A Population Diversity-Based Artificial Bee Colony Algorithm for Assembly Hybrid Flow Shop Scheduling with Energy Consumption

Abstract

Assembly shop scheduling and energy-efficient scheduling have attracted much attention in the past decades; however, energy consumption is often ignored in assembly hybrid flow shop scheduling. Neglecting energy consumption will greatly diminish the progress of sustainable manufacturing. In this study, an assembly hybrid flow shop scheduling problem considering energy consumption (EAHFSP) is investigated, and a population diversity-based artificial bee colony algorithm (DABC) is proposed to minimize the makespan and total energy consumption (TEC) simultaneously. Diversified search strategies based on rank value are introduced to the employed bee phase; a novel probability selection method in the onlooker bee phase is designed to control the selection pressure; moreover, a diversity control strategy is applied to improve the diversity of food sources and avoid falling into stagnation. A number of experiments based on 44 extended benchmark instances from the literature and a real case are conducted to test the performance of the DABC algorithm. The statistical results show that the DABC algorithm is superior to the other four state-of-the-art algorithms on over 70% of the instances corresponding to metrics $IGD$ and $c$, which means that the DABC algorithm is effective and competitive in solving the considered EAHFSP.

1. Introduction

Production scheduling is imperative in the decision-making procedure of the manufacturing system [1,2,3]. With the shift in production patterns from unified to customized production, assembly shop scheduling has been extensively utilized in various manufacturing scenarios [4,5,6,7]. Meanwhile, with the increasing awareness of global warming and the energy crisis, green manufacturing that integrates production efficiency and environmental sustainability has become an inevitable trend [8,9]. Based on this background, research on assembly shop scheduling problems from the perspective of energy efficiency is of great practical and engineering significance.

As a typical assembly scheduling problem, the assembly hybrid flow shop scheduling problem (AHFSP), consisting of a hybrid flow shop manufacturing stage and an assembly stage, is widely encountered in modern manufacturing systems. It was proved to be strongly NP-hard and has been well studied since the pioneering work of Lee et al. [10] and Potts et al. [11]. Throughout the previous work, studies on the AHFSP mainly focused on single-objective problems. Yokoyama [12] investigated the AHFSP and presented a branch and bound (B&B) method to optimize the total completion time. Lagodimos et al. [13] and Fattahi et al. [14] utilized a series of heuristic rules and algorithms to minimize the makespan. Later, Fattahi et al. [15] and Hosseini et al. [16] improved the model of the AHFSP by adding time and maintenance-related constraints. For a two-stage AHFSP with makespan minimization, Nikzad et al. [17] designed a hybrid imperialist competitive algorithm, and Komaki et al. [18] proposed an artificial immune algorithm. Considering lot streaming in the AHFSP, Nejati et al. [19] presented a genetic algorithm and simulated annealing algorithm to optimize the sum of weighted completion time. Recently, Cai et al. [20] proposed a novel shuffled frog leaping algorithm with reinforcement for the distributed AHFSP. Regarding a multi-objective AHFSP, Lei and Zheng [21] presented a novel neighborhood search to minimize the total tardiness, makespan, and maximum tardiness in the AHFSP with key objectives. Fan et al. [22] used a mutant firefly algorithm to solve a bi-objective AHFSP with the simultaneous rate and on-time delivery rate. Li et al. [23] modified the genetic algorithm for the AHFSP with common due dates to minimize the total waiting time and the total earliness/tardiness.

In recent years, energy-efficient scheduling has often been adopted in manufacturing systems. Regarding energy-efficient assembly shop scheduling problems, Ren et al. [24] proposed a novel heuristic algorithm for energy-aware assembly flexible job shop scheduling problems. Deng et al. [25] applied a hybrid estimation of distribution algorithm to an energy-efficient, three-stage, integrated scheduling problem. For energy-efficient distributed assembly flow-shop scheduling problems, the cooperative memetic algorithm [26] and brainstorm optimization algorithm [27] were designed. Meanwhile, the energy-efficient hybrid flow shop scheduling problem (EHFSP) was also considered. Li et al. [28] raised a Q-learning and general variable neighborhood search-driven non-dominated sorting genetic algorithm II for EHFSP with uniform machines. Tang et al. [29] proposed an improved particle swarm optimization algorithm for an energy-efficient dynamic HFSP. Utama et al. [30] applied a novel hybrid Aquila optimizer to an EHFSP. Zuo et al. [31] presented a novel multi-population artificial bee colony to solve an EHFSP with variable speeds.

As stated above, studies on the AHFSP only consider the efficiency-related objectives and no existing paper involves energy-oriented criteria. In fact, the energy crisis and environmental problems caused by production activities are increasingly serious. Energy-oriented criteria are as important as the efficiency-related objectives in realizing green manufacturing. Energy consumption has been considered in hybrid flow shop, assembly flow shop, and assembly flexible job shop scheduling problems. However, it is often ignored in AHFSPs. The considered EAHFSP is more general and complex than AHFSPs, EHFSPs, and other energy-related shop scheduling problems. This means that EAHFSP is also NP-hard. Therefore, it is essential to develop an effective and efficient algorithm to investigate EAHFSPs, in which energy costs and total production costs are regarded equally important.

The artificial bee colony (ABC) algorithm is an excellent metaheuristic algorithm that simulates the foraging behavior of bee colonies in nature [32]. Its extraordinary features, such as effective global search ability and fast convergence capability, have been proven in many complicated optimization problems [33,34,35,36]. Moreover, the ABC algorithm also has many successful applications in various scheduling problems. Li et al. [37] applied a multi-objective ABC to a flexible job shop scheduling with maintenance. Gong et al. [38] proposed a hybrid ABC algorithm for blocking lot-streaming flow shop scheduling. Li et al. [39] developed an improved ABC algorithm for hybrid flow shop scheduling. Li et al. [40] presented a novel ABC algorithm for parallel batching distributed flow-shop scheduling. Lei et al. [41] proposed an ABC algorithm with division to investigate distributed parallel machine scheduling. The ABC algorithm is also utilized to solve assembly shop scheduling [42] and energy-efficient shop scheduling [43]. However, the ABC algorithm is rarely used to investigate EAHFSPs. The excellent characteristics of the ABC in scheduling problems encourage us to study EAHFSPs by improving the ABC algorithm.

In this study, an AHFSP considering energy consumption is investigated, which widely exists in production but without enough attention. Moreover, this is the first time that a green scheduling objective and an adjustable processing speed set are added to an AHFSP. The main features are as follows. (1) A multi-objective EAHFSP is investigated, and a mathematical model is built. (2) A novel DABC is proposed to minimize the makespan and total energy consumption simultaneously. In the employed bee phase, diversified search strategies are constructed, and the suitable search strategy is adaptively selected based on the rank value of food sources. In the onlooker bee phase, a novel probability selection method is introduced to select the tracking objects, and the number of neighborhood searches is different for each food source; a diversity enhancement strategy is utilized to maintain the diversity of food sources and avoid prematurity. (3) Extensive comparative experiments are executed to demonstrate the effectiveness of DABC in solving the EAHFSP. The novelties of this paper are summarized in

Table 1. Difference between this work and the previous literature.

Difference	The Previous Literature	This Work
Problem model	The existing studies on AHFSP only consider efficiency-oriented objectives ([10,11,12,13,14,15,16,17,18,19,20,21,22,23]); energy-related criteria are considered in other assembly scheduling problems ([24,25,26,27]) and HFSP ([28,29,30,31]).	A problem model of EAHFSP is built to optimize both efficiency-dependent and energy-related objectives. It is the first time that energy-saving criterion is investigated in AHFSP, which is more general and complex than other energy-efficient assembly scheduling problems and EHFSP.
Optimization algorithm	Most of existing ABCs for shop scheduling problems are often modified by integrating other optimization algorithms; two phases: the employed bee phase and the onlooker bee phase are implemented in the same way ([37,38,39,40,41,42,43]).	A novel ABC based on population diversity is proposed to control the selection pressure and improve the diversity of food sources.

.

The remainder of this paper is organized as follows. A description of the problem is provided in Section 2. A description of the ABC algorithm is given in Section 3. A DABC method for EAHFSPs is proposed in Section 4. Computational experiments using the DABC algorithm are provided in Section 5. The conclusions and prospects are reported in Section 6.

2. Problem Description

An illustration of the EAHFSP considered in this study is provided in Figure 1.

The EAHFSP is briefly described as follows.

There are $n$ products $P_1,P_2,\cdots ,P_k,\cdots ,P_n$, each product is produced by assembling a set of components $PP=\left\{P{P}_1,P{P}_2,\cdots ,P{P}_r,\cdots ,P_w\right\}$, $N_k$ indicates a set of the required components of product $P_k$ and $w={\displaystyle {\sum }_{k=1}^n\left|N_k\right|}$ denotes the total number of components. For each product $P_k$, the production process mainly contains the manufacturing stage and the assembly stage.

At the manufacturing stage, each component $P{P}_r$ is processed in a hybrid flow shop with $m$ stages, and each processing stage consists of $\left|S_i\right|\ge 2$ unrelated parallel machines with a set of speeds $V=\left\{v_1,v_2,\cdots v_l,\cdots ,v_d\right\}$. In the processing mode, $p_{rijl}$ indicates the processing time of component $P{P}_r$ on an unrelated parallel machine $M_{ij}$ with speed $v_l$, where $M_{ij}$ is the $jth$ machine at processing stage $i$.

Once the manufacturing of $\left|N_k\right|$ components required by product $P_k$ is finished, these components are transferred to the final assembly operation. There is only one assembly machine $M_a$ at the assembly stage, and the speed of $M_a$ remains constant during the assembly operation. $A_k$ indicates the assembly time for product $P_k$, and ${\mathrm{C}}_k$ is the completion time for $P_k$.

The following assumptions are considered in the EAHFSP:

All components and machines are available and independent at any time.

The handling and transportation time of the components are ignored.

No preemption or interruption is allowed in the processing and assembly process.

Each component should be manufactured or assembled on one machine at most.

Each machine processes no more than one component at a time.

The manufacturing of each component and the assembly of each product must be conducted in sequence.

In this study, the energy consumption of products during manufacturing and assembly processes is considered. Machines at the manufacturing and assembly stage exist in two modes: the working mode and the stand-by mode. Therefore, the total energy consumption (TEC) can be divided into four modules: energy consumption of manufacturing machines in the working mode (${EC}_1$), energy consumption of manufacturing machines in the stand-by mode (${EC}_2$), energy consumption of assembly machines in the working mode (${EC}_3$), and energy consumption of assembly machines in the stand-by mode (${EC}_4$).

${EC}_1$ is the energy consumption of all manufacturing machines during processing, and ${EC}_1$ can be calculated as

$${EC}_1={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{\left|S_i\right|}{\displaystyle {\int }_0^{C_{max}}{\displaystyle \sum _{l=1}^d{E}_{ijl}{y}_{ijl}\left(t\right)}dt}}}$$

(1)

where $C_{max}$ is the maximum completion time; $E_{ijl}$ indicates the energy consumption per unit time when $M_{ij}\in S_i$ works at speed $v_l$; and $y_{ijl}\left(t\right)$ is a binary variable, where $y_{ijl}\left(t\right)=1$ if the manufacturing machine $M_{ij}\in S_i$ is in the working mode at time $t$; otherwise, $y_{ijl}\left(t\right)$ is equal to 0.

${EC}_2$ is the energy consumption of all manufacturing machines during the stand-by period, and ${EC}_2$ can be calculated as

$${EC}_2={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{\left|S_i\right|}{\displaystyle {\int }_0^{C_{max}}{\displaystyle \sum _{l=1}^d\mathrm{S}{E}_{ij}{z}_{ij}\left(t\right)}dt}}}$$

(2)

where $S{E}_{ij}$ indicates the energy consumption per unit time when $M_{ij}\in S_i$ is in the stand-by mode; and $z_{ij}\left(t\right)$ is a binary variable, where $z_{ij}\left(t\right)$ is 1 if $M_{ij}\in S_i$ is in the stand-by mode at time $t$; otherwise, $z_{ij}\left(t\right)=0$.

${EC}_3$ is the energy consumption of an assembly machine during the assembly process, and ${EC}_3$ can be calculated as

$${EC}_3={\displaystyle \sum _{k=1}^n{\displaystyle {\int }_0^{C_{max}}A{E}_k{u}_k\left(t\right)dt}}$$

(3)

where ${AE}_k$ is the energy consumption per unit time when assembly machine $M_a$ is in the assembly mode; and $u_k\left(t\right)$ is a binary variable, where $u_k\left(t\right)=1$ if the assembly machine $M_a$ is in working mode at time $t$; otherwise, $u_k\left(t\right)=0$.

${EC}_4$ is the energy consumption of an assembly machine during the idle time, and ${EC}_4$ can be calculated as

$${EC}_4={\displaystyle {\int }_0^{C_{max}}ASE{\mu }_k\left(t\right)dt}$$

(4)

where $ASE$ is the energy consumption per unit time when $M_a$ is in the stand-by mode; and ${\mu }_k\left(t\right)$ is a binary variable, where ${\mu }_k\left(t\right)=1$ if $M_a$ is idle at time $t$; otherwise, ${\mu }_k\left(t\right)=0$.

The EAHFSP contains three sub-problems: the scheduling sub-problem, for determining the processing and operation sequence for components and products; the machine assignment sub-problem, for allocating suitable machines for each operation; and the speed selection sub-problem, for selecting an appropriate speed for the selected machine.

The EAHFSP aims to minimize the makespan and TEC simultaneously. The mathematical model is given as follows

$$f_1=C_{max}=\underset{k=1,2,\cdots ,n}{max}\left\{C_k\right\}$$

(5)

$$f_2={TEC=EC}_1+E{C}_2+E{C}_3+E{C}_4$$

(6)

subject to

$$a_{ri}\ge 0, \forall r,i$$

(7)

$$b_{ri}-a_{ri}={\displaystyle \sum _{j=1}^{\left|{\mathrm{S}}_i\right|}{\displaystyle \sum _{l=1}^d{p}_{rijl}\times y_{rijl}}, \forall r,i}$$

(8)

$$a{a}_k\ge \underset{r\in N_k}{max\left\{b_{rm}\right\}} \forall k$$

(9)

$$a_{r,i+1}-b_{r,i}\ge 0, \forall i$$

(10)

$$x_{r{r}^{\prime }ij}+X_{r^{\prime }rij}\le 1, \forall r,r^{\prime },i,j$$

(11)

$$x_{k{k}^{\prime }}+X_{k^{\prime }k}\le 1, \forall k,k^{\prime }$$

(12)

$${\displaystyle \sum _{j=1}^{\left|S_i\right|}{\displaystyle \sum _{l=1}^d{y}_{rijl}=}1}, \forall r,i$$

(13)

where $C_k$ is the completion time of product $P_k$, which is the finishing time of the final assembly operation; $a_{ri}$ indicates the starting time of processing stage $i$ for component ${PP}_r$; $b_{ri}$ is the finishing time of processing stage $i$ for component ${PP}_r$; $a{a}_k$ indicates the starting time of the assembly operation on product $P_k$; and $y_{rijl}$ is a binary variable, where $y_{rijl}=1$ if $M_{ij}\in S_i$ is working at speed $v_l$; otherwise, $y_{rijl}=0$. $X_{r{r}^{\prime }ij}\in \left\{0,1\right\}$, where $X_{r{r}^{\prime }ij}=1$ if component $r$ is processed before $r^{\prime }$ on $M_{ij}$; and $X_{k{k}^{\prime }}\in \left\{0,1\right\}$, $X_{k{k}^{\prime }}=1$ if product $P_k$ is assembled before $P_{k^{\prime }}$ on $M_a$.

Objective Function (5) is the makespan, and Objective Function (6) indicates the total energy consumption during the manufacturing and assembly process. Constraint (7) indicates that each component is available after zero time; Constraint (8) ensures that there is no preemption or interruption during manufacturing and assembly period; Constraint (9) guarantees that the assembly operation is later than the fabrication of all components belonging to a product; Constraint (10) explains the precedence constraint between operations; Constraints (11) and (12) demonstrate that the manufacturing and assembly machine can only process one operation at a time; and Constraint (13) ensures that a single processing speed is selected at a time.

In this model, the EAHFSP is a multi-objective optimization problem with the minimization of two equally important objectives. Several concepts are defined below.

(1) Pareto dominance. $x\succ y$ if $f_i\left(x\right)\le f_i\left(y\right)$ for $\forall i\in \left\{1,2\right\}$ and $f_i\left(x\right)\le f_i\left(y\right)\exists i\in \left\{1,2\right\}$.

(2) Non-dominated solution. For a set $\mathrm{\Omega }$ and a solution $x\in \mathrm{\Omega }$, if $x$ is not dominated by other solutions in $\mathrm{\Omega }$, then $x$ is a non-dominated solution.

(3) Pareto optimal. If a solution $x$ is not dominated by other solutions in the search space, then $x$ is Pareto optimal.

In this case, $x\succ y$ means that $x$ dominates $y$.

An illustrative example of this problem is described in

Table 2. An illustrative example of EAHFSP.

Products	Parts	Stage 1		Stage 2		Assembly
		${\mathit{M}}_{\mathbf{11}}$	${\mathit{M}}_{\mathbf{12}}$	${\mathit{M}}_{\mathbf{21}}$	${\mathit{M}}_{\mathbf{22}}$
$P_1$	$J_1$	18	15 ($v_3$)	11	18 ($v_1$)	12
	$J_2$	12	15 ($v_5$)	20	13 ($v_1$)
$P_2$	$J_3$	17	16 ($v_1$)	13	17 ($v_2$)	19
	$J_4$	19 ($v_3$)	10	15 ($v_1$)	18
	$J_5$	11	19 ($v_1$)	11	14 ($v_4$)
$P_3$	$J_6$	10 ($v_1$)	12	10	15 ($v_5$)	18
	$J_7$	12 ($v_4$)	10	10 ($v_1$)	16
$P_4$	$J_8$	17	12 ($v_4$)	15	19 ($v_5$)	18
	$J_9$	19 ($v_1$)	20	15 ($v_1$)	19
	$J_{10}$	20	12 ($v_2$)	11	16 ($v_1$)

.

3. Introduction to the ABC

An intelligent optimization algorithm is an effective method for solving various optimization problems [44,45]. The ABC algorithm is one of the most promising meta-heuristics inspired by the foraging behavior of bee colonies [32,33,34,35,36,37]. In the ABC, the location of a food source represents a solution in population $P$, and the quality of the food source corresponds to the fitness value of the solution.

The steps of the ABC algorithm are shown in Algorithm 1 and a flowchart is given in Figure 2.

Algorithm 1 ABC
Input: $N$, $Limit$ Output: $x_{optimal}$
1:	Initialize population $P$ with $N$ solutions
2:	while the termination condition is not met do
3:	the employed bee phase: exploit a food source
4:	the onlooker bee phase: wait and choose a food source to follow
5:	the scout bee phase: randomly produce a new solution if $tria{l}_i>Limit$
6:	end while
7:	output the optimal solutions

After the initial population $P$ with $N$ solutions is randomly generated, the ABC consists of three phases: the employed bee phase, the onlooker bee phase, and the scout bee phase.

In the employed bee phase, for each solution $x_i\in P$, a new solution $y_i$ is generated as

$$y_i=x_i+\phi \left(x_i-x_j\right)$$

(14)

where $\phi \in \left[-1,1\right]$ is a random real number and $x_j\in P$ is another random solution, $x_j\ne x_i$.

Greedy selection is used to update $x_i$: if $fit\left(y_i\right)>fit\left(x_i\right)$, then $y_i$ replaces $x_i$, where $fit\left(y_i\right)$ indicates the fitness of $x_i$.

In the onlooker bee phase, roulette selection is applied to help each onlooker bee choose a food source, and the selection probability is defined as

$$p_i=fit\left(x_i\right)/{\displaystyle \sum _{k=1}^{N}fit\left(x_k\right)}$$

(15)

where $p_i$ is the selection probability of solution $x_i$.

Once solution $x_i$ is selected, a new solution $y_i$ is generated using Equation (14), and greedy selection is also used to decide whether $x_i$ can be updated by $y_i$.

In the scout bee phase, the number of consecutive searches $tria{l}_i$ for each solution $x_i$ is computed. Initially, $tria{l}_i=0$. If solution $x_i$ is not renewed by $y_i$, $tria{l}_i=tria{l}_i+1$; otherwise, $tria{l}_i=0$. If $tria{l}_i>Limit$, the corresponding employed bee will become a scout. The scout bee randomly generates a new food source to update the old one, where $Limit$ is the search threshold. When the termination condition is met, the algorithm outputs the non-dominated solutions $x_{optimal}$.

4. The DABC Algorithm for an EAHFSP

In basic ABCs [38,39,40,41,42,43], updating the food source location is stochastic and inefficient, but the selection pressure of food sources and the diversity of food sources are difficult to control. Diversified search strategies in the employed bee phase will help improve the search efficiency; a novel probability selection method according to fitness and the order of food sources is helpful to control the selection pressure. Meanwhile, a diversity enhancement strategy is critical to control the diversity of food sources. Based on the above analysis, a novel algorithm named DABC is proposed for an EAHFSP.

4.1. Initialization

For an EAHFSP with $n$ products and $w={\displaystyle {\sum }_{k=1}^n\left|N_k\right|}$ components, which are manufactured in $m$ processing stages and an assembly stage, the part-based encoding and decoding method [21] is used. A solution can be represented as a component scheduling string $\left[{\theta }_1,{\theta }_2,\cdots ,{\theta }_w\right]$, a machine assignment string $\left[{\pi }_{11},\cdots ,{\pi }_{1m},\cdots ,{\pi }_{w1},\cdots ,{\pi }_{wm}\right]$, and a speed selection string $\left[z_{11},\cdots ,z_{1m},\cdots ,z_{w1},\cdots ,z_{wm}\right]$, where ${\theta }_r\in \left\{1,2,\cdots ,w\right\}$, ${\pi }_{ri}\in S_i$ is the unrelated parallel machines assigned for the processing of component ${\theta }_r$ at stage $i$, and $z_{ri}\in V$ indicates the processing speed selected by processing machine ${\pi }_{ri}$.

Decoding is described as follows: Beginning with the first component ${\theta }_1$, select the corresponding processing machine and processing speed. The first $m$ machines are assigned to the $m$ stages of component ${\theta }_1$, then the second $m$ machines are arranged for the processing of the $m$ stages of component ${\theta }_2$, and so on. Once the processing of all required components of product $P_k$ is completed and the assembly machine is ready, product $P_k$ can be assembled until all products are finished.

To guarantee the diversity of solutions, the initial population $P$ with $N$ solutions is randomly generated. For each solution $x_i$, the fitness value $F_i$ is computed as

$$F_i=max\left\{ran{k}_i\right\}-ran{k}_i+crow{d}_i/{\displaystyle {\sum }_{k\in {\mathrm{\Theta }}_{ran{k}_i}}crow{d}_k}$$

(16)

where $ran{k}_i$ is the rank value after executing non-dominated sorting [46] on $x_i$, $\mathrm{c}row{d}_i$ is the crowding distance defined in [47], and ${\mathrm{\Theta }}_{ran{k}_i}$ is a set of solutions with the rank value of $ran{k}_i$.

For the manufacturing example in Section 2, a solution consists of a scheduling string $\left[9,10,2,7,6,4,1,5,3,8\right]$, a machine assignment string $\left\{M_{11},M_{21},M_{12}\right.,M_{22},M_{12},M_{22},M_{11},M_{21},$ $\left.M_{11},M_{22}{,M}_{11},M_{21}{,M}_{12},M_{22}{,M}_{12},M_{22}{,M}_{12},M_{22}{,M}_{12},M_{22}\right\}$, and a speed selection string $\left[v_1,v_1,v_2,v_1,\right.v_5,$ $\left.v_1,v_4,v_1,v_1,v_5,v_3,v_1,v_3,v_1,v_1,v_4,v_1,v_2,v_4,v_5\right]$. $V=\left\{1.0,1.3,1.55,1.8,2.0\right\}$, $E_{ijl}=4\times v_l^2$, $S{E}_{ij}=1$, ${AE}_k=4$, and $ASE=1$. Figure 3 displays the schedule plan of this example. $f_1=121.59$. At the manufacturing stage, the total energy consumption of machines in the processing mode is $E{C}_1=1631.21$; the energy consumption of the processing machines in the stand-by mode is $E{C}_2=32.16$. At the assembly stage, the energy consumption in the assembly mode is $E{C}_3=268.02$; the energy consumption in the stand-by mode is $E{C}_4=54.59$, and $f_2=1985.97$.

4.2. Employed Bee Phase

In basic ABCs, employed bees exploit food sources through a random crossover operation, and the search strategies used by all employed bees are the same. This stochastic and homogeneous search strategy is often inefficient. In this study, diversified search strategies are designed, and the suitable search strategy is adaptively selected according to the rank of food sources.

The detailed search procedure in employed the bee phase is described in Algorithm 2, where $su{b}_1$, $su{b}_2$, and $su{b}_3$ are divided as follows: Calculate the rank of non-dominated sorting for all food sources, then allocate food sources with $ran{k}_i=1$ to $su{b}_1$, assign food sources with $ran{k}_i=max\left\{ran{k}_i\right\}$ to $su{b}_3$, and distribute the remaining food sources to $su{b}_2$. $\delta$ is a random number, $\beta$ is a real number, and $\beta =0.5$, as determined with experiments.

Algorithm 2 Employed bee phase
Input: $N$, $R$, $\delta$ Output: $x_i, 1\le i\le N$
1:	Calculate the rank of non-dominated sorting for all food sources
2:	Divide the population $P$ into three groups: $su{b}_1$, $su{b}_2$, and $su{b}_3$
3:	for $i=1$ to $N$ do
4:	if $x_i\in {sub}_1$, then
5:	execute neighborhood search on $x_i$ and obtain a new food source $x_i{}^{\prime }$
6:	else if $x_i\in {sub}_2$, then
7:	stochastically select a food source $x_j\in {sub}_1$
8:	conduct global search between $x_i$ and $x_j$, produce a new food source $x_i{}^{\prime }$
9:	else if $x_i\in {sub}_3$, then
10:	if $\delta <\beta$, then
11:	randomly choose a food source $x_j\in {sub}_1$
12:	else
13:	stochastically select a food source $x_j\in {sub}_2$
14:	end if
15:	conduct global search between $x_i$ and $x_j$, generate a new food source $x_i{}^{\prime }$
16:	end if
17:	if $x_i{}^{\prime }$ is not dominated by $x_i$, then
18:	update $x_i$ with $x_i{}^{\prime }$
19:	end if
20:	end for

The global search is executed based on three crossover operators [21]. For two food sources $x_i$ and $x_j$, the crossover operation is performed on the scheduling string, machine assignment string, and speed selection string in sequence. A detailed description of the three crossover operators is given in [21].

The neighborhood search is conducted based on three neighborhood structures $insert$, $swap$, and $change$, which are named $N_1$, $N_2$, and $N_3$, respectively. A description of the neighborhood search is as follows: Let $t=1$, $g=1$, when $N_g$ is executed on food source $x_i$, repeat the following steps until $t=R$: obtain $x_i{}^{\prime }\in N_g\left(x_i\right)$, $t=t+1$, if $x_i{}^{\prime }$ is not dominated by $x_i$, then renew $x_i$ with $x_i{}^{\prime }$; otherwise, $g=g+1$, let $g=1$ if $g=4$, where $R$ is an integer, and $N_g\left(x_i\right)$ represents a set of solutions after conducting the neighborhood search on $x_i$ with $N_g$. A detailed description of $insert$, $swap$, and $change$ is given in [48].

As shown in Algorithm 2, all food sources are divided into three groups according to the rank value. Food sources in different groups can be improved using diversified search strategies, which is beneficial to improve search quality and increase population diversity.

4.3. Onlooker Bee Phase

Generally, onlooker bees select a food source to follow depending on fitness, and the search times for all food sources are the same. To reduce the impact of following the super-individuals and control the selection pressure, a novel probability selection method is provided for onlooker bees to select the following objects. Meanwhile, the search times for different food sources are newly defined according to the rank value, which can help improve the search quality and avoid wasting computing resources.

The novel probability selection method is described as follows: Calculate the fitness and sort all food sources in descending order. Suppose that $F_1>F_2>\cdots >F_N$, then compute the selection probability $p\left(x_i\right)$ of food source $x_i$, conduct the roulette selection, and select a food source $x_j$ to follow based on the selection probability $p\left(x_i\right)$.

$$p\left(x_i\right)=\frac{1}{N}+sin\left(\frac{F_i}{max\left\{F_i\right\}}\times \frac{\pi }{2}\right)\times \frac{N+1-2\times i}{N\times \left(N+1\right)},i=1,2,\cdots ,N$$

(17)

where $F_i$ is the fitness value of food source $x_i$.

In this method, the selection probability is determined by the fitness and the order of food sources, which not only avoids the control of selection pressure by super-individuals with the biggest fitness but also avoids wasting more computing resources on poor food sources.

The search procedure in the onlooker bee phase is shown in Algorithm 3, where $\eta$ is a random number and $\tau$ is a real number, where $\tau =0.5$, as determined using experiments. $R^{\prime }$ is the number of neighborhood searches for food sources selected in the onlooker bee phase. $R^{\prime }$ is defined as

$$R^{\prime }=round\left(R\times \left(F_i/max\left(ran{k}_i\right)\right)\right)$$

(18)

The definition of neighborhood search times can dynamically adjust the neighborhood search range, increase the diversity of the population, and effectively balance the development and exploration ability of the algorithm.

Algorithm 3 Onlooker bee phase
Input: $N$, $R$, $\eta$ Output: $x_i, 1\le i\le N$
1:	for $i=1$ to $N$ do
2:	calculate the rank value $ran{k}_i$ and fitness $F_i$ for all food sources
3:	select a food source $x_j$ to follow by the novel probability selection method
4:	if $x_j\in {sub}_1\cup {sub}_2$, then
5:	calculate the neighborhood searches $R^{\prime }$ for $x_j$
6:	execute neighborhood search on $x_j$ for $R^{\prime }$ times and obtain ${x^{\prime }}_j$
7:	else if $x_j\in {sub}_3$, then
8:	if $\eta <\tau$, then
9:	stochastically choose a food source $x_k\in {sub}_1$
10:	else
11:	randomly select a food source $x_k\in {sub}_2$
12:	end if
13:	execute global search between $x_j$ and $x_k$, produce a new food source $x_j{}^{\prime }$
14:	end if
15:	if $x_j{}^{\prime }$ is not dominated by $x_j$, then
16:	renew $x_j$ with $x_j{}^{\prime }$
17:	end if
18:	end for

4.4. Scout Bee Phase

In the scout bee phase, if a food source $x_i$ has not been renewed after $Limit$ consecutive trials, the corresponding employed bee will turn into a scout bee. The traditional update method of a scout is to generate a random food source instead of the old one. However, this method is inefficient and a waste of computing resources. The new scout bee phase is shown in Algorithm 4, where $tria{l}_i$ is the number of consecutive trials, $Limit$ is the threshold value of trials, and the neighborhood search and global search are the same as in the employed bee phase.

Algorithm 4 Scout bee phase
Input: $N$, $R$, $Limit$ Output: $x_i, 1\le i\le N$
1:	for $i=1$ to $N$ do
2:	if $tria{l}_i>Limit$ then,
3:	calculate the rank value $ran{k}_i$ for $x_i$
4:	if $ran{k}_i=1$, then
5:	conduct the neighborhood search and obtain a new food source $x_j$ to update $x_i$
6:	else
7:	randomly choose a food source $x_k\in {sub}_1$
8:	conduct a global search on $x_i$ and $x_k$, produce a new food source $x_j$ to update $x_i$
9:	end if
10:	end if
11:	end for

4.5. Diversity Control Strategy

In the evolutionary process, the diversity of food sources will gradually decrease, and the similarity in food sources will increase. However, the diversity control strategy lacks the traditional ABC.

For two food sources $x_i$ and $x_j$, the similarity ${\mathrm{\Delta }}_{ij}$ is defined as

$${\mathrm{\Delta }}_{ij}={‖sgn\left(\left|I_i-I_j\right|\right)‖}_1$$

(19)

where $I_{i\left(j\right)}=\left({\theta }_1^{i\left(j\right)},\cdots ,{\theta }_w^{i\left(j\right)},{\pi }_{11}^{i\left(j\right)},\cdots ,{\pi }_{wm}^{i\left(j\right)},z_{11}^{i\left(j\right)},\cdots ,z_{wm}^{i\left(j\right)}\right)$, ${\theta }_r$, ${\pi }_{ri}$, and $z_{ri}$ are elements of the scheduling string, machine assignment string, and speed selection string, respectively.

In Equation (19), ${\mathrm{\Delta }}_{ij}\ne 0$ means that there are differences between the three strings in food sources $x_i$ and $x_j$, then $x_i$ and $x_j$ can be called independent individuals. A larger ${\mathrm{\Delta }}_{ij}$ means a greater difference between $x_i$ and $x_j$.

Define the diversity of food sources as $D=\gamma /N$, where $\gamma$ indicates the number of independent individuals. A comparison between $D$ and the diversity threshold $\xi$ is used to determine whether the diversity enhancement strategy is needed. The detailed diversity control process is shown in Algorithm 5, where $\xi$ is a real number.

Algorithm 5 Diversity control strategy
Input: $N$, $\xi$ Output: $x_i, 1\le i\le N$
1:	calculate the diversity of food sources $D=\gamma /N$
2:	if $D<\xi$ then
3:	set $\mathrm{\Psi }=\varnothing$
4:	for $x_i\in su{b}_1\cup su{b}_2$ do
5:	for $x_k\in su{b}_1\cup {sub}_2$ do
6:	compute the similarity ${\mathrm{\Delta }}_{ij}$ between food sources $x_i$ and $x_j$
7:	if ${\mathrm{\Delta }}_{ij}\ne 0$, then
8:	$\mathrm{\Psi }=\mathrm{\Psi }\cup x_i\cup x_j$
9:	end if
10:	end for
11:	end for
12:	for each $x_i\in \mathrm{\Psi }$, do
13:	conduct neighborhood search on $x_i$ for $R^{\prime }$ times, generate a new solution $x_i{}^{\prime }$
14:	if $x_i{}^{\prime }$ is not dominated by, then
15:	update $x_i$ with $x_j{}^{\prime }$
16:	end if
17:	end for
18:	end if

As shown in Algorithm 5, the number of independent individuals is important to compute the diversity of food sources and determine whether to conduct the diversity enhancement strategy. When the diversity of food sources is lower than a threshold value, the use of a diversity control strategy can increase the diversity gradually, which means larger differences among food sources and a greater probability of generating new solutions. Thus, a diversity control strategy is beneficial to maintain the diversity of food sources and avoid falling into stagnation.

4.6. Algorithm Description

The steps of the DABC are given in Algorithm 6 and a flowchart is shown in Figure 4, where the number of iterations is also provided.

Algorithm 6 DABC
Input: $N$, $R$, $Limit$,$\xi$ Output : $x_{optimal}$
1:	Initialization
2:	while the stopping condition is not met, then
3:	conduct the employed bee phase with diversified search strategies
4:	execute the onlooker bee phase with a novel probability selection method
5:	perform a novel scout bee phase
6:	calculate the diversity of food sources $D=\gamma /N$
7:	if $D<\xi$, then
8:	seek out the solutions ${\mathrm{\Delta }}_{ij}=0$ between food sources in ${sub}_1\cup su{b}_2$, construct $\mathrm{\Psi }$
9:	execute the diversity control strategy on $x_i\in \mathrm{\Psi }$
10:	end if
11:	$Iter=Iter+1$
12:	end while
13:	Output the non-dominated solutions $x_{optimal}$

The features of the DABC are concluded as follows: Diversified search strategies are provided for employed bees and the suitable search strategy is adaptively selected according to the rank value of food sources; a novel probability selection method in the onlooker bee phase is designed to control the selection pressure, and different neighborhood search times are conducted on the selected food source. A diversity control strategy is applied to improve the diversity of food sources and avoid falling into prematurity.

5. Computational Results

Numerical experiments based on 44 extended benchmark instances from the literature and a real case are executed to evaluate the performance of the DABC algorithm for the considered EAHFSP. All experiments are compiled with MATLAB and run on 4.0 G RAM 2.0 GHz CPU PC.

5.1. Test Instances and Comparative Algorithms

As stated above, EAHFSPs are rarely investigated, and there are no existing benchmark instances, so we extend the benchmark instances of AHFSPs [21] by adding energy consumption information from the manufacturing and assembly stage. The test set consists of 44 randomly generated instances based on $w=\left\{20,30,40,50,60,70,80,90,100,110,120\right\}$ and $m=\left\{2,4,6,8\right\}$. The parallel machines at each stage $S_j$ and components needed for each product $N_k$ are from the literature [21]. ${\eta }_{ij}\in \left[10,20\right]$, $v_l=\left\{1.0,1.3,1.55,1.8,2.0\right\}$, and $A_k\in \left[10,20\right]$. The added energy consumption information is as follows: $E_{ijl}=\zeta \times v_l^2$, $S{E}_{ij}=1$, ${AE}_k\in \left[3,5\right]$, and $ASE=\left[1,2\right]$.

Two metrics $IGD$ and $c$ are utilized to test the algorithm performance in this study.

Metric $IGD$ [49] is often used to evaluate the convergence of algorithms and the distribution of non-dominated solutions, which is a comprehensive performance metric. It is the minimum distance between each point in a reference set ${\mathrm{\Delta }}^{*}$ and points in a non-dominated set ${\mathrm{\Delta }}_l$ gained by a certain algorithm.

$$IGD\left({\mathrm{\Delta }}_l,{\mathrm{\Delta }}^{*}\right)=\frac{\left|{\displaystyle {\sum }_{\delta \in {\mathrm{\Delta }}^{*}}d\left(\delta ,{\mathrm{\Delta }}_l\right)}\right|}{\left|{\mathrm{\Delta }}^{*}\right|}$$

(20)

where $d\left(\delta ,{\mathrm{\Delta }}_l\right)$ is the minimum distance between point $\delta$ in a reference set ${\mathrm{\Delta }}^{*}$ and points in a non-dominated set ${\mathrm{\Delta }}_l$ gained by a certain algorithm, and $\left|{\mathrm{\Delta }}^{*}\right|$ represents the number of non-dominated solutions on the Pareto front. The smaller the $IGD$, the better the convergence and distribution quality.

Metric $c$ [50] is often utilized to evaluate the mutual coverage of two algorithms.

$$c\left(A,B\right)=\frac{\left|\left\{x\in B\left|\exists \right.y\in A:y\succ x\right\}\right|}{\left|B\right|}$$

(21)

where $A$ and $B$ are two non-dominated solution sets and $y\succ x$ represents that solution $x$ is dominated by $y$. If $c\left(A,B\right)>c\left(B,A\right)$, algorithm $A$ is better than $B$.

In this study, the ABC, TICA [48], HPSO [51], and CCA [52] are selected as comparative algorithms.

The ABC algorithm is chosen to demonstrate the effectiveness of new strategies. It is conducted using the traditional method.

Li [48] designed a TICA for an energy-efficient HFSP, with the relative importance of objectives to minimize the makespan, total tardiness, and total energy consumption. After the newly defined Pareto dominance is removed, the TICA can be directly used to solve EAHFSPs. Ding [51] investigated an HPSO to optimize the electric power costs and total tardiness in an energy-aware flexible flow shop including time-of-use and variable processing speeds. Karimi [52] introduced a CCA for a multi-objective hybrid flow shop problem. The CCA can be applied to solve EAHFSPs after adding the assembly operation. So, we choose these algorithms as the comparative algorithms.

5.2. Parameter Settings

Our experiments indicate that DABC can converge fully in all instances after $0.3\times n\times m$ seconds; moreover, all comparative algorithms also converge well when this termination time is used. All parameters except the above termination condition are chosen from the literature.

There are four key parameters in the DABC: $R$, $N$, $Limit$, and $\xi$. The Taguchi method is used to decide the settings of these four parameters. The levels of each parameter are listed in

Table 3. The levels of four parameters.

Parameter	Factor Level
	1	2	3	4
$R$	6	8	10	12
$N$	30	40	50	60
$\mathit{Limit}$	7	8	9	10
$\xi$	0.2	0.3	0.4	0.5

.

Table 4. The orthogonal array $L_{16}\left(4^4\right)$.

Number	Factor				$\mathit{A}\mathit{R}\mathit{V}$
	R	N	Limit	ξ
1	1	1	1	1	0.314
2	1	2	2	2	0.295
3	1	3	3	3	0.184
4	1	4	4	4	0.320
5	2	1	2	3	0.084
6	2	2	1	4	0.137
7	2	3	4	1	0.377
8	2	4	3	2	0.447
9	3	1	3	4	0.159
10	3	2	4	3	0.432
11	3	3	1	2	0.237
12	3	4	2	1	0.176
13	4	1	4	2	0.190
14	4	2	3	1	0.170
15	4	3	2	4	0.273
16	4	4	1	3	0.175

defines the orthogonal array and $L_{16}\left(4^4\right)$, where $ARV$ denotes the average value of the DABC with each combination independently run 10 times, for instance, $80\times 8$. The rank of each parameter is given in

Table 5. The rank of each parameter.

Level	$\mathit{R}$	$\mathit{N}$	$\mathit{L}\mathit{i}\mathit{m}\mathit{i}\mathit{t}$	$\mathit{\xi }$
1	0.278	0.319	0.276	0.213
2	0.197	0.250	0.207	0.279
3	0.251	0.255	0.265	0.185
4	0.266	0.168	0.245	0.317
Delta	0.081	0.151	0.069	0.132
Rank	3	1	4	2

.

The main effect plot for metric $IGD$ is shown in Figure 5. It is concluded that the best settings are $R=8$, $N=60$, $Limit=8$, and $\xi =0.4$.

5.3. Results and Comparative Analysis

The DABC, ABC, TICA, HPSO, and CCA were independently run 10 times for each instance. The reference set ${\mathrm{\Delta }}^{*}$ contains the non-dominated solutions of ${\mathrm{\Delta }}_1\cup {\mathrm{\Delta }}_2\cup {\mathrm{\Delta }}_3\cup {\mathrm{\Delta }}_4\cup {\mathrm{\Delta }}_5$, where ${\mathrm{\Delta }}_1$, ${\mathrm{\Delta }}_2$, ${\mathrm{\Delta }}_3$, ${\mathrm{\Delta }}_4$, and ${\mathrm{\Delta }}_5$ indicate the optimal solutions obtained using the DABC, ABC, TICA, HPSO, and CCA methods, respectively.

Table 6. Results of all algorithms on metric $IGD$.

Instance	DABC	ABC	TICA	HPSO	CCA	Instance	DABC	ABC	TICA	HPSO	CCA
20 × 2	0.058	0.131	0.059	0.132	0.194	70 × 6	0.163	0.387	0.176	0.135	0.376
20 × 4	0.033	0.486	0.280	0.522	0.594	70 × 8	0.102	0.596	0.184	0.183	0.571
20 × 6	0.149	0.381	0.244	0.201	0.574	80 × 2	0.080	0.415	0.089	0.105	0.362
20 × 8	0.000	0.638	0.343	0.413	0.658	80 × 4	0.063	0.654	0.411	0.265	0.584
30 × 2	0.053	0.175	0.070	0.187	0.228	80 × 6	0.232	0.706	0.086	0.174	0.558
30 × 4	0.135	0.829	0.152	0.209	0.729	80 × 8	0.048	0.228	0.086	0.204	0.261
30 × 6	0.082	0.349	0.224	0.180	0.452	90 × 2	0.070	0.487	0.312	0.097	0.573
30 × 8	0.007	1.169	0.579	0.701	1.170	90 × 4	0.121	0.388	0.160	0.110	0.431
40 × 2	0.096	0.346	0.165	0.169	0.334	90 × 6	0.054	0.428	0.196	0.186	0.331
40 × 4	0.044	0.514	0.205	0.191	0.601	90 × 8	0.079	0.543	0.127	0.128	0.304
40 × 6	0.089	0.488	0.210	0.290	0.431	100 × 2	0.109	0.479	0.325	0.228	0.482
40 × 8	0.155	0.436	0.225	0.130	0.487	100 × 4	0.114	0.781	0.334	0.200	0.344
50 × 2	0.081	0.189	0.059	0.237	0.285	100 × 6	0.146	0.459	0.107	0.282	0.223
50 × 4	0.072	0.727	0.213	0.517	0.599	100 × 8	0.383	1.553	0.116	0.966	1.128
50 × 6	0.020	0.385	0.169	0.184	0.378	110 × 2	0.204	0.533	0.199	0.117	0.453
50 × 8	0.061	0.788	0.156	0.423	0.373	110 × 4	0.076	0.333	0.169	0.081	0.296
60 × 2	0.213	0.423	0.161	0.039	0.533	110 × 6	0.128	0.451	0.233	0.092	0.325
60 × 4	0.124	0.685	0.282	0.327	0.660	110 × 8	0.084	0.819	0.341	0.100	0.564
60 × 6	0.000	0.952	0.584	0.937	0.776	120 × 2	0.079	0.452	0.152	0.160	0.343
60 × 8	0.175	1.505	0.099	0.468	1.017	120 × 4	0.069	0.518	0.337	0.174	0.499
70 × 2	0.048	0.265	0.081	0.110	0.261	120 × 6	0.120	0.531	0.193	0.231	0.400
70 × 4	0.231	1.069	0.512	0.063	0.867	120 × 8	0.209	0.652	0.340	0.000	0.387

and

Table 7. Results of all algorithms on metric $c$.

Instance	$\mathit{c}\left(\mathit{A},\mathit{B}\right)$	$\mathit{c}\left(\mathit{B},\mathit{A}\right)$	$\mathit{c}\left(\mathit{A},\mathit{C}\right)$	$\mathit{c}\left(\mathit{C},\mathit{A}\right)$	$\mathit{c}\left(\mathit{A},\mathit{D}\right)$	$\mathit{c}\left(\mathit{D},\mathbf{A}\right)$	$\mathit{c}\left(\mathit{A},\mathit{E}\right)$	$\mathit{c}\left(\mathit{E},\mathit{A}\right)$
20 × 2	0.800	0.111	0.538	0.111	0.727	0.000	0.400	0.000
20 × 4	1.000	0.000	0.714	0.000	0.500	0.000	0.875	0.000
20 × 6	0.833	0.000	0.833	0.000	0.750	0.000	0.714	0.000
20 × 8	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
30 × 2	0.800	0.000	0.429	0.400	0.429	0.200	0.556	0.000
30 × 4	1.000	0.000	0.571	0.250	0.333	0.250	1.000	0.000
30 × 6	1.000	0.000	0.667	0.000	0.600	0.000	1.000	0.000
30 × 8	1.000	0.000	0.667	0.333	1.000	0.000	1.000	0.000
40 × 2	1.000	0.000	0.600	0.250	0.400	0.500	0.857	0.000
40 × 4	1.000	0.000	0.667	0.143	0.800	0.000	1.000	0.000
40 × 6	1.000	0.000	0.667	0.000	1.000	0.000	1.000	0.000
40 × 8	1.000	0.000	0.000	0.833	0.300	0.333	1.000	0.000
50 × 2	1.000	0.000	0.462	0.167	0.667	0.000	1.000	0.000
50 × 4	1.000	0.000	0.900	0.000	1.000	0.000	1.000	0.000
50 × 6	1.000	0.000	0.800	0.167	0.800	0.000	1.000	0.000
50 × 8	1.000	0.000	0.667	0.200	1.000	0.000	0.800	0.000
60 × 2	0.800	0.000	0.500	0.000	0.000	0.600	0.833	0.000
60 × 4	1.000	0.000	0.833	0.400	0.833	0.000	1.000	0.000
60 × 6	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
60 × 8	1.000	0.000	0.000	0.600	0.800	0.000	1.000	0.000
70 × 2	0.833	0.100	0.500	0.400	0.727	0.000	0.500	0.000
70 × 4	1.000	0.000	0.500	0.200	0.000	0.600	0.750	0.000
70 × 6	1.000	0.000	0.500	0.375	0.000	0.750	0.833	0.125
70 × 8	1.000	0.000	0.500	0.250	0.200	0.250	1.000	0.000
80 × 2	1.000	0.000	0.636	0.250	0.500	0.250	1.000	0.000
80 × 4	1.000	0.000	1.000	0.000	0.250	0.000	1.000	0.000
80 × 6	1.000	0.000	0.143	0.750	0.200	0.750	1.000	0.000
80 × 8	1.000	0.000	0.000	0.455	0.800	0.182	0.667	0.182
90 × 2	1.000	0.000	1.000	0.000	0.636	0.200	1.000	0.000
90 × 4	0.750	0.286	0.000	0.571	0.200	0.571	0.500	0.000
90 × 6	1.000	0.000	0.750	0.000	0.800	0.400	1.000	0.000
90 × 8	1.000	0.000	0.500	0.000	0.714	0.000	1.000	0.000
100 × 2	1.000	0.000	0.889	0.000	0.667	0.000	1.000	0.000
100 × 4	1.000	0.000	0.500	0.000	0.500	0.667	0.833	0.167
100 × 6	1.000	0.000	0.000	0.714	0.000	0.429	0.556	0.286
100 × 8	1.000	0.000	0.000	0.750	0.750	0.250	1.000	0.000
110 × 2	1.000	0.000	0.286	0.444	0.000	0.778	1.000	0.000
110 × 4	0.875	0.000	0.000	0.400	0.375	0.200	0.571	0.100
110 × 6	1.000	0.000	0.750	0.000	0.636	0.000	1.000	0.000
110 × 8	1.000	0.000	1.000	0.000	0.250	0.200	1.000	0.000
120 × 2	1.000	0.000	0.625	0.000	0.500	0.375	0.875	0.000
120 × 4	1.000	0.000	1.000	0.000	0.500	0.200	1.000	0.000
120 × 6	1.000	0.000	0.400	0.500	0.000	0.500	1.000	0.000
120 × 8	1.000	0.000	0.800	0.400	0.000	1.000	0.333	0.200

provide the computational results of five algorithms for metric $IGD$ and $c$, in which “A”, “B”, “C”, “D”, and “E” represent “DABC”, “ABC”, “TICA”, “HPSO”, and “CCA”. Figure 6 shows the distribution curves for the optimal solutions of all algorithms for six instances, including $20\times 8$, $30\times 2$, $50\times 4$, $60\times 6$, $70\times 2$, and $120\times 2$. Box plots are provided to further analyze the stability of the DABC algorithm. Figure 7 shows the box plots for five algorithms corresponding to $IGD$ and $c$.

As stated in Table 6 and Table 7, $IGD$ of the DABC algorithm is less than that of the ABC algorithm in 44 instances, and $c\left(A,B\right)$ exceeds $c\left(B,A\right)$ in 44 instances. Moreover, $c\left(A,B\right)$ is equal to 1 in 37 instances, that is, all non-dominated solutions of the ABC are dominated by those of the DABC in 37 instances. Thus, the DABC algorithm provides better results than the ABC in most instances. Similarly, the DABC generates a smaller $IGD$ than CCA in all instances, and it also gains a smaller $c\left(C,A\right)$ than $c\left(A,C\right)$ in all instances. Moreover, $c\left(A,C\right)=1$ is gained in 27 instances, that is, all solutions of CCA are dominated by the non-dominated ones of the DABC. On the other hand, the DABC obtains a smaller $IGD$ than TICA and HPSO in 37 and 35 instances; $c\left(A,D\right)$ is larger than $c\left(D,A\right)$ in 34 instances; and $c\left(A,E\right)$ is bigger than $c\left(E,A\right)$ in 31 instances. Therefore, it is concluded that the DABC algorithm performs notably better than the ABC algorithm, TICA, HPSO, and CCA in over 70% of the instances corresponding to two metrics. This conclusion can also be drawn from the distribution curves for the five algorithms in Figure 6.

According to the box plots in Figure 7, the lower quartile (Q₁), the median (Q₂), the upper quartile (Q₃), and the quartile difference (IQR) between Q₁ and Q₃ of the DABC for metric $IGD$ are smaller than those of the four compared algorithms. Similarly, the DABC algorithm produces a greater value for Q₁, Q₂, Q₃, and IQR for metric $c$. These comparisons and analyses demonstrate that the stability of the DABC algorithm is superior to the ABC algorithm, TICA, HPSO, and CCA.

To analyze the statistical difference in the experimental results, a Wilcoxon test is conducted. The significant level is $\alpha =0.05$, that is, if the test value is $p<0.05$, then a significant difference exists between the comparative algorithms. The Wilcoxon test results are shown in

Table 8. Results of the Wilcoxon test.

Pair	Metric	R+	R−	$\mathit{p}$	$\mathit{\alpha } = 0.05$
DABC vs. ABC	$IGD$	0	44	7.61 × 10−9	Yes
	$c$	44	0	4.31 × 10−10	Yes
DABC vs. TICA	$IGD$	7	37	1.10 × 10−5	Yes
	$c$	34	10	8.41 × 10−4	Yes
DABC vs. HPSO	$IGD$	9	35	5.80 × 10−5	Yes
	$c$	31	13	2.05 × 10−3	Yes
DABC vs. CCA	$IGD$	0	44	7.61 × 10−9	Yes
	$c$	44	0	7.60 × 10−9	Yes

, where “R⁺” indicates that the DABC algorithm is better than the comparative algorithm, “R⁻” indicates that the DABC is worse than the comparative algorithm, and “Yes” means that there is a significant difference between the DABC and the other algorithms. From Table 8, it is concluded that the DABC algorithm has more significant advantages than other algorithms.

To further verify the significant difference among all algorithms, Friedman’s test is applied. The mean ranks generated using Friedman’s test are shown in

Table 9. Results of Friedman’s test.

Algorithm	Mean Rank
DABC	1.36
ABC	4.57
TICA	2.30
HPSO	2.50
CCA	4.27

. It is found that the mean rank of the DABC algorithm is the smallest among all experimental algorithms. Thus, it is concluded that the DABC is significantly superior to the other four compared algorithms when used on the EAHFSP.

The DABC algorithm performs better than the other four comparative algorithms on the EAHFSP, mainly depending on the diversified search strategies in the employed bee phase, the novel probability selection method in the onlooker bee phase, and the utilization of the diversity enhancement control strategy. These features help the DABC improve search efficiency, increase population diversity, and avoid falling stagnation. Therefore, the DABC algorithm obtains a better performance on the EAHFSP.

5.4. A Real-Life Example

Roll is a very important tool in a steel rolling production line. In this study, we choose small and medium-sized rolls produced in a workshop to test the efficiency of the DABC. The processing of rolls includes turning, milling, grinding, and assembly. There are three machines for turning ($M_{11}-M_{13}$), three machines for milling ($M_{21}-M_{23}$), four machines for grinding ($M_{31}-M_{34}$), and one assembly machine ($M_a$). The manufacturing and assembly process is a typical AHFSP.

There are 30 components that will be manufactured and assembled into 12 products. The DABC runs 10 times randomly for the real-life example. The non-dominated solutions generated using the DABC algorithm are given in

Table 10. The non-dominated solutions generated using DABC.

Solutions	Makespan	TEC
1	192.88	5555.96
2	213.37	5391.84
3	178.26	6021.80
4	202.60	5449.45
5	178.78	5610.60

. A Gantt chart for the optimal solution is illustrated in Figure 8.

6. Conclusions

In this study, a novel DABC algorithm is proposed to investigate an EAHFSP with makespan and TEC minimization. In the employed bee phase, diversified search strategies are designed, and the suitable search strategy for each food source is adaptively selected to improve the search quality. In the onlooker bee phase, a novel probability selection method is introduced to control the selection pressure, and a diversity control strategy is applied to increase the population and avoid the algorithm falling into stagnation. Finally, a series of comparative results demonstrate that the DABC owns excellent performance on the considered EAHFSP.

Research using the DABC algorithm for EAHFSPs can assist in selecting the optimal scheduling plan and achieving a balance between economic efficiency and environmental effects. Thus, it is of great significance to managers and manufacturers. However, there are some limitations in this study, for example, some actual processing constraints and deteriorating effects are seldom considered. In the future, we will focus on an EAHFSP with more realistic constraints and a more complex processing environment, such as transportation time, preventive maintenance, uncertainty in processing, deteriorating effects, and multi-factory manufacturing. In addition, the improvements in the meta-heuristic algorithm, including adding more excellent strategies and improving other optimization algorithms will also be considered in our future study.