion time and the delivery date of the workpiece, and the workpiece tends to be delivered on time. The optimization objectives of the present invention can be expressed as:
f1＝min(maxjCjK) (5)
wherein f is1、f2Respectively representing a function aimed at minimizing the maximum completion time and the mean delay deviation, CjKIndicating that workpiece j is at the lastCompletion time of processing stage K, djIndicating the delivery date of workpiece j.
According to the characteristics of the actual production problem, the problem model should satisfy the following constraints:
constraint (7) indicates that a piece of equipment can only process one workpiece at a time, and constraint (8) indicates that a workpiece can only be processed once at each stage. Constraint (9) indicates that the part cannot be processed before it reaches the production system. The constraints (10) ensure that the discrete processing apparatus can only process one workpiece at a time. Constraint (11) indicates that the next process can only be started if the previous stage is completed. The constraint (12) defines the completion time of the workpiece at stage k. The constraint (13) represents the actual machining time of the workpiece at the k-th stage. The constraint (14) indicates that the run time of a batch on the batch processing tool is equal to the maximum value of the run times of the workpieces in the batch. Constraints (15) ensure that a workpiece is assigned to only one lot. Constraints (16) specify that the batch size cannot be less than the minimum batch requirement and cannot exceed the batch processing equipment capacity. Constraint (17) indicates that workpieces assigned to the same batch have the same start time in the batch phase. Constraint (18) indicates that a workpiece j can only belong to one product type i, and constraint (19) indicates that a batch of workpieces can only come from one product type.
Aiming at a multi-target model taking minimized maximum completion time and average delay deviation as optimization targets, considering the characteristics of incompatible parallel batch processing of workpiece families, further providing an improved multi-target model solving method based on a gene expression programming algorithm in the step S4, wherein the step S3 comprises the following steps:
s3-1, constructing an endpoint set and a function set based on the classical scheduling rule. Taking the classical scheduling rule as an endpoint set, selecting the operators { +, -/√ v, ()2The GEP algorithm represents different scheduling rules by linear combination of endpoint set and function set as elements of function set.
Encoding based on the combined scheduling rules, S3-2. And according to a three-section real number coding mode, the dispatching rules of the equipment selection process, the job sorting process and the batch sorting process are respectively expressed.
S3-3, decoding the chromosome to generate a set of dispatch rule function expressions containing device selection rules, job ordering rules, and lot ordering rules.
S3-4-genetic manipulation based on effective length. Defining effective length of gene, designing genetic operator based on effective length, and carrying out genetic operation on elements in effective length.
S3-5: aiming at the defects that the algorithm is prematurely converged, easily falls into local optimum and is easy to generate repeated individuals, an algorithm improvement strategy is provided: 1) a repeated individual elimination algorithm is provided, and the duplication elimination is carried out on the population after genetic operation so as to ensure the diversity of the population; 2) two neighborhood structures are provided to improve the local searching capability of the algorithm; 3) and introducing an adaptive genetic operator to adjust the genetic probability.
The details of the function set and endpoint set construction involved for step S3-1 are as follows:
(1) function set
The GEP algorithm represents different scheduling rules through linear combination of an endpoint set and a function set, the search space of the GEP algorithm grows exponentially with the increase of the number of set elements, and in order to improve the algorithm efficiency while reducing the search space, the set needs to be designed appropriately. By integrating the results of the related research, the present invention selects the operators { +, -/()2As elements of the function set, where the division is a protective division, i.e. returns 1 when divided by 0, otherwise returns the normal quotient, and if the square root parameter is negative, the result is
Non-linear and linear combinations of parameters can be achieved by these operators.
(2) Endpoint set
Different from the traditional research that the use environment and the system parameters are directly used as the endpoint set, the invention uses the widely applied classic scheduling rules in the literature as the elements in the endpoint set and outputs the functional relationship among the classic scheduling rules. In combination with the problem features of the present invention, the following table 2 is used for the device selection process, job sorting process, and batch process, respectively.
TABLE 2 classical scheduling rules
The step S3-2 adopts a three-stage real number encoding method to represent the dispatching rules of the equipment selection process, the job sorting process and the lot sorting process respectively. Based on the fixed-length coding principle of the GEP algorithm, each gene segment is divided into a head part and a tail part, the head region of each gene segment is randomly selected from a function set or an end point set of the function set, and the tail region is only selected from the end point set of the function set.
The gene length is the sum of the head length h and the tail length t, and the relationship satisfies t ═ hx (n-1) +1, where n represents the maximum value of the number of operators in the function set. The gene generated in this manner is referred to as a K expression. Using the operators in the function set to correspond to the numbers {2,2,2, 1,1}, so that n is 2, setting the lengths of the heads of the three gene segments to be {4,5,4}, the lengths of the tails to be {5,6,5}, and the lengths of the genes to be {9,11,9}, constructing a chromosome as shown in fig. 3, where the numbers in the chromosome correspond to the elements in the end point set, and Q represents the square.
The chromosome decoding process in step S3-3 is illustrated by FIG. 4.
Step 1: the K Expression is converted into a K Expression Tree (ET) by using a Depth First Search (DFS) algorithm. And taking the first element of the gene segment as a root node of the tree structure, sequentially traversing from left to right according to the number of the K expression until all the nodes of the last layer are the symbols in the terminator set, and ending the iteration.
Step 2: and traversing the K-expression tree by adopting a middle-order traversal algorithm to obtain a function expression of the scheduling rule. The left sub-tree is traversed firstly, then the root node is visited, finally the right sub-tree is traversed, when the left sub-tree and the right sub-tree are traversed, the left sub-tree is still traversed firstly, the root node is visited, the right sub-tree is traversed, and after the traversal is finished, a scheduling rule function expression based on relevant elements in the production process can be generated.
And S3-4 fitness calculation, namely decoding the chromosome to generate a group of dispatching rule function expressions comprising an equipment selection rule, a job sorting rule and a batch sorting rule, applying the mined dispatching rule to an experimental scene, constructing a dispatching rule evaluation simulation model, and calculating individual fitness values according to the selected 2 individual performance indexes. In order to avoid the problem that the characteristic parameter dimension in the endpoint set is not uniform and the function operation cannot be carried out, the normalization processing is carried out on the relevant parameter values of the workshop according to a formula (20).
In the formula, thetamaxAnd thetaminRespectively the maximum value and the minimum value of the workshop related parameter value theta, and theta is the value of the parameter theta of the workpiece or equipment. RθThe normalized parameter value is used as the priority of the workpiece determined according to the parameter theta.
The effective length of the defined gene in the step S3-4 is mainly defined by four modes of a crossover operator, a mutation operator, an RIS insertion operator and a non-dominant sequencing and selection operator. The GEP algorithm is easier to realize genetic operation, but a phenomenon that partial codes do not work in a final practical expression occurs, so that invalid elements appear at the tail of a gene sequence. The effective iteration of the population cannot be guaranteed by performing genetic operation on invalid elements, so that the effective length of a gene is defined, and genetic operators based on the effective length are designed to perform genetic operation on the elements in the effective length.
Definition 1: the effective length L of the gene. And the number of nodes of the tree structure formed after the chromosome is converted into the K expression tree from the K expression. For example, the effective lengths of the three gene segments shown in FIG. 3 are {6,9,6 }.
(1) Crossover operator
Three crossover operators are set for two individuals to be crossed: single point crossover, two point crossover, and gene crossover. Single-point forward crossing tends to cross the functional combinatorial relationship between two individual terminal elements, two-point crossing tends to cross a certain substructure in two individual tree structures, and genetic crossing is used to simulate different combinations of three types of rules. And the chromosomes are fully transformed by combining three types of crossing modes to increase the population diversity and enhance the global search capability of the algorithm.
The single-point crossing and two-point crossing operations are performed for each gene segment of two individuals to be crossed. Firstly, determining the minimum value C of the effective lengths of the current gene segments of two individuals to be crossedmin＝min{Li,LjAnd maximum value Cmax＝max{Li,Lj}. For single point crossing, at random [1, C ]min]To select a node for forward single point crossing as shown in fig. 5 (a). For two-point crossing, randomly at [0, Cmin]One node is selected as an initial node Cross1In [ Cross ]1,Cmax]Randomly selects a node as an end node Cross2The elements between the two are exchanged as shown in FIG. 5(b), wherein the yellow segment is the effective length part of the gene segment. The gene crossover operation was performed for a certain gene fragment among the three gene fragments as shown in FIG. 5 (c).
(2) Mutation operator
Setting a single point mutation operator for each gene segment of an individual to be mutated is shown in fig. 6. The smaller value of the effective length of the gene fragment i and the length of the head is taken as the maximum range of the variation position, namely, the value is in [0, min { M {i,h}]And randomly selecting a variation position, if the position is positioned at the head, randomly selecting other elements which are different from the existing elements of the variation position from the function set and the union set of the endpoint set, and if the variation position is positioned at the tail, randomly selecting from the endpoint set.
(3) RIS insert string operator
RIS(Root Insertion SequAnd (ce) the skewing operator is a genetic operator specific to GEP, and can effectively destroy the original structure of an individual genotype and an individual phenotype. The RIS insertion string operation performed on each gene fragment of an individual is shown in fig. 7. In [1, h]Randomly selecting one element in the range as the position R of the operator in the function set1In [ R ]1,h+t]Randomly selecting a position R within range2Will [ R ]1,R2]The fragment is inserted into the head of the fragment, the original element is correspondingly moved backwards, and a gene segment with the same length as the inserted fragment is deleted from the tail of the original head so as to ensure the legality of the chromosome structure.
(4) Non-dominant sorting and selection operator
Introducing a Non-dominant sorting method of an NSGA-II (Non-dominant sorting genetic algorithm-II) with elite strategies, finding out all Non-inferior solutions from the current population according to the fitness evaluation result, calculating the crowding distance of the Non-inferior solutions, assigning grades to the Non-inferior solutions according to the sequence, and removing the found Non-inferior solutions from the population until the size of the population is 0. The child selection is performed according to the non-inferior solution grade, and the non-inferior solution of the same grade is selected according to the congestion distance of the non-inferior solution. To avoid premature convergence of the algorithm, each iteration is based on a parent population P of size NiGenerating an offspring population Q with the same size as the parent populationiA 1 is to PiAnd QiAnd merging the two generations of populations, and generating the next generation of population on the basis.
The improvement strategy for the algorithm described in S3-5 is detailed below:
(1) repeated individual elimination algorithm
Because the GEP algorithm uses fixed-length codes to represent the characteristics of tree structures with different sizes, although the genetic operation based on the effective length controls the difference of individuals before and after the genetic operation, the situation that two individuals with different genotypes but the same phenotype type and the same generated scheduling rule function expression are generated in a combined population cannot be avoided. The pair of individuals having identical episomes of each gene in the effective length is defined as a repetitive individual. The repeated individuals generated by genetic operation have two sources, including that an offspring individual and a parent individual are mutually repeated individuals, and the inside of an offspring population is mutually repeated individuals.For example, as shown in FIG. 8, the cross-over operation of P1 and P2 individuals resulted in C1 individuals and P of the parent populationiThe individuals are mutually repetitive individuals; to PiC after individual RIS inserting string operationiIndividual and pair PjC produced by individual after single point mutationjThe individuals are mutually repetitive individuals.
In order to avoid that the algorithm is prematurely converged to be in local optimum due to the occurrence of repeated individuals, before non-dominated sorting and filial generation selection, the repeated individuals occurring in the GEP iterative evolution process are removed. The method comprises the following specific steps:
step 1: traversing the merged population, and searching for repeated individuals.
Step 2: and determining an individual to be operated. And if the two individuals which are mutually repeated are respectively positioned in the parent population and the offspring population, taking the parent individuals corresponding to the repeated individuals in the offspring population as the individuals to be operated. And if the two individuals which are mutually repeated are both positioned in the offspring population, randomly selecting one individual corresponding to the parent individual as the individual to be operated.
Step 3: and (5) repeatedly removing individuals. And repeating the variation and RIS string insertion operation on the individuals to be operated until the generated individuals are not repeated with other individuals in the combined population.
After repeated individuals are removed, all the individuals in the combined population after the duplication removal are subjected to non-dominated sorting, and the optimal N individuals are selected as a new generation population p according to the quality degree of the individuals in the populationi+1。
(2) Variable neighborhood search algorithm
The global search capability of the algorithm is enhanced to a great extent through genetic operation based on effective length and selection operation based on repeated individual elimination, and the diversity of the population is ensured. By changing the Neighborhood structure near the current local optimal solution, the Variable Neighborhood Search algorithm (VNS) can continuously Search for a better local optimal solution, and can effectively improve the local Search capability of the GEP algorithm.
Two neighborhood structures are designed according to the structural characteristics of the problem, the offspring population generated after repeated individual removal is subjected to non-domination sorting, and the individuals in the non-domination stage are subjected to variable neighborhood searching, so that the local searching capability of the algorithm is improved. The neighborhood structure definition includes: randomly scrambling and rearranging the neighborhood, namely scrambling and rearranging all end point set elements in the effective length of the current gene segment; and (3) forward insertion neighborhood, namely randomly selecting the positions of two elements belonging to the endpoint set in the effective length part of the current gene segment, inserting the element corresponding to the latter position into the front position, and sequentially moving the elements of the other endpoint sets backwards. The variable neighborhood search process is illustrated in fig. 9.
(3) Adaptive genetic operator
When an evolutionary Algorithm such as Genetic Algorithm (GA) and GEP is used for solving the problem, the higher the Genetic operation probability is, the faster the speed of generating a new individual is, the higher the fitness individual structure is, and the search process tends to random search; if the genetic operation probability is too small, the search process becomes slow or even stagnates, the population diversity cannot be maintained, and the local optimum is easy to fall into. In order to avoid the disadvantages caused by using the fixed genetic probability, the invention introduces an adaptive strategy, dynamically controls cross operation, mutation operation and RIS string insertion probability, ensures that the population can be rapidly converged, simultaneously prevents the diversity of the population from being damaged, improves the adaptability of the search to the space change, and has a specific setting method as shown in a formula (21).
In the formula, PmaxAnd PminThe maximum value and the minimum value of the genetic operation probability are respectively, f is the fitness value of the current individual to be operated, and for the crossover operator, f is the larger value of the fitness of the two individuals to be crossed. f. ofmaxAnd favgThe maximum value and the average value of the fitness of all individuals in the population to be operated are respectively. Aiming at the multi-objective optimization problem, the performance targets are subjected to non-dimensionalization by adopting a max-min standardization method, and the sum of the standardized targets is taken as the fitness value of the current individual. The self-adaptive genetic operator is adopted to endow poorer individuals with higher genetic operation probability and accelerateThe individual is updated and evolved, and a good individual is endowed with lower genetic operation probability, so that the good individual is protected.
Example 1
An improved gene expression programming algorithm is applied. Design example related parameters, the number of production stages obeys uniform distribution U (3,7), batch processing stage BkIs located at the first
In the stage, the capacity B of the batch processor is subjected to uniform distribution U (2,6), the number of workpiece types is subjected to uniform distribution U (2,6), the scale of workpieces is subjected to uniform distribution U (300,1000), and the utilization rate of equipment is subjected to uniform distribution U (0.5, 0.9); the delivery date tension factor obeys uniform distribution U (4,8), and the invention sets 3 levels according to the distribution range of the calculation parameters.
And setting relevant parameters for each calculation example, and respectively setting operation time distribution for discrete steps of the batch processing procedure. Batch process processing time pbSubject to a uniform distribution U (100,200), discrete process processing times pdUniform distribution U (1,30) is obeyed. The workpiece weight setting method adopts a 4:2:1 method, wherein a very important workpiece accounts for 20%, the weight is given to 4, a more important workpiece accounts for 60%, the weight is given to 2, a less important workpiece accounts for 20% as well, and the weight is given to 1. 18 orthogonal examples of different sizes were generated based on the IBM SPSS Statistics 19 platform.
The algorithm parameters of the present invention relating to the improved GEP algorithm and the comparison algorithm (GEP algorithm, plant parameter GEP algorithm and classical scheduling rules algorithm) are shown in Table 3.
TABLE 3 GEP Algorithm and comparison Algorithm parameter settings
The improved GEP algorithm, the workshop parameter GEP algorithm and the classical scheduling rule algorithm are adopted to solve all the example problems, 5 independent simulation experiments are carried out, the average values of the trans-generation distance and the distributivity index delta in 5 running times are counted and shown in the table 4, wherein the value shown in bold is the optimal value obtained by calculation of the 4 algorithms. As can be seen from the table, for most of the test problems, the Inverse Genetic Distance (IGD) and the distributivity index Δ of the improved GEP algorithm are superior to those of the GEP algorithm, the plant parameter GEP and the classical scheduling rule, and as can be seen from the Total result in the last row in the table, the improved GEP algorithm is superior to the other three comparative algorithms in terms of convergence, diversity and distributivity of non-inferior solutions for all the test problems.
Table 4 comparative algorithm inverse distance IGD mean results
Fig. 10 is a box plot of the relative coverage of the modified GEP algorithm versus the three other comparative algorithms. Each box plot has different examples on the abscissa and relative coverage on the ordinate, representing the relative coverage C (a, B) between row associated algorithm a and column associated algorithm B. The box plot as the top right corner represents the ratio of the non-dominant solution generated by the modified GEP algorithm to cover the solution in the classical scheduling rule. As can be seen from the figure, the solutions generated by the classical scheduling rules are almost all dominated by the non-inferior solutions generated by the improved GEP algorithm, which proves that the improved GEP algorithm is improved in both the maximum completion time and the average delay deviation, and furthermore, the non-inferior solutions generated by the improved GEP algorithm can dominate the vast majority of non-columns generated by the plant parameter GEP algorithm, while, as a whole, C (improved GEP, GEP) > C (GEP, improved GEP). Therefore, the effectiveness of the algorithm improvement strategy provided by the invention, the effectiveness of the strategy that the terminal element adopts the classical scheduling rule and the effectiveness of the scheduling rule generated by adopting the intelligent algorithm can be proved.
Examples adopt Sn1Bn2Fn3Nn4U _ C denotes a symbol having n1 stagesThe flexible flow shop with the capacity of n2 in batch processing stage has the processing tasks including n3 n4 workpieces of product types, the shop utilization rate is U, and the delivery date tension factor of the workpieces is C. In order to show the process of the scheduling rule directing the output of the scheduling scheme, a small-scale example of s3b4f2n30_0.8_2 is designed for description, which means that the application scenario is a flexible flow shop with 3 stages and a batch processing stage capacity of 4, the processing tasks include 30 workpieces of 2 product types, the shop utilization rate is 0.8, and the workpiece delivery time tension factor is 2. Fig. 11 is a gantt chart corresponding to a non-dominant solution (i.e., a combined dispatching rule including a device selection rule, a job sorting rule, and a lot sorting rule) generated by the GEP algorithm according to this example, and the combined dispatching rule formed after decoding the non-dominant solution is shown in table 5. FIG. 12 is a Gantt chart generated using the SP-ATC-RMSTB classic schedule rule for the s3b4f2n30_0.8_2 algorithm. Wherein the blue-based workpiece and the orange-based workpiece respectively represent a product type 1 and a product type 2. It can be seen that under the classical scheduling rules approach, the batch processing phase completes 10 batches, each containing an average of 3 workpieces, for a total completion time of 2150. In the scheduling scheme formed by improving the scheduling rule generated by the GEP algorithm, 14 batch jobs are completed in the batch processing stage, each batch contains 2.5 workpieces on average, and the total completion time is smaller than that of the scheduling scheme generated by the classical scheduling rule. It can be shown that the production line can be more balanced by reasonable batch decision. The effectiveness of the improved GEP algorithm proposed by the present invention is also demonstrated.
TABLE 5 non-dominated solution of s3b4f2n30_0.8_2 example obtained by improving GEP algorithm
In summary, aiming at the batch scheduling problem of the flexible flow shop, the performance and the production efficiency of the combined scheduling rule including the equipment selection rule, the operation sequencing rule and the batch sequencing rule generated by the improved GEP algorithm are greatly improved. Compared with a workshop parameter GEP algorithm, the scheduling rule formed by combining functions of the classical scheduling rule is proved to have better performance and can effectively guide the scheduling process compared with the scheduling rule based on the workshop parameter.
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present invention.
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