A Mixed-Integer Linear Programming Model for Addressing Efficient Flexible Flow Shop Scheduling Problem with Automatic Guided Vehicles Consideration

Abstract

With the development of Industry 4.0, discrete manufacturing systems are accelerating their transformation toward flexibility and intelligence to meet the market demand for various products and small-batch production. The flexible flow shop (FFS) paradigm enhances production flexibility, but existing studies often address FFS scheduling and automated guided vehicle (AGV) path planning separately, resulting in resource competition conflicts, such as equipment idle time and AGV congestion, which prolong the manufacturing cycle time and reduce system energy efficiency. To solve this problem, this study proposes an integrated production–transportation scheduling framework (FFSP-AGV). By using the adjacent sequence modeling idea, a mixed-integer linear programming (MILP) model is established, which takes into account the constraints of the production process and AGV transportation task conflicts with the aim of minimizing the makespan and improving overall operational efficiency. Systematic evaluations are carried out on multiple test instances of different scales using the CPLEX solver. The results show that, for small-scale instances (job count ≤10), the MILP model can generate optimal scheduling solutions within a practical computation time (several minutes). Moreover, it is found that there is a significant marginal diminishing effect between AGV quantity and makespan reduction. Once the number of AGVs exceeds 60% of the parallel equipment capacity, their incremental contribution to cycle time reduction becomes much smaller. However, the computational complexity of the model increases exponentially with the number of jobs, making it slightly impractical for large-scale problems (job count > 20). This research highlights the importance of integrated production–transportation scheduling for reducing manufacturing cycle time and reveals a threshold effect in AGV resource allocation, providing a theoretical basis for collaborative optimization in smart factories.

1. Introduction

Discrete manufacturing accounts for around 62.1% of China’s manufacturing market and is widely employed in fields like machinery and equipment, automotive parts and semiconductors [1,2]. The discrete manufacturing system is a complicated process and contains numerous elements; transportation is crucial for maintaining the production process. Traditional transportation relies heavily on manual methods, which are inefficient and costly and cannot meet modern manufacturing needs for fast and flexible material distribution. AGVs, as advanced automated transporters, enhance material transportation efficiency, reduce manual intervention and lower transportation costs. Their implementation in discrete manufacturing has witnessed exponential growth, particularly in flexible flow shop (FFS) environments, where the co-optimization of production scheduling and AGV coordination (denoted as FFSP-AGV) has emerged as a critical research frontier for industrial applications.

FFS is a highly flexible and productive manufacturing system that can adapt to multi-variety and small-batch production. However, traditional FFS scheduling research mostly overlooks the impact of transportation on production and is mainly confined to traditional manual transportation. This may lead to resource competition conflicts, such as idle equipment and AGV congestion, which can prolong the production cycle and reduce the system’s energy efficiency. In turn, this gives rise to deviations between production planning and actual execution. Therefore, studying the integrated scheduling problem of FFS and AGV is important for enhancing the overall performance of discrete manufacturing systems.

In this paper, the coupling mechanism and integrated scheduling problem of FFS and AGV (FFSP-AGV) are investigated by applying system engineering ideas. This approach is expected to fully utilize the advantages of integrated scheduling, promote the efficient operation of the entire discrete manufacturing system and generate more economic and social benefits for enterprises.

When addressing a fresh scheduling problem, especially for an integrated scheduling problem (FFSP-AGV), it is important to establish a feasible and effective mixed-integer linear programming (MILP) model for it [3]. This is because such a feasible MILP model can further examine the inherent information of scheduling problems and be employed as a reference to construct more approximate approaches. When formulating MILP models for addressing shop scheduling problems, four modeling ideas are usually adopted: sequence-based idea, position-based idea, time-based idea and adjacent sequence-based idea. The differences among them lie in their approach for determining the job sequence [4]. Of these four ideas, the adjacent sequence-based idea is more preferred and will be used to model the FFSP-AGV in this paper due to its resulting lower model complexity and higher solution efficiency.

Compared to previous related studies, the paper offers the following four main contributions:

(1)

An integrated scheduling problem of FFS and AGV, called FFSP-AGV, is investigated by considering the interaction between the shop production link and the AGV transportation link.

(2)

A MILP model that minimizes the production cycle, i.e., makespan, is developed based on the adjacent sequence-based modeling idea.

(3)

The feasibility and model complexity (size complexity and computation complexity) of the MILP model are verified and evaluated through a series of specially designed test instances.

(4)

Some key parameters in FFSP-AGV are further analyzed, and the marginal effect of AGV numbers on the production cycle is clearly revealed.

The rest of this paper is organized as follows. Section 2 reviews the existing works from three aspects (FFSP, AGV scheduling and MILP modeling) and summarizes the research gap, respectively. Section 3 introduces the problem description of FFSP-AGV and several necessary assumptions. In Section 4, a MILP model is built using the concept of adjacent sequence modeling idea. Numerical experiments are executed, and experimental results are presented in Section 5. The financial and operational insights of this work are summarized in Section 6. Finally, Section 7 concludes our work and shows some future research directions.

2. Literature Review

2.1. Related Works on FFSP

For the FFSP, the current studies mainly focus on application background and optimization strategies. Optimization strategies vary with respect to different application scenarios and production requirements. Based on the count of scheduling objectives, studies on the FFSP can be further divided into two categories: single-objective FFSP and multi-objective FFSP.

For the single-objective FFSP, researchers employ population-based intelligent algorithms, evolution-based intelligent algorithms and reinforcement learning algorithms as the optimization strategies. When dealing with the single-objective FFSP, minimizing the maximum completion time (i.e., makespan) is usually selected as the optimization objective.

By considering the transportation time in a distributed manufacturing environment, Tang et al. [5] established a distributed FFSP scheduling model aimed at minimizing the maximum completion time and proposed an improved cuckoo algorithm for solving the model. Xuan et al. [6] investigated an FFSP that encompassed multiple unrelated parallel machines (UPMs) for each process and considered the transportation time among processes. They constructed an integer programming model with the objective of minimizing the total weighted completion time and proposed an improved greedy genetic algorithm due to the model’s NP-hard property. To address an FFSP involving limited buffers and two distinct process routes, Zheng et al. [7] proposed a discrete whale algorithm that integrated five variational operators and a deduplication strategy. To address the FFSP with UPM, Shi et al. [8] proposed an enhanced gray wolf algorithm with new formulations for the key algorithmic control parameters. Xu et al. [9] proposed a hybrid genetic algorithm combining Nawaz–Enscore–Ham (NEH) heuristic, local search and adaptive genetic algorithms for addressing an FFSP with UPM and batch production type. In this hybrid algorithm, NEH heuristic was used to generate initial solutions to improve the population quality. Owing to the potential and random occurrence of machine failures during the actual production process, Malekpour et al. [10] employed the Hopfield neural network algorithm to simulate the annealing principle and the Nash bidding model to solve the FFSP. By comparing them with the genetic algorithm and the compact genetic algorithm on different instances, the effectiveness of the proposed algorithm was validated.

In addition to makespan, other objectives (such as energy consumption, total delay time, and so on) are also simultaneously considered in the FFSP. When addressing such multi-objective FFSPs, researchers primarily employ population-based intelligent algorithms, evolution-based intelligent algorithms and human-behavior-based intelligent algorithms.

Yue et al. [11] investigated the two-stage FFSP in a circuit board discrete manufacturing system and developed a multi-objective optimization model aimed at minimizing both energy consumption and weighted delay time. Dai et al. [12] regarded a capacitor discrete manufacturing shop as their research subject, constructed a multi-level FFS scheduling system architecture in an internet of things environment and designed an improved distribution estimation algorithm. Song et al. [13] proposed an improved ant colony optimization (ACO) algorithm for solving the FFSP with the objectives of maximizing the processing revenue and minimizing total carbon emission. The solution space is explored in depth by using three types of neighborhood structures to enhance the quality of the solutions. The adaptive construction probability was utilized to adjust the ant colony generation path. Considering that the production scheduling and process planning in a flexible flow shop were independent, Li et al. [14] developed a multi-objective optimization model aimed at minimizing the maximum completion time and the processing cost. They also proposed an enhanced artificial bee colony algorithm with segmentation to fully utilize the machine idle time and improved the algorithm’s local search capability by dynamically triggering the domain mechanism. Song et al. [15] proposed a method supporting five different decoding rules to represent the solutions for an FFSP with UPM. They created a greedy displacement algorithm to reduce energy consumption without changing the makespan and used a weight matching strategy to accelerate the convergence rate of the algorithm. Liu et al. [16] considered multiple time factors, including waiting time, processing time and transportation time, and developed a new hybrid production model. They proposed a new guideline for determining the optimal transportation sequence and then solved the FFSP using an evolutionary algorithm. Kong et al. [17] established a sustainable production model for an FFS with the optimization objectives of minimizing the makespan, total energy consumption and production cost. Based on different application scenarios, three scheduling models were proposed: the efficiency model, the energy-saving model and the economic model. To prioritize energy consumption in the FFSP, Schulz et al. [18] prioritized three basic strategies (i.e., reducing energy consumption, reducing energy cost and balancing the load configuration) by integrating them into a single model and proposed a multi-stage iterative local search algorithm as the solution strategy.

2.2. Related Works on AGV Scheduling Problem

As mentioned above, an AGV is a driverless transportation vehicle that is widely used for material transportation and controlled by computers. Related studies on AGV scheduling problem usually focus on specific application scenarios, such as discrete manufacturing system or logistics. For instance, Li [19] added an induced factor to the state transfer rule to guide AGVs to avoid conflicts while optimizing the paths and waiting times and proposed an induced ant colony particle algorithm to solve the conflict problem in the scheduling of multiple AGVs. Han et al. [20] proposed a multi-AGV scheduling method based on an improved genetic algorithm, which can reduce the overall path distance of all AGVs, as well as the longest path distance of a single AGV. Xie [21] utilized FlexSim simulation to gradually increase the number of AGVs, examined the hourly and average utilization rates of AGVs under different design schemes and developed an AGV configuration scheme to meet the actual production requirement.

In recent years, with the widespread dissemination of the concept of intelligent manufacturing, AGVs have been increasingly applied to material transportation in manufacturing shops due to the outstanding features of easy operation, rapid response and high efficiency. Many scholars adopted intelligent optimization algorithms, such as the genetic algorithm, pollen propagation algorithm, particle swarm algorithm, and so on, to solve the shop scheduling problems with AGVs additionally involved in the transportation link. Zhou et al. [22] synchronized the three processes of production, transportation and warehousing for optimization, set the coupling term when building the model and used the objective cascade method. Zheng et al. [7] studied the joint scheduling problem of machines and AGVs in a flexible manufacturing system and built a MILP model with the objective of minimizing the makespan. A tabu search algorithm was designed as the solution strategy, and simulation experiments showed that the algorithm could obtain an optimal solution within reasonable time. Rahman et al. [23] studied assembly line balancing and the AGV transportation problem and proposed a layered planning approach with the objective of minimizing the processing time and the delay time of AGV transportation. A particle swarm algorithm was used in the first layer for solving the mixed-integer programming model of the assembly line balancing sub-problem. The obtained task sequencing and start times were input into the second layer in which a heuristic was designed for solving the AGV transportation sub-problem. Li et al. [24] designed an improved whale optimization algorithm that incorporated population initialization, crossover and mutation operators for the integrated scheduling problem of AGVs and flexible job shops. The improved algorithm used a process-sequencing-machine selection-based encoding scheme and a first-come-first-served-based AGV scheduling strategy. Liu et al. [25] considered an integrated AGV job shop scheduling problem and constructed a bi-objective scheduling model that took cycle time and AGV utilization rate as the optimization objectives. Its shortcoming was that the count of AGVs was small, and it did not consider emergency issues, such as equipment failure, AGV conflicts, AGV congestion, and so on. Li et al. [26] studied the integrated scheduling problem of FFS production and AGV transportation and proposed an improved genetic algorithm based on the tabu search algorithm. Its shortcoming was that a fixed heuristic rule was chosen for task allocation, and the initial population was randomly generated.

2.3. Related Works on MILP Model Formulation

As mentioned in Section 1, four modeling ideas are usually adopted when formulating MILP models for shop scheduling problems: the sequence-based idea, the position-based idea, the time-based idea and the adjacent sequence-based idea [4].

The concept behind the sequence-based modeling idea is to manage the sequential dependencies between tasks or activities. This is achieved by introducing continuous variables to represent key parameters and ensuring that tasks are completed in the correct order while meeting other time and resource limitations/constraints. Recently, Zeng et al. [27] introduced a MILP model to address the UPM scheduling problem with batch processing. For the non-permutation flow shop problem with the consideration of order acceptance and weighted tardiness, Xiao et al. [28] created a MILP model by using the sequence-based modeling idea. However, they found that medium- and large-scaled problems cannot be solved well. In addition, Bozorgirad and Logendranan [29] constructed a MILP model for minimizing the total completion time and total tardiness in a hybrid flow shop.

The second modeling idea separates a workstation into several processing positions and decides which of these positions can be used for executing an operation. A MILP model for reducing the maximum completion time of a flexible job shop was initially presented by Fattahi et al. [30] based on this idea. Later, Naderi and Azab [31] introduced a new MILP model by using the position-based modeling idea in a distributed scheduling system. Using the position-based modeling idea, Zhang et al. [32] constructed a MILP model for an FFSP that took various machines’ energy use ratios into account. Finally, the experimental results demonstrated that there existed a trade-off relation between completion time and energy consumption. In the same way, a MILP model for an energy-efficient flexible job shop scheduling problem (FJSP) was further developed by Zhang et al. [33].

The time-based modeling idea optimizes scheduling problems by dividing the total processing time into multiple periods or intervals, determining whether each operation should be executed within a specific period. This approach has been widely applied to various shop scheduling problems. For instance, Yavari et al. [34] developed a MILP for a two-phase assembly scheduling problem, introducing time-indexed decision variables to represent the start times of operations in both the process phase and the assembly phase. Similarly, Pei et al. [35] proposed a MILP model for a proportional two-phase no-wait FJSP, noting that the solution time using CPLEX software (v.12.6.3) increased significantly with a wide range of job processing times. Gicquel et al. [36] introduced a hybrid flow shop scheduling model using discrete-time representation, accounting for finite waiting times and zero intermediate capacity constraints. Wu and Chien [37] developed a MILP model for semiconductor final testing, extending the FJSP with finite and unstable resources.

The last modeling idea involves expressing the relative sequential links between jobs in a flexible manner utilizing binary variables, time indices and restrictions. With exact order requirements, production scheduling issues can be resolved effectively. The blocking FFSP was addressed by Qin et al. [38] using a MILP model, where the relationships between task families were clearly identified through decision variables. Zheng et al. [39] developed a MILP model that minimized the weighted total completion time and total processing cost for a UPM scheduling problem. They incorporated a binary variable to indicate whether a machine processed two distinct tasks sequentially. A novel MILP model was formulated by Fanjul-Peyro et al. [40] to solve the machine assignment and job sequencing sub-problems using two kinds of decision variables. To reduce the energy consumption and maximum completion time of a two-machine flow shop, Mansouri and Aktas [41] employed three binary variables to build its corresponding MILP model.

3. Problem Description of FFSP-AGV

The FFSP-AGV investigated in this paper is defined by the following: s (s > 1) successive stage processing is required for n (n > 1) jobs, following the same production flow as stage 1, stage 2, …, stage s. Moreover, each stage j (j = 1, 2, …, s) has l_j (l_j ≥ 1) UPM, and at least one stage has two or more parallel machines. Each operation O_ij (i = 1, 2, …, n) is executed without pre-emption by one machine k (k = 1, 2, …, l_j) chosen from l_j machines, and the processing time of each operation is deterministic and known in advance. Let transportation task T_ij represent the transport of job i to the machine executing O_ij. Obviously, before beginning the operation, Tij should be carried out by one AGV chosen from a set of r (r ≥ 2) identical AGVs. Note that an AGV may need to execute empty/null travel from its current location to the location where the job must be picked up, unless the two locations are the same. It should be noted that the transport times between one UPM and another UPM or between a UPM and the load area (LA) are distance-dependent and are exactly the same for each AGV, regardless of the job being transported.

The schematic diagram of the studied problem is shown in Figure 1. At the beginning, all jobs are ready to be transported at the LA from where each of them must be transported by one AGV to the machine in stage 1 to execute their first operations Oi1. For each of its following operations, each job will be picked up via an AGV from the machine in which its previous operation was executed to the machine processing the current operation. A job arriving at a machine is transferred to its buffer area, and thus, the AGV can proceed to its next vehicle assignment without delay. Moreover, the machine layouts in the shop and AGV paths between the LA and the UPM or among UPMs are known in advance.

The objective of the FFSP-AGV is to minimize the makespan, i.e., the completion time of the last operation. The scheduling is composed of three sub-problems: (1) sequencing the jobs in each stage, (2) assigning the machines to jobs in each stage and (3) assigning the AGVs to transport jobs in each stage. In addition, the problem requires the following additional assumptions:

(i)

The buffer area is sufficiently large, such that jobs can wait either for the machines or for the AGVs.

(ii)

Each machine can only execute one operation at a time at most.

(iii)

Each AGV can only execute one transport task at a time at most.

(iv)

Failure and preventive maintenance activities of machines are not considered.

(iv)

Once a job starts to be processed or transported, the task cannot be interrupted.

(v)

Cross conflict and congestion conflict of AGVs are not considered.

4. MILP Model Formulation

To analyze the integrated scheduling problem mathematically, a MILP model is developed on the basis of the adjacent sequence modeling idea (for more details on the idea, please refer to Section 2.3). The reasons for selecting the adjacent sequence-based modeling approach can be attributed to two main aspects.

On the one hand, the adjacent sequence-based modeling approach has several advantages. (1) It can represent job sequences more naturally and efficiently by focusing on the relative positions of adjacent jobs, which is crucial for accurately modeling the constraints and objective functions of scheduling problems. (2) It generally leads to lower model complexity, requiring fewer variables and constraints to represent the same scheduling problem, thus improving solution efficiency. This advantage is especially significant for large-scale scheduling problems, where computational resources and time are critical considerations. (3) It shows high flexibility and adaptability in handling various scheduling scenarios, including those with complex constraints and multiple objectives. It can effectively deal with different types of scheduling problems, such as those in flexible flow shops with multi-variety and small-batch production. (4) The experimental results from related studies indicate that adjacent-sequence-based MILP models often outperform other types of MILP models in characterizing and solving scheduling problems. For example, in [4], this model was more effective than position-based MILP models in solving the three-stage remanufacturing system scheduling problem (3T-RSSP) due to its ability to more accurately capture the intrinsic characteristics of scheduling problems, leading to more optimal solutions.

On the other hand, compared with other modeling methods, (1) sequence-based approaches can represent job sequences and their precedence relationships, but they usually require more variables and constraints, leading to higher model complexity and longer solution times, especially for large-scale problems. (2) Position-based approaches focus on the specific positions of jobs in the schedule but are less flexible and more complex to implement than adjacent-sequence-based approaches. They may be inefficient in handling certain scheduling constraints and objectives. (3) Time-based approaches model scheduling problems by considering the time intervals of job processing. Although they provide detailed timing information, they often result in complex models with numerous variables and constraints, making them computationally challenging to solve, particularly for problems with many jobs and resources.

As noted above, the parallel machines in each processing stage are unrelated, but the AGVs in the shop are identical, and therefore, we can model the problem without explicitly considering the AGVs. In other words, we can avoid the utilization of an additional index for AGVs, which can lead to a remarkable reduction in the number of decision variables and constraints. Next, we introduce some of the notations used, followed by the MILP model.

4.1. MILP Model

4.1.1. Optimization Objective

In the MILP model, Equation (1) refers to the optimization objective of minimizing the makespan.

$$\left[\mathrm{P}\right] \mathrm{Minimize} C_{\max }$$

(1)

4.1.2. Constraints

The constraints in the MILP model are divided into the following two parts.

$$C_{\max }\ge C_{is{l}_s},\forall i\in I$$

(2)

$${\displaystyle \sum _{i\in I}{X}_{ihjk}}+X{F}_{hjk}={\mathrm{A}}_{hjk},\forall h\in I;j\in J,k\in M_j$$

(3)

$${\displaystyle \sum _{k\in M_j}{\mathrm{A}}_{hjk}}=1,\forall h\in I;j\in J$$

(4)

$$C_{ijk}\ge A{T}_{ij}^k+{\mathrm{A}}_{ijk}\times P{T}_{ijk},\forall i\in I,j\in J,k\in M_j$$

(5)

$${\displaystyle \sum _{i\in I}{X}_{ihjk}}+X{F}_{hjk}-{\displaystyle \sum _{b\in I}{X}_{hbjk}}-X{L}_{hjk}=0,\forall h\in I;j\in J;k\in M_j$$

(6)

$$C_{hjk}-{\mathrm{A}}_{hjk}\times P{T}_{hjk}+L\times \left(1-X_{ihjk}\right)\ge C_{ijk},\forall i,h\in I;j\in J;k\in M_j$$

(7)

$${\displaystyle \sum _{i\in I}X{L}_{ijk}}\le 1,\forall j\in J,k\in M_j$$

(8)

Inequality (2) ensures the correct makespan value. Constraint (3) aims to establish the relation among the decision variables set for job sequence and machine assignment sub-problems. Constraint (4) ensures that each operation is executed on one machine. Constraint (5) ensures that the completion time of a specific operation on a machine must exceed the job’s arrival time. Constraint (6) ensures that, for a specific machine, each job only has a single predecessor (which will transform into a dummy/null job for the first job in each machine) and a single successor (which will transform into a dummy/null job for the last job in each machine). Constraint (7) prevents any two jobs from overlapping on the same machine in any processing stages. Constraint (8) ensures that each machine in any stage can only process one job at a time at most.

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{Y}_{ij}^F}}\le r$$

(9)

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{Y}_{ij}^F}}={\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{Y}_{ij}^L}}$$

(10)

$${\displaystyle \sum _{h\in I}{\displaystyle \sum _{f\in J}{Y}_{ij}^{hf}}}+Y_{ij}^L=1,\forall i\in I;j\in J$$

(11)

$${\displaystyle \sum _{h\in I}{\displaystyle \sum _{f\in J}{Y}_{hf}^{ij}}}+Y_{ij}^F=1,\forall i\in I;j\in J$$

(12)

$$\begin{array}{l}A{T}_{hf}^{k^{\prime }}-A{T}_{ij}^k-T{C}_k^{k^{″}}-T{C}_{k^{″}}^{k^{\prime }}+L\times \left(4-Y_{ij}^{hf}-{\mathrm{A}}_{h,f-1,k^{″}}-{\mathrm{A}}_{hf{k}^{\prime }}-{\mathrm{A}}_{ijk}\right)\ge 0,\\ \forall i,h\in I;j\in J;f\in J\backslash \left\{1\right\};k\in M_j;k^{\prime }\in M_f;k^{″}\in M_{f-1}\end{array}$$

(13)

$$\begin{array}{l}A{T}_{h1}^{k^{\prime }}-A{T}_{ij}^k-T{C}_k^{LU}-T{C}_{LU}^{k^{\prime }}+L\times \left(3-Y_{ij}^{h1}-{\mathrm{A}}_{h1k^{\prime }}-{\mathrm{A}}_{ijk}\right)\ge 0,\\ \forall i,h\in I;j\in J;k\in M_j;k^{\prime }\in M_1\end{array}$$

(14)

$$\begin{array}{l}A{T}_{hf}^k-C_{h,f-1,k^{\prime }}-T{C}_{k^{\prime }}^k+L\times \left(2-{\mathrm{A}}_{hfk}-{\mathrm{A}}_{h,f-1,k^{\prime }}\right)\ge 0,\\ \forall h\in I;f\in J\backslash \left\{1\right\};k\in M_f;k^{\prime }\in M_{f-1}\end{array}$$

(15)

$$A{T}_{h1}^k-T{C}_{LU}^k+L\times \left(2-{\mathrm{A}}_{h1k}-Y_{h1}^F\right)\ge 0,\forall h\in I;k\in M_1$$

(16)

$$X_{ihjk},X{F}_{ijk},X{L}_{ijk},{\mathrm{A}}_{ijk}\in \left\{0,1\right\},\forall i,h\in I;j\in J;k\in M_j$$

(17)

$$Y_{ij}^{hf},Y_{ij}^F,Y_{ij}^L\in \left\{0,1\right\},\forall i,h\in I;j,f\in J$$

(18)

$$C_{\max },C_{ijk},A{T}_{ij}^k\ge 0,\forall i\in I;j\in J;k\in M_j$$

(19)

Constraint (9) and Constraint (10) ensure that the count of the first transport tasks is at most the number of AGVs and the same as the count of the last transport tasks. Constraint (11) ensures that each transport task is either the first task of an AGV, or it is immediately preceded by another transport task. Constraint (12) mandates that each transport task is either the last task of an AGV, or it is followed by another transport task. Constraint (13) ensures that for the operation Ohf executed, except for the first processing stage (f ≠ 1), if the AGV had transported another job immediately prior to the current one, then it must do the following: (i) deliver such job to its corresponding machine; (ii) pick up the current job from where its previous Oh(f-1) was processed; (iii) deliver it to the machine processing the current Ohf. Constraint (14) provides a supplement to Constraint (13) when f equals one. The difference is that the current job will be picked up from the LA, since it is the first operation. Constraint (15) and Constraint (16) aim to establish the relation between the arrival time and completion time. Constraint (16) describes a special case where the current operation is the first operation. Constraint (17) and Constraint (18) define the binary variables. Constraint (19) ensures that these variables are non-negative.

4.2. Theoretical Analysis

The problem [P] expressed in Equation (1) is of the NP-hard type, since its special case is the FFSP with no AGVs when r = 0. Since the FFSP that minimizes the makespan is already of the NP-hard type, we can easily observe that the problem [P] is of the NP-hard type.

5. Experimental Results

5.1. Feasibility of the MILP Model

To test the feasibility of the MILP model formulated in Section 4.2, an example is first presented for analysis. Consider an FFS with two processing stages (s = 2), where three and two parallel machines are installed. For the convenience of analysis, in this example, we assume that the parallel machines in each stage are identical. Five jobs (n = 5) are prepared to be processed by the machines and transported by two AGVs (r = 2). Detailed data are shown below in a matrix form from Equations (20)–(22). The MILP model is programmed and executed using CPLEX commercial software. The experimental outcome in the CPLEX software is presented in Figure 2 and Figure 3.

In detail, Figure 2 plots the optimal makespan of the FFSP-AGV. Obviously, it is 63 s. Figure 3 plots the running process of the MILP model in CPLEX. The optimal solution is found when the red solid line intersects with the green dashed line. Figure 4 and Figure 5 show the values of binary decision variables and continuous decision variables in the MILP model, respectively. The Gantt chart of the optimal schedule obtained via CPLEX is displayed in Figure 6. It can be seen that with the help of binary/continuous decision variables in Figure 4 and Figure 5, the job sequence, machine assignment and AGV assignment are reasonably and uniquely determined. Therefore, according to these experimental results, the feasibility of the MILP model is verified.

$$T{T}_k^{k^{\prime }}=\left[\begin{array}{cccccc}& {\mathrm{M}}_{11}& {\mathrm{M}}_{12}& {\mathrm{M}}_{13}& {\mathrm{M}}_{21}& {\mathrm{M}}_{22}\\ {\mathrm{M}}_{11}& 0& 1& 2& 4& 6\\ {\mathrm{M}}_{12}& 1& 0& 3& 2& 2\\ {\mathrm{M}}_{13}& 2& 3& 0& 5& 6\\ {\mathrm{M}}_{21}& 4& 2& 5& 0& 4\\ {\mathrm{M}}_{22}& 6& 2& 6& 4& 0\end{array}\right]$$

(20)

$$T{T}_{LU}^k=\left[\begin{array}{ccccc}{\mathrm{M}}_{11}& {\mathrm{M}}_{12}& {\mathrm{M}}_{13}& {\mathrm{M}}_{21}& {\mathrm{M}}_{22}\\ 5& 3& 4& 2& 1\end{array}\right]$$

(21)

$$P{T}_{ijk}=\left[\begin{array}{cc}10& 15\\ 20& 8\\ 20& 30\\ 5& 25\\ 30& 10\end{array}\right]$$

(22)

In the actual production, there are often practical constraints, such as collision avoidance, real-time AGV scheduling and idle time analysis optimization. For collision avoidance, a priority-based strategy can be introduced to assign priority levels to AGVs, allowing those with higher priority to pass the critical paths first. Speed adjustment strategies can also be used to prevent collisions by adjusting AGV speeds when potential collisions are detected. For real-time AGV scheduling, wireless communication networks and positioning technologies can be leveraged to enable real-time communication between the AGVs and the scheduling center, promptly updating AGV statuses and task completion. For idle time analysis optimization, the idle time includes both equipment idle time and AGV idle time, as shown in the Gantt chart in Figure 6. Drawing on the equipment-driven comprehensive scheduling algorithm for adjusting equipment idle periods, analyze the time sequences of processing and transportation tasks to identify potential idle periods and take optimization measures. For example, schedule subsequent tasks or perform maintenance for equipment about to become idle, and promptly assign new transportation tasks or recharge idle AGVs. Additionally, optimize the production processes and transportation routes to reduce waiting times between tasks, lowering the likelihood of idle time.

5.2. Test Instances Generation

To further evaluate the performance of the formulated MILP model, a series of test instances are generated by using the combinatorial nomenclature method. For convenience, the test instances are named J[n]-S[s]-M[u]-A[r]. Recall that n, s and r represent the counts of jobs, processing stages and AGVs, respectively. u represents the maximum count of UPMs in each processing stage, i.e., $u=\underset{j}{\max }\left\{l_j\right\}$. For example, J3-S2-M3-A2 means that there are three jobs to be scheduled (Job), two processing stages (Stage), at most three UPMs (Machine) in each stage and two AGVs. Moreover, the job processing time $P{T}_{ijk}$ obeys a uniform distribution on [6,29], and the job transportation time $T{T}_k^{k^{\prime }}$ obeys a uniform distribution on [1,7]. Finally, a set of 25 instances are randomly generated.

5.3. Complexity Analysis of the MILP Model

Model complexity directly determines the ability of the mathematical model to process data for computational solution. After verifying the feasibility of the MILP model, it is necessary to analyze its model complexity. Generally speaking, model complexity encompasses two aspects: size complexity and computation complexity.

Regarding the size complexity of the MILP model formulated in Section 4.2, three evaluation indices are utilized: (i) the number of binary decision variables (NBV), (ii) the number of constraints (NC) and (iii) the number of continuous decision variables (NCV). Regarding the computation complexity, two evaluation indices are employed: (iv) the optimal production cycle (FV) and (v) the CPU time (Times). The computation complexity generally depends on the algorithm, which determines the ability of the model to efficiently and accurately identify the optimal solution. All of these five evaluation indices can be obtained easily and directly from the property box of the CPLEX interface. The solution time of CPLEX is set to 600 s, and the corresponding experimental results are shown in

Table 1. Size complexity and computation complexity analysis of the MILP model.

Test Instances	NBV	NC	NCV	FV	Times
J3-S2-M3-A2	138	553	31	38	0.33
J3-S4-M5-A2	348	2187	61	91	4.25
J6-S2-M3-A2	438	1999	61	55	7.68
J6-S4-M5-A2	1164	8322	121	132	600
J9-S2-M3-A2	900	4345	91	76	600
J9-S4-M5-A2	2488	18417	181	-	600

Note: “-” indicates failure to obtain a feasible solution within 600 s.

.

Note that, except for the first three test instances, none of the other test instances could obtain the optimal solution/schedule within the prefixed period. Therefore, the FV values from the test are actually the result (i.e., the sub-optimal solution/schedule) when CPLEX is set to 600 s.

As can be seen in Table 1, the size complexity of the MILP model varies with the test instances. Specifically, the NBV, NC and NCV indices tend to non-linearly increase with the size growth of the test instances, with NC and NCV presenting the largest and smallest increase rates, respectively. As illustrated by the test instances J3-S2-M3-A2 and J6-S2-M3-A2, when the number of jobs increases by 1.00, the NBV, NC and NCV indices increase by 3.17, 3.61 and 2, respectively.

As demonstrated in Table 1, the first three test instances can rapidly and precisely determine the optimal solution within a relatively brief period, whereas the last three test instances cannot obtain the corresponding optimal target value of the CPLEX within 600 s. For example, instance J9-S4-M5-A2 fails to obtain a feasible sub-optimal solution, which is denoted by the “-” symbol. This phenomenon indicates that as the problem/instance size increases, the difficulty of solving the model increases exponentially, which is reflected in the escalation of computation complexity. In fact, the size complexity index, i.e., NBV, exhibits a strong correlation with the computation complexity index, i.e., Times, of the formulated MILP model.

As indicated in Table 1, instances like J6-S2-M3-A2, where NBV is below 1200, can be solved within 10 min, suggesting the model’s suitability for daily scheduling in SMEs. When handling larger-scale problems, we need to consider the following approaches: (1) Setting time limits: According to the actual demand, CPLEX’s solution/time limit can be flexibly set. Increasing it for large-scale problems allows the model more time to find better solutions, but it also requires a balance between the time cost and solution quality; (2) Model simplification and decomposition: By simplifying the model or decomposing it into smaller sub-problems for separate solutions and coordination, we can reduce its complexity; (3) Proof-based approximations: Advance termination when the optimality gap is under 5% can balance quality and runtime; (4) Use of heuristic and approximation algorithms: For large-scale problems, heuristic or approximation algorithms can be adopted.

5.4. Performance of FFSP-AGV Parameters

5.4.1. The Number of Jobs

In this sub-section, we investigate the effect of the number of jobs (i.e., n) on model complexity while ensuring that all the other parameters (such as s, u, r and other time-related parameters) are the same. To this end, we generate eight sets of test instances by varying the number of jobs (from two to nine) based on the existing instance J3-S2-M3-A2 reported in Table 1. The solution time of CPLEX is set to 600 s, and the experimental results are shown in

Table 2. Experimental results for the effect of change in the number of jobs on model complexity (time limit: 600 s).

Test Instances	NBV	NCV	NC
J2-S2-M3-A2	74	21	271
J3-S2-M3-A2	138	31	553
J4-S2-M3-A2	220	41	935
J5-S2-M3-A2	320	51	1417
J6-S2-M3-A2	438	61	1999
J7-S2-M3-A2	574	71	2681
J8-S2-M3-A2	728	81	3463
J9-S2-M3-A2	900	91	4345

below.

In Table 2, it can be seen that when the number of scheduled jobs n increases, the following conclusions can be drawn: (1) the NBV, NC and NCV indices of the corresponding MILP model also increase; (2) the NBV and NC indices of the MILP model show the same and sharp change trend, while NCV changes more slowly by comparison.

5.4.2. The Number of AGVs

AGV quantity determination is one of the most prioritized key issues to be considered in the production activities of AGV participation scheduling, which is also the focus of this paper. For a production system driven only by AGV scheduling, if the number of AGVs is too small, the transportation of jobs will be slow, which will lead to a lot of time and resources being wasted, and the timeliness of the production tasks will be difficult to ensure. However, the number of AGVs should not be too large because too many AGVs are not only costly but may also increase energy consumption when traveling empty and waiting for tasks, which will reduce the efficiency of the whole production system.

Therefore, investigating the most appropriate number of AGVs is one of the most important ways for manufacturing companies to reduce costs and increase efficiency. For this reason, in this section, instances J6-S2-M3- and J9-S2-M3- from Table 1 are used as benchmark data to investigate the performance of AGV count in the FFSP-AGV integration scheduling by varying the number of AGVs from one to eight.

As mentioned earlier, in the MILP model constructed for the FFSP-AGV, the processing machines in each production stage are UPMs, which take different amounts of time in each parallel machine to complete specific operations. To be fair, the processing machines in each processing stage are also set as identical parallel machines (IPMs), and the processing time is averaged to cope with different production situations. The results of the test instances are recorded in

Table 3. Experimental results for the effect of AGV numbers on makespan under instance set J6-S2-M3 (time limit: 600 s).

Test Instances	UPM	IPM
J6-S2-M3-A1	68	80
J6-S2-M3-A2	51	74
J6-S2-M3-A3	51	74
J6-S2-M3-A4	51	74
J6-S2-M3-A5	51	74
J6-S2-M3-A6	51	74
J6-S2-M3-A7	51	74
J6-S2-M3-A8	51	74

and

Table 4. Experimental results for the effect of AGV numbers on makespan under instance set J9-S2-M3 (time limit: 600 s).

Test Instances	UPM	IPM
J9-S2-M3-A1	114	123
J9-S2-M3-A2	77	105
J9-S2-M3-A3	68	100
J9-S2-M3-A4	68	99
J9-S2-M3-A5	67	98
J9-S2-M3-A6	67	99
J9-S2-M3-A7	68	98
J9-S2-M3-A8	66	98

and plotted in Figure 7 and Figure 8.

In Table 3 and Table 4 and Figure 7 and Figure 8, it can be observed that, whether in a UPM or an IPM environment, with the number of participating AGVs increasing, the optimization target makespan shows a decreasing trend. In other words, for a certain size of FFS, using more AGVs can effectively shorten the production cycle time and realize the improvement of whole production efficiency. At the same time, when the number of AGVs increases from one to two, the decrease in makespan is the largest. As the number of AGVs continues to increase, this decrease is gradually reduced. With the number of jobs being six, when the number of AGVs reaches two, the makespan no longer changes; meanwhile, in another group, with a job count of nine, when the number of AGVs reaches four, the makespan is not reduced, maintaining a near constant value. Analysis of the data shows that once the number of AGVs exceeds 60% of the capacity of parallel equipment for the same number of jobs, their incremental contribution to cycle time reduction becomes much smaller.

Moreover, the two sets of data in Table 3 and Table 4 conform to the law of marginal effect. For example, in Table 3, for the given production conditions in instance J6-S2-M3-, the optimal number of AGVs is three. When the number of participating AGVs is less than three, by increasing the number of AGVs, a faster transportation of jobs can be achieved. But when the number of AGVs is more than three, the makespan is mainly limited by the production shop itself, such as the number of processing machines, the processing time of operations, the transportation time between stages, and so on. In the other group with instance J9-S2-M3- in Table 4, the optimal number of AGVs is four. In other words, after the number of AGVs reaches the optimal number, the processing of machines will reach saturation status. When the distance factor is excluded, the ratio of the processing time of each machine will be infinitely close to 1. If the shop manager wants to further improve production efficiency, he/she must adopt other methods, e.g., increasing the number of processing machines in each stage, reducing the distance between adjacent stages, increasing the running speed of AGVs, and so on.

In addition, it can be found in Figure 7 and Figure 8 that the number of AGVs reaching the optimal configuration increases with the number of jobs to be scheduled in the FFSP-AGV system. Note that for instance J9-S2-M3- in Table 4 and Figure 8, the makespan increases rather than decreases with the increase in the number of AGVs, which is not consistent with common sense. This may be due to the following reasons: (a) unlike the traditional FFSP, the FFSP-AGV examined in this paper is an integrated scheduling problem for coordinating the processing link and the transportation link, which has more complex problem features and constraints, and the computation complexity of the MILP model increases as the number of jobs increases, which leads to the difficulty in quickly calculating the final optimal schedule; (b) there may exist randomness in the process of setting the processing times and transportation times, resulting in the formation of bottlenecks that are difficult to optimize and cause blockage in the FFSP-AGV, even though the number of AGVs is sufficiently large.

5.4.3. The Type of Parallel Machines in the Shop

It can be found in Table 3 and Table 4 and Figure 7 and Figure 8 that, in most cases, the makespan of the UPM is usually smaller than that of the IPM. For example, in the case of J6-S2-M3 in Table 3, the total completion time for the UPM is 51, while the total completion time for the IPM is 74. Similarly, in the case of J9-S2-M3 in Table 4, the makespan of the UPM is 67, while the makespan of the IPM is 98. The potential reasons are summarized as follows: (a) Machine flexibility: In UPMs, since the time required for each machine to process each job is independent, the shop manager can allocate jobs based on the different processing times. This flexibility allows for more optimal scheduling, which may reduce the makespan. (b) UPMs provide more optimization space for scheduling. In UPMs, the capacity differences among machines are known and fixed; each machine has different processing times for different jobs, and the scheduler can choose the most suitable machine according to the characteristics of different jobs, thus improving scheduling efficiency and reducing unnecessary waste of time.

In the actual production, deciding whether the machines are unrelated or whether they can be deemed identical involves the following three aspects:

(1)

Machine processing time differences: Machines with minimal time differences for the same operation can be roughly identical, simplifying the model if the differences fall within an acceptable error range.

(2)

Production demand: Standardized products prefer IPMs for minor scheduling deviations and resource balance; customized production requires UPMs to shorten completion time and boost AGV utilization.

(3)

Model simplification needs: Machines with similar characteristics may be treated the same in actual applications to simplify model complexity, aiding scheduling and optimization.

In addition, the two distinct machine types, i.e., UPM and IPM, both have a marginal effect. Namely, as the number of AGVs increases, the production cycle time gradually stabilizes around a fixed value. Based on this effect or phenomenon, it is realistic for a company to focus on the machines’ preventive maintenance activities and minimize the capacity gap among machines in the processing stage, given the limited number of AGVs.

5.5. A Short Discussion

In Section 5.1, we first verified the feasibility of the MILP model by testing a small test instance containing five pending jobs. Then, in Section 5.2 and Section 5.3, we defined a series of test instances and analyzed the size complexity and computational complexity of the MILP model. Finally, in Section 5.4, the impact of some key parameter settings (i.e., the number of jobs, the number of AGVs and the parallel machine types) on the makespan was further explored, and the marginal effect between the makespan and AGV count was found. However, as the number of jobs increased, the computational complexity of the model grew exponentially, and when dealing with large-scale problems (number of jobs greater than 20), a pure MILP model may not be feasible for rendering results in a reasonable amount of time. To bridge this gap and maintain the quality of the solution, the following approaches can be considered: (1) Adopting a hybrid MILP–heuristic framework utilizing the decomposition approach (Benders’ decomposition, column generation) and a hierarchical framework (decomposition of a problem into sub-problems) to deal with large-scale scheduling problems to improve the solution efficiency and the quality of the solution. This approach combines the accuracy of MILP with the flexibility of heuristics and can find a near-optimal solution in a reasonable amount of time; (2) Approximation algorithms for real-time scheduling are algorithms for solving complex scheduling problems, and common approximation algorithms include reinforcement learning (RL) agents and graph neural network (GNN) scheduling. They can significantly improve the solution efficiency while ensuring certain solution quality and are suitable for large-scale and dynamically changing scheduling scenarios.

6. Financial and Operational Insights

From a financial point of view, using the model results in cost savings, optimized resource utilization and a shorter payback period. By efficiently utilizing automated guided vehicles (AGVs) and reducing equipment idle time, the model helps reduce the operating costs and lower the unit production costs, which in turn improves profitability. In addition, it optimizes resource allocation by ensuring efficient use of AGVs and equipment, reducing equipment wear and tear and lowering maintenance costs. Increased productivity and shorter production cycles also help accelerate return on investment.

On the operational side, the model improves productivity, supply chain responsiveness and system stability. Through an integrated scheduling of the production and transportation processes, cycle time and idle time are reduced, resulting in increased productivity. This also makes the supply chain more responsive and flexible, helping companies quickly adapt to market changes and customer demand. In addition, by alleviating AGV congestion and equipment idling, the model improves system stability and reliability and reduces uncertainty and risk in the production process.

In short, through integrated production–transportation scheduling, the model brings significant advantages to the enterprise in both financial and operational aspects. It helps reduce costs, improve efficiency, optimize resource utilization and enhance supply chain flexibility and stability.

7. Conclusions

In this paper, to achieve higher production efficiency in discrete manufacturing systems, we examine an integrated scheduling problem FFSP-AGV that combines the FFS production link and the AGV transportation link. To this end, based on the adjacent sequence-based modeling idea, a MILP model that minimizes the makespan (i.e., maximum completion time) is formulated. In the model, two distinct decision variables, i.e., binary decision variables and continuous decision variables, are used to determine the job sequence, machine assignment and AGV assignment. The MILP model is coded and operated using CPLEX software. The feasibility of the MILP model is evaluated using a small example with five jobs to be scheduled, and the complexity of the MILP model is evaluated using a series of test instances. The latter is analyzed both in terms of size complexity and computation complexity aspects. Some key settings, i.e., the number of jobs, and the type of parallel machines contained in the FFSP-AGV are further evaluated.

The experimental results show that (1) the formulated MILP model is feasible and can accurately seek the optimal schedule for the investigated efficiency-oriented FFSP-AGV; (2) it will take more time for the MILP model to work out the optimal/sub-optimal schedule in the CPLEX software if the instances are on a large scale; (3) the size complexity NBV index exhibits a strong correlation with the computation complexity Times index; (4) there exists a law of marginal effect between the makespan and the number of AGVs when addressing the FFSP-AGV.

Future research directions can be extended in the following aspects: (1) attempting other MILP modeling ideas to formulate the FFSP-AGV; (2) considering other optimization objectives related to AGVs; (3) exploring heuristic or approximation methods to handle large-scale scheduling scenarios, where a pure MILP model may not be feasible for rendering results in a reasonable amount of time [42,43].