Energy Saving in Single-Machine Scheduling Management: An Improved Multi-Objective Model Based on Discrete Artificial Bee Colony Algorithm

Abstract

Green manufacturing, which takes environmental effect and production benefit into consideration, has attracted increasing concern with the target of carbon peaking and carbon neutrality proposed. As a critical process in the manufacturing system, shop scheduling is also an important method for enterprises to achieve green manufacturing. Therefore, it is necessary to consider both production benefits and environmental objectives in shop scheduling, which are symmetrical and equally important. In addition, noise pollution has become an important environmental issue that cannot be ignored in the manufacturing processes, but which has been paid less attention in previous studies. Thus, the MODABC algorithm, with the optimization objectives of simultaneously minimizing lead-time/tardiness cost and job-shop noise pollution emission is proposed in this paper. We designed a discrete permutation-based two-layer encoding mechanism to generate the initial population. Then, three crossover methods were used to perform nectar update operations in the employed bee search phase, and three neighbourhood structures were used to improve the onlooker bee search operations. Finally, the MODABC algorithm was compared with other classical MOEAs. The results demonstrate that MODABC can provide non-dominated solution set with good convergence and distribution, and show significant superiority in solving green single-machine multi-objective scheduling problems.

1. Introduction

Single-machine scheduling is the most basic type of job-shop scheduling. It is commonly seen in production and serves as the basis for studying more complex job-shop scheduling problems. Mouzon and Yildirim [1] were the first to research the job-shop green scheduling problem. They explored the problem of minimizing the total energy consumption and total tardiness in a single-machine environment. Moreover, they proposed a multi-objective optimization framework using a greedy randomized adaptive search meta-heuristic algorithm to obtain an approximate Pareto front solution and determined the optimal scheduling solution using an analytic hierarchy process. Lu and Zhang et al. [2] considered a green job-shop scheduling problem with variable machining speeds (JSPVMS), established a new mixed-integer linear programming model that simultaneously minimized both total energy consumption and the makespan, and then, developed a new knowledge-based multi-objective memetic algorithm (MOMA) to address this problem. Shrouf et al. [3] proposed a mathematical model for minimizing the cost of energy consumption in the single-machine production scheduling process by considering the variation of energy prices during a day by comparing the analytical and heuristic solutions using a genetic algorithm and demonstrated that avoiding high energy price cycles significantly reduced energy costs. Liu and Yang et al. [4] studied a single-machine system with deterministic product arrival time and first-come, first-served processing rules. They developed a multi-objective model that simultaneously minimized both total CO₂ emissions and completion time and proposed a non-dominated sorting genetic algorithm (NSGA)-II with good performance to solve the abovementioned problem. Wang and Liu [5] studied a dual-objective single-machine batch scheduling problem with minimized completion time and minimized total energy cost under different operation sizes, time-of-use tariffs and different energy consumption rates and developed an integer programming model with an exact constraint method to obtain the exact Pareto front. For the large-scale problem, two heuristics for approximating the Pareto front were proposed based on the decomposition. Che and Zeng et al. [6] studied the single-machine scheduling problem under a time-of-use tariff with the optimization objective of minimizing the total power cost in a given completion time, developed a continuous-time mixed-integer linear programming (MILP) model, and proposed an efficient greedy insertion heuristic algorithm. Yin and Li et al. [7] proposed a single-machine scheduling model with controllable processing and order-dependent preparation times that can simultaneously minimize the total advance/delay, cost and energy consumption, and proposed a multi-objective evolutionary algorithm with a multi-local search strategy. Furthermore, Che and Wu [8] considered a single-machine scheduling problem with a power reduction mechanism and established a MILP model that simultaneously minimizes both total energy consumption and maximum delay. They proposed a basic ε-constraint method that greatly reduces the solution space using cluster analysis. Rubaiee et al. [9] studied the single-machine no-priority scheduling problem by building a mixed-integer multi-objective mathematical planning model. They developed several new genetic algorithms and showed the possibility to optimize the production processes, improve energy efficiency and manage environmental challenges through operation management techniques. Moreover, they showed the application of the computation results of the multi-objective model in decision making.

From the above literature review, there is currently scarce research literature on green single-machine scheduling problems. Existing research targeted energy consumption or carbon emission as the main optimization objective, and mainly focused on small-scale scheduling problems. Few researchers have studied multi-objective, large-scale scheduling problems with noise emission as the green optimization index. Most solution methods use heuristics or mathematical planning algorithms, while a few use group intelligence algorithms. Shop scheduling is considered as an important way for enterprises to achieve green manufacturing, so it is necessary to consider both production benefits and environmental objectives. Furthermore, noise pollution, which has a negative effect on employees’ health and emotion, has become an important environmental issue that cannot be ignored in the manufacturing processes [10,11]. Therefore, considering both production benefits and environmental objectives in single-machine scheduling, a multi-objective discrete ABC algorithm, MODABC, with the optimization objectives of simultaneously minimizing lead-time/tardiness cost and job-shop noise pollution emission is proposed, aiming to coordinate and optimize the economic performance and environmental objectives, which were considered symmetrical and equally important.

The contributions of this paper are summarized as follows: (1) MODABC is proposed with the optimization objectives of simultaneously minimizing lead-time/tardiness cost and job-shop noise pollution emission. (2) A discrete permutation-based two-layer encoding mechanism is designed to generate the initial population using four methods: EDD, SPT, LPT and random generation. Then, three crossover methods, PMX, OX and PBX, are used to perform nectar update operations in the employed bee search phase. Moreover, three neighbourhood structures, swap, reverse order and insertion, are used to improve the onlooker bee search operations. (3) The parameters of the algorithm are calibrated by using the Taguchi method, and the MODABC algorithm is compared with other classical MOEAs to verify its effectiveness in solving the green single-machine multi-objective scheduling problem.

The structure of this paper is as follows. Section 2 establishes a mathematical model for MODABC. Section 3 elaborates a new optimization approach for this MODABC. Section 4 provides numerical experiments to verify the effectiveness of the proposed algorithm. Section 5 concludes this work.

2. Problem Description

2.1. Green Single-Machine Multi-Objective Scheduling Problem Description

The green single-machine multi-objective scheduling problem is defined as follows: $n$ individual jobs are scheduled to be processed on a single machine; both the machine and job are available at time zero; the machine processes continuously; at most one job can be processed at one time without interruption; the processing order between jobs is not predetermined, and each processing parameter is independent of its position in the processing order. The machine is available with different processing speeds, each corresponding to different noise emissions. From this processing characteristic, the processing speed of the job can be adjusted by selecting the processing speed of the machine. The higher the processing speed, the higher the noise emission, and vice versa [12,13]. By sequencing the jobs to be processed on the machine, both objectives of minimizing lead time/tardiness costs and job-shop noise emissions are simultaneously achieved.

2.2. Green Single-Machine Multi-Objective Scheduling Model Establishment

The symbols and decision variables involved in this paper are shown below.

2.2.1. Symbols

• $n$: The number of jobs.
• $J_j$: Job $j$; $1\le j\le n$.
• $p:$ The number of processing position on the machine; the machine sets $n$ processing positions, i.e., $1\le p\le n$.
• $u:$ The number of available processing speeds of the machine.
• $v:$ The order number of machine speeds; $v=1,\dots ,u$.
• $V_v:$ The $v$th processing speed of the machine.
• $l_j:$ The processing load of job j.
• $d_j$: The delivery time of job j.
• $C_j:$The completion time of job j.
• $S_j$: The start time of job j.
• $E_j:$ The early completion time of job j. $E_j=max\{d_j-C_j,0\}$.
• $T_j:$ The tardiness of job j. $T_j=max\{C_j-d_j,0\}$.
• ${\alpha }_j:$ The penalty cost of the delivery time of job j.
• ${\beta }_j:$ The penalty cost of the tardiness of job j.
• $P{N}_v:$ The noise emitted by the machine during processing at speed $V_v$.
• $t_{jv}:$ Time required by the machine to process job j at speed $V_v$; $t_{jv}=l_j/V_v$.
• $M$: An infinite positive number.

2.2.2. Decision Variables

• $x_{jpv}=\{\begin{array}{c}1,\mathrm{when} \mathrm{job} j \mathrm{is} \mathrm{being} \mathrm{processed} \mathrm{by} \mathrm{the} \mathrm{machine} \mathrm{at} \mathrm{speed} Vv \mathrm{at} \mathrm{position} p\\ 0,\mathrm{otherwise}\end{array}$
• $b_l:$ The start time of processing of the job at position l.

This problem is an extended version of the traditional single-machine scheduling problem defined as a multi-objective mathematical model, which is represented as follows.

Objective function:

$$min{f}_1={\displaystyle \sum }_{j=1}^n({\alpha }_j·E_j+{\beta }_j·T_j)$$

(1)

$$min{f}_2=10·log\frac{{\sum }_{j=1}^n{\sum }_{p=1}^n{\sum }_{v=1}^u{10}^{0.1·P{N}_v}·\frac{l_j}{V_v}·x_{jpv}}{{\sum }_{j=1}^n{\sum }_{p=1}^n{\sum }_{v=1}^u\frac{l_j}{V_v}·x_{jpv}}$$

(2)

Constraint conditions:

$${{\displaystyle \sum }}_{l=1}^n{{\displaystyle \sum }}_{v=1}^u{x}_{jpv}=1,j=1,\dots ,n$$

(3)

$${{\displaystyle \sum }}_{j=1}^n{{\displaystyle \sum }}_{v=1}^u{x}_{jpv}\le 1, p=1,\dots ,n$$

(4)

$$b_{p+1}-b_p={{\displaystyle \sum }}_{j=1}^n{{\displaystyle \sum }}_{v=1}^u{x}_{jpv}·t_{jv}, p=1,\dots ,n-1$$

(5)

$$E_j\ge d_j-\left(b_p+x_{jpv}·t_{jv}\right)-\left(1-x_{jpv}\right)·M, v=1,\dots ,u; j,p=1,\dots ,n$$

(6)

$$E_j\ge 0, j=1,\dots ,n$$

(7)

$$T_j\ge b_p+x_{jpv}·t_{jv}-\left(1-x_{jpv}\right)·M-d_j, v=1,\dots ,u; j,p=1,\dots ,n$$

(8)

$$T_j\ge 0, j=1,\dots ,n$$

(9)

$$x_{jpv}\in \{0,1\}, v=1,\dots ,u, j,p=1,\dots ,n$$

(10)

Equations (1) and (2) define the optimization objective of the problem as minimizing the total lead-time/tardiness cost and job-shop noise emissions. Equation (3) denotes that each job can be scheduled for processing at only one position on the machine at a set speed. Equation (4) denotes that, at most, one job can be scheduled for processing at any position of the machine. Equation (5) defines the processing time of the machine at position $p.$ Equations (6) and (7) calculate the early completion time $E_j$ of job $j$; if the pth position of the machine is not scheduled to process job j with speed v, then $x_{jpv}=0$, since $-M$ is negative, the equation is automatically satisfied. However, when $x_{jpv}=1$, $E_j$ will reflect the difference between the delivery time and completion time of job $j.$ Equations (8) and (9) calculate the tardiness $T_j$ of job $j$. As mentioned above, when $x_{jpv}=0$, the equation is automatically satisfied. Otherwise, when $x_{jpv}=1$, $T_j$ will reflect the difference between the completion and delivery times of job $j.$ Equation (10) is used to limit the range of values of the variables.

Compared with other job-shop scheduling types, this scheduling problem is relatively simple. However, it is still an NP-hard problem [7]. Therefore, it is important to further explore and study this problem.

3. MODABC for the Green Single-Machine Multi-Objective Scheduling Problem

3.1. Standard Artificial Bee Colony Algorithm

The Turkish scholar Karaboga first proposed the artificial bee colony (ABC) algorithm in 2005 by studying the foraging behaviour of bees and published the research results [14,15] of this algorithm in 2007. His works have developed into one of the most popular research branches in group intelligence. The algorithm was first used in single-objective optimization problems, where it demonstrated good performance with its simple principle, fast convergence, good robustness and few adjustment parameters.

The standard ABC algorithm performs optimal solution search by simulating the foraging behaviour of different bees, thereby classifying bees into three types according to their respective functions: employed, onlooker and scout bees. Furthermore, it divided the optimal solution search process into four basic phases, namely, initialization, employed bee search, onlooker bee search and scout bee search phases. The algorithm generates the initial nectar source using a random method, and then iteratively performs the search in the employed, onlooker and scout bee phases, for population evolution until the termination condition is satisfied. The basic elements of the algorithm are as follows.

3.1.1. Initialization

The standard ABC algorithm represents the solution of the problem to be optimized using the location of the nectar source, given the number of nectar sources is denoted as $N$. In the initialization phase, the algorithm randomly generates $N$ random solutions, and the exploitation level of each nectar source is recorded as $Trai{l}_i$, which is set to 0. Assuming that the dimension of the problem to be optimized is $D$ and the range of values of each dimension is $\left[l{b}_j,u{b}_j\right]$, then the initialization equation is expressed as follows.

$$x_{i,j}=l{b}_j+rand\left(0,1\right)\cdot \left(u{b}_j-l{b}_j\right),i=1,2,\cdots ,N;j=1,2,\cdots ,D$$

(11)

where $x_{i,j}$ is the value of the jth dimension of the ith nectar source in the current population, and $rand\left(0,1\right)$ is a random number on the interval [0, 1].

3.1.2. Employed Bee Search

In the employed bee search phase, each nectar source is exploited once to generate new nectar sources, and their merits are determined by judging the value of the fitness function of the old and new nectar sources. The following equation is used to generate new solutions in this phase:

$$x_{i,j}^{\prime }=x_{i,j}+{\phi }_{i,j}\cdot \left(x_{i,j}-x_{k,j}\right), i,k=1,2,\cdots ,N;j=1,2,\cdots ,D$$

(12)

where $x_{i,j}^{\prime }$ is the value of the jth dimension of the variance solution $x_i$ of the ith nectar source $x_i^{\prime }$ in the current population and ${\phi }_{i,j}$ is a random number in the interval [0, 1]. Additionally, $x_k$ denotes a nectar source in the current population different from $x_i$. If the update is successful, then $Trai{l}_i$ is reset to 0; otherwise,$Trai{l}_i\leftarrow Trai{l}_i+1$.

3.1.3. Onlooker Bee Search

During the onlooker bee search phase, a portion of the nectar in the current population is exploited, using the same nectar update formula as the employed bee search, as detailed in (12). Here, it is necessary to ensure that $x_i$ is different from $x_k$. Similarly, the nectar source $i$ is selected by the ABC algorithm using a roulette wheel from the current N nectar sources. For instance, in the minimization problem, the probability value of each nectar source being selected is as follows.

$$P_i=\frac{fi{t}_i}{{\sum }_{j=1}^{N}fi{t}_j}$$

(13)

$$fi{t}_i=\{\begin{array}{cc}1/\left(1+f_i\right)& if·f_i\ge 0\\ 1+abs\left(f_i\right)& if·f_i<0\end{array}$$

(14)

where $f_i$ denotes the objective function value of $x_i$; $fi{t}_i$ denotes the adaptation degree of the ith nectar source $x_i$. Note that the value of $fi{t}_i$ is directly linked and inversely proportional to $f_i$.

3.1.4. Scout Bee Search

This phase checks the exploitation level $Trai{l}_i$ of each nectar source $x_i$ in the current population, and if the value exceeds the predetermined maximum exploitation level $Limit$, i.e., $Trai{l}_i>Limit$, then the nectar source $x_i$ is discarded and a new nectar source is randomly created. When the new nectar source is generated, the corresponding scout bee is transformed into an employed bee. At the most, there is one scout bee in each generation of the population.

The standard ABC algorithm flowchart is shown in Figure 1.

3.2. MODABC Algorithm Design and Implementation

3.2.1. A Decomposition-Based Multi-Objective Algorithmic Framework

The standard ABC algorithm is mainly used for solving single-objective continuous optimization problems and cannot be directly applied to solving the green single-machine multi-objective scheduling problem. Meanwhile, this scheduling problem is an NP-hard problem, and finding an effective solution to the large-scale single-machine scheduling problem within an acceptable computational time using traditional analytical mathematical methods is difficult. However, the meta-heuristic algorithm can be used to obtain an approximate or even an optimal solution, within an effective duration. Therefore, this paper designs a MODABC algorithm to solve the green single-machine multi-objective scheduling problem.

The MODABC algorithm decomposes the multi-objective optimization problem into several single-objective subproblems [16,17]. This approach first generates N uniformly distributed weight vectors and then, constructs N single-objective subproblems using these weight vectors. By solving these N subproblems simultaneously, we obtained the set of non-inferior solutions to the problem to be optimized. Let ${\lambda }^1,{\lambda }^2,\cdots ,{\lambda }^N$ to be a set of weight vectors and ${\lambda }^j=\left({\lambda }_1^j,{\lambda }_2^j\right)$ denote the first- and second-dimensional objective functions of the $j\mathrm{th}$ weight vector, the weighted sum approach for the aggregation function (as shown in Figure 2) of the $j$th subproblem is set as follows.

$$f(x|{\lambda }^j)={\lambda }_1^j\cdot {\stackrel{-}{f}}_1\left(x\right)+{\lambda }_2^j\cdot {\stackrel{-}{f}}_2\left(x\right)$$

(15)

where x is a solution to the problem to be optimized, and ${\stackrel{-}{f}}_1\left(x\right)$ and ${\stackrel{-}{f}}_2\left(x\right)$ are values of the normalized objective functions (1) and (2) of solution $x$. The normalization is used to minimize the negative impact of the different objective function magnitudes on the optimization effect.

By simultaneously solving for the weight vectors $\lambda$ and ${\lambda }^{\prime }$ of the optimal solution, two non-inferior solutions of this multi-objective problem (MOP), i.e., the intersection of the weight vector and the Pareto front, are obtained. Moreover, simultaneously solving for the solutions of N weight vectors of a uniform design guaranty that a set of uniform non-inferior solutions can be obtained. The weight vectors are designed uniformly using a mixture, and for the $j\mathrm{th}$ subproblem, the weight vector ${\lambda }^j$ is as follows.

$${\lambda }^j=\left(\frac{j-1}{N-1},\frac{N-j}{N-1}\right)$$

(16)

In the MODABC algorithm, the Euclidean distance between the vectors serves as a criterion, which determines the neighbourhood of each subproblem, where $B_i$ denotes the index set of the weight vector of the neighbourhood corresponding to the ith subproblem.

3.2.2. Encoding and Decoding Methods

The primary problem of using the multi-objective evolutionary algorithm to solve the scheduling problem lies in encoding and decoding; MODABC designed a discrete two-layer encoding and decoding method using permutation. The green single-machine multi-objective scheduling problem must simultaneously solve two sub-tasks: job sorting and processing speed selection, for which the discrete two-layer encoding method using permutation is designed. This encoding method contains two layers of information. The first layer uses permutation encoding, which is the encoded position of the job index, to indicate the job processing order. The second layer indicates the processing speed selected on the machine when the job is being processed.

Take the single-machine scheduling including nine jobs with three processing speeds as an example. Figure 3 gives one encoding example for this problem, and the corresponding scheduling scheme is as follows: job 6 is processed at speed 3, job 3 is processed at speed 1, and so on until job 9 is processed at speed 3 at the last time. This discrete two-layer encoding method has three advantages: easy generation of feasible solutions, simple encoding structure and effective reduction of information redundancy compared with random key encoding.

3.2.3. Population Initialization

The merits of the initial solution have a significant impact on both the speed and quality of the solution the algorithm produces. In this paper, we propose a combination of heuristics and random generation to generate and improve the quality of the initial solution. As mentioned earlier, the algorithm consists of job ordering and a corresponding array of processing speed selections. The MODABC algorithm uses three heuristics (estimated delivery date, shortest processing time and longest processing time) to each generate 10% of the initial population. In producing the initial solution using heuristics, the processing speed of each job is randomly generated, while the job ordering is determined using heuristic rules. The remaining 70% of individuals are randomly generated using the proposed encoding mechanism to improve the diversity of solutions.

3.2.4. Employed Bee Search

To ensure the feasibility of the solution, we redefined the nectar source update formula. Since the first layer of the encoding uses the permutation encoding and the second layer of the encoding is the result of the processing speed decision of the corresponding job, the job-speed combination at the same position in the first and second layers of the encoding is considered as a whole for the nectar source update transformation. The nectar source update method of the MODABC algorithm consists of three crossovers: partial map crossover, order crossover and position-based crossover. The nectar source update methods are described below.

(1)

Partial Map Crossover (PMX)

Partial map crossover ensures that genes in each chromosome occur only once, and is used to code for ordering problems. The specific steps are described below.

Step 1: Randomly select the starting and ending positions of several genes in a pair of paternal chromosomes, with the same position selected on both chromosomes, which is shown in Figure 4.

Step 2: Swap the positions of both sets of genes, as shown in Figure 5.

Step 3: Perform conflict detection. A mapping relationship is established using both sets of genes exchanged, as shown in the Figure 6a,b. Take the mapping relationship 1–6–3 as an example, there are two genes 1 in temporary child nectar source 1 in the second step result, which are then transformed them into gene 3 using the mapping relationship until there is no more conflict. In the end, all conflicting genes are mapped to ensure that the new pair of child genes formed is free from conflicts.

(2)

Order Crossover (OX)

Step 1: The first step is the same as that of PMX, where the starting and ending positions are randomly selected in a pair of parental chromosomes, and the same position is selected for both chromosomes, which is shown in Figure 7.

Step 2: Generate a child gene and ensure that the position of the selected gene in the child is the same as that of the parent, which is shown in Figure 8.

Step 3: First, find the position of the gene selected in the first step in the parent nectar source 2, and then fill in the child nectar source 1 with the remaining genes of the parent nectar source 2 in order, as shown in Figure 9.

This algorithm will also generate two children. The generation process of the other child is the same, only both parent chromosomes need exchange positions, and the position of the genotype in the first step is the same. The generation process of the other child in this example is described in Figure 10.

(3)

Position-based Crossover (PBX)

Unlike OX, PBX selects for discontinuous genotypic positions. Other than this, the rest is essentially identical to OX. The specific steps are described as follows.

Step 1: Randomly select several genes in a pair of paternal chromosomes, with the same position selected on both chromosomes, as shown in Figure 11.

Step 2: Generate a child gene and ensure that the position of the selected gene in the child is the same as that of the parent, as shown in Figure 12.

Step 3: First, find the position of the gene selected in the first step in the other parent, and then fill in the temporary child nectar source 1 with the remaining genes of the parent nectar source 2 in order, as shown in Figure 13.

This PBX will also generate two children. The generation process of the other child in this example is described in Figure 14.

Based on the above description, the nectar source update method of the employed bee search is summarized as Algorithm 1.

Algorithm 1: Nectar source update method of Employed Bee Search

Input: Weight vector $\lambda$;     Candidate solution $x$ of the single-objective subproblem to be optimized;     Set of neighbouring solutions $H$ for the current solution $x$

Output: Improved solution $\tilde{x}$

Step 1. Randomly generate a random permutation of PMX, OX and PBX $\pi$, let $\tilde{x}\leftarrow x$, move to step 2.

Step2. Choose a random neighbouring solution $x^{\prime }$from set $H$, and use the first crossover operator for the perturbation of $\tilde{x}$ in the reference permutation $\pi ,$ which generates the variational solution $x^{″}$. If $f(x^{″}|\lambda )<f(\tilde{x}|\lambda )$, then let $\tilde{x}\leftarrow x$. Continue to step 3.

Step 3. Choose a random neighbouring solution $x^{\prime }$ from set $H$, and use the second crossover operator for the perturbation of $\tilde{x}$ in the reference permutation $\pi ,$ which generates the variational solution $x^{″}$. If $f(x^{″}|\lambda )<f(\tilde{x}|\lambda )$, then let $\tilde{x}\leftarrow x$. Continue to step 4.

Step 4. Choose a random neighbouring solution $x^{\prime }$ from set $H$, and use the third crossover operator for the perturbation of $\tilde{x}$ in the reference permutation $\pi ,$ which generates the variational solution $x^{″}$. If $f(x^{″}|\lambda )<f(\tilde{x}|\lambda )$, then let $\tilde{x}\leftarrow x$. Continue to step 5.

Step 5. Terminate the nectar source update operation and output the improved solution $\tilde{x}$.

From the above nectar source update approach, the employed bee search process is defined as Algorithm 2.

Algorithm 2: Employed Bee Search Flow

Input: Weight vectors ${\lambda }^1,{\lambda }^2,\cdots ,{\lambda }^N$;     Current population $P_L$;     Neighbours of each subproblem (i.e., each weight vector) $B_i=\left\{i_1,i_2,\cdots ,i_T\right\}$

Output: New population $P_L{}^{\prime }$

For $i=1,2,\cdots ,N$do

Step 1. Construct a new solution: use $x_k$ as a candidate solution, select three indices $k_1$,$k_2$ and $k_3$ from neighbour $B_i$. Use $f(\cdot |{\lambda }^i)$ as the evaluation function and combine the above nectar update method using nectar sources $x_i$, $x_{k1}$, $x_{k2}$ and $x_{k3}$ to generate a new solution $\tilde{x}$.

Step 2. Update the extreme values of each dimensional objective function for normalisation.

Step 3. Update the nearest neighbour solution: for each $j\in B_i$such that $f(\tilde{x}|{\lambda }^j)<f(x_j|{\lambda }^j)$, replace $x_j$ with $\tilde{x}$, the exploitation level $Trai{l}_j\leftarrow 0$. Otherwise, the exploitation level $Trai{l}_j\leftarrow Trai{l}_j+1$.

End

3.2.5. Onlooker Bee Search

To improve the local search capability of the algorithm, this section uses a Variable Neighborhood Search (VNS) mechanism to improve the search operation of the onlooker bee. The basic idea of the VNS algorithm is to systematically change the neighbourhood structure of the current solution during the search process to expand the search range and find the local optimal solution using the local search algorithm. The above process is then repeated using this local optimal solution to finally achieve convergence after several iterations. Here, MODABC uses three neighbourhood structures:

(1)

N₁ (Swap): select two arbitrary positions on the genes of the nectar source and swap the job-speed gene pairs at both positions.

(2)

N₂ (Reverse order): select two arbitrary positions on the genes of the nectar source and arrange the job-speed gene pairs between both positions in reverse order.

(3)

N₃ (Insertion): select an arbitrary position on the genes of the nectar source and randomly insert the job-speed gene pair at that position at another position on the nectar source.

The schematic representation of the execution process of the three neighbourhood structures is summarized in Figure 15:

The flow of the VNS implementation is defined as Algorithm 3:

Algorithm 3: VNS Local Search

Input: Weight vector $\lambda$;     Candidate solution $x$;     Neighbourhood structures $N_k \left(k=1,2,3\right)$;     Number of local search iteration $ite{r}_{max}$;     The total number of variational solutions generated for each neighbourhood structure $L$

Output: Improved solution $\tilde{x}$

Step 1. Let $iter\leftarrow 1$, $\tilde{x}\leftarrow x$, and $k\leftarrow 1$. Continue to Step 2.

Step 2. Generate $L$ variational solutions in the neighbourhood $N_k$. Evaluate each solution with $f(\cdot |\lambda )$ as the evaluation function and compare with the solution $\tilde{x}$, and then update $\tilde{x}$. Continue to Step 3.

Step 3. If $k<3$, let $k\leftarrow k+1$, and continue to Step 2; otherwise, continue to Step 4.

Step 4. If $iter<ite{r}_{max}$, let $iter\leftarrow iter+1$, and continue to Step 2; otherwise, continue to Step 5.

Step 5. Terminate the nectar source update operation and output the improved solution $\tilde{x}$.

The roulette wheel rule is used in the basic ABC algorithm to select candidate nectar sources for the population update. Its core idea revolves around a higher probability of selecting high-quality nectar sources for the population update. Note that in multi-objective optimization, the non-inferior solution of the current population is superior to the remaining other dominated nectar individuals, whereas for a set of non-inferior solutions, individual objectives are mutually constrained and the superiority of the solution cannot be intuitively identified. Therefore, the MODABC algorithm updates the population in the onlooker bee search phase using the non-inferior solution of the current population.

The implementation flow of the onlooker bee search is summarized as Algorithm 4.

Algorithm 4: Onlooker Bee Search Flow

Input: Weight vectors ${\lambda }^1,{\lambda }^2,\cdots ,{\lambda }^N$;     Current population $P_L$ of the non-inferior solution set E;     Neighbours of each subproblem (i.e., each weight vector) Bi = {i1, i2, …, iT}.

Output: New population

For $i=1,2,\cdots ,N$,do

Step1. Construct a new solution: select $x_k$ from the non-inferior solution set E as the candidate solution. Use $f(\cdot |{\lambda }^i)$ as the evaluation function and the above VNS local search flow to generate a new solution $\tilde{x}$.

Step 2. Update the extreme values of each dimensional objective function for normalization.

Step3. Update the nearest neighbour solution: for each $j\in B_i$ with $f(\tilde{x}|{\lambda }^j)<f(x_j|{\lambda }^j)$, replace $x_j$ with $\tilde{x}$, the exploitation level $Trai{l}_j\leftarrow 0$. Otherwise, the exploitation level $Trai{l}_j\leftarrow Trai{l}_j+1$.

End

3.2.6. Scout Bee Search

A scout bee search, i.e., a reinitialisation operation, is required during the iterations of the MODABC algorithm, when the exploitation level of the nectar source $i$ is greater than the pre-set maximum exploitation level ($Trai{l}_i>Limit$). To ensure the quality and diversity of the solutions, the MODABC algorithm uses three heuristic rules: EDD, SPT and LPT, and a randomized approach for reinitialisation to ensure the generation of high-quality initial solutions. In generating the initial solution using heuristics, the processing speed of each job is randomly generated, while the job ordering is determined by the heuristic rules and randomization method.

3.2.7. Algorithm Description

In conclusion, the whole procedure of MODABC is shown as Algorithm 5.

Algorithm 5: MODABC Flow

Input: MOP to be optimized;     Control parameters;     Stop condition.

Output: Non-inferior solutions set $P_E$

Step1: Initialization

Step 1.1: Generate the initial nectar source for each weight vector.

Step 1.2: Update the extreme values of each dimensional objective function for normalization.

Step 1.3: Load the non-inferior solution of the current population in $P_E$.

Step2: Update

Step 2.1: Employed bee search, perform nectar update operations using three crossover methods, PMX, OX and PBX.

Step 2.2: Onlooker bee search, update each subproblem in the second round using non-inferior solutions of the current population and VNS local search.

Step 2.3: Scout bee search, reset the nectar source using reinitialisation operation when the nectar source exploitation level is lower.

Step 2.4: Update $P_E$.

Step3: Is the algorithm termination condition satisfied? If the termination condition is satisfied, the iteration is stopped and the non-inferior solution is outputted. Otherwise, go to Step 2.

Observe that MODABC achieves local search improvement through adaptive local search, VNS in the employed and onlooker bee search phases, and global search in the scout bee search. Combining the two search strategies, MODABC achieves a relatively balanced search performance.

4. Numerical Experiments

4.1. Test Problems

To systematically evaluate the optimization performance of different multi-objective evolutionary algorithms (MOEAs), numerical examples of different scales are configured. However, obtaining real data from real applications is difficult. Alternatively, since it is difficult to obtain test datasets similar to the single-machine scheduling problem proposed in this paper from existing literature, the test numerical examples in this paper are constructed using the test data generation method for machine scheduling problems proposed by Hall and Posner [18]. These examples are considered representative of real-world data in the manufacturing industry. The job delivery time are generated from a discrete uniform distribution $\left[\overline{D}\left(1-R\right),\overline{D}\left(1+R\right)\right]$, where $R$ is a parameter that controls the range of job delivery times; $R\in \{0.1,0.2,0.3,\dots ,0.9,1.0\}$. The mean value of the job delivery time $\overline{D}$ is an estimate of the job completion time and machine characteristics and equals half of the average total processing load per machine divided by the average machine processing speed, as follows.

$$\overline{D}=\frac{1}{2}·{{\displaystyle \sum }}_{j=1}^n{l}_j/\frac{{\sum }_{v=1}^u{V}_v}{u}$$

(17)

There are two types of test numerical examples constructed in this paper, small and medium scale, with a total of 16.

Table 1. Data set distribution.

Input Variable	Distribution
n: Total number of jobs	10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 105, 110, 115, 120, 125, 130
$l_j$: The processing load of job j	Discrete uniform [10, 100]
${\alpha }_j$: The penalty cost of the delivery time of job j	${\alpha }_jϵ\left\{0.1,0.2,0.3,0.4,0.5\right\}$
${\beta }_j$: The penalty cost of the tardiness of job j	${\beta }_jϵ\left\{0.6,0.7,0.8,0.9,1.0\right\}$
$d_j$: The delivery time of job j	$\mathrm{Discrete} \mathrm{uniform} \left[\overline{D}\left(1-R\right),\overline{D}\left(1+R\right)\right]$
$V_v$$:$th processing speed of the machine	Discrete uniform [1, 10]
$P{N}_v$: The noise emitted by the machine during processing at speed $V_v$.	Discrete uniform [50, 100] dB

shows the parameters of the numerical examples defined randomly, and there are 16 different quantities of jobs in these problems. The processing load of the job, the processing speed of the machine and the noise emitted by the machine during processing are generated from a discrete uniform distribution. The test numerical examples are constructed as follows.

4.2. Parameter Verification

All algorithms in this paper are implemented in Matlab. Meanwhile, the comparison experiments are conducted on the same platform R2015a with Intel Core i3-7100, 3.90 GHz, 4 GB RAM, and Windows 10 OS.

Parameters of the MODABC algorithm includes the number of nectar sources $N$, size of the neighbourhood $T$, number of variational solutions generated by a local search of neighbourhood structure $L$, number of local search iterations $ite{r}_{max}$ and maximum number of scout bee exploitation $Limit$, making a total of five parameters. The parameter calibration is conducted using the Taguchi method [19,20] in two parts: small-scale and medium-to-large-scale parameter calibration. Here, the Taguchi method is illustrated using the medium-to-large-scale parameter calibration as an example. Taguchi experiments of five parameters and three levels are conducted, and the values of each parameter at different levels are shown in

Table 2. Parameters and their levels setting.

Level	Parameter
	N	T	L	$\mathit{i}\mathit{t}\mathit{e}{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}$	$\mathit{L}\mathit{i}\mathit{m}\mathit{i}\mathit{t}$
1	60	10	3	5	40
2	80	15	5	10	60
3	100	20	7	15	80

. The experimental design is performed using Minitab. Table 2 shows 27 groups of the experimental setup.

The numeric example with 70 jobs was selected for the parameter calibration. For a certain combination of parameters, the mean values of the generation distance [21,22], inverse generation distance [23] and interval distribution [24] indexes were calculated for the results of 15 experiments. After 27 experiments, the normalized values of the three test indices were calculated, and the values of the three normalized mean indices $Sum$ were solved accordingly for each row. The following transformation formula was used to calculate the response variable [25].

$$S/N=-10{log}_{10}(Sum)$$

(18)

Test results are shown in

Table 3. Orthogonal table and test results.

Test	Parameter Level					Mean Values of Test Results			Sum	S/N
	$\mathit{N}$	$\mathit{T}$	$\mathit{L}$	$\mathit{i}\mathit{t}\mathit{e}{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}$	$\mathit{L}\mathit{i}\mathit{m}\mathit{i}\mathit{t}$	GD	Spread	IGD
1	1	1	1	1	1	4.92 × 10−2	6.30 × 10−1	7.25 × 10−2	1.82	−2.60
2	1	1	1	1	2	4.68 × 10−2	6.66 × 10−1	7.14 × 10−2	1.78	−2.51
3	1	1	1	1	3	4.72 × 10−2	6.09 × 10−1	7.36 × 10−2	1.45	−1.60
4	1	2	2	2	1	4.44 × 10−2	6.41 × 10−1	7.20 × 10−2	1.26	−1.00
5	1	2	2	2	2	4.46 × 10−2	6.29 × 10−1	7.21 × 10−2	1.18	−0.71
6	1	2	2	2	3	4.61 × 10−2	6.47 × 10−1	6.50 × 10−2	0.93	0.34
7	1	3	3	3	1	4.67 × 10−2	6.78 × 10−1	6.96 × 10−2	1.71	−2.33
8	1	3	3	3	2	4.62 × 10−2	6.34 × 10−1	7.63 × 10−2	1.80	−2.56
9	1	3	3	3	3	4.78 × 10−2	6.78 × 10−1	7.09 × 10−2	1.98	−2.97
10	2	1	2	3	1	4.71 × 10−2	5.97 × 10−1	7.16 × 10−2	1.14	−0.58
11	2	1	2	3	2	4.54 × 10−2	6.63 × 10−1	6.88 × 10−2	1.32	−1.22
12	2	1	2	3	3	4.28 × 10−2	6.97 × 10−1	7.04 × 10−2	1.48	−1.71
13	2	2	3	1	1	4.97 × 10−2	6.72 × 10−1	7.06 × 10−2	2.14	−3.31
14	2	2	3	1	2	4.77 × 10−2	6.66 × 10−1	7.20 × 10−2	1.94	−2.88
15	2	2	3	1	3	4.62 × 10−2	6.39 × 10−1	7.56 × 10−2	1.80	−2.54
16	2	3	1	2	1	4.74 × 10−2	6.74 × 10−1	6.88 × 10−2	1.70	−2.32
17	2	3	1	2	2	4.79 × 10−2	6.41 × 10−1	7.56 × 10−2	2.03	−3.08
18	2	3	1	2	3	5.05 × 10−2	6.26 × 10−1	7.54 × 10−2	2.21	−3.45
19	3	1	3	2	1	4.96 × 10−2	6.17 × 10−1	6.93 × 10−2	1.45	−1.62
20	3	1	3	2	2	4.69 × 10−2	6.51 × 10−1	7.16 × 10−2	1.65	−2.18
21	3	1	3	2	3	4.57 × 10−2	6.41 × 10−1	7.55 × 10−2	1.75	−2.43
22	3	2	1	3	1	4.75 × 10−2	6.94 × 10−1	7.20 × 10−2	2.20	−3.42
23	3	2	1	3	2	4.68 × 10−2	6.67 × 10−1	7.00 × 10−2	1.66	−2.20
24	3	2	1	3	3	4.67 × 10−2	6.55 × 10−1	7.10 × 10−2	1.61	−2.08
25	3	3	2	1	1	4.82 × 10−2	6.51 × 10−1	7.32 × 10−2	1.97	−2.94
26	3	3	2	1	2	4.92 × 10−2	6.63 × 10−1	7.03 × 10−2	1.96	−2.92
27	3	3	2	1	3	4.96 × 10−2	6.58 × 10−1	7.52 × 10−2	2.39	−3.79

.

Based on the test results shown in Table 3, the extreme difference and variance analysis are conducted and the significance level of variance analysis is 0.05. In addition, the Shapiro–Wilk method is used to make the normality test of the data, and the result shows that the significance level of each parameter is greater than 0.05 (p ≥ 0.05), which indicates that it is suitable for analysis of variance. Finally, the extreme differences analysis of each parameter is shown in

Table 4. Extreme difference analysis of Taguchi method.

Level	Parameter
	N	T	L	$\mathit{i}\mathit{t}\mathit{e}{\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}$	$\mathit{L}\mathit{i}\mathit{m}\mathit{i}\mathit{t}$
1	−1.77	−1.83	−2.58	−2.79	−2.23
2	−2.34	−1.98	−1.61	−1.83	−2.25
3	−2.62	−2.93	−2.53	−2.12	−2.25
Extreme difference	0.85	1.10	0.97	0.96	0.02
Rank	4	1	2	3	5

, the variance analysis of each parameter is shown in

Table 5. Variance analysis of Taguchi method.

Source	SS	df	MS	F	Prob > F
$N$	3.3652	2	1.6826	5.06	0.0198
$T$	6.4073	2	3.20364	9.64	0.0018
$L$	5.3613	2	2.68067	8.07	0.0038
$ite{r}_{max}$	4.374	2	2.18698	6.58	0.0082
$Limit$	0.0013	2	0.00067	0	0.998
Error	5.3176	16	0.33235
Total	24.8268	26

, and the influence trend for each parameter on the performance of the algorithm is shown in Figure 16.

From Table 4, the parameter neighbourhood size T has the largest extreme difference, indicating that the parameter neighbourhood size T has the largest impact on the optimization result of the algorithm. This is attributed to the influence of the neighbourhood size on the local search ability and convergence speed of the algorithm: the larger the neighbourhood size T, the larger the local search capability and the slower the convergence speed, and vice versa. Thus, a balance between the local search capability and convergence speed of MODABC must be achieved. Through the experimental verification, as shown in Figure 16, when T = 10, the performance of the algorithm is optimal. Observe that the number of variational solutions generated by the local search of a neighbourhood structure $L$ has the second-largest impact because when $L$ is too small, it is not conducive to deep mining the local optimal solution, and when it is too large, it is not conducive for the bee colony to jump out of the local minima. Thus, we observed that the performance of the algorithm is optimal when $L$ = 5. The extreme differences in the remaining three parameters are smaller than that of the parameter T, indicating they have little effect on the optimization result. From Table 5, p values of parameters T, L, $ite{r}_{max}$ and $N$ are all smaller than 0.05, indicating that all four parameters significantly impact the algorithm test. This is consistent with the results of the extreme difference analysis in Table 4.

From the above analysis, the influence for each parameter on the optimization results is $L$>$T>ite{r}_{max}$>$N$>$Limit$ in descending order. In this paper, the optimize efficiency could be combined with the quality, and MODABC parameters were set as: $N$ = 60, $T$= 10,$L$= 5,$ite{r}_{max}$= 10, $Limit$= 40.

In MOEAs, different parameter configurations affect the performance of the algorithm, and several experiments have demonstrated that both population and external document sizes are sensitive to the performance of the algorithm for problems of different sizes to a certain extent [26,27]. To further validate the performance of MODABC and ensure the fairness of algorithm comparison, the selected MOEAs are parameter-checked for different problem sizes, and the best algorithm parameter settings are obtained, as shown in

Table 6. Parameters setting.

Parameters	MODABC		NSGA-II		SPEA2		MOEA/D
	Small-Scale	Medium-Scale	Small-Scale	Medium-Scale	Small-Scale	Medium-Scale	Small-Scale	Medium-Scale
Population size	30	60	30	60	30	60	30	60
External document size	40	80	40	80	40	80	40	80
Crossover probability			0.6	0.9	0.6	0.9	0.6	0.9
Mutation probability			0.1	0.3	0.1	0.3	0.1	0.3
Neighbourhood size	6	10					6	10
Number of variational solutions	3	5
Number of local search iterations	5	10
Maximum number of scout bee exploitation	20	40

. Here, the parameters are used for both MODABC and MOEAs for the comparison test experiments in the next section.

4.3. Comparison of Test Results

All algorithms in the paper used the programming language, computing platform and experimental environment described in the previous section. To compare different algorithms fairly, all experimental algorithms used the same maximum function evaluation times, set to 20,000 and 40,000 for small-scale and medium-scale problems, respectively. MODABC is compared with classical algorithms such as NSGA-II [24], SPEA2 [28] and MOEA/D [29]. The three indices of generation distance (GD), distribution index spread and inverse generation distance (IGD) were tested and compared. Each MOEA ran 30 times independently. The optimal results are highlighted in bold.

Table 7. Mean and Standard deviation of GD index of four test algorithms.

Scale	NSGA-II		SPEA2		MOEA/D		MODABC
	Mean Values	Standard Deviation	Mean Values	Standard Deviation	Mean Values	Standard Deviation	Mean Values	Standard Deviation
10	1.22 × 10−2	2.98 × 10−3	3.34 × 10−2	9.16 × 10−3	1.73 × 10−2	6.19 × 10−3	1.03 × 10−2	1.54 × 10 − 3
20	5.20 × 10−2	9.30 × 10−3	7.37 × 10−2	5.64 × 10−1	2.57 × 10−2	3.81 × 10−3	1.53 × 10 − 2	3.09 × 10 − 3
30	5.22 × 10−2	3.99 × 10−2	2.40 × 10−2	6.64 × 10−3	1.75 × 10−1	6.13 × 10−1	2.01 × 10 − 2	1.71 × 10 − 3
40	1.39 × 10−1	2.57 × 10−2	4.11 × 10−2	3.57 × 10−3	6.80 × 10−2	8.15 × 10−3	2.97 × 10 − 2	1.97 × 10 − 3
50	2.07 × 10−1	6.65 × 10−2	6.32 × 10−2	9.34 × 10−3	1.41 × 10−1	2.55 × 10−2	2.90 × 10 − 2	8.87 × 10 − 3
60	5.91 × 10−2	6.77 × 10−3	2.12 × 10 − 2	2.03 × 10 − 3	1.07 × 10−1	2.14 × 10−2	2.70 × 10−2	4.68 × 10−3
70	1.87 × 10−1	2.21 × 10−2	8.64 × 10−2	1.42 × 10−3	1.74 × 10−1	1.44 × 10−3	4.28 × 10 − 2	1.29 × 10 − 3
80	1.13 × 10−1	8.27 × 10−3	2.28 × 10−2	8.15 × 10−3	1.23 × 10−1	7.00 × 10−2	1.35 × 10 − 2	5.34 × 10 − 3
90	1.63 × 10−1	4.62 × 10−2	8.05 × 10−2	9.92 × 10−3	6.80 × 10−2	8.46 × 10−3	1.93 × 10 − 2	1.64 × 10 − 3
100	6.24 × 100	4.12 × 10−1	3.74 × 100	2.49 × 10−1	1.35 × 100	2.26 × 10−1	3.62 × 10 − 1	1.40 × 10 − 2
105	9.14 × 10−2	1.32 × 10−1	1.54 × 10−1	2.29 × 100	1.27 × 10−1	3.55 × 10−1	2.61 × 10 − 2	1.08 × 10 − 1
110	2.24 × 10−1	2.47 × 100	3.66 × 10−1	4.05 × 100	2.83 × 10−1	3.96 × 100	5.77 × 10 − 2	1.53 × 10 − 1
115	7.10 × 10−1	6.99 × 100	5.45 × 10−1	5.04 × 100	5.97 × 10−1	6.20 × 100	8.62 × 10 − 2	1.33 × 10 − 1
120	2.15 × 10−1	2.42 × 100	3.03 × 10−1	4.17 × 100	1.58 × 10−1	2.02 × 100	3.26 × 10 − 2	1.12 × 10 − 1
125	3.44 × 10−1	3.92 × 100	1.63 × 10−1	1.83 × 100	4.76 × 10−1	5.47 × 100	6.30 × 10 − 2	1.12 × 10 − 1
130	1.48 × 10−1	1.94 × 100	2.27 × 10−1	3.21 × 100	1.30 × 10−1	1.54 × 100	5.73 × 10 − 2	2.04 × 10 − 1
win rate	0/16		1/16		0/16		15/16

shows the statistical results of the performance index of GD. From Table 7, the MODABC algorithm outperforms other MOEAs in most problems and achieves optimal results in 15 of the 16 problems. SPEA2 only achieved optimal results in one problem, and the other two algorithms failed to obtain optimal results. Therefore, in the green single-machine multi-objective scheduling problem, MODABC is superior to SPEA2 in order of convergence performance, while SPEA2 is slightly superior to NSGA-II and PAES. This indicates that MODABC has the best convergence performance among these MOEAs, which is mainly because it updates the population using non-inferior solutions of the current population in the onlooker bee search phase. The update is performed by three VNS algorithms: swap, reverse order and insertion. During the search process, the set of neighbourhood structures of the current solution is systematically modified to expand the search range, and the local optimal solution is obtained using the local search algorithm. Furthermore, the above process is iterated using the local optimal solution, and convergence is finally achieved after several iterations, indicating a higher convergence speed.

Table 8. Mean and Standard deviation of Spread index of four test algorithms.

Scale	NSGA-II		SPEA2		MOEA/D		MODABC
	Mean Values	Standard Deviation	Mean Values	Standard Deviation	Mean Values	Standard Deviation	Mean Values	Standard Deviation
10	8.27 × 10−1	4.52 × 10−2	6.67 × 10−1	2.90 × 10−2	8.67 × 10−1	5.22 × 10−2	4.95 × 10 − 1	2.64 × 10 − 2
20	7.49 × 10−1	4.96 × 10−2	6.39 × 10−1	3.91 × 10−2	6.23 × 10−1	3.81 × 10−2	3.69 × 10 − 1	3.08 × 10 − 2
30	9.86 × 10−1	4.15 × 10−2	7.76 × 10−1	2.49 × 10−2	1.09 × 100	1.64 × 10−1	4.74 × 10 − 1	1.85 × 10 − 2
40	7.12 × 10−1	1.80 × 10−2	8.73 × 10−1	3.51 × 10−2	8.48 × 10−1	3.02 × 10−2	5.52 × 10 − 1	1.58 × 10 − 2
50	7.08 × 10−1	3.18 × 10−2	9.77 × 10−1	8.43 × 10−2	9.00 × 10−1	4.94 × 10−2	5.53 × 10 − 1	2.49 × 10 − 2
60	8.67 × 10−1	3.83 × 10−2	8.89 × 10−1	4.37 × 10−2	9.93 × 10−1	8.65 × 10−2	5.87 × 10 − 1	3.78 × 10 − 2
70	6.86 × 10−1	1.97 × 10−2	8.72 × 10−1	7.36 × 10−2	7.29 × 10−1	2.86 × 10−2	5.97 × 10 − 1	1.82 × 10 − 2
80	1.06 × 100	1.25 × 10−1	1.07 × 100	2.02 × 10−1	9.59 × 10−1	8.53 × 10−2	9.33 × 10 − 1	5.10 × 10 − 2
90	7.22 × 10 − 1	2.57 × 10 − 2	1.01 × 100	2.78 × 10−1	9.99 × 10−1	8.11 × 10−2	9.73 × 10−1	3.51 × 10−2
100	1.00 × 100	1.32 × 10−1	1.02 × 100	3.76 × 10−1	1.04 × 100	4.17 × 10−1	7.77 × 10 − 1	7.28 × 10 − 2
105	8.86 × 10−1	3.18 × 10−2	8.64 × 10−1	2.38 × 10−2	1.03 × 100	3.22 × 10−1	8.27 × 10 − 1	1.54 × 10 − 2
110	6.96 × 10−1	1.65 × 10−2	8.63 × 10−1	2.12 × 10−2	8.80 × 10−1	6.79 × 10−2	6.44 × 10 − 1	1.07 × 10 − 2
115	7.89 × 10−1	2.76 × 10−2	7.54 × 10−1	2.23 × 10−2	9.14 × 10−1	7.42 × 10−2	7.45 × 10 − 1	1.32 × 10 − 2
120	9.51 × 10−1	9.76 × 10−2	8.64 × 10−1	7.27 × 10−2	6.97 × 10−1	6.00 × 10−2	6.80 × 10 − 1	4.22 × 10 − 2
125	8.15 × 10−1	2.47 × 10−2	8.92 × 10−1	2.87 × 10−2	9.09 × 10−1	3.01 × 10−2	7.13 × 10 − 1	1.67 × 10 − 2
130	1.01 × 100	7.14 × 10−1	9.99 × 10−1	7.58 × 10−2	1.00 × 100	6.79 × 10−1	9.65 × 10 − 1	1.57 × 10 − 2
win rate	1/16		0/16		0/16		15/16

shows the statistical results of the distribution performance (Spread index) of the algorithm, for which MODABC attains better competitive performance than the competitor algorithms. Specifically, for the Spread index, MODABC achieved optimal results for 15 problems, NSGA-II achieved optimal results for one problem, while MOEA/D and SPEA2 failed to achieve optimal results for any problem. This indicates that as MODABC continuously approximates the optimal Pareto solution set, its solution distributivity is also guaranteed to some extent.

Table 9. Mean and Standard deviation of IGD index of four test algorithms.

Scale	NSGA-II		SPEA2		MOEA/D		MODABC
	Mean Values	Standard Deviation	Mean Values	Standard Deviation	Mean Values	Standard Deviation	Mean Values	Standard Deviation
10	5.42 × 10−2	9.16 × 10−3	4.19 × 10−2	6.19 × 10−3	3.53 × 10−2	2.98 × 10−3	1.71 × 10 − 2	1.54 × 10 − 3
20	1.09 × 10−1	5.64 × 10−2	6.79 × 10−2	3.81 × 10−3	6.86 × 10−2	9.30 × 10−3	1.73 × 10 − 2	3.09 × 10 − 3
30	9.49 × 10−2	6.13 × 10−3	1.92 × 10−1	6.64 × 10−2	1.17 × 10−1	1.71 × 10−2	2.17 × 10 − 2	3.99 × 10 − 3
40	1.68 × 10−1	2.57 × 10−2	1.85 × 10−1	3.57 × 10−2	1.67 × 10−1	1.97 × 10−2	6.90 × 10 − 2	8.15 × 10 − 3
50	2.33 × 10−1	6.65 × 10−2	1.56 × 10−1	2.55 × 10−2	3.43 × 10−1	9.34 × 10−2	5.03 × 10 − 2	8.87 × 10 − 3
60	1.67 × 10−1	2.14 × 10−2	1.54 × 10−1	2.03 × 10−2	3.09 × 10−1	4.68 × 10−2	6.37 × 10 − 2	6.77 × 10 − 3
70	1.77 × 10−1	1.44 × 10−2	1.48 × 10−1	1.42 × 10−2	1.77 × 10−1	2.21 × 10−2	6.50 × 10 − 2	1.29 × 10 − 3
80	3.00 × 10−1	8.27 × 10−2	1.26 × 10−1	7.00 × 10−2	1.58 × 10−1	8.15 × 10−2	5.57 × 10 − 2	5.34 × 10 − 3
90	2.68 × 10−1	4.62 × 10−2	5.68 × 10−1	9.92 × 10−2	2.74 × 10−1	8.46 × 10−2	1.59 × 10 − 1	1.64 × 10 − 2
100	5.88 × 100	4.12 × 10−1	3.87 × 100	2.49 × 10−1	8.51 × 10−1	2.26 × 10−2	1.46 × 10 − 1	1.40 × 10 − 2
105	2.29 × 10−1	3.69 × 10−2	1.32 × 10−1	2.94 × 10−2	3.55 × 10−1	5.93 × 10−2	1.08 × 10 − 1	1.43 × 10 − 2
110	4.05 × 10−1	7.27 × 10−2	2.47 × 10−1	4.38 × 10−2	3.96 × 10−1	6.06 × 10−2	1.53 × 10 − 1	2.10 × 10 − 2
115	5.04 × 10−1	3.56 × 10−2	6.99 × 10−1	8.36 × 10−2	6.20 × 10−1	6.88 × 10−2	1.33 × 10 − 1	1.96 × 10 − 2
120	4.17 × 10−1	9.32 × 10−2	2.42 × 10−1	7.86 × 10−2	2.02 × 10−1	6.12 × 10−2	1.12 × 10 − 1	2.12 × 10 − 2
125	1.83 × 10−1	1.87 × 10−2	3.92 × 10−1	5.75 × 10−2	5.47 × 10−1	6.27 × 10−2	1.12 × 10 − 1	1.54 × 10 − 2
130	1.54 × 100	1.37 × 10−1	1.94 × 100	3.66 × 10−1	3.21 × 10−1	3.07 × 10−2	2.04 × 10 − 1	2.70 × 10 − 2
win rate	0/16		0/16		0/16		16/16

shows the statistical results of the performance index for IGD. Here, MODABC outperforms other MOEAs in all problems. Additionally,

Table 10. ANOVA of IGD.

Source	SS	df	MS	F	Prob > F
Columns	21.406	3	7.13517	3.81	0.0145
Error	112.452	60	1.8742
Total	133.857	63

shows that MODABC significantly outperforms its competitor algorithms in the performance index for IGD with a 95% confidence level, indicating that MODABC has excellent comprehensive performance among these MOEAs. This is mainly because the algorithm applies the basic framework of the ABC algorithm, redesigns the neighbourhood search method of the algorithm, improves the individual encoding and new individual generation methods, and optimally calibrates the algorithm parameter settings to adapt to the characteristics of the green single-machine multi-objective scheduling model.

A one-way analysis of variance (ANOVA) was used to analyze the IGD of the test results. The test level of ANOVA was 0.05 and the concomitant probability was 0.0145 < 0.05, indicating that MODABC was significantly superior to the other three comparative algorithms in terms of comprehensive index IGD.

Figure 17 shows the Pareto front ends obtained by different MOEAs on nine randomly selected problems, which correspond to the optimal anti-generation distance IGD obtained by different MOEAs in 30 independent operations, to show the performance of these MOEAs. These diagrams confirm the conclusion from the numerical analysis that MODABC can provide a non-dominated solution set with better convergence and distribution than NSGA-II, MOEA/D and SPEA2.

5. Conclusions

This paper investigated the green single-machine multi-objective scheduling problem. At first, a scheduling model was established. Then, a multi-objective discrete ABC algorithm, MODABC, with the optimization objectives of simultaneously minimizing lead-time/tardiness cost and job-shop noise pollution emission, was proposed, which considers production benefits and environmental objectives symmetrical and equally important. Furthermore, a discrete permutation-based two-layer encoding mechanism was designed to fit the problem characteristics, and the initial population was generated using four methods: EDD, SPT, LPT and random generation. Moreover, in the employed bees search phase, three crossover methods, PMX, OX and PBX, were used to perform nectar update operations. In addition, in the onlooker bee search phase, three neighbourhood structures, swap, reverse order and insertion, were used to improve the search operation. Finally, the parameters of the algorithm were calibrated by using the Taguchi method, and the MODABC algorithm is compared with other classical MOEAs to verify the effectiveness of the proposed algorithm in solving the green single-machine multi-objective scheduling problem. Hopefully, the conclusions can provide an effective theoretical support and policy-making reference to enterprises for green manufacturing, help enterprises to improve production efficiency and reduce noise pollution, and aim to achieve the coordinated development of the economy and the environment. This study does not consider the application scenarios of MODABC beyond the single-machine scheduling problem, and subsequent work will continue to improve the algorithm, enhance the search efficiency of the algorithm, and further investigate the application of the algorithm in other production scheduling problems.