Efficient Multi-Start Gray Wolf Optimization Algorithm for the Distributed Permutation Flowshop Scheduling Problem with Preventive Maintenance

The distributed permutation flowshop scheduling problem (DPFSP) has received increasing attention in recent years, which always assumes that the machine can process without restrictions. However, in practical production, machine preventive maintenance is required to prevent machine breakdowns. Therefore, this paper studies the DPFSP with preventive maintenance (PM/DPFSP) aiming at minimizing the total flowtime. For solving the problem, a discrete gray wolf optimization algorithm with restart mechanism (DGWO_RM) is proposed. In the initialization phase, a heuristic algorithm that takes into consideration preventive maintenance and idle time is employed to elevate the quality of the initial solution. Next, four local search strategies are proposed for further enhancing the exploitation capability. Furthermore, a restart mechanism is integrated into algorithm to avert the risk of converging prematurely to a suboptimal solution, thereby ensuring a broader exploration of potential solutions. Finally, comprehensive experiments studies are carried out to illustrate the effectiveness of the proposed strategy and to verify the performance of DGWO_RM. The obtained results show that the proposed DGWO_RM significantly outperforms the four state-of-the-art algorithms in solving PM/DPFSP.

SECTION 1 Introduction

The permutation flowshop scheduling problem (PFSP) was first defined by Johnson^[1]. As an extension of PFSP, the distributed permutation flowshop scheduling problem (DPFSP) has garnered significant attention in the research community in recent years^{[2], [3]}. In the context of the fourth industrial revolution, the multi-factory production has become more prevalent in manufacturing field to improve manufacturing productivity. In DPFSP, there are $f$ factories, each of which has $m$ flow line machines that are identical. $n$ jobs must be handled in one of the $f$ factories.

The manufacturing industry is steadily advancing towards sustainable manufacturing as technology becomes more sophisticated. The shop scheduling problem has been studied extensively, for example, cascaded flowshop joint scheduling^[4], distributed assembly permutation flowshop scheduling^[5], serial distributed permutation flowshop scheduling^[6]. In these studies, all machines work continuously under ideal conditions, regardless of failures. In reality, machines can fail due to wear and tear during the production process, leading to uncertainty in production. Therefore, it is necessary to add maintenance operation to the shop scheduling problem. Maintenance is divided into preventive maintence (PM) and corrective maintenance^[7]. Corrective maintenance is the operation of a machine after it has failed, while PM is the operation of a machine before it broke down. Compared to corrective maintenance, PM has advantages in terms of maintenance costs and productivity. Therefore, this paper employs PM operation to prevent machine failures and unscheduled interruptions. PM enhances the stability of production systems by reducing the risk of catastrophic failure^[8].

The total flowtime (TFT) is the sum of the makespan for all jobs on the final machine and is widely used as an evaluation indicator for flowshop scheduling problems. A smaller TFT can reduce the overall cycle time and work in process inventory, so it is necessary to minimize the TFT by determining the factory assignment and the order in which jobs are processed in each factory. The gray wolf optimization algorithm (GWO) has been applied to path planning^[9], joint scheduling of AGVs and parallel machines^[10]. Based on the good performance of GWO and the characteristics of distributed permutation flowshop scheduling problem with preventive maintenance (DPFSP/PM), this paper proposes an improved discrete gray wolf optimization algorithm with restart mechanism (DGWO_RM) to minimize TFT. This paper contributes are as follows:

1. A heuristic algorithm considering the preventive maintenance and idle time is proposed to generate high quality initial population.

2. A combination of five local search methods is proposed to further optimize the generated initial solution and the path relinking method is used to bring the population closer to the head wolf.

3. A restart mechanism based on the problem knowledge is employed to avert the risk of converging prematurely to a suboptimal solution.

4. The superiority of DGWO_RM algorithm is verified through experiments on 225 cases using four advanced algorithms.

The remainder of this paper has the following structure. A comprehensive review of the relevant literature is presented in Section 2, and the problem formulation is introduced in Section 3. Section 4 describes the proposed meta-heuristic algorithm in detail. Section 5 presents a comprehensive comparison with existing state-of-the-art algorithms. Finally, Section 6 provides a concise summary and future outlook.

SECTION 2 Literature Review

This section reviews the relevant works, including (1) DPFSP based on TFT criterion, (2) DPFSP with preventive maintenance, and (3) application of the discrete gray wolf optimization algorithm.

2.1 DPFSP Based on TFT

The DPFSP was proposed by Naderi and Ruiz^[11] in 2010. TFT and makespan are the two most widely used optimization objectives currently. Different algorithms have been used by many scholars to study the problem based on makespan: Gao et al.^[12] combined a neighborhood generation method of swapping subsequence and an enhanced local search method into a taboo algorithm. Naderi and Ruiz^[13] improved the scatter search algorithm by adding a reference set update and restart strategy suitable for the problem, which enhanced the algorithm performance. Bargaoui et al.^[14] introduced a chemical reaction optimization algorithm with single-point crossover and greedy strategies for solving 720 benchmark instances and Ali et al.^[15] developed a mixed linear integer programming model to help the tabu search algorithm find the optimal solution. Lin et al.^[16] proposed a modified iterated greedy algorithm which specifies the number of jobs removed during the destruction phase and the temperature in the simulated annealing as variables. On the basis of iterated greedy algorithm, Li et al.^[17] proposed four neighborhood structures and used simulated annealing to improve the detection and processing of jobs delivered to machines by robots in factories. Huang et al.^[18] improved the discrete artificial bee colony algorithm by combining six composite neighbourhood operators and local search based on problem characteristics.

However, the literature on DPFSP based on TFT is relatively limited. Femandez-Viagas et al.^[19] introduced 18 heuristic algorithms and proposed an improved evolutionary algorithm to further optimize the solution sequence for minimizing the TFT. Meng and Pan^[3] proposed an improved artificial bee colony algorithm with inter-individual collaboration mechanism and restart strategy. Pan et al.^[20] designed three efficient heuristics using Liu-Reeves and N awaz-Enscore-Ham (NEH), as well as four metaheuristics based on problem features. Guo et al.^[21] introduced a discrete fruit fly optimization algorithm with a combination update mechanism to enhance the overall performance of the algorithm.

From the above reviews, it can be seen that DPFSP has been the focus of many researchers. However, most of the existing literature on DPFSP is targeted on makespan. TFT is important in optimization problems and deserves further research. Therefore, we study the DPFSP based on TFT.

2.2 Preventive Maintence of Machines

There was a lot of literature on preventive maintenance, for example, Wang and Liu^[22] proposed a heuristic based genetic algorithm to effectively solve the PM/DPFSP aiming at minimizing the makespan. Ruiz et al.^[23] used a modern ant colony algorithm and a genetic algorithm to effectively solve the PM/DPFSP. Afterwards, Mao et al.^[7] developed a multi-start iterated greedy algorithm to optimize solution sequences for minimizing makespan. Then Mao et al.^[24] improved the maintenance strategy and developed the memetic algorithm based on the hash map technology to solve the DPFSP/PM with TFT. In addition, Wang and Yu^[25] introduced a heuristic algorithm that utilized filtered beam search to solve the flexible job-shop scheduling problem with PM under multi-objective constraints. Wang et al.^[26] introduced a multi-objective hybrid optimization algorithm, combining the genetic algorithm and differential evolution algorithm with a multi-area division sampling strategy to address the flexible job-shop scheduling problem with PM, aiming to minimize total energy consumption and makespan.

In recent years, both DPFSP and PM have garnered significant attention, but there is a relative lack of literature on the PM/DPFSP. Since preventive maintenance of machines can save time, reduce costs, and keep a facility running efficiently, it is valuable to develop algorithms that are both effective and efficient to solve the PM/DPFSP.

2.3 Application of Discrete Gray Wolf Optimization Algorithm to Optimization

The GWO algorithm is designed to mimic the hunting strategy and hierarchical leadership of gray wolves in nature. It has been extensively applied in various optimization domains, including scheduling, network coverage, and energy optimization. Lu et al.^[27] introduced a multi-objective cellular GWO algorithm to address the hybrid flowshop scheduling problem, combining the diversification of metacellular automata with the intensification benefits of variable neighborhood search to balance exploration and exploitation. Zhang et al.^[28] integrated simulated annealing with GWO to improve wireless sensor network coverage. Miao et al.^[29] developed a hybrid GWO algorithm to optimize parameters in proton exchange membrane fuel cells, enhancing the search process through specific operators. Guo et al.^[30] introduced a tracking-based GWO algorithm for function optimization, incorporating the lion optimizer algorithm to add disturbance factors, enhancing the wolves' search ability. The dynamic weights were incorporated into the position update process of the grey wolves to prevent the algorithm from getting trapped in a local optimum. Lu et al.^[31] proposed a topological cytology GWO algorithm for benchmark and engineering problems, which limited interactions to each wolf topological neighbors and used information diffusion to maintain population diversity. Emary et al.^[32] designed a binary GWO variant that uses two different binary execution methods to select the optimal feature subset. Dey et al.^[33] introduced hybrid Variegated GWO algorithms to reduce operating costs and transmission losses in power system optimization. Finally, P. Jangir and N. Jangir^[34] proposed a non-dominated sorting GWO algorithm for multi-objective optimization, using a crowding distance mechanism to guide wolves towards dominant areas in the search space.

Based on the above review, it becomes evident that the GWO has garnered significant popularity in the realm of optimization. Given the excellent convergence ability of GWO and coupled with its fewer parameters and simple implementation, we have improved the algorithm by taking advantage of its intrinsic strengths in the study. DGWO-RM is proposed to address DPFSP/PM.

2.4 Research Gap

From the above, despite the significant progress made in the field of optimization in recent years both in terms of DPFSP and in terms of PM, there is still a relative paucity of literature on the DPFSP/PM problem. There exists ample room for enhancing the algorithms employed in addressing the problem, suggesting that further optimization is possible. For the two important research objectives of TFT and makespan, the existing literature mainly focuses on the makespan, for example, Han et al.^[6] proposed a collaborative multi-swap iterative greedy algorithm to minimise makespan, which used an adaptive destruction method and four swapping operators to improve the search capability. The TFT is relatively less explored. By optimising the TFT, we can significantly reduce waiting and delays of jobs throughout the production process. This, in turn, greatly enhances the overall efficiency of the production line, and ensures a smoother and more efficient workflow. Consequently, prioritizing the optimization of TFT is paramount to achieving operational excellence and maximizing production outcomes. Therefore, this study aims to investigate the DPFSP/PM problem in depth through the DGWO_RM method.

SECTION 3 Problem Description and Model

This section begins with an introduction to the problem, followed by an exposition of the mathematical constraints. Finally, an example is used to make the process of jobs machining and the calculation of the TFT clearer.

3.1 Problem Description

According to the perspective of Naderi and Ruiz^[11], the DPFSP can be comprehensively characterized as: there are $n$ jobs $(J=\{J_{1},J_{2},\ldots,J_{n}\})$ that need to be processed in $f$ factories $(F=\{F_{1},F_{2},\ldots,F_{f}\})$. All factories are homogeneous. Each job $J_{j}\in J$ can be processed by any of $F_{f}\in F$ factories. Each factory contains $m$ machines $(M=\{M_{1},M_{2},\ldots,M_{m}\})$ deployed in series. The processing sequence of each job in each factory cannot be changed. The time for performing PM operations on machine $m$ is represented by $T_{i}$ and the maintenance interval of machine $M_{j}$ is represented by $L_{j}$. Before starting processing one job on the machine, check if the remaining maintenance interval $L_{i}$ of the machine is sufficient to handle the job. The objective is to minimize TFT.

Before the study, the following assumptions are first declared:

1. A machine cannot process multiple jobs simultaneously, and a single job cannot be handled by more than one machine at the same time.

2. Job $J_{6}$ cannot be stopped on machine $M_{i}$ before completing the process.

3. Each machine and each job can start at time 0.

4. If the remaining maintenance level of the machine is greater than the processing time of the job on that machine, the job is processed directly; otherwise, the PM is carried out on that machine first, and then the job is processed.

3.2 Problem Model

Parameters:

$J:$	$n$ jobs to be processed, $J=\{J_{1},J_{2},\ \ldots,J_{n}\}$.
$M$:	$m$ machines with preventive maintenance, $M=\{M_{1},M_{2},\ldots,M_{m}\}$.
$F:$	$f$ identical factories, $F=\{F_{1},F_{2},\ldots,F_{f}\}$.
$j$:	job index number.
$i$:	machine index number.
$k$:	factory index number.
$H_{i,j}$:	time required for job $J_{i}$ to be processed on machine $M_{i}, J_{j}\in J, M_{i}\in M$.
$PMT_{i}$:	duration required for carrying out preventative maintenance tasks on machine $M_{i}$.
$C_{i,k,p}$:	time taken to complete the job located at position $p$ on the machine $M_{i}$ in the factory $F_{k}$.
$\text{OS}_{i,k,p}$,:	maintenance level of machine $M_{i}$ in factory $F_{k}$ when it is about to start processing job located at position $p$.
$\text{ML}_{i}$:	the maximum maintenance interval for machine $M_{i}$.
MN:	a large enough number.
TFT:	total flowtime.

Decision variables:

$X_{i,k,p}$:	if the job located at position $p$ is to be processed on machine $M_{i}$ in factory $F_{k}$ before this machine requires PM operation, then $X_{i,k,p}=1$; otherwise., $X_{i,k,p}=0$.
$Y_{i,k,p}$:	if the job $J_{j}$ is assigned to the position $p$ of the factory $F_{k}$ for processing, then $Y_{i,k,p}=1$; otherwise, $Y_{i,k,p}=0$.

Objective:

$$\begin{equation*}\operatorname{Min} \text{TFT}=\sum_{k=0}^f \sum_{p=0}^n\left(C_{m, k, p} \times \sum_{j=0}^n Y_{j, k, p}\right)\tag{1}\end{equation*}$$

Subject to:

$$\begin{gather*}\sum_{p=1}^n \sum_{k=1}^j Y_{j, k, p}=1, \quad \forall j\tag{2} \\\sum_{j=1}^n \sum_{k=1}^j Y_{j, k, p}=1, \quad \forall p\tag{3}\\ C_{i, k, p} \geqslant C_{i, k, p-1}+\sum_j^n Y_{j, k, p} \times H_{i, j}+X_{j, k, p} \times \text{PMT}_i, \quad \forall i, k, p > 1\tag{4} \\\quad C_{i, k, p} \geqslant C_{i-1, k, p}+\sum_{j=1}^n Y_{j, k, p} \times H_{i, j}, \quad \forall i > 1, k, p\tag{5}\\\text{OS}_{i, k, p} \geqslant \sum_{j=1}^n Y_{j, k, p} \times H_{i, j}, \quad \forall i, k, p \tag{6}\\\text{OS}_{i, k, p}=\left\{\begin{array}{ll}\text{OS}_{i, k, p}-\sum_{j=1}^n Y_{j, k, p} \times H_{i, j}=0, & X_{i, k, p}=0; \\\text{ML}_i, & X_{i, k, p}=1\end{array}\right.\\\forall i, k, p > 1\tag{7}\end{gather*}$$ $$\begin{gather*}\text{OS}_{i, k, p}-\left(\text{OS}_{i, k, p-1}-\sum_{j=1}^n Y_{j, k, p-1} \times H_{i, j}\right) \geqslant-\text{MN}\left(X_{i, k, p-1}\right), \\\forall i, k, p > 1\tag{8} \\\text{OS}_{i, k, p}-\left(\text{OS}_{i, k, p-1}-\sum_{j=1}^n Y_{j, k, p-1} \times H_{i, j}\right) \leqslant \operatorname{MN}\left(X_{i, k, p-1}\right),\\\forall i, k, p > 1\tag{9} \\\text{OS}_{i, k, p}-\text{ML}_i \geqslant-\text{MN}\left(1-X_{i, k, p-1}\right), \quad \forall i, k, p > 1\tag{10} \\\text{OS}_{i, k, p}-\text{ML}_i \leqslant \text{MN}\left(1-X_{i, k, p-1}\right), \quad \forall i, k, p > 1\tag{11}\\ Y_{j, k, p} \in\{0,1\}, \quad \forall i, k, p \tag{12}\\ X_{j, k, p} \in\{0,1\}, \quad \forall i, k, p \tag{13}\\ C_{i, k, p} \geqslant 0, \quad \forall i, k, p\tag{14}\end{gather*}$$

Equation (1) is the objective function TFT. Constraints (2) and (3) ensure that each job can only be processed at a fixed location in a factory and each location in a factory must be occupied by a job. Constraint (4) requires that before a job located at location $p$ in factory $F_{k}$ is processed on machine $M_{i}$, it is ensured that the previous job and PM operation have been processed on the machine. Constraint (5) ensures that job $J_{j}$ in factory $F_{k}$ is processed on machine $M_{i}$ only after machine $M_{i-1}$ has finished processing it. Constraint (6) indicates that if remaining maintenance interval of the machine is insufficient to process a job, a PM operation is performed on the machine first. Constraint (7) indicates that the maintenance level decreases as the jobs are machined and returns to its maximum value after the PM operation is performed. Constraints (8)-(11) convert Constraint (7) into a linear format, thereby guaranteeing that the maintenance level surpasses zero at all times. The decision variables are defined by the set of Constraints (12)–(14).

3.3 Illustrative Example

Taking eight jobs, two machines, and two factories as an example to show the process of jobs. The detail data including the processing time of jobs, the maintenance time, and interval of the machines are displayed in Table 1. The optimal solution without considering the PM operation is shown in Fig. 1. The processing sequence for factory $F_{0}$ is $\{J_{3},J_{1},J_{4},J_{0}\}$ and for factory $F_{1}$ is $\{J_{7},J_{2},J_{6},J_{5}\}$. Combined with Eq. (1), the total flowtime is the sum of the time that the jobs in all factories are processed on the last machine, from which we can calculate the TFT of Factory 0 in Fig. 1 to be $C_{1,0,0+C_{1,0,1}}+C_{1,0,2}+C_{1,0,3}=73$ and the total flowtime of Factory 1 to be $C_{1,1,0}+C_{1,1,1}+C_{1,1,2}+C_{1,1,3}=63$. In Fig. 1, its total flowtime is $73+63=136$. Figure 2 shows the solution considering the PM operator, it can be seen that the total flowtime is prolonged, where the TFT of Factory 0 is $c_{1,0_{2}0+C_{10,1}},+C_{1,0,2}+C_{1,0,3}=133$, the TFT of Factory 1 is $C_{1,1,0}+C_{1,1,1}+C_{1,1,2}+C_{1,1,3}= 103$, and thus the total flowtime of the entire sequence of jobs is $133+103=236$. Obviously, the solution shown in Fig. 2 is not optimal and needs to be further optimized.

SECTION 4 Proposed Algorithm

The GWO algorithm originally proposed by Mirjalili et al.^[35], which is a simple and easy to implement metaheuristic. It has been used in various fields^{[36]–​[39]}. The basic GWO algorithm includes the following four steps. In the initialization stage, the gray wolf population is initialized. The fitness of each individual is assessed. The wolves are divided into four classes based on fitness, from the highest to lowest. $\alpha$ represents the wolf with the optimal fitness value. $\beta$ and $\delta$ are wolves with the second and third optimal fitness values, respectively. The remaining wolves in the population are indicated by $\omega\{i\}$. In the encircling prey phase, candidate solution positions are updated based on prey positions. In the hunting prey phase, the $\omega\{i\}$ wolves in the population are guided by $\alpha, \beta$, and $\delta$ gray wolves in the direction of the probable position of the target prey. The fundamental DGWO encounters a challenge in its initial population distribution, which can be skewed or comprise a significant proportion of suboptimal solutions. This imbalance can hinder the iterative process, potentially prolonging the convergence towards the global optimal solution. In extreme cases, prevent the algorithm from reaching it altogether. Furthermore, the standard DGWO lacks robust local search and learning mechanisms that are crucial for refining and enhancing the quality of the discovered solutions. The following improvements are made in this study to overcome these shortcomings.

Fig. 1

Gantt chart of optimal solution sequences applicable to DPFSP.

Fig. 2

Gantt chart after adding PM operations to the sequence of optimal solutions for DPFSP.

Table 1 Data for the instance.

4.1 Solution Representation

In DPFSP/PM, a two-dimensional matrix is employed in this paper to encapsulate the entire solution, which is denoted $\Pi=\{\pi_{1},\pi_{2}, \ldots,\pi_{f}\}$, where $\pi_{k}(k=1,2,\ \ldots,f)$ represents the sequence in which the jobs are processed in the k-th factory. The first dimension represents the arrangement of multiple factories. The second dimension represents the jobs permutation allocated to the factory. For example, the solution shown in Fig. 1 can be represented as $\Pi=\{\{J_{3},J_{1},J_{4},J_{0}\},\{J_{7},J_{2},J_{6},J_{5}\}\}$, which indicates that the first factory processes jobs $J_{3},J_{1},J_{4}$, and $J_{0}$ in order and the second factory processes jobs $J_{7},J_{2},J_{6}$, and $J_{5}$ in order.

4.2 Population Initialization Phase

Pan et al.^[20] proposed the distributed Liu-Reeves (DLR) heuristic for multiple factories and combined it with the NEH heuristic algorithm for solving DPFSP with TFT. Mao et al.^[24] proposed the distributed Liu-Reeves combined with improved Nawaz-Enscore-Ham heuristic (DLR-INEH) algorithm for solving DPFSP with preventive maintenance. Distributed Liu-Reeves combined with enhanced Nawaz-Enscore-Ham heuristic (DLR-DNEH) and DLR-INEH have obtained good performance, hence, this paper further improves the DLR and NEH and proposes improved distributed Liu-Reeves combined with enhanced Nawaz-Enscore-Ham heuristic (IDLR-SNEH) based on the characteristics of preventive maintenance.

IDLR-SNEH heuristic algorithm is proposed based on the improved DLR heuristic algorithm and improved NEH2_en^[40]. The DLR heuristic adds jobs sequentially with the minimum makespan to the end of the factory $k^{*}$. The sequence of $n_{k^{*}}$ jobs that have been assigned to the factory is $\pi_{k^{*}}$ and the job that induces the smallest idle time $(\text{IT}_{j,n_{k}^{*}})$ of the machine is selected. Considering the PM operations of the machine, in IDLR, we choose the job $j$ that has the smallest $\text{IT}_{j,n_{k^{*}}}$ or causes the least number of PM operations of the machine. $\text{IT}_{j,n_{k}^{*}}$ is given by the following equation:

$$\begin{equation*}\text{IT}_{j, n_{k^*}}=\sum_{i=2}^m \frac{m \times\left\{C_{i, k^*, n_{k^*+1}}-C_{i, k^*, n_{k^*}}, 0\right\}}{i+n_{k^*} \times(m-i) /(n / f-2)}\tag{15}\end{equation*}$$

We use an indexing function $\text{IF}_{j,n_{k^{*}}}$ to select the job $j$ to be assigned to factory $k^{*}$. The formula is shown below:

$$\begin{equation*}\text{IF}_{j, n_{k^*}}=\left(\frac{n}{f}-n_{k^*}-2\right) \times \text{IT}_{j, n_{k^*}}+C_{m, k^*, n_{k^*}+1}\tag{16}\end{equation*}$$

We propose an enhanced version (SNEH), building upon the foundation of NEH2_en. NEH2_en arranges all jobs in non-descending order based on their total processing durations. After the jobs have been inserted one by one into the best position, the previous or next job is then inserted in the optimal position across all factories. Based on the IDLR and the SNEH, the IDLR-SNEH heuristic algorithm is proposed, which consists of $d$ jobs scheduled by the SNEH and the remaining $n-d$ jobs are scheduled by the IDLR. According to Mirjalili et al.^[35], set $d$ to $0.1n$. In the proposed initialization method, the IDLR-SNEH generates two solutions, while the remaining ones are randomly produced. The IDLR-SNEH pseudocode is shown in Algorithm 1.

4.3 Hierarchy and Hunting Prey Phase

During the gray wolf social hierarchy stage, $\alpha, \beta$, and $\delta$ gray wolves with the highest fitness are selected in order. The remaining gray wolves are $\omega\{i\}$. During the hunting prey phase, each wolf hovers in its original position and updates its position based on the location of its prey. Since $\alpha, \beta$, and $\delta$ gray wolves exhibit heightened awareness of the probable whereabouts of their prey, they are used to determine the location of their prey while directing other individual gray wolves to update their own location. In this study, we use the path relinking method proposed by Pan et al.^[41] to guide population wolves towards the head wolf and keep approaching the prey. First, select $\alpha$ wolf and sequentially select one of the wolves in $\omega\{i\}$. The path relinking method is employed to manipulate the selected two wolves, aiming to bring the wolf in $\omega\{i\}$ closer to the potential location of the prey. Suppose that $\alpha$ wolf is $ff_{\alpha}=\{\{J_{1},J_{6},J_{7},J_{0}\},\ \{J_{3},\ J_{2},J_{5},J_{4}\}\}$, and $\Pi_{\omega(i)}= \{\{J_{3},J_{1},J_{4},J_{0}\},\{J_{7},J_{2},J_{6},J_{5}\}\}$. Select two jobs $J_{a}$ and $J_{b}$ with the same position in $Jif_{\omega\{i\}}$ and in $Jf_{\alpha}$ in turn for comparison. If they are the same, skip them, otherwise, find a job in $\Pi_{\omega\{i\}}$ that is the same as $J_{b}$ and swap it with $J_{a}$. Each swap yields a temporary solution. As the number of comparisons increases, the temporary solution obtained becomes more and more similar to $\Pi_{\alpha}$. The temporary solution with the minimum TFT is preserved. The exchange process continues until all positions are compared. If the minimum TFT is not reached within the limit, the best solution encountered so far is retained as the temporary solution. The process retains the single best temporary solution encountered so far. This reduces memory usage and computational load, allowing the algorithm to focus on optimising the most promising solutions. The specific operation is shown in Fig. 3.

Algorithm 1 IDLR-SNEH

4.4 Local Search Phase

We propose four effective neighborhood search strategies and improve the hybrid reference local search (HRLS) method proposed by Mao et al.^[24]. The proposed five local search strategies are described in detail as follows:

Random Insertion (RI)

One factory is chosen at random from a set of $f$ factories. Subsequently, a job is picked randomly from that factory and placed in the most suitable position among all the factories. The process is repeated $Q$ times. $Q$ is set to 15 in this paper.

Random Swapping (RS)

First, select a factory by the tournament selection method. It is that two factories are randomly selected. The factory with the larger TFT is selected. Then, extract one job in sequence, swap it with all other jobs, and find the optimal swap position $P$. Assess the jobs preceding and succeeding the one at position $P$, choose the one with the longer processing duration, and swap it with every job across all factories to find the optimal position, as shown in Fig. 4a.

Fig. 3

Specific process of path relinking.

Fig. 4

Three operator diagrams.

Job Block Insertion (JBI)

The factory that has the largest TFT is chosen as the key factory. First, Block, consecutive jobs are selected at random as a block from the key factory, and attempts are made to insert the block into every viable position within the factory. Finally, the optimal position is selected, as shown in Fig. 4b.

Special Factory Insertion (SFI)

Identify a job with the maximum processing duration from a key factory and attempt to place it in every possible location within the factory that has the shortest TFT. The best position is selected, as shown in Fig. 4c.

Improved HRLS (HRLS_pm)

In HRLS, a random seed permutation is generated, all jobs are tried to insert sequentially into the best position in all factories. If the best neighboring solution generates less TFT than the original solution, the original solution is replaced. The process is reiterated until no further enhancement can be achieved. In HRLS_PM, assuming that the position that produces the smallest TFT after inserting the job is $P^{*}$. Then we randomly extract a job from $P^{*}+1$ and $P^{*}=1$ and swap it with jobs in all factories, and the best one swap is chosen. The pseudo code is shown in Algorithm 2.

4.5 Overall Framework of DGWO RM

DGWO_RM uses the IDLR-SNEH heuristic algorithm to generate two solutions during the initialization phase, and the remaining solutions are generated randomly. First, during the hunting phase, the algorithm employs random selection of different operators by five separate operators, allowing for varied search paths in each iteration round. This approach facilitates broader exploration across different regions of the search space, preventing the algorithm from becoming confined to a specific area. Second, in both the hounding and attacking phases, the sequential execution of the five operators systematically advances the exploration of the solution space to increase the comprehensiveness of the search. A path relinking method is applied to move other wolves closer to the leading wolf. This layered approach, transitioning from exploration to exploitation, allows the algorithm to achieve optimization goal more efficiently and accurately by progressively narrowing in on the optimal solution. Finally, the algorithm is reinitialized to escape local optima with a restart mechanism, which enabling further exploration for even more optimal solutions. The restart mechanism is employed to avert the algorithm from becoming trapped in a local minimum. If the evolution is still not obtained in $r$ iterations, the solution is discarded and reinitialized. The above process is repeated until the stopping criteria are satisfied. To make the algorithmic process easier to understand, Fig. 5 shows the overall algorithmic flow.

Algorithm 2 HRLS_PM (II)

Figure 2 is further optimized using DGWO_RM to obtain the optimal solution sequence and the solution is shown in Fig. 6. The processing order of the jobs in factory $F_{0}$ and factory $F_{1}$ is $\{J_{1},J_{6},J_{7},J_{0}\}$ and $\{J_{3},J_{2},J_{5},J_{4}\}$, respectively. The TFT of Factory 0 is $c_{1,0,0+C_{1,0,1}}+C_{1,0,2}+C_{1,0,3}=88$ and the total flowtime of Factory 1 to be $C_{1,1,0}+C_{1,1,1}+C_{1,1,2}+C_{1,1,3}=102$. Thus, the total flowtime of the entire sequence of jobs is $88+102=190$. The TFT has been improved and reduced to 190. From Figs. 1, 2, and 6, it can be seen that DGWO RM considering PM operation significantly reduces the number of machine maintenance.

SECTION 5 Experimental Result and Comparison

To validate the effectiveness of DGWO_RM, the parameters used in the algorithm are firstly calibrated through orthogonal experiments to select the optimal set of parameters. Secondly, the strategy proposed in the algorithm is verified through comparative experiments, and finally, the algorithm is compared with existing state-of-the-art algorithms in terms of analysis of variance (ANOV A) and convergence analysis. The proposed algorithm is encoded and implemented in Microsoft Visual Studio 2022 using C++. All experiments are run on 12th Gen Intel (R) Core (TM) with 32 GB of RAM i7-12 700@2.10 GHz under the Windows 10 operation system. Performance indicator is measured by relative percentage deviation (RPD). The termination condition of the algorithm is defined by the consumed CPU time, i.e., $T=n\times m\times C$ (ms)^[40] and $C$ is set to 100 for all experiments. For the RPD results of all algorithms, ANOV A is used to assess the statistical significance of variations among different samples. The calculation formula for RPD is as follows:

$$\begin{equation*}\text{RPD}=\frac{\left(\text{TFT}-\text{TFT}^*\right)}{\text{TFT}^*} \times 100 \%\tag{17}\end{equation*}$$ where TFT represents the optimal solution generated from a given algorithm on a particular instance, and TFT^* represents the optimal TFT found from the compared algorithms on the same instance.

The test instances used in this paper are from Mao et al.^[24] The instance size includes {100, 200,300,400, $500\}\times\{5,8,10\}\times\{2,4,6\}$, resulting in a total of 45 combinations. Each combination contains five instances and $45\times 5=225$ instances are generated. The parameters related to the instance are organized in Table 2. The minimum value of $T_{i}$ is set between 50 and 100 to ensure that maintenance time has a meaningful impact on scheduling. Furthermore, the average $T_{i}$ value is deliberately calibrated to be twice that of the average $P$ value. As for $L_{i}$, its range is strategically set between 50% and 75% of the average total machining time per machine, guaranteeing that each machine is assigned only one PM operation throughout the entire production cycle, thereby optimizing maintenance intervals.

Fig. 5

Overall flowchart of DGWO_RM algorithm.

Fig. 6

Gantt chart of optimal solution sequences applicable to PM/DPFSP.

Table 2 Parameters of the instance.

5.1 Parameter Setting

According to Mao et al.^[24], the value of the parameter $x$ for IDLR_SNEH is configured as 0.1. The three key parameters of DGWO_RM are as follows: the size of the population (PS), the size of job block $(\text{Block}_{k})$, and number of consecutive unimproved $(r)$. Table 3 displays the results for varying sets of these three parameters. We use orthogonal experiments to find the best combination of parameters. Table 4 shows the RPD values. Table 5 presents the response metrics of evaluating the effect of each parameter. From Table 5, $\text{Block}_{k}$ has the most obvious influence, followed in order by PS and $r$. Different trends of each parameter are depicted according to Table 3, as shown in Fig. 6. To make the gap more pronounced, multiply the resulting RPD value by 100 and denote it as rpd_1 00 in this study. From Fig. 7, it is clear to see that GWO_PM performs well when PS is 10, $\text{Block}_{k}$ is 5, and $r$ is 150.

5.2 Analysis of the Effectiveness of the Proposed Method

In this section, several comparison experiments are set to validate and analyze the effectiveness of the restart mechanism and the proposed local search. In order to guarantee the impartiality of the experiments, all comparison experiments are tested on a base of 225 instances, each of which is repeated five times. Several comparison tests are designed for analyzing the effectiveness of the proposed algorithm. The first comparison test is DGWO_RM and DGWO_RMI. DGWO_RMI represents the algorithm without RI strategy. From Fig. 8a, the RPD of DGWO_RM is significantly lower than that of DGWO_RMI because RI introduces randomness into the solution space, which helps to avoid the algorithm falling into local optima at early stages and ensures a wider range of solutions to explore. Therefore, the RI strategy is pivotal to the overall functionality of the algorithm.

Fig. 7

Comparison of different parameter combinations.

Table 3 Different parameter combinations.

Table 4 RPD values from different combinations of parameters.

Table 5 Average RPD value for each parameter.

The second comparison test is DGWO RM and DGWO_RM2. DGWO_RM2 represents the algorithm without RS strategy. In Fig. 8b, the RPD of DGWO_RM is significantly lower than that of DGWO_RM2, because the RS strategy selects the factory through a roulette method, which is not only more targeted than complete randomization, but also possesses stochasticity compared to selecting only the key factory. Therefore, the RS strategy is effective. The third comparison test is DGWO_RM and DGWO_RM3. The fourth comparison test is DGWO RM and DGWO RM4. DGWO RM3 and DGWO_RM4 represent the algorithms without JBI strategy and without SFI strategy, respectively. From Fig. 8c and Fig. 9a, as can be observed, DGWO_RM significantly outperforms DGWO_RM3 and DGWO_RM4. JBI improves the efficiency of the algorithm by adjusting for successive blocks of jobs, which avoids the computational complexity associated with adjusting each job individually. SFI helps to quickly adjust the structure of the global solution by globally adjusting the positions of long time jobs so that the algorithm can approach the global optimum faster. Therefore, JBI and SFI are effective.

The fifth comparison test is DGWO RM and DGWO_RM5. DGWO_RM5 represents the algorithm without HRLS_PM strategy. In Fig. 9b, DGWO_RM and DGWO_RM5 are significantly different within the 95% confidence interval. The HRLS_PM policy enhances the depth of local optimization and improves the diversity of solutions through a combination of insertion and swapping. This confirms the effectiveness of HRLS PM.

Fig. 8

Validating the effectiveness of RI, RS, and JBI strategies.

Fig. 9

Validating the effectiveness of SFI, HRLS_PM strategies, and restart mechanism.

The sixth comparison test is DGWO RM and DGWO_RM6. DGWO_RM6 represents the algorithm without restart mechanism. Figure 9c shows that DGWO RM achieves better results. This validates the effectiveness of the restart mechanism as it helps the algorithm to jump out of the local optimum. Thus, the restart mechanism plays an important role for the whole algorithm.

5.3 Comparative Analysis with Existing Algorithms

This section assesses the performance of DGWO_RM through a comparative analysis with four existing state-of-the-art algorithms including the hash map-based memetic algorithm (HMMA)^[24], the discrete fruit fly optimization (DFFO) algorithm^[21], the iterated greedy (IG) algotithm^[20], and the genetic algorithm (GA)^[42]. The problems solved by the above algorithms are strongly related to DPFSP/PM, however, in order to satisfy the PM constraint, we adapt these algorithms to DPFSP/PM in this paper. The parameters in these comparative algorithms are shown in Table 6.

Table 7 shows the results obtained by all the algorithms, grouped by job (n), machine (m), and factory (f). In addition, to emphasise the effectiveness of the modifications made by the DGWO algorithm, we compare the improved version with the basic version. The best experimental results are bolded.

Table 6 Algorithm parameters.

Obviously, DGWO_RM algorithm consistently produces better results with different groupings. The overall performance of these algorithms in descending order is RPD(DFFO)=4.215490, RPD(GA)=3.951 700, RPD(DGWO)=1.655473, RPD(IG)=0.508583, RPD(HMMA)=0.235825, and RPD(DGWO_RM)= 0.150 433. Figure 9a clearly shows the strengths and weaknesses of the overall results of the five algorithms. Among them, HMMA and IG perform relatively well, so DGWO_RM, HMMA, and IG are compared separately. Figure 9b shows that DGWO_RM has the smallest RPD value. In summary, from Fig. 10, the DGWO_RM algorithm has better overall results.

Figures 11–​13 are interaction plots of the algorithm with f, m, and $n$ for $C$ = 100. Figure 11 illustrates the RPD values for different number of jobs. Figure 12 shows the RPD values for different number of factories. Figure 13 shows the RPD values for different number of machines in each factory. These three plots demonstrate that DGWO_RM has good RPD values for instances of different sizes, and the trend is relatively flat to the point where results are more stable. Since HMMA solves the same problem and objective of PM/DPFSP, HMMA is the closest in effect to DGWO_RM, but HMMA is inferior relative to DGWO_RM.

Fig. 10

Comparison chart of average RPD values. (a) Comparison chart of average RPD values of all compared algorithms. (b) Comparison of average RPD values of DGWO_RM, HMMA, and IG algorithms.

Fig. 11

Means plots for the interaction between algorithm type and n.

Fig. 12

Means plots for the interaction between algorithm type and f.

Fig. 13

Means plots for the interaction between algorithm type and m.

Table 7 Average RPD grouped by factory (f), machine (m), and job (n).

In addition, to distinctly demonstrate the convergence trend of the proposed algorithm, we carefully selected HMMA and IG algorithms with performance closer to DGWO-RM for comparison, highlighting the superiority of DGWO-RM algorithm.

F our instances of different sizes were chosen to verify the convergence advantage of DGWO_RM. The four instances are: 1 $30\times 5\times 2,200\times 8\times 4,300\times 8\times 6$, and $500\times 10\times 6$. The convergence graph of the total flowtime for each instance is shown in Fig. 14, which shows that all three algorithms exhibit a clear convergence trend as time increases. HMMA and IG have similar convergence performance, but the quality of the initial solution is better for HMMA. As can be seen from Fig. 14, the initialization method proposed by the DGWO_RM algorithm for the characteristics of the DPFSP/PM problem gives it a good initial solution quality. DGWO_RM has better convergence performance on any scale of instances.

SECTION 6 Conclusion and Future Work

To tackle the PM/DPFSP incorporating TFT, this paper introduces DGWO_RM. First, based on the characteristics of preventive maintenance, DLR and NEH are improved and the IDLR-SNEH method is devised for crafting the initial solution. In the selection stage, $\alpha, \beta$, and $\delta$ gray wolves are greedily selected, and path relinking method is adopted to bring w{i} wolves closer to potential prey locations guided by $\alpha, \beta$, and $\delta$ gray wolves. A local search framework based on four local search strategies and an HRLS_PM are proposed to enhance the quality of the solution. Moreover, we introduce a restart mechanism to evade potential stagnation of the algorithm within a local optimum. Performance tests are conducted on 225 instances of different sizes, and we compare the proposed DGWO_RM with current state-of-the-art algorithms. Based on the experimental outcomes, the DGWO RM algorithm exhibits superior performance.

Preventive maintenance is incorporated into the production scheduling plan, which improves the reliability and stability of the equipment, reduces the maintenance cost of the equipment compared to the maintenance of the equipment after the failure, and allows for a more rational arrangement of the processing tasks and equipment maintenance time to improve the production efficiency. However, in actual production, more factors need to be considered. For example, consider the likelihood of a machine blocking^{[43], [44]} and more adapted to the industrial situation to consider sequence-dependent setup time constraints^{[45]–​[47]}. In addition, due to the complexity of the problem, traditional scheduling algorithms are often ineffective in solving such problems, more efficient algorithms need to be designed to cope with such complex scheduling problems. For example, combine some reinforcement learning techniques with intelligent optimization algorithms to strengthen the learning ability of individuals^{[48], [49]}.

Fig. 14

Convergence graph of the three algorithms.

Conflict of Interest

The authors declare no conflict of interest.

ACKNOWLEDGMENT

This work was supported by the National Natural Science Foundation of China (Nos. 62473186 and 62273221), Natural Science Foundation of Shandong Province (No. ZR2024MF017), and Discipline with Strong Characteristics of Liaocheng University Intelligent Science and Technology (No. 319462208).