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Article

Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search

1
School of Manufacturing Institute, Nanyang Institute of Technology, Nanyang 473004, China
2
State Key Laboratory of High-End Compressor and System Technology, Hefei General Machinery Research Institute, Hefei 230031, China
*
Author to whom correspondence should be addressed.
Processes 2025, 13(8), 2325; https://doi.org/10.3390/pr13082325
Submission received: 10 June 2025 / Revised: 16 July 2025 / Accepted: 16 July 2025 / Published: 22 July 2025
(This article belongs to the Section Process Control and Monitoring)

Abstract

A green scheduling problem is proposed in this work, where both constraints on intermediate storage capacity and job transportation requirements are simultaneously considered. An improved discrete pathfinder algorithm (IDPFA) with multi-neighborhood local search is proposed to minimize the maximum completion time and total energy consumption. The algorithm addresses the green flow shop scheduling problem with limited buffers and automated guided vehicle (GFSSP_LBAGV). Firstly, based on the machine speed constraints, the transportation time for moving jobs by the automated guided vehicle (AGV) is incorporated to establish a mathematical model. Secondly, the core idea of the pathfinder algorithm (PFA) is applied to the evolutionary process of the discrete PFA, where three different crossover operations are used to replace the exploration process of the pathfinder, the influence of the pathfinder on the followers, and the mutual learning among the followers. Then, a multi-neighborhood local search is employed to conduct a detailed exploration of high-quality solution spaces. Finally, extensive standard test sets are used to verify the effectiveness of the proposed IDPFA in solving GFSSP_LBAGV.

1. Introduction

Green manufacturing not only requires the consideration of environmental impacts but also ensures the production efficiency of enterprises. Green shop scheduling, which is oriented toward green manufacturing, aims to achieve low-carbon emission reduction, efficiency improvement, and resource saving by optimizing production processes, resource allocation, and operational modes. In recent years, many research papers on this topic have been published by international engineering societies and journals, mainly focusing on the following three aspects:
  • Optimizing energy consumption indicators such as carbon emissions [1,2,3];
  • Reducing machine idle time by utilizing startup and shutdown constraints [4,5];
  • Adjusting machine speeds to achieve energy savings and emission reductions without compromising economic performance [6].
With growing global emphasis on low-carbon environmental protection and sustainable development, energy-saving and emission-reduction technologies in the construction machinery sector have become a research hotspot. Optimizing energy management strategies (EMSs) involves both system-level configuration improvements and innovative control algorithm design. Hybrid construction machinery (HCM) has gained prominence for reducing greenhouse gas emissions and fossil fuel consumption, yet optimizing energy management strategies remains critical to enhance efficiency, cost savings, and durability, necessitating a comprehensive review of engine and fuel cell-based HCM configurations, optimization techniques, and future challenges [7]. Phan [8] develops an adaptive disturbance observer (ADO)-based nonlinear tracking control for a PEMFC-powered mini-excavator, addressing oxygen starvation under load variations and model uncertainties to enhance fuel cell efficiency, mitigate energy waste, and ensure stable air feed regulation.
For the green shop scheduling problem, Kemmoe [9] proposed a greedy randomized adaptive search algorithm with the primary optimization objective of minimizing the maximum completion time. The local search mechanism was further improved, and the experimental comparison showed that the proposed method was superior to the existing algorithms in solving similar problems. Tian [10] studied the dynamic flexible job shop scheduling problem for multi-variety, small-batch aerospace industrial production. An energy-efficient shop floor model was established aiming to minimize total energy consumption, completion time, and processing costs, and a multi-objective dual-population differential artificial bee colony algorithm was proposed to solve the model. Wu [11] addressed the green hybrid flow shop scheduling problem with sequence-dependent setup times and transportation times. A mixed integer programming model with the dual objectives of minimizing the maximum duration and total energy consumption was formulated, and an improved memetic algorithm was proposed according to the characteristics of the problem. Li [12] investigated the green scheduling problem for flexible job shop systems, considering energy consumption and worker learning effects. The problem was modeled as a mixed-integer linear multi-objective optimization problem aiming to simultaneously minimize makespan and total carbon emissions. An improved multi-objective sparrow search algorithm was proposed to find optimal solutions. Zhou [13] studied a low-carbon flexible job shop scheduling problem and proposed a gray wolf optimization algorithm with the objective of minimizing the sum of carbon emission costs and completion time costs. A Q-learning-based multi-objective particle swarm optimization algorithm was introduced to solve the distributed flexible job shop scheduling problem, aiming to minimize both makespan and total energy consumption [14].
Currently, while pursuing the intelligence of production equipment, enterprises are also delving deeply into the intelligence of manufacturing processes. Particularly with the proposal and construction of “smart factory” strategies in industries such as chemical pharmaceuticals, semiconductor manufacturing, and steel production, production scheduling with limited intermediate storage capacity has once again become a hot research topic among scholars, that is, the flow shop scheduling problem with limited buffers (FSSP_LB). It has been proven that even FSSP_LB with only two machines is an NP-hard problem [15]. For FSSP_LB, Wang [16] proposed a hybrid genetic algorithm with multiple genetic operators to solve the permutation flow shop scheduling problem with limited buffer, aiming at minimizing the maximum completion time, which effectively balanced the relationship between global search and local search. Qian [17] discretized the differential evolution algorithm to solve single-objective and multi-objective FSSP_LB by the LOV rule and integrated the local search strategy of the structural nature of the problem into the algorithm. Aiming at FSSP_LB with limited production time, an improved discrete empire competition algorithm was proposed to solve it with the objective of minimizing the completion time [18]. A hybrid shuffled frog leading algorithm was proposed to minimize the maximum completion time, and the performance of the proposed algorithm was evaluated by a benchmark test [19]. Dong [20], aiming at the multi-objective flow shop scheduling problem with limited buffers, presented a hybrid pathfinder algorithm to minimize the electricity cost and the maximum tardiness. Lu [21] is the first attempt to study this distributed permutation flow-shop problem with limited buffers (DPFSP-LB) with objectives of minimizing makespan and total energy consumption. To solve this energy-efficient DPFSP-LB, a Pareto-based collaborative multi-objective optimization algorithm was proposed.
With the deepening advancement of intelligent manufacturing and the continuous development of artificial intelligence technologies, the construction of smart factories is vigorously underway across industries. The automated guided vehicles (AGVs), as a critical tool in workshop production systems, present a central challenge in smart factory development: how to achieve seamless integration with manufacturing operations and enhance collaborative coordination mechanisms. Therefore, the green integration scheduling problem (GISP) is highly concerned and widely studied by combining production scheduling with logistics distribution and automated guided vehicles and introducing green evaluation indicators such as low carbon and energy saving. For the integrated scheduling problem of workshop production and AGV, the existing literature is mostly combined with job shop scheduling. Aiming at the machine and AGV scheduling problem in a flexible manufacturing system, a Dijkstra algorithm based on the time window was proposed for solving the issue, under the constraint of a limited number of automated guided vehicles, with the optimization objective of minimizing the maximum completion time [22]. To tackle the green integrated scheduling problem of flexible job shops and crane transportation, Liu [23] developed a mixed-integer programming model aimed at minimizing the total energy consumption cost and reducing the maximum completion time. For fuzzy integrated cell formation and production scheduling considering automated guided vehicles and human factors, Alireza [24] designed a mixed-integer linear programming model with the optimization objectives of minimizing the completion time and intercellular movements. A hybrid algorithm combining a genetic algorithm and whale optimization algorithm was proposed to solve the problem. In the framework of production and logistics integration in a machining workshop based on industrial internet, Zhang [25] achieved the intelligent modeling of critical manufacturing resources and investigated the mechanisms of self-organizing configuration. For the green scheduling of flexible manufacturing cells with auto-guided vehicle transportation, a bi-objective optimization model was established to achieve the minimization of the maximum completion time and the total energy consumption. An improved bi-objective salp swarm algorithm based on decomposition was proposed and applied to the problem [26].
As the FSSP_LB serves as a scheduling model for numerous production processes and possesses the unique characteristic of limited buffer sizes, the application of AGVs may exist both between machines and buffers as well as between machines themselves. In this paper, a GISP scheduling model is established for the FSSP_LB with automated guided vehicles. Based on the existing literature review, it has been found that there is currently no related research on solving the GISP. Therefore, the study of this problem holds significant academic value and engineering importance.
The pathfinder algorithm (PFA) is a swarm intelligence algorithm proposed by Yapici in 2019 [27]. Inspired by the hunting behavior of group animals, the algorithm divides individuals in a population into pathfinders (leaders) and followers. The optimization process of the pathfinder algorithm simulates the foraging behavior of the population, achieving optimization through the interaction between the two different roles of pathfinders and followers. Meanwhile, during the evolutionary process, it also enhances the retention of superior information across three generations of the population. The pathfinder algorithm boasts advantages such as ease of understanding and simple implementation. Furthermore, its search performance surpasses that of the particle swarm optimization and teaching–learning-based optimization algorithms, which are already widely applied in combinatorial optimization problems [27]. In general, the standard pathfinder algorithm cannot be applied directly to solve job shop scheduling problems (discrete problems). It requires certain methods to map between the real number domain and the discrete domain, with the largest-order-value (LOV) rule being one such method. However, evaluating solutions using the LOV rule can further increase the evaluation time of solutions. Therefore, many scholars have opted to directly discretize similar algorithms, such as the artificial bee colony algorithm and the teaching–learning-based optimization algorithm, to address discrete problems [28,29]. Drawing from the experience of other discrete optimization algorithms, this article discretizes the pathfinder algorithm to further enhance its search performance.
The standard pathfinder algorithm selects the individual with maximum dispersion in the Pareto front as the pathfinder when solving multi-objective problems. However, since multiple optimal solutions (non-dominated solutions) typically exist in multi-objective optimization, employing only one pathfinder inevitably limits the algorithm’s search breadth. To address this limitation, the improved discrete pathfinder algorithm (IDPFA) proposes three key enhancements: firstly, all non-dominated solutions in the current population are designated as pathfinders to guide the optimization process. These pathfinders undergo position updates through multi-neighborhood local search operations, enabling extensive exploration within their neighborhoods. Secondly, the follower update mechanism incorporates a distance-based selection strategy, where each follower selects the nearest pathfinder for guidance. The position updates integrate two distinct crossover operations to enhance search diversity. Thirdly, to prevent followers from being trapped in local optima during updates, a random regeneration strategy is implemented to replace stagnant individuals, thereby enriching population diversity. Experimental results demonstrate that these improvements collectively enhance the algorithm’s search capability significantly.
This paper proposes a GFSSP_LBAGV scheduling model that comprehensively considers the maximum completion time and total energy consumption and discretizes the original PFA to design an improved discrete pathfinder algorithm (IDPFA) for the solution. The GFSSP_LBAGV scheduling model not only takes into account the machine speed constraints but also integrates the scheduling of transportation times between machines and AGVs in buffer zones. Investigating this problem helps to improve the resource utilization of machines, AGVs, and buffer zones, achieving the effect of collaborative optimization in workshop production.

2. Problem Statement

2.1. Problem Description of FSSP_LB

The FSSP_LB can be described as follows: (1) The processing sequence of all jobs on m machines is unchanged, and the processing time for each operation is fixed. (2) At all hours, each machine can handle at most one job, and each job can be handled on at most one machine. (3) On each machine, all jobs are processed on the machine in the same order. (4) The job needs to enter the buffer zone when the current operation is completed, but the machine where the next operation is located is in the processing state. If the buffer is full, it will be blocked on the current machine until the buffer is empty. (5) When no job can be stored in the buffer and the downstream machine is idle, the job proceeds directly to the next machine for processing. If the buffer can store multiple jobs, the jobs will enter the buffer in turn and be taken out in the order of entry; that is, it is agreed that all jobs will obey the First In First Out (FIFO) rule in the buffer. The mathematical notations and definitions used in this paper are summarized in Table 1.

2.2. Mathematical Model of GFSSP_LBAGV

The GFSSP_LBAGV can be described as follows: the processing of n jobs J = { 1 , 2 , , n } on m machines M = { 1 , 2 , , m } in a production workshop, with g AGVs W = { 1 , 2 , , g } transporting the jobs. The production process of all jobs still adheres to the constraints of FSSP_LB. At the same time, after a job is processed on the current machine, it will be transported by an AGV to the next machine or a buffer zone. All AGVs start from the starting station (IN) and eventually reach the ending station (OUT) to complete their tasks. Additionally, the following assumptions are made for the production scheduling process:
  • The time for jobs to arrive at the first machine (transportation time) is not considered;
  • An AGV can only transport one job at a time;
  • The speed of the AGV is unaffected by whether it is carrying a load or running empty;
  • Collisions and failures during AGV transportation are not considered;
  • The transportation time of the AGV is solely related to the current transportation distance;
  • The number of AGVs is sufficient to ensure that a job is immediately transported to the designated location after being offloaded.
Let π = ( π ( 1 ) , π ( 2 ) , , π ( n ) ) denote the processing sequence of jobs, B j represent the buffer size between machines j and j + 1 , D π ( i ) , j , g indicate the completion time of job π ( i ) on machine j with g AGVs, and P i , j signify the processing time of job i on machine j . T j , j + 1 π ( i ) denotes the transportation time of the job π ( i ) between machines, T j , B j π ( i ) denotes the transportation time of the job π ( i ) from machine j to buffer B j , and T B j , j π ( i ) denotes the transportation time of the job π ( i ) from buffer B j to buffer machine j .
The example of 4 × 4 Gantt chart (4 jobs processed on 4 machines) GFSSP_LBAGV is shown in Figure 1. In Figure 1, the numbers represent job IDs, dashed lines indicate the transition instants when jobs enter corresponding states, green boxes denote buffer occupancy periods, and gray hatched boxes show blocking time intervals. In the chart, all buffer sizes are set to one, and AGVs are required to transport the jobs, with the transportation time solely determined by the distance of the current transfer and already specified.
In Figure 1, job 1 is not constrained by the buffer or machine limitations, and it is immediately transported by an AGV to the next machine for processing once its current operation is completed. Let t 0 denote the start time of job1 on Machine M 1 , t 1 represent both the completion time of job1 on M 1 and the initiation time of AGV transportation, and define the material handling time T 1 , 2 1 as the time difference between t 2 and t 1 ( T 1 , 2 1 = t 2 t 1 ). Job 2 must satisfy the machine constraints during production. After completing processing on machine M 1 , it needs to enter the buffer B 1 to wait for machine M 2 to become available. The processes of entering and leaving the buffer are also carried out by AGVs. The material handling time T 1 , B 1 2 for transporting job 2 from M 1 to buffer B 1 is defined as the time difference between t 4 and t 3 ( T 1 , B 1 2 = t 4 t 3 ), while the handling time T B 1 , 2 2 for transporting from buffer B 1 to M 2 equals the time difference between t 7 and t 6 ( T B 1 , 2 2 = t 7 t 6 ). Job 3 must satisfy buffer constraints during processing. At time t 5 , upon completion on machine M 1 , since machine M 2 is processing job 1 and buffer B 1 is occupied by the job 2, job 3 will experience a blockage period t 5 - t 6 . Similarly, due to buffer constraints, job 4 also forms a blockage segment t 10 - t 11 . Moreover, the transportation time of AGVs must be taken into consideration when determining whether a job should enter a buffer. For instance, at time t 8 , after the completion of job 3 on machine M 2 , it is directly transported to machine M 3 for subsequent processing, incorporating the AGV transportation time into the decision. Consequently, the inherent complexity of the GFSSP_LBAGV model significantly increases the difficulty of solving this problem.
In general, the transportation of jobs also takes into account the loading and unloading time of AGVs. Therefore, the time T j , j + 1 π ( i ) required to transport a job from the current machine to the next machine should be less than the total time required to first transport the job to buffer B j and then to the next machine. This constraint can be expressed as
T j , j + 1 π ( i ) < T j , B j π ( i ) + T B j , j + 1 π ( i )
At this juncture, due to the presence of this constraint, a special scenario arises where the difference between the off-machine time D π ( i 1 ) , j , g of a job on the current machine and the off-machine time D π ( i ) , j 1 , g of the preceding operation on the current job satisfies Equation (2). In such circumstances, the job is directly transported to the next machine to await subsequent processing.
T j , j + 1 π ( i ) < D π ( i 1 ) , j , g D π ( i ) , j 1 , g < T j , B j π ( i ) + T B j , j + 1 π ( i )
In industrial production processes, machines are typically categorized into two operational states: processing and standby. Since machine equipment consumes significant energy during startup/shutdown transitions, and given that machines generally operate continuously in practical production scenarios, this study does not consider the energy consumption associated with power cycling. Furthermore, within the FSSP_LB model, the blocking state where jobs await processing without being actively machined is also classified as a standby state.
The processing state of the machine is also distinguished by its rotational speed, which is adjusted by setting the machine’s gear (rotational speed). Let V π ( i ) , j denote the processing speed of job π ( i ) on machine j , where V π ( i ) , j S , S = V 1 , V 2 , , V s . Here, S denotes the set of machine speed levels (gears), and s represents the number of distinct speed levels. The actual processing time is inversely proportional to the selected machine gear. Additionally, let C max ( π ) denote the maximum completion time of processing sequence π . Based on the above description, the mathematical model of GFSSP_LBAGV can be formulated as follows:
D π ( 1 ) , 1 , g = P π ( 1 ) , 1 / V π ( 1 ) , 1
D π ( 1 ) , j , g = D π ( 1 ) , j 1 , g + T j 1 , j π ( i ) + P π ( 1 ) , j / V π ( 1 ) , j , j = 2 , 3 , , m
D π ( i ) , 1 , g = D π ( i 1 ) , 1 , g + P π ( i ) , 1 / V π ( i ) , 1 , i = 2 , 3 , , B 1 + 1
D π ( i ) , j , g = max D π ( i 1 ) , j , g , D π ( i ) , j 1 , g + T j 1 , j π ( i ) + P π ( i ) , j / V π ( i ) , j , i = 2 , 3 , , B j + 1 , j = 2 , 3 , , m 1
D π ( i ) , 1 , g = max D π ( i 1 ) , 1 , g + P π ( i ) , 1 / V π ( i ) , 1 , D π ( i B 1 1 ) , 2 , g T B 1 , 2 π ( i ) , i > B 1 + 1
D π ( i ) , j , g = max { D π ( i 1 ) , j , g + P π ( i ) , j / V π ( i ) , j , D π ( i ) , j 1 , g + T j 1 , j π ( i ) + P π ( i ) , j / V π ( i ) , j , D π ( i B j 1 ) , j + 1 , g T B j , j + 1 π ( i ) } , i > B j + 1 , j = 2 , 3 , , m 1
D π ( i ) , m , g = max D π ( i 1 ) , m , g , D π ( i ) , m 1 , g + T m 1 , m + P π ( i ) , m / V π ( i ) , m , i = 2 , 3 , , n
C max ( π ) = D π ( n ) , m , g
Equations (3) and (4) indicate that the first job can be processed sequentially on all machines without considering machine or buffer availability. Equations (5) and (7) describe the processing status of jobs on the first machine, while Equations (6) and (8) specify that job blocking on machines need not be considered when the number of jobs does not exceed B j + 1 ; otherwise, buffer size and machine constraints must be accounted for. For the last machine, as shown in Equation (9), completed jobs exit the production line, and subsequent jobs either follow immediately or enter the last machine after completion on preceding machines, with inter-machine transportation handled by g AGVs. In Equation (10), C max ( π ) denotes the makespan (maximum completion time) of processing sequence π , including both machining and transportation time. Evidently, when i < B j + 1 , the problem reduces to the classical permutation flow shop scheduling problem with infinite buffer capacity. Conversely, when B j = 0 , it corresponds to the blocking flow shop scheduling problem.

2.3. Energy Consumption of GFSSP_LBAGV

In general, enterprises tend to increase the rotational speed of machinery with the aim of enhancing economic efficiency. While elevating the machine’s operational gear can indeed reduce the processing time of corresponding operations, it concurrently escalates the energy consumption associated with those operations [30], which can be expressed as
V π ( i ) , p > V π ( i ) , q p , q 1 , 2 , , s
P π ( i ) , j / V π ( i ) , p < P π ( i ) , j / V π ( i ) , q
P P j , V π ( i ) , p P π ( i ) , j / V π ( i ) , p > P P j , V π ( i ) , q P π ( i ) , j / V π ( i ) , q
Here, P P j , V π ( i ) , p denotes the energy consumption per unit time for machine j when running at gear level V π ( i ) , p . To better characterize this property, a processing instance with four machines and four jobs is presented in Equations (14)–(16). The model incorporates gear selection as shown in Equation (15), where the machine’s gear S = 1 , 1.5 , 2 . Consequently, the processing time of jobs changes from P π ( i ) , j in Equation (14) to P π ( i ) , j in Equation (16).
P π ( i ) , j = 2 4 1 4 1 3 3 6 2 1 1 4 4 2 2 1
V π ( i ) , j = 1 2 1 2 1 1.5 1.5 2 2 1 1 2 2 1 1 1
P π ( i ) , j = 2 2 1 2 1 2 2 3 1 1 1 2 2 2 2 1
Let X j , V π ( i ) , j ( t ) denote that machine j operates at gear V π ( i ) , j at time t , and Y j ( t ) denote that machine j is in standby mode at time t . X j , V π ( i ) , j ( t ) and Y j ( t ) are decision variables (0 or 1) that can be expressed as follows:
X j , V π ( i ) , j ( t ) = 1 , machine   j   operates   at   gear   V π ( i ) , j   at   time   t 0 , others
Y j ( t ) = 1 , machine   j   is   standby   mode   at   time   t 0 , others
In GFSSP_LBAGV, the energy consumption of the production process and the processing time of jobs are influenced by the machine’s rotational speed, which in turn affects the maximum completion time and maximum delay time of the entire production process. Unlike traditional models, GFSSP_LBAGV introduces AGVs, and the total energy consumption of the production process must also account for the energy consumed during job transportation.
Let T E C denote the total energy consumption of GFSSP_LBAGV during the entire production and transportation process; T E C M represent the energy consumption of machines during the production process; and T E C A represent the energy consumption of AGVs when transporting jobs between different machines or buffer zones. T E C can be expressed as follows:
T E C ( π ) = T E C M ( π ) + T E C A ( π )
T E C A ( π ) = P 0 i = 1 n ( j = 1 m T j , j + 1 π ( i ) + j = 1 m T j , B j π ( i ) + j = 1 m T B j , j π ( i ) )
T E C M ( π ) = 0 C max ( j = 1 m ( V π ( i ) , j = 1 s P P j , V π ( i ) , j X j , V π ( i ) , j ( t ) + S P j Y j ( t ) ) ) d t
In Equation (20), P 0 represents the average power consumption during the transportation process of the AGV. In Equation (21), S P j represents the energy consumption per unit time when the machine is in standby mode. Theoretically, the energy consumption of a machining process is the product of machining power and machining time. Both machining power and machining time are determined by the processing speed V π ( i ) , j of the machine, thereby establishing the energy consumption per unit time P P j , V π ( i ) , p under different machine speed settings.
In GFSSP_LBAGV, the introduction of AGV transportation reduces labor costs, but the transportation time inevitably affects the completion time of each job. In summary, the optimization objectives of the GFSSP_LBAGV production scheduling model proposed in this paper are as follows:
  • Minimize the maximum completion time C max :
f 1 ( π ) = min C max π
  • Minimize the total energy consumption T E C :
f 2 ( π ) = min T E C π
The optimization objective is to find a solution in the set of all permutations Ω that simultaneously minimizes both the makespan C max and the total energy consumption T E C , which can be expressed as follows:
M i n π Ω [ f 1 ( π ) , f 2 ( π ) ]
Although increasing machine rotation speed reduces job processing time, it simultaneously leads to a significant surge in energy consumption during that time period. Consequently, there exists an inherent trade-off between the two optimization objectives: minimizing C max and minimizing T E C . Furthermore, any permutation solution must satisfy the GFSSP_LBAGV constraints; otherwise, it constitutes an infeasible solution.
( i 1 , 2 : f i ( π 1 ) f i ( π 2 ) ) ( j 1 , 2 : f j ( π 1 ) < f j ( π 2 ) )
Consider two feasible solutions π 1 and π 2 . If they satisfy the relationship given in Equation (25), then solution π 1 dominates solution π 2 (or equivalently, solution π 2 is dominated by solution π 1 ), denoted as π 1 π 2 . A solution π 1 is termed a non-dominated solution if no other solution exists that dominates π 1 . The collection of all such non-dominated solutions constitutes the non-dominated solution set.

3. IDPFA for Solving GFSSP_LBAGV

3.1. Population Initialization

A high-quality initial population plays a crucial role in the algorithm’s solving process. This article employs the Nawaz–Enscore–Ham (NEH) heuristic to generate a portion of individuals, while the remainder are randomly created [31,32]. The initial population is randomly generated to maintain dispersion and diversity within the population.

3.2. The Encoding and Decoding

The position vector of the individual in the standard PFA is a real number in the continuous domain, so a largest-order value (LOV) rule is used to transform the position vector of pathfinder individuals l = ( l 1 , l 2 , , l n ) into the arrangement of jobs π = ( π 1 , π 2 , , π n ) .
The detailed steps of the LOV rule are as follows:
Step 1: l = ( l 1 , l 2 , , l n ) are arranged in descending order to obtain a sequence φ i , which represents the serial number of the corresponding job arranged in descending order.
Step 2: The sequence π of the job is calculated by the following formula:
π φ i = i
An example of the LOV rule is provided in Table 2. Assuming that the individual is l i = ( 1.28 , 2.09 , 3.17 , 0.78 , 2.66 ) , since l 1 is ranked fourth in descending order, φ 1 = 4 ; similarly, l 2 is ranked third, so φ 2 = 3 ; the other values follow accordingly, yielding φ i = ( 4 , 3 , 1 , 5 , 2 ) . Then, according to Equation (26), π φ 1 = π 4 = 1 , and π φ 2 = π 3 = 2 , and π i = ( 3 , 5 , 2 , 1 , 4 ) can be obtained in the same manner.
By invoking the LOV rule, the positional information of individuals in the population can be transformed into processing sequences in the GFSSP_LBAGV model, establishing a one-to-one correspondence between real-valued vectors and job permutations. Furthermore, when the job sequence changes, the RLOV (Reverse Largest-Order-Value) rule can be employed to adjust the individuals’ positional information accordingly.

3.3. Pathfinder Location Update

In the pathfinder algorithm, the population individuals are categorized into pathfinders and follower members, collectively forming the population group. The pathfinder individual serves as the group leader who guides the global search direction of the algorithm, while the follower members in the population move along the direction indicated by the pathfinder. The movement of individuals within the population causes changes in their position vectors, consequently altering their corresponding piecework permutations, which constitutes the population update process.
The location of each individual in the PFA represents a solution. In the process of updating the population, the pathfinder is the explorer in the direction of the population and moves ahead of the follower. The updating methods are shown in (27) and (28).
X p K + 1 = X p K + 2 r 1 X p K X p K 1 + A
A = u 1 e 2 K K max
where K denotes the current iteration number of the algorithm, K max is the maximum number of iterations, X p K is the position vector of the contemporary pathfinder, X p K 1 is the position of the pathfinder from the previous generation and X p K + 1 represents the pathfinder’s updated position vector. The step size factor r 1 , governing the pathfinder’s movement, follows a uniform distribution over the interval [ 0 , 1 ] . A indicates the multi-directional and random movement of the pathfinder. The directionality is determined by the value of u 1 , which is a random number within the range [ 1 , 1 ] , while the randomness of the step size depends on 2 K K max and correlates with the number of algorithm iterations.
Moreover, owing to the multidirectionality and stochasticity inherent in the exploration process, the position discovered by the pathfinder may prove inferior to the original position. Consequently, an elitism-preserving operation is incorporated following the pathfinder’s update: if the updated position demonstrates worse performance, the pathfinder reverts to its original position.
Upon completion of the pathfinder update, the followers within the population undergo position updates based on the pathfinder’s location as follows:
X i K + 1 = X i K + R 1 X j K X i K + R 2 X p K X i K + ε , i 2
R 1 = α r 2
R 2 = β r 3
ε = 1 K K max u 2 D i j
D i j = X i X j
where K denotes the current iteration count of the algorithm, X i K represents the current position of the follower, and X i K + 1 indicates the follower’s updated position. The movement of followers depends not only on the pathfinder X p K but is also influenced by other followers X j K , where α denotes the interaction coefficient among followers, β represents the attraction coefficient from the pathfinder to followers, and r 2 , r 3 represent the step-size factors for movements toward other followers and the pathfinder, respectively, both being random numbers uniformly distributed in the interval 0 , 1 . Similarly, as with the pathfinder, ε characterizes the randomness in follower movement, where u 2 and D i j determine the direction and step size of random motion, while D i j represents the distance between the current follower and other followers.
As demonstrated above, the random step sizes of both pathfinders and followers are correlated with the algorithm’s iteration count, exhibiting a decreasing trend as generations progress. This design originates from the algorithmic requirement for extensive exploration in the expansive solution space during initial phases to identify promising regions, followed by intensified local search in later stages for precise optimization. Such a mechanism effectively maintains the equilibrium between global exploration and local exploitation in the pathfinder algorithm.

3.4. Improved Discrete Pathfinder Location Update

The improved PFA essentially discretizes the algorithmic updates while preserving the evolutionary framework of the standard PFA. The population still consists of pathfinders and followers, with the pathfinder updates preceding follower updates to guide the global search direction. The key distinction lies in the pathfinder position update mechanism: instead of executing (27) and (28), the discrete version implements the update formula specified in (34) for pathfinders.
X p K + 1 = C r o s s o v e r ( X p K , X p K - 1 ) , r m I n s e r t ( X p K ) , I n t e r c h a n g e ( X p K )
The pathfinder X p K position update consists of two distinct phases:
Phase I involves mutual learning between the current pathfinder and its predecessor from the previous generation X p K 1 , implemented through a two-point crossover operation C r o s s o v e r . This interaction generates new candidate individuals X p K as formally specified in (35).
Phase II constitutes a self-exploration process for the newly generated individuals X p K , achieved through dual neighborhood operations ( I n s e r t and I n t e r c h a n g e ). The distinct neighborhood structures of these operators inherently produce different step sizes during X p K exploration, necessitating the introduction of probability factors r m for controlled implementation, as mathematically described in (36).
X p K = C r o s s o v e r ( X p K , X p K 1 ) , X p K X p K - 1 X p K , X p K 1 = X p K
X p K + 1 = I n s e r t ( X p K ) , i f   r a n d [ 0 , 1 ] r m I n t e r c h e a n g e ( X p K ) , or
In the first stage of learning, since X p K 1 is an individual selected from the non-dominated solution set of the previous generation, it may potentially represent the same solution as the current pathfinder X p K . The two-point crossover operation C r o s s o v e r is executed as follows (Figure 2), the arrows indicate the selected jobs in the processing sequence: Firstly, two random integers a and b (where 1 a < b n ) are generated. A part of the sequence X ( a , b ) of X p K can be inherited by X p K . Subsequently, the remaining positions in X p K are iteratively populated with job indices from X p K 1 that are non-redundant with those in X ( a , b ) , thereby constructing the new individual X p K .
In the second phase of the exploration process, operations I n t e r c h a n g e and I n s t e r can be described as follows:
Operation I n t e r c h a n g e : Randomly select two adjacent positions e and f (where f = e + 1 ) in the processing sequence and swap the jobs at these positions. For example, given a processing sequence π = ( π ( 1 ) , π ( 2 ) , π ( 5 ) , π ( 4 ) , π ( 3 ) ) , if positions e = 2 and f = 3 are selected, the I n t e r c h a n g e operation yields the resulting sequence as π = ( π ( 1 ) , π ( 5 ) , π ( 2 ) , π ( 4 ) , π ( 3 ) ) .
Operation I n s t e r : Randomly select two distinct positions e and f in the processing sequence. If e > f , perform a forward insertion (move the job at position e to immediately before position f ); otherwise, perform a backward insertion (move the job at position e to immediately after position f ). For processing sequence π = ( π ( 1 ) , π ( 2 ) , π ( 5 ) , π ( 4 ) , π ( 3 ) ) , selecting positions e = 3 and f = 1 for the I n s t e r operation yields resultant sequence π = ( π ( 5 ) , π ( 1 ) , π ( 2 ) , π ( 4 ) , π ( 3 ) ) . Conversely, selecting positions e = 1 and f = 3 produces the resultant sequence π = ( π ( 2 ) , π ( 5 ) , π ( 1 ) , π ( 4 ) , π ( 3 ) ) .
The pseudo-code for the neighborhood operations designed in this study is presented as follows:
Step 1: According to the LOV rule, the position vector X i of individuals in the population is converted into a sequence π 0 ;
Step 2: Randomly performing domain search on the permutation π 0 to obtain π 1 ;
Step 3: Make l o o p = 1
repeat
k = 1 , max _ k = 2
While k < max _ k do
Random select m _ k , m _ k 1 , 2 .
Randomly select u and v, where u , v 1 , n and u v ;
If m _ k = 1 , π 2 = I n t e r c h a n g e ( π 1 , u , v ) ;
If m _ k = 2 , π 2 = I n s e r t ( π 1 , u , v ) ;
If π 2 π 1 then π 1 = π 2 else k = k + 1 ;
End while
l o o p = l o o p + 1
Until l o o p = max _ l o o p
If π 1 π 0 then π 0 = π 1 ;
Step 4: Transform the permutation π 0 into X i according to the RLOV rule.
After completing the corresponding operation, the new pathfinder position X p K + 1 is obtained. At this point, it is necessary to make a dominant judgment on X p K + 1 and X p K , and the non-dominant solution among them is regarded as the final X p K + 1 .

3.5. Improved Discrete Followers Location Update

GFSSP_LBAGV is a multi-objective optimization problem where no single optimal solution (approximate optimal solution) exists. Instead, the optimal (approximately optimal) solutions consist of multiple Pareto solutions (non-dominated solutions) distributed across different regions of the solution space. Clearly, when solving such problems, the algorithm should strive for a dispersed search. Most existing pathfinder algorithms for multi-objective optimization primarily select the most widely dispersed non-dominated solution from the non-dominated solution set as the pathfinder [25]. However, this approach struggles to achieve broad-range exploration, thereby limiting the algorithm’s performance. To address this issue, the IDPFA treats every individual in the current non-dominated solution set as a pathfinder during the evolutionary process, driving the algorithm to explore a wider region of the solution space. Additionally, to ensure that the vicinity of each pathfinder receives sufficient exploration, a distance-based selection mechanism is adopted to determine the followers for each pathfinder, followed by their update:
X i X p d
The update of the follower positions occurs after the movement of the pathfinder, which corresponds to the local exploration phase of the algorithm. Similarly, the update mechanism for the followers in the improved PFA follows the same principle as Equation (29). The generation of a new individual X i K + 1 is influenced by both the pathfinder X p K and other followers X j K . The specific update formula is as follows:
X i K + 1 = O B X ( X p K , X i K ) , B S E C ( X i K , X j K )
In Equation (38), O B X and S E C represent two distinct crossover operations. O B X (Order-Based Crossover) is a permutation-based crossover method. Its procedure can be described as follows: Randomly select l positions (where 1 < l < n ) in the permutation X i K , and extract the corresponding job numbers into a set Ω . Copy the job numbers from X p K that are not in Ω directly into their original positions in X i K . Fill the remaining positions in X i K sequentially with the leftover job numbers from X i K . Figure 3 illustrates the O B X crossover operation when l = 4 .
B S E C ( X i K , X j K ) represents the mutual influence factor among followers, where the S E C (Subtour Exchange Crossover) operation can be described as follows:
Randomly select another contemporary follower X j K to form a new parent pair with the individual X i K obtained after the O B X operation. Similar to the C r o s s o v e r operation, a random interval [ a , b ] is selected, and all job numbers within this interval are marked. Unmarked job numbers in both parent permutations remain unchanged. The marked job numbers are exchanged between the two parent permutations according to their original occurrence order, thereby generating offspring individuals X i K and X i K . The specific procedure is illustrated in Figure 4.
After obtaining X i K and X i K , it is necessary to perform a dominance comparison between the two individuals, with the non-dominated solution among them being selected as the final X i K + 1 .
After all pathfinders and followers complete their position updates, the non-dominated solutions in the population are reselected as the new pathfinders. To mitigate the algorithm’s tendency to converge to local optima during optimization, 10% of the followers are randomly selected, and a new solution is regenerated stochastically as replacement followers to maintain population diversity throughout the search process.

3.6. Multi-Neighborhood Local Search

Neighborhood Structure N 1 : Insert neighborhood ( I n s e r t ).
The I n s e r t neighborhood is one of the most representative structures among several neighborhood types. When combined with the algorithm’s global search capability, it demonstrates strong optimization performance.
Neighborhood Structure N 2 : Partial sequence inversion ( I n v e r s e ).
Similar to the two-point crossover method described earlier, a random interval [ a , b ] is first selected, and then the permutation within this interval undergoes the I n v e r s e operation.
Neighborhood Structure N 3 : Fragment sequence insert ( I n s e r t )
A randomly selected segment of length g ( g = r a n d o m 3 , 4 , 5 ) is extracted from the solution sequence and then inserted into a randomly chosen position within the sequence, thereby generating a new sequence configuration.
The concrete steps of the multi-neighborhood local search strategy are as follows: Sequentially select a solution X from the non-dominated solution set S of the IDPFA. Perform neighborhood searches N 1 , N 2 and N 3 on X in sequence, generating a new solution X after each operation. Conduct a dominance comparison between X and X : If X dominates X ( X X ), update X = X . If they are mutually non-dominated, add X to the non-dominated solution set S and update S . If the size of S changes, restart the local search strategy from the beginning. Otherwise, proceed to the next neighborhood structure ( N 3 ) until all neighborhood searches are completed.

3.7. Algorithm Scheme

The algorithm scheme of the IDPFA for solving GFSSP_LBAGV can be described as follows: firstly, the NEH heuristic operation is employed to randomly generate the initial population, ensuring both dispersion and diversity of the population. Secondly, the IDPFA designates all non-dominated solutions in each generation as pathfinders and performs intensive exploitation-oriented search through self-learning multi-neighborhood operations (including C r o s s o v e r , I n s e r t , and I n t e r c h a n g e ) on the updated pathfinders, thereby refining the non-dominated solution set and guaranteeing pathfinder quality. Subsequently, in the follower update phase, a distance-based selection mechanism is introduced, enabling each follower to track its nearest pathfinder to ensure thorough exploration-oriented search around each pathfinder’s vicinity. Hybrid crossover operations ( O B X and S E C ) are applied for position updates, while partial followers are randomly reset to maintain population diversity. Furthermore, the IDPFA dynamically updates the non-dominated solution set during iterations to ensure solution quality. By synergistically combining the broad guidance of pathfinders with the diversified search of followers, the IDPFA effectively balances global exploration and local exploitation capabilities, ultimately yielding a high-quality and uniformly distributed Pareto front solution set.
According to the above description, the flow chart of the IDPFA for solving GFSSP_LBAGV is shown in Figure 5.

4. Experimental Simulation and Comparison

4.1. Experimental Setup

All algorithms and tests in the simulation experiments were implemented using Delphi 2010 and executed on a computer system with the following specifications: Intel Core i7-1165G7 CPU @ 2.8 GHz and 16 GB RAM. The experimental data were all sourced from standard benchmark test sets (Reeves benchmark instances) for flow shop scheduling problems in the international research community [33,34]. A total of six datasets with different scales and 18 problem types were adopted to evaluate the performance of the algorithms.
In this study, the machine speed levels are configured as follows: S = 1 , 1.1 , 1.2 , 1.3 , 1.4 , P P j , V π ( i ) , p = 4 × V π ( i ) , p 2 , and S P j = 1 [35].

4.2. Comparison Between DPFA and Standard PFA

To verify the effectiveness of the proposed DPFA in solving discrete problems, this section conducts comparative experiments between DPFA and the standard PFA. The model used for solving is the single-objective ( C max ) of FSSP_LB, with a unified buffer size setting of Buffer = 1. Both DPFA and the standard PFA were configured with identical population size ( P o p s i z e = 50 ) and iteration count ( g e n _ max = 200 ), and neither algorithm incorporated additional local search operations. Table 3 presents the minimum (Min), maximum (Max), and average values (Avg) obtained from 20 independent runs of each algorithm on FSSP_LB, along with their respective computation times (Time).
Comparative data between PFA and DPFA under buffer size B j = 1 are presented in Table 3. The optimal experimental data are emphasized using bold formatting for enhanced visibility. The performance of DPFA and PFA was evaluated and compared using four key metrics: Min, Max, Avg, and Time:
(1)
Minimum makespan (Min): The smallest C max value obtained from multiple independent algorithm runs. This metric reflects the algorithm’s optimal performance, representing the most compact scheduling arrangement achievable.
(2)
Maximum makespan (Max): The largest C max value observed across multiple runs. This metric indicates the algorithm’s stability in worst-case scenarios, demonstrating the upper bound of solution quality even under suboptimal conditions.
(3)
Average makespan (Avg): The arithmetic mean of C max values over multiple runs. As a core indicator of algorithm reliability, it measures overall performance. When the average value approximates the minimum while remaining substantially lower than the maximum, the algorithm demonstrates strong robustness with minimal result fluctuations.
(4)
Computational time (Time): The elapsed time from algorithm initiation to termination. This metric evaluates computational efficiency, where high-performance algorithms should consistently obtain high-quality solutions within reasonable time frames.
As can be seen from the comparative results in Figure 6a,b, the proposed DPFA achieves smaller minimum values than the standard PFA for all test problems. Moreover, the average values over 18 runs are also lower than those of the standard PFA in the vast majority of problem types. Meanwhile, as shown in Figure 6c regarding algorithm runtime, the DPFA requires significantly less time than the standard PFA for 200 generations of optimization as the problem scale increases—in some problem types, even only half the time of PFA. However, in terms of comparing the maximum values in Figure 6d, the standard PFA consistently outperforms DPFA, indicating that DPFA has a broader global search range but is more prone to falling into local optima. The comprehensive analysis demonstrates that DPFA can be effectively applied to discrete problem solving, exhibiting a wider global search range and stronger convergence. Its search performance within the same time frame is significantly superior to standard PFA. Nevertheless, a jump-out mechanism needs to be incorporated to prevent DPFA from becoming trapped in local optima.

4.3. Comparison Between DPFA and Other Algorithms

To verify the effectiveness of the proposed IDPFA in solving GFSSP_LBAGV, this paper compares IDPFA with other renowned multi-objective algorithms, namely, INSGA-II [36] and SFLA [35]. Among them, Wang et al. proposed an improved non-dominated sorting genetic algorithm (INSGA-II), which incorporates a local search strategy. Additionally, SFLA was introduced by Lei et al. as a multi-objective algorithm for solving the green flexible job shop scheduling problem, and its strong search performance was demonstrated in reference [35].
All simulation experiments were conducted using internationally recognized standard test sets. Under identical time constraints, each algorithm was independently executed 20 times. The non-dominated solution sets obtained from each run were used for comparative analysis. The performance of these solution sets was evaluated using two widely adopted multi-objective optimization metrics [37]: the quality of non-dominated solutions in the set of non-dominated solutions ( R N D S ) and the number of non-dominated solutions in the set of non-dominated solutions ( O N S N ).
Let S denote the collection of K non-dominated solution sets, i.e., S = S 1 S 2 S K . The R N D S for a given solution set S j is defined as the proportion of solutions in S j that are not dominated by any solution from other algorithms’ non-dominated solution sets. This ratio ranges between 0 and 1, where a higher value indicates better solution quality and convergence performance of the algorithm. Mathematically, R N D S can be expressed as follows:
R N D S = S j x S j y S : y x S j
For multi-objective problems, in addition to solution quality as a key metric, the number of non-dominated solutions ( O N S N ) in the non-dominated solution set is another crucial indicator, which can be formulated as Equation (40). A higher O N S N value indicates a greater number of non-dominated solutions, reflecting better algorithm performance.
O N S N = S j x S j y S : y x
Following the same datasets as previously described, a total of 6 groups comprising 18 datasets of varying scales were employed for simulation experiments. When the buffer capacity ( B j = 0 ), the problem reduces to the blocking flow shop scheduling problem, which typically models serial automated production lines. In this scenario, only transportation times between machines exist and can often be integrated into the processing times of jobs on respective machines, exhibiting certain particularities. Therefore, this study focuses on evaluating algorithm performance under buffer capacities B j = 1 , B j = 2 , and B j = 4 . Table 4, Table 5 and Table 6 present comparative results of different algorithms for solving GFSSP_LBAGV under these three buffer configurations, respectively. The optimal experimental data are emphasized using bold formatting for enhanced visibility.
As evidenced by the experimental data in Table 4, Table 5 and Table 6, the proposed IDPFA demonstrates significant advantages in solving GFSSP_LBAGV for the majority of test instances.
As shown in Figure 7, (a), (b), and (c) represent the comparative results of the O N S N under buffer capacities B j = 1 , B j = 2 , and B j = 4 , respectively. The IDPFA significantly outperforms INSGA-II and SFLA across all problem scales, indicating that IDPFA possesses a broader search range and is capable of discovering more non-dominated solutions. While SFLA also exhibits competitive performance compared to INSGA-II, it still lags considerably behind IDPFA. This further highlights the superior exploration ability and solution diversity achieved by the proposed IDPFA.
As can be seen from Figure 8a–c, regarding the comparison of R N D S , as the problem scale increases, SFLA demonstrates enhanced solving capability. However, IDPFA exhibits superior performance under constrained buffer conditions:
(1)
For buffer size B j = 1 , 78% of non-dominated solutions in IDPFA outperforms SFLA;
(2)
At B j = 2 and B j = 4 , this advantage increases to 82%;
(3)
Notably, 67% of IDPFA’s solutions show significantly better quality than SFLA’s solutions.
Through its multi-neighborhood local search strategy and distance-based selection mechanism, IDPFA achieves more precise approximation to the Pareto front for small-to-medium scale problems, demonstrating stronger optimization capability. Notably, INSGA-II yields R N D S value of nearly zero in these cases, indicating that its obtained non-dominated solutions are almost entirely dominated by those from other algorithms. The R N D S results reveal that IDPFA successfully achieves a balance between exploration breadth and exploitation depth. This dual capability enables IDPFA to effectively address the critical trade-off in multi-objective optimization between solution quantity and solution quality.
A comprehensive comparison of the data in the three tables reveals that, while the performance of the SFLA algorithm improves with increasing problem scale, it still exhibits a noticeable gap compared to IDPFA. It can be concluded that IDPFA achieves a balanced trade-off between search breadth and depth when solving GFSSP_LBAGV. Therefore, the simulation experiments and comparative analysis demonstrate that IDPFA holds a distinct advantage in solving GFSSP_LBAGV and can effectively address this problem.

5. Conclusions

This paper proposes a green integrated scheduling model for the flow shop with limited buffers and automated guided vehicles, aiming to optimize both makespan and total energy consumption. To solve this problem, an improved discrete pathfinder algorithm is developed. Building upon machine speed constraints, the model further incorporates transportation times for AGV-based job handling. The core principles of the pathfinder algorithm are adapted for discrete optimization by replacing the traditional exploration process with three distinct crossover operations: the pathfinder’s exploration, its influence on followers, and mutual learning among followers. Additionally, a multi-neighborhood local search strategy is employed to enable refined exploration of high-quality solution spaces. Finally, extensive experiments on standard benchmark datasets are conducted to validate the effectiveness of the IDPFA in solving GFSSP_LBAGV problems.

Author Contributions

Conceptualization, X.L. and S.W.; methodology, Y.D.; software, Y.D.; validation, X.L. and S.W.; formal analysis, Y.D.; investigation, Y.D., S.W. and X.L.; resources, X.L.; data curation, S.W.; writing—original draft preparation, Y.D.; writing—review and editing, Y.D., X.L. and S.W.; visualization, Y.D. and S.W.; supervision, X.L.; project administration, X.L.; funding acquisition, X.L. and S.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (Grant No. 12102118), National Programs for Science and Technology Development of Henan Province (Grant No. 242102241036), and Key Research Projects in Higher Education Institutions in Henan Province (Grant No. 23A470015).

Data Availability Statement

The original contributions presented in this study are included in this article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:
GFSSP_LBAGVGreen flow shop scheduling problem with limited buffers and automated guided vehicle
FSSP_LBFlow shop scheduling problem with limited buffers
IDPFAImproved discrete pathfinder algorithm
AGVAutomated guided vehicle
PFAPathfinder algorithm
NEHNawaz–Enscore–Ham
LOVLargest-order-value
OBXOrder-based crossover
SECSubtour exchange crossover
RNDSQuality of non-dominated solutions in the set of non-dominated solutions
ONSNNumber of non-dominated solutions in the set of non-dominated solutions

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Figure 1. Gantt chart of 4 × 4 GFSSP_LBAGV (4 jobs processed on 4 machines).
Figure 1. Gantt chart of 4 × 4 GFSSP_LBAGV (4 jobs processed on 4 machines).
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Figure 2. Schematic diagram of two-point crossover operation.
Figure 2. Schematic diagram of two-point crossover operation.
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Figure 3. Schematic diagram of the O B X crossover operation.
Figure 3. Schematic diagram of the O B X crossover operation.
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Figure 4. Schematic diagram of the S E C crossover operation.
Figure 4. Schematic diagram of the S E C crossover operation.
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Figure 5. The algorithm flowchart of the IDPFA.
Figure 5. The algorithm flowchart of the IDPFA.
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Figure 6. PFA vs. DPFA performance comparison ( B j = 1 ): (a) Comparison of minimum values between PFA and DPFA; (b) Comparison of average values between PFA and DPFA; (c) Comparison of running time between PFA and DPFA; (d) Comparison of maximum values between PFA and DPFA.
Figure 6. PFA vs. DPFA performance comparison ( B j = 1 ): (a) Comparison of minimum values between PFA and DPFA; (b) Comparison of average values between PFA and DPFA; (c) Comparison of running time between PFA and DPFA; (d) Comparison of maximum values between PFA and DPFA.
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Figure 7. Comparison results of O N S N among IDPFA, SFLA, and INSGA-II: (a) Comparison results of O N S N among IDPFA, SFLA, and INSGA-II in buffer B j = 1 . (b) Comparison results of O N S N among IDPFA, SFLA, and INSGA-II in buffer B j = 2 . (c) Comparison results of O N S N among IDPFA, SFLA, and INSGA-II in buffer B j = 4 .
Figure 7. Comparison results of O N S N among IDPFA, SFLA, and INSGA-II: (a) Comparison results of O N S N among IDPFA, SFLA, and INSGA-II in buffer B j = 1 . (b) Comparison results of O N S N among IDPFA, SFLA, and INSGA-II in buffer B j = 2 . (c) Comparison results of O N S N among IDPFA, SFLA, and INSGA-II in buffer B j = 4 .
Processes 13 02325 g007aProcesses 13 02325 g007b
Figure 8. Comparison results of R N D S among IDPFA, SFLA, and INSGA-II: (a) Comparison results of R N D S among IDPFA, SFLA, and INSGA-II in buffer B j = 1 . (b) Comparison results of R N D S among IDPFA, SFLA, and INSGA-II in buffer B j = 2 . (c) Comparison results of R N D S among IDPFA, SFLA, and INSGA-II in buffer B j = 4 .
Figure 8. Comparison results of R N D S among IDPFA, SFLA, and INSGA-II: (a) Comparison results of R N D S among IDPFA, SFLA, and INSGA-II in buffer B j = 1 . (b) Comparison results of R N D S among IDPFA, SFLA, and INSGA-II in buffer B j = 2 . (c) Comparison results of R N D S among IDPFA, SFLA, and INSGA-II in buffer B j = 4 .
Processes 13 02325 g008aProcesses 13 02325 g008b
Table 1. Mathematical notations and the definitions used in this paper.
Table 1. Mathematical notations and the definitions used in this paper.
Mathematical NotationsDefinitions
n the number of jobs
m the number of machines
g the number of AGVs
π = ( π ( 1 ) , π ( 2 ) , , π ( n ) ) the processing sequence of jobs
B j the buffer size between machines j   and   j + 1
D π ( i ) , j , g the completion time of job π ( i ) on machine j with g AGVs
P i , j the processing time of job i on machine j
T j , j + 1 π ( i ) the transportation time of the job π ( i ) between machines
T j , B j π ( i ) the transportation time of the job π ( i ) from machine j to buffer B j
T B j , j π ( i ) the transportation time of the job π ( i ) from buffer B j to buffer machine j
V π ( i ) , j the processing speed of job π ( i ) on machine j
C max ( π ) the maximum completion time of processing sequence π
P P j , V π ( i ) , p the energy consumption per unit time for machine j when running at gear level V π ( i ) , p
X j , V π ( i ) , j ( t ) the machine j operates at gear V π ( i ) , j at time t
Y j ( t ) the machine j is in standby mode at time t
T E C the total energy consumption
T E C M the energy consumption of machines during the production process
T E C A the energy consumption of AGVs when transporting jobs between different machines or buffer zones
P 0 the average power consumption during the transportation process of the AGV
S P j the energy consumption per unit time when the machine is in standby mode
K the current iteration number of the algorithm
K max the maximum number of iterations
X p K the position vector of the contemporary pathfinder
X p K 1 the position of the pathfinder from the previous generation
X p K + 1 the pathfinder’s updated position vector
r the step size factor
A the multidirectional and random movement of the pathfinder
X i K the current position of the follower
X i K + 1 the follower’s updated position
α the interaction coefficient among followers
β the attraction coefficient from the pathfinder to followers
ε the randomness in follower movement
Table 2. Representation of solutions in the LOV rule.
Table 2. Representation of solutions in the LOV rule.
Dimension12345
l i 1.282.093.170.782.66
φ k 43152
π i 35214
Table 3. Comparison results between PFA and DPFA with buffer size B j = 1 .
Table 3. Comparison results between PFA and DPFA with buffer size B j = 1 .
Problem
n , m
B j = 1
PFADPFA
MinMaxAvgTimeMinMaxAvgTime
rec0120, 5128413701336.8050.80124916001351.7542.95
rec0320, 5111912241161.9046.15111114241142.3042.15
rec0520, 5126013321294.7046.85124516211280.1540.60
rec0720, 10158617031642.0555.50158420001640.1054.70
rec0920, 10158317351649.2054.65156021291664.7054.75
rec1120, 10148716621561.1056.15144420331553.4553.20
rec1320, 15202721672082.5059.85195724922038.8568.25
rec1520, 15203321492077.9560.90196524312021.7570.40
rec1720, 15197221072058.0068.85194323822001.2071.00
rec1930, 10221923732297.4091.50215326032229.6568.70
rec2130, 10211122822203.2589.05205026702143.6073.45
rec2330, 10215223112225.8589.95206928242241.6571.00
rec2530, 15268628482771.55100.20259531682701.4588.85
rec2730, 15253927142653.85100.70244332112536.4590.70
rec2930, 15245826672574.1098.55237431562492.7591.25
rec3150, 10333335063436.25189.15321239203366.70110.85
rec3350, 10329635403439.55186.00317139983303.10107.00
rec3550, 10340435783491.50188.95330039193407.30112.6
Table 4. Comparison results of IDPFA with SFLA, INSGA-II in buffer B j = 1 .
Table 4. Comparison results of IDPFA with SFLA, INSGA-II in buffer B j = 1 .
Problem n , m B j = 1
IDPFASFLAINSGA-II
R N D S O N S N R N D S O N S N R N D S O N S N
rec0120, 50.585.600.683.200.080.25
rec0320, 50.819.000.592.300.000.00
rec0520, 50.695.850.652.900.000.00
rec0720, 100.898.450.340.950.000.00
rec0920, 100.738.150.481.700.000.00
rec1120, 100.847.700.381.050.000.00
rec1320, 150.645.200.060.200.250.80
rec1520, 150.756.300.150.300.381.00
rec1720, 150.534.050.330.800.461.10
rec1930, 100.705.800.722.050.000.00
rec2130, 100.716.400.671.400.000.00
rec2330, 100.796.400.612.100.000.00
rec2530, 150.837.250.320.750.000.00
rec2730, 150.927.850.260.650.050.30
rec2930, 150.828.000.571.350.000.00
rec3150, 100.423.050.982.950.000.00
rec3350, 100.524.250.922.500.000.00
rec3550, 100.472.900.861.800.000.00
Table 5. Comparison results of IDPFA with SFLA, INSGA-II in buffer B j = 2 .
Table 5. Comparison results of IDPFA with SFLA, INSGA-II in buffer B j = 2 .
Problem n , m B j = 2
IDPFASFLAINSGA-II
R N D S O N S N R N D S O N S N R N D S O N S N
rec0120, 50.697.650.592.200.030.10
rec0320, 50.879.300.412.150.000.00
rec0520, 50.766.600.722.300.000.00
rec0720, 100.927.550.190.450.030.05
rec0920, 100.948.850.321.100.000.00
rec1120, 100.9410.000.240.700.030.05
rec1320, 150.735.650.200.650.280.40
rec1520, 150.585.400.170.300.341.75
rec1720, 150.644.900.270.800.510.90
rec1930, 100.736.450.541.350.000.00
rec2130, 100.755.800.481.200.050.10
rec2330, 100.766.700.481.200.000.00
rec2530, 150.898.150.300.850.000.00
rec2730, 150.857.700.461.050.000.00
rec2930, 150.818.200.260.600.070.45
rec3150, 100.493.850.903.050.000.00
rec3350, 100.443.350.842.250.000.00
rec3550, 100.533.150.931.600.000.00
Table 6. Comparison results of IDPFA with SFLA, INSGA-II in buffer B j = 4 .
Table 6. Comparison results of IDPFA with SFLA, INSGA-II in buffer B j = 4 .
Problem n , m B j = 4
IDPFASFLAINSGA-II
R N D S O N S N R N D S O N S N R N D S O N S N
rec0120, 50.777.100.461.900.020.05
rec0320, 50.9011.400.241.250.000.00
rec0520, 50.675.700.692.550.000.00
rec0720, 100.988.400.200.450.010.05
rec0920, 100.949.000.371.100.000.00
rec1120, 100.919.400.130.300.000.00
rec1320, 150.514.150.050.050.751.35
rec1520, 150.847.150.250.800.050.05
rec1720, 150.837.700.240.600.150.45
rec1930, 100.655.850.762.000.000.00
rec2130, 100.706.600.521.500.000.00
rec2330, 100.776.350.481.400.000.00
rec2530, 150.847.150.320.900.000.00
rec2730, 150.919.050.481.050.100.10
rec2930, 150.828.700.250.550.150.20
rec3150, 100.483.250.912.650.000.00
rec3350, 100.413.150.932.800.000.00
rec3550, 100.523.050.751.950.000.00
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Dong, Y.; Wang, S.; Liu, X. Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search. Processes 2025, 13, 2325. https://doi.org/10.3390/pr13082325

AMA Style

Dong Y, Wang S, Liu X. Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search. Processes. 2025; 13(8):2325. https://doi.org/10.3390/pr13082325

Chicago/Turabian Style

Dong, Yuming, Shunzeng Wang, and Xiaoming Liu. 2025. "Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search" Processes 13, no. 8: 2325. https://doi.org/10.3390/pr13082325

APA Style

Dong, Y., Wang, S., & Liu, X. (2025). Flow Shop Scheduling with Limited Buffers by an Improved Discrete Pathfinder Algorithm with Multi-Neighborhood Local Search. Processes, 13(8), 2325. https://doi.org/10.3390/pr13082325

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