Low-carbon scheduling and estimating for a flexible job shop based on carbon footprint and carbon efficiency of multi-job processing

Abstract

With the enhancement of people’s environmental awareness, low-carbon and energy efficiency in manufacturing industry have been drawing much attention due to the huge consumption of raw materials and energy during machining processes. But as one of the approaches to reduce carbon emission, manufacturing shop scheduling strategies have historically emphasized the makespan, machine workload, and so on and neglected energy and environmental factors in most cases. This article presents a model of low-carbon scheduling of the flexible job shop, which considers both factors of production (i.e. makespan and machine workload) and environmental influence (i.e. carbon emission). A carbon footprint model of multi-job processing is established to quantify the carbon emission of different scheduling plans, and three carbon efficiency indicators are put forward to estimate the carbon emission of parts and machine tools, that is, processing carbon efficiency, part carbon efficiency, and machine tool carbon efficiency. To solve the proposed model, a hybrid non-dominated sorting genetic algorithm II which combines the original non-dominated sorting genetic algorithm II with a local search algorithm based on neighborhood search is proposed. Finally, test of some well-known benchmark instances is carried out to verify the effectiveness of the proposed algorithm, and an actual case is studied to demonstrate the feasibility and applicability of the proposed model.

Keywords

Carbon footprint carbon efficiency flexible job-shop scheduling problem Pareto optimality neighborhood search

Introduction

The growing energy and resource consumption has led to concerns about the economic development in countries. According to the International Energy Outlook 2010 on world energy statistic compiled by the US Energy Information Administration, the global energy-related carbon dioxide (CO₂) emissions are estimated to be 43% higher in 2035 than the level in 2007, assuming no new policies were imposed.¹ And the increasing emission of CO₂ makes a crucial contribution to globe warming.

Manufacturing, as the backbone of industrialized society, is one of the main energy consumers and greenhouse gas (GHG) contributors.² Therefore, the study related to reducing carbon emission (CE) in the manufacturing industry has gradually been a focus. And the concept of low-carbon manufacturing which is referred to the manufacturing process that generates low CE intensity and utilizes energy and resources efficiently and effectively during the process was proposed as well. To reduce CE of a workshop, a lot of research has been conducted, for example, Overcash et al.³ studied the effects of process parameters, cutting tools, and material of workpieces on the processing energy consumption by building the statistical model of specific consumed energy.

The flexible job-shop scheduling problem (FJSP) is an extension of the classical JSP for flexible manufacturing systems, which allows each machine to perform more than one type of operation. And it consists of two sub-problems: the routing sub-problem that assigns each operation to a machine among a set of capable machines and the scheduling sub-problem that sequences the assigned operations on all the machines to obtain a feasible schedule with optimized objectives.⁴ In terms of the objectives of the FJSP, many researchers have considered the factors of production such as the makespan, the workload of machines, and the cost,^4–6 but neglected the environmental influence. Though Fang et al.⁷ presented a new mathematical programming model of the flow shop scheduling problem which considers peak power load, energy consumption, and associated carbon footprint, it only involved CE of the energy of machine tools but neglected that generated by material consumption, logistic process, and so on. In this article, most of these factors will be taken into consideration to evaluate CE of multi-job processing.

Literature review

To reduce CE of manufacturing processes, a significant number of research studies on CE evaluation are observed in the literature which can be roughly viewed under two perspectives, namely, “process” level and “plant” level. Some researchers have focused on the individual equipment at the process level within a production system. For instance, the effects of process parameters, cutting tools, and material of workpieces on the processing energy consumption were studied.³ Based on that, Deshpande et al.⁸ took other phases during machining a workpiece into consideration, such as the start-up process, idle hours, and tool-change phase. Narita et al.⁹ introduced the influence of the peripheral devices of a machine tool, the spindle and the servo motors, the coolant, the lubricant oil, the cutting tool, and the metal chips on global warming in detail by using life cycle assessment (LCA). On the other hand, at the plant level, in order to meet the increasing requirement of practical low-carbon thinking in manufacturing, Shi and Meier¹⁰ proposed a systematic approach to assess CE in production systems by using a hybrid emission analysis model. And Seow and Rahimifard¹¹ concentrated on energy consumed by facilities and other upper level services which are responsible for maintaining the required production conditions, for example, lighting, heating, and cooling within a facility. Overall, most researchers mainly studied CE assessment of machine tools or a plant, but few of them applied these models to production scheduling.

As for FJSP, it is considered as one of the most difficult problems in combinatorial optimization in both academic and application fields.¹² Gomes et al.¹³ presented a new integer linear programming (ILP) model to schedule the flexible job shop. Pezzella et al.¹⁴ proposed a genetic algorithm for the FJSP which integrates different strategies for generating the initial population, selecting the individuals for reproduction and reproducing new individuals. Zhang et al.⁶ presented a particle swarm optimization (PSO) algorithm which is combined with a tabu search (TS) algorithm to solve the multi-objective FJSP with several conflicting and incommensurable objectives including the makespan, the maximal machine workload, and the total workload of machines. Recently, Wang et al.⁴ proposed an effective Pareto-based estimation of distribution algorithm (P-EDA) in which the fitness evaluation based on Pareto optimality is employed and a probability model is built with the Pareto superior individuals for estimating the probability distribution of the solution space. Besides, Sun et al.¹⁵ applied non-cooperative game theory with complete information to build new scheduling model in order to solve FJSP subject to machine breakdown.

With respect to the methods to solve the multi-objective flexible job-shop scheduling problem (MFJSP), they can be roughly classified into two types: weighting approach and Pareto-based approach.⁴ The weighting approach usually solves the FJSP by transforming the multi-objective problem to a single-objective problem through assigning a different weight for each objective. On the contrary, the Pareto-based approach solves the MFJSP based on the Pareto optimality concept to generate a set of Pareto optimal solutions. And as a Pareto-based approach, non-dominated sorting genetic algorithm II (NSGA-II)¹⁶ has been an effective method to solve various multi-objective optimization problems recently. Yang et al.¹⁷ proposed a hybrid algorithm of NSGA-II with local search (LS) to solve the multi-objective optimization of facility planning for energy-intensive companies. Bandyopadhyay and Bhattacharya¹⁸ put forward a modified NSGA-II with a new mutation algorithm for a parallel machine scheduling problem and compared the computational results with the original NSGA-II and Strength Pareto Evolutionary Algorithm-2 (SPEA2) to prove the effectiveness of the proposed algorithm. In addition, in order to address a multi-objective order scheduling problem in production planning under a complicated production environment with the consideration of multiple plants, multiple production departments, and multiple production processes, Guo et al.¹⁹ developed a Pareto optimization model which combines a NSGA-II-based optimization process with an effective production process simulator.

As a whole, there are few articles in the literature dealing with low-carbon scheduling especially for the FJSP, and in this article, a CE estimation model of multi-job processing is established to quantify CE of different scheduling plan. Besides, a new hybrid NSGA-II, which combines the NSGA-II with an LS algorithm based on neighborhood search, is proposed to solve the FJSP for CE reduction.

The remaining sections of this article are organized as follows: in section “Problem formulation,” the FJSP for CE reduction is formulated. In section “The hybrid NSGA-II for solving LC-FJSP,” the framework of the hybrid NSGA-II for solving the new FJSP is proposed. Some benchmark instances are tested to prove the effectiveness of the proposed algorithm in section “Case study,” and an actual case is also studied to verify the feasibility and applicability of the proposed model. Finally, some conclusions are drawn in section “Conclusion.”

Problem formulation

To describe the problem conveniently later, a definition is given first.

Definition 1. FJSP for CE reduction or low-carbon FJSP (LC-FJSP) is an expansion of FJSP, which not only considers some common objectives, such as the makespan, the maximal machine workload, and the cost, but also evaluates CE of different scheduling schemes by using LCA or carbon footprint in order to reduce the influence of scheduling plans on environment.

In a manufacturing workshop, since many activities are related to CE, such as energy consumption, cutting tool wear, and transportation process, different scheduling plans will generate different amount of CE. In this section, first a carbon footprint model of multi-job processing is established to quantify CE of manufacturing processes. Based on the carbon footprint model, three carbon efficiency indicators are proposed to evaluate CE utilization rate of parts and machine tools. And then, the mathematical programming model of LC-FJSP which considers the makespan of the jobs, the makespan interval of different jobs (MIDJ), the maximal workload of machine tools, and the total CE of machining processes is presented.

Carbon footprint model of multi-job processing

In a workshop, the inputs are raw materials or semi-finished products, while the outputs include end products and effluent. In between, many activities will generate CE, such as machining processes, assembling, spraying, transporting, storage, and so on. To calculate CE of a workshop accurately, machining processes of a part are drawn based on LCA, which is shown in Figure 1. This diagram could be used to recognize the carbon emission activities (CEAs), thus no activity will be left out.

Figure 1.

Machining processes of a part.

Based on the analysis of CEAs shown in Figure 1, CE of a single-part processing mainly comes from four parts, that is, energy consumption of machining processes, auxiliary material consumption, tool wear, and transportation process, which is shown in equation (1)

C E_{i} = C E_{i}^{p r} + C E_{i}^{a u} + C E_{i}^{t o o l} + C E_{i}^{l o g}

(1)

In terms of energy consumption of the ith job, an empirical relationship between specific energy consumption (SEC) and material removal rate (MRR) of a machine tool is adopted,²⁰ as shown in equation (2)

SEC = C_{0} + \frac{C_{1}}{MRR}

(2)

where $C_{0}$ and $C_{1}$ are machine-specific coefficients, and they are related to workpiece material, tool geometrics, spindle drive characteristics, and machine tool, and they are usually obtained through experiments.

And CE of energy consumption of the ith job can be calculated by multiplying the removal volume of the part by SEC and $E F^{el}$ , as shown in equation (3)

C E_{i}^{p r} = (\sum_{j = 1}^{n_{i}} V_{i, j}^{p r} \cdot S E C_{i, j}^{p r}) \cdot E F^{e l}

(3)

During the processing, machine tools consume some auxiliary materials, especially coolant and lubricant oil. Coolant is generally circulated by a coolant pump and will decrease bit by bit until it is updated because some of the coolant is adhered to metal chips. Hence, coolant is supplied for compensation after a period of time.⁹ In terms of lubricant oil, it is mainly used for a spindle and a slide way of a machine tool, and minute amount of oil is infused to the spindle part and the slide way in decided intervals. To simplify the complexity of the model, the time of tools changing and the recovery processing of lubricant oil and coolant are ignored although they can also influence CE of auxiliary materials. Hence, the following equation (4) is adopted to calculate CE due to coolant and lubricant oil consumption

C E_{i}^{a u} = \sum_{j = 1}^{n_{i}} (\frac{t_{i, j}}{T_{i, j}^{c o o l}} \cdot I C_{i, j}^{c o o l} \cdot E F_{i, j}^{c o o l} + \frac{t_{i, j}}{T_{i, j}^{l u}} \cdot L O_{i, j}^{l u} \cdot E F_{i, j}^{l u}) = \sum_{j = 1}^{n_{i}} t_{i, j} \cdot (\frac{I C_{i, j}^{c o o l} \cdot E F_{i, j}^{c o o l}}{T_{i, j}^{c o o l}} + \frac{L O_{i, j}^{l u} \cdot E F_{i, j}^{l u}}{T_{i, j}^{l u}}) = \sum_{j = 1}^{n_{i}} t_{i, j} \cdot ω_{i, j}^{a u}

(4)

CE of cutting tools is estimated from the viewpoint of tool life, which is shown in equation (5). And some cutting tools, particularly those for a solid end mill, are recovered by regrinding after reaching their life limit. In this study, CE of a cutting tool is estimated by comparing machining time of a process with tool life and considering the regrinding process

\begin{matrix} {CE}_{i}^{tool} = \sum_{j = 1}^{n_{i}} \frac{t_{i, j}}{T_{i, j} \cdot (N_{i, j}^{gr} + 1)} \cdot \\ ({CE}_{i, j}^{tool} + N_{i, j}^{gr} \cdot P_{i, j}^{gr} \cdot E F^{el}) = \sum_{j = 1}^{n_{i}} t_{i, j} \cdot ω_{i, j}^{tool} \end{matrix}

(5)

CE of transportation processes in the workshop is connected with modes of transport and the distance of transportation. Different transportation machines will consume different resources, for example, conveyer belts or cranes need electric energy and forklifts may use diesel which will generate CO₂ directly. So CE of transportation processes contains the direct CE and CE of energy consumption, which is expressed in equation (6) and equation (7)

C E_{i}^{l o g} = \sum_{j = 1}^{n_{i}} L_{i, j}^{f f} \cdot K_{i, j}^{f f} \cdot E F^{f f} + \sum_{j = 1}^{n_{i}} L_{i, j}^{t r} \cdot K_{i, j}^{t r} \cdot E F^{e l}

(6)

L_{i, j}^{ff} = L_{i, j}^{tr} = | x_{i, j} - x_{i, j - 1} | + | y_{i, j} - y_{i, j - 1} |

(7)

Referring to equations (1)–(7), CE of multi-part processing is the sum of the aforementioned CE and the stand-by energy consumption of machine tools, as shown in equation (8)

C E_{total} = \sum_{i = 1}^{n} C E_{i} + C E^{s} = \sum_{i = 1}^{n} C E_{i} + \sum_{k = 1}^{m} P_{k}^{s} \cdot t_{k} \cdot E F^{el}

(8)

Carbon efficiency indicators of parts and machine tools

Although many CEAs generate CE, only CE of machining processes will add the value of a part, and other CEAs just play a supplementary role. To evaluate the carbon efficiency of different parts and machine tools, three indicators are proposed, that is, processing carbon efficiency (PRCE), part carbon efficiency (PACE), and machine tool carbon efficiency (MTCE).

To compare the carbon efficiency between different processes, it is necessary to put forward an indicator, which does not depend on the total CE of processes. PRCE is such an indicator, which can be defined as the ratio of CE of a machining process to the total CE of the process

η_{PR} = \frac{C E^{pr}}{C E^{pr} + C E^{au} + C E^{tool}}

(9)

Similar to PRCE, PACE is defined as shown in equation (10), which is the ratio of CE of all the machining processes to the total CE of the part, and it can be used to compare the carbon efficiency of different parts to find the least efficient part

η_{PA} = \frac{{CE}_{i}^{pr}}{C E_{i}}

(10)

To evaluate the carbon efficiency of a machine tool, MTCE is defined as the ratio of CE of machining processes on a machine to the total CE of the machine in a scheduling plan, which is shown in equation (11)

η_{M T} = \frac{\sum_{j}^{m} C E_{j}^{p r}}{\sum_{j}^{m} (C E_{j}^{p r} + C E_{j}^{a u} + C E_{j}^{t o o l}) + C E^{s}}

(11)

Multi-objective optimization model of the LC-FJSP

FJSP could be formulated as follows. There are $n$ jobs $J = {J_{1}, J_{2}, \dots, J_{n}}$ to be processed on $m$ machines $M = {M_{1}, M_{2}, \dots, M_{m}}$ . Each job $J_{i}$ consists of a sequence of $n_{i}$ operations ${O_{i, 1}, O_{i, 2}, \dots, O_{i, n_{i}}}$ and each operation $O_{i, j} (i = 1, 2, \dots, n; j = 1, 2, \dots, n_{i})$ has to be processed on one machine $M_{k}$ out of a set of given compatible machines $M_{i, j} \subseteq M$ . FJSP is to determine both an assignment and a sequence of operations on the machines to satisfy the given criteria. And LC-FJSP will consider CE of different scheduling schemes through the above evaluation model of CE.

Before describing the formulation, some assumptions are made as follows:

Each operation cannot be interrupted during its performance (non-preemptive condition).

Each machine can perform at most one operation at any time (resource constraint).

The precedence constraints of the operations in a job can be defined for any pair of operations.

Setting up time of machines and tool change are negligible.

Machines are independent from each other.

Jobs belong to a component or a machine and will perform the assembly process once their machining processes are completed. Although machining processes of different jobs are independent from each other, the MIDJ will influence the inventory cost (energy or inventory space) because one job will be suspended for waiting other jobs in the inventory if it completes ahead of other jobs. This usually happens in the small-batch production, for example, the production of a concave printer.

Here, the multi-objective LC-FJSP is formulated with the following four objectives to be minimized: (1) the makespan of all the jobs: $C_{\max}$ ; (2) the maximal MIDJ: $MID J_{\max}$ , which is to reduce the service time of the inventory; (3) the maximal workload of machines: $M L_{\max}$ ; and (4) the total CE of all the jobs: $C E_{total}$ .

Mathematically, LC-FJSP optimization can be formulated as follows

Minimize C_{\max} = ma x_{1 \leq i \leq n} {C_{i, n_{i}}}

(12)

M i n i m i z e M I D J_{m a x} = {max}_{1 \leq i \leq n} {C_{i, n_{i}}} - {min}_{1 \leq i \leq n} {C_{i, n_{i}}}

(13)

Minimize M L_{\max} = ma x_{1 \leq k \leq m} {\sum_{i = 1}^{n} \sum_{j = 1}^{n_{i}} (α_{i, j, k} \cdot t_{i, j, k})}

(14)

Minimize C E_{total} = \sum_{i = 1}^{n} C E_{i}

(15)

Subject to

C_{i, j} - C_{i, j - 1} \geq \sum_{k \in U_{i, j}} (α_{i, j, k} \cdot t_{i, j, k}), j = 2, \dots, n_{i}, \forall i

(16)

\begin{matrix} (C_{i, j} - C_{p, q} - t_{i, j, k}) \cdot (C_{p, q} - C_{i, j} - t_{p, q, k}) \\ \cdot α_{i, j, k} \cdot α_{p, q, k} \leq 0, \forall (i, j), (p, q), k \end{matrix}

(17)

\sum_{k \in U_{i, j}} α_{i, j, k} = 1, \forall i, j

(18)

\sum_{r = 1}^{N_{k}} β_{i, j, k, r} = α_{i, j, k}, \forall i, j

(19)

α_{i, j, k}, β_{i, j, k, r} \in {0, 1}, \forall i, j, k, r

(20)

t_{i, j, k}, S_{i, j} \geq 0, \forall i, j, k

(21)

where equation (16) ensures the precedence relationships between operations of a job, equation (17) explains each machine can be assigned to only one operation at each time, equation (18) states that only one machine could be selected from the set of available machines for each operation, and equation (19) forces each operation to be processed at one priority on one machine.

The hybrid NSGA-II for solving LC-FJSP

To solve LC-FJSP, a hybrid NSGA-II which combines NSGA-II with an LS algorithm based on neighborhood search is proposed in this article. NSGA-II is selected because of its population based on nature and non-domination sort which assigns a rank and a crowding distance (CD) to each individual (chromosome) in the population. Besides, NSGA-II is the most widely applied Multi-objective Evolutionary Algorithm as observed in the literature.^17,18,21 Based on the original NSGA-II, a modified dynamic crowding distance (DCD) is applied in the non-domination sort process to maintain the uniform diversity of the Pareto front. To increase the number of non-domination solutions, an LS algorithm based on neighborhood search is used just when ranks of all the individuals in the population are equal to 1. Figure 2 shows the framework of the proposed hybrid NSGA-II which will continue to execute till the maximum number of generations. The main components of the hybrid NSGA-II are summarized below.

Figure 2.

The framework of the hybrid NSGA-II.

Solution representation and encoding

In the population, every individual is a solution of the LC-FJSP, which is a combination of operation sequence and machine assignment. Thus, an individual can be expressed by processing sequence of operations on machines and machine assignment of each operation. Here, an individual consists of two parts corresponding to the two sub-problems of FJSP, that is, operation sequence genes and machine assignment genes, which is illustrated in the left side of Figure 3.

Figure 3.

The representation of a solution and its Gantt chart.

For operation sequence genes, the number of genes is the total number of operations $N_{O}$ . Each job number represents a certain operation of the job, and the ith appearance of a job number refers to the ith operation in the processing sequence of the job. For machine assignment genes, each gene donates the selected machine for each operation, so the number is $N_{O}$ . There is an example to explain the solution representation, which considers a problem with three jobs and four machines, and the Gantt chart corresponding to the representation of a feasible solution is also illustrated, as shown in the right side of Figure 3. The operation sequence and machine assignment of the solution in Figure 3 can be interpreted as follows: $(O_{2, 1}, M_{2}), (O_{1, 1}, M_{1}), (O_{2, 2}, M_{1}), (O_{3, 1}, M_{4}), (O_{1, 2}, M_{3}), (O_{3, 2}, M_{2}), (O_{2, 3}, M_{3}), (O_{3, 3}, M_{1}), (O_{1, 3}, M_{4}), (O_{3, 4}, M_{3})$ .

To calculate makespan and workload of machines, the representation of solutions needs to be encoded. And the completion time $C_{i, j}$ of operation $O_{i, j}$ can be deduced according to the following equation (22)²¹

C_{i, j} = {\begin{matrix} t_{hp} + t_{i, j, k}, j = 1 \\ \max {C_{i, j - 1}, t_{hp}} + t_{i, j, k}, j \geq 2 \end{matrix}

(22)

where $t_{hp}$ denotes the completion time of the predecessor operation of $O_{i, j}$ on the machine $M_{k}$ . If $O_{i, j}$ is the first operation of the job $i$ , $t_{h p} = 0$ , and if not, the maximum of $t_{hp}$ and $C_{i, j - 1}$ need to be considered.

Population initialization

For NSGA-II, the quality of the initial solutions is important for the efficiency of the algorithm. So, some different strategies are used in a hybrid way to generate initial solutions with a certain quality and diversity at the beginning of the evolution.

First, the following two rules are applied to generate initial machine assignments. (1) Random rule: randomly select a machine for each operation from the set of candidate machines and then place it at the position in the machine assignment genes. (2) Minimum processing CE rule: machines with lower SEC are chosen preferentially for each operation. In our algorithm, there are 60% of solutions generated by the random rule, while the other solutions are done by the minimum processing CE rule for initial machine assignments.

Once machines are assigned to all the operations, the following two rules are applied to generate initial operation sequences. (1) Random rule: randomly sort the operation sequence genes, for example, “1, 1, 1, 2, 2, 2, 3, 3, 3, 3” in the left of Figure 3 will be sorted randomly to generate initial operation sequences. (2) Most number of remaining operations rule:¹⁴ jobs with most remaining operations unprocessed have a high priority to be selected. And for initial operation sequences, 60% of solutions in the population are generated by the random rule, while the remaining ones follow the second rule.

Non-dominated sorting based on DCD

Before selection operation is performed, each individual is assigned a rank based on non-domination: all non-dominated individuals are classified into one level. For the individuals in the same level, the CD is calculated to keep a diverse front by making sure that each member stays a CD apart, as is shown in equation (23)

C D_{i} = \frac{1}{r} \sum_{k = 1}^{r} (\frac{| f_{i + 1}^{k} - f_{i - 1}^{k} |}{f_{\max}^{k} - f_{\min}^{k}})

(23)

where $r$ is the number of objectives, $f_{i + 1}^{k}$ is the kth objective of the (i + 1)th individual, $f_{i - 1}^{k}$ is the kth objective of the (i + 1)th individual, and $f_{\max}^{k}$ and $f_{\min}^{k}$ are the maximum and the minimum of the kth objective, respectively.

But CD is lack of uniform diversity in the obtained non-dominated solutions, and to overcome this problem, the DCD method is recently suggested.²² In this approach, one individual with the lowest DCD value is removed every time and DCD is recalculated for the remaining individuals. The DCD of individuals is calculated as follows

DC D_{i} = \frac{C D_{i}}{\log (1 / V_{i})}

(24)

V_{i} = \frac{1}{r} {\sum_{k = 1}^{r} (\frac{| f_{i + 1}^{k} - f_{i - 1}^{k} |}{f_{m a x}^{k} - f_{m i n}^{k}} - C D_{i})}^{2}

(25)

where $V_{i}$ is the variance of CDs of individuals which are neighbors of the ith individual. $V_{i}$ can give information about the difference variations of CD in different objectives.

Crossover and mutation operations

Once chromosomes for reproduction have been selected, the crossover and mutation operations are applied to produce offsprings. Crossover operation is applied to pairs of chromosomes, while mutation operator is for single individual.

Since chromosomes consist of two parts, operation sequence genes and machine assignment genes, two crossover strategies are applied in this article. For the first part, the improved precedence operation crossover (IPOX) is adopted,²¹ which is shown in Figure 4. At first, all the jobs are divided randomly into two sets: $J_{1}$ and $J_{2}$ . Then, the first offspring $Of f_{1}$ will inherit the operations in $J_{1}$ from $Pa r_{1}$ and the remaining positions will inherit from the operations in $J_{2}$ from $Pa r_{2}$ in the same sequence. The generating way of the second offspring $Of f_{2}$ is similar to $Of f_{1}$ .

Figure 4.

IPOX on the operation sequence genes.

For machine assignment genes, the multi-point preservative crossover (MPX) is chosen to generate the offsprings.^21,23 During the crossover operation, there are three crossover operators: $P_{c 1}$ , $P_{c 2}$ , and $P_{c 3}$ , which represent the probability of using IPOX alone, MPX alone, and combination of IPOX and MPX, respectively.

Besides, the swap mutation is used in the operation sequence genes of a chromosome, while the two-point mutation is applied to ensure the diversity of the machine assignment genes. There are also three mutation operators: $P_{m 1}$ , $P_{m 2}$ , and $P_{m 3}$ , which represent the probability of using swap mutation alone, two-point mutation alone, and combination of these two methods, respectively.

LS based on neighborhood search

To improve the algorithm’s effectiveness and to increase the number of the non-dominated solutions, an LS algorithm based on neighborhood search is applied in the NSGA-II when ranks of all chromosomes in the population are equal to one, which is illustrated in Figure 5. The procedure of the proposed LS algorithm is defined as follows:

Figure 5.

The flowchart of the local search algorithm.

When ranks of all chromosomes in the parent population equal 1, divide the population into $N_{obj}$ subpopulations, and $N_{obj}$ is the number of optimization objectives.

For each subpopulation, operation sequence genes and machine assignment genes of its chromosomes are adjusted just for a single objective. For example, for the first subpopulation $S P_{1}$ , operation sequence genes of its chromosomes will be adjusted based on neighborhood search which will select $γ$ genes randomly and sequence these genes to obtain its whole neighborhood chromosomes; for their machine assignment genes, $α$ genes will be selected and replaced with the more efficient machine tools; and then, the chromosome with the shortest makespan will be selected from all the neighborhood chromosomes and form a new subpopulation $NS P_{1}$ . Similarly, the neighborhood search and method of low-load machine tool selection are adopted for the operation sequence genes and machine assignment genes of the chromosomes in $S P_{2}$ , respectively. In $S P_{3}$ , only machine assignment genes are altered through the method of low-load machine tool selection to reduce the maximal workload of machines, and in $S P_{4}$ , machine tools with lower SEC will be selected to reduce the total CE.

The new subpopulations $NS P_{1}$ , $NS P_{2}$ , $NS P_{3}$ , and $NS P_{4}$ will form a new population, which will be combined with the parent population and sorted based on non-domination rank and DCD again to generate the offspring population.

Case study

The proposed hybrid NSGA-II is coded in MATLAB and runs on a Dell OptiPlex 360 with a 2.93 GHz processor and 2 GB RAM. Because of the particularity of the problem described above, the case study includes two parts: to test the performance of the algorithm, four widely used benchmarks from Kacem instances²⁴ are carried out; a case from an actual production workshop is simulated to verify the rationality of the model of the LC-FJSP.

Parameter setting

It is generally accepted that the difficulty in solving the MFJSP is closely associated with problem size. Therefore, the maximum number of generations is set as Gen = 10·n·m and the population size is set as Pop = 5·n·m.⁴ In addition, the proposed hybrid NSGA-II contains several other key parameters: the crossover probabilities, that is, $P_{c 1}, P_{c 2}, P_{c 3}$ , the mutation probabilities, that is, $P_{m 1}, P_{m 2}, P_{m 3}$ , and the probabilities of performing LS $γ$ and $α$ . First, the general crossover probability and the general mutation probability are set 90% and 10%, respectively, so the sum of $P_{c 1}, P_{c 2}, P_{c 3}$ is 0.9 and the sum of $P_{m 1}, P_{m 2}, P_{m 3}$ is 0.1. In order to simplify the complexity of parameter selection, two assumptions are made that $P_{c 1} = P_{c 2}$ and $P_{m 1} = P_{m 2}$ . To investigate the influence of these parameters on the performanceof our algorithm, the Taguchi method of design of experiment is implemented by using a moderate scale instance Problem 8 × 8.²⁴ Factor levels of these parameters are listed in Table 1.

Table 1.

Factor levels of parameters.

Parameters	Factor levels
	1	2	3
$P_{c 1}$	0.2	0.3	0.4
$P_{m 1}$	0.02	0.03	0.04
$γ$	0.2	0.3	0.4
$α$	0.3	0.4	0.5

According to the number of parameters and the number of factor levels, the orthogonal array $L_{9} (3^{4})$ is chosen, which is listed in Table 2. For each parameter combination, the proposed hybrid NSGA-II is run 10 times independently. The non-dominated solutions among all the solutions obtained by each experiment are compared. Clearly, the more non-dominated solutions obtained by the combination of parameters, the better the combination is. Thus, the average percentage of non-dominated solutions (APNS) for each combination is used to measure the effectiveness of the algorithm. The obtained APNS values are also listed in Table 2.

Table 2.

Orthogonal array and APNS values.

No.	Factor levels				APNS (%)
	$P_{c 1}$	$P_{m 1}$	$γ$	$α$
1	1	1	1	1	31.25
2	1	2	2	2	51.56
3	1	3	3	3	43.75
4	2	1	2	3	39.06
5	2	2	3	1	46.88
6	2	3	1	2	46.88
7	3	1	3	2	45.31
8	3	2	1	3	45.31
9	3	3	2	1	39.06

APNS: average percentage of non-dominated solutions.

According to the APNS values, the trend of each factor level is illustrated in Table 3. It can be clearly seen that a good choice of parameter combination is suggested $P_{c 1} = P_{c 2} = P_{c 3} = 0.3$ , $P_{m 1} = P_{m 2} = 0.03$ , $P_{m 3} = 0.04$ , $γ = 0.4$ , and $α = 0.4$ .

Table 3.

Factor-level trend of the hybrid NSGA-II.

Parameters	Factor-level trend (%)
	1	2	3
$P_{c 1}$	42.19	44.27	43.23
$P_{m 1}$	38.54	47.92	43.23
$γ$	41.15	43.23	45.31
$α$	39.06	47.92	42.71

Test of benchmark instances

Here, four Kacem instances (Problem 4 × 5, Problem 8 × 8, Problem 10 × 10, Problem 15 × 10) are tested, and the results are listed in Table 4. And the proposed hybrid NSGA-II is also compared with the existing weighted approaches including AL + CGA,²⁴ PSO + TS,⁶ and Pareto-based approaches multiple-objective genetic algorithm (MOGA)²⁵ and multi-objective particle swarm optimization (MOPSO) + LS,²⁶ the results of which are also listed in Table 4 and all the solutions dominated by others are marked with underlines. In Table 4, $C_{M}$ , $W_{T}$ , and $W_{M}$ stand for makespan, the total workload of machines, and the maximal machine workload, respectively; S₁, S₂, S₃, and S₄ represent non-dominated solutions; for example, for the AL + CGA,²⁴ it only obtains one non-dominated solution S₁(16, 34, 10) when solving Problem 4 × 5.

Table 4.

Results of four Kacem instances.

Problem	Obj.	AL + CGA²⁴		PSO + TS⁶		MOGA²⁵				MOPSO + LS²⁶				Hybrid NSGA-II
		$S_{1}$	$S_{2}$	$S_{1}$	$S_{2}$	$S_{1}$	$S_{2}$	$S_{3}$	$S_{4}$	$S_{1}$	$S_{2}$	$S_{3}$	$S_{4}$	$S_{1}$	$S_{2}$	$S_{3}$	$S_{4}$
$4 \times 5$	$C_{M}$	16	–	11	–	11	11	12	–	–	–	–	–	11	12	13	11
	$W_{T}$	34	–	32	–	32	34	32	–	–	–	–	–	32	32	33	34
	$W_{M}$	10	–	10	–	10	9	8	–	–	–	–	–	10	8	7	9
$8 \times 8$	$C_{M}$	15	16	14	15	15	15	16	–	15	16	14	16	16	15	16	15
	$W_{T}$	79	75	77	75	81	75	73	–	75	73	77	78	73	75	78	74
	$W_{M}$	13	13	12	12	11	12	13	–	12	13	12	11	13	12	11	13
$10 \times 10$	$C_{M}$	7	–	7	–	8	7	8	–	8	7	8	7	8	7	7	8
	$W_{T}$	45	–	43	–	42	42	41	–	41	42	42	43	41	42	43	42
	$W_{M}$	5	–	6	–	5	6	7	–	7	6	5	5	7	6	5	5
$15 \times 10$	$C_{M}$	23	24	11	–	11	12	11	–	12	11	–	–	11	11	–	–
	$W_{T}$	95	91	93	–	91	95	98	–	93	91	–	–	91	93	–	–
	$W_{M}$	11	11	11	–	11	10	10	–	10	11	–	–	11	10	–	–

PSO: particle swarm optimization; TS: tabu search; MOGA: multiple-objective genetic algorithm; MOPSO: multi-objective particle swarm optimization; LS: local search; NSGA-II: non-dominated sorting genetic algorithm II.

From Table 4, it can be seen that the proposed hybrid NSGA-II is effective in solving these four Kacem instances. With regard to Problem $4 \times 5$ , the hybrid NSGA-II obtains more non-dominated solutions than the MOGA and PSO (TS), and its results are similar to the MOPSO (LS) for Problem $4 \times 5$ and Problem $10 \times 10$ , which all get four non-dominated solutions. When solving Problem $15 \times 10$ , the hybrid NSGA-II obtains two solutions and so does the MOPSO (LS), but its solution (12, 93, 10) is dominated by (11, 93, 10) obtained by our algorithm. Though the MOGA obtains three for Problem $15 \times 10$ , its solutions (12, 95, 10) and (11, 98, 10) are both dominated by (11, 93, 10).

In a word, the proposed hybrid NSGA-II is an effective algorithm to solve the multi-objective FJSP, so it is also used to solve the proposed LC-FJSP.

Simulation based on an actual case

To verify the rationality of LC-FJSP, a case from a flexible manufacturing shop in a printing machine manufacturing factory in Shaanxi Province is simulated. There are six jobs and five machines in the workshop (Problem $6 \times 5$ ), and the relative data including material removal volume $V_{i, j} (c m^{3})$ and processing time $t_{i, j} (s)$ of each process, machining parameters $SEC (kJ / c m^{3})$ , and tool parameters $ω^{tool} (kg C O_{2} - e / s)$ are shown in Table 5.

Table 5.

Relative data of machining processes.

Job	$O_{i, j}$	$V_{i, j}$	$M_{1}$			$M_{2}$			$M_{3}$			$M_{4}$			$M_{5}$
			$t$	$SEC$	$ω^{tool}$	$t$	$SEC$	$ω^{tool}$	$t$	$SEC$	$ω^{tool}$	$t$	$SEC$	$ω^{tool}$	$t$	$SEC$	$ω^{tool}$
$J_{1}$	$O_{1, 1}$	240	21	5.13	0.082	15	5.49	0.025	–	–	–	57	6.67	0.073	52	3.89	0.027
	$O_{1, 2}$	400	–	–	–	24	4.26	0.064	45	3.32	0.024	–	–	–	24	3.75	0.054
	$O_{1, 3}$	180	14	4.57	0.016	–	–	–	56	5.33	0.039	24	6.98	0.069	33	2.93	0.021
$J_{2}$	$O_{2, 1}$	360	50	6.68	0.029	22	7.26	0.127	97	3.56	0.025	–	–	–	20	3.10	0.039
	$O_{2, 2}$	420	57	8.20	0.060	115	2.83	0.071	–	–	–	54	3.02	0.140	81	2.60	0.056
	$O_{2, 3}$	300	–	–	–	54	3.94	0.057	17	4.33	0.014	27	2.85	0.015	54	5.42	0.069
	$O_{2, 4}$	180	13	3.85	0.078	20	4.44	0.026	19	3.22	0.048	70	3.38	0.150	82	4.01	0.075
$J_{3}$	$O_{3, 1}$	620	130	3.09	0.009	–	–	–	244	6.16	0.067	47	7.93	0.030	145	5.97	0.045
	$O_{3, 2}$	300	72	4.73	0.161	18	7.25	0.045	17	7.38	0.020	114	6.31	0.059	–	–	–
	$O_{3, 3}$	400	29	6.66	0.075	132	5.98	0.090	–	–	–	43	7.82	0.058	41	4.45	0.116
	$O_{3, 4}$	140	–	–	–	11	6.17	0.037	11	2.75	0.147	25	5.53	0.099	27	5.02	0.032
$J_{4}$	$O_{4, 1}$	520	34	4.88	0.013	67	4.20	0.056	43	7.18	0.072	54	6.84	0.037	60	4.08	0.034
	$O_{4, 2}$	340	68	7.54	0.030	–	–	–	21	6.79	0.170	–	–	–	20	7.07	0.084
$J_{5}$	$O_{5, 1}$	140	18	4.27	0.031	11	3.18	0.071	10	2.24	0.070	11	7.20	0.019	–	–	–
	$O_{5, 2}$	260	23	2.47	0.070	28	4.80	0.059	–	–	–	26	2.37	0.117	30	4.48	0.070
	$O_{5, 3}$	300	43	7.97	0.062	–	–	–	20	5.88	0.129	19	6.47	0.044	23	6.67	0.061
	$O_{5, 4}$	420	48	4.07	0.130	24	2.91	0.176	28	5.15	0.067	–	–	–	50	2.89	0.013
	$O_{5, 5}$	260	34	4.28	0.020	18	6.33	0.094	22	6.45	0.152	69	5.19	0.050	–	–	–
$J_{6}$	$O_{6, 1}$	440	–	–	–	66	5.54	0.016	38	3.14	0.083	167	9.24	0.048	50	4.48	0.025
	$O_{6, 2}$	320	43	4.91	0.029	30	5.45	0.035	–	–	–	41	2.89	0.069	26	5.18	0.057
	$O_{6, 3}$	160	17	5.26	0.039	–	–	–	26	4.63	0.054	19	4.92	0.065	12	6.76	0.014

SEC: specific energy consumption.

Referring to environmental report, technical report, and industrial table of China,^27,28 the emission factors of electrical energy, lubricant oil, coolant, and diesel are $2.41 kg C O_{2} - e / kW h$ , $0.469 kg C O_{2} - e / L$ , $3.782 kg {CO}_{2} - e/L$ , and $2.68 kg C O_{2} - e / L$ , respectively. Then, relative data of each machine tool which includes workshop layout $WL (m)$ , auxiliary material consumption $ω^{au} (kg C O_{2} - e / s)$ , and stand-by power $P^{s} (kW)$ are listed in Table 6. In the workshop, two diesel forklift trucks are used to transport the six jobs from one machine to another, and the specific diesel consumption of a truck is about 0.004 L/m. Since transportation time is much shorter than machining time, the influence of transportation time of jobs on the scheduling process is ignored.

Table 6.

Relative data of machine tools.

Machines	$M_{1}$	$M_{2}$	$M_{3}$	$M_{4}$	$M_{5}$
$WL (m)$	$(0, 0)$	$(20, 0)$	$(0, 10)$	$(23, 10)$	$(0, 18)$
$ω^{au} (kg C O_{2} - e / s)$	0.095	0.062	0.061	0.060	0.059
$P^{s} (kW)$	3.77	4.00	1.75	1.42	2.43

Results and discussions

Due to the complexity of the problem, the population sizes and the numbers of generation are set to 100 and 900, respectively, and other parameters still refer to section “Parameter setting.” By using the proposed algorithm to optimize the above Problem $6 \times 5$ , a set of non-domination solutions with larger DCD is finally found which includes 15 solutions, as shown in Table 7.

Table 7.

A set of non-domination solutions.

No.	$C_{\max}$	$MID J_{\max}$	$M L_{\max}$	$C E_{total}$	$C E^{pr}$	CE ratio $C E^{pr} / C E_{total} (%)$
1	137	39	101	28.35	22.53	79.48
2	131	31	107	27.85	21.46	77.07
3	149	11	133	27.48	21.26	77.35
4	357	249	357	22.91	17.29	75.48
5	131	20	125	29.53	22.84	77.34
6	357	249	357	22.91	17.29	75.48
7	324	173	324	23.09	17.49	75.76
8	357	206	357	22.95	17.29	75.34
9	324	216	324	23.05	17.49	75.89
10	282	107	277	23.16	17.51	75.60
11	285	48	277	23.30	17.51	75.16
12	256	148	244	23.21	17.71	76.30
13	282	174	277	23.08	17.51	75.87
14	272	164	272	23.20	17.69	76.25
15	333	36	324	23.59	17.49	74.14

MIDJ: makespan interval of different jobs; CE: carbon emission.

From Table 7, it can be seen that the No. 1, No. 2, No. 3, and No. 4 have the minimum $M L_{\max}$ , $C_{\max}$ , $MID J_{\max}$ , and $C E_{total}$ , respectively, which have been marked with underlines. At different times or in different occasions, decision-makers will choose different solutions. For example, if jobs are urgently demanded and the laws and regulations are strict with CE of the plant, solutions with the shorter $C_{\max}$ and less $C E_{total}$ will be chosen, while $MID J_{\max}$ and $M L_{\max}$ will be regarded as secondary considerations, such as No. 1 or No. 2 solution. Therefore, compared to other traditional multi-objective optimization algorithms such as weighted sum method, the LC-FJSP based on our hybrid algorithm can get a Pareto optimal set which includes all possible optimal solutions, and managers can make the final decision up to the practical situation and his preference.

In addition, although $C_{\max}$ of No. 2 solution is the least, its CE is $27.85 kg C O_{2} - e$ , which is higher than other solutions, and its CE ratio is lower than that of No. 1 solution, which is 79.48%. To analyze and increase the carbon efficiency of No. 2 solution further, the Gantt chart is obtained, as shown in Figure 6.

Figure 6.

The Gantt chart of the No. 2 solution.

First, MTCE is employed to evaluate the carbon efficiency of each machine tool, which is shown in Table 8. It can be clearly seen that $M_{5}$ generates the most amount of CE, which is $5.54 gC O_{2} - e$ , but its MTCE is not the highest. While $M_{1}$ generates the least CE and its MTCE is up to 87.90%, which means that $M_{1}$ is the highest efficient among these machine tools. Furthermore, it can be clearly seen that $M_{1}$ , $M_{2}$ , $M_{3}$ , and $M_{5}$ are in the state of waiting to process jobs during the interval time between jobs, especially $M_{2}$ and $M_{5}$ , which have a longer interval time than others, as shown in Figure 6. In that situation, if the machine tool is shutdown, the energy consumption will be reduced.

Table 8.

MTCE of each machine tool.

	${CE}_{total}^{mach}$	$C E^{s}$	$C E^{pr}$	MTCE (%)
M1	4.22	0.0757	3.71	87.90
M2	5.02	0.091	4.24	84.54
M3	5.49	0.0305	4.71	85.91
M4	4.93	0	4.14	84.02
M5	5.54	0.0293	4.66	84.11

CE: carbon emission; MTCE: machine tool carbon efficiency.

On the other hand, PACE of each part is obtained to analyze the carbon efficiency, as shown in Table 9. The processing of $J_{3}$ generates the most CE which is up to $7.49 gC O_{2} - e$ because its processing time is much longer than others, as shown in Figure 6. By contrast, $J_{1}$ generates the least CE and has a higher PACE. And PACE of $J_{2}$ is the lowest which is only 65.94%. To search for the key process which affects the carbon efficiency of $J_{2}$ , PRCE of $J_{2}$ is acquired in Table 10. From Table 10, it can be seen that the process $p_{2}$ of $J_{2}$ generates the most amount of CE and its PRCE is only 48.75%, which means more than half of its CE comes from non-value activities, such as auxiliary materials and the transportation process. So, some methods should be taken to reduce CE of these non-value activities, for example, changing its cutting tool or machining parameters.

Table 9.

PACE of each part.

	${CE}_{total}^{part}$	$C E^{pr}$	PACE (%)
$J_{1}$	3.05	2.57	84.26
$J_{2}$	4.44	2.93	65.94
$J_{3}$	7.49	6.23	83.10
$J_{4}$	3.76	3.31	88.02
$J_{5}$	5.39	3.83	70.96
$J_{6}$	3.48	2.60	74.61

CE: carbon emission; PACE: part carbon efficiency.

Table 10.

PRCE of each process of $J_{2}$ .

	${CE}_{total}^{mach}$	$C E^{pr}$	PRCE (%)
$p_{1}$	0.85	0.75	87.44
$p_{2}$	1.74	0.85	48.75
$p_{3}$	1.17	0.87	74.63
$p_{4}$	0.68	0.46	68.07

CE: carbon emission; PRCE: processing carbon efficiency.

In summary, by means of the proposed model and algorithm, a set of optimal solutions can be obtained, and the final decision is up to practical situation and the preference of decision-makers. At the same time, the three indicators PRCE, PACE, and MTCE are used to evaluate the carbon efficiency of different parts, processes, and machine tools and find the key processes and machine tools which have the most influence on environment.

Conclusion

A new scheduling model LC-FJSP is put forward in this article, which considers both factors of production (i.e. makespan, MIDJ, and machine workload) and environmental influence (i.e. CE). In the model, CE of the manufacturing processes is classified into five types, that is, energy consumption of the processing, auxiliary material consumption, tool wear, CE of transportation process, and stand-by energy consumption of machine tools, and a carbon footprint model of multi-job processing is established to quantify CE of different scheduling plans. And to evaluate the carbon efficiency of different parts and machine tools, three indicators are put forward, that is, process carbon efficiency, PACE, and MTCE.

To solve the proposed LC-FJSP, a hybrid NSGA-II is proposed which combines NSGA-II with an LS algorithm based on neighborhood search. Since CD of the original NSGA-II is lack of uniform diversity in the obtained non-dominated solutions, the DCD method is applied to overcome its shortcoming. And it proves the effectiveness of the proposed algorithm by comparing its results with another four multi-objective optimization algorithms through the test of four benchmark instances.

Finally, a use case from an actual manufacturing workshop is studied to demonstrate the feasibility and applicability of the proposed LC-FJSP model on finding the low-carbon scheduling plan and searching for the low-carbon efficiency of machines and parts.

Future research in this area will include the modification and amelioration of the LC-FJSP because in the model some factors are ignored, such as CE of the inventory, the time of tool changing, and recovery of the auxiliary materials. Furthermore, the LC-FJSP and carbon efficiency indicators can be extended to other more complex aspects, such as dynamic job-shop scheduling or batch scheduling. And the carbon footprint model is also suitable for other problems of a workshop, such as facility layout and process planning.

Footnotes

Appendix 1 Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was financially supported by the Leading Talent Project of Guangdong Province and the National Natural Science Foundation of China (grant no. 51275396).

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