Dynamic Events in the Flexible Job-Shop Scheduling Problem: Rescheduling with a Hybrid Metaheuristic Algorithm

Abstract

In the real world of manufacturing systems, production planning is crucial for organizing and optimizing various manufacturing process components. The objective of this paper is to present a methodology for both static scheduling and dynamic scheduling. In the proposed method, a hybrid algorithm is utilized to optimize the static flexible job-shop scheduling problem (FJSP) and dynamic flexible job-shop scheduling problem (DFJSP). This algorithm integrates the genetic algorithm (GA) as a global optimization technique with a simulated annealing (SA) algorithm serving as a local search optimization approach to accelerate convergence and prevent getting stuck in local minima. Additionally, variable neighborhood search (VNS) is utilized for efficient neighborhood search within this hybrid algorithm framework. For the FJSP, the proposed hybrid algorithm is simulated on a 40-benchmark dataset to evaluate its performance. Comparisons among the proposed hybrid algorithm and other algorithms are provided to show the effectiveness of the proposed algorithm, ensuring that the proposed hybrid algorithm can efficiently solve the FJSP, with 38 out of 40 instances demonstrating better results. The primary objective of this study is to perform dynamic scheduling on two datasets, including both single-purpose machine and multi-purpose machine datasets, using the proposed hybrid algorithm with a rescheduling strategy. By observing the results of the DFJSP, dynamic events such as a single machine breakdown, a single job arrival, multiple machine breakdowns, and multiple job arrivals demonstrate that the proposed hybrid algorithm with the rescheduling strategy achieves significant improvement and the proposed method obtains the best new solution, resulting in a significant decrease in makespan.

1. Introduction

The flexible job-shop scheduling problem (FJSP) is an expanded version of the traditional job-shop scheduling problem (JSSP), with additional flexibility and complexity in the scheduling process [1]. The FJSP involves determining the operation sequence for executing operations with satisfied constraints and the selection of suitable machines to carry out these operations, considering factors such as compatibility, efficiency, and resource utilization [2]. Consequently, the FJSP emerges as a highly demanding problem, characterized by its status as an extremely nondeterministic polynomial-time hard (NP-hard) problem that demands intensive computational effort to derive optimal solutions [3]. Scheduling has received a lot of research attention, but most of it has focused on static scheduling. In real-world smart factories, production environments must deal with uncertainty, as various unpredictable and dynamic occurrences frequently arise. In industry, such events are referred to as dynamic scheduling problems [4]. These include issues such as machine breakdowns [5], unexpectedly arriving jobs [6], and order cancellations [7]. Such unpredictable events add complexity to the scheduling and coordination of operations, highlighting the need for scheduling methodologies to address the dynamic flexible job-shop scheduling problem (DFJSP), which deals with two types of datasets: single-purpose machine datasets and multi-purpose machine datasets. The method that can help to solve the dynamic event problem is rescheduling; this approach involves rescheduling when a new event occurs [8].

The GA is widely used for scheduling and optimization problems due to its ability to explore the solution space and find the near-optimal solution, but it has a low convergence rate and is capable of falling into the local optima, which can result in longer computational times and may impact performance [9,10]; to avoid these conditions, many studies have proposed the hybrid algorithms explained above. Therefore, this study suggests using a hybrid genetic algorithm, simulated annealing, and variable neighborhood search (GASAVNS) algorithms to enable strong local search ability with global optimization. Another contribution is the influence of the hybrid algorithm on various dynamic scenarios observed in the context of flexible job-shop scheduling. The method that is used for dynamic scheduling is rescheduling with the hybrid GASAVNS algorithm by extracting the static scheduling data. All observations for dynamic events will be carried out on the obtained datasets and datasets from [11]. The simulation results show that the proposed method works well in both scheduling and rescheduling for the FJSP and DFJSP, respectively.

The structure of the remaining manuscript is as follows: A literature review is presented in Section 2. Section 3 gives a brief description of the FJSP formulation of both datasets, namely the single-purpose machine and multi-purpose machine datasets, along with the required assumptions to initiate the FJSP. Section 4 explains the proposed method, the hybrid GASAVNS algorithm, while Section 5 elaborates on the proposed rescheduling method for dynamic events such as machine breakdown and job arrival. Section 6 presents the experimental results and analysis for the FJSP and various dynamic events using both datasets. The conclusion is provided in Section 7.

2. Literature Review

Over the last few years, in response to the strong NP-hard FJSP, researchers have introduced various heuristic approaches aimed at providing effective and practical solutions that involve complex combinatorial optimization and multiple constraints. Some dispatching rules like first-in–first-out (FIFO), shortest processing time (SPT), and longest processing time (LPT) are easy to use, but they are not accurate enough for solving tough scheduling problems. Heuristic approaches are proposed to find the near-optimal solution for the static FJSP. Giovanni et al. [12] proposed an improved genetic algorithm (GA) to solve distributed and flexible job-shop scheduling. Lim et al. [13] implemented a hyperheuristic-based simulated annealing (SA) method to solve the FJSP. This method utilized single-point selection that maintains a single candidate solution from the search space. Saidi-Mehrabad et al. [14] utilized the tabu search (TS) algorithm to tackle the FJSP. This heuristic approach implements a sequence-dependent setup and is implemented using the visual fortune language. Han et al. [15] adapt the reinforcement learning algorithm for solving the FJSP. A variety of improved and hybrid methods have emerged to enhance optimization objectives. These approaches integrate diverse techniques, aiming to achieve superior performance in scheduling. For example, Gao et al. [16] presented an effective hybrid approach by combining GA with the local improvement capacity of TS. Their study focused on addressing the FJSP with the objective of minimizing the makespan. Escamilla-Serna et al. [17] made a hybrid GA-RRHC algorithm using a GA and random-restart hill-climbing (RRHC) to perform a local search. Additionally, the authors used cross-over and mutation operators with a cellular automaton (CA)-inspired neighborhood to perform a global search, with implementation carried out using MATLAB. Tang et al. [18] introduced a hybrid algorithm combining chaos particle swarm optimization and a GA, addressing the optimization mechanism of the FJSP.

The dynamic job-shop scheduling problem has recently captured significant research attention even though it involves uncertain events. Zhang et al. [19] proposed a hybrid tabu search and genetic algorithm combination to address the dynamic job-shop scheduling problem, which involves the difficulties of machine breakdowns and job arrivals. Wang et al. [11] proposed a variable interval rescheduling strategy to deal with the dynamic flexible job-shop scheduling problem; this approach involves the improvement of a genetic algorithm to obtain an effective solution. A heuristic approach for a dynamic flexible job-shop scheduling problem considering variable processing times was proposed by Shahgholi et al., inspired by the artificial bee colony algorithm, that aimed to achieve a near-optimal solution [20]. Zhang et al. [21] proposed a framework of dynamic scheduling based on an improved gene expression programming algorithm combined with effective neighborhood structures. Fattahi et al. [22] considered two objectives, efficiency and stability, for dynamic scheduling in flexible job-shop systems, including various strategies for handling arrival disturbances using a genetic algorithm. Kundakci et al. [23] combined a genetic algorithm and tabu search to address the optimization problem of the dynamic job-shop scheduling problem with machine breakdown for a parallel machine environment. Wei et al. [24] proposed a hybrid genetic algorithm with a simulated annealing algorithm for flow-shop scheduling with makespan criteria. The implementation of a hybrid genetic algorithm (GA) and a simulated annealing (SA) approach was employed by Al-milli et al. to tackle timetabling challenges under specific constraints, showing promising results [25]. The study by Shady et al. [26] introduces the integration of GA with the NSGA-II to automatically generate dispatching rules for the DJSP with machine breakdown, achieving the minimizing mean flow time and makespan simultaneously. Industry 4.0 is witnessing a transformative shift where manufacturing systems are becoming increasingly intelligent and capable of optimization [27,28]. Researchers have explored the application of scheduling; for example, Hadi et al. [29] used flexible flow-shop scheduling in the domain of a waste-to-energy system, which reduced the total cost significantly in the equipment-driven industry, and Soroush et al. [30] investigated a dynamic scheduling problem within a job-shop robotic cell for material handling. This study proposes the utilization of a hybrid approach comprising genetic algorithms, simulated annealing, and variable neighborhood search (GASAVNS) to enhance local search capabilities within the context of global optimization.

3. Problem Formulation

In the Problem Formulation Section, the two types of datasets encountered within manufacturing environments are described and formulated: single-purpose machine datasets and multi-purpose machine datasets. Foundational assumptions are established under the FJSP framework. These constraints ensure the feasibility of a solution. The primary objective of the FJSSP is to minimize the maximum completion time expressed in Equation (1)

Objective = Min (Cmax)

(1)

Here, Cmax represents the maximum completion time required to complete jobs among all jobs processed by machines.

3.1. Single-Purpose Machine Dataset

Considering the FJSP for the single-purpose machine, a set of ‘n’ jobs is represented by J = {J₁, J₂, J₃, …, J_n} and a set of ‘g’ machine groups is represented by MG = {MG₁, MG₂, MG₃, …, MG_g}. Each machine group can have m machines MG_g = {M₁, M₂, …, M_m}, and every operation has ‘m’ machine choices. Each group of machines is dedicated to one unique operation, and the number of operations will be equal to the number of machines. This type of dataset has single-purpose machines (SPMs). Each operation in the job can be processed by any one machine out of a dedicated machine group. Each job consists of a sequence of operations represented as O = (O_i1, O_i2, …, O_ik,… O_i,ni). Each machine Mi becomes available at time AMi ≥ AM = {AM₁, AM₂,…, AM_m}. Each operation needs to be completed by a functional machine within a particular time T = {T_i1, T_i2, …, T_ik}. The processing time varies depending on the resources used for processing operations. Each machine has a different execution time, and every operation has a different resource usage time. Therefore, the processing time of machines, even within the same group, differs. In other words, different machines may have different processing times for the same operation; some machines are faster, and some are slower.

Considering the textile factory environment, many products are involved in production, such as pants (Product 1), shirts (Product 2), and t-shirts (Product 3), which correspond to J1, J2, and J3, respectively. Each product undergoes a series of operations, such as sizing, weaving, knitting, dying, and printing [31,32].

Table 1. Processing information of FJSP with single purpose-machine.

	Ops	OP1 (MG1)				OP2 (MG2)		OP3 (MG3)			OP4 (MG4)
Jobs
J1		M1	M2	M3	M4	M5	M6	M7	M8	M9	M10	M11	M12
		8	7	8	9	4	5	-	-	-	-	-	-
J2		M1	M2	M3	M4	M5	M6	M7	M8	M9	M10	M11	M12
		-	-	-	-	8	7	-	-	-	5	8	3
J3		M1	M2	M3	M4	M5	M6	M7	M8	M9	M10	M11	M12
		7	4	6	5	-	-	9	7	6	9	6	7

shows the initial information for the FJSP. There are three jobs (J1, J2, J2), and each job has multiple operations. As shown in Table 1 there are four machine groups (MG₁, MG₂, MG₃, MG₃), and every group performs a specific operation, which means all machines present in MG1 perform the same operation. An operation OP₁₁ has a choice of four machines, M₁, M₂, M₃, and M₄, whereas J1 has two operations (OP₁₁, OP₁₂). Depending on the product specifications, some of these operations may not be required. The symbol “-” indicates that the corresponding operation does not apply to that job. Each operation has a machine group to perform that operation; OP₁₁ (job1, operation1) is processed on machine group 1 (MG₁), and MG₁ has a choice of four machines: M1 (8-unit processing time), M2 (7-unit processing time), M3 (8-unit processing time), and M4 (9-unit processing time). OP₁₂ (Job1, operation2) is processed on machine group 2 (MG₂), and MG₂ has a choice of two machines, i.e., M5 and M6, with processing times of 4 and 5, respectively. Operations 3 (OP₁₃) and 4 (OP₁₄) are not available for J1. On other hand, J2 has two operations (OP₂₂, OP₂₄) and job 3 has three operations (OP₃₁, OP₃₃, OP₃₄).

3.2. Multi-Purpose Machine Dataset

Considering the FJSP for multi-purpose machines, there is set of n jobs J = {J₁, J₂, J₃, …., Jn}, and each J_i job consists of n_i operations O = (O_i1, O_i2, …, O_ik, … O_i,ni). There is a set of m multi-purpose machines M = {M₁, M_2,…, M_m}, and one machine is capable of processing multiple operations, but at the same time, only one operation can proceed [33]. Each machine can perform p_m operations at a given time P_{m, i} = {P_m,₁, P_m,₂, … P_mpm}. Each machine Mi is available at time AMi => AM= {AM₁, AM_2, …, AM_m}. Each operation needs to be processed by a functional machine within a particular time T = {T_i1, T_i2, …, T_ik}.

As shown in

Table 2. Processing information of FJSP with multi-purpose machine.

Jobs	Operations	Processing Time
		M1	M2	M3
J1	O11	7	2	3
	O12	3	4	5
	O13	3	-	6
J2	O21	2	-	5
	O22	4	4	-
	O23	2	-	5
J3	O31	4	2	3
	O32	3	-	-
	O33	2	8	5

“-” shows that the machine cannot proceed with the corresponding operation.

, there are three jobs {J₁, J₂, J₃} and three machines {M₁, M₂, M₃}. Each job has three operations that need to be performed, and every operation has many machine choices. Operation O₁₁ from J₁ can be followed by a choice of three machines: M₁ with a processing time of 7 units, M₂ with a processing time of 2 units, or M₃ with a processing time of 3 units. The third operation of J₁ has a choice of two machines, M₁ and M₃. On the other hand, J₂ and J₃ have three operations each, with their number of machine choices mentioned in Table 2.

3.3. Assumptions for Static Scheduling

The hypotheses or assumptions considered for the static initial scheduling are as follows [11]:

• Every machine in the system is available for the scheduling operation at time zero.

  ∀i∈m, Ami = 0

  (2)
• Every job within the system can be started at the initial time point, which is time zero, as mentioned in equation [34].

  C_ik = T_ik(i,k) ∈ O and k = 1

  (3)
• A machine can perform one operation at a time and is not capable of performing many operations at once. The scheduling process needs to ensure that operations assigned to a particular machine do not overlap in time.

  ∀i∈m, p_i = 1

  (4)
• An operation on a machine cannot be stopped or interrupted once it has started and must continue until finished.
• Due dates and release times are unspecified.
• Job setup time and transportation time are ignored.

4. Proposed Hybrid GASAVNS Algorithm

In this paper, a hybrid algorithm is used which includes the GA, SA, and the variable neighborhood search (VNS) algorithm. The SA is inserted into the procedure of GA for local searching. The GA can fall into the local optimal solution, which is why the proposed hybrid algorithm uses the local search algorithm, i.e., SA, to avoid being trapped in the local minima and accelerate the convergence speed. To diversify the solution, the VNS is used with SA with the aim to escape the local optimum and reach a global optimum. VNS works on the principle of using two or more neighborhood structures and systematically changing the neighborhood within the local search [35]. Figure 1 shows the workflow of the proposed hybrid algorithm. The following is a description of the GASA and VNS steps that are used in the proposed hybrid algorithm. The ‘1’ in Figure 1 flowchart represents the link between the scheduling and rescheduling methods allowing for retrieval of scheduling data for use in the rescheduling process.

4.1. Encoding and Decoding

In the flexible job-shop scheduling problem (FJSP), the following scheme is utilized for the selection of available machines and the scheduling of operations based on job numbers. Each job number is linked with the available machines, options, and operations, along with their corresponding processing times. The assignment of operations and machines is then carried out according to the feasible solution to satisfy the constraints [36,37].

SEQUENCE = (Job, Machine, Operation, Option, Processing Time)

4.2. Initial Population

Creating an initial population of solutions is an essential step at the beginning of a genetic algorithm. To form the initial population, we must generate a set of solutions by randomly selecting an optional machine from a set of machines and a job from a set of jobs. In this method, 300 numbers of random populations are created, each of which represents a potential solution to the problem.

4.3. Selection

The selection operator used in the GA to select the individual plays a crucial role in determining which members of a population will be selected for further generations. In this paper, two selection operators are adopted, with the capacity for either to be utilized in a single iteration. One is tournament selection. In tournament selection, the tournament size is initialized to 5. We must select the two individual sequences (chromosomes) with the highest fitness (minimum makespan) from the b randomly selected sequences from the current population. The other is random selection. In random selection, two sequences are randomly selected without considering their fitness, which allows any sequence to participate in further steps to produce offspring and allows for the possibility of discovering diverse solutions.

4.4. Cross-Over

In this research, two cross-over operators are adopted with the ability to be applied in an iteration. One is two-point cross-over. In this case, the two selected parents make a cross-over with each other and produce two offspring sequences. In a two-point cross-over, we exchange the position of two randomly selected operations (genes) between the two parents and generate two offspring. The other is multi-point cross-over. In multi-point cross-over, more than two operations are selected from two selected parents to exchange the position, which will generate two offspring.

4.5. Repair

It is possible that after cross-over, the generated offspring will violate the problem constraints; to repair this kind of problem, a repair operator is used to make sure that the newly generated offspring maintain the validity of the solution.

4.6. Local Search

In this paper, the heuristic method of SA is used as a local search algorithm. The SA can be successfully applied to the optimization scheduling problem [38]. SA will be applied to the output obtained from the cross-over in each iteration. SA is run through several iterations until the stopping requirement is satisfied.

The procedure is described as follows:

• Set the initial parameters: In this proposed method, the initial values set for the parameters of SA are an initial temperature of 50.0, a cooling rate of 0.90, and a final temperature of 10.
• Neighborhood structures: For neighborhood structures, we use the VNS. The idea behind VNS is to use two or more neighborhood structures and systematically mutate them in a neighborhood within a local search [39]. In each iteration of the local search, one neighborhood structure is chosen at random and applied to the cross-over output sequence (offspring).

The following are the six types of neighborhood structures that are based on VNS:

• Swap neighborhood: Two positions are randomly selected from the sequence, and their positions are swapped.
• Insert neighborhood: Two elements are chosen at random and the latter element is placed in front of the earlier element.
• Inverse neighborhood: The elements are rearranged between the two randomly chosen positions in reverse order.
• Four positions are randomly selected from the sequence and their positions shuffled (rearrangement).
• Adjacent neighborhood: A position is randomly selected and switched to its adjacent position.
• Inverse neighborhood: Two positions are chosen from the sequence randomly, and then, the element order between those two positions is reversed.

3.

Acceptance criteria and update: first, we find the delta makespan, i.e., the difference between the neighborhood makespan and the current makespan. Acceptance probability functions are crucial for accepting and rejecting new solutions based on the makespan difference and current temperature. If the new solution is better, i.e., the makespan is less than the current makespan, it is accepted; otherwise, it is rejected.

4.

Cooling schedule: This uses an exponential cooling scheme. The temperature steadily and gradually drops with the cooling rate (temperature*cooling rate).

5.

Final temperature: The final temperature is a stopping condition. When the current temperature is equal to the final temperature, the local search loop stops, and we obtain the new solution. After obtaining the new solution, we update the population and control go to the HA next iteration.

4.7. Termination Criteria

The proposed hybrid GASAVNS algorithm terminates when the number of generations reaches the maximum iteration value; in this study, the iteration value is set to 300, and we obtain the most feasible solution with the optimal makespan.

5. Proposed Rescheduling Methods

In this section, rescheduling approaches are presented to address dynamic events in the FJSP. Specifically, we explore two distinct scenarios.

5.1. Rescheduling for Machine Breakdown (Dynamic Event)

Figure 2 shows the workflow for machine breakdown. A machine breakdown, referred to as a dynamic event, occurs when a machine fails unexpectedly at a specific time of scheduling. When it comes to flexible job-shop scheduling, if there is sudden (unplanned) unavailability of a machine, it can have a significant impact on the production process and lead to a delay in processing. In the proposed method, initially, basic information on machine breakdown is required, like the broken machine number and breakdown time of the machine. At the time of machine breakdown, all upcoming operations are rescheduled, and the starting time of operations is defined immediately after the breakdown time with interruption of the operations that had already started to process on the broken machine. After eliminating the machine breakdown, with the help of extracted data from static scheduling, the remaining operations that need to take place are rescheduled.

The conditions to filter out the operation that participates in rescheduling are as follows:

The operations involved in the machine breakdown:

• Operations that end before breakdown time (i.e., stop time of operation < breakdown time) will not be involved in the rescheduling strategy.
• Operations that end after breakdown time (i.e., stop time operation > breakdown time) will be involved in the rescheduling strategy.

The operations that are not involved in the machine breakdown:

• Operations that had already started (i.e., start time of operation > breakdown time) will not be involved in the rescheduling strategy.
• Operations that started after the breakdown time (i.e., start time operation > breakdown time) will be involved in the rescheduling strategy.

After eliminating the non-required operation, we must reschedule all remaining operations using the proposed hybrid optimization method with a new start time for the machines to obtain the optimization solution for rescheduling the sequence.

5.2. Rescheduling for Job Arrival (Dynamic Event)

Figure 3 shows the workflow for the job arrival dynamic event. In this dynamic event, there can be the arrival of one or multiple urgent jobs. At the start of this method, the job arrival dataset and job arrival timing (i.e., rescheduling timing) are required. In the next step, we filter out all operations into operations that have already taken place and operations that need to be rescheduled with newly arrived jobs.

The conditions to filter out the operations that participate in rescheduling are as follows:

• Operations that already started before the rescheduling time (i.e., the start time of operation < rescheduling time) will not be involved in the rescheduling strategy.
• Operations that did not start before the rescheduling time (i.e., the start time of operation > rescheduling or breakdown time) will be involved in the rescheduling strategy.

After filtering out the operations, we remove unwanted operations from the initial static scheduling sequence. The remaining operations that participate in rescheduling are added to the operations from the new job arrival dataset. Then, we produce a population of 300 for the new operation sequence and optimize the solution with the proposed hybrid algorithm. After applying the proposed algorithm, we obtain the best solution with the minimum makespan.

6. Results and Discussion

The result of this study shows both the static and dynamic event results, along with their respective analyses.

6.1. Static Scheduling Experiment Results

To illustrate the effectiveness and performance of the proposed hybrid GASAVNS algorithm on a static scheduling dataset, this study utilizes the benchmark problem dataset adapted from [40].

Table 3. Comparative results for different JSSP instances.

Instances	SPT	LPT	DQN	SA	GA	Proposed
La01 (10 × 5)	920	889	666	889	717	666
La02 (10 × 5)	901	894	655	871	751	655
La03 (10 × 5)	770	748	597	788	677	606
La04 (10 × 5)	916	848	609	804	658	609
La05 (10 × 5)	827	787	593	785	593	593
La06 (15 × 5)	1369	1105	926	1020	946	926
La07 (15 × 5)	1128	1145	890	1125	1007	890
La08 (15 × 5)	1168	1061	863	1089	992	863
La09 (15 × 5)	1289	1105	951	1129	980	951
La10 (15 × 5)	1345	1136	958	1065	963	958
La11 (20 × 5)	1654	1476	1222	1543	1880	1222
La12 (20 × 5)	1352	1222	1047	1402	1286	1039
La13 (20 × 5)	1747	1298	1151	1466	1239	1150
La14 (20 × 5)	1757	1360	1292	1485	1315	1292
La15 (20 × 5)	1476	1510	1221	1551	1432	1219
La16 (10 × 10)	1588	1238	980	1230	1135	1000
La17 (10 × 10)	1094	1157	799	1291	947	794
La18 (10 × 10)	1259	1264	859	1264	1063	859
La19 (10 × 10)	1339	1140	872	1256	1089	860
La20 (10 × 10)	1331	1293	924	1375	1107	924
La21 (15 × 10)	1707	1545	1162	1672	1458	1132
La22 (15 × 10)	1257	1409	1021	1489	1327	1000
La23 (15 × 10)	1522	1330	1053	1417	1423	1034
La24 (15 × 10)	1554	1472	1029	1500	1336	1000
La25 (15 × 10)	1624	1382	1067	1429	1355	1061
La26 (20 × 10)	2137	1616	1327	1696	1742	1277
La27 (20 × 10)	2048	1776	1397	1863	1837	1345
La28 (20 × 10)	2034	1668	1386	1748	1746	1305
La29 (20 × 10)	2048	1649	1323	2048	1691	1290
La30 (20 × 10)	2081	1783	1417	1848	1806	1370
La31 (30 × 10)	2379	2394	1854	2415	2360	1784
La32 (30 × 10)	2823	2571	1900	2694	2587	1850
La33 (30 × 10)	2487	2372	1782	2445	2348	1719
La34 (30 × 10)	2500	2425	1880	2515	2402	1748
La35 (30 × 10)	2440	2514	1941	2514	2499	1888
La36 (15 × 15)	2070	1884	1355	1821	1806	1395
La37 (15 × 15)	2075	1940	1540	1983	1964	1504
La38 (15 × 15)	1944	1841	1348	1826	1832	1392
La39 (15 × 15)	1790	2064	1357	1902	1730	1281
La40 (15 × 15)	2003	1829	1336	1800	1814	1300

The bolded makespan in the table corresponds to the minimum makespan compared to the others.

shows the results of the experiment, providing a comparison of the makespan obtained using our algorithm with that obtained using other algorithms. The proposed algorithm is compared with the following DRL-based methods: DDQN [15], SPT, LPT, SA, and GA. The results indicate that the proposed hybrid GASAVNS algorithm shows the best result for 38 instances out of 40 compared to the other algorithms. This means that the proposed method is highly efficient for solving the static FJSP.

In Figure 4, a Gantt chart illustrates the scheduling solution for problem instance LA28. It displays 20 jobs allocated across 10 machines. Each color represents a specific job; for example, all green sections indicate job number 17. On machine 1, the green-colored section represents job number 17, indicated within brackets as (17,1,1) (75). Here, 17 denotes the job number, 1 signifies the operation number, 1 indicates the chosen machine option from the set of machines, and 75 denotes the processing time. The maximum completion time (makespan) for scheduling all operations is 1305 units.

6.2. Machine Breakdown (Dynamic Event)

In this event, machine breakdown is considered a dynamic event.

Table 4. Processing information for 5-job and 6-job machine FJSP.

Jobs	Operations	Processing Time
		M1	M2	M3	M4	M5	M6
J1	O11	2	3	4	-	-	-
	O12	-	3	-	2	4	-
	O13	1	4	5	-	-	-
	O14	4	-	-	3	5	-
	O15	-	6	8	7	6	9
J2	O21	3	-	5	5	2	-
	O22	4	3	-	-	6	-
	O23	-	-	4	4	7	11
	O24	-	5	-	-	-	5
	O25	4	5	7	7	5	-
J3	O31	5	6	-	-	-	-
	O32	-	4	3	3	5	-
	O33	-	-	-	-	9	12
	O34	6	5	4	4	8	-
	O35	8	6	-	-	7	8
J4	O41	9	-	7	9	-	-
	O42	-	6	-	4	-	5
	O43	1	-	3	-	-	3
	O44	6	-	9	7	5	4
	O45	-	8	7	8	8	-
J5	O51	4	3	7	9	3	6
	O52	5	6	-	4	-	5
	O53	6	-	4	-	-	3
	O54	-	5	-	7	-	7
	O55	7	-	8	7	8	-

shows the processing data adapted from [11] to demonstrate this event. These data come under the multi-purpose machine dataset type. Figure 5 displays a Gantt chart for static scheduling (before the machine breakdown) obtained using the proposed HA, with a makespan of 29 units. Then, one machine fails, i.e., machine 6 at a time of 20 units. Figure 6 shows a Gantt chart after rescheduling (before the machine breakdown) using the proposed HA strategy for the dynamic event, resulting in a makespan of 28 units, which is less than the comparative result [17,32]. The comparative results are shown in

Table 5. Comparison of results for rescheduling.

Events	Method Proposed in [41]	Method Proposed in [11]	Proposed Method
Failure time	20	20	20
Machine repair time	0	0	0
Ideal makespan	37	35	29
Actual makespan	37	35	28

. In the processes of rescheduling, Operations O14, O34, O45, and O54, which were not on the broken machine and already started before breakdown time, are completed without any changes in rescheduling. All remaining operations participate in the rescheduling process, forming a new sequence.

6.3. Urgent Job Arrival

In this dynamic event, an urgent job arrives at a specific time.

Table 6. Processing information of a job arrival event.

Job	Operations	Processing Time
		M1	M2	M3	M4	M5	M6
J6	O61	6	7	3	6	2	5
	O62	6	7	2	6	7	2
	O63	2	2	5	5	2	3
	O64	3	4	7	7	2	3
	O65	3	3	5	7	5	5

shows the processing information of the arrival of a single job, i.e., job 6, consisting of five operations and six machines. The data from Table 4 are utilized for static scheduling. Job 6 arrives at a time of 20 units. Figure 7 shows a Gantt chart of the static scheduling of five jobs before job six arrives, with a makespan of 29. Figure 8 illustrates a Gantt chart of rescheduling after arriving at a new job without optimizing, with a makespan 37. After applying the proposed rescheduling HA strategy, the makespan is minimized to 31, as illustrated in the Gantt chart in Figure 9.

As shown in

Table 7. Comparison of job arrival events.

Events	Method Proposed in [11]	Proposed HA Method
Int. scheduling makespan	35	29
Makespan before optimization (rescheduling)	50	37
Makespan after optimization (rescheduling)	37	31

, the proposed hybrid algorithm demonstrated better performance in both scheduling and rescheduling when compared to the other methods.

6.4. Multiple Machine Breakdowns

To demonstrate this dynamic event, the single-purpose machine dataset mentioned in

Table 8. Single-purpose machine data with 8 jobs and 5 machine groups.

	Ops	OP1 (MG1)				OP2 (MG2)		OP3 (MG3)			OP4 (MG4)			OP5 (MG5)
Jobs
J1		-	-	-	-	44	42	33	25	36	37	29	27	35	31	33	32
J2		46	41	42	40	18	11	-	-	-	46	43	42	18	28	17	19
J3		22	15	14	19	45	44	41	37	33	43	41	49	-	-	-	-
J4		18	21	24	13	25	23	11	12	21	-	-	-	40	34	38	30
J5		43	38	37	45	37	43	-	-	-	10	15	21	28	40	31	41
J6		46	41	48	38	25	19	-	-	-	13	14	15	28	26	34	27
J7		32	28	21	20	-	-	23	28	26	39	46	41	48	38	39	40
J8		16	17	13	12	22	17	-	-	-	49	42	45	25	33	29	27

“-” shows that the machine cannot proceed with the corresponding operation.

is utilized, consisting of eight jobs and five machine groups, with each machine group allocated a specific set of machines. The first machine group offers four machine choices, the second group provides two choices, the third group presents three choices, the fourth group offers three machine choices, and the fifth machine group provides a choice of four machines. Each machine group is exclusively assigned to execute a specific operation.

In this dynamic event, multiple machines become unavailable due to service or material unavailability. Then, multiple machines need to break down at a time. Specifically, machines numbers 5, 10, and 15 will be unavailable at the time of 80 units. Figure 10 shows a Gantt chart of static scheduling with a makespan of 163 units, and after rescheduling with the proposed HA, the makespan is 200 units, as demonstrated in the Gantt chart mentioned in Figure 11. Operations O₅₂ and O₇₃, which already started on the broken machine before the breakdown time, proceed again following rescheduling.

6.5. Multiple Job Arrivals

In this dynamic event, an urgent batch of multiple jobs arrives at a specific time.

Table 9. Single-purpose machine data with 5 jobs and 5 machine groups (job arrivals).

	Ops	OP1 (MG1)				OP2 (MG2)		OP3 (MG3)			OP4 (MG4)			OP5 (MG5)
Jobs
J9		32	35	29	30	46	47	21	16	17	17	16	11	-	-	-	-
J10		19	17	24	22	-	-	29	38	30	47	34	43	13	20	14	16
J11		46	34	44	43	-	-	12	13	14	14	21	18	27	29	39	28
J12		44	47	46	41	29	31	-	-	-	46	40	49	21	32	30	33
J13		40	47	44	49	28	23	28	23	26	26	24	33	15	16	11	19

shows the processing data of the arrival of multiple jobs consisting of five jobs and five machine groups. The data from Table 8 are utilized for static scheduling. The batch of five jobs arrives on time with 80 units. Figure 12 shows a Gantt chart of static scheduling with HA of eight jobs before a new batch of jobs arrives, with a makespan of 160 units. Figure 13 illustrates a Gantt chart of rescheduling (without optimization) after the arrival of a new batch of five jobs, with a makespan of 379 units. After applying the proposed rescheduling HA strategy, the makespan is minimized to 217 units, as shown in Figure 14, reaching a makespan from 379 to 217 units over iteration. Figure 14 shows the minimum makespan after every iteration. Figure 15 shows a Gantt chart of the final optimized rescheduling sequence.

7. Conclusions

The DFJSP plays a crucial role in the operation of real-world manufacturing systems. This manuscript proposed a rescheduling method to address dynamic scheduling in the FJSP with the goal of minimizing the makespan. The GA, SA, and VNS were used to obtain optimized solutions. To demonstrate that the proposed hybrid GASAVNS algorithm is efficient, 40 static benchmark datasets were utilized. Out of 40 instances, 38 perform better than the other algorithms mentioned in the study. This study confirms that the proposed rescheduling hybrid GASAVNS algorithm works efficiently for a single machine breakdown, multiple machine breakdowns, a single job arrival, and multiple job arrivals, regardless of whether single-purpose machine or multi-purpose machine datasets are used. For single-machine breakdown, the makespan after scheduling is 29 units. Then, after the dynamic event triggers rescheduling using the proposed method, the makespan decreases to 28 units. This reduction is significant when compared to the other algorithm, which results in a makespan of 35 units for static scheduling and remains at 35 units after rescheduling. For urgent job arrival, the makespan before optimization for rescheduling is 37 units. After being optimized by the proposed algorithm, it reduces to 31 units, which is a better result compared to the other algorithm, where the makespan decreases from 50 units before optimization to 37 units after optimization for rescheduling. For multiple machine breakdowns, the results demonstrate a dynamic event involving the breakdown of three machines. After rescheduling, the makespan is 200 units. The experiment shows that after the breakdown, other jobs efficiently change their positions to achieve an optimized solution. For multiple job arrivals, the makespan after static scheduling is 160 units. Then, after the dynamic event (five jobs arrival) triggers rescheduling, the makespan increases to 379 units before optimization. However, after applying the proposed rescheduling algorithm, the makespan is significantly reduced to 217 units. As a result, the objective of minimizing the makespan is significantly achieved after applying the proposed rescheduling algorithm for the DFJSP.