Complex System Modeling and Simulation Complex System Modeling and Simulation

: Smart manufacturing in the “Industry 4.0” strategy promotes the deep integration of manufacturing and information technologies, which makes the manufacturing system a ubiquitous environment. However, the real-time scheduling of such a manufacturing system is a challenge faced by many decision makers. To deal with this challenge, this study focuses on the real-time hybrid flow shop scheduling problem (HFSP). First, the characteristic of the hybrid flow shop in a smart manufacturing environment is analyzed, and its scheduling problem is described. Second, a real-time scheduling approach for the HFSP is proposed. The core module is to employ gene expression programming to construct a new and efficient scheduling rule according to the real-time status in the hybrid flow shop. With the scheduling rule, the priorities of the waiting job are calculated, and the job with the highest priority will be scheduled at this decision time point. A group of experiments are performed to prove the performance of the proposed approach. The numerical experiments show that the real-time scheduling approach outperforms other single-scheduling rules and the back-propagation neural network method in optimizing most objectives for different size instances. Therefore, the contribution of this study is the proposal of a real-time scheduling approach, which is an effective approach for real-time hybrid flow shop scheduling in a smart manufacturing environment.


Introduction
With the introduction of the concept of Industry 4.0, there has recently been an emphasis on advancing manufacturing technologies in developed and developing countries [1] .The development of smart manufacturing is considered a key measure to establish a competitive advantage in manufacturing [2] .Smart manufacturing takes advantage of advanced information and manufacturing technologies to achieve a flexible, smart, green [3,4] , and reconfigurable manufacturing process [5] to respond to the dynamic market [6] .
A smart shop floor is the specific carrier of smart manufacturing.A large number of intelligent sensing devices, such as radio frequency identification (RFID) tags, are deployed in smart shop floors, and various devices can collect real-time data, such as the status of machines or jobs.On a real shop floor, various kinds of disturbances dynamically happen, such as a new job arrival, machine failure, and processing time change.In conventional shop floors, without the support of data collection and interaction technologies, it is difficult to quickly make optimal decisions to deal with these disturbances in real time.However, under a smart manufacturing environment, the production data can be collected in real time and further utilized to make scheduling decisions to deal with those disturbances.In the new manufacturing, however, production planning and scheduling still play a crucial role in improving production efficiency [7] .As such, there is a need to develop a real-time scheduling approach for smart manufacturing.
Making scheduling decisions with the support of real-time data is a challenge.The completely reactive scheduling strategy, represented by the rule-based scheduling method, has a low computational cost and fast response.Compared with the predictive (or robust) scheduling and predictive-reactive scheduling strategy, this strategy is more suitable for shop floors with frequent production status changes.Developing efficient scheduling rules is the key to a perfect schedule performance.With the support of smart manufacturing technology, how to choose an appropriate scheduling rule according to the real-time production status has become a primary challenge.
Many authors have attempted to use artificial intelligence based methods to dynamically select dispatching rules according to the changes in system status.For example, Refs.[8−12] used a neural network to solve different scheduling problems.Reference [13] adopted reinforcement learning based approaches to multi-objective dynamic scheduling.Reference [14] presented an inductive learning method based real-time scheduling mechanism for reentrant hybrid flow shops.Reference [15] used decision trees to select scheduling rules to solve scheduling problems.
Many rule-learning methods have been studied in the existing studies, but the best method in a specific environment remains unknown.Some researchers study in other directions, which automatically generate new and effective rules for a given scheduling environment.Gene expression programming (GEP) is one of the machine learning methods and has been applied in many fields [16,17] , but it is seldom used to solve dynamic scheduling [18] .References [19−22]  constructed scheduling rules with genetic programming and GEP methods for different scheduling problems.GEP can flexibly construct scheduling rules, which adapts to shop floors with frequent status changes.
The hybrid flow shop scheduling problem (HFSP) is more complex than the traditional flow shop problem.On the dynamic HFSP, most relevant publications focused on the traditional manufacturing environment.Reference [23] considered two-stage hybrid flow shops with a dynamic order arrival and estimated the flow time of each order at the arrival time using four common scheduling rules.Reference [24] developed a priority-based hybrid parallel genetic algorithm with a predictive reactive complete rescheduling strategy for an energy-efficient dynamic flexible flow shop scheduling problem.Reference [25] proposed an improved particle swarm optimization method to address the dynamic flexible flow shop scheduling problem considering new job arrivals and machine breakdowns.A perusal of the existing literature concludes that most of the existing publications focused on the traditional shop floor environment, in which the real-time data of the production status cannot be used to aid scheduling decision making.To the best of our knowledge, few publications have investigated the application of GEP into real-time HFSP.Accordingly, this paper proposes a real-time scheduling approach to automatically constructing efficient scheduling rules in real time for HFSP in a smart manufacturing environment.
The main contributions of this study are as follows: (1) The difference between the real-time scheduling problem in a smart manufacturing environment and the traditional scheduling is discussed in detail, and its scheduling task is described.
(2) A real-time scheduling approach for different types of shop floors under a smart manufacturing environment is developed.When the shop floor status changes, an optimal scheduling rule suitable for the current status can be obtained through the GEP module, then the processing priority for each job using the obtained rule can be calculated, and thus an efficient scheduling decision can be made.The proposed framework is specifically applied to real-time HFSP under the smart manufacturing environment, and GEP is designed in detail according to the characteristics of the HFSP.
The second is also the novelty of this work.Though there are some applications of GEP in other fields, its performance in solving real-time scheduling has not been evaluated yet.This gap motivates this study.
The remainder of the paper is organized as follows.Section 2 presents the HFSP in a ubiquitous environment.Section 3 proposes the solution and mechanism for the scheduling problem.Section 4 provides the experiments and results.Section 5 concludes the paper and discusses future work.

Hybrid Flow Shop Real-Time Scheduling
Problem in Smart Manufacturing

RFID-based manufacturing shop floor
The hybrid flow shop scheduling in a ubiquitous environment is different from the conventional scheduling.It is equipped with a large number of active intelligent devices, such as RFID tags, RFID readers, wireless networks, and Bluetooth devices [9] .These data collection devices are deployed on shop floors.Such advanced data collection technologies provide a channel for real-time information interaction and communication among various manufacturing resources.In the production scheduling process, the processing information of a shop floor can be collected through active sensing technology.Moreover, the job task to be processed and processing machines are bound with RFID electronic tags, so they become carriers of real-time production status information and have capabilities of information exchange and feedback.The RFID network is used to collect data about a job in process and the machine status information in real time.Electronic tags can perform a real-time data collection of manufacturing processes.These fixed readers equipped on the machine act as a fixed carrier for production data, record the running statuses of machines in real time, and provide machine operators with theoretical technical assistance about the production processing technology.A job buffer is placed at each production stage, and an RFID reader is installed at the entrance and exit of the buffer area to collect the real-time information of the job reaching and leaving the buffer area.Each machine operator is equipped with an RFID card for confirming the corresponding work content.When the machine operator places the RFID card on the stationary reader, the reader will display the job that needs processing on the machine, and then the operator will go to the buffer to take the corresponding job for processing.Each job follows rigid modes from the first stage to the last stage and must be processed on a certain machine at each stage.Figure 1 shows a manufacturing shop floor equipped with RFID tags in a ubiquitous environment [9] .

Real-time scheduling problem description
The real-time HFSP under a smart manufacturing environment can be described as follows [9] : New orders successively arrive at the shop floor with various kinds of information, such as arrival time, due time, and processing time at different stages of jobs.There is a series of parallel machines at each stage.Each job must go through all stages and be processed on one of the machines at each stage.At each stage, a buffer with infinite capacity is used for storing the jobs from the upstream stage.The stage manager needs to make scheduling decisions according to the status information of the buffer area and machine obtained from the RFID device.With the support of such information, the stage manager can select a job from the waiting queue in the buffer area and transfer it to the idle machine.At each decision point, the scheduling rule that should be used to select a proper job to optimize the scheduling performance is the primary concern of real-time scheduling.
The scheduling rule usually refers to the method of evaluating the priority of scheduling tasks in the production process, by which the next system element (usually machine or job) that should be processed can be determined in order.For example, the shortest processing time (SPT), first input first output (FIFO), and earliest due date are the classical scheduling rules frequently employed in scheduling jobs.
In the real-time HFSP, the scheduling rule is the method of calculating the priority value for each job in the queue of the buffer area.Then, the stage manager can make a scheduling decision and process the job with the highest priority.Scheduling rules can greatly affect scheduling performance.Therefore, it is critically important to choose an appropriate scheduling rule at each decision point according to the real-time status of the shop floor.

Assumption
Some assumptions in this study are presented as follows: (1) Each machine can only process one job at a certain moment.
(2) The transportation time between the buffer and machine can be neglected.
(3) Once started, the process cannot be interrupted.(4) A job can only be processed by one of the available machines at a certain stage.

Notation
The notations are listed in Table 1.

Real-Time Flow Shop Scheduling Approach
Existing research on the applications of machine learning in the manufacturing research community has evolved into various branches.In the new manufacturing environment, searching for new and suitable algorithms for real-time scheduling problems has become an important research direction.GEP, proposed by Ref. [26], is an evolutionary algorithm (EA) for effectively searching for accurate decision rules [27] .Compared with other EAs (e.g., genetic algorithm), the major difference in GEP lies in the form of the solution provided.The solutions provided by GEP are heuristic rules.GEP can automatically select elements from a database and then optimally combine them to form a new heuristic rule [28] .However, in other EAs, the solutions are usually a vector of values, providing optimal or best parameters for a predetermined rule structure.Due to this characteristic, GEP can be used to generate decision rules to schedule jobs in real time.Therefore, we propose a GEP-based real-time flow shop scheduling approach in this study.

Real-time flow shop scheduling approach with GEP
Considering the characteristics of a hybrid flow shop in a ubiquitous manufacturing environment, a real-time scheduling approach is proposed, as shown in Fig. 2. The online scheduling process steps are as follows: (1) The jobs dynamically arrive at the production shop floor.
(2) The intelligent device collects the information of jobs waiting to be processed at each stage buffer and the idle machine status information in the current stage in real time.
(3) When there are scheduling decision points for several jobs waiting for processing, the required production status information is collected in real time using intelligent devices.Then, the optimal scheduling rule obtained with the GEP module is used to determine the job with the highest priority in the buffer and arrange it to the idle machine for processing.
(4) The job selection and processing of all scheduling decision points are completed.
The details are discussed in the following sections.

GEP module
The best scheduling rule is obtained by the GEP evaluation module with the environmental configuration and scheduling performance indicators in shop floors.

GEP step
There are two genetic components in GEP: Terminal and function. (

1) Terminal
The position of a terminal is at the end of the system.It is characterized as an information element and does not deal with other information elements.Corresponding to its tree structure, terminals represent the leaf nodes in a tree.When the system is running, the values corresponding to such leaf nodes may have various sources, such as the inputs outside the system, Table 1 Notations.

Notation Description t
The decision point time K The number of stage i The job index j The operation index The processing time of Job i at Stage j The Stage j at decision point time t The job set waiting in the buffer of stage at decision point time t The idle machine set in the buffer of stage at decision point time t Their roles are to provide information to the system for use.Typically, a terminal set is a collection of several terminals, which include various input data and noargument functions in GEP [26] .When scheduling a manufacturing system, the input data generally include to-be-scheduled tasks and available machines.In this study, we expect GEP to choose a job to be scheduled at each decision-making time point.Hence, the terminal elements should be job-related information.
The terminal set chosen in this study, which comprises the attributes of jobs, is listed in Table 2.
Normally, the symbol is used to represent the terminal set.The terminal elements in the terminal set are represented by t.The relationship between t and is .

(2) Function
The position of the function in GEP includes the intermediate structure formed by any non-terminal symbol and all the non-leaf nodes corresponding to the tree structure.A function set typically includes operators associated with a certain problem or a program component of a programming language.The function is responsible for the transfer and calculation of node information.The purpose of GEP is mainly to search for formulas and rules, so the included functions are basically common operators, as shown in Table 3.In this study, we just needed arithmetic to calculate each job's priority.Hence, the function set used in our method includes four functions: "+", "−", "×", and "/".In GEP, the phenotype of an individual is generally an arithmetic expression, such as .A tree structure is usually used to describe the individual phenotype of GEP, which is called an expression tree.The leaf nodes in the tree are made up of elements in a terminal set, and the elements in a function set form the non-leaf nodes in the tree structure.Here the elements in the terminal set are represented by letters, and the expression tree can be expressed as For example, suppose that = , = .Accordingly, the expression tree of the mathematical expression is shown in Fig. 3. (

4) K-expression
In GEP, K-expression is obtained by traversing all the elements of an expression tree according to the level, which will construct the gene coding of the chromosome.
Definition 1 (K-expression): Traversing all the nodes in an expression tree in order from top to bottom and from left to right, the resulting linearized sequence is called the K-expression corresponding to the expression tree [26] .
Two methods can be used to obtain a K-expression: width-first traversal [26] and depth-first traversal [29] .The width-first traversal accesses each node in the expression tree from top to bottom and from left to right.By contrast, the depth-first traversal accesses each sub-tree whose node is a root node in the expression tree from top to bottom and from left to right.The K-expression obtained by these methods is shown in Fig. 4.
Because the sequence obtained by the depth-first traversal method preserves the original sub-tree structure of the expression tree, it is more suitable for the genetic evolution to generate high-quality solutions.

Xiuli Wu et al.: Real-Time Hybrid Flow Shop Scheduling Approach in Smart Manufacturing Environment 339
In this case, the depth-first traversal method is employed.By applying the reverse operation of this method, the K-expression can be easily transformed into an expression tree.The first gene of the Kexpression corresponds to the root node of the expression tree.If the node of the expression tree belongs to , then the algorithm produces several branches whose number is equal to the max number of parameters required by the function.If the node belongs to , then it is placed in the vacant leaf node branched from the root node of the sub-tree according to the principles "first left and then right" and "first bottom and then top".The process of converting a K-expression into an expression tree using the formula b + a × c -a / d is shown in Fig. 5.
The relationship between the K-expressions and expression trees is one to one.In GEP, because the gene is derived from the GA pattern, the length of the gene is fixed.To provide the gene with the ability to express various expression trees of various shapes, the start points of the gene and K-expression are set to be the same, whereas the end point of the K-expression does not necessarily coincide with the end point of the gene.Therefore, for each individual gene to fully express an expression tree, the gene must have special characteristics, which will be described in detail in the next section.
(5) Gene structure In GEP, the gene encoding of a chromosome consists F GEP of a head and tail.The gene satisfies two conditions: First, the elements contained in the head can be any terminal set elements or function set elements.Second, the tail can only include terminals.The head length h is determined according to the specific problem, and the tail length is a function of h and n, where n represents the number of parameters of the function element with the most parameters required in the set , and the value of the tail is obtained by Eq. ( 2) [30] .
The purpose of constructing genes through the rule is to ensure that each gene can be transformed into a complete K-expression and that no illegal individuals will be produced in the evolutionary operation.Because the end point of the K-expression does not necessarily coincide with the end of the gene, there will be a portion of the non-coding region at the end of the gene, which can be easily implemented to transform genes into various expression trees and simultaneously maintain individual genes' legality.
Continuing the example given earlier, = { }; = {a, b, c, d}, n = 2, h = 5, Tail = , the length of the gene is 11.A chromosome and its expression tree are shown in Fig. 6.In Fig. 6, "cc" is a non-coding area, which does not make contribution to construct the expression tree.

Flow chart of GEP
The process of GEP is presented as follows: Step (1) Generate an initial population.
Step (3) If the termination condition is met, then the algorithm ends; otherwise, go to Step (4).
Step (4) Reserve the elite individual in the population.
Step (5) Select a new population from the current population as the current population.
Step (6) Perform mutation according to the mutation probability.
Fig. 5 Steps of converting a K-expression into an expression tree.
Step (7) Perform shifting according to the shifting probability.
Step (8) Perform recombination according to the recombination probability.
Step (9) Generate the offspring population and return to Step (2).
The flow chart is shown in Fig. 7.
The key details are described as follows: (1) Coding Existing studies have found that scheduling rules can not only be a single traditional rule but also be constructed by combining system status attributes, which are generally a set of attributes of the processing T GEP tasks or machines.consists of some system status attributes, which contribute to forming a new dispatching rule.
According to the specification of a chromosome, the head contains terminals or functions, and the tail only contains terminals.Here, a typical example is given, assuming h = 5, and the chromosome composition is shown in Fig. 8. (

2) Decoding
The chromosome is decomposed through the depthfirst traversal method [29] , as introduced earlier, and the chromosome is transformed into a corresponding expression tree.That is, an expression corresponding to the scheduling rule is obtained, and the process is shown in Fig. 9.
(3) Selection Roulette wheel selection has been widely employed in the literature [3] , so we also used it in this study to select parents.The basic idea of roulette wheel selection is that the probability that each individual is selected is proportional to the value of its fitness function.The better the fitness, the larger the area occupied by the individual and the greater the probability of being selected into the next generation.Figure 10 shows a schematic diagram of the roulette selection.The times to rotate the wheel are determined by the population size, which ensures that the total number of offspring individuals is equal to that of

Xiuli Wu et al.: Real-Time Hybrid Flow Shop Scheduling Approach in Smart Manufacturing Environment 341 parent individuals. (4) Mutation
Mutation occurs with a small probability and can occur anywhere on a chromosome.The algorithm randomly generates arbitrary positions, replaces the genes in these positions, and obtains the mutated chromosome.However, to ensure the legality of individuals after mutation, the mutation operator needs to follow two rules: • First, if the mutation occurs in the head part of a chromosome, then the replaced gene can be elements in either the terminal set or function set.
• Second, if the mutation occurs in the tail part of a chromosome, then the replaced gene can be the elements in the terminal set.For example, an example of a mutation is shown in Fig. 11."×" in the head gene is replaced by "Wt" and "Lt" in the tail genes is replaced by "Ar" (5) Shifting The process of shifting is presented as follows: Randomly select a gene fragment (e.g., Dd, Pt, Lt) of an arbitrary length in the chromosome, copy it, and insert the copied fragment into a head's random position (e.g., 3) except the first position.To ensure the fixed length of the head, we delete the extra genes (e.g., Ar ×) from the head behind that position.An example of how shifting works is illustrated in Fig. 12.

(6) Recombination
Recombination is an operation used to generate an offspring population.Here, we select a random position in two parent chromosomes, by which each parent is divided into two parts.Then, we exchange the parts of the parents behind the position to obtain two offspring chromosomes.An example of how recombination works is illustrated in Fig. 13.
(7) Fitness evaluation The fitness function is used to evaluate the performance of individuals.For a flow shop scheduling problem, the fitness function is the scheduling objective, such as makespan, objectives based on the flow time, or objectives based on the due date.Here, seven scheduling performance objectives are selected, as shown in Table 4.A scheduling solution is generated through the hybrid flow shop scheduling.As such, each scheduling objective of the corresponding dispatching rule is obtained.For each chromosome, it can obtain several performance values on different objectives.Assuming that Q is the total number of scheduling instances, the performance of each chromosome on a certain objective can be expressed with Eq. ( 3), where is the y-th instance of the K-th objective: After various scheduling objectives are obtained, Average flow time we calculate and ; and calculate the fitness value f i with Eq. ( 4), where n represents the population size, is the average value of the and ε is a small constant.The fitness of each chromosome calculated by Eq. ( 4) will fall into the interval [0, 1].The better the performance chromosome, the more the value tends to 1.

Real-time scheduling for hybrid flow shops
After the best scheduling rule is obtained through GEP, it can be used for the real-time scheduling of hybrid flow shops.The GEP-based scheduling rule calculates the priority of each job in the waiting buffer in real time according to the status of the current production system at the scheduling decision time point and then selects the job with the highest priority for the next processing.
The most critical step in the online scheduling process is to calculate the scheduling rule values and to select the job according to the real-time production status through the GEP module.
For ease of understanding, an example is illustrated below.Assume that the production system is currently at a certain scheduling decision point t = 173.41s.At this time, there are three jobs in the buffer waiting for processing.Their attributes are shown in Table 5.The optimal scheduling rule obtained by GEP is shown in Formula (5).The value of the scheduling rule is calculated by the attributes of each job, and the results are shown in the last column of Table 5.
As shown in Table 5, the maximum value of the obtained priority result is 304.48, and the corresponding task is Job 3. Therefore, Job 3 will be extracted from the buffer for processing.Similarly, all scheduling decision points that appear in online scheduling will be intelligently determined in this way.

Data
We used numerical examples to simulate the real-time online scheduling scenario of the hybrid flow shop in a smart manufacturing environment.The real-time scheduling framework proposed in the preceding sections was used to optimize the best scheduling rule.Instances of three scales were used, which correspond to the number of jobs: Small, medium, and large.All of the shop floor parameters were randomly generated within a predefined distribution.The values of experimental parameters are shown in Table 6.For different numbers of jobs, ten scheduling instances were tested based on randomly generated shop floor parameters, and each instance represents a different processing situation, such as the processing time and job arrival interval.The unit of the processing time is second (s).

Parameter setting
The approach was compiled and run in Intel Core i5-3210, 2.50 GHz CPU, 4.00 G RAM, Win7 32-bit operating system, and MATLAB2013b programming environment.
To determine the experimental parameters, an orthogonal experiment was designed.The parameters that have a significant influence on the GEP algorithm are head length, mutation probability, and recombination probability [19] .For each parameter, we tested three levels (Table 7).We used the orthogonal experimental design to determine the settings of parameters.The orthogonal table is L 9 (3 4 ).
We tested the instance (N = 10) to determine the parameter setting and ran it 30 times for each setting.The average makespan of the 30 runs is reported in Table 8.
Hence, the best setting for the three parameters is head length (Level 2), mutation probability (Level 1), and recombination probability (Level 3).Among them, the most important parameter is the recombination probability.As to the population size and iteration number, the larger, the better.However, a higher computing cost will be taken.Therefore, we chose 100 and 1000 for the two parameters, respectively.In sum, the experimental parameters of the GEP algorithm are shown in Table 9.

Aim
The goal of the experiment is to verify the effectiveness and efficiency of the proposed method.The proposed method is compared with two types of other methods.One is the existing scheduling rules, and the other is the back-propagation neural network (BPNN) method, which is the most common neural network model based on error back-propagation learning.The BPNN includes an input layer, one or more hidden layers, and an output layer.The adjacent layers achieve full connectivity between neurons, whereas there is no connection between neurons in a layer [31] .The study of Ref. [9]    for a specific objective in the existing research.The proposed approach will be compared with these scheduling rules.
(2) Comparison with the BPNN method For real-time scheduling, scheduling rules are changed in real time by scheduling knowledge obtained by the BPNN.The main technology is to obtain the relationship between various system statuses and reasonable rules under a certain status by the BPNN and then conducting real-time scheduling decisions using the results from the BPNN learning.The proposed approach will also be compared with the BPNN method.

Results and discussion
There are three types of experimental results for each job number scale.For each scheduling objective, the results of 10 instances (where N = 10), which were randomly generated, are shown in Table For each instance, the performance of three methods, i.e., the scheduling rules, BPNN, and GEP, were compared.In Table 11, the data include the relative results and their ranking.The relative results in Tables 11-13 were obtained according to the following steps: Take the results for optimizing the maximum completion time as an example.
Step (1) We tested ten instances with N = 10.The maximum completion time for each instance is listed in each row.
Step (2) Divide each value by the minimum in the row, and the results can be converted to a relative result.Evidently, if the relative value is closer to 1, then the method performs better.
Step (3) Calculate the average value among the ten instances to obtain the average relative value for the maximum completion time objective.
Similarly, the average relative values for other objectives are listed in the second to seventh rows in Tables 11−13.According to the average relative values, their rankings are given in the right of the symbol "/" For example, "1.17/3" means that the average relative value is 1.17, and its ranking is the third for optimizing the maximum completion time objective.To further visualize the results of the comparison, the average values of the groups in Table 11 are ranked and plotted as a column chart, as shown in Fig. 14.The X-axis shows the scheduling objectives, and the Y-axis shows the ranking.The shorter the column in Fig. 14, the higher its ranking.The comprehensive ranking, which is obtained by the average value of the ranking on various scheduling objectives for each rule, reflects the overall performance of the scheduling rules.The results show that the proposed approach has excellent performance stability compared with other scheduling rules.
As shown in Fig. 14, various rules have different performances on different scheduling objectives.For the average completion time objective, average flow time objective, and maximum tardiness objective, the GEP-based rule outperforms others.On the objective of the maximum flow time, the BPNN-based rule outperforms others, and the GEP-based rule has the second best performance.On the average tardiness objective, the GEP-based rule performs best.In sum, in terms of the overall performance, the GEP-based rule is the best for most instances, and its comprehensive ranking is far better than other rules.This finding shows that the GEP-based rule has excellent performance for small-scale scheduling problems.
Similarly, we obtained the comparisons on the different objectives for N = 30 and N = 100 instances, as shown in Tables 12 and 13, respectively.Figures 15 and 16 report them through histograms, respectively.
In Fig. 15, for the maximum completion time objective, the SPT rule is the best, followed by the GEP-based rule.For the average completion time objective, average flow time objective, maximum lateness objective, and maximum flow time, the GEPbased rule outperforms the others.For the maximum tardiness objective, the BPNN-based rule is the best, followed by the GEP-based rule.Clearly, in terms of the overall performance, the GEP-based rule is the best, and its comprehensive ranking is far better than other rules.This finding indicates that the GEP-based rule performs excellently for medium-scale scheduling problems.
As shown in Fig. 16, for the average completion time objective, maximum lateness objective, maximum flow time objective, maximum tardiness objective, and average tardiness objective, the GEP-based rule ranks first, followed by the BPNN-based rule.For the average flow time objective, the SPT rule ranks first, followed by the GEP-based rule.For the maximum completion time objective, the BPNN-based rule ranks first, followed by the GEP-based rule.Hence, in terms of the overall performance, the GEP-based rule is also the best, and its comprehensive ranking is more prominent than the other rules, although it does not always have the best performance on all objectives.Therefore, the GEP-based rule is effective for large- Moreover, in Fig. 16, almost all classical scheduling rules are unstable on various scheduling objectives as the job number changes, while the GEP-based rule relatively has stable performance.When the optimization objective is to consider all the objectives mentioned above in a comprehensive way, the GEPbased rule has clear advantages on different job number scale scheduling problems as compared with other scheduling rules.As the GEP-based rule considers the status attributes of production systems, the scheduling rule constructed by GEP is highly robust, whereas the scheduling objectives' performance is outstanding.
To explain this directly, we print the scheduling Gantt charts generated by the different rules or methods in Fig .17  To conclude this study from the perspective of statistical analysis, a Wilcoxon test was performed.We set the confidence level , and the results are reported in Table 14.For the small-sized and mediumsized instances, GEP significantly outperforms other rules, except the BPNN.Based on the value of p, if the confidence level increases to 0.1, then GEP is the best.For the large-sized instances, GEP significantly outperforms the other rules, except SPT and BPNN.If the confidence level increases to 0.1, then GEP is also better than the SPT rule.

Conclusion
Developing an effective and efficient scheduling algorithm for smart manufacturing is always of great importance.Accordingly, this study aims at the realtime scheduling problem of hybrid flow shops in a  Our future work will consider more complex situations in the hybrid flow shop problem to improve the applicability of our proposed approach.
the function set, and the elements in the function set are represented by f.The relationship between f and is .(3) Expression tree a + b × c
for a small-sized instance (N = 10).The rectangles with the same color indicate the operations belonging to the same job, in which the number is the job index.The makespan objective is reported at the top of each chart.α = 0.05

Table 2 Terminal set for scheduling a manufacturing system.
i − t − K ∑

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Table 14 Statistical analysis between GEP and the other rules or methods.
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