Research on Flexible Job Shop Scheduling Method for Agricultural Equipment Considering Multi-Resource Constraints

Abstract

The agricultural equipment market has the characteristics of rapid demand changes and high demand for machine models, etc., so multi-variety, small-batch, and customized production methods have become the mainstream of agricultural machinery enterprises. The flexible job shop scheduling problem (FJSP) in the context of agricultural machinery and equipment manufacturing is addressed, which involves multiple resources including machines, workers, and automated guided vehicles (AGVs). The aim is to optimize two objectives: makespan and the maximum continuous working hours of all workers. To tackle this complex problem, a Multi-Objective Discrete Grey Wolf Optimization (MODGWO) algorithm is proposed. The MODGWO algorithm integrates a hybrid initialization strategy and a multi-neighborhood local search to effectively balance the exploration and exploitation capabilities. An encoding/decoding method and a method for initializing a mixed population are introduced, which includes an operation sequence vector, machine selection vector, worker selection vector, and AGV selection vector. The solution-updating mechanism is also designed to be discrete. The performance of the MODGWO algorithm is evaluated through comprehensive experiments using an extended version of the classic Brandimarte test case by randomly adding worker and AGV information. The experimental results demonstrate that MODGWO achieves better performance in identifying high-quality solutions compared to other competitive algorithms, especially for medium- and large-scale cases. The proposed algorithm contributes to the research on flexible job shop scheduling under multi-resource constraints, providing a novel solution approach that comprehensively considers both workers and AGVs. The research findings have practical implications for improving production efficiency and balancing multiple objectives in agricultural machinery and equipment manufacturing enterprises.

1. Introduction

As the main manifestation of agricultural productivity, agricultural mechanization equipment is an important material foundation for ensuring food security in the context of reducing agricultural labor and achieving agricultural modernization in the context of complex crops. Agricultural mechanization equipment is an important material foundation for the development of modern agriculture. Promoting the development of agricultural mechanization equipment is an objective requirement for improving agricultural labor productivity, land output rate, and resource utilization rate. It is a practical need to support the development of agricultural mechanization, the transformation of agricultural development mode, and the quality and efficiency of agriculture. The production of agricultural mechanization equipment is greatly influenced by market demand, with fast updates and replacements of models and configurations. The demand is highly targeted, and different models need to be selected for different regions, environments, and operational needs [1]. At the same time, the agricultural operating environment varies greatly, and the busy farming hours also vary with climate change, resulting in rapid changes in market demand for various agricultural equipment and diverse demands for machine models. Due to the complexity of the market environment, compared with other mechanical equipment manufacturing industries, agricultural machinery and equipment enterprises have the characteristics of more model selection and unstable market demand cycles in production. Therefore, multi-variety, small-batch, and customized production methods have become the mainstream of agricultural machinery enterprises, and at present, in order to meet the diversified needs of the market, the production tasks have the characteristics of multi-variety and small-batch production [2].

In order to meet the diverse demands of the agricultural mechanization equipment production market and achieve flexible production goals of multiple varieties and small batches, manufacturing enterprises can adopt flexible production lines for production. Flexible job shop (FJS) is an advanced form of production organization that combines high flexibility and customized production capabilities to adapt to the changing market demands and customer orders in modern manufacturing. It can improve production efficiency, expand production capacity, and meet the assembly needs of multiple types of materials. It also has a certain degree of flexibility, and the products it produces can meet the different needs of customers and can make corresponding adjustments according to the changes in different customer needs.

The flexible manufacturing shop of agricultural mechanization equipment belongs to the flexible job shop, and its corresponding workshop scheduling problem can be attributed to FJSP. FJSP was first introduced in reference [3], accompanied by a polynomial algorithm tailored for addressing small-scale problem instances. Since Brucker and Schlie [3] first delved into FJSP, it has garnered significant attention from numerous scholars. Consequently, many methods have been proposed to tackle this complex issue. The FJSP has found widespread application in various industrial fields, including automotive assembly, agricultural machinery equipment production, semiconductor manufacturing, and so on. Mahmoodjanloo et al. [4] addressed FJSP with reconfigurable machine tools (RMTs) and configuration-dependent setup times. A self-adaptive DE algorithm with a Nelder–Mead mutation strategy (SADE-NMMS) was introduced to efficiently solve the problem. Fan et al. [5] addressed FJSP with lot-streaming and machine reconfigurations (FJSP-LSMR) to minimize the total weighted tardiness. A matheuristic method combining genetic algorithm (GA) and variable neighborhood search (VNS) was proposed. Guevara et al. [6] presented a case study on a flexible job shop scheduling problem (FJSSP) in a real enterprise, focusing on minimizing the makespan by optimizing worker task schedules while considering precedence constraints. This study highlighted the effectiveness of GA in solving complex scheduling problems in industrial settings. Barak et al. [7] presented a novel multi-objective mathematical model for resource-constrained Flexible Manufacturing System (FMS) scheduling, considering machine loading/unloading, operation scheduling, machine assignment, and AGV scheduling. A Modified Multi-Objective Particle Swarm Optimization (MMOPSO) algorithm was developed and outperformed the classic MOPSO. Ayyoubzadeh et al. [8] addressed FJSP under environmental uncertainties, considering energy tax regulations and machine breakdowns. A bi-objective mathematical model was developed to minimize tax costs on surplus energy consumption and the total costs of job lateness based on soft time-windows. An improved NSGA-II algorithm with a floating scheduling operator was proposed to solve the problem efficiently.

The extended FJSP which considers the worker–resources–constraint can be seen as a type of dual-resource constrained flexible job shop scheduling problem (DRCFJSP) [9]. Wu et al. [10] addressed a DRCFJSP that incorporated worker learning with the goal of minimizing the makespan. They devised a hybrid genetic algorithm integrating variable neighborhood search (VNS) to explore a wide range of potential solutions. Computational tests showed that their algorithm was effective in solving the problem. Kress, Müller, and Nossack [11] studied a DRCFJSP considering setup times and worker skills. They aimed to minimize makespan and tardiness using exact and heuristic methods by splitting the problem into routing and worker assignment sub-problems. Gong et al. [12] studied a double flexible job shop scheduling problem (DFJSP) with the goal of minimizing makespan and maximal total worker cost. They proposed a new hybrid genetic algorithm (NHGA) to solve this problem which outperformed the non-dominated sorting genetic algorithm II (NSGA-II) in terms of accuracy and efficiency. Vital-Soto et al. [13] tackled the challenge of simultaneously determining machines, worker assignments, and sequencing flexibility in a flexible job shop setting. Their objective was to minimize the makespan, worker workload, and weighted tardiness. They developed a multi-objective optimization model and applied a unique variation in the elitist non-dominated sorting genetic algorithm (NSGA-II) to solve the complex problem they faced.

Collaborative scheduling in workshops involves simultaneously allocating processing machines and AGVs for tasks, aiming to optimize production and transportation resources in a specific workshop environment. Raman et al. [14] first introduced the scheduling of AGVs in the flexible job scheduling problem (FJSPT) using a mixed integer programming model to minimize completion time. They treated transportation like operations, with AGVs as machines and travel times as processing times, applying traditional flexible scheduling methods to allocate both machines and AGVs. Yan et al. [15] studied the FJSP under limited transportation conditions in a digital twin workshop and designed an improved genetic algorithm to minimize makespan. Hu et al. [16] combined deep reinforcement learning with hybrid AGV scheduling rules for real-time scheduling in flexible workshops. Upon task completion or job arrival, a DQN-based scheduler selects a scheduling rule to allocate the most suitable AGV, aiming to minimize makespan and delay rates. Umar et al. [17] proposed a hybrid multi-objective genetic algorithm to obtain a Pareto solution to the problem of minimizing multiple conflicting objectives, including total makespan, AGV travel time, and delay cost. Similar studies include the literature [18,19]. Tan et al. [18] investigated the low-carbon joint scheduling problem in a flexible open-shop environment with constrained AGVs (LCJS-FOSCA). A mixed-integer programming model was formulated to minimize total carbon emissions and makespan. Berterottière et al. [19] addressed the FJSP with transportation resources (FJSPT), proposing a disjunctive graph model and a tabu search algorithm. The model integrated production and transportation operations considering travel times and vehicle assignments.

The existing algorithms for FJSP have made significant contributions to the field, but they often face limitations when applied to real-world scenarios with multi-resource constrained flexible job shop scheduling problem (MRCFJSP). Many existing algorithms focus primarily on either machine constraints or worker constraints, but rarely both simultaneously. This limitation becomes evident when dealing with complex production environments where multiple resources need to be optimized together. Some algorithms, such as certain genetic algorithms and particle swarm optimization methods, struggle with scalability. They may perform well on small-scale problems but fail to efficiently handle medium- and large-scale instances due to increased computational complexity. In many studies, the continuous working hours of workers are not considered, leading to schedules that may be optimal in terms of makespan but impractical in terms of worker fatigue and productivity. To address these limitations, we propose a MODGWO algorithm. Our method integrates a hybrid initialization strategy and a multi-neighborhood local search to effectively balance exploration and exploitation.

The main contributions of this paper are listed as follows: (1) Unlike previous studies that primarily focused on either worker constraints or automated guided vehicle (AGV) constraints individually, our research comprehensively integrates both workers and AGVs along with machines into a unified scheduling model. (2) In order to solve MRCFJSP, we propose the MODGWO algorithm, which has a hybrid initialization strategy and a discrete solution update mechanism, integrating the principles of the Jaya algorithm to improve the balance of exploration and development. (3) Taking the continuous working time of workers as the optimization objective not only enhances the authenticity of the scheduling model, but also conforms to the modern people-oriented production concept, ensuring efficient production while providing sufficient rest for workers to maintain high productivity and reduce safety risks. (4) Extensive experiments using extended Brandimarte test cases and real-world data from an agricultural machinery manufacturing enterprise show that the proposed method can effectively deal with MRCFJSP, and it has generality and superiority compared to other methods.

2. Materials and Methods

2.1. Problem Description

The MRCFJSP considering multiple resource constraints of machines, workers, and AGVs can be described as follows:

In this study, there are a total of 4 data sets, as follows:

(1)

Jobs and Operations

There are n jobs J = {J₁, J₂, …, J_n} to be processed, each consisting of a different number of operations. Job J_i has r_i operations, denoted as O_i1, O_i2, …, O_iri. The processing sequence of operations for each job is predetermined and must be followed strictly.

(2)

Machines

There are m machines M = {M₁, M₂, …, M_m} available for processing the operations. Each operation O_ij can be processed on one or more machines, and the set of compatible machines for operation O_ij is denoted as Ω_ij ⊆ M. The processing time of operation O_ij on machine M_k operated by worker W_s is denoted as P_ijks.

(3)

Workers

There are l workers W = {W₁, W₂, …, W_l} available to operate the machines. Each worker can operate only one machine at a time, and each machine can be operated by only one worker at a time.

(4)

AGVs

There are w AGVs A = {A₁, A₂, …, A_w} responsible for transporting jobs to the machines. AGVs are responsible for transporting jobs between the loading/unloading (LU) area and the machines, with each AGV capable of transporting only one job at a time.

The MRCFJSP needs to solve 5 sub-problems:

(1)

Machine Allocation: Assign each operation to a machine that can process it.

(2)

Worker Allocation: Assign each operation to a worker who can operate the selected machine.

(3)

AGV Allocation: Assign each job to an AGV for transportation to the designated machine.

(4)

Operation Sequence: Determine the order in which operations are processed on each machine.

(5)

Transportation Sequence: Determine the order in which AGVs transport jobs between the LU area and machines.

The optimization objectives are as follows:

(1)

Minimize Makespan: The maximum completion time of all jobs.

(2)

Minimize Maximum Continuous Working Hours of Workers: The longest continuous working period for any worker.

The problem is subject to the following constraints:

(1)

Each machine can process at most one operation at a time.

(2)

A worker can only operate one machine at a time.

(3)

Only one machine can be selected for each operation and one AVG can be selected for job J_i before operation O_ij, and they cannot be interrupted during processing.

(4)

All the jobs, machines, and workers are available at zero time. All the jobs and AGVs are located in the loading/unloading(LU) area at zero time.

(5)

The processing operations for the same job shall meet the constraints of the operation sequence.

(6)

Disregarding the transfer time of workers between machines, a worker can only operate one machine at a time.

(7)

Machine and AVG failures are not considered and the AVGs are sufficiently charged, and the loading and unloading time of the material on the machine is not considered.

(8)

An AVG can only load one job at a time.

(9)

After the current transportation task is completed, the AGV will not return to the loading and unloading area, but will go to the machine where the next transportation task is to be performed.

(10)

Each machine has a buffer zone that can be used to park AGVs and store operations with sufficient buffer capacity.

Figure 1 illustrates a specific scenario. AGV, workers, and machines are regarded as three different resources in the production workshop. AGV is responsible for transporting raw materials from the loading area to various machines, thus establishing an important logistics chain. Starting from the loading area, the AGV picks up tasks one by one and transports them to the designated machine, as shown in the loaded AGVs in Figure 1. Then, the machine is operated by a worker to start the processing task. Alternatively, the AGV can run empty to a certain machine and transport the semi-finished products to the machine to be processed in the next process, such as unloaded AVGs. Workers operate machines when they have processing tasks assigned, and they can take a break in the resting area when not assigned to any tasks. This continuous cycle process ensures efficient workflow and sustained productivity in the workshop.

2.2. Mathematical Model

The notations required for the mathematical model are listed in

Table 1. Variable definitions and symbol descriptions.

Notation	Description
Indexes and set
i, i′, h	index of jobs, i, i′, h ∈ {1, 2,…, n}
j, j′, g	index of operations, j, j′, g ∈ {1, 2,…, ri}
k, q	index of machines, k, q ∈ {1, 2,…, m}
s, r	index of workers, s, r ∈ {1, 2,…, l}
v	index of AVGs, v ∈ {1, 2,…, w}
Ji	The ith job
Mk	The kth machine
Ws	The sth worker
Av	The vth AVG
Oij	The jth operation of Ji
Ωij	The set of compatible machines of Oij
λk	The set of compatible workers of Mk
Parameters
n	Number of total jobs
ri	Number of operations of Ji
m	Number of total machines
l	Number of total workers
w	Number of total AVGs
Pijks	The processing time of Oij on Mk operated by Ws
STij	The start time of Oij
CTij	The completion time of Oij
Ci	The completion time of the last operation of Ji
STijv	The start time of AGV Vv when transporting Oij
CTijv	The completion time of AGV Vv when transporting Oij
tgk	Delivery time of AGV Vv from machine Mg to machine Mk
Ts	Maximum continuous working hours for worker Ws
L	a sufficiently large positive integer
Decision variables
xijk	a binary variable: if Oij is assigned on Mk, xijk = 1, otherwise xijk = 0
xijks	a binary variable: if Oij is assigned on Mk operated by Ws, xijks = 1, otherwise xijks = 0
yiji′j′ks	a binary variable: if Oij is ahead of Oi′j′ assigned on Mk operated by Ws, yiji′j′ks = 1, otherwise yiji′j′ks = 0
ziji′j′k	a binary variable: if Oij is ahead of Oi′j′ assigned on Mk, ziji′j′k = 1, otherwise ziji′j′k = 0
αijv	a binary variable: if Oij is transported by AVG Vv, αijv = 1, otherwise αijv = 0
βiji′j′v	a binary variable: if Oij is ahead of Oi′j′ transported by AVG Vv, βiji′j′v = 1, otherwise βiji′j′v = 0

.

Based on the above problem definition, constraints, assumptions, and notations, the mathematical model of MRCFJSP is established as follows:

$$\min C_{\max }=\underset{1\le i\le n}{\max }(C_i)$$

(1)

$$\min T_{\max }=\underset{1\le s\le l}{\max }(T_s)$$

(2)

This paper considers two optimization objectives: Equation (1) represents minimizing the maximum completion time of the last job (makespan). The second objective is to minimize the maximum continuous working hours of all workers, which is shown in Equation (2).

It is subject to the following:

$$C_i=C{T}_{i(ri)},i=1,2,\dots n$$

(3)

$$C{T}_{ij}=S{T}_{ij}+{\displaystyle \sum _{\mathrm{k}=1}^m{x}_{ijk}}{P}_{ijk\mathrm{s}},i=1,2,\dots n;j=1,2,\dots r_i;s=1,2,\dots ,l$$

(4)

$$T_s\le {\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^{r_i}{\displaystyle \sum _{k=1}^m{x}_{ijks}}}}{P}_{ijks},s=1,2,\dots \dots l$$

(5)

$$S{T}_{ij}\ge C{T}_{i(j-1)},i=1,2,\dots n;j=2,3,\dots ,r_i$$

(6)

$$C{T}_{ij}{\displaystyle \sum _{k=1}^m{x}_{ijk}}+L(1-{\displaystyle \sum _{k=1}^m{z}_{ij{i}^{\prime }{j}^{\prime }k}})\le S{T}_{i^{\prime }{j}^{\prime }}{\displaystyle \sum _{k=1}^m{x}_{i^{\prime }{j}^{\prime }k}},i=1,2,\dots n;j=1,2,\dots ,ri;j^{\prime }=1,2,\dots r_{i^{\prime }}$$

(7)

$$C{T}_{ij}{\displaystyle \sum _{s=1}^l{x}_{ijks}}+L(1-{\displaystyle \sum _{s=1}^l{y}_{ij{i}^{\prime }{j}^{\prime }s}})\le S{T}_{i^{\prime }{j}^{\prime }}{\displaystyle \sum _{s=1}^l{x}_{i^{\prime }{j}^{\prime }ks}},i,i^{\prime }=1,2,\dots n;j=1,2,\dots ,r_i;j^{\prime }=1,2,\dots ,r_{i^{\prime }};\forall k$$

(8)

$$C{T}_{ijv}{\displaystyle \sum _{v=1}^w{\alpha }_{ijv}}+L(1-{\displaystyle \sum _{v=1}^w{\beta }_{ij{i}^{\prime }{j}^{\prime }v}})\le S{T}_{i^{\prime }{j}^{\prime }v}{\displaystyle \sum _{v=1}^w{\beta }_{ij{i}^{\prime }{j}^{\prime }v}},i=1,2,\dots n;j=1,2,\dots ,r_i;j^{\prime }=1,2,\dots ,r_{i^{\prime }}$$

(9)

$$C{T}_{i^{\prime }{j}^{\prime }}-C{T}_{ij}+L(1-z_{ij{i}^{\prime }{j}^{\prime }k})\ge {\displaystyle \sum _{k=1}^m{P}_{ijk\mathrm{s}}{x}_{ijk}};i,i^{\prime }=1,2,\dots n;j==1,2,\dots r_i;j^{\prime }=1,2,\dots r_{i^{\prime }};k=1,2,\dots ,m;s=1,2,\dots ,l$$

(10)

$$\begin{array}{l}[(S{T}_{ij}-S{T}_{hg})\times x_{hgks}\times x_{ijks}\ge 0]\vee [(S{T}_{hg}-S{T}_{ij})\times x_{hgks}\times x_{ijks}\ge 0]\\ \forall k=1,2,\dots ,m;\forall s=1,2,\dots ,l;i,h=1,2\dots ,n;j=1,2,\dots r_i;g=1,2,\dots r_h;(i\ne h)\vee (j\ne g);\end{array}$$

(11)

$${\displaystyle \sum _{k=1}^m{x}_{ijk}=1},i=1,2,\dots n;j=1,2,\dots r_i$$

(12)

$${\displaystyle \sum _{s=1}^l{x}_{ijks}=1},i=1,2,\dots n;j=1,2,\dots r_i;k=1,2,\dots m$$

(13)

$${\displaystyle \sum _{v=1}^w{\alpha }_{ijv}=1},i=1,2,\dots n;j=1,2,\dots r_i$$

(14)

$$x_{ijk}\in \{0,1\},i=1,2,\dots n;j=1,2,\dots r_i;k=1,2,\dots m$$

(15)

$$x_{ijks}\in \{0,1\},i=1,2,\dots n;j=1,2,\dots r_i;k=1,2,\dots m;s=1,2,\dots l$$

(16)

$$y_{ij{i}^{\prime }{j}^{\prime }s}\in \{0,1\},i,i^{\prime }=1,2,\dots n;j,j^{\prime }=1,2,\dots r_i,s=1,2,\dots l$$

(17)

$$z_{ij{i}^{\prime }{j}^{\prime }k}\in \{0,1\},i,i^{\prime }=1,2,\dots n;j,j^{\prime }=1,2,\dots r_i,k=1,2,\dots m$$

(18)

$${\alpha }_{ijv}\in \{0,1\},i=1,2,\dots n;j=1,2,\dots r_i;v=1,2,\dots w$$

(19)

$${\beta }_{ij{i}^{\prime }{j}^{\prime }v}\in \{0,1\},i,i^{\prime }=1,2,\dots n;j,j^{\prime }=1,2,\dots r_i,v=1,2,\dots w$$

(20)

where Equation (3) represents that the processing end time of the job is the end time of the last operation of the job. Equation (4) ensures that the completion time of each job is strictly after its start time. This is a fundamental constraint in scheduling problems, as it enforces the logical sequence that a job must start before it can be completed. Equation (5) ensures that the maximum continuous working time of each worker does not exceed their total working time. Equation (6) enforces the processing sequence constraint of a job. It ensures that operations within a job are processed in a specific order. For example, operation O_ij must be completed before operation O_{i (j+1)} can start. Equations (7)–(9) represent the processing time constraints. These constraints ensure that an operation cannot start until the assigned machine, worker, and AGV are ready. Specifically, Equation (7) ensures that the operation cannot start until the assigned machine is available. Equation (8) ensures that the operation cannot start until the assigned worker is available. Equation (9) ensures that the operation cannot start until the assigned AGV is available. Equations (10) and (11) ensure that each machine can process at most one operation at a time and that each worker can operate only one machine at a time. These constraints prevent the overlapping of operations on the same machine and ensure that workers are not assigned to multiple machines simultaneously. Equations (12)–(14) ensure that each operation can be assigned to only one machine, one worker, and one AGV, respectively. These constraints ensure that the resources are uniquely assigned to each operation. Equations (15)–(20) indicate that all the decision variables are binary variables. This is essential for maintaining the integrity of the scheduling model and ensuring that the assignments are clear and unambiguous.

The Grey Wolf Optimization (GWO) [20] algorithm effectively balances exploration and exploitation through its social hierarchy and hunting behavior, enabling it to navigate the complex and highly constrained search space of MRCFJSP. Its simplicity and ease of implementation, with fewer control parameters compared to other metaheuristics, make it practical for real-world applications. Additionally, GWO’s robustness and scalability allow it to handle both small- and large-scale instances efficiently. By incorporating non-dominated sorting and crowding distance, GWO can generate a diverse set of non-dominated solutions, providing valuable trade-off options for decision-makers.

2.3. Grey Wolf Optimization Algorithm

The GWO algorithm was initially proposed by Australian scholars Seyedali Mirjalili et al. in 2014 for continuous real parameter optimization problems, and has been improved to handle some practical engineering problems [21,22,23,24]. As shown in Figure 2, the gray wolf pack has a very strict social hierarchy similar to the pyramid. The first three hierarchical categories are Alpha (α), Beta (β), and Delta (δ) as the best solutions to lead the rest of the wolves named Omega (ω) wolves toward promising areas in order to find the global solution.

A wolf (i) evaluates its proximity to the three best solutions based on Equations (21)–(26) and subsequently updates its position using Equation (25).

$$\overrightarrow{A}=2\overrightarrow{a}{\overrightarrow{r}}_1-\overrightarrow{a}$$

(21)

$$\overrightarrow{B}=2\times {\overrightarrow{r}}_2$$

(22)

$$\left\{\begin{array}{c}{\overrightarrow{D}}_{\alpha }=\left|{\overrightarrow{C}}_1{\overrightarrow{X}}_{\alpha }-\overrightarrow{X}\right|\\ {\overrightarrow{D}}_{\beta }=\left|{\overrightarrow{C}}_2{\overrightarrow{X}}_{\beta }-\overrightarrow{X}\right|\\ {\overrightarrow{D}}_{\delta }=\left|{\overrightarrow{C}}_3{\overrightarrow{X}}_{\delta }-\overrightarrow{X}\right|\end{array}\right.$$

(23)

$$\left\{\begin{array}{c}{\overrightarrow{X}}_1={\overrightarrow{X}}_{\alpha }-A_1{\overrightarrow{D}}_{\alpha }\\ {\overrightarrow{X}}_2={\overrightarrow{X}}_{\beta }-A_2{\overrightarrow{D}}_{\beta }\\ {\overrightarrow{X}}_3={\overrightarrow{X}}_{\delta }-A_3{\overrightarrow{D}}_{\delta }\end{array}\right.$$

(24)

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

(25)

where $\overrightarrow{X}$_i, $\overrightarrow{X}$_α, $\overrightarrow{X}$_β, and $\overrightarrow{X}$_δ, indicate the position vectors of wolf i, α, β, and δ; $\overrightarrow{X}$_i(t + 1) is the location of the wolf i in the next iteration; $\overrightarrow{r}$₁ and $\overrightarrow{r}$₂ are random vectors in [0, 1]; and $\overrightarrow{A}$ and $\overrightarrow{B}$ are two coefficient vectors where the vector $\overrightarrow{a}$ is linearly decreased from 2 to 0 over the course of iterations and can be calculated by Equation (26).

$$a=2-{\displaystyle \frac{2\times t}{MaxIter}}$$

(26)

where t is the present number of iterations and MaxIter represents the maximum number of iterations.

2.4. Multi-Object Discrete Grey Wolf Optimization Algorithm for MRCFJSP

For continuous optimization problems, the gray wolf individual position can be updated by Equations (21)–(25); however, MRCFJSP is a discrete problem and cannot be directly solved using the above formulas. This paper proposed a MODGWO algorithm for MRCFJSP based on the characteristics of the problem and combined with cross-operation in the genetic algorithm. First, an encoding/decoding method is introduced, followed by a method for initializing a mixed population. Then, the Discrete Grey Wolf iteration operators are presented for global search. Finally, an effective VNS with four neighborhood structures is introduced to improve the local search ability.

To clarify the proposed algorithm, a 3 × 3 × 2 × 2 instance involving three jobs, three machines, two workers, and two AVGs is provided in

Table 2. Processing time.

Job	Operation	Processing Time
		M1	M2	M3
J1	O11	2	3	-
	O12	2	-	1
	O13	3	4	3
J2	O21	1	3	-
	O22	-	2	2
J3	O31	5	-	7
	O32	-	8	6

and

Table 3. Worker-operated machines.

	W1	W2
M1	√	-
M2	-	√
M3	√	√

. Table 2 outlines the fundamental processing durations, while Table 3 displays the information on machines that workers can operate. The symbol ‘-’ signifies that a machine is incompatible with a particular operation or a worker is unable to operate a specific machine. In addition, by default, each AGV can transport all jobs.

2.4.1. Four-Vector Encoding

The efficiency of encoding and decoding techniques has a substantial influence on algorithm performance. Utilizing effective encoding and decoding methods can facilitate algorithm operations and enhance computational performance. The MRCFJSP needs to solve four sub-problems: operation scheduling, machine selection, worker selection, and AVG selection. Therefore, in this subsection, we introduce an encoding scheme where each individual comprises four distinct vectors: the operation sequence (OS) vector, which denotes the order of operation execution; the machine selection (MS) vector, which specifies the assignment of machines; the worker selection (WS) vector, which outlines the allocation of workers; and the AVG selection (AS) vector, which indicts the assignment of AVGs. Collectively, these four vectors constitute a viable solution for the MRCFJSP. In the four-layer coding method, we borrowed the idea of the MOSO coding method proposed by Ho et al. [25]. Figure 3 shows the four-vector representation approach of the examples in Table 1 and Table 2, where green indicates job 1, yellow indicates job 2, and blue indicates job 3.

(1)

Operation sequence vector

An OS vector consists of an array of integers where the number of array elements equals the total number of operations in all the jobs. Each operation is denoted by its corresponding job index. Specifically, the jth occurrence of index i signifies the jth operation within job i, and the count of index i repetitions is equivalent to the total number of operations in job i.

For example, {3, 2, 1, 2, 3, 1, 1} is a legal vector or the aforementioned instance shown in Table 2 and Figure 3a. The sequence of operational decoding for this OS vector is O₃₁→O₂₁→O₁₁→O₂₂→O₃₂→O₁₂→O₁₃.

(2)

Machine selection vector

An MS vector consists of an array of integers, the length of which equals the length of the OS vector and each element indicates an available machine for the corresponding operation.

An MS vector {1, 3, 2, 1, 3, 1, 2} means that for the operation set {O₁₁, O₁₂, O₁₃, O₂₁, O₂₂, O₃₁, O₃₂}, the machine selection indicated by the MS vector is {M₁, M₃, M₂, M₁, M₃, M₁, M₂} correspondingly shown in Figure 3b.

(3)

Worker selection vector and AGV selection vector

The WS vector and AS vector are encoded in a similar way to the MS vector. As shown in Figure 3c, the first number 1 in the WS vector represents that the O₁₁ is processed by worker W₁ on machine M₁. The WS vector {1, 2, 2, 1, 1, 1, 2} indicates that the operations {O₁₁, O₁₂, O₁₃, O₂₁, O₂₂, O₃₁, O₃₂} are processed by workers {W₁, W₂, W₂, W₁, W₁, W₁, W₂} on machines {M₁, M₃, M₂, M₁, M₃, M₁, M₂}, respectively. Similarly, the AS vector {1, 1, 2, 1, 2, 2, 1} shown in Figure 3d represents that the operations{O₁₁, O₁₂, O₁₃, O₂₁, O₂₂, O₃₁, O₃₂} are transported by AGVs{A₁, A₁, A₂, A₁, A₂, A₂, A₁}, respectively.

2.4.2. Decoding

There are mainly four approaches to decoding a solution represented by the four-vector encoding scheme: non-delay, semi-active, active, and hybrid schedules. In this paper, the active schedule is utilized to decode all the solutions. The decoding procedure is as follows: (1) Pick out the operations from the OS vector one by one from left to right, and assign the corresponding machine, worker, and AGV to the current job in sequence. (2) Determine the starting time of the current job from the earliest available start time of the corresponding machine, worker, and AVG, until the last job is processed. (3) Arrange the operations at their earliest feasible time. (4) Calculate the maximum completion time and maximum worker duration, which are the values of the two objective functions.

2.4.3. Population Initialization

The standard of initial solutions significantly influences how well an algorithm performs in producing high-quality solutions. Recognizing the significant impact of population initialization on the algorithm’s performance, we propose several adaptive strategies aimed at enhancing the quality of the initial population. To achieve a balance between the quality and diversity of initial solutions within the search space, we utilize a hybrid approach that combines random generation with strategic selection to form the initial population. Since the problem in this paper is divided into four sub-problems: operation sequencing, machine selection, worker selection, and AVG selection, and the population initialization is also carried out in four stages. Firstly, the initialization of OS vectors employs a hybrid approach [26] according to the following three rules:

(1)

Most work remaining rule [27]: The job that has the maximum total processing time remaining is prioritized for execution first.

(2)

Most number of operations remaining rule [26]: The job that has the most remaining operations is processed first.

(3)

Random rule: Operation sequencing is randomly generated.

The aforementioned three rules are utilized in the generation of OS vectors, and their respective proportions are 30%, 30%, and 40%. The initialization of MS vectors is initialized employing the following three rules:

(1)

Global minimum processing time rule [28]: Select the machine with the shortest processing time for the current operation.

(2)

Workload considered rule [29]: Assign machines according to the machine workload.

(3)

Random rule: Randomly assign machines.

The machine selection of the population is sequentially generated using the first and second rules in the proportion of 30% and the remaining 40% by random rule. The initialization process of the WS vectors and the AS vectors is similar to that of MS vectors, and the only difference is that only worker and AVG loads, as well as random allocation, are used, with the respective proportions of 50% and 50%.

2.4.4. Solution-Updating Mechanism

Since the basic GWO cannot be directly applied to the discrete scheduling problem, this paper designs a discrete version of the solution-updating mechanism for the machine selection, AGV selection, and operation sequence. The gray wolf individual crosses with wolf α, wolf β, and wolf δ according to a certain probability, and thus obtains a new individual, as shown in Equation (27).

$$X_k(t+1)=\left\{\begin{array}{l}Cross(X_k(t),X_{\alpha }(t)),rand\le {\displaystyle \frac{1}{3}}\\ Cross(X_k(t),X_{\beta }(t)),{\displaystyle \frac{1}{3}}<rand<\\ Cross(X_k(t),X_{\delta }(t)),rand\ge {\displaystyle \frac{2}{3}}\end{array}\right.{\displaystyle \frac{2}{3}}$$

(27)

where X_k is the position vector of gray wolf t, X_α, X_β, and X_δ are the position vectors of gray wolf α, β, and δ, respectively, and rand is a random number that follows a uniform distribution between 0 and 1. Cross is the crossover operator, and the specific crossover method for each vector is described in detail below.

In basic GWO, new individuals are generated based on the information of the three individuals α, β, and δ that are currently serving as the decision-level individuals. To determine α, β, and δ individuals within a population, this paper adopts a method based on non-dominated level and crowding distance to obtain decision-level individuals, that is, sorting individuals according to their non-dominated level and crowding distance in the population, and the top three individuals have the opportunity to become decision level individuals. For any two individuals, the individual with the lower level ranks is selected first, and if two individuals have the same level, we compare their crowding distance, and the individual with the larger crowding distance ranks first.

Meanwhile, we integrate the ideas of the Jaya algorithm into the proposed MODGWO. The Jaya algorithm is a novel population-based metaheuristic recently proposed by Rao [30] which is based on the principle of continuous improvement, and improves the quality of the solution by moving the individuals closer to the best solution while moving away from the worst solution. The Jaya algorithm iteratively evolves through Equation (28) to obtain new solutions.

$$X_k(t+1)=X_k(t)+r_1\times (X_{best}-\left|X_k(t)\right|)-r_2\times (X_{worst}-\left|X_k(t)\right|)$$

(28)

where X_k (t + 1) is the updated solution corresponding to the initial solution X_k (t). X_best and X_worst represent the best and worst solutions for the current population, respectively. r₁ and r₂ are uniformly distributed. In the improved MODGWO, one of X_α, X_β, and X_δ is randomly chosen as X_best using Equation (27), while the worst individual X_worst is selected from the last rank having the minimum crowding distance value. The detailed procedures for operation sequencing, machine selection, worker selection, and AVG selection are summarized below:

(1)

Operation sequencing

This paper draws on the method in the literature [31] to delete sequences in the current solution that are identical to the worst solution and replace them with the best solution. The specific steps are as follows:

• Step 1. Identify the worst individual (X_worst) and choose one form X_α, X_β, and X_δ randomly as X_best (assuming that X_β is selected) from the population.
• Step2. Identify similar assignments of operations in the individual X_i and X_worst.
• Step 3. Eliminate similar assignments from X_i and transfer the remaining assignments to a new solution X_inew.
• Step 4. Beginning from the initial assignment of X_β, eliminate the assignments corresponding to the remaining assignments of individual X_inew.
• Step 5. Beginning from the initial assignment of X_β, transfer the remaining assignments of X_β in the vacant elements of individual X_inew in the corresponding order.

Figure 4 illustrates the operation sequencing vector-updating mechanism.

(2)

Machine selection and AGV selection

The solution-updating mechanism for the machine selection and AVG selection is the same and similar to that of operation sequencing. Only the detailed steps for updating machine selection are as follows:

• Step 1. Identify the best individual X_best (Assuming it is X_α) and worst individual X_worst from the population.
• Step 2. Identify similar assignments of machines assigned to the same machines in the individuals X_i and X_worst.
• Step 3. Eliminate similar assignments from the individual X_i and transfer the remaining assignments to a new solution X_inew.
• Step 4. Transfer the corresponding machine assignments of X_α to the vacant elements of individual X_inew.

Figure 5 illustrates the machine selection vector-updating mechanism.

(3)

Worker selection

The updating mechanism of worker selection is the same as the previous steps of machine selection and AVG selection, eliminating elements from the current individual that correspond to the position of the worst solution, and filling the eliminated elements with elements corresponding to the position of the best solution. Finally, it is necessary to conduct feasible checks on new individuals. Check for the feasibility of each assignment of X_inew based on the capable worker set of the corresponding machine in the updated machine selection vector. If it is infeasible, randomly allocate an integer value within the capable worker set range.

2.4.5. Local Search

To enhance the local search capability, variable neighborhood search operations are performed on the OS, MS, WS, and AS of the optimal individuals after each iteration, respectively. Variable neighborhood search (VNS) can expand the search space of the algorithm by changing the domain structure based on problem features, preventing the algorithm from falling into the local optimal solution. Firstly, the initial solution that requires a neighborhood search is given. Secondly, the neighborhood structure is used for neighborhood search to obtain the neighborhood structure solution. If the target value of the neighborhood solution is better than the current solution, the current solution is updated and the weights are updated. The search returns to the first neighborhood structure and starts again until a better solution cannot be found. Then, the next neighborhood structure is reached for search until all the neighborhood structures have been searched. Based on the characteristics of the MRCFJSP, the following 4 domain structures are constructed:

(1)

VNS1 (Insert, for OS vectors): Within the length range of the OS vector, randomly generate two positions r1 and r2 (r1 < r2), insert the element corresponding to r2 into position r1, and move the elements after position r1 backwards in sequence. For the OS vector in Figure 3, the two generated random numbers r1 and r2 are 2 and 5, respectively. The solution process of VNS1 is shown in Figure 6.

(2)

VNS2 (Choose the minimum processing time, for MS vectors): Randomly generate r ∈ [1, N] random numbers, where N is the total number of operations, and randomly select r positions from the MS vector. For each selected position corresponding to an operation, select the machine with the shortest processing time from its corresponding optional machine set for replacement. After replacement, check whether the workers corresponding to r positions in WS can operate the newly selected equipment. If it is not feasible, randomly select a worker from the operable worker set of the new equipment. For the solution generated in Figure 7, MS is 1,122,312, and r = 3 is randomly generated. Three positions are randomly selected, such as positions 2, 3, and 7. The solution process of VNS2 is shown in Figure 7.

(3)

VNS3 (Flip, for WS vectors): Within the length range of the WS vector, randomly generate a position r1. In the MS vector, position r1 should correspond to more than one available processing worker for the processing equipment, and then select a new worker from the set of available workers for equipment replacement. The solution process of VNS3 is shown in Figure 8. From Figure 8, it can be observed that: Machine M₃ can be operated by either workers W₁ or W₂. Initially, worker W₂ was selected for operating machine M₃. After VNS3, worker W₁ is chosen to operate machine M₃ instead.

(4)

VNS4 (swap, for AS vectors): Within the length of the AS vector, two randomly generated positions r1 and r2 (r1 ≠ r2) are exchanged for the elements at the r1 and r2 positions. The solution process of VNS4 is shown in Figure 9.

3. Results

The proposed approach is coded in Python 3.9 on an Intel i7 processor with a speed of 2.6 GHz on a Windows 11 operating system with 16 GB RAM.

3.1. Performance Metrics

To assess the efficacy of the Pareto front derived from various perspectives, we employ two metrics: IGD [32] and Hypervolume [33]. The aforementioned metrics encompass various facets concerning the quality of non-dominated solutions, particularly focusing on those derived from Pareto-based approaches. The detailed explanation of these two metrics is summarized below:

(1)

Inverse Generational Distance (IGD): Measure the average distance from a predefined set of uniformly spaced points along the true or ideal Pareto front (denoted as PF*) to the nearest solution in the obtained PF. Essentially, IGD assesses how well the obtained solutions approximate the true Pareto front by calculating these distances and averaging them.

The formula for the IGD is typically expressed as follows:

$$IGD={\displaystyle \frac{1}{\left|P{F}^{\ast }\right|}}{\displaystyle \sum _{i=1}^{\left|P{F}^{\ast }\right|}\underset{j\in PF}{\min }d(P{F}_i^{\ast },P{F}_j)}$$

(29)

where

• |PF*| is the number of uniformly spaced points along the true Pareto front.
• PF*_i represents the ith point in PF*.
• PF is the set of obtained solutions (Pareto front).
• PF_j represents the jth solution in PF.
• d(PF*_i, PF_j) is the Euclidean distance between point PF*_i and the closest solution PF_j in PF.

A lower IGD value indicates that the obtained PF is closer to the true Pareto front, suggesting better performance of the optimization algorithm in identifying high-quality solutions. Since the actual PF* of MRCFJSP in this paper is unknown, the non-dominated solutions from multiple runs of all the algorithms are first integrated into a hybrid solution set, and then the non-dominated solutions in the hybrid solution set are used as PF*.

(2)

Hypervolume (HV): HV is the volume of a hyper polyhedron enclosed by non-dominated solutions and reference points in the search space, calculated as follows:

$$HV(P,F)=\underset{p\in P}{\overset{P}{\cup }}v(p,F)$$

(30)

where P represents the Pareto front approximate non-dominated solution set obtained by the algorithm, F represents the reference points of the solution set, and v is the volume of a hypercube enclosed by the pth solution and the reference points. The target vectors corresponding to all the Pareto solutions are normalized and the reference point for calculating HV is set to (1, 1).

3.2. Test Instances Generation

This paper extends the classic Brandimarte test case [27] in the flexible job shop scheduling problem by randomly adding worker and AVG information for solving the MRCFJSP. The number of workers in each workshop is calculated by multiplying the number of machines in the workshop by 0.6 and rounding up, and the probability that a worker can operate a certain machine is 0.5. For each test case, the ratio of the number of AGVs to the number of machines is considered to be 1:1, 1:2, and 1:3, respectively. Taking MK01 as an example, this example has 10 jobs and six machines. In the generated new example, the number of workers is four, and the number of AVGs is six, three, and two, respectively. The names of the extended examples are MK01_4_6, MK01_4_3, and MK01_4_2. In this way, 45 instances for MRCFJSP are used to test in this study, and

Table 4. Information of the instances.

Instance ID	Scale (n × m)	Number of Workers	Number of AVGs
MK01	10 × 6	4	2, 3, 6
MK02	10 × 6	4	2, 3, 6
MK03	15 × 8	5	3, 4, 8
MK04	15 × 8	5	3, 4, 8
MK05	15 × 4	3	1, 2, 4
MK06	10 × 10	6	3, 5, 10
MK07	20 × 5	3	2, 3, 5
MK08	20 × 10	6	3, 5, 10
MK09	20 × 10	6	3, 5, 10
MK10	20 × 15	9	5, 8, 15
MK11	30 × 5	3	2, 3, 5
MK12	30 × 10	6	3, 5, 10
MK13	30 × 10	6	3, 5, 10
MK14	30 × 15	9	5, 8, 15
MK15	30 × 15	9	5, 8, 15

shows the relevant information, where n represents the number of jobs and m represents the number of machines.

3.3. Effectiveness Verification for a Hybrid Initialization Method

To verify the effectiveness of the hybrid initialization method and local search proposed procedure in this paper, it is compared with the random initialization method and method without local search, and the comparison algorithms are as follows:

(1)

MODGWO represents the method proposed in this paper.

(2)

RI (random initialization) indicates that MODGWO uses a random initialization method to obtain the initial solution.

(3)

WLS (without local search) indicates that MODGWO does not use a local search procedure, but only a single tracking mode for searching.

Each method was independently run 20 times based on the expanded 45 test cases. The average values of IGD and HV were calculated based on the non-dominated solution sets obtained by the three methods and calculated results are shown in

Table 5. Performance comparison between MODGWO, RI, and WLS.

Instance ID	IGD			HV
	MODGWO	RI	WLS	MODGWO	RI	WLS
MK01_4_2	0.0919	0.1085	0.2587	0.6601	0.6194	0.3813
MK01_4_3	0.0341	0.0258	0.1066	0.7038	0.7255	0.3099
MK01_4_6	0.0298	0.0595	0.3416	0.7943	0.7536	0.5588
MK02_4_2	0.0249	0.0456	0.2040	0.6404	0.7204	0.3349
MK02_4_3	0.0091	0.0146	0.1039	0.7466	0.7176	0.5290
MK02_4_6	0.0693	0.0941	0.3108	0.7093	0.7023	0.3623
MK03_5_3	0.1085	0.0994	0.2941	0.7244	0.7582	0.4103
MK03_5_4	0.0585	0.0672	0.3206	0.7638	0.6609	0.3661
MK03_5_8	0.0873	0.0611	0.3032	0.7939	0.7022	0.4098
MK04_5_3	0.0838	0.1041	0.4440	0.7756	0.7047	0.5579
MK04_5_4	0.0208	0.0385	0.2965	0.7724	0.7018	0.4028
MK04_5_8	0.0604	0.0403	0.3118	0.6237	0.6935	0.4174
MK05_3_1	0.0496	0.0621	0.3322	0.6158	0.5725	0.2878
MK05_3_2	0.0653	0.0586	0.4328	0.5828	0.4430	0.3592
MK05_3_4	0.0220	0.0385	0.2061	0.6333	0.5146	0.3680
MK06_6_3	0.1065	0.1311	0.3859	0.6906	0.6298	0.4645
MK06_6_5	0.0987	0.1197	0.4437	0.6058	0.4608	0.2340
MK06_6_10	0.0565	0.0694	0.2122	0.7321	0.6329	0.4171
MK07_3_2	0.0926	0.1123	0.3348	0.6305	0.5279	0.3358
MK07_3_3	0.0585	0.0670	0.3647	0.5685	0.4399	0.2629
MK07_3_5	0.0966	0.1033	0.4429	0.7409	0.7828	0.5083
MK08_6_3	0.1008	0.1176	0.3291	0.7952	0.7656	0.4415
MK08_6_5	0.0770	0.0977	0.2344	0.7493	0.6188	0.3711
MK08_6_10	0.0792	0.0952	0.3526	0.6171	0.6046	0.2311
MK09_6_3	0.0457	0.0538	0.3173	0.7160	0.6342	0.3227
MK09_6_5	0.0359	0.0412	0.3259	0.7495	0.6690	0.4145
MK09_6_10	0.0155	0.0355	0.3259	0.7033	0.6923	0.4733
MK10_9_5	0.1242	0.1395	0.3674	0.6663	0.6284	0.3925
MK10_9_8	0.0426	0.0680	0.2490	0.6864	0.6599	0.3530
MK10_9_15	0.0416	0.0628	0.2571	0.6888	0.7554	0.3711
MK11_3_2	0.0171	0.0426	0.3503	0.6513	0.6407	0.3360
MK11_3_5	0.0441	0.0513	0.4000	0.7777	0.6842	0.5382
MK11_3_3	0.0694	0.0941	0.2872	0.7257	0.6272	0.5121
MK12_6_3	0.0785	0.0930	0.2593	0.6782	0.5767	0.3459
MK12_6_5	0.0808	0.1079	0.4347	0.6425	0.5650	0.4217
MK12_6_10	0.1113	0.1365	0.4819	0.7708	0.7509	0.4145
MK13_6_3	0.0533	0.0611	0.2760	0.5823	0.6512	0.2344
MK13_6_5	0.0670	0.0943	0.3321	0.7700	0.6634	0.5513
MK13_6_10	0.0135	0.0234	0.3950	0.7075	0.6767	0.3673
MK14_9_5	0.0080	0.0196	0.2478	0.6811	0.5930	0.3006
MK14_9_8	0.0237	0.0390	0.2242	0.6731	0.5487	0.4462
MK14_9_15	0.0100	0.0373	0.3446	0.6845	0.6180	0.3912
MK15_9_5	0.1029	0.1132	0.2900	0.7163	0.7033	0.3343
MK15_9_8	0.0745	0.0859	0.2558	0.7524	0.6647	0.4298
MK15_9_15	0.0874	0.1023	0.3954	0.6601	0.5603	0.3834

Bold text indicates the optimal values achieved by different methods for the corresponding test cases.

. The results from Table 5 are shown as box plots in Figure 10.

The experimental results show that MODGWO can obtain better initial populations compared to RI which benefited from the effective diversity strategy introduced in their initialization. Compared with RI, MODGWO achieved a total of 40 minimum values on IGD and 38 maximum values on HV, indicating that the diversified initialization method proposed in this paper can obtain better quality and more Pareto solutions. And, the five best IGD values obtained from the RI method are all concentrated in smaller-scale cases, indicating that for medium- and large-scale cases, a diverse population initialization method is needed to obtain better solutions.

To assess the effectiveness of the proposed local search procedure, we compare WLS to MODGWO and RI for all the problem instances. From Table 5 and Figure 10, it is seen that MODGWO and RI outperform WLS for all the data sets for the IGD and HV metrics. The poor performance of WLS concerning the IGD and HV metrics can be attributed to the algorithm’s absence of the local search procedure. On the whole, the introduced local search procedure plays a crucial role within the proposed methodology and has a substantial impact on enhancing the convergence performance of MODGWO. This indicates that the local search mechanism is integral to the overall effectiveness and efficiency of the MODGWO algorithm in finding optimal or near-optimal solutions to MRCFJSP.

Figure 10 presents box plots of the IGD and HV metrics based on the average values from Table 5, comparing the performance of the MODGWO, RI, and WLS algorithms. It can be seen that MODGWO consistently achieves the lowest IGD values and highest HV values, indicating superior solution quality and diversity with high stability across different instances. In contrast, RI exhibits higher variability and poorer performance in both metrics due to its random initialization strategy. WLS performs the worst, with significant variability and poor median values, highlighting the importance of local search in finding high-quality solutions. This indicates that the hybrid initialization method and local search techniques proposed in this paper significantly enhance the algorithm’s ability to generate high-quality and diverse solutions for the MRCFJSP.

3.4. Comparison with Other Algorithms

There have been many studies on flexible job shop scheduling under dual resource constraints for workers alone [34,35], as well as flexible job shop scheduling under AGV constraints alone [36,37]. However, there is a lack of research on flexible job shop scheduling under multi-resource constraints that comprehensively consider both workers and AGVs. Therefore, this paper chooses MOPSO proposed by Zhang et al. [38], MOJAYA proposed by Rylan H et al. [39], and NSGA II which is commonly used to solve multi-objective flexible job shop scheduling problems as comparative algorithms. These three algorithms have been often used to solve FJSP under resource constraints, which are most similar to the problem studied in this paper.

The MODGWO algorithm proposed in this paper involves only a few parameters, which only include population size, total iteration times, local search termination iteration times for VNS1 and VNS4, and the mutation probability for VNS1 and VNS4. By conducting multiple experiments to set the parameters, the population size is 150, the total number of iterations is 800, the number of local search termination iterations for VNS1 and VNS4 is 30, and the mutation probability for VNS1 and VNS4 is 0.3. For the MOJAYA algorithm, set the parameters according to reference [39], that is, the population size is 100, the probability of executing the local search is 0.8, and the mutation probability is 0.3. For the MOPSO algorithm, the method described in reference [38] was used to set the parameters as follows: population size of 100, initial temperature of 3, final temperature of 0.01, annexing rate of 0.9, local search threshold parameters of 0.6 and 0.8, maximum adaptive probability of 0.8, and minimum adaptive probability of 0.2. For the NSGA II algorithm, the population size is 120, the crossover probability is 0.8, and the mutation probability is 0.2. The total number of iterations for MOJAYA, MOPSO, and NSGA II is the same as MODGWO, which is 800.

The experimental results for the two performance indicators have been organized in

Table 6. Performance comparison between MOJAYA, MOPSO, NSGAII, and MODGWO.

Instance ID	MOJAYA		MOPSO		NSGA-II		MODGWO
	IGD	HV	IGD	HV	IGD	HV	IGD	HV
MK01_4_2	0.2132	0.5653	0.3544	0.4671	0.2536	0.5295	0.0973	0.6601
MK01_4_3	0.1785	0.5690	0.3518	0.5474	0.3222	0.6038	0.0829	0.7038
MK01_4_6	0.1640	0.7512	0.2513	0.6470	0.2306	0.6849	0.0231	0.7943
MK02_4_2	0.1466	0.5988	0.2687	0.4327	0.2679	0.5365	0.0532	0.6404
MK02_4_3	0.1840	0.5989	0.3560	0.5562	0.2893	0.6248	0.0829	0.7466
MK02_4_6	0.0953	0.7322	0.4035	0.5626	0.2837	0.5925	0.1083	0.7093
MK03_5_3	0.0765	0.7054	0.2851	0.5825	0.1938	0.6067	0.0142	0.7244
MK03_5_4	0.0907	0.6745	0.2520	0.6323	0.1835	0.6237	0.0075	0.7638
MK03_5_8	0.0726	0.7174	0.3120	0.5879	0.2304	0.6448	0.0143	0.7939
MK04_5_3	0.1170	0.6352	0.3066	0.6282	0.2536	0.6404	0.0374	0.7756
MK04_5_4	0.1431	0.7325	0.3598	0.6113	0.2431	0.6251	0.0628	0.7724
MK04_5_8	0.1822	0.6874	0.2891	0.4046	0.3145	0.4919	0.0812	0.6237
MK05_3_1	0.0851	0.5472	0.2400	0.4069	0.2147	0.4681	0.0260	0.6158
MK05_3_2	0.0438	0.5199	0.2927	0.4252	0.2233	0.4365	0.0571	0.5828
MK05_3_4	0.0844	0.5955	0.2237	0.4312	0.2121	0.5284	0.0220	0.6333
MK06_6_3	0.1255	0.6431	0.3127	0.5452	0.2074	0.5504	0.0490	0.6906
MK06_6_5	0.1203	0.6384	0.3374	0.4279	0.3816	0.4952	0.1372	0.6058
MK06_6_10	0.1263	0.7197	0.2949	0.5633	0.3194	0.6240	0.0700	0.7321
MK07_3_2	0.0812	0.6016	0.2850	0.4016	0.1756	0.4938	0.0097	0.6305
MK07_3_3	0.1020	0.6207	0.2697	0.4417	0.1989	0.4586	0.0116	0.5685
MK07_3_5	0.0893	0.6456	0.3473	0.5429	0.3292	0.6204	0.1033	0.7409
MK08_6_3	0.1645	0.7556	0.2837	0.5688	0.2773	0.6824	0.0457	0.7952
MK08_6_5	0.2029	0.6224	0.3517	0.5749	0.2783	0.6244	0.0785	0.7493
MK08_6_10	0.2067	0.5198	0.3019	0.4323	0.3225	0.4896	0.0787	0.6171
MK09_6_3	0.0852	0.5868	0.2973	0.5102	0.1896	0.5907	0.0336	0.7160
MK09_6_5	0.2139	0.6444	0.3963	0.5363	0.2742	0.6025	0.1080	0.7495
MK09_6_10	0.1990	0.5699	0.3195	0.5767	0.2683	0.5964	0.0918	0.7033
MK10_9_5	0.1711	0.6909	0.3173	0.5237	0.3204	0.5457	0.1032	0.6663
MK10_9_8	0.2454	0.6700	0.3531	0.4643	0.2680	0.5742	0.1164	0.6864
MK10_9_15	0.1575	0.6744	0.3280	0.5062	0.2698	0.5660	0.0805	0.6888
MK11_3_2	0.1010	0.5984	0.2786	0.5153	0.2866	0.5125	0.0435	0.6513
MK11_3_5	0.2013	0.6342	0.3677	0.5999	0.2878	0.6703	0.0720	0.7777
MK11_3_3	0.0638	0.6879	0.3226	0.5954	0.2754	0.5845	0.0909	0.7257
MK12_6_3	0.1458	0.5345	0.2969	0.4730	0.1998	0.5669	0.0361	0.6782
MK12_6_5	0.1470	0.5558	0.2521	0.4972	0.2236	0.4950	0.0342	0.6425
MK12_6_10	0.1055	0.6647	0.3062	0.6239	0.2711	0.6259	0.0295	0.7708
MK13_6_3	0.1662	0.6192	0.3400	0.4477	0.2445	0.4810	0.0567	0.5823
MK13_6_5	0.1504	0.6436	0.2801	0.6118	0.2972	0.6589	0.0750	0.7700
MK13_6_10	0.1830	0.6003	0.2417	0.5868	0.2545	0.5952	0.0392	0.7075
MK14_9_5	0.0883	0.5527	0.3827	0.5578	0.2664	0.5641	0.0989	0.6811
MK14_9_8	0.1366	0.5366	0.2568	0.5266	0.2056	0.5357	0.0142	0.6731
MK14_9_15	0.1614	0.6520	0.3658	0.5071	0.3027	0.5745	0.0995	0.6845
MK15_9_5	0.1697	0.7319	0.3046	0.5157	0.2661	0.6026	0.0473	0.7163
MK15_9_8	0.1618	0.6735	0.3407	0.5709	0.3061	0.6273	0.0604	0.7524
MK15_9_15	0.1838	0.6515	0.3467	0.4704	0.2242	0.5548	0.0639	0.6601

Bold text indicates the optimal values achieved by different algorithms for the corresponding test cases.

. Regarding the IGD metric detailed in Table 6, MODGWO demonstrates a notable superiority over its rival algorithms across all the datasets. In 45 cases, MODGWO achieves 39 minimum values, while MOJAYA achieves 6 minimum values. It indicates that MODGWO exhibits superior exploration and exploitation capabilities. In comparison to MOPSO and NSGA-II, MOJAYA exhibits superior performance. Additionally, NSGA-II displays a competitive performance that is comparable to MOPSO. As for the HV metric, a similar trend is observed in Table 6, and MODGWO demonstrates a significantly better performance as compared to the other algorithms. In 45 cases, the MODGWO achieves 38 optimal values, and the MOJAYA achieves 7 optimal values. MOJAYA offers a better performance in terms of the obtained metric values and the number of significantly better instances.

This indicates that both MODGWO and MOJAYA can obtain good solutions when solving MRCFJSP, with MODGWO yielding better results than MOJAYA. From Equation (26), it can be seen that the Jaya algorithm drives the entire population to converge quickly to the global optimal solution through the optimal solution at each iteration, while from Equation (25), the GWO algorithm drives the population to converge to the global optimal solution through the first three optimal solutions. This ensures the diversity of solutions and prevents the population from falling into local optimal solutions earlier. Therefore, MODGWO proposed in this paper combines the advantages of the GWO algorithm and Jaya algorithm, which can effectively solve MRCFJSP.

To gain a deeper understanding of MODGWO’s performance, especially regarding its convergence and diversity in tackling optimization challenges of varying scales, we compare its effectiveness based on small, medium, and large problem sizes. Benchmark instances are classified based on their jobs, machines, and operations. This detailed classification can more thoroughly evaluate the capabilities of the algorithms. The performance of the compared algorithms in terms of IGD and HV metrics for the selected instances are shown in Figure 11, Figure 12 and Figure 13.

Figure 11 depicts the comparative performance of the algorithms on smaller-scale instances. In Figure 11a, the IGD metric is used to assess the proximity and distribution of the obtained solutions to the true Pareto front. The results show that MODGWO consistently outperforms the other algorithms, achieving the lowest IGD values for most instances. For example, in instances MK01_4_2 and MK01_4_3, MODGWO demonstrates superior performance, indicating that its solutions are more accurate and well distributed compared to MOJAYA, MOPSO, and NSGA-II. However, there are a few exceptions where MOJAYA slightly outperforms MODGWO, such as in instances MK02_4_6 and MK05_3_2. Figure 11b illustrates the performance of the algorithms in terms of the Hypervolume (HV) metric. The results show that MODGWO achieves the highest HV values for most small-scale instances, demonstrating its superior ability to generate high-quality and diverse solutions. For instance, in instances MK01_4_2 and MK01_4_6, MODGWO’s HV values are significantly higher than those of MOJAYA, MOPSO, and NSGA-II. This suggests that MODGWO not only finds solutions that are closer to the true Pareto front but also ensures wide coverage of the objective space. MOJAYA also shows competitive performance in some instances, while MOPSO and NSGA-II exhibit similar performance levels, indicating comparable diversity and convergence in their respective Pareto solutions.

The comparison of medium-scale instances is shown in Figure 12. As seen in Figure 12a, in terms of the IGD metrics, MODGWO generally achieves the lowest IGD values across most medium-scale instances, demonstrating superior performance in terms of solution accuracy and distribution. For example, MODGWO outperforms MOJAYA, MOPSO, and NSGA-II in instances such as MK07_3_2 and MK07_3_3. However, there are a few exceptions where MODGWO does not achieve the lowest IGD values, such as in instances MK06_6_5 and MK07_3_5. In these cases, MOJAYA shows slightly better performance. Overall, MODGWO’s low IGD values highlight its strong ability to balance exploration and exploitation in medium-scale problems. Regarding the HV metric depicted in Figure 12b, the results show that MODGWO achieves the highest HV values for most medium-scale instances, such as MK08_6_3 and MK09_6_5. This indicates that MODGWO not only finds solutions that are closer to the true Pareto front but also ensures wide coverage of the objective space. MOJAYA also shows competitive performance in some instances, while MOPSO and NSGA-II exhibit similar performance levels, indicating comparable diversity and convergence in their respective Pareto solutions. The similarity between MOPSO and NSGA-II suggests that both algorithms are effective in medium-scale problems but may not match the performance of MODGWO in terms of solution quality and diversity.

Figure 13 presents a comparison of algorithms on large-scale instances. In Figure 13a, MODGWO demonstrates a strong advantage, achieving the lowest IGD values across nearly all the large-scale instances. For example, MODGWO significantly outperforms MOJAYA, NSGA-II, and MOPSO in instances such as MK13_6_10 and MK14_9_8, indicating its superior ability to generate solutions that are both highly accurate and well distributed. This suggests that MODGWO is particularly effective at navigating the complexity of large-scale problems, maintaining high solution quality even as the problem size increases. However, there are a few instances (e.g., MK11_3_3 and MK14_9_5) where MOJAYA shows comparable performance, highlighting that while MODGWO is generally dominant, MOJAYA can still offer competitive results in specific scenarios. In Figure 13b, it is observed that the performance of MODGWO demonstrates a similar trend as compared to the IGD metric. MODGWO exhibits a superior performance compared to the other three algorithms. Meanwhile, MOJAYA, NSGA-II, and MOPSO exhibit relatively similar performance, indicating that while they can produce satisfactory results, they are less effective than MODGWO in optimizing the trade-offs between different objectives for large-scale problems. This confirms that MODGWO efficiently balances maintaining diversity in the search process with improving the convergence of solutions along the Pareto front.

3.5. Scheduling Gantt Chart

To demonstrate the internal scheduling situation of the production workshop, select a scheduling result from MK05_3_4 and draw its scheduling Gantt chart. This case involves 15 jobs, four machines, three workers, and four AGVs, where the set of machines that can be operated by the workers is shown in

Table 7. The set of machines that can be operated by workers in case MK05_3_4.

Worker Number	The Set of Machines Operated by Workers
W1	{M1, M2, M4}
W2	{M2, M3}
W3	{M1, M3, M4}

, and the transportation time of AGVs between each machine and the LU area is based on the Layout1 proposed by Bilge and Ulusoy [40]. The Pareto optimal solution for MK05_3_4 obtained by MODGWO is [501, 32], [381, 54], [427, 37], [404, 48], [411, 44], [451, 36], and [412, 40], and the Gantt chart obtained by selecting [501, 32] is shown in Figure 14.

Figure 14 presents the scheduling Gantt chart for the instance MK05_3_4, which involves 15 jobs, four machines, three workers, and four AGVs. The Gantt chart visually illustrates the scheduling outcomes obtained using the MODGWO algorithm, highlighting the allocation of operations to machines and the transportation of jobs by AGVs. The horizontal axis represents the processing time of operations on machines and the transportation time of jobs by AGVs. Each bar in the chart corresponds to a specific operation or transportation task, with the length of the bar indicating the duration of the task. The vertical axis lists the machines (M1–M4) and AGVs (AGV1–AGV4). The bars for machines represent the processing operations, while the bars for AGVs represent the transportation of jobs between the loading/unloading (LU) area and the machines. The numbers in the rectangular boxes corresponding to machines are in the format “x_y_z”, where x is the job number, y is the operation number within the job, and z is the worker number assigned to the operation. For example, “3_1_3” on M4 indicates that the first operation of job 3 is processed by worker 3 on machine M4. The numbers in the rectangular boxes corresponding to AGVs are in the format “x_y”, where x is the job number, and y is the operation number within the job. For example, “4_1” on AGV2 indicates that job 4 before operation O₄₁ is transported by AGV2.

The Gantt chart in Figure 14 clearly shows how the scheduling respects the constraints of machines, workers, and AGVs. For instance, “5_3_1” on M4 must wait for AGV2 to transport “5_3” to M4 and for the completion of the previous operation “O₅₂”. It also needs to wait for worker 1 to finish the previous task “10_2_1” on M1. The chart demonstrates the efficiency of the MODGWO algorithm in balancing the workload across machines and AGVs. The operations are scheduled in a way that minimizes the makespan while ensuring that all the resource constraints are met. The Gantt chart also highlights the utilization of workers. Each worker is assigned to specific operations, and their tasks are scheduled to avoid conflicts and ensure smooth operation. The transportation tasks of AGVs are also optimized. AGVs are assigned to transport jobs between the LU area and machines, ensuring that the transportation time is minimized and does not become a bottleneck in the scheduling process.

At the same time, we conducted a scheduling scheme that only satisfies the single-objective constraint of makespan and compared it with the scheduling scheme that considers the continuous working hours of workers under the dual-objective optimization, as shown in Figure 15. The horizontal axis represents the time, showing the duration of each operation and the overall makespan. The vertical axis lists the workers (W1–W3) and their assigned tasks. The rectangular boxes represent the tasks assigned to each worker with the format “x_y” where x is the job number and y is the operation number within the job.

Figure 15 provides a detailed comparison between two distinct scheduling approaches for a production scenario, highlighting the differences between single-objective optimization focused solely on makespan (Figure 15a) and dual-objective optimization that balances makespan with workers’ continuous working hours (Figure 15b). In the single-objective plan, the goal is to minimize the makespan, resulting in a total completion time of 359 time units. However, this approach leads to long continuous working periods for workers, which can significantly reduce work efficiency and increase the risk of errors and safety incidents due to fatigue. The Gantt chart in Figure 15a shows that workers are assigned tasks without considering their continuous working hours, leading to extended periods of labor without adequate rest.

In contrast, the dual-objective plan in Figure 15b takes a more balanced approach by optimizing both the makespan and the maximum continuous working hours of workers. This plan achieves a makespan of 501 time units, which is longer than the single-objective plan but offers substantial benefits in terms of worker well-being. The longest continuous working time for any worker is limited to 32 time units (e.g., from “3_7” to “5_6” for worker W2), ensuring that workers have sufficient rest periods to maintain high productivity and reduce safety risks. The Gantt chart in Figure 15b illustrates how tasks are distributed among workers to avoid long continuous shifts, promoting a healthier and more sustainable working environment.

The comparison between these two scheduling plans underscores the importance of considering workers’ continuous working hours alongside the makespan. While the single-objective plan prioritizes production speed, it neglects the human factor, which can ultimately lead to inefficiencies and safety issues. The dual-objective plan, on the other hand, not only improves the overall production efficiency by maintaining a balanced workload but also enhances worker well-being by ensuring adequate rest periods. This approach aligns with modern people-oriented production concepts, emphasizing the importance of balancing productivity with worker health and safety. By adopting the dual-objective optimization strategy, the production process becomes more sustainable, efficient, and aligned with the needs of both the workforce and the manufacturing goals.

3.6. Examples of Agricultural Machinery and Equipment Production

The products of agricultural machinery and equipment manufacturing enterprises are involved in various aspects of agricultural production such as cultivation, planting, fertilization, irrigation, harvesting, transportation, and processing. The main processing techniques for its product components include “cutting”, “turning”, “boring”, “milling”, “drilling”, “rolling”, “inserting”, and “heat treatment”. Due to its wide range of product models, many of which are personalized for customers, the product structure, standards, and process routes are complex and varied. As a result, it is very difficult to prepare the operation plan by manually operating the computer.

In this section, the actual production data of a large-scale seeding equipment manufacturing enterprise is used as an example, and the proposed MODGWO algorithm and other comparative algorithms are utilized to solve this problem in order to verify and compare the effectiveness and performance of the proposed algorithm in solving practical problems. The actual production data were obtained from a batch production task during a week in April 2024 in a cotton seeding equipment production workshop in the Aksu region, Xinjiang, China. For simplicity, scheduling was performed according to individual parts to verify the effectiveness of the proposed algorithm in this paper. Finally, the actual production scheduling of the workshop was calculated by multiplying the scheduling plan for individual parts by the number of parts in each batch. From the company’s products, five jobs with 18 operations were processed on 11 machines, and the number of AVGs was five. The detailed information on the machines is shown in

Table 8. Machine information.

Machine Name	Machine Number
CNC lathe	M1
	M2
Laser cutting machine	M3
Automatic straightening machine	M4
Spray painting robot	M5
	M6
CNC milling machine	M7
CNC grinding machine	M8
High-frequency quenching machine	M9
Forging machine	M10
	M11

, and the detailed information on the operation processing is shown in

Table 9. Operation processing information.

Job	Operation	Operation Name	Machine	Processing Time
Shaftless soil conveyor pig iron roller pin	O11	Milling	M7	50
	O12	Grinding	M8	70
	O13	High-Frequency Hardening	M9	120
Soil-covered guide frame suspension rod	O21	Cutting	M3	150
	O22	Sandblasting	M5/M6	80/65
	O23	Straightening	M4	40
Extinguisher Traction Column	O31	Forging	M10/M11	50/40
	O32	Turning	M1/M2	130/110
	O33	Grinding	M8	70
	O34	Sandblasting	M5/M6	75/60
Short Wheel tail pin	O41	Forging	M10/M11	20/10
	O42	CNC Turning	M1/M2	90/70
	O43	CNC Grinding	M8	65
	O44	Sandblasting	M5/M6	50/35
Long four-bar linkage pin	O51	Forging	M10/M11	35/25
	O52	Rough Machining	M1/M2	80/60
	O53	Fine Machining	M1/M2	60/40
	O54	High-Frequency Hardening	M9	100

. The transportation time of AGVs between each machine and the LU area is shown in

Table 10. The transportation time between each machine and the LU area.

	LU	M1	M2	M3	M4	M5	M6	M7	M8	M9	M10	M11
LU	0	10	20	30	25	35	45	15	25	35	30	40
M1	10	0	5	30	50	60	70	40	50	60	100	110
M2	20	5	0	30	50	60	70	40	50	60	100	110
M3	30	30	30	0	20	10	20	50	60	70	110	120
M4	25	50	50	20	0	30	40	70	80	90	130	140
M5	35	60	60	10	30	0	5	80	90	100	140	150
M6	45	70	70	20	40	5	0	90	100	110	150	160
M7	15	40	40	50	70	80	90	0	10	20	60	70
M8	25	50	50	60	80	90	100	10	0	10	70	80
M9	35	60	60	70	90	100	110	20	10	0	80	90
M10	30	100	100	110	130	140	150	60	70	80	0	5
M11	40	110	110	120	140	150	160	70	80	90	5	0

, and the set of machines that can be operated by the workers is shown in

Table 11. The set of machines that can be operated by workers in the enterprise.

Worker Number	The Set of Machines Operated by Workers
W1	{M1, M2, M5, M6, M10, M11}
W2	{M1, M2, M5, M6, M7}
W3	{M1, M2, M3, M4, M5, M6}
W4	{M3, M4, M8, M10, M11}
W5	{M3, M4, M9, M10, M11}
W6	{M1, M2, M4, M8, M10, M11}
W7	{M1, M2, M5, M6, M9}

.

The instance is solved using MODGWO proposed in this paper and the other three algorithms in Section 3.4, and the performance indicators are shown in Table 11. From

Table 12. Running results.

Algorithms	IGD	HV
MODGWO	0.0824	0.6457
MOJAYA	0.1513	0.6011
NSGA-II	0.2932	0.5246
MOPSO	0.3828	0.4876

Bold text indicates the optimal values achieved by different algorithms for the corresponding test cases.

, it can be seen that both indicator values of MODGWO are superior to the other three algorithms, indicating that MODGWO still has advantages compared to the other comparative algorithms in solving enterprise instances.

In order to clearly display the results obtained by the different algorithms,

Table 13. Pareto solutions for four algorithms.

Algorithms	Number of Solutions	Pareto Solutions
MODGWO	2	[610, 120], [625, 100]
MOJAYA	4	[610, 170], [625, 120], [640, 110], [670, 100]
NSGA-II	4	[635, 120], [630, 160], [660, 110], [675, 100]
MOPSO	5	[645, 120], [640, 170], [680, 115], [790, 110], [915, 100]

lists the Pareto solution sets obtained by the four algorithms. The first value in each Pareto solution represents the maximum completion time, i.e., makespan, and the second value represents the maximum continuous working hours of all the workers (MCWH). It can be seen from the table that MODGWO obtained the highest quality solution.

The set of Pareto solutions in Table 13 is plotted as a scatter plot as shown in Figure 16. Figure 16 presents a scatter plot comparing the Pareto solutions obtained by MODGWO, MOJAYA, NSGA-II, and MOPSO for the MRCFJSP. The plot illustrates the trade-off between makespan (X-axis) and the maximum continuous working hours of workers (Y-axis). MODGWO’s solutions are concentrated in the lower-left part of the plot, indicating superior performance in balancing both objectives. These solutions, such as [610, 120] and [625, 100], show shorter makespans and lower worker fatigue compared to other algorithms. MOJAYA’s solutions are more spread out, with examples like [610, 170] and [670, 100], indicating a wider range of trade-offs but generally dominated by MODGWO. NSGA-II and MOPSO exhibit higher variability and less optimal trade-offs, with solutions like [635, 120], [675, 100] for NSGA-II and [645, 120], [915, 100] for MOPSO, which are less efficient in balancing the objectives. Overall, MODGWO demonstrates the best balance between makespan and worker well-being, resulting in higher-quality Pareto solutions and more sustainable scheduling outcomes.

4. Discussion

The research presented in this paper addresses the MRCFJSP by proposing a MODGWO algorithm. This problem is particularly complex due to the simultaneous consideration of machine, worker, and AGV constraints, aiming to optimize both the makespan and workers’ continuous working time.

The MODGWO algorithm introduced in this study incorporates several innovations to effectively tackle the MRCFJSP. Firstly, an encoding/decoding method and a mixed population initialization strategy are designed to create a diverse set of initial solutions. This initialization process is crucial as it impacts the subsequent search performance of the algorithm. The results indicate that MODGWO, with its effective diversity strategy, outperforms a random initialization method, achieving better quality and more Pareto solutions.

Furthermore, the MODGWO algorithm integrates a solution-updating mechanism tailored for discrete scheduling problems. This mechanism is inspired by the Grey Wolf Optimizer but adapted to include cross-operation principles from genetic algorithms. By incorporating the Jaya algorithm’s principles of continuous improvement, MODGWO further enhances its solution quality. The integration of these algorithms allows MODGWO to effectively explore and exploit the search space, leading to superior performance compared to other metaheuristic algorithms such as MOJAYA, MOPSO, and NSGA-II.

The experimental results demonstrate MODGWO’s superiority across various performance indicators, including the Inverted Generational Distance (IGD) and Hypervolume (HV). MODGWO achieves the best results in most instances, particularly in medium- and large-scale problems, highlighting its robustness and scalability. In addition to quantitative results, this study also presents scheduling Gantt charts to visually demonstrate the internal scheduling situation of the production workshop. These charts illustrate the differences between single-objective and dual-objective optimization approaches, emphasizing the importance of considering workers’ continuous working time alongside makespan. By doing so, MODGWO contributes to improving work efficiency and reducing production safety risks.

Despite the promising results, several areas for future research can be identified. First, exploring additional constraints and objectives within the MRCFJSP framework could further enrich the problem’s complexity and practical relevance. For instance, incorporating energy consumption or machine maintenance schedules could provide a more holistic view of the scheduling problem. Second, hybridizing MODGWO with other advanced metaheuristic algorithms or machine learning techniques may yield even better performance. Lastly, implementing MODGWO in real-world industrial settings and evaluating its practical impact would be a valuable contribution to the field.

In conclusion, this paper proposes a novel MODGWO algorithm for solving the MRCFJSP, demonstrating its effectiveness and superiority over existing algorithms. By considering both makespan and workers’ continuous working time, MODGWO contributes to more efficient and sustainable production scheduling. The research findings not only advance the theoretical understanding of complex scheduling problems but also offer practical insights for industrial applications.

5. Conclusions

Agricultural machinery and equipment production is greatly affected by market demand and seasons, and is characterized by fast production conversion and strong dynamics. The method of flexible job shop scheduling fully utilizes the flexible characteristics of various resources, which can greatly improve the production efficiency of enterprises when applied to agricultural machinery and equipment production enterprises. This paper proposes a MODGWO algorithm that integrates a hybrid initialization strategy and multi-neighborhood local search for flexible job shop scheduling problems under multi-resource constraints of integrated workers and AGVs. On the basis of an in-depth analysis of the actual operation of various resources, a flexible job shop scheduling model was constructed, which includes four production factors: jobs, machines, AGVs, and workers. The effectiveness of the hybrid initialization method and the multi-neighborhood local search is verified by comparison based on several instances. The results compared with various advanced algorithms show that the proposed algorithm can obtain better scheduling solutions when solving this type of scheduling problem.

The method proposed in this paper has been validated in the actual manufacturing of seeding locomotives. The scheduling scheme proposed in this paper not only improves production efficiency, but also balances the two goals of completion time and workers’ continuous working hours. In the future, based on the current research, we will further construct a multi-objective multi-resource constrained scheduling model, study efficient multi-objective optimization methods, and utilize the current popular deep reinforcement learning method for solving.