A Genetic Regulatory Network-Based Method for Dynamic Hybrid Flow Shop Scheduling with Uncertain Processing Times

The hybrid flow shop is a typical discrete manufacturing system. A novel method is proposed to solve the shop scheduling problem featured with uncertain processing times. The rolling horizon strategy is adopted to evaluate the difference between a predictive plan and the actual production process in terms of job delivery time. The genetic regulatory network-based rescheduling algorithm revises the remaining plan if the difference is beyond a specific tolerance. In this algorithm, decision variables within the rolling horizon are represented by genes in the network. The constraints and certain rescheduling rules are described by regulation equations between genes. The rescheduling solutions are generated from expression procedures of gene states, in which the regulation equations convert some genes to the expressed state and determine decision variable values according to gene states. Based on above representations, the objective of minimizing makespan is realized by optimizing regulatory parameters in regulation equations. The effectiveness of this network-based method over other ones is demonstrated through a series of benchmark tests and an application case collected from a printed circuit board assembly shop.


Introduction
The Hybrid Flow Shop (HFS) is a typical discrete manufacturing system in which a set of jobs passes through a series of production stages to complete required operations. The stages are composed of parallel machines to protect the job flow from being blocked by a single machine [1][2][3]. The operation processing time varies with different machines because capacities of parallel machines are normally unrelated at each stage [4,5]. This type of workshop exists in various industries, which include Printed Circuit Board (PCB) assembly, textile production and automobile assembly [6][7][8][9].
Since the first HFS problem was described by Salvador [10], vast amounts of academic work have been carried out on HFS scheduling due to its complexity and practical relevance [11][12][13][14][15]. Different kinds of methods (e.g., exact methods, heuristics and metaheuristics) were proposed to minimize a variety of objectives, which include the maximum completion time, the maximum flow time, the number of late jobs [16][17][18][19]. In terms of minimizing the maximum completion time, i.e., makespan, Mirsanei et al. [20] proposed a simulated annealing algorithm to solve the HFS scheduling problem featured with sequence-dependent setup times. Wang et al. [21] developed an efficient dispatching rule together with several local search heuristics in a semiconductor manufacturing system. Wang et al. [22] proposed a branch-and-bound algorithm to investigate the two-stage HFS scheduling problem in a no-wait environment. Komaki et al. [23] developed several algorithms featured with a new lower In a HFS scheduling problem, the binary decision variable is similar to the state of a gene. The decision variable interaction in constraints and objectives has an analogous form with gene regulations. Therefore, genes in a GRN are used to express decision variables, and gene regulations are adopted to describe constraints and objectives. In this way, a gene expression procedure depending on the regulations appropriately can obtain a feasible solution, in which the state of a gene indicates the assigned value of a related decision variable. In general, the differential equation method is most suitable for such a GRN because it provides the detailed description of gene regulations in a quantitative way [41]. In such an equation, undetermined regulatory parameters can be further optimized to obtain a near-optimal solution within the reasonable computational time because the number of these parameters is limited. Based on the mapping relationship illustrated in Figure 2, a GRN-based method is thus proposed for HFS rescheduling.  In a HFS scheduling problem, the binary decision variable is similar to the state of a gene. The decision variable interaction in constraints and objectives has an analogous form with gene regulations. Therefore, genes in a GRN are used to express decision variables, and gene regulations are adopted to describe constraints and objectives. In this way, a gene expression procedure depending on the regulations appropriately can obtain a feasible solution, in which the state of a gene indicates the assigned value of a related decision variable. In general, the differential equation method is most suitable for such a GRN because it provides the detailed description of gene regulations in a quantitative way [41]. In such an equation, undetermined regulatory parameters can be further optimized to obtain a near-optimal solution within the reasonable computational time because the number of these parameters is limited. Based on the mapping relationship illustrated in Figure 2, a GRN-based method is thus proposed for HFS rescheduling. , 7, 23 3 of 19 gene z; ε1, ε2 and ε3 are regulatory parameters in gene regulation equations. At the beginning, all genes are in the unexpressed state. Based on the inhibition coefficients determined by regulatory parameters, the gene expression procedure converts certain genes to the expressed state at discrete moments, and finally ends with the gene states (0,1,1). In a HFS scheduling problem, the binary decision variable is similar to the state of a gene. The decision variable interaction in constraints and objectives has an analogous form with gene regulations. Therefore, genes in a GRN are used to express decision variables, and gene regulations are adopted to describe constraints and objectives. In this way, a gene expression procedure depending on the regulations appropriately can obtain a feasible solution, in which the state of a gene indicates the assigned value of a related decision variable. In general, the differential equation method is most suitable for such a GRN because it provides the detailed description of gene regulations in a quantitative way [41]. In such an equation, undetermined regulatory parameters can be further optimized to obtain a near-optimal solution within the reasonable computational time because the number of these parameters is limited. Based on the mapping relationship illustrated in Figure 2, a GRN-based method is thus proposed for HFS rescheduling.  In general, the predictive-reactive scheduling strategy based on an event-driven rescheduling mechanism is applicable for HFS scheduling with uncertain processing times. However, it is difficult to obtain a high-quality solution in real time when reactive scheduling is required. This paper thus proposes a GRN-based method to solve this dynamic scheduling problem and uses computational experiments to validate this method. The rest of this paper is organized as follows. Section 2 presents the HFS scheduling problem with uncertain processing times. Section 3 introduces the event-driven rescheduling strategy. In Section 4, the GRN-based rescheduling method is presented. Section 5 gives computational experiments and discussions. The conclusion is outlined in Section 6.

Hybrid Flow Shop Dynamic Scheduling Problem
The HFS scheduling problem is to determine the sequence of jobs entering the first stage and assign jobs to alternative machines at every stage. Its objective is to minimize the makespan of jobs in order to increase machine utilization and guarantee on-time job delivery simultaneously. The following assumptions are taken into consideration in this problem:

1.
Each job and each machine are available at the initial time.

2.
Each job passes through multiple production stages to complete operations. 3.
One or more parallel machines are available at each stage.

4.
Parallel machines require different processing times for the same operation.

5.
Each machine is not able to process more than one job at the same time, and cannot be interrupted until the operation on this job has been completed.

6.
A job can enter the next stage if its operation at current stage has been accomplished. 7.
Processing times are uncertain, and their actual values may be different from the expected ones. 8.
A machine requires changeover time if it needs to process two jobs of different types consecutively. 9.
Operations of a job have no effect on those of other jobs. Table 1 lists the notations used in this problem. In this table, t mki represents the operation processing time, which is uncertain in an actual environment. Erlang distribution is a common way to construct the distribution of processing times based on the queuing theory. Based on this distribution, the possibility density function of a processing time t mki is as follows: where t represents the actual processing time, f tmi (t) represents the v-order Erlang distribution of t mki , E(t) = 1/λ. y rm and x rki are decision variables representing the sequence of jobs entering the first stage and the processing equipment of each job at every stage. The dynamic scheduling should find appropriate decision variable values to minimize the makespan when the processing times are constantly changing throughout the HFS production.

Notations Definitions
Sets {1, · · · , m, · · · , M} Set of job types {1, · · · , i, · · · , I} Set of production stages {1, · · · , k, · · · , K i } Set of parallel machines at stage i {1, · · · , r, · · · , R} Set of waiting processing jobs Parameters t mki Operation processing time of job type m on the kth machine at stage i c mm ki Changeover time required by the kth machine at stage i to operate on job type m after job type m d m Production volume of job type m, R = ∑ M m=1 d m Table 1. Cont.

Notations Definitions
Variables y rm Binary variable: 1, if the rth job entering the first production stage belongs to job type m; 0, otherwise x rki Binary variable: 1, if the rth job entering the first production stage is processed on the kth machine at stage i; 0, otherwise a rki The time instant the kth machine of production stage i to be available for the rth job b rki The job type processed by the kth machine of production stage i before a rki

Event-Driven Rescheduling Strategy
The predictive-reactive strategy is adopted to realize the dynamic scheduling process in a HFS. According to this strategy, a predictive plan is first developed based on the expected value of processing times. However, there will be a difference between the actual situation and the predictive plan in terms of operation processing times. The event-driven rescheduling mechanism evaluates the disturbance caused by these events and judges the necessity of a reactive scheduling. As shown in Figure 3, taking the objective of minimizing makespan into consideration, the judgment is based on whether delivery time deviation of jobs exceeds a specific tolerance. Delivery time deviation of jobs is calculated as follows: where C nmI and rC nmI represent expected and actual delivery time of the nth batch of job type m, respectively. If the deviation satisfies δ nmI > δ max , a rescheduling is required, otherwise, the predictive plan is kept. A large δ max causes the reactive scheduling insensitive to dynamic events, whereas a small δ max leads to frequent rescheduling. Thereupon, delivery deviation tolerance δ max is an important parameter in the rescheduling strategy. The job type processed by the k th machine of production stage i before rki a

Event-Driven Rescheduling Strategy
The predictive-reactive strategy is adopted to realize the dynamic scheduling process in a HFS. According to this strategy, a predictive plan is first developed based on the expected value of processing times. However, there will be a difference between the actual situation and the predictive plan in terms of operation processing times. The event-driven rescheduling mechanism evaluates the disturbance caused by these events and judges the necessity of a reactive scheduling. As shown in Figure 3, taking the objective of minimizing makespan into consideration, the judgment is based on whether delivery time deviation of jobs exceeds a specific tolerance. Delivery time deviation of jobs is calculated as follows: where CnmI and rCnmI represent expected and actual delivery time of the n th batch of job type m , respectively. If the deviation satisfies δnmI > δmax, a rescheduling is required, otherwise, the predictive plan is kept. A large δmax causes the reactive scheduling insensitive to dynamic events, whereas a small δmax leads to frequent rescheduling. Thereupon, delivery deviation tolerance δmax is an important parameter in the rescheduling strategy. Moreover, the reactive scheduling is performed within an operation-based rolling window because the processing times are related to operations. As illustrated in Figure 4, the window keeps removing the completed operations and adding the waiting processing ones based on real-time shop information. S(l) represents the set of p operations within the lth window. CS(l) represents the set of completed operations. ES(l) is the set of waiting processing operations. The rolling window forms Moreover, the reactive scheduling is performed within an operation-based rolling window because the processing times are related to operations. As illustrated in Figure 4, the window keeps removing the completed operations and adding the waiting processing ones based on real-time shop information. S(l) represents the set of p operations within the lth window. CS(l) represents the set of completed operations. ES(l) is the set of waiting processing operations. The rolling window forms the rescheduling problem based on S(l) once the reactive scheduling is regarded as necessary. The rescheduling algorithm solves this problem without interrupting the ongoing operations and then generates a new predictive plan based on CS(l) and ES(l), as shown in Figure 3. For each window, a larger p makes the rescheduling problem more complex, but enables better global optimization. On the contrary, a smaller p decreases the computational effort by compromising solution quality. Consequently, the number of operations within each window is also a key parameter in the rescheduling strategy.
Appl. Sci. 2017, 7, 23 6 of 19 the rescheduling problem based on S(l) once the reactive scheduling is regarded as necessary. The rescheduling algorithm solves this problem without interrupting the ongoing operations and then generates a new predictive plan based on CS(l) and ES(l), as shown in Figure 3. For each window, a larger p makes the rescheduling problem more complex, but enables better global optimization. On the contrary, a smaller p decreases the computational effort by compromising solution quality. Consequently, the number of operations within each window is also a key parameter in the rescheduling strategy.

Genetic Regulatory Network-Based Rescheduling Method
Based on the event-driven mechanism, a rescheduling problem is to be addressed if the delivery time deviation of jobs is larger than the deviation tolerance. A GRN-based method is proposed to solve this problem: Step (1) Genes are generated to represent the decision variables. In terms of decision variables yrm and xrki in Table 1, two kinds of genes (i.e., The gene rm π denotes that the r th job entering the HFS belongs to job type m , whereas the gene rki π denotes that the r th job entering the HFS is processed on the k th machine at the i th production stage. Step (2) Regulation equations are developed to describe the constraints and objectives: where Z represents the number of genes in a GRN, ( ) z n μ is a binary variable that is equal to 1 if gene z π is in the expressed state at the n th iteration, otherwise, ( ) z n μ is equal to 0, z σ is the inhibition coefficient that describes the inhibitory effects on gene z π quantitatively;  θ θ θ 1 2 , , , E are regulatory parameters; and : R E z f is a nonlinear function related to workshop conditions. Step (3) Gene expression procedures are designed to determine solutions. At the beginning of such a procedure, the set of related genes is first confirmed based on operations within the rolling window. If a reactive scheduling is necessary, all these genes are initialized to the unexpressed state. At each iteration (i.e., As shown in Figure 5, the regulation equation and expression procedure of genes

Genetic Regulatory Network-Based Rescheduling Method
Based on the event-driven mechanism, a rescheduling problem is to be addressed if the delivery time deviation of jobs is larger than the deviation tolerance. A GRN-based method is proposed to solve this problem: Step (1) Genes are generated to represent the decision variables. In terms of decision variables y rm and x rki in Table 1, two kinds of genes (i.e., {π rm |r = 1, 2, · · · , R; m = 1, 2, · · · , M} and {π rki |r = 1, 2, · · · , R; k = 1, 2, · · · , K i ; i = 1, 2, · · · , I}) are generated. The gene π rm denotes that the rth job entering the HFS belongs to job type m, whereas the gene π rki denotes that the rth job entering the HFS is processed on the kth machine at the ith production stage.
Step (2) Regulation equations are developed to describe the constraints and objectives: where Z represents the number of genes in a GRN, µ z (n) is a binary variable that is equal to 1 if gene π z is in the expressed state at the nth iteration, otherwise, µ z (n) is equal to 0, σ z is the inhibition coefficient that describes the inhibitory effects on gene π z quantitatively; ϑ 1 , ϑ 2 , · · · , ϑ E are regulatory parameters; and f z : R (Z+E) → R is a nonlinear function related to workshop conditions. Step (3) Gene expression procedures are designed to determine solutions. At the beginning of such a procedure, the set of related genes is first confirmed based on operations within the rolling window. If a reactive scheduling is necessary, all these genes are initialized to the unexpressed state. At each iteration (i.e., n = 1, 2, 3, · · · , N), some of these genes are converted to the expressed state based on the regulation equations. When n > N, genes in the expressed state are confirmed, and their corresponding decision variable values are equal to 1 in the rescheduling solution.

Regulation Equation and Expression Procedure of Gene rm π
In HFS scheduling, each machine prefers to operate on the same job type in order to reduce the setup time. For instance, the jobs of type m1 are operated on the 1st machine (i.e., the jobs on this machine are thus "m1-m1-m1-m1-…"), and those of m2 and m3 are assigned to the 2nd machine and the 3rd machine, respectively. In this case, the job sequence entering this production stage will be "m1-m2-m3-m1-m2-m3-…" because the parallel machines perform their tasks simultaneously. Thereupon, each job type would appear in the job sequence cyclically, and the related cycle time should be accorded with the number of parallel machines at each stage. However, the greedy nature of rule-based sequencing methods will probably keep the HFS scheduling choosing the "easy" jobs, which means jobs of a "hard" type will be left to the last positions of the job sequence. For instance, four jobs of type m4 might be left for the last four positions of a job sequence, which is not preferred because there is no scheduling flexibility when assigning these jobs to parallel machines. On the contrary, it is much better to leave four jobs m1, m2, m3 and m4 for remaining positions because the job assigning procedure can choose machines with shorter processing time, less changeovers or earlier availability for each job to minimize the makespan further. For this reason, it is necessary to keep the stable production rate of each job type in the job sequence in order to avoid the near sightedness of rule-based sequencing methods. In terms of these facts, following rules should be obeyed: 1. A job type cannot be selected if its cycle of entering the HFS is not accord with the number of parallel machines at each stage; 2. A job type cannot be selected if its production ratio differs from its demand ratio.
No model sequence could satisfy all these rules completely, and each unsatisfied case might increase the makespan. The regulation equation of gene rm π is thus developed as follows: where rm u represents the inhibition coefficient to gene rm π , lm d represents the number of job type m entering the HFS within the l th window, W represents the number of jobs having entered the HFS before the l th window, 1 ε and 2 ε are regulatory parameters, and ( ) H x is a step function that satisfies . The first two terms of the  presented in Sections 4.1 and 4.2, respectively. The regulatory parameter optimization procedure is given in Section 4.3.

Regulation Equation and Expression Procedure of Gene rm π
In HFS scheduling, each machine prefers to operate on the same job type in order to reduce the setup time. For instance, the jobs of type m1 are operated on the 1st machine (i.e., the jobs on this machine are thus "m1-m1-m1-m1-…"), and those of m2 and m3 are assigned to the 2nd machine and the 3rd machine, respectively. In this case, the job sequence entering this production stage will be "m1-m2-m3-m1-m2-m3-…" because the parallel machines perform their tasks simultaneously. Thereupon, each job type would appear in the job sequence cyclically, and the related cycle time should be accorded with the number of parallel machines at each stage. However, the greedy nature of rule-based sequencing methods will probably keep the HFS scheduling choosing the "easy" jobs, which means jobs of a "hard" type will be left to the last positions of the job sequence. For instance, four jobs of type m4 might be left for the last four positions of a job sequence, which is not preferred because there is no scheduling flexibility when assigning these jobs to parallel machines. On the contrary, it is much better to leave four jobs m1, m2, m3 and m4 for remaining positions because the job assigning procedure can choose machines with shorter processing time, less changeovers or earlier availability for each job to minimize the makespan further. For this reason, it is necessary to keep the stable production rate of each job type in the job sequence in order to avoid the near sightedness of rule-based sequencing methods. In terms of these facts, following rules should be obeyed: 1. A job type cannot be selected if its cycle of entering the HFS is not accord with the number of parallel machines at each stage; 2. A job type cannot be selected if its production ratio differs from its demand ratio.
No model sequence could satisfy all these rules completely, and each unsatisfied case might increase the makespan. The regulation equation of gene rm π is thus developed as follows: where rm u represents the inhibition coefficient to gene rm π , lm d represents the number of job type In HFS scheduling, each machine prefers to operate on the same job type in order to reduce the setup time. For instance, the jobs of type m 1 are operated on the 1st machine (i.e., the jobs on this machine are thus "m 1 -m 1 -m 1 -m 1 -. . . "), and those of m 2 and m 3 are assigned to the 2nd machine and the 3rd machine, respectively. In this case, the job sequence entering this production stage will be "m 1 -m 2 -m 3 -m 1 -m 2 -m 3 -. . . " because the parallel machines perform their tasks simultaneously. Thereupon, each job type would appear in the job sequence cyclically, and the related cycle time should be accorded with the number of parallel machines at each stage. However, the greedy nature of rule-based sequencing methods will probably keep the HFS scheduling choosing the "easy" jobs, which means jobs of a "hard" type will be left to the last positions of the job sequence. For instance, four jobs of type m 4 might be left for the last four positions of a job sequence, which is not preferred because there is no scheduling flexibility when assigning these jobs to parallel machines. On the contrary, it is much better to leave four jobs m 1 , m 2 , m 3 and m 4 for remaining positions because the job assigning procedure can choose machines with shorter processing time, less changeovers or earlier availability for each job to minimize the makespan further. For this reason, it is necessary to keep the stable production rate of each job type in the job sequence in order to avoid the near sightedness of rule-based sequencing methods. In terms of these facts, following rules should be obeyed:

1.
A job type cannot be selected if its cycle of entering the HFS is not accord with the number of parallel machines at each stage; 2.
A job type cannot be selected if its production ratio differs from its demand ratio.
No model sequence could satisfy all these rules completely, and each unsatisfied case might increase the makespan. The regulation equation of gene π rm is thus developed as follows: where u rm represents the inhibition coefficient to gene π rm , d lm represents the number of job type m entering the HFS within the lth window, D l = ∑ M m=1 d lm , W l represents the number of jobs having entered the HFS before the lth window, ε 1 and ε 2 are regulatory parameters, and H(x) is a step function that satisfies H(x) = 0 (x ≤ 0) and H(x) = +∞ (x > 0). The first two terms of the right side of Equation (4) represent the inhibition strength to gene π rm owing to Rules 1 and 2, respectively. The last two terms ensure the job sequence to satisfy the predetermined quantity of each job type.
In the expression procedure of gene π rm , all the operations are arranged in an ascending sequence in terms of their starting time in the predictive plan. p operations with their starting time later than Appl. Sci. 2017, 7, 23 8 of 18 the rescheduling instant are selected consecutively in the operation sequence. Assuming that d lm represents the number of operations processed on the first production stage for job type m, there are D l = ∑ M m=1 d lm jobs to be arranged for the job sequence within the lth window (from the W l th position to the (W l + D l )th position). The genes {π rm |r = W l , W l + 1, · · · , W l + D l ; m = 1, 2, · · · , M} are thus initialized to the unexpressed state in the rescheduling problem, whereas the gene states {y rm |r = 1, 2, · · · , W l ; m = 1, 2, · · · , M} and {y rm |r = W l + D l + 1, W l + D l + 2, · · · , R; m = 1, 2, · · · , M} are given values based on the predictive plan. The gene expression procedure deals with the gene states {y rm |r = W l , W l + 1, · · · , W l + D l ; m = 1, 2, · · · , M}. At each iteration n = α (α ∈ {1, 2, · · · , D l }), the inhibition coefficient u rm is calculated for genes π (W l +α)m |m = 1, 2, · · · , M and the gene with minimum u rm is converted to the expressed state. When n > D l , the job sequence within the lth window is rescheduled based on {y rm |r = W l , W l + 1, · · · , W l + D l ; m = 1, 2, · · · , M}. Appendix A presents this gene expression procedure within the lth window. presented in Sections 4.1 and 4.2, respectively. The regulatory parameter optimization procedure is given in Section 4.3.

Regulation Equation and Expression Procedure of Gene rm π
In HFS scheduling, each machine prefers to operate on the same job type in order to reduce the setup time. For instance, the jobs of type m1 are operated on the 1st machine (i.e., the jobs on this machine are thus "m1-m1-m1-m1-…"), and those of m2 and m3 are assigned to the 2nd machine and the 3rd machine, respectively. In this case, the job sequence entering this production stage will be "m1-m2-m3-m1-m2-m3-…" because the parallel machines perform their tasks simultaneously. Thereupon, each job type would appear in the job sequence cyclically, and the related cycle time should be accorded with the number of parallel machines at each stage. However, the greedy nature of rule-based sequencing methods will probably keep the HFS scheduling choosing the "easy" jobs, which means jobs of a "hard" type will be left to the last positions of the job sequence. For instance, four jobs of type m4 might be left for the last four positions of a job sequence, which is not preferred because there is no scheduling flexibility when assigning these jobs to parallel machines. On the contrary, it is much better to leave four jobs m1, m2, m3 and m4 for remaining positions because the job assigning procedure can choose machines with shorter processing time, less changeovers or earlier availability for each job to minimize the makespan further. For this reason, it is necessary to keep the stable production rate of each job type in the job sequence in order to avoid the near sightedness of rule-based sequencing methods. In terms of these facts, following rules should be obeyed: 1. A job type cannot be selected if its cycle of entering the HFS is not accord with the number of parallel machines at each stage; 2. A job type cannot be selected if its production ratio differs from its demand ratio.
No model sequence could satisfy all these rules completely, and each unsatisfied case might increase the makespan. The regulation equation of gene rm π is thus developed as follows: where rm u represents the inhibition coefficient to gene rm π , lm d represents the number of job type The major objective of assigning jobs to alternative machines is to avoid machine idle time because the makespan is decreased mainly by increasing the utilization of machines. For this reason, the rule of assigning each waiting processing job to the earliest available machine is widely adopted. In addition, each machine prefers to operate on the same job type in order to reduce changeover activities while setup times are taken into consideration. Therefore, assigning a job to parallel machines at each stage should comply with following rules:

1.
A machine cannot be selected if the waiting time of a job on this machine is longer than that on another machine.

2.
A machine cannot be selected if it requires setup time for a job.
It is almost impossible to satisfy these rules completely, and each unsatisfied case might increase the objective function value. In addition, the 5th assumption should also be obeyed. The regulation equation of gene π rki is thereby developed as follows: where w rki represents the inhibition coefficient to gene π rki , a rki and b rki represent the end time and the job type of last operation on the kth parallel machine when the rth job enters the ith production stage, m 0 represents the type of the rth job (i.e., m 0 = ∑ M m=1 my rm ), h 1 and h 2 are regulatory parameters, and H(x) is a step function that satisfies H(x) = 0 (x ≤ 0) and H(x) = +∞ (x > 0). The first two terms of the right side of Equation (5) represent the inhibition strength to gene π rki owing to Rules 1 and 2, respectively. The last term describes the constraint originated from the 5th assumption. The variable b rki satisfies: and the variable a rki satisfies: Assuming ϕ li jobs (from the σ li th position to the (σ li + ϕ li )th position in the job sequence) enter the ith production stage based on the p = ∑ I i=1 ϕ li operations selected for the lth window, the genes π (σ li +β i )ki |β i = 1, 2, · · · , ϕ li ; k = 1, 2, · · · , K i ; i = 1, 2, · · · , I} are initialized to the unexpressed state, whereas other gene states are in accordance with the predictive plan. In the expression procedure of gene π rki , at each iteration n = β i + ∑ i−1 v=1 ϕ lv (i = 1, 2, · · · , I, β i ∈ {1, 2, · · · , ϕ li }), the inhibition coefficient w rki is calculated for genes π (σ li +β i )ki |k = 1, 2, · · · , K i and the gene with minimum w rki is converted to the expressed state. When n > p, the production plan within the lth window is rescheduled based on gene states x (σ li + β i )ki |β i = 1, 2, · · · , ϕ li ; k = 1, 2, · · · , K i ; i = 1, 2, · · · , I}. The pseudo codes of this procedure are presented in Appendix B.

Regulatory Parameter Optimization
Based on values of regulatory parameters ε 1 and ε 2 , the gene expression procedure governed by Equation (4) determines the gene states {y rm |r = 1, 2, · · · , R; m = 1, 2, · · · , M} within the lth window, each of which represents whether the rth job entering the HFS belongs to job type m. Moreover, the regulatory parameters h 1 and h 2 specify the gene regulation in Equation (5) and further determine the gene states {x rki |r = 1, 2, · · · , R; k = 1, 2, · · · , K i ; i = 1, 2, · · · , I} within the lth window. Each gene state x rki represents whether the rth job is manufactured on the kth machine at the ith production stage. In this way, a solution to the rescheduling problem is obtained based on regulatory parameter values (i.e., ε 1 , ε 2 , h 1 and h 2 ). Figure 6 illustrates this procedure in the rescheduling of a specific HFS. Appl

Strategy Parameter Analysis
As discussed in Section 3, delivery deviation tolerance and operation-based window size are important parameters in the event-driven rescheduling strategy. To determine appropriate parameter values, the numeric tests listed in Table 2 are presented.  A parameter optimization procedure is further implemented to minimize the makespan. According to the regulation equations, a gene receives inhibition if its expression breaks scheduling rules, and the inhibition strength is weighted by related regulatory parameters. These parameters have different values because each rule plays its distinctive role in determining an optimal solution. For example, decreasing setup times is important if machines require comparatively longer durations for job type changeovers, and the parameter h 2 should have a large value. The sensitivity analysis on makespan can evaluate the importance weights and thus realize regulatory parameter optimization [42]. Alternatively, machining learning methods can also optimize these parameters if there is enough historical data. For instance, a neural network with workshop conditions can be trained to recommend appropriate parameters.
Apart from these analytical and machine learning methods, random searching algorithms (e.g., genetic algorithms and immune learning algorithms) are also alternative to optimize parameters [43]. For instance, each individual in the genetic algorithm can represent a specific value set of regulatory parameters, and the fitness value of this individual can be evaluated based on related makespan. Through a series of genetic operations (i.e., evaluation, crossover and mutation), the best individual in this algorithm determines the minimum makespan. Moreover, a real-coded algorithm should be used in the parameter optimization procedure because the regulatory parameters are real variables.

Strategy Parameter Analysis
As discussed in Section 3, delivery deviation tolerance and operation-based window size are important parameters in the event-driven rescheduling strategy. To determine appropriate parameter values, the numeric tests listed in Table 2 are presented. For these numeric tests, a static scheduling result is first obtained by taking the whole planning horizon as a special rescheduling window and using the GRN-based method. A real-coded genetic algorithm is specifically used in this method to optimize regulatory parameters, in which the population size is 200, the maximum generation is 50, the crossover possibility is 0.8, and the mutation possibility is 0.1. Figure 7 illustrates the Gantt chart of static scheduling results (makespan = 392 s). In this diagram, white rectangles represent processing times and black ones denote changeover times. Moreover, the numbers within each rectangle represents the batch of a job type. For instance, "1, 3" represents the third batch of the first job type. Assuming that the processing times follow a 4-order Erlang distribution, the rescheduling strategies with different deviation tolerances and window sizes are then used. Table 3 lists the makespan, rescheduling times and computational time of strategies with different deviation tolerances while the window size is 20 operations. In this table, the strategy with the delivery time deviation tolerance of 0.125 minimizes the makespan. Table 4 thus lists dynamic scheduling results with different window sizes while the delivery time deviation tolerance is 0.125. Based on these Assuming that the processing times follow a 4-order Erlang distribution, the rescheduling strategies with different deviation tolerances and window sizes are then used. Table 3 lists the makespan, rescheduling times and computational time of strategies with different deviation tolerances while the window size is 20 operations. In this table, the strategy with the delivery time deviation tolerance of 0.125 minimizes the makespan. Table 4 thus lists dynamic scheduling results with different window sizes while the delivery time deviation tolerance is 0.125. Based on these results, the delivery deviation tolerance of 0.125 and the window size of 77 operations are adopted to generate dynamic scheduling results, as shown in Figure 8.

Comparative Experiments
Nine benchmarks from Qin et al. [44] are further used to validate the effectiveness of the GRN-based method, as shown in Table 5. Assuming that actual processing times follow the Erlang distribution, these benchmarks are solved by the GRN-based method and the Improved Ant Colony

Comparative Experiments
Nine benchmarks from Qin et al. [44] are further used to validate the effectiveness of the GRN-based method, as shown in Table 5. Assuming that actual processing times follow the Erlang distribution, these benchmarks are solved by the GRN-based method and the Improved Ant Colony Algorithm (IACO) introduced in [44], respectively. An Intel ® Core TM i7-2720QM CPU @ 2.20 GHz and 8.00 GB RAM based notebook computer (Dell Inc., Xiamen, China) is adopted to test these experiments. Table 6 lists the experimental results. Because both the IACO and the GRN-based method are based on random search procedures, all these results are averaged values over 20 replications.  As shown in Table 6, the IACO method achieves better results than the GRN-based methods for the benchmarks "6 × 2", "30 × 2" and "100 × 2". Because these benchmarks are featured with a small-scale solution space, the IACO method is possibly to search out the optimal solutions via its global searching procedure, whereas the GRN-based method might fail to find an optimal one owing to the predetermined rules embedded in its regulation equations. When the problem scale increases in the benchmarks "6 × 4" to "100 × 8", the GRN-based method obtains smaller makespans than the IACO method. The IACO can hardly search out optimal solutions, or even near-optimal ones, when the solution space is increased, whereas the GRN-based method ensures a good solution owing to its embedded rules and can further find a better one by optimizing regulatory parameters. Moreover, the GRN-based method optimizes four parameters, rather than all the decision variables in the IACO method. This proposed method can thus save the CPU time and demonstrate better response capability to processing time variations in HFS.
Thereupon, the GRN-based method is validated as an effective and efficient method for dynamic scheduling in HFS.

Case Study
A specific PCB assembly shop composed of four production stages (SMT chip processing stage, plug processing stage, welding processing stage, and test stage) are also investigated. At each production stage, several production lines are alternative for PCB assembly (S1, S2 and S3 at the 1st stage; M1 and M2 at the 2nd stage; A1 and A2 at the 3rd stage; and T1 and T2 at the 4th stage). Ten types of PCBs are assembled in these lines. The requirement for each PCB type is 1000. The processing times and setup times are listed in Tables 7 and 8, respectively.  Figure 9 illustrates the Gantt chart of static scheduling results obtained by using the GRN-based method. As shown in this figure, all PCB products are divided into 100 batches, and the makespan is 3308 s. Because the starting time of operations is directly determined by the end time of the former operation on the same machine or that of the same job, the processing time variations have cumulative impacts on the completion time of jobs. Based on static scheduling results, the IACO method is used to deal with processing time variations. Figure 10 is the Gantt chart obtained by the IACO method, in which the makespan is 3415.77 s. The GRN-based method is also used to solve the dynamic scheduling problem and achieves a makespan of 3261.98 s, as shown in Figure 11. By integrating job sequencing rules and job assigning rules in a reasonable manner, the GRN-based method ensures waiting jobs to be in-time assigned to an idle machine with comparatively shorter setup time. The regulatory optimization procedure realizes the tradeoff between shorter job waiting time and less changeover activities to minimize the makespan. However, the IACO method fails to realize these targets for some rolling windows because its global search procedure can hardly search out optimal solutions for a real-time scheduling. Taking the first production stage for example, the Garnet chart in Figure 10 realizes higher utilization of Machine 1 and shorter job waiting time on Machine 3 than that in Figure 11. Consequently, the GRN-based method is a more effective scheduling method than the IACO method. time and less changeover activities to minimize the makespan. However, the IACO method fails to realize these targets for some rolling windows because its global search procedure can hardly search out optimal solutions for a real-time scheduling. Taking the first production stage for example, the Garnet chart in Figure 10 realizes higher utilization of Machine 1 and shorter job waiting time on Machine 3 than that in Figure 11. Consequently, the GRN-based method is a more effective scheduling method than the IACO method.   time and less changeover activities to minimize the makespan. However, the IACO method fails to realize these targets for some rolling windows because its global search procedure can hardly search out optimal solutions for a real-time scheduling. Taking the first production stage for example, the Garnet chart in Figure 10 realizes higher utilization of Machine 1 and shorter job waiting time on Machine 3 than that in Figure 11. Consequently, the GRN-based method is a more effective scheduling method than the IACO method.

Conclusions
This paper solves the HFS scheduling problem with uncertain processing times based on the predictive-reactive strategy. For the rescheduling problems in response to processing time variation

Conclusions
This paper solves the HFS scheduling problem with uncertain processing times based on the predictive-reactive strategy. For the rescheduling problems in response to processing time variation events, a novel GRN-based method is developed to minimize the makespan. The critical factor is the employment of GRN to describe the HFS scheduling problem and some scheduling rules. This enables the regulatory parameter optimization procedure to generate near-optimal rescheduling solutions within the reasonable computational time. The effectiveness of this method over the IACO method was demonstrated by a series of benchmark tests and the case study in a PCB assembly shop.
This paper investigates the dynamic event of processing time variations in HFS scheduling, however, the machine breakdowns are also common in real environments. These events will cause the machines to be unavailable for a certain duration known as Mean Time To Repair (MTTR). To deal with these situations, the dynamic scheduling should first generate a predictive plan with minimum makespan based on the assumptions that all the machines are reliable and that the processing times take their expectation values. The job delivery deviation caused by dynamic events of machine breakdowns as well as processing time variations is then monitored during the execution of this plan. If the deviation is beyond a tolerance, the GRN-based method reschedules the production plan within a rolling window to respond to these events. This method will have very complex regulation equations because the unavailability of failed machines needs to be involved in the calculation of job waiting time and machine idle time. Thereupon, the HFS scheduling will be more challenging when the dynamic events of machine breakdowns are taken into account and it will also have better practical values at the same time. Therefore, in our further work, we will develop an enhanced rescheduling method that deals with the dynamic events of machine breakdowns as well as processing time variations by using the GRN. In addition, we will extend this method to multi-objective scheduling that minimizes the due date of jobs, the idle time of machines, the scheduling adjustment cost, etc.