1. Introduction
In light of global economic, population, and societal changes, the occurrence of environmental pollution, climate warming, and energy scarcity has become increasingly frequent. Consequently, there is a growing focus on green and low-carbon initiatives. Given the manufacturing industry’s high energy consumption and emissions, it is imperative to explore a low-carbon transformation path that transcends the previously prioritized economic and technical cost indicators, such as makespan and total machine load.
Moreover, the development of artificial intelligence (AI) has ushered in a range of new technologies, including reinforcement learning, which has significantly propelled the advancement of intelligent manufacturing in the context of Industry 4.0 [
1]. Coupled with the challenges posed by the green and low-carbon movement, the manufacturing industry is progressively transitioning towards a sustainable Industry 5.0, emphasizing the consideration of human factors and the environment, as well as a shift towards diverse product offerings and small-batch personalized production [
2].
One approach that aligns with this paradigm shift is the adoption of Flexible Job Shop (FJS) scheduling, which accounts for multiple jobs, machines, and uncertainties regarding job processing machines. FJS enhances production flexibility and partially adheres to the personalized production mode characterized by multiple varieties and small batches. Destouet et al. [
2] conducted a review survey of flexible job shop scheduling problems (FJSP) involving human and environmental factors. In this article, the authors selected 135 papers from 2013 onwards, of which 101 considered environmental factors, 100 of 101 concerned energy consumption and 22 concerned carbon emissions. This observation underscores the increasing prominence of research pertaining to production modes that account for green indicators such as energy consumption and carbon emissions within the manufacturing industry.
Duan and Wang [
3] proposed a heuristic non-dominated sorting genetic algorithm (NSGA-II) embedded with batch transportation rules for dynamic FJSP with machine breakdowns, comprehensive consideration of machine on/off, and machine multi-speed selection, aiming at makespan and energy consumption. Galdeira et al. [
4] considered the FJSP of new job arrivals and machine on/off comprehensively and designed an improved backtracking search algorithm embedded in an insertion rescheduling strategy with the objectives of makespan, energy consumption, and instability. Gong et al. [
5] realized energy efficient by controlling machine on/off, and designed a two-stage memetic algorithm aiming to makespan, energy consumption and number of machine restarts. Lei et al. [
6] researched makespan and total tardiness under energy consumption threshold constraints and proposed a two-stage meta-heuristic algorithm integrating an imperialist competitive algorithm and variable neighborhood search. Wei et al. [
7] considered the problem of energy consumption for machines with variable machining speeds, developed energy-aware model for different states of the machine, and designed multiple energy-saving scheduling measures. Wu et al. [
8] similarly designed NSGA-II targeting makespan and energy consumption for the FJSP with variable machining speeds. Zhang et al. [
9] developed a multi-objective hybrid algorithm for solving the FJSP considering makespan, setup time and energy consumption. Jiang and Deng [
10] proposed an improved bi-population discrete cat swarm algorithm using the sum of machine energy consumption cost and earliness/tardiness cost as the objective. Ning et al. [
11] proposed a quantum bacterial foraging optimization algorithm targeting makespan, total machine load and total energy consumption. Yin et al. [
12] proposed a genetic algorithm based on a simplex lattice for solving the FJSP with makespan, total energy consumption, and noise emissions as the objectives. Many scholars consider not only energy consumption indicators, but also carbon emissions as a green indicator by analyzing the level of carbon emission factors.
Zhu et al. [
13] proposed a memetic algorithm with the four neighborhood structures based on the Low Carbon FJSP for Worker Learning (LFJSP-WL) with makespan, total carbon emissions and total worker cost as the objectives. Seng et al. [
14] effectively demonstrated that decoding based on low carbon emissions can be compatible with the strengths of decoding based on the makespan and energy consumption. Piroozfard et al. [
15] designed a multi-objective genetic algorithm for FJSP with carbon footprint and total tardiness as the objectives, and demonstrated the effectiveness of the algorithm through multi-dimensional indicators. Workshop production as an entirety, some scholars take a holistic view of the whole life cycle, considering all carbon emissions produced by the jobs from entering the workshop until machining is completed, i.e., the carbon footprint. Jiang et al. [
16] developed a low-carbon flexible job shop scheduling model by comprehensively considering the carbon emissions produced from machine energy consumption, tool wear and cutting fluid. Liu et al. [
17] designed an improved NSGA-II aimed at makespan and carbon footprint by considering the carbon emissions caused by machines, jobs transportation, machining swarf, and machining materials. Wen et al. [
18] proposed an improved NSGA-II based on the N5 neighborhood search, considering the carbon emissions produced from the machine, job setup time, coolant, and lubricant consumption.
The above-mentioned studies considered energy consumption and carbon emissions in various scenarios from the machine level, but rarely the transport of jobs, which has been increasingly considered between machines in order to better reproduce the actual production situation.
Li and Lei [
19] proposed a feedback imperialist competitive algorithm with the objectives of makespan, tardiness, and energy consumption by integrating the FJSP, which considers transportation, sequence-dependent setup time. Jiang et al. [
20] investigated the FJSP with a simultaneous job transportation and deterioration effect, aiming at total energy consumption. Li et al. [
21] comprehensively considered the transportation resources and job setup time constraints, and proposed an improved artificial bee colony algorithm with makespan and total energy consumption as the objectives. Dai et al. [
22] constructed a multi-situational energy-aware model for FJSP with AGV transportation resource constraints and demonstrated that the transportation time was positively correlated to the overall objective. Wang et al. [
23] developed an improved NSGA-II for FJSP with job transportation constraints, aiming at makespan, total tardiness and energy consumption.
While the studies mentioned above considered the transport of jobs between machines, the models have frequently been simplified by making assumptions such as infinite amounts of transport equipment and that the front or back operation of the jobs are not involved in the transport, which may to a certain extent be a deviation from actual production. Li et al. [
24] creatively embed machine layout rearrangement, transport equipment allocation and worker assignment in the study of job shop scheduling problem (JSP). Zhang et al. [
25] similarly embedded limited AGV transport in the study of energy-efficient FJSP with contingencies.
In summary, it can be seen that studies on FJSP need to consider not only traditional economic and technical cost indicators such as makespan, but also environmental indicators such as energy consumption. However, there are not many studies that consider the carbon emissions and carbon footprint, and the research focus is mostly on the machine level, while studies focusing on the finite amounts of transport equipment allocation and jobs machining processes are still insufficient.
Hence, based on previous studies, the major contributions of this paper are as follows:
- (1)
In this paper, a Green Low-Carbon Flexible Job Shop Scheduling Model with Multiple Transport Equipment (GFJSP-MT) is constructed integrating machine layout, limited transport equipment allocation, job transport time, job setup time and various types of machining material consumption with the objectives of makespan, total machine load and total carbon footprint.
- (2)
AGV allocation strategy is developed based on various mixed scenarios in the job machining process.
- (3)
A heuristic strategy NSGA-II is designed to address the inadequacy of the traditional NSGA-II elite solution selection strategy, which has been embedded to obtain new individuals to replace duplicate individuals with the crowding distance of zero using cross-mutation between elite pool solutions.
- (4)
In response to the need for model, this paper has been developed on the basis of previous research, and the case data set is expanded to form a model benchmark case set for further research by scholars.
In this paper, the research contents are arranged as follows:
Section 2 constructs a mathematical model of the integrated multi-factor GFJSP-MT and develops an AGV allocation strategy.
Section 3 is focused on improved operational operators for NSGA-II.
Section 4 is a detailed experimental case study. Conclusions and outlooks are given in
Section 5.
2. Model Construction
2.1. Problem Description
Before formulating the problem, some symbols are defined as follows in
Table 1.
There are a set of jobs and a set of machines and a set of AGVs . Each contains of operations . Each operation can be processed on any one machine of a set of available machines and transported by an AGV to the machine . During the processing of the job, it is necessary to take into account the clamping and disassembly time before and after the processing of the operation . Simultaneously, job swarf is produced and lubricants and coolants are consumed.
For this GFJSP-MT model, some assumptions are made as follows:
- (1)
Both the jobs and the AGVs are simultaneously available at zero time and located in the material center.
- (2)
All machines are powered on at zero time until all process jobs on that machine are completed and then the machine is powered off.
- (3)
Allow the machine to have no jobs, then the machine will be in idle mode until all jobs have been processed.
- (4)
Each AGV can only transport one job at a time.
- (5)
Each job can only be processed by one machine, and the machine can only process one job, and processing cannot be interrupted.
- (6)
Jobs and machines are independent, i.e., there is no interdependency between different jobs or machines. However, precedence relationships or technological sequences between operations of the same job must be considered.
- (7)
When the job is transported by the AGV to machine , if the previous job of the machine has not yet been completed, the job is stored in the pending processing area of the machine, and the AGV is released at that time.
- (8)
When all jobs are completed, the jobs are transported by AGV to product staging area, then all AGVs return to the material center.
2.2. AGV Allocation Strategy
In this model, AGVs perform a significant role, undertaking on the transport of the entire scheduling task. In order to correctly allocate AGVs to the operation , this paper formulates an AGV allocation strategy based on the following principles.
- (1)
When the job is not machined for the first time, it is necessary to consider whether the machine is machining operation and continuously, i.e., when , there is no need to allocate AGVs, while correcting for and .
- (2)
When , it means that adjacent machining tasks of machine are not the same job , then it is necessary to determine whether and are processed on the same machine, i.e., whether are equal. If the condition is true, no AGVs should be allocated to operation .
Besides the above, other operations
need to transport the job by AGV before and after processing, as shown in
Figure 1.
For each AGV, the magnitude of the relationship between the sum of the end time of the previous transport task () and the idle time from the current machine to machine () and the completion time () of operation is determined.
When the sum of and of all AGVs is greater than , i.e.,, at which point, for each AGV, operation has been already completed processing or just completed processing when the AGV arrives at machine without load, and the job is in the machine staging area waiting to be transported. When the above conditions are true, select the AGV with the . Otherwise, to improve the utilization of the AGVs without increasing the waiting transportation time of the job, select the AGV with the .
2.3. GFJSP-MT Model
Naturally, in this GFJSP-MT model, the carbon footprint is divided into five parts, i.e., carbon emissions from machine process (PCE) and idle (ICE), transport (TCE), swarf (SCE), lubricants (LCE) and coolants (CCE).
2.3.1. Carbon Footprint
1. Carbon emissions from processing/idling
Carbon emissions from machines in this paper are derived from the electrical energy consumed by machines during processing and idling. Where PCE is the carbon emission created by the processing of
, as shown in Equation (1).
As for ICE it is necessary to consider the machine idle waiting times, such as the transport, clamping and disassembly times of
and the presence of unused machines, so the calculation is presented in Equation (2).
2. Carbon emissions from transport
There are three main parts to the carbon emissions caused by AGVs transport, as shown in Equation (3).
The first part is the carbon emission from the AGV’s load transport . As Assumption (8) stipulates that the job has to be transferred to the product staging area after completed, so the job requires an additional transport task to the product staging area, i.e., . The second part is the carbon emission from the AGV’s no-load transport at the current machine and machine . The third part is the carbon emission from the AGVs return to the material center.
3. Carbon emissions from swarf
After
has been machined, there will be a loss of quality, i.e., swarf, and this part of the swarf will also cause carbon emissions. Assuming that the swarf in machining does not vary with the machine, then this carbon emission will be a constant, as shown in Equation (4).
4. Carbon emissions from lubricants and coolants
Lubricants and coolants are consumed during
machining. Assuming
machining, the amount of lubricant and coolant consumed per unit time varies with the machine. Therefore, the carbon emissions from lubricant and coolant consumption are shown in Equations (5) and (6), respectively.
Summing up, the .
2.3.2. Makespan
Previously it has been described how to account for the carbon footprint in the GFJSP-MT model. However, calculation of various types of carbon emissions is subject to time constraints such as and .
As both the setup time and transport time of the job and the end of the AGV current transport task are considered in the GFJSP-MT model, so multiple comparisons are required to determine the
, as shown in Equation (7).
Equation (7) discussed in two parts, where the first part was required AGV transportation, and the second part was not required AGV transportation. As an example, taking the first case of the first part, when the u-th AGV has finished the previous transport task and without load to machine , the processing of has not yet been completed, i.e., , whereas when the has been transported by the AGV from machine to machine , the previous machining task on machine has not yet been completed, i.e., . At this point, . Similarly, the other cases are not elaborated in this paper.
Based on the discussion of the multiple cases of
and the model assumptions,
and
can be further obtained as shown in Equations (8) and (9), respectively.
Note that
and
in Equation (8) have to be corrected forward according to
Figure 1, i.e., when
, there is no need to allocate AGV for
, at which point
and
.
In summary,
and
are given in Equation (10).
2.3.3. Total Machine Load
The total machine load (
) refers to the total time the job has been on the machine and reflects the overall machine utilization, as shown in Equation (11).
2.3.4. Comprehensive Optimization Model
A multi-objective optimization model for GFJSP-MT is shown in Equation (12), where carbon footprint is used a green indicator and completion time and total machine load are used as economic indicators.
Equation (13) indicates that job processing is subject to process priority, Equation (14) is the decision variable indicating that the same job can only be processed by one machine at the same time, Equation (15) show that the same job can only be transported by one AGV at the same time, Equation (16) means that only one job can be processed by the same machine at the same time, Equation (17) indicates that only one job can be transported by one AGV at the same time, Equations (18)–(21) are the constraints of the decision variable.
3. An Improved NSGA-II for Solving GFJSP-MT
Non-Dominated Sorting Genetic Algorithm (NSGA-II) was proposed by Deb et al. [
26] as an improvement on NSGA, which has been widely used in the field of multi-objective job shop scheduling due to its ingenious mechanism of non-dominated sorting and crowding distance, which allows for an excellent robustness and solving ability. Nevertheless, there are still inadequacies in NSGA-II, such as the diversity and quality deficits of Pareto front solutions due to its elite solution selection strategy.
On this basis, an improved NSGA-II solving GFJSP-MT is proposed in this paper, and the algorithm framework is shown in
Figure 2 following which the key components are described.
3.1. Encoding and Decoding
In this paper, the GFJSP-MT is encoded in a real number format, which specifically includes two parts: operation sort (OS) and machine selection (MS), as shown in
Figure 3.
Where the real number in OS represents
, such as 3 for
, and the number of occurrences represents the number of operations corresponding to the job, for example, the first 3 indicates the first operation of
, i.e.,
, and the second 2 indicates the second operation of
, i.e.,
. Real number in MS represents the machine corresponding to the operation, for example, the set of available machines
for
is
and
is randomly selected to
. Additionally,
in
Figure 3. is the total number of operations, resulting in individuals with a chromosome coding length of
.
In this paper, a multi-layer decoding matrix is designed for the special AGV allocation strategy in GFJSP-MT to determine the and on machine and calculate the individual fitness values, the steps are as follows:
Step 1: For each individual in the population, the coded values are obtained from left to right, and a series of indicators such as the and are determined.
Step 2: Based on the AGV allocation strategy in
Figure 1, assign AGVs to each operation, forming a 3-layer decoding matrix consisting of OS, MS, and AS. Taking two AGVs as an example, combined with
Figure 3, there can be a 3-layer decoding matrix as shown in
Figure 4.
Where AS is the AGV code value assigned to each operation, and the number indicates which AGV is responsible for transporting ; for example, is transported by the 1st AGV. AS can be seen, there is no need to assign AGVs when adjacent machining tasks on the machine are the same job; for example, sequentially processes two operations of , thus no AGV needs to be assigned to , denoted by 0. In addition, when adjacent operations of the same job are processed on the same machine, there is no need to assign AGVs, such as .
Step 3: Based on a 3-layer decoding matrix, metrics such as , , , setup time, transport time and material consumption are embedded to calculate the individual fitness values by a multi-layer decoding matrix.
3.2. Selection
In this paper, the frequently used tournament selection operation in the NSGA-II algorithm is adopted. This operation is performed by setting up groups of population size (), randomly selecting two different individuals to form a group, and then comparing them to select the better solution. Limited by space, this paper will not elaborate more specifically.
3.3. Crossover
Based on the individual as a hybrid encoding form of operations and machines, this paper introduces the precedence operation crossover (POX) operator proposed by Zhang et al. [
27]. As the POX operator only targets the operation level, this paper simultaneously adjusts the machines to achieve mixed operations and crossover on the basis of operations crossover with the following steps:
Step 1: Randomly select two different individuals of the parent generation, designated as and .
Step 2: Randomly divide the set into sub-sets, designated as and , where and .
Step 3: contains the jobs of , whose operations and machines are replicated in the offspring according to their position, and contains the jobs of , whose operations and machines are replicated in the offspring according to their position.
Step 4: contains the jobs of , whose operations and machines are replicated sequentially in the offspring , and contains the jobs of , whose operations and machines are replicated sequentially in the offspring .
3.4. Mutation
Mutation operation can expand the population search to a degree that allows the population to break out of the partial optimum; however, different mutation operators bring different degrees of change to individuals. Therefore, the mixed mutation operation will be used in this paper, i.e., one of the three mutation operators with different degrees of mutation will be chosen randomly, as shown in
Figure 6, which are multiple spots interchange mutation, segment interchange mutation and reverse order mutation.
Figure 6a illustrates a random selection of four distinct gene points from the parental generation F1’s OS encoding. The operation codes corresponding to gene points 1 and 3 are swapped, as are the operation codes of gene points 2 and 4. In
Figure 6b, another random selection of four distinct gene points from the parental generation F1’s OS encoding is presented. Here, the operation between gene point 1 and gene point 2 is interchanged with the operation between gene point 3 and gene point 4.
Figure 6c showcases yet another random selection of two distinct gene points from the parental generation F1’s OS encoding. Subsequently, the operation segment between these selected gene points is reversed.
These diverse mutation operations give rise to a new offspring generation, C1, with an updated operation encoding section. Lastly, it is imperative to thoroughly inspect and rectify the MS and AS encoding of the offspring generation C1 to ensure its feasibility as an independent entity.
3.5. Adaptive Operator
During the early stages of the algorithm, there is a requirement to expand the search to obtain as many high-quality individuals as possible, while in the later stages, there is a requirement to ensure that the individuals are stable towards optimal, meanwhile being able to break out of the local optimum. Therefore, this paper introduces an adaptive operator to control the crossover and mutation probabilities by the number of
[
28], as shown in Equations (22) and (23).
3.6. Heuristic Strategy
In traditional NSGA-II, after the parent and offspring are merged to obtain the population (), there is a tendency to select individuals directly to form the new generation of parent population based on rank values and crowding distance. However, as the elite selection strategy of traditional NSGA-II tends to result in duplicate individuals with a crowding distance of 0 within the population, which not only predisposes the population towards a local optimum, but also reduces the diversity and quality of the Pareto solution.
Therefore, in this paper, a heuristic strategy is designed to update the population
before selecting individuals from the population
to ensure the non-duplication of individuals, as shown in
Figure 7 and the steps are as follows:
Step 1: Determine whether there are individuals with a crowding distance of 0 in population P2, if there are, go to Step 2, if not, end.
Step 2: Obtain the elite individuals for .
Step 3: Determine whether there is only one elite solution. If this condition is met, new individuals are obtained by crossover and mutation between elite and non-elite individuals. Otherwise, the new individuals are obtained by crossover and mutation between elite individuals.
Step 4: Duplicate individuals are replaced by optimal new individuals.
Step 5: After all duplicate individuals have been replaced, return to Step 1.
Step 6: After
Step 5 is completed, the population
has been updated. At this point it is required to select
individuals from
to form a new population
. The number of individuals allowed to be retained under each rank is constrained by Equations (24) and (25).
where
indicates the number of solutions allowed to be retained for
,
indicates the number of actual available solutions under each
, Pareto Archiving Coefficients is the archival retaining coefficient for the elite solutions of the
, R is the number of total ranks, and
denotes the number of solutions allowed to be retained for the overflow of
.
Step 7: Each sequence selects solutions with maximum crowding distance by tournament.
Step 8: If the number of retained individuals is more than , then remove the individuals with the highest rank value and the smallest crowding distance by tournament selection again until the number of retained individuals is .
3.7. Non-Dominated Sorting and Crowding Distance
Non-dominated sorting has been central in NSGA-II, and as all the objective functions in this paper are minimized, the dominated relationship between two individuals can be determined according to Equation (26).
where
and
are two different and non-sorted individuals, respectively, and
is the objective function. When Equation (26) is established, i.e., there exists an objective function
of individual
that is optimal for individual
and none of the objective functions of individual
is inferior to individual
, then individual
dominates
. Conversely,
is not dominated by
and comparisons have to be made until no other individual in the population can dominate
, at which point
can be given a dominance rank.
Individual crowding distance (
) is determined by individual rankings and objective functions. Namely, all individuals under each rank value and their corresponding objective functions are obtained in turn, and then for each objective function
, a descending sort is performed, and the index of the individuals is retained. When the value of the objective function
of individual
is minimum or maximum, its crowding distance is set to
, otherwise, each objective function
of
is considered comprehensively and
is calculated based on the neighboring individuals under each objective function of
, as shown in Equation (27).
5. Conclusions
The implementation of sustainable and cost-effective automated production has been a significant focus of research in the field of manufacturing intelligence. Numerous factors, including job transportation strategies and the optimal number of transport equipment, have posed limitations on this endeavor. Hence, the study described in this paper holds great value and significance. Building on this basis, the paper introduces an allocation strategy for finite Automated Guided Vehicles (AGVs) by simulating various mixed processing states of jobs using the Flexible Job Shop (FJS) benchmark. After multi-dimensional analysis, the optimal number of AGVs in the case was determined and the effectiveness of the strategy was verified. Furthermore, a detailed mathematical formula for each objective was constructed with this strategy, such as the carbon footprint awareness estimation model for different bodies under the whole life cycle and job makespan model under multiple scenarios, etc. To form the GFJSP-MT, which has integrated multiple factors such as finite AGVs transport, machine layout, job setup time and job processing material consumption, and has given the relevant case data set.
Regarding the algorithm, this paper proposed an NSGA-II embedded in a heuristic strategy of replacing duplicate individuals using cross-mutation between elite solutions to address the shortcomings of the traditional NSGA-II, and the effectiveness and advantages of the proposed algorithm are demonstrated through comparative analysis of the convergence curves of fitness values and Pareto quality and diversity. At the same time, the optimal scheduling scheme for each sub-objective was discussed and it was found that the scheduling scheme based on the optimal carbon footprint could be adopted more frequently.
As this paper studied the static GFJSP-MT, which did not consider the effect of AGVs route planning on the AGVs allocation strategy and scheduling result, it also lacked dynamic factors such as machine breakdowns and new job insertions. Therefore, there may be a certain degree of deviation from the actual production.
The next steps will be in-depth research as follows:
- (1)
To further integrate the impact of time and route factors on AGVs allocation, and develop an AGVs allocation strategy that is more in line with actual production.
- (2)
Based on static scheduling, further dynamic green shop scheduling with limited AGVs transport will be explored, taking into account dynamic factors.