Research on Flexible Job Shop Scheduling with Work-Piece Handling and Machine Prevetive Maintenance

Abstract

Conventional research on flexible job shop scheduling (FJSP) often overlooks critical factors such as workpiece handling, machine preventive maintenance, and variable machining speeds, resulting in scheduling schemes with limited practicality and suboptimal performance. To tackle these issues, this study establishes a Flexible Job Shop Scheduling Problem with Workpiece Handling and Machine Preventive Maintenance (WHMPM-FJSP) model, aiming to minimize both makespan and total energy consumption. An Improved Multi-Objective Discrete Grey Wolf Optimization (IMOD-GWO) algorithm is proposed to solve this model. The algorithm incorporates three key innovations: (1) A tri-level encoding structure that integrates machine assignments, operation sequences, and machining speed selection, tailored to the problem’s characteristics. (2) Multiple effective population initialization strategies combined with novel individual update mechanisms. (3) Implementation of distributed computing methods to enhance search efficiency within limited timeframes. To verify the rationality and efficacy of the model and the algorithm, comparative experiments were conducted using benchmark instances of varying scales against existing multi-objective optimization algorithms. The experimental results show that in medium- to large-scale cases, IMOD-GWO outperforms other methods, demonstrating significant advantages and highlighting its enhanced global search capability in solving WHMPM-FJSP problems. The proposed model and algorithm effectively solve the scheduling problem in flexible workshops with integrated processing and maintenance, demonstrating strong performance and practicality.

1. Introduction

Production planning constitutes a critical challenge in modern industrial systems [1]. Optimizing production plans can significantly improve manufacturing efficiency, specifically by eliminating scheduling conflicts, shortening production cycles, increasing resource utilization, and reducing production disruptions on the factory floor. As the core component of production planning, the job shop scheduling problem (JSP) persistently confronts multifaceted challenges [2]. Notably, conventional JS approaches often fail to ensure task-specific resource flexibility, rendering them inadequate for evolving industrial demands—particularly given the increasing adoption of flexible manufacturing systems and widespread automation [3,4]. Unlike traditional job shop scheduling problems, the flexible job shop scheduling problem (FJSP) introduces greater flexibility in machine selection, capacity allocation, and task processing order. These advancements are of great significance for improving corporate competitiveness and operational feasibility. Since its formal introduction in 1990 [5], the flexible job shop scheduling problem (FJSP) has attracted widespread attention from researchers worldwide due to its closer alignment with actual scheduling needs. Given that FJSP represents a classic NP-hard problem [6], where obtaining exact solutions within polynomial time is generally infeasible, heuristic and metaheuristic algorithms have emerged as practical approaches for identifying feasible solutions in real-world applications. Recent algorithmic advancements include the following: Wang et al. developed a Pareto-based dual-population cooperative evolutionary algorithm (CEA-TP), which decouples operation sequencing and machine assignment subproblems while incorporating a crowding similarity mechanism for Pareto archive maintenance, thereby enhancing non-dominated solution quality in multi-objective FJSP [7]. He et al. proposed an improved African vulture optimization algorithm (IAVOA) featuring tri-level encoding and dynamic neighborhood search operations, effectively optimizing dual-resource-constrained scheduling scenarios [8]. Meng et al. designed an improved genetic algorithm (IGA) addressing collaborative optimization challenges in multi-AGV flexible job shop scheduling [9]. Lv et al. introduced an enhanced Harris hawk optimization algorithm (GNHHO) that integrates elite preservation strategies and chaotic perturbation mechanisms to mitigate premature convergence in conventional approaches [10]. Long et al. devised a dynamic self-learning artificial bee colony algorithm (DSLABC) employing Q-learning for adaptive search dimension adjustment, thereby improving both efficiency and robustness in dynamic FJSP environments [11]. Pablo et al. presented an enhanced memetic algorithm (EMA) combining evolutionary computation with tabu search to optimize energy consumption in fuzzy FJSP scenarios [12]. Liu et al. established a hybrid salp swarm algorithm (MHSSA) incorporating Lévy flight strategies to enhance solution set performance for green scheduling in double-flexible job shops [13]. Boudjemline et al. developed a genetic algorithm-based multi-objective framework (GA-FJSSP) utilizing steady-state selection and hybrid crossover strategies for efficient FJSP resolution [14]. Qu et al. proposed a variable-speed flexible job shop scheduling strategy considering operation-dependent energy consumption, constructed a multi-energy consumption fusion model, and proposed a multi-objective evolutionary algorithm (MOEA/D-DDWA) with dynamic diffusion weight adjustment to effectively optimize the completion time and total energy consumption [15]. Bezoui et al. proposed a preference-integrated multi-objective genetic algorithm based on a non-compensatory reference level model to guide the search toward solutions better aligned with decision-maker aspirations in flexible job shop scheduling [16].

In real-world FJSP scenarios, scheduling models that consider machine availability constraints are more representative of actual production environments. This consideration also improves the stability of production plans. Recent research progress on FJSP problems that include machine preventive maintenance includes the following: Pal et al. proposed a multi-agent system (MAS) enabling decentralized decision-making through bidding mechanisms between job and machine agents, explicitly embedding maintenance windows into the bidding process [17]. Yan et al. designed a deep Q-network (DQN)-based agent framework that dynamically adjusts maintenance windows and learns optimal job assignment-maintenance timing policies via a decaying ε-greedy strategy [18]. Wöcker et al. formulated a mixed-integer programming (MIP) model with local search algorithms to optimize joint scheduling of parallel machines [19]. Baykasoğlu et al. developed a GRASP (Greedy Randomized Adaptive Search Procedure)-based framework, integrating local reordering to minimize average tardiness, schedule instability, and makespan [20]. Gupta et al. employed response surface methodology to analyze interactions between stochastic machine failures and preventive maintenance on scheduling performance [21]. Zhao et al. established two mixed-integer linear programming (MILP) models based on machine positioning and operation sequencing, complemented by a QVNS-Q algorithm to enhance large-scale problem-solving efficiency [22]. An et al. constructed a multi-objective optimization framework accounting for imperfect preventive maintenance (PM) models and accelerated convergence using an enhanced NSGA-III/ARV algorithm [23]. Soofi et al. introduced a robust fuzzy stochastic programming model hybridizing genetic algorithms with vibration-damping optimization to address dual-resource-constrained scheduling [24]. Although these studies have advanced the optimization of flexible job shop scheduling problems by considering material handling factors, they have failed to adequately address the interdependencies between machine preventive maintenance factors and scheduling dynamics.

In actual manufacturing processes, workpiece machining is not entirely completed on a single machine; therefore, workpiece handling between different machines is necessary during production. For flexible job shop scheduling problems (FJSP) that include workpiece handling constraints, recent methodological advancements include the following: Wu et al. proposed an adaptive population NSGA-III (APNSGA-III) algorithm with dual control strategies, optimizing energy consumption and weight parameters in multi-objective flexible flow shop scheduling. Their model explicitly accounts for energy expenditure during transportation, setup, unloading, and idle phases, along with job weight considerations [25]. Zhang et al. developed a hybrid brain storm optimization (BSO) and Q-learning algorithm for distributed flexible flow shop scheduling, enhancing job allocation through variable neighborhood search refinement [26]. Yunusoğlu et al. established a constraint programming (CP)-based multi-objective scheduling model addressing flexible flow shop sub-batch scheduling with setup and transportation resource constraints, augmented by CP-guided large neighborhood search iterations [27]. Zhang et al. designed a learning-driven multi-objective cooperative artificial bee colony algorithm (L-MCABC) for distributed flexible flow shops integrating preventive maintenance and transportation operations [28]. Wang et al. introduced a multi-domain stratified sampling MOGA-DE (MDSS-MOGA-DE) algorithm, specifically optimizing scheduling with maintenance and transportation processes through hybrid genetic and differential evolution strategies [29]. Xuan et al. formulated an artificial immune differential evolution (AIDE) algorithm to resolve sequence-dependent setup times and transportation scheduling in distributed heterogeneous flexible flow shops [30]. Li et al. proposed a bi-population balanced multi-objective evolutionary algorithm (BPBMO) for steel production systems, addressing distributed flexible flow shop scheduling with fuzzy processing times and crane transportation [31]. Although these studies have made significant progress, they have overlooked the impact of variations in processing speed levels on energy consumption, which is crucial for the optimization results of FJSP.

In summary, although the existing literature has investigated the FJSP from various perspectives, few studies have integrated constraints such as machining speeds, machine preventive maintenance plans, and workpiece handling into FJSP research. In practical workshop scheduling, equipment failures and maintenance activities can disrupt the smooth execution of scheduling schemes, leading to resource conflicts, increased energy consumption, and extended makespan. To reduce equipment failures in real-world production, preventive maintenance must be performed periodically or conditionally on machine tools. Although preventive maintenance activities consume planned cycle time, they can reduce or avoid machine failures, thereby improving the stability and reliability of the production system and enhancing the feasibility of production plans. With the implementation of Just-In-Time (JIT) production and enterprises’ increasing emphasis on order delivery punctuality, the requirement for material punctuality within workshops is increasingly heightened. If the production scheduling scheme ignores workpiece transportation time and only considers processing time, workpieces or materials may not arrive at the corresponding workstations on time during the production process, thus hindering the progress of the entire production line. Furthermore, modern machine tools offer multiple speed levels. Selecting the appropriate processing speed during workpiece processing can shorten the production cycle and reduce energy consumption. To solve these issues, this study proposes a green scheduling method for FJSP that considers workpiece handling and machine pre-maintenance. By constructing a multi-objective optimization model incorporating constraints such as preventive maintenance plans and workpiece handling, this method rationally schedules operation sequences, machine assignments, and machining speed selection to minimize maximum makespan and total energy consumption. An Improved Multi-Objective Discrete Grey Wolf Optimization (IMOD-GWO) algorithm is proposed to solve this FJSP problem. Finally, experiments validate the efficacy of the proposed method.

This study innovatively introduces factors such as workpiece handling time, equipment preventive maintenance, and variable processing speed into the traditional flexible job shop scheduling problem. A multi-objective scheduling model is established with optimization goals of minimizing maximum makespan (Cmax) and total energy consumption (Etotal). To effectively solve this complex problem, an Improved Multi-Objective Discrete Grey Wolf Optimization (IMOD-GWO) algorithm is proposed, which integrates the Jaya, introduces mutation operators, and employs distributed computing.

The rest of this paper is arranged as follows:

Section 2: Problem description and mathematical model for WHMPM-FJSP. This chapter introduces the flexible job shop scheduling problem and formulates a mathematical model with constraints on workpiece handling and machine preventive maintenance, aiming to minimize makespan and total energy consumption.

Section 3: Algorithm design. This chapter presents the proposed IMOD-GWO algorithm, detailing its workflow, key components, and operating mechanisms.

Section 4: Experimental analysis. This chapter conducts experiments on benchmark instances, including parameter tuning, ablation studies, and comparative analyses to verify the effectiveness and performance of the algorithm.

Section 5: Conclusions. This chapter summarizes the research contributions, discusses limitations, and outlines potential directions for future work.

2. Problem Description and Mathematical Model

2.1. Problem Description

The studied problem is formulated as a planning-level abstract model for a workshop containing n workpieces and m machines. Each workpiece comprises n_i operations. The standard processing time for each operation is predetermined and must be executed on a capable machine. Each machine supports multiple speed levels, with energy consumption per unit time varying across speeds. The actual processing time of an operation equals its standard processing time divided by the selected machine speed. All machines operate under predetermined preventive maintenance schedules, during which no workpiece processing is allowed. The workshop is assumed to have sufficient transportation resources, and the transportation times between machines are known. The scheduling objectives are to minimize makespan and total energy consumption. Research assumptions:

(1)

All workpieces and machines are idle and available at time zero;

(2)

Each machine can only process one operation at a time;

(3)

A single workpiece cannot be processed on multiple machines simultaneously;

(4)

Operations must be processed continuously without interruption;

(5)

Operations of the same workpiece must follow their predefined processing sequence;

(6)

Switching between machining speeds on a machine incurs no time delay.

These simplifications facilitate modeling and solution design while capturing essential scheduling constraints; in practice, factors such as stochastic transport delays or condition-based maintenance could be incorporated in extensions.

2.2. Notation of Parameters

The variables and their definitions involved in this study are listed in

Table 1. Notation and definitions.

Symbol	Definition
n	Quantity of workpieces
m	Number of machines
h,i,x	Workpiece index
g,j,y	The index of the operation
l,k	Machine index
r	Speed level index
nh	The number of operations on workpiece h
ml	The number of speed levels on machine l
PMl	Number of preventive maintenance tasks for machine l
Ohg	g-th operation of workpiece h
spthg	Ohg’s standard processing time
Shglr	Machining speed of Ohg on machine l at speed level r
pthglr	Actual processing time of Ohg on machine l at speed level r
sthgl	Start time of Ohg on machine l
cthgl	Completion time Ohg on machine l
Ch	Completion time of workpiece i
mstlq	Start time of the q-th maintenance on machine l
metlq	End time of the q-th maintenance on machine l
DTlq, MT	Duration of the q-th maintenance on machine l
ETlq, ST	Earliest start time of the q-th maintenance on machine l
LTlq, CT	Latest start time of the q-th maintenance on machine l
ttkl	Transportation time from machine k to l
PE	Total processing energy consumption of all machines
FE	Total idle energy consumption of all machines
TE	Total transportation energy consumption
TEC	Total energy consumption
pel	Energy consumption for processing per unit of time of machine l
fel	Energy consumption for idling per unit of time for machine l
te	Energy consumption for transportation per unit time
M	A sufficiently large positive constant
xhgl	Binary variable: It takes a value of 1 when Ohg is processed on machine l; otherwise, it takes a value of 0
yhgijk	Binary variable: It takes a value of 1 when Ohg is processed after Oij on machine l; otherwise, it takes a value of 0
Zhglr	Binary variable: It takes a value of 1 when Ohg is processed on machine k at speed r; otherwise, it takes a value of 0
αhkl	Binary variable: It takes a value of 1 when workpiece h is transported from machine k to l; otherwise, it takes a value of 0

.

2.3. Mathematical Model

Based on the above problem description and assumptions, this paper establishes the following mathematical model to describe the scheduling problem of a flexible job shop considering workpiece transportation and machine preventive maintenance, with the scheduling optimization objectives being the minimization of the makespan and the total energy consumption.

$$PE={\displaystyle \sum _{h=1}^{\mathrm{n}}{\displaystyle \sum _{g=1}^{n_h}{\displaystyle \sum _{l=1}^{\mathrm{m}}p}}}{e}_l\times p{t}_{hgl}\times x_{hgl}$$

(1)

$$FE={\displaystyle \sum _{h=1}^{\mathrm{n}}{\displaystyle \sum _{g=1}^{n_h}{\displaystyle \sum _{i=1}^{\mathrm{n}}{\displaystyle \sum _{j=1}^{n_j}{\displaystyle \sum _{l=1}^m{\mathit{fe}}_l}}}}}\times (s{t}_{hgl}-c{t}_{ijl})\times y_{hgijl}$$

(2)

$$TE={\displaystyle \sum _{k=1}^m{\displaystyle \sum _{l=1}^m{\displaystyle \sum _{h=1}^{n}t}}}e\times t{t}_{kl}\times a_{hkl}$$

(3)

$$min{C}_{max}=min(max(C_h))$$

(4)

$$minTEC=min(PE+TE+FE)$$

(5)

Equations (1)–(3) are used to calculate the processing energy consumption (PE), idle energy consumption (FE), and transportation energy consumption (TE), respectively. Equation (4) defines the minimization of the makespan, where the makespan is determined by the maximum completion time among all machines; Equation (5) defines the minimization of total energy consumption, where the total energy consumption is the sum of each machine’s PE, FE, and TE.

The following are the constraints for the WHMPM-FJSP:

$${\displaystyle \sum _{l=1}^{\mathrm{m}}{x}_{hgl}}=1$$

(6)

$${\displaystyle \sum _{h=1}^n{\displaystyle \sum _{g=1}^{{\mathrm{n}}_h}{\displaystyle \sum _{l=1}^{\mathrm{m}}{y}_{hgijl}}}}⩽1$$

(7)

$${\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^{n_h}{\displaystyle \sum _{l=1}^{\mathrm{m}}{y}_{hgijl}}}}⩽1$$

(8)

$${\displaystyle \sum _{l=1}^m{\displaystyle \sum _{r=1}^{m_l}{Z}_{hglr}}}⩽1$$

(9)

$$p{t}_{hglr}=sp{t}_{hg}/S_{hglr}\times Z_{hglr}$$

(10)

$$c{t}_{hgl}⩾s{t}_{hgl}+p{t}_{hglr}\times x_{hgl}$$

(11)

$$s{t}_{hg+1l}⩾c{t}_{hgk}+t{t}_{kl}\times {\alpha }_{hkl}$$

(12)

$$C_h⩾c{t}_{hgl}$$

(13)

$$c{t}_{ijk}⩾c{t}_{hgk}+y_{hgijl}\times p{t}_{hgl}-M\times (1-y_{hgijl})$$

(14)

$$E{T}_{lq}⩽ms{t}_{lq}$$

(15)

$$ms{t}_{lq}⩽L{T}_{lq}$$

(16)

$$m\mathrm{e}{t}_{lq}⩾ms{t}_{lq}+D{T}_{lq}$$

(17)

Specifically, Equation (6) guarantees that precisely one machine handles each operation of every workpiece. Equations (7) and (8) stipulate that every operation can have, at most, a single operation that immediately precedes it or follows it. Equation (9) restricts each operation to be executed at a single machining speed on one machine. Equation (10) calculates the actual processing time of operations. Equation (11) imposes constraints on operation processing times. Equation (12) guarantees sequential processing of operations within the same workpiece. Equation (13) defines the completion time of workpieces. Equation (14) stipulates that a machine cannot process multiple operations simultaneously. Equations (15)–(17) describe temporal relationships between machine maintenance activities. It is crucial to point out that the traditional FJSP has been demonstrated to be NP-hard. This complexity is further exacerbated in our study by the introduction of additional constraints such as workpiece handling and machine preventive maintenance, rendering traditional operations research methods inadequate for effective resolution. To address this challenge, this study proposes the Improved Multi-Objective Discrete Grey Wolf Optimization (IMOD-GWO) algorithm, offering a novel methodological framework for solving such complex combinatorial optimization problems.

3. Improved Multi-Objective Discrete Grey Wolf Optimization

The GWO algorithm solves optimization problems by dividing the gray wolf population into different hierarchical groups (α, β, δ, and ω wolves). The α, β, δ, and ω wolves guide the search process, while the subordinate ω wolves follow their instructions. This algorithm primarily simulates three hunting behaviors of gray wolves: surrounding the target, pursuing the prey, and attacking the prey. The distance formula and position formula for grey wolves searching for prey:

$$D=|C\cdot X_{\mathrm{P}}(t)-X(t)|$$

(18)

$$X(t+1)=X_P(t)-A\cdot D$$

(19)

In Equations (18) and (19), the variable t stands for the present iteration count. A and C are vectors of coefficients. X_p and X signify the position vectors of the prey and the grey wolf, respectively.

$$A=2a\cdot r_1-a$$

(20)

$$C=2\cdot r_2$$

(21)

In Equations (20) and (21), the convergence factor, denoted as a, undergoes a linear reduction from 2 to 0 across multiple iterations; r₁ and r₂ are random vectors with components uniformly distributed in the interval [0, 1].

$$\left\{\begin{array}{l}{D}_{\alpha }=|C_1\cdot X_{\alpha }-X|\hfill \\ D_{\beta }=|C_2\cdot X_{\beta }-X|\hfill \\ D_{\delta }=|C_3\cdot X_{\delta }-X|\hfill \end{array}\right.$$

(22)

In Equation (22), D_α, D_β, and D_δ signify the distances between the α, β, and δ wolves and their subordinate counterparts, respectively. Meanwhile, X_α, X_β, and X_δ indicate the locations of the α, β, and δ wolves.

$$\left\{\begin{array}{l}{\overrightarrow{X}}_1={\overrightarrow{X}}_{\alpha }-{\overrightarrow{A}}_1\overrightarrow{D}\hfill \\ {\overrightarrow{X}}_2={\overrightarrow{X}}_{\beta }-{\overrightarrow{A}}_2\overrightarrow{D}\hfill \\ {\overrightarrow{X}}_3={\overrightarrow{X}}_{\delta }-{\overrightarrow{A}}_3\overrightarrow{D}\hfill \end{array}\right.$$

(23)

$$\overrightarrow{X}(t+1)={\displaystyle \frac{{\overrightarrow{X}}_1+{\overrightarrow{X}}_2+{\overrightarrow{X}}_3}{3}}$$

(24)

Within the Grey Wolf Optimization (GWO) algorithm, the pursuit and capture of prey are steered by the three most well-adapted dominant wolves (α, β, and δ). These dominant wolves then lead and guide the subordinate wolves (ω) throughout these processes. Specifically, Equation (21) defines the step size and direction for ω wolves moving toward the leaders, while Equation (22) determines their updated positions. Through dynamic parameter adjustments during iterations, the wolf population ultimately converges to optimal solutions.

The traditional GWO algorithm is primarily used for single-objective continuous optimization problems. To apply it to discrete job shop scheduling problems, this study proposes an IMOD-GWO. The key steps of IMOD-GWO include encoding, decoding, population initialization, individual updating, and Pareto ranking. The workflow of IMOD-GWO is illustrated in Figure 1. The algorithm features four key innovations: (1) Hybrid population initialization: Multiple strategies are combined to generate high-quality initial solutions. (2) Nonlinear worst solution avoidance: Inspired by Jaya, a nonlinear avoidance factor dynamically adjusts exploration to escape local optima. (3) Mutation operators: Introduced to enhance population diversity during iterations. (4) Distributed parallel computation: Three subpopulations cooperate to balance exploration and exploitation, preventing premature convergence.

3.1. Coding

This study adopts a three-layer encoding scheme based on operation sequence (OS), machine assignment (MA), and machining speed selection (SS) to represent solutions, where the total encoding length corresponds to the number of operations. The OS layer consists of workpiece indices, where each occurrence of a workpiece index i represents a specific operation of that workpiece. A feasible OS encoding requires that the number of occurrences of any workpiece i equals its total number of operations n_i. The MA layer encodes machine indices assigned to each operation in the order of ascending workpiece numbers, directly mapping each operation to its designated machine. Similarly, the SS layer encodes speed-level indices for each operation on its assigned machine, specifying the selected machining speed. For example, as shown in

Table 2. WHMPM-FJSP example.

Job	Operation	Machine
		M1	M2	M3	M4
J1	O11	(12, 6, 3, 2.4)	-	(12, 3, 4)	-
	O12	-	(2.6, 3.25, 13, 4.33)	(13, 3.25, 4.33)	-
J2	O21	-	(2.6, 3.25, 13, 4.33)	-	(2.6, 4.33, 3.25, 6.5)
	O22	(17, 8.5, 4.25, 3.4)	(3.4, 4.25, 17, 5.67)	-	(3.4, 5.67, 4.25, 8.5)
J3	O31	-	-	(17, 4.25, 5.67)	(3.4, 5.67, 4.25, 8.5)
	O32	(12, 6, 3, 2.4)	(2.4, 3, 12,4)	-	(2.4, 4, 3, 6)
	O33	-	(2.8, 4.67, 3.5, 7)	(14, 3.5, 4.67)	(2.8, 3.5, 14, 4.67)
J4	O41	(15, 7.5, 3.75, 3)	(3, 3.75, 15, 5)	-	(3, 5, 3.75, 7.5)
	O42	-	(3.6, 4.5, 18, 6)	(18, 4.5, 6)	(3.6, 6, 4.5, 9)
	O43	-	-	(16, 4, 5.33)	(3.2, 5.33, 4, 8)

, machine M₁ offers four speed levels (12, 6, 3, 2.4) for processing operation O₁₁ of workpiece J₁. The resulting encoding in Figure 2 demonstrates an OS sequence of (O₂₁, O₁₁, O₃₁, O₃₂, O₄₁, O₁₂, O₂₂, O₄₂, O₃₃, O₄₃), with the MA layer indicating that O₁₂ is assigned to M₃ and O₂₂ to M₄, while the SS layer specifies speed level 2 (S₃₂) for O₁₂ on M₃ and speed level 1 (S₃₂) for O₂₂ on M₄. This three-layer encoding effectively integrates operation sequencing, machine assignment, and speed selection, providing a structured and easily understandable solution representation method for subsequent decoding and optimization processes.

3.2. Decoding

The decoding process is a crucial step in converting the encoded solution into an executable scheduling plan. This study uses operating system-based encoding to sequence the operations while simultaneously utilizing the MA layer and SS layer to determine the processing equipment and speed level for each operation. To accommodate machine preventive maintenance constraints, all operations must ensure that maintenance tasks are completed within their predefined time windows. The specific decoding rules are as follows:

(1)

First operation on a newly used machine. If operation O_i1 (the first operation of workpiece J_i) is assigned to machine M_l and M_l is used for the first time, O_i1 starts processing immediately on M_l;

(2)

Non-first operation on a newly used machine. If operation O_ij (not the first operation of workpiece J_i) is assigned to machine M_l and M_l is newly used, O_ij is transported to M_l immediately after its predecessor O_ij−1 completes and then begins processing on M_l;

(3)

First operation on a reused machine. If operation O_i1 (the first operation of workpiece J_i) is assigned to machine M_l, which has been previously used in the schedule, O_i1 must wait until M_l finishes its prior tasks before starting;

(4)

Non-first operation on a reused machine. If operation O_ij (not the first operation of workpiece J_i) is assigned to machine M_l, which has been used before, O_ij is transported to M_l after its predecessor O_ij−1 completes and must wait until M_l finishes its prior operations before starting processing.

This decoding method rigorously enforces preventive maintenance constraints, ensuring that maintenance tasks are initiated within their time windows (e.g., ET_kq ≤ mst_kq ≤ LT_kq). The generated schedules satisfy both processing sequence requirements and machine reliability criteria, significantly enhancing feasibility and practicality. To ensure machine preventive maintenance during processing operations, the following scenarios are addressed:

(1)

The first scenario for determining machine preventive maintenance time windows during processing operations is illustrated in Figure 3.

$$\begin{array}{l}CT⩽C_{h,g};\\ S_{h,g}-C_{i,j}⩾MT.\end{array}$$

(25)

In this scenario, operations O_ij and O_hg are processed on machine M_l, with their start and completion times satisfying Constraint (25). The preventive maintenance of machine M_l begins at time C_ij, which represents the finish time of operation O_ij.

(2)

The second scenario for determining machine preventive maintenance time windows during processing operations is illustrated in Figure 4.

$$\begin{array}{l}CT⩽C_{x,y};\\ S_{h,g}-C_{i,j}⩽MT;\\ C_{h,g}⩽CT.\end{array}$$

(26)

In this scenario, operations O_ij, O_hg, and O_xy are processed on machine M_l, with their start and completion times satisfying Constraint (26). The preventive maintenance of machine M_l begins at time C_hg, which represents the finish time of operation O_hg.

(3)

The third scenario for determining machine preventive maintenance time windows during processing operations is illustrated in Figure 5.

$$C_{i,j}⩽CT$$

(27)

In this scenario, operation O_ij is processed on machine M_l, and O_ij is the final operation assigned to M_l. The completion time of O_ij satisfies Constraint (27), and the preventive maintenance of machine M_l begins at time C_ij, which represents the finish time of operation O_ij.

(4)

The fourth scenario for determining machine preventive maintenance time windows during processing operations is illustrated in Figure 6.

$$C_{i,j}⩽ST$$

(28)

In this scenario, operation O_ij is processed on machine M_l, and O_ij is the final operation assigned to M_l. The completion time of O_ij satisfies Constraint (28), and the preventive maintenance of machine M_l begins at time ST, which represents the scheduled earliest start time of maintenance.

3.3. Population Initialization

An initial population of superior quality is of great significance in enhancing the convergence rate and optimization effectiveness of the algorithm. In relation to the optimization goals of minimizing the maximum completion time and the overall energy consumption investigated in this research, the population initialization approach put forward by Zhang et al. is utilized to create the initial population [32]. The article covers random selection, global selection, and local selection. In random selection, the OS, MA, and SS are generated randomly within the feasible solution space under technological constraints to enhance population diversity; global selection selects machines and speed levels based on shop-floor-level global information, such as machine workload balance, processing efficiency, and energy consumption characteristics, aiming to construct individuals with superior global performance, shorter makespan, and lower total energy consumption; local selection focuses on individual operations by preferentially selecting machines and speed levels that minimize local processing time or energy consumption, emphasizing the exploitation of local optimal solutions.

In this method, the OS layer uses random encoding, while the MA layer is created by integrating three different strategies: global selection, local selection, and random selection. However, this study improves upon the original method by introducing the selection of machine speed levels. Specifically, the operating system layer is still generated randomly. The machine allocation layer and the SS layer are jointly generated using a hybrid strategy of global selection, local selection, and random selection. This refined approach not only guarantees the variety of the initial population but also efficiently integrates features specific to the problem, offering top-notch initial solutions for the subsequent optimization procedures. As a result, it substantially boosts the overall efficiency of the algorithm.

3.4. Individual Update

The individual update mechanisms described in Equations (18)–(24) are suitable for continuous optimization challenges. However, they cannot be directly applied to discrete optimization problems. To achieve effective individual updates, this study proposes the following customized update strategies:

(1)

Select the individual within the population that has the worst fitness, denoted as X_worst. Leveraging the “avoiding worst solutions” principle from the Jaya, X_i wolf is steered away from X_worst wolf by performing reverse crossover between X_i and X_worst using POX and TPX to generate new individuals. Under this method, the update mechanisms for the OS, MA, and SS layers are illustrated in Figure 7, Figure 8 and Figure 9, respectively.

The reverse crossover operation for the OS layer involves the following steps: First, the workpieces are partitioned into two groups in a random manner, Job₁ and Job₂. The workpiece indices of X_i corresponding to positions within Job₂ are retained. Meanwhile, the remaining positions are filled in reverse order by utilizing the workpiece indices of X_worst from Job₁. This method ensures precedence constraints are maintained while introducing diversity into the operation sequence through reverse crossover.

For the MA layer reverse crossover, the machine assignments at corresponding positions of X_i and X_worst are compared. If they are identical, a different feasible machine (excluding the one assigned by X_worst) is randomly selected; if only one feasible machine exists, X_worst’s assignment is retained. If they are different, X_i’s machine assignment is preserved. This mechanism enhances diversity while ensuring solution feasibility and alignment with machine capabilities.

For the SS layer reverse crossover, the speed levels at corresponding positions of X_i and X_worst are compared. If they are identical, a different feasible speed level (excluding the one selected by X_worst) is randomly chosen; if the machine has only one available speed level, X_worst’s selection is retained. If they are different, X_i’s speed level is preserved. This approach ensures diversity in speed selection while maintaining energy efficiency and adhering to machine-specific speed constraints.

This method enhances the GWO framework by incorporating the “worst solution avoidance” strategy from the Jaya algorithm, thereby promoting population diversity during the early iterations. However, to ensure convergence in later stages, a nonlinear worst solution avoidance factor is introduced to dynamically regulate the avoidance intensity. The factor is defined as Equation (29).

$$a=-a_{\max }\tanh ((4/ite{r}_{\max })iter-4)$$

(29)

In Equation (29), a_max denotes the maximum probability of the avoidance factor, iter_max denotes the overall quantity of iterations, and iter indicates the present iteration.

Figure 10 illustrates the variation curve of the nonlinear worst solution avoidance factor. During the initial iterations, the slope of the nonlinear avoidance factor curve is small, with a higher avoidance factor value, indicating that the algorithm prioritizes global exploration by maintaining population diversity and searching broadly across the solution space for an extended period. In later iterations, the curve exhibits a steep slope and a lower avoidance factor value, signifying a shift to local exploitation, where the algorithm rapidly converges within a refined search region.

(2)

Select the fittest solutions (α, β, and δ wolves), then randomly choose one of the three fittest solutions (α, β, or δ wolf) as X_best. Guide $X_i^1$ wolf toward X_best using POX and TPX to generate new individuals $X_i^2$ and $X_i^3$. The update mechanisms for the OS, MA, and SS layers are detailed in Figure 11, Figure 12 and Figure 13, respectively.

(3)

The fittest individual $X_i^{\prime }$ is selected from $X_i^2$ and $X_i^3$. A mutation operation is then applied to $X_i^{\prime }$ with a predefined probability, generating a new individual $X_j^{\prime }$. The mutation mechanisms for the OS and MA/SS layers are illustrated in Figure 14 and Figure 15, respectively.

In the OS layer’s mutation operation, two positions within individual $X_i^{\prime }$ are randomly chosen. Subsequently, the encoding sequence between these two selected positions is reversed.

For the MA and SS layers, a random multi-point mutation is applied to individual $X_i^{\prime }$. The mutation selects machines or speed levels with the minimum processing time for the corresponding operations. If multiple machines or speed levels have the same minimum processing time, one is randomly chosen.

3.5. Pareto Sort

In the context of a combinatorial optimization problem featuring n minimization goals, a solution x is considered to outperform another solution y (represented as x < y) when the subsequent conditions are satisfied:

(1)

$f_j(x)⩽f_j(y)$ for all objectives j ∈ {1,2,…,n};

(2)

$f_j(x)<f_j(y)$ for at least one objective j ∈ {1,2,…,n}.

f_j(x) and f_j(y), respectively, denote the values of the j-th objective function for solutions x and y. A solution is referred to as non-dominated when it is not dominated by any other solution in the population. The non-dominated sorting method is a useful way to establish dominance relations in multi-objective optimization. It categorizes all solutions in the population into multiple non-dominated ranks (fronts). Solutions of a higher level have superiority over at least one solution in lower levels. Conversely, solutions of a lower level are outperformed by at least one solution in higher levels. Solutions that belong to the same level do not dominate one another (no solution in the rank dominates another).

3.6. Distributed Computing Framework

This study adopts a tri-population parallel cooperation strategy, in which three independent IMOD-GWO optimize their respective populations simultaneously. The strategy involves five key steps: initialization, independent evolution, periodic migration, integration and refinement, and termination. During the initialization phase, each subpopulation generates individuals that satisfy technological constraints while taking into account the conditions specific to the problem to ensure the quality of the initial solutions and maintain population diversity.

Subsequently, each subpopulation undergoes independent evolution according to the IMOD-GWO rules, including position updating, leader selection, and adaptive parameter adjustment, allowing exploration of different regions of the solution space and effectively preventing premature convergence. After a predefined number of iterations, periodic migration is performed among subpopulations, in which selected individuals (high-quality) are exchanged. The migrated individuals are then integrated into the recipient populations for integration and refinement, replacing low-quality individuals and expanding population size, followed by continued iterative optimization to improve overall solution quality.

The evolution and migration process continues until the termination condition is met, such as reaching the maximum number of iterations or achieving a target fitness threshold. This design combines the computational efficiency of parallel processing with the global search advantages of multi-population cooperation. Independent evolution of subpopulations prevents premature convergence, while periodic migration promotes the propagation of high-quality solutions, thereby enhancing global exploration capability and convergence speed.

By preserving the simplicity of the standard IMOD-GWO, this strategy significantly improves optimization performance through intelligent inter-population interactions. Moreover, it can be easily extended to distributed computing environments for large-scale optimization tasks, making it particularly suitable for complex multimodal optimization problems.

3.7. Optimal Scheme Selection Method

In the multi-objective optimization problems explored in this study, conflicting objectives such as minimizing completion time and total energy consumption often make it impossible to find a single optimal solution. Instead, a set of non-dominated solutions forms the Pareto front. Among methods for choosing a final solution from the Pareto front, the normalized weighted sum method is widely adopted in production scheduling and other multi-criteria decision-making domains due to its simplicity and efficiency. This approach transforms multi-objective optimization into a single-objective problem by normalizing and weighting individual objectives. For a problem with n objectives, the mathematical formulation is defined as

$$U(x)={\displaystyle \sum _{j=1}^{\mathrm{n}}{w}_j{f}_j^{\prime }(x)}$$

(30)

In Equation (30), w_j represents the weight coefficients for each normalized objective $f_j^{\prime }\left(x\right)$. The normalized value of the j-th objective function f_j(x), denoted as $f_j^{\prime }\left(x\right)$, is calculated by Equation (30).

$$f_j^{\prime }(x)={\displaystyle \frac{f_j(x)-f_{j,\min }}{f_{j,\max }-f_{j,\min }}}$$

(31)

In Equation (31), f_j,max and f_j,min represent the maximum value and minimum value of the objective f_j, respectively.

This paper uses the Fuzzy Analytic Hierarchy Process (FAHP) to determine the weight values of each objective. FAHP is a multi-attribute decision-making method that was developed by introducing fuzzy mathematics into the traditional Analytic Hierarchy Process (AHP). Its core objective is to address the problem of imprecise evaluation caused by the fuzziness and uncertainty of human judgment in the decision-making process, thus making the resulting judgments more accurate and reliable.

The FAHP method, based on fuzzy consistency matrices, mainly involves the following calculation steps:

(1)

Establish the fuzzy complementary judgment matrix A.

Experts performed pairwise comparisons of the indicators, resulting in a fuzzy judgment matrix A = (a_ij)_n×n, where the fuzzy values a_ij in the matrix are based on a 0.1–0.9 scale. The scale and definition of the fuzzy value a_ij are shown in

Table 3. Scale and definition.

Scale	Definition	Explanation
0.5	Equally important	The two factors are equally important when compared to each other
0.6	Slightly important	Comparing the two factors, one factor is slightly more important than the other
0.7	Clearly important	Comparing the two factors, one factor is clearly more important than the other
0.8	Very important	Comparing the two factors, one factor is far more important than the other
0.9	Extremely important	Comparing the two factors, one factor is significantly more important than the other
0.1, 0.2, 0.3, 0.4	Counter-comparison	If factor ai is compared with factor aj, it results in judgment rij, Then, the judgment obtained by comparing factor aj with factor ai is rji = 1 − rij

.

(2)

Calculate the weight vector W.

$$W_i={\displaystyle \frac{{\displaystyle \sum _{j=1}^n{a}_{ij}}+\frac{n}{2}-1}{n(n-1)}},i=1,2,\cdots ,n$$

(32)

Among them, i and j represent row coordinates and column coordinates, respectively.

(3)

Calculate the feature matrix W*.

Let W = (W₁,W₂,···,W_n)^T be the weight vector of the fuzzy judgment matrix A, and the sum of W_i in W is 1. Let

$$W_{ij}={\displaystyle \frac{W_i}{W_i+W_j}}(\forall i,j=1,2,\cdots ,n)$$

(33)

After calculating the data at each position of the n × n matrix using the values in the weight vector W, an n-order matrix is obtained:

$$W^{*}={\left(W_{\mathrm{ij}}\right)}_{\mathrm{n}\times \mathrm{n}}$$

(34)

W* is called the characteristic matrix of judgment matrix A.

(4)

Calculate compatibility index I.

Let matrices A = (a_ij)_n×n and B = (b_ij)_n×n both be fuzzy judgment matrices, denoted as

$$I\left(A,B\right)={\displaystyle \frac{1}{n^2}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n|a_{ij}{+\mathrm{b}}_{ij}-1|}}$$

(35)

Compatibility index for A and B.

(5)

Consistency check.

Calculate the compatibility index I(A,W*) between the fuzzy judgment matrix A and its feature matrix W. If I(A,W*) ≤ α, the consistency test is considered to have passed. The smaller the value of α, the higher the requirement for consistency in the fuzzy judgment matrix by the decision maker. Generally, α = 0.1 can be taken.

In solving multi-objective decision-making problems, decision-makers typically evaluate the relative importance of each objective based on practical requirements and assign corresponding weights to reflect their priorities. To determine the best solution from multi-objective optimization results, this study proposes a two-stage decision-making method integrating Pareto front filtering and normalized weighted summation. Specifically, firstly, based on the Pareto front elimination method, solutions with unsatisfactory performance are removed from the Pareto solution set, narrowing down candidate solutions. Secondly, for the remaining Pareto solutions, the normalized weighted sum method is applied to compute a comprehensive score by normalizing objectives and summing their weighted values, thereby selecting the solution with optimal balanced performance. This method reduces decision-making complexity while ensuring the final solution aligns with practical requirements and achieves a balanced trade-off between conflicting objectives. By combining elimination and aggregation in two stages, it provides an efficient and practical framework for multi-objective optimization in industrial scheduling.

4. Experiment

4.1. Experimental Data

Since there are no existing test instances suitable for the problem under study, this paper extends the classic Brandimarte instance to generate 15 test instances. The scale of these test instances is detailed in

Table 4. Scale of test cases.

Name of Experiment	Quantity of Workpieces	Number of Machines
MK01	10	6
MK02	10	6
MK03	15	8
MK04	15	8
MK05	15	4
MK06	10	15
MK07	20	5
MK08	20	10
MK09	20	10
MK10	20	15
MK11	30	5
MK12	30	10
MK13	30	10
MK14	30	15
MK15	30	15

.

As shown in

Table 5. Ranges of parameters.

Symbol	Range
fek	1
te	2
ttlk	[1, 5]
Sijkr	[1, 5]
pek	3 + Sijkr
ST	[0, 30]
CT	[5, 40]
MT	[3, 10]

, the table defines the parameter ranges for generating each experiment. In this table, each range indicates that the value of the corresponding parameter will be randomly generated within the given interval. The experiments assume that preventive maintenance tasks are required for 30% of the total machines.

4.2. Parameter Experimentation

Algorithm parameters significantly affect performance; therefore, it is necessary to select appropriate parameter configurations for the IMOD-GWO algorithm. To this end, this study employs a five-level orthogonal experimental design method.

Table 6. Source data of orthogonal experiment.

Level	Factor
	NP	RIPR	MPAWS	MR
1	50	20%	0.6	0.01
2	100	30%	0.7	0.05
3	150	40%	0.8	0.1
4	200	50%	0.9	0.2
5	300	60%	1.0	0.3

lists the factors studied and their corresponding levels. The factors in Table 6 include the following parameters: population size (NP), Random Initialization Population Ratio (RIPR), Maximum Probability of Avoiding Worst Solutions (MPAWS), and Mutation Rate (MR). Based on the parameter settings in Table 6, an orthogonal experimental plan comprising 25 parameter combinations is designed, with detailed results presented in

Table 7. Result of parameter experimentation.

Index	Factor				Average Response Value
	NP	RIPR	MPAWS	MR
1	50	20%	0.6	0.01	0.2966
2	50	30%	0.7	0.05	0.0291
3	50	40%	0.8	0.1	0.0000
4	50	50%	0.9	0.2	0.0000
5	50	60%	1.0	0.3	0.2446
6	100	20%	0.7	0.2	0.4316
7	100	30%	0.8	0.3	0.0000
8	100	40%	0.9	0.01	0.0000
9	100	50%	1.0	0.05	0.0000
10	100	60%	0.6	0.1	0.6780
11	150	20%	0.8	0.05	0.0000
12	150	30%	0.9	0.1	0.0000
13	150	40%	1.0	0.2	0.0000
14	150	50%	0.6	0.3	0.1105
15	150	60%	0.7	0.01	0.1016
16	200	20%	0.9	0.1	0.0000
17	200	30%	1.0	0.2	0.0000
18	200	40%	0.6	0.3	0.0000
19	200	50%	0.7	0.01	0.0000
20	200	60%	0.8	0.05	0.2649
21	300	20%	1.0	0.3	0.0000
22	300	30%	0.6	0.01	0.5283
23	300	40%	0.7	0.05	0.0166
24	300	50%	0.8	0.1	0.0000
25	300	60%	0.9	0.2	0.0000

.

To validate the effectiveness and superiority of the IMOD-GWO, the parameter experiments are conducted using the medium-scale benchmark instance Mk09. Each parameter group in Table 7 is executed 20 times with a 10 min runtime per iteration. All programs are implemented in Python 3.12 on a hardware platform equipped with 32 GB of RAM and an Intel Core™ i5-12400F CPU (2.50 GHz). The Intel Core™ i5-12400F processor was sourced from Intel Corporation, with the manufacturing location in the United States. The 32 GB of RAM is sourced from Kingbank Silver Knight, DDR4 3200, and the manufacturing location is Guangdong Province, China.

After each algorithm run, a set of reference solutions that are non-dominated is produced. Subsequently, the non-dominated solution subsets associated with all 25 parameter groups are combined into a comprehensive non-dominated solution set. The evaluation of the performance for each parameter group is evaluated using the average response value (AvgRV), as shown in Table 6. A greater AvgRV implies that the parameter combination has better performance. The calculation of the AvgRV is carried out in the following manner:

$$\mathrm{Avg}R{V}_{\mu }=\left({\displaystyle \sum _{\theta =1}^{20}N}subnd{s}_{\mu }(\theta )/Nrnds(\theta )\right)/20$$

(36)

In Equation (36), AvgRV_μ denotes the average response value for the μ-th parameter group. Nrnds(θ) denotes the quantity of elements within the non-dominated reference set that is derived from the θ-th run of the IMOD-GWO, while Nsubnds_μ(θ) indicates the contribution (number of solutions) of the μ-th parameter group to the reference set during the θ-th run.

Based on the data in Table 7,

Table 8. Performance at each level of each factor.

Level	Factor
	NP	RIPR	MPAWS	MR
1	0.1140	0.1456	0.3226	0.1853
2	0.2219	0.1114	0.1157	0.0621
3	0.0424	0.0033	0.0529	0.1356
4	0.0529	0.0221	0.0000	0.0863
5	0.1619	0.2578	0.0489	0.0710
Range	0.1795	0.2545	0.3226	0.1232

summarizes the algorithm’s performance at each factor level, and the corresponding trends are shown in Figure 16. The results indicate that the optimal parameter configuration for the algorithm is PSet_1 = {NP = 100, RIPR = 60%, MPAWS = 0.6, MR = 0.01}. However, the results in Table 7 show that PSet_2 (Group 10 parameters) achieves the highest performance. Furthermore, as demonstrated in Table 8, the maximum probability of MPAWS exhibits the most significant impact on algorithm performance, indicating that the algorithm is most sensitive to variations in this parameter. Thus, MPAWS requires prioritized attention during parameter optimization.

To refine the parameter configuration for the IMOD-GWO, supplementary experiments were conducted by incorporating PSet_1 into the original 25 parameter groups. The results reveal that PSet_2 (AvgRV = 0.6780) significantly outperforms PSet_1 (AvgRV = 0.3348). Based on these findings, we recommend adopting PSet_2 as the standard parameter configuration for the IMOD-GWO.

In this study, coverage (C), inverted generational distance (IGD), and error rate (ER) are used to systematically evaluate the effectiveness and superiority of the improved algorithm.

(1)

Coverage (C): The coverage metric measures the proportion of solutions in one algorithm’s Pareto set dominated by solutions from another algorithm’s Pareto set.

$$C(P{F}_1,P{F}_2)={\displaystyle \frac{\left|\right\{y\in P{F}_2|\exists x\in P{F}_1:x>y\}|}{\left|P{F}_2\right|}}$$

(37)

In Equation (37), PF₁ and PF₂ denote the non-dominated solution sets generated by two algorithms. A higher C(PF₁, PF₂) indicates that PF₁ is superior. C(PF₁, PF₂) = 1 indicates that all solutions in PF₂ are dominated by some solutions in PF₁. C(PF₁, PF₂) = 0 indicates that no solution in PF₁ dominates any solution in PF₂.

(2)

Inverted generational distance (IGD): The core mechanism of IGD index is to realize the comprehensive quantitative evaluation of the convergence and distribution characteristics of the solution set by calculating the spatial distance between the approximate front (PF) and the reference front (PF*).

$$IGD(P{F}_1,P{F}^{*})={\displaystyle \frac{1}{\left|P{F}^{*}\right|}}{\displaystyle \sum _{x\in P{F}^{*}}\underset{y\in P{F}_1}{\min }dis(x,y)}$$

(38)

In Equation (38), PF* denotes the non-dominated solution set. |PF*| denotes the number of solutions in PF*. dis(x,y) denotes the Euclidean distance between solutions x and y in the multi-objective space. A smaller IGD(PF₁, PF*) indicates better performance of PF₁. In this research, PF* for each test case is formed by aggregating non-dominated solution sets obtained from 20 independent runs of each algorithm.

(3)

Error rate (ER): The error rate reflects how closely a Pareto set approximates the true Pareto front.

$$IGD(P{F}_1,P{F}^{*})={\displaystyle \frac{1}{\left|P{F}^{*}\right|}}{\displaystyle \sum _{x\in P{F}^{*}}\underset{y\in P{F}_1}{\min }dis(x,y)}$$

(39)

$$e_a\left\{\begin{array}{l}0 \mathrm{x} \mathrm{is} \mathrm{a} \mathrm{member} \mathrm{of} P{F}^{*}\hfill \\ 1 \mathrm{x} \mathrm{not} \mathrm{is} \mathrm{a} \mathrm{member} \mathrm{of} P{F}^{*}\hfill \end{array}\right.$$

(40)

A smaller ER value indicates that the algorithm provides more non-dominated solutions aligned with PF*, with a lower proportion of “useless” solutions (not in PF*).

This article assumes the existence of three experts who provide fuzzy complementary judgment matrices regarding the maximum completion time and energy consumption, namely A₁, A₂, and A₃.

$$A_1=\left(\begin{array}{cc}0.5000& 0.6000\\ 0.4000& 0.5000\end{array}\right)$$

$$A_2=\left(\begin{array}{cc}0.5000& 0.7000\\ 0.3000& 0.5000\end{array}\right)$$

$$A_3=\left(\begin{array}{cc}0.5000& 0.4000\\ 0.6000& 0.5000\end{array}\right)$$

According to Equation (32), [W₁,W₂]₁ = [0.5500,0.4500], [W₁,W₂]₂ = [0.6000,0.4000], and [W₁,W₂]₃ = [0.4500,0.5500].

Next, obtain $W_1^{\mathrm{*}}$, $W_2^{\mathrm{*}}$, and $W_3^{\mathrm{*}}$ through Equations (33) and (34).

$$W_1^{*}=\left(\begin{array}{cc}0.0000& 0.0500\\ 0.0500& 0.0000\end{array}\right)$$

$$W_2^{*}=\left(\begin{array}{cc}0.0000& 0.1000\\ 0.1000& 0.0000\end{array}\right)$$

$$W_3^{*}=\left(\begin{array}{cc}0.0000& 0.0500\\ 0.0500& 0.0000\end{array}\right)$$

Finally, by using Equation (35), I(A₁,$W_1^{\mathrm{*}}$) = 0.0250, I(A₂,$W_2^{\mathrm{*}}$) = 0.2000, and I(A₃,$W_3^{\mathrm{*}}$) = 0.0250 are obtained. The obtained I(A,W*) values are all less than α = 0.1, proving that all three sets of fuzzy judgment matrices have passed the consistency test. The final weight is calculated by taking the arithmetic mean of the weights of the three groups to obtain W₁ = 0.5333 and W₂ = 0.4667.

4.3. Ablation Experiment

To verify the effectiveness of the improved strategies in the IMOD-GWO algorithm, this paper systematically combined different permutations of these strategies, resulting in the 16 algorithm variants listed in

Table 9. Sixteen algorithms for different strategies.

Algorithm Name	Strategy
	INIT	JAYA	MUT	DIST
GWO	×	×	×	×
GWO-I	√	×	×	×
GWO-J	×	√	×	×
GWO-M	×	×	√	×
GWO-D	×	×	×	√
GWO-IJ	√	√	×	×
GWO-IM	√	×	√	×
GWO-ID	√	×	×	√
GWO-JM	×	√	√	×
GWO-JD	×	√	×	√
GWO-MD	×	×	√	√
GWO-IJM	√	√	√	×
GWO-IJD	√	√	×	√
GWO-IMD	√	×	√	√
GWO-JMD	×	√	√	√
IMOD-GWO	√	√	√	√

Note: √ means there is such a strategy; × means there is no such strategy.

. The strategies include the following: INIT is hybrid population initialization, JAYA is Jaya-based worst solution avoidance strategy, MUT is a mutation strategy, and DIST is distributed computing strategy.

In this study, PSet_2 = {NP = 100, RIPR = 60%, MPAWS = 0.6, MR = 0.1} will be set as the parameters of all algorithms. The 16 algorithms are run 20 times on the medium-scale MK09 example, each run for 10 min, and the union of each generation of solutions of each algorithm is obtained. The C value of each algorithm and IMOD-GWO, IGD, and ER values of each algorithm are calculated. The index C1 is C(X, IMOD-GWO) of the corresponding algorithm X and IMOD-GWO, C2 is C(IMOD-GWO, X), the index IGD is the reverse generation distance of the corresponding algorithm X and IMOD-GWO, ER is the error rate of the union of each generation of solutions of each algorithm to the total solution set, and the IGD and ER values in bold are the best among the 16 algorithms.

As can be seen from

Table 10. Results of the ablation experiment.

Algorithm Name	Metrics
	C1	C2	ER	IGD
GWO	0.0000	1.0000	1.0000	42.8488
GWO-I	0.0000	1.0000	1.0000	19.0705
GWO-J	0.0000	1.0000	1.0000	11.5339
GWO-M	0.0000	1.0000	1.0000	11.4326
GWO-D	0.0000	1.0000	1.0000	38.9878
GWO-IJ	0.0000	0.8000	1.0000	14.1590
GWO-IM	0.2142	0.7368	0.7619	7.9915
GWO-ID	0.0000	1.0000	1.0000	19.6623
GWO-JM	0.0000	1.0000	1.0000	6.2092
GWO-JD	0.0000	1.0000	1.0000	15.1345
GWO-MD	0.0000	1.0000	1.0000	10.1229
GWO-IJM	0.0000	1.0000	1.0000	7.1687
GWO-IJD	0.2142	0.3333	0.7619	11.1771
GWO-IMD	0.2142	0.6470	1.0000	6.2330
GWO-JMD	0.0000	1.0000	1.0000	9.1336
IMOD-GWO	-	-	0.4761	3.1127

Note: Bold indicates the optimal value of each indicator.

, under the extended MK09 example, most algorithms yielded a C1 value of 0, and most algorithms yielded a C2 value of 1. This indicates that the solution obtained by the IMOD-GWO algorithm is superior. A dominance relationship exists between the GWO-IM, GWO-IJD, and GWO-IMD algorithms, as well as the IMOD-GWO algorithm, but the IMOD-GWO algorithm dominates the others to a greater extent. This means that the Pareto front solutions of IMOD-GWO can cover the Pareto front solutions of the other 15 algorithms, and its Pareto front is superior to the Pareto fronts obtained by the other 15 algorithms. The performance comparison of different GWO variants and component combinations demonstrates the effectiveness of the proposed IMOD-GWO framework. Single-component enhancements (I, J, M, D) significantly reduce IGD values compared to the GWO, indicating improved Pareto front approximation. Combining two or three components further decreases IGD, although some combinations show minor trade-offs in coverage indicators (C1/C2). Among all tested variants, IMOD-GWO achieves the lowest IGD (3.1127) and a substantially reduced error ratio (ER = 0.4761), highlighting its superior ability to balance convergence and diversity. These results suggest that while individual components contribute to performance, the integrated design and coordinated operation of multiple components, including the tri-population strategy, is the key factor driving overall algorithmic effectiveness under identical time constraints, confirming the advantage of the proposed approach in solving the WHMPM-FJSP. The ER and IGD of IMOD-GWO are smaller than those of other algorithms, which means that IMOD-GWO’s Pareto frontier is closer to the real Pareto frontier.

4.4. Comparison Experiments

To verify the performance of the IMOD-GWO algorithm in the FJSP considering workpiece handling and machine preventive maintenance, an extended Brandimarte example was selected for the experiment, and the experimental results were compared with those of the NSGA-II algorithm, the multi-objective Jaya algorithm, and the MOWPA [33] algorithm. All three comparison algorithms were executed using their default settings, without a multi-population framework, and each algorithm was run for 10 min. The comparison results are presented in

Table 11. Results of comparison experiments.

Experiment	NSGA-II				MO-Jaya				MOWPA				IMOD-GWO
	C1	C2	IGD	ER	C1	C2	IGD	ER	C1	C2	IGD	ER	IGD	ER
MK01	0.0000	1.0000	1.7528	1.0000	0.0000	1.0000	1.0562	1.0000	0.0000	1.0000	4.4054	1.0000	0.0000	0.0000
MK02	0.0000	0.7777	1.7644	0.8750	0.1176	0.6666	0.9140	0.7500	0.0000	1.0000	9.2295	1.0000	0.5858	0.3750
MK03	0.0000	1.0000	1.5871	1.0000	0.0000	1.0000	1.8979	1.0000	0.0000	1.0000	3.0573	1.0000	0.0000	0.0000
MK04	0.0833	0.5000	3.2465	0.9166	0.0000	1.0000	1.8183	1.0000	0.0000	1.0000	4.9470	1.0000	0.1303	0.0833
MK05	0.0000	1.0000	2.0046	1.0000	0.0000	1.0000	0.9235	1.0000	0.0000	1.0000	4.8420	1.0000	0.0000	0.0000
MK06	0.0000	1.0000	1.9491	1.0000	0.0000	1.0000	5.2117	1.0000	0.0000	1.0000	6.4198	1.0000	0.0000	0.0000
MK07	0.0000	1.0000	3.2039	1.0000	0.0000	1.0000	1.8975	1.0000	0.0000	1.0000	8.9801	1.0000	0.0000	0.0000
MK08	0.0000	1.0000	0.2917	1.0000	0.0000	1.0000	0.7388	1.0000	0.0000	1.0000	0.6332	1.0000	0.0000	0.0000
MK09	0.0000	1.0000	6.0051	1.0000	0.0000	1.0000	13.0815	1.0000	0.0000	1.0000	11.9028	1.0000	0.0000	0.0000
MK10	0.0000	1.0000	1.1872	1.0000	0.0000	1.0000	3.6877	1.0000	0.0000	1.0000	4.0314	1.0000	0.0000	0.0000
MK11	0.0000	1.0000	0.4697	1.0000	0.0000	1.0000	0.7884	1.0000	0.0000	1.0000	0.7565	1.0000	0.0000	0.0000
MK12	0.0000	1.0000	0.2997	1.0000	0.0000	1.0000	0.5409	1.0000	0.0000	1.0000	0.5214	1.0000	0.0000	0.0000
MK13	0.0000	1.0000	3.1940	1.0000	0.0000	1.0000	9.5085	1.0000	0.0000	1.0000	9.6855	1.0000	0.0000	0.0000
MK14	0.0000	1.0000	7.2617	1.0000	0.0000	1.0000	13.6204	1.0000	0.0000	1.0000	12.1600	1.0000	0.0000	0.0000
MK15	0.0000	1.0000	2.2903	1.0000	0.0000	1.0000	10.3418	1.0000	0.0000	1.0000	7.2910	1.0000	0.0000	0.0000

Note: Bold indicates the optimal value of each indicator.

. Among them, the variable n stands for the quantity of workpieces, and the variable m stands for the number of machines. The set of non-dominated solutions for IGD is formed by combining the Pareto solutions obtained from four algorithms running 20 times on each example.

As is evident from Table 11, (1) the results of C(NSGA-II,IMOD-GWO), C(MO-Jaya,IMOD-GWO), and C(MOWPA,IMOD-GWO) in most of the examples are 0, and the results of C(IMOD-GWO,NSGA-II), C(IMOD-GWO,NSGA-II), and C(IMOD-GWO, MOWPA) in most of the examples are 1, which shows that the IMOD-GWO’s Pareto frontier is significantly better than the Pareto frontier obtained by the NSGA-II, the MO-Jaya and the MOWPA in quality. (2) The results of IGD(IMOD-GWO) in all examples are smaller than those of IGD(NSGA-II), IGD(MO-Jaya), and IGD(MOWPA), which shows that the IMOD-GWO’s Pareto frontier is closer to the real Pareto frontier. (3) The results of ER(IMOD-GWO) in all examples are smaller than those of ER(NSGA-II), ER(MO-Jaya), and ER(MOWPA), which shows that IMOD-GWO obtains fewer invalid solutions and obtains better solutions. (4) Comparing the results of C, IGD, and ER calculations, the IMOD-GWO performs more stably in the test case, which shows that the IMOD-GWO put forward in this research can effectively solve the scheduling problem of flexible job shops considering workpiece handling and machine pre-maintenance.

The main reasons why the IMOD-GWO is put forward in this research to obtain a better Pareto frontier are as follows: (1) High-quality initialization population. This paper uses global, local, and random selection strategies to generate encodings for the machine layer and processing speed layer. The process layer encoding is generated using a random selection strategy to optimize the maximum completion time. Simultaneously, the randomly generated individuals can increase the diversity of the population. (2) Improved individual update mechanism. The idea of “staying away from the worst solution” in Jaya is introduced. By adding the factor away from the worst solution and distributed computing, IMOD-GWO effectively avoids becoming trapped in the problem of local optimal solutions. In addition, a mutation factor is incorporated to ensure that the population maintains its variety during the iteration process, thereby further improving the algorithm’s global exploration capacity.

The calculation results of the three evaluation indicators C, IGD, and ER show that the IMOD-GWO designed in this study is able to acquire a more optimized scheduling scheme than the three comparison algorithms, NSGA-II, MO-Jaya, and MOWPA, when solving the scheduling problem of flexible job shops considering workpiece handling and machine pre-maintenance. To more intuitively compare the performance of various algorithms, this paper uses the extended case MK01 as an example and employs the fuzzy hierarchical analysis method to determine the weights of the maximum completion time and total energy consumption, which are 0.5333 and 0.4667, respectively. The machine pre-maintenance time is configured to 0 for the earliest time of machine 1, 5 for the latest time, and 7 for the maintenance time; the optimal scheduling scheme is selected from the NSGA-II’s Pareto frontiers, MO-Jaya’s Pareto frontiers, and IMOD-GWO’s Pareto frontiers at one time, and the Gantt charts shown in Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23 and Figure 24 are drawn. In the Gantt chart of machine and time, the label in the maintenance color block is machine pre-maintenance, and the label in other color blocks is workpiece–process; in the Gantt chart of workpiece and time, the label in the transfer color block is machine before handling–handling target machine, and the label in other color blocks is process–processing machine–processing speed. Comparing the Gantt chart, we can see that the comparison algorithm can obtain similar pre-maintenance schedules, but the completion period is significantly longer than the scheduling scheme obtained by IMOD-GWO in this research. The maximum completion time and total energy consumption of the optimal scheduling scheme of each algorithm are shown in Table 11, where the bold indicates the best among the four algorithms.

As can be seen from

Table 12. Optimal scheduling scheme of algorithm.

Experiment	NSGA-II		MO-Jaya		MOWPA		IMOD-GWO
	Cmax	Etotal	Cmax	Etotal	Cmax	Etotal	Cmax	Etotal
MK01	43.0000	1505.1000	42.2000	1482.0000	46.6000	1600.7999	37.1999	1480.9500
MK02	38.2000	1419.2000	35.4000	1404.7333	44.6000	1546.1666	35.8000	1399.7333
MK03	135.5999	4365.7500	105.7333	4549.6666	135.1000	4819.9000	91.2000	4267.4666
MK04	94.0000	2897.3499	78.3500	2733.8999	93.1000	2845.1000	83.0000	2654.1499
MK05	112.8000	2835.2499	94.4000	2834.4000	103.5500	3041.4000	91.5999	2814.4499
MK06	109.2500	4251.5999	122.1333	4723.5000	131.5333	5094.9500	102.1999	4171.1500
MK07	81.8000	2616.7499	71.5000	2580.8999	77.1000	2799.7000	67.6000	2580.1000
MK08	166.2500	6859.8500	187.1666	7345.0666	179.3166	7224.5000	166.2500	6772.6666
MK09	162.4000	7171.5333	180.5000	7977.1000	170.6500	7831.1500	141.6000	7033.7000
MK10	150.1000	6568.0000	137.6333	7352.0833	145.3500	7549.0000	122.6000	6522.7333
MK11	145.2000	4851.4666	150.6666	4993.4833	144.7833	4841.1499	144.3333	4810.9166
MK12	171.0000	6014.1999	171.0000	6220.6833	171.0000	6199.5166	171.0000	5921.0499
MK13	161.6000	6817.9666	132.1666	7244.3333	143.6000	7278.1666	113.3333	6728.1166
MK14	201.1999	9061.7166	182.2999	9660.8666	163.1000	9581.9833	126.6666	8887.6500
MK15	182.6000	8792.9666	159.1666	10007.9000	161.6333	9402.5333	121.9999	8618.5500

Note: Bold indicates the optimal value of each indicator.

, in most extended examples, the optimal scheduling scheme obtained by the IMOD-GWO algorithm is superior to the other three algorithms in terms of both maximum completion time and total energy consumption. However, in small-scale examples, the optimal scheme obtained by the IMOD-GWO algorithm is not entirely superior to the schemes obtained by the other three algorithms. In a small part of small-scale examples, the optimal solution obtained by the other three algorithms will be smaller than the optimal solution obtained by the IMOD-GWO in terms of the optimization goal of maximum completion time, but in medium- and large-scale examples, the optimal solution obtained by the IMOD-GWO is completely better than the optimal solution obtained by the other three algorithms. In all medium- and large-scale examples, the optimal solution obtained by the IMOD-GWO is better than the optimal solution obtained by the other three algorithms in terms of the two optimization goals of maximum completion time and total energy consumption. This shows that the IMOD-GWO is more efficient in solving the scheduling problem of flexible job shops, considering workpiece handling and machine pre-maintenance, especially when dealing with medium and large-scale problems, and the IMOD-GWO is capable of yielding superior solutions.

5. Conclusions

In this study, the traditional FJSP is extended by innovatively considering factors such as workpiece handling, machine preventive maintenance, and variable processing speeds. A multi-objective scheduling model is constructed with the optimization objectives of minimizing the maximum completion time and total energy consumption. In order to effectively solve this complex problem, an IMOD-GWO is proposed. IMOD-GWO has the following innovative features:

(1)

The multi-objective FJSP that considers workpiece handling, machine preventive maintenance, and processing speed selection is studied, and an applicable optimization model is constructed;

(2)

A hybrid population initialization approach employing multiple strategies is devised to enhance the quality and variety of the initial solution;

(3)

A nonlinear factor based on the Jaya is introduced to stay away from the worst solution, which enhances the algorithm’s ability to break free from the local optimum;

(4)

The mutation operator of the genetic algorithm is integrated to maintain population diversity;

(5)

Distributed computing is used to improve the efficiency of the algorithm’s solution.

To verify the effectiveness of the IMOD-GWO algorithm and its various components, this paper conducted 16 sets of ablation experiments by combining different components. Additionally, we performed 15 sets of comparative experiments with varying scales. Based on these experimental data, the following conclusions can be drawn:

(1)

The proposed multi-strategy population initialization method, genetic algorithm mutation operator, Jaya away from the worst solution strategy, and distributed computing significantly improved the overall optimization capabilities of the algorithm;

(2)

In the 16 groups of comparative experiments, the IMOD-GWO showed better performance than the other 15 comparative algorithms, which fully verified the collaborative effectiveness of the combination of various algorithm modules;

(3)

In 15 experiments of different scales, the IMOD-GWO algorithm showed significant advantages over NSGA-II, MO Jaya, and MOWPA algorithms, especially in dealing with large-scale complex WHMPM-FJSP problems in benchmark cases, demonstrating better solution quality and computational efficiency.

However, devising algorithms capable of solving real-time scheduling problems while taking into account more comprehensive scheduling is still the subject of further research in this paper. In the future, the subsequent matters will be considered:

(1)

Manufacturing processes in production enterprises rely on various energy sources, including traditional and renewable energy. It is worth noting that the price level, market demand, and supply stability of these energy sources will change significantly over time and with the change of seasons. Based on this reality, it is of great research value to explore the production scheduling optimization problem in the background of multiple energy supplies. This research direction is not only of academic significance, but also can provide new solutions for manufacturing enterprises to effectively control production costs;

(2)

Implement differentiated parameter strategies in the distributed optimization framework, optimize each sub-population independently by using different parameter configurations, and then realize information sharing and co-evolution by designing an effective population interaction mechanism. This parameter diversity strategy contributes to maintaining the population’s global exploration capacity and generating higher quality candidate solutions throughout the iterative procedure, thereby significantly improving the probability of the algorithm obtaining the optimal solution and the overall solution quality.