An Adaptive Two-Class Teaching-Learning-Based Optimization for Energy-Efficient Hybrid Flow Shop Scheduling Problems with Additional Resources

Abstract

Energy-efficient scheduling problems with additional resources are seldom studied in hybrid flow shops. In this study, an energy-efficient hybrid flow shop scheduling problem (EHFSP) with additional resources is studied in which there is asymmetry in the machine. An adaptive two-class teaching-learning-based optimization (ATLBO) which has multiple teachers is proposed to simultaneously minimize the makespan and the total energy consumption. After two classes are formed, a teacher phase is first executed, which consists of teacher self-learning and teacher training. Then, an adaptive learner phase is presented, in which the quality of two classes is used to adaptively decide the learner phase or the reinforcement search of the temporary solution set. An adaptive formation of classes is also given. Extensive experiments were conducted and the computational results show that the new strategies are effective and that ATLBO was able to provide better results than comparative algorithms reported in the literature in at least 54 of 68 instances.

1. Introduction

Hybrid flow shop scheduling problems (HFSPs) occur frequently in many manufacturing industries, such as chemical engineering, metallurgy, textiles, petrochemicals, airplane engines, and semiconductors [1]. Compared with a flow shop, a hybrid flow shop has at least two stages, including at least one stage with at least two parallel machines. The redundancy of the machines used may enable dealing with jobs flexibly and avoid bottlenecks [2]. In recent decades, HFSPs have been extensively investigated and a number of results have been obtained [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].

With the development and application of green manufacturing, companies and society as a whole are paying increased attention to reducing energy consumption in the manufacturing process and improving the rates of energy utilization. In recent years, as a typical meansd of scheduling in green manufacturing, EHFSPs have attracted much attention with a number of investigations being reported. Yan et al. [3] developed a multi-level optimization method to optimize the total energy consumption and the makespan. Zeng et al. [4] investigated the conflicting relationship between total energy consumption and the makespan and employed the non-dominated sorting genetic algorithm-II (NSGA-II). Li et al. [5] presented a two-level imperialist competitive algorithm (TICA). Zuo et al. [6] designed a novel multi-population artificial bee colony (ABC) algorithm. Song [7] constructed a mixed integer linear programming (MILP) model and developed a hybrid multi-objective teaching-learning-based optimization (TLBO) based on decomposition to minimize the makespan and energy consumption. Yue et al. [8] designed a hybrid Pareto spider monkey optimisation algorithm for solving problems, minimizing the total energy consumption and total time taken. Chen et al. [9] reported an improved NSGA-II including optimization strategies. Qin et al. [10] presented a mathematical model of a blocking EHFSP and proposed a modified iterative greedy algorithm based on a swap strategy. Zuo et al. [11] designed a population diversity-based ABC algorithm to minimize the makespan and the total energy consumption simultaneously. Wang et al. [12] considered a fuzzy EHFSP with variable machine speed and proposed use of the NSGA-II algorithm to minimize the fuzzy makespan and total fuzzy energy consumption. Li et al. [13] proposed use of the NSGA-II algorithm combined with Q-learning and general variable neighborhood search. Wang et al. [14] presented an improved multi-objective firefly algorithm.

In scheduling problems, the additional resources involved mainly consist of machine operators, tools, pallets, dies, industrial robots, and automated guided vehicles. The additional resources may be renewable or non-renewable, discrete or continuous, and be used in the processing or setup [18]. When addressing scheduling problems, there are additional resources constraints, that is, the sum of each additional resource used by all machines cannot exceed a given limit at any moment. The existing published studies mainly concern unrelated parallel machine scheduling problems (UPMSPs) with additional resources [19,20,21,22,23,24,25,26,27].

With respect to UPMSPs with additional resources, Zheng and Wang [19,20] presented a two-stage adaptive fruit fly optimization algorithm using a heuristic and knowledge-guided search, and also described use of a collaborative multi-objective fruit fly optimization algorithm. Yepes-Borrero et al. [21] presented three heuristics and a greedy randomized adaptive search procedure for UPMSPs with setup times and additional limited resources in the setup. Fanjul-Peyro et al. [22] constructed two integer linear programming models and proposed three matheuristics. Fanjul-Peyro [23] focused on a UPMSP with a processing resource, setup resource, and a shared resource, and presented a MILP model and a three-phase algorithm. Pinar and Seyda [24] solved a multi-resource-constrained UPMSP with sequence-dependent setup times, a precedence relation, machine eligibility, and release dates using a constraint programming model and two branching strategies. Bitar et al. [25] investigated a UPMSP with auxiliary resources and sequence-dependent and machine-dependent setup times in the photolithography workshop of a semiconductor plant. A mimetic algorithm combined with an integer linear programming model was used to minimize the makespan. Lei and He [26] developed an adaptive ABC approach for addressing a problem with additional resources involving preventive maintenance. An ABC method with adaptive competition was also applied to solve the above problem [27].

Unlike UPMSPs with additional resources, HFSPs with additional resources are seldom considered. For a two-stage HFSP with limited additional resources, Figielska [28,29] proposed a heuristic with several rules for job selection and two heuristic algorithms by combining a column generation technique with a genetic algorithm or column generation with simulated annealing.

As stated above, many works have been published on EHFSPs and UPMSPs with additional resources. Only one paper has considered solving HFSPs with additional resources, while EHFSPs with additional resources in which the additional resource, machine resource, and energy are dealt with simultaneously, have been barely investigated. An additional resource constraint is needed to be handled at each stage and multiple conflicting objectives are optimized simultaneously in EHFSPs with additional resources. As a result, optimization difficulties increase greatly. Moreover, the reasonable usage of additional resources and the optimization of energy consumption will result in EHFSPs with additional resources being close to the realistic situation of a hybrid flow shop. The corresponding optimization results will have high application value, so it is important to consider EHFSPs with additional resources.

TLBO simulates the teaching process of teachers and the learning process of learners in real classes, which consists of a teacher phase and a learner phase [30]. Compared with other algorithms, TLBO has a simple structure, fewer parameters, and is both easy to understand and implement. It has become an effective method for addressing production scheduling problems [7,31,32,33,34,35,36,37,38]. Lei et al. [34] proposed a novel TLBO for EHFSPs. Li et al. [35] applied a discrete TLBO to solve an HFSP with five types of disruption events and makespan minimization. Xu et al. [36] reported an effective TLBO for HFSPs with fuzzy processing time. Buddala and Mahapatra [37] presented a two-stage TLBO to solve an HFSP involving unexpected machine breakdowns. Lei and Xi [38] proposed a Q-learning-based TLBO for two-stage fuzzy HFSPs.

The above studies show that TLBO has strong search capabilities and advantages for solving HFSPs and EHFSPs. EHFSPs with additional resources and EHFSPs have strong similarities. For example, the same coding can be used for two problems. The strong search abilities of TLBO for HFSPs and EHFSPs show that TLBO possesses potential advantages in solving EHFSPs with additional resources, so TLBO is chosen here, and how to design an efficient TLBO for EHFSPs with additional resources is investigated.

In this study, energy-efficient scheduling in a hybrid flow shop is considered. The main contributions are summarized as follows:

• An EHFSP with additional resources is considered.
• An adaptive two-class teaching-learning-based optimization (ATLBO) is presented to minimize the makespan and total energy consumption simultaneously. To produce high quality solutions, a teacher phase with teacher self-learning and teacher training are presented and an adaptive learner phase is reported which uses the quality of two classes to decide adaptively the learner phase or the reinforcement search of the temporary solution set. An adaptive formation of classes is also performed.
• The performance of the ATLBO is tested through a number of experiments. The effectiveness of the new strategies of the ATLBO is validated and the search advantages of the ATLBO for solving the EHFSP with additional resources is also demonstrated.

The rest of the paper is arranged as follows: Section 2 describes the considered EHFSP with additional resources. Section 3 shows the detailed steps of the ATLBO for the EHFSP with additional resources. The computational experiments undertaken are descibed in Section 4. The conclusions and future topics are reported in Section 5.

2. Problem Description

The EHFSP with additional resources is described as follows: There are n jobs $J_1,J_2,\cdots ,J_n$ and a hybrid flow shop with m stages. Each stage l has $S_l$ unrelated parallel machines $M_{l1},M_{l2},\cdots ,M_{l{S}_l}$. Each job $J_i$ is processed according to the same production flow: stage 1, stage 2, ⋯, stage $m\ge 2$. $p_{ilk}$ indicates the processing time of job $J_i$ on $M_{lk}$. One additional resource is needed for each stage. For job $J_i$ processed on $M_{lk}$, it needs $r_{ilk}$ units of the additional resource. At most, $R_l$ units of additional resource can be used at any time on each machine at stage l. Machine $M_{lk}$ has two modes: a processing mode and an idle mode. $e_{lk}$ and $i{e}_{lk}$ indicate the energy consumption per unit time when $M_{lk}$ is in the processing mode and the idle mode, respectively.

An additional resource constraint is that the sum of each additional resource used by all the machines cannot exceed a given limit at any moment. If there are no additional resources, symmetry exists between any two adjacent jobs on the same machine; after additional resources are considered, asymmetry often exists for adjacent jobs on a machine.

The following constraints on jobs and machines are considered:

Each job can be processed on only one machine at a time.

Each machine handles at most one job at a time.

The operations cannot be interrupted.

All the machines are available at all times.

The goal is to to minimize the makespan and the total energy consumption.

$$f_1=min{C}_{max}=max\left\{C_i\left|i=1,2,\cdots ,n\right.\right\}$$

(1)

$$f_2=TEC=\sum _{l=1}^m\sum _{k=1}^{S_l}\sum _{i=1}^n{\int }_0^{C_{max}}{w}_{ilkt}\times e_{lk}dt+\sum _{l=1}^m\sum _{j=1}^{S_l}{\int }_0^{C_{max}}{z}_{lkt}\times i{e}_{lk}dt,$$

(2)

where n indicates n jobs, $C_i$ indicates the completion time of job $J_i$, $C_{max}$ is the maximum completion time of all jobs, m indicates the m stages, $S_l$ indicates that there are $S_l$ machines at stage l, and $e_{lk}$ and $i{e}_{lk}$ indicate the energy consumption per unit time when $M_{lk}$ is in the processing mode and the idle mode, respectively. $w_{ilkt}$ is 1 if job $J_i$ is processed on $M_{lk}$ at time t and 0 otherwise. $z_{ilkt}$ is 1 if machine $M_{lk}$ is free at time t and 0 otherwise. $f_1$ is the minimized makespan. $f_2$ is the total energy consumption. $TEC$ is an acronym for the total energy consumption.

The Pareto domination is a fundamental concept in multi-objective optimization problems [39]. For the above problem, if x and y meet $\forall i\in \left\{1,2\right\}, f_i\left(x\right)\le f_i\left(y\right)$ and $\exists i\in \left\{1,2\right\}$, $f_i\left(x\right)<f_i\left(y\right)$, then x dominates y; $x\succ y$ denotes x dominates y.

An example of an EHFSP with additional resources, two stages and eight jobs is given in

Table 1. Processing time of the example.

${\mathit{J}}_{\mathit{i}}$	${\mathit{J}}_1$	${\mathit{J}}_2$	${\mathit{J}}_3$	${\mathit{J}}_4$	${\mathit{J}}_5$	${\mathit{J}}_6$	${\mathit{J}}_7$	${\mathit{J}}_8$
$M_{11}$	21(2)	25(5)	115(8)	20(5)	110(9)	88(7)	16(1)	27(2)
$M_{12}$	26(6)	36(7)	111(3)	17(4)	112(6)	89(8)	22(5)	17(3)
$M_{21}$	54(9)	51(2)	44(3)	60(5)	41(5)	49(5)	56(1)	47(7)
$M_{22}$	61(7)	70(5)	77(9)	64(7)	59(4)	70(5)	76(4)	73(4)

. Every stage has two machines. In Table 1, the number outside the bracket is the processing time and the data in the bracket are $r_{ilk}$. $R_l=10,(l=1,2)$, $e_{11}=10$, $e_{12}=10$, $e_{21}=3$, $e_{22}=2$, $i{e}_{lk}=1$. Figure 1 gives a schedule of the example, where $f_1=652$, $f_2=2589$.

3. ATLBO for EHFSP with Additional Resources

TLBO usually has only one class. A multi-class TLBO is considered to improve the performance of TLBO for scheduling problems [32,38]. These algorithms have more than two classes and the number of classes is the algorithm parameter. In this study, a two-class TLBO is presented in which the number of classes is fixed to two and an adaptive learner phase is used.

3.1. Initialization and Formation of Two Classes

A two-string representation is applied. For an EHFSP with n jobs and m stages, its solution is composed of a scheduling string $[q_1,q_2,\cdots ,q_n]$ and a machine assignment string $[M_{1h_1},M_{1h_2},\cdots ,M_{1h_n}]$, where $M_{1h_1}$ is the assigned machine for job $J_i$ at stage 1 and $q_1\in \left\{0,1\right\}$ is a real number for $J_i$.

The decoding procedure is shown below.

Step 1: For stage $l=1$, assign all jobs on each machine $M_{1k},k=1,2,\cdots ,S_1$ according to the machine assignment string; then, on each machine $M_{1k}$, sort all assigned jobs on $M_{1k}$ in the ascending order of their $q_i$; obtain a permutation, start with the first job of the permutation. For each job $J_i$ in the permutation, process $J_i$ on machine $M_{1k}$ when the additional resource constraint is met.

Step 2: For stage $l=2,3,\cdots ,m$, sort all jobs in the ascending order of their completion time at stage $l-1$; then, obtain a permutation of all jobs $[{\pi }_1,{\pi }_2,\cdots ,{\pi }_n]$. Start with the first job of the permutation; for each job ${\pi }_i$, choose the machine $M_{lk}$ with the smallest beginning time and process ${\pi }_i$ on $M_{lk}$ when the additional resource constraint is met.

For job $J_i$, when it is processed on machine $M_{lk}$ at stage l, the handling method [26] for the additional resource is directly applied. $J_i$ can be inserted into the idle period means that $J_i$ can be started and completed during the idle time and at most $R_l$ units of the additional resource can be used at any time on all machines at stage l.

A solution of the example in Table 1 is a machine assignment string $[0.07,0.48,0.05,0.20,$$0.37,0.23,0.69,0.32]$ and a scheduling string $[1,1,2,2,1,2,1,2]$. At stage 1, jobs $J_1,J_2,J_5,J_7$ are assigned on machine $M_{11}$, a permutation of jobs $[1,5,2,7]$ is obtained on $M_{1k}$ in the ascending order of $q_1=0.07,q_2=0.48,q_5=0.37,q_7=0.69$, $J_7$ can be processed in the idle period $[0,111]$, and the additional resource can also be met, $J_7$ is inserted into idle period $[0,111]$. At stage 2, obtain a permutation of all jobs $[7,3,4,1,6,8,5,2]$ in the ascending order of their completion time at stage 1, start with the first job $J_7$ of the permutation, then all the jobs are processed in order. $J_5$ can be inserted into the idle period $[77,191]$.

An initial population P with N solutions is randomly generated. Let $\lambda =0$, $\lambda$ is an integer parameter. The best b solutions from population P are selected as teachers, which constitute the set of teachers $\Theta =\left\{x_{teacher}^1,x_{teacher}^2,\cdots ,x_{teacher}^b\right\}$. Then, the $N-b$ solutions as learners are arranged randomly for two classes $Cl{s}_1,Cl{s}_2$.

A formation of two classes is described as follows:

Step 1: Perform non-dominated sorting and compute the crowding distance [40] on all the solutions in P.

Step 2: Compute the fitness $c\left(x_i\right)$ of each solution $x_i$, sort all solutions in the descending order of $c\left(x_i\right)$, suppose that $c\left(x_1\right)\le c\left(x_2\right)\le \cdots \le c\left(x_N\right)$, then select the best b solutions as teachers.

Step 3: For solutions $x_{b+1},x_{b+2},\cdots ,x_N$, let $l=b+1,\tau =1$; repeat the following steps until $l>N$: $x_l$ is assigned to $Cl{s}_{\tau },l=l+1,\tau =\tau +1$. If $\tau =3$, then let $\tau =1$.

$$c\left(x_i\right)=\underset{l=1,2,\cdots ,N}{max}\left\{{rank}_l\right\}-{rank}_i+{dist}_i/{\sum }_{j\in {\theta }_{{rank}_i}}{dist}_{j,}$$

(3)

where $ran{k}_i$ and $dis{t}_i$ indicate the $rank$ value and the crowding distance of the solution $x_i$.

3.2. Teacher Phase

The teacher phase is depicted below: for each teacher $x_{teacher}^r$, execute a multiple neighborhood search on it, then randomly assign $(N-b)/b$ learners, which are included into a set ${\Lambda }_r$. For each learner $x_i\in {\Lambda }_r$, perform a global search between $x_i$ and $x_{teacher}^r$, obtain a new solution z. If $z\succ x_i$ or z and $x_i$ are non-dominated over each other, then replace $x_i$ with z.

A multiple neighborhood search is executed. For solution x, let $g=1$. Repeat the following steps until $g=4$: produce a new solution $z\in {\mathcal{N}}_g\left(x\right)$, if $z\succ x$ or z and x is non-dominated over each other, then replace x with z and let $g=4$; otherwise, $g=g+1$. Where ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ are $swap$ and $insert$, respectively, applied on the scheduling string, ${\mathcal{N}}_3$ is $change$ executed on the machine assignment string. The descriptions on $swap$, $insert$, $change$ can be found in [34]. ${\mathcal{N}}_i\left(x\right)$ is the set of neighborhood solutions generated by ${\mathcal{N}}_i$.

The global search is described below. For two solutions $x,y$, if a random number $rand<\mu$, then order crossover [41] is executed on the scheduling string of $x,y$; otherwise, a two-point crossover [42] is applied on the machine assignment string, where $\mu$ follows a uniform distribution on [0, 1], and $rand$ is a real number.

Generally, the computational complexity of a scheduling sub-problem is notably greater than that of the machine assignment. To obtain the best solution to address the problem efficiently, more computing resource is assigned for scheduling the sub-problem, so we set $\mu >0.5$ to adapt to the above features of the two sub-problems.

In the teacher phase, each teacher is first improved by self-learning implemented with a multiple neighborhood search. Then, some learners are randomly assigned for each teacher and teaching activities are conducted between each teacher and its assigned learners. The self-learning and teaching take full advantage of high quality solutions. As a result, higher quality solutions can be obtained with high probability; moreover, the global search ability can be improved because of the random assignment of learners.

3.3. Adaptive Learner Phase

The detailed steps of the adaptive learner phase are shown below.

Step 1: Execute a multiple neighborhood search on each learner $x_i\in (Cl{s}_1\cup Cl{s}_2)$.

Step 2: Compute $rank$ and the crowding distance of learner $x_i\in (Cl{s}_1\cup Cl{s}_2)$ [40]; then calculate $Ev{o}_i$ for each class $Cl{s}_i$.

Step 3: If $Ev{o}_g<\beta ,Ev{o}_{3-g}\ge \beta ,g\in \{1,2\}$, then let $\lambda =0$; for each learner $x_i\in Cl{s}_g$, produce a random number $rand$.

If $rand\ge Ev{o}_{3-g}/\left(Ev{o}_1+Ev{o}_2\right)$, then randomly select $y,z\in Cl{s}_{3-g}$; execute global search between y and z, produce a new solution $z^{\prime }$. If $z^{\prime }\succ x_i$ or $z^{\prime }$ and $x_i$ are non-dominated over each other, then replace $x_i$ with $z^{\prime }$; otherwise, execute a multiple neighborhood search on $x_i$.

If $rand<Ev{o}_{3-g}/\left(Ev{o}_1+Ev{o}_2\right)$, then randomly select $z\in Cl{s}_{3-g}$. Execute a global search between $x_i$ and z, produce a new solution $z^{\prime }$. If $z^{\prime }\succ x_i$ or $z^{\prime }$ and $x_i$ are non-dominated over each other, then replace $x_i$ with $z^{\prime }$; otherwise, carry out a multiple neighborhood search on $x_i$.

Step 4: If $Ev{o}_1<\beta ,Ev{o}_2<\beta$, then let $\lambda =\lambda +1$, a set of solutions $\Phi$, which is composed of the updated solutions in Step 1 and all teachers in the population P. For each $x\in \Phi$, randomly select $y\in \Phi$, execute global search between x and y, produce a new solution z. If $z\succ x$ or z and x are non-dominated over each other, then replace x with z; otherwise, perform a multiple neighborhood search on x, produce a new solution $z^{\prime }$. If $z^{\prime }\succ x$ or $z^{\prime }$ and x are non-dominated over each other, then replace x with $z^{\prime }$; otherwise, replace the worst solution of $Cl{s}_1\cup Cl{s}_2$ with $z^{\prime }$. If $\lambda \le 3$, then return each $x\in \Phi$ to the original class; otherwise, perform an adaptive formation of classes.

Where $\beta$ is a real number, and $Ev{o}_i$ is the evaluation index of class $Cl{s}_i(i=1,2)$ and computed by

$$Ev{o}_i={\sum }_{x\in Cl{s}_i}c\left(x\right)/\frac{N-b}{2}$$

(4)

The detailed steps of the adaptive formation of classes are shown below.

Step 1: Perform non-dominated sorting and crowding distance [40] on all solutions in P.

Step 2: Compute the fitness $c\left(x_i\right)$ of each solution $x_i$. Sort all the solutions in the descending order of $c\left(x_i\right)$. Suppose that $c\left(x_1\right)\le c\left(x_2\right)\le \cdots \le c\left(x_N\right)$; the best b solutions are selected as teachers.

Step 3: If $\left|\Phi \setminus \Theta \right|\ge (N-b)/2$, then select the $(N-b)/2$ best solutions in $\Phi \setminus \Theta$ and put them into $Cl{s}_1$. All the remaining solutions are put into $Cl{s}_2$; otherwise, assign all solutions $x\in \Phi \setminus \Theta$ to $Cl{s}_1$ first. All the remaining solutions are randomly assigned to two classes, ensuring equal numbers in both classes.

Where $\Phi \setminus \Theta$ is a set of all the updated solutions in Step 1 of the adaptive learner phase.

Usually, the learner phase is about mutual learning between learners. If the learners in a class have poor solution quality, then their mutual learning will have low efficiency; as a result, waste of computing resources occurs. To avoid the above case, an adaptive learner phase is used. If the conditions of Step 3 are met, mutual learning is just applied for class with higher $Ev{o}_i$; if the conditions of Step 4 are met, then no mutual learning occurs and a reinforcement search of the temporary set is executed to update the worst solution of the class. If $Ev{o}_1>\beta ,Ev{o}_2>\beta$, then the classes evolve well in the teacher phase, so the learner phase is skipped directly.

3.4. Algorithm Description

The detailed steps of ATLBO are shown below.

Randomly produce an initial population P with N solutions, let $\lambda =0$, compute $c\left(x_i\right)$ for each solution.

While the stopping condition is not met, do

Form two classes.

Execute the teacher phase.

Perform the adaptive learner phase.

End While

Step 6: Output the non-dominated in P.

A flowchart of ATLBO is shown in Figure 2.

Unlike the existing TLBOs [32,38], ATLBO just uses two classes. After two classes are formed, a teacher phase with self-learning and teaching is executed on multiple teachers. Then, an adaptive learner phase is presented in which the quality of the two classes is used to decide adaptively the learner phase or the reinforcement search of the temporary solution set, and an adaptive formation of classes is used. Resources are given to better solutions to produce high quality solutions and to save computing resources so that the search efficiency can be improved.

4. Computational Experiments

Extensive experiments were conducted to test the performance of ATLBO for an EHFSP with additional resources. All experiments were implemented using Microsoft Visual C++ 2019 and run on 16.0 G RAM 2.00 GHz CPU PC.

4.1. Test Instances, Metrics, and Comparative Algorithms

A total of 68 instances are used [26], which are the combinations of $n\in$ [8, 16, 30, 50, 150, 250, 350] and $m\in [2, 4, 6, 8]$. The detailed descriptions of their data, such as $p_{ilk}$ and $r_{ilk}$, can be obtained directly from the following link: http://soa.iti.es/ (accessed on 10 January 2024). $R_l=5S_l$. Each instance is described as $n\times m\times \mathit{No}$. There is one way to produce the processing time and three ways to generate the additional resources. There are three combinations of processing time and additional resources. $\mathit{No}$ is defined as a combination of the one way of producing the processing time and the $\mathit{No}$-th way of generating the additional resources.

The following three metrics are chosen in this study.

The metric $\mathcal{C}$ [43] is applied to measure the dominant relationship between the non-dominated solution sets of the two algorithms.

$$\mathcal{C}(L,B)=\frac{\left|\right\{b\in B:\exists h\in L, h\succ b\left\}\right|}{\left|B\right|}$$

(5)

$\mathcal{C}(L,B)$ measures the fraction of members of B that are dominated by members of L.

The metric $\rho$ [44] is the ratio of $\{x\in {\Omega }_A\mid x\in {\Omega }^{*}\}$ to $\left|{\Omega }^{*}\right|$, where ${\Omega }_A$ is the non-dominated set of algorithm A. The reference set ${\Omega }^{*}$ consists of the non-dominated solutions in the union of the non-dominated sets of all the algorithms.

$$\rho \left({\Omega }_A\right)=\frac{\left|\left\{x\in {\Omega }_A\mid x\in {\Omega }^{*}\right\}\right|}{\left|{\Omega }^{*}\right|}$$

(6)

The metric $D{I}_R$ [45] measures the non-dominated solution set ${\Omega }_A$ of an algorithm with the reference set ${\Omega }^{*}$.

$$D{I}_R\left({\Omega }_A\right)=\frac{1}{\left|{\Omega }^{*}\right|}\sum _{y\in {\Omega }^{*}}min\left\{{\sigma }_{xy}\mid x\in {\Omega }_A,\right\}$$

(7)

where ${\sigma }_{xy}$ is the distance between a solution x and a reference solution y in the normalized objective space.

Meta-heuristics [46,47,48,49] are often applied to solve complicated optimization problems. In this study, three meta-heuristics are selected, which include an improved multi-objective firefly algorithm (IMOFA) [14], a two-level imperialist competitive algorithm (TICA) [5], and a hybrid multi-objective teaching-learning-based optimization (HMOTLBO) [7]. These algorithms can be used to solve an EHFSP with additional resources after minor revisions are made, so they are chosen as the comparative algorithms.

Related steps on additional resources are added in three comparative algorithms to deal with the considered problem. For IMOFA, the speed determination matrix and its related crossover operators and neighborhood searches are deleted; only the makespan and total energy consumption were chosen as objectives. For TICA, a three-string representation was changed to only a scheduling string and a machine assignment string. Only two crossovers on the scheduling strings and machine assignment string are used. HMOTLBO can be directly used.

For TLBO, the teacher phase and the learner phase are executed by global search. Comparisons between ATLBO and TLBO are performed to show the effect of the new strategies of ATLBO.

4.2. Parameter Settings

ATLBO has the following parameters: N, b, $\mu$, $\beta$, $\lambda$ and a stopping condition. With respect to the stopping condition, ATLBO converges well when $0.15\times n\times m$ seconds CPU time is reached; moreover, when $0.15\times n\times m$ seconds CPU time is applied, IMOFA, TICA, and HMOTLBO also converge well, so the above CPU time is given as the stopping condition for all the algorithms.

The Taguchi method [50] is used to decide the settings for the other parameters by using, for instance, $50\times 2$.

Table 2. Parameters and their levels.

Parameters	Factor Level
	1	2	3
N	80	100	120
b	3	5	7
$\mu$	0.65	0.7	0.75
$\lambda$	2	3	4
$\beta$	8.5	9	9.5

gives the levels of each parameter. The orthogonal array $L_{27}\left(3^5\right)$ is tested. ATLBO with each combination runs 10 times independently, for instance, $50\times 2$.

The results of the $D{I}_R$ and $S/N$ ratio are given in Figure 3, where the $S/N$ ratio is $-10lo{g}_{10}\left({D{I}_R}^2\right)$. As shown in Figure 3, ATLBO with the combination $N=100$, $b=5$, $\mu =0.7$, $\lambda =3$ and $\beta =9$ can obtain better results than ATLBO with other combinations, so the above parameter settings are used.

TLBO has two parameters $N=100$, $\mu =0.7$. Experiments show that the parameter settings of IMOFA, TICA, and HMOTLBO are still effective, so they continue to be adopted.

4.3. Results and Discussion

The ATLBO, IMOFA, TICA, HMOTLBO, and TLBO algorithms are compared. Each algorithm runs randomly 10 times for each instance.

Table 3. Results of ATLBO, IMOFA, TICA, HMOTLBO, and TLBO on $D{I}_R$.

Instance	ATLBO	IMOFA	TICA	HMOTLBO	TLBO	Instance	ATLBO	IMOFA	TICA	HMOTLBO	TLBO
8 × 2 × 1	0.109	0.615	0.429	0.436	0.553	150 × 2 × 3	0.000	0.992	0.688	0.644	0.732
8 × 2 × 2	0.141	0.288	0.623	0.212	0.359	150 × 4 × 1	0.000	0.073	0.722	0.694	0.951
8 × 4 × 1	0.000	0.366	0.586	0.732	1.098	150 × 4 × 2	0.000	0.609	0.433	0.618	0.997
8 × 4 × 2	0.281	0.363	0.499	0.441	0.547	150 × 4 × 3	0.000	0.502	0.392	0.560	0.705
8 × 6 × 1	0.067	0.355	0.585	0.210	0.381	150 × 6 × 1	0.000	0.942	0.441	0.515	0.655
8 × 6 × 2	0.144	0.243	0.603	0.310	0.468	150 × 6 × 2	0.000	0.864	0.560	0.669	0.698
8 × 8 × 1	0.145	0.286	0.271	0.282	0.637	150 × 6 × 3	0.000	0.589	0.451	0.650	1.078
8 × 8 × 2	0.131	0.045	0.355	0.545	0.533	150 × 8 × 1	0.000	0.586	0.714	0.702	1.088
16 × 2 × 1	0.000	0.500	0.378	0.540	0.720	150 × 8 × 2	0.000	1.032	0.380	0.477	0.624
16 × 2 × 2	0.283	0.559	0.095	0.252	0.404	150 × 8 × 3	0.000	1.008	0.606	0.694	0.733
16 × 4 × 1	0.000	0.451	0.654	0.915	0.801	250 × 2 × 1	0.000	0.492	0.331	0.572	1.205
16 × 4 × 2	0.614	0.410	0.152	0.336	0.518	250 × 2 × 2	0.000	0.592	0.296	0.842	0.826
16 × 6 × 1	0.000	0.380	0.419	0.463	0.577	250 × 2 × 3	0.000	1.348	0.475	0.418	0.580
16 × 6 × 2	0.000	0.545	0.438	0.542	0.725	250 × 4 × 1	0.000	1.019	0.643	0.946	0.784
16 × 8 × 1	0.222	0.150	0.562	0.416	0.301	250 × 4 × 2	0.000	0.548	0.437	0.623	1.025
16 × 8 × 2	0.120	0.159	0.656	0.259	0.418	250 × 4 × 3	0.000	0.592	0.594	0.729	0.972
30 × 2 × 1	0.000	0.644	0.320	0.422	0.676	250 × 6 × 1	0.000	0.960	0.549	0.917	0.655
30 × 2 × 2	0.000	0.360	0.549	0.729	0.994	250 × 6 × 2	0.000	1.111	0.510	0.458	0.575
30 × 4 × 1	0.445	0.529	0.278	0.515	0.747	250 × 6 × 3	0.000	0.670	0.223	1.005	0.980
30 × 4 × 2	0.743	0.478	0.468	0.516	0.587	250 × 8 × 1	0.000	0.501	0.597	0.972	1.041
30 × 6 × 1	0.460	0.369	0.335	0.288	0.465	250 × 8 × 2	0.000	0.672	0.499	0.756	1.008
30 × 6 × 2	0.000	0.610	0.557	0.715	0.930	250 × 8 × 3	0.000	1.046	0.466	0.631	0.853
30 × 8 × 1	0.000	0.458	0.525	0.635	0.881	350 × 2 × 1	0.000	0.578	0.359	0.848	0.976
30 × 8 × 2	0.000	0.585	0.358	0.570	0.763	350 × 2 × 2	0.000	1.085	0.648	0.863	0.980
50 × 2 × 1	0.000	0.642	0.520	0.682	0.857	350 × 2 × 3	0.000	1.040	0.610	0.678	0.907
50 × 2 × 2	0.000	0.662	0.570	0.812	0.880	350 × 4 × 1	0.000	0.590	0.493	0.520	0.425
50 × 4 × 1	0.000	0.869	0.432	0.463	0.625	350 × 4 × 2	0.000	1.064	0.834	1.404	1.414
50 × 4 × 2	0.000	0.887	0.387	0.511	0.487	350 × 4 × 3	0.000	0.686	0.515	0.857	1.030
50 × 6 × 1	0.000	0.552	0.413	0.653	0.640	350 × 6 × 1	0.000	0.509	0.832	0.524	0.641
50 × 6 × 2	0.000	0.553	0.494	0.553	0.637	350 × 6 × 2	0.000	0.898	0.517	0.730	0.677
50 × 8 × 1	0.000	0.843	0.282	0.580	0.593	350 × 6 × 3	0.000	0.711	0.338	0.862	1.013
50 × 8 × 2	0.000	0.299	0.472	0.535	0.788	350 × 8 × 1	0.000	0.736	0.589	0.884	1.031
150 × 2 × 1	0.000	0.800	0.470	0.431	0.984	350 × 8 × 2	0.000	0.751	0.319	0.877	0.541
150 × 2 × 2	0.000	0.193	0.184	0.681	1.042	350 × 8 × 3	0.000	0.822	0.460	1.056	0.822

,

Table 4. Results of ATLBO, IMOFA, TICA, HMOTLBO, and TLBO on $\mathcal{C}$.

Instance	$\mathcal{C}(\mathit{A},\mathit{I})$	$\mathcal{C}(\mathit{I},\mathit{A})$	$\mathcal{C}(\mathit{A},\mathit{T})$	$\mathcal{C}(\mathit{T},\mathit{A})$	$\mathcal{C}(\mathit{A},\mathit{H})$	$\mathcal{C}(\mathit{H},\mathit{A})$	$\mathcal{C}(\mathit{A},\mathit{TL})$	$\mathcal{C}(\mathit{TL},\mathit{A})$
8 × 2 × 1	1.000	0.000	0.571	0.000	1.000	0.000	1.000	0.000
8 × 2 × 2	1.000	0.000	0.700	0.000	0.900	0.000	1.000	0.000
8 × 4 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
8 × 4 × 2	1.000	0.000	0.875	0.000	1.000	0.000	1.000	0.000
8 × 6 × 1	1.000	0.000	0.500	0.000	0.600	0.000	1.000	0.000
8 × 6 × 2	1.000	0.000	0.800	0.000	1.000	0.000	1.000	0.000
8 × 8 × 1	1.000	0.000	0.600	0.000	0.900	0.000	1.000	0.000
8 × 8 × 2	0.429	0.000	0.500	0.000	1.000	0.000	1.000	0.000
16 × 2 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
16 × 2 × 2	1.000	0.000	0.500	0.000	1.000	0.000	1.000	0.000
16 × 4 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
16 × 4 × 2	1.000	0.000	0.778	0.000	1.000	0.000	1.000	0.000
16 × 6 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
16 × 6 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
16 × 8 × 1	0.333	0.000	1.000	0.000	1.000	0.000	1.000	0.000
16 × 8 × 2	0.500	0.000	0.800	0.000	1.000	0.000	1.000	0.000
30 × 2 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
30 × 2 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
30 × 4 × 1	1.000	0.000	0.857	0.000	1.000	0.000	1.000	0.000
30 × 4 × 2	0.000	1.000	0.300	0.000	0.400	0.000	0.500	0.000
30 × 6 × 1	1.000	0.000	1.000	0.000	0.900	0.000	1.000	0.000
30 × 6 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
30 × 8 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
30 × 8 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
50 × 2 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
50 × 2 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
50 × 4 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
50 × 4 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
50 × 6 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
50 × 6 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
50 × 8 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
50 × 8 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 2 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 2 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 2 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 4 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 4 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 4 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 6 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 6 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 6 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 8 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 8 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
150 × 8 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 2 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 2 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 2 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 4 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 4 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 4 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 6 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 6 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 6 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 8 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 8 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
250 × 8 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 2 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 2 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 2 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 4 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 4 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 4 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 6 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 6 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 6 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 8 × 1	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 8 × 2	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
350 × 8 × 3	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000

and

Table 5. Results of ATLBO, IMOFA, TICA, HMOTLBO, and TLBO on $\rho$.

Instance	ATLBO	IMOFA	TICA	HMOTLBO	TLBO	Instance	ATLBO	IMOFA	TICA	HMOTLBO	TLBO
8 × 2 × 1	0.25	0.00	0.75	0.00	0.00	150 × 2 × 3	1.00	0.00	0.00	0.00	0.00
8 × 2 × 2	0.25	0.00	0.75	0.00	0.00	150 × 4 × 1	1.00	0.00	0.00	0.00	0.00
8 × 4 × 1	1.00	0.00	0.00	0.00	0.00	150 × 4 × 2	1.00	0.00	0.00	0.00	0.00
8 × 4 × 2	0.50	0.00	0.50	0.00	0.00	150 × 4 × 3	1.00	0.00	0.00	0.00	0.00
8 × 6 × 1	0.17	0.00	0.83	0.00	0.00	150 × 6 × 1	1.00	0.00	0.00	0.00	0.00
8 × 6 × 2	0.33	0.00	0.67	0.00	0.00	150 × 6 × 2	1.00	0.00	0.00	0.00	0.00
8 × 8 × 1	0.17	0.00	0.67	0.17	0.00	150 × 6 × 3	1.00	0.00	0.00	0.00	0.00
8 × 8 × 2	0.20	0.80	0.00	0.00	0.00	150 × 8 × 1	1.00	0.00	0.00	0.00	0.00
16 × 2 × 1	1.00	0.00	0.00	0.00	0.00	150 × 8 × 2	1.00	0.00	0.00	0.00	0.00
16 × 2 × 2	0.33	0.00	0.67	0.00	0.00	150 × 8 × 3	1.00	0.00	0.00	0.00	0.00
16 × 4 × 1	1.00	0.00	0.00	0.00	0.00	250 × 2 × 1	1.00	0.00	0.00	0.00	0.00
16 × 4 × 2	0.23	0.00	0.77	0.00	0.00	250 × 2 × 2	1.00	0.00	0.00	0.00	0.00
16 × 6 × 1	1.00	0.00	0.00	0.00	0.00	250 × 2 × 3	1.00	0.00	0.00	0.00	0.00
16 × 6 × 2	1.00	0.00	0.00	0.00	0.00	250 × 4 × 1	1.00	0.00	0.00	0.00	0.00
16 × 8 × 1	0.33	0.67	0.00	0.00	0.00	250 × 4 × 2	1.00	0.00	0.00	0.00	0.00
16 × 8 × 2	0.25	0.25	0.50	0.00	0.00	250 × 4 × 3	1.00	0.00	0.00	0.00	0.00
30 × 2 × 1	1.00	0.00	0.00	0.00	0.00	250 × 6 × 1	1.00	0.00	0.00	0.00	0.00
30 × 2 × 2	1.00	0.00	0.00	0.00	0.00	250 × 6 × 2	1.00	0.00	0.00	0.00	0.00
30 × 4 × 1	0.50	0.00	0.50	0.00	0.00	250 × 6 × 3	1.00	0.00	0.00	0.00	0.00
30 × 4 × 2	0.00	0.50	0.50	0.00	0.00	250 × 8 × 1	1.00	0.00	0.00	0.00	0.00
30 × 6 × 1	0.50	0.00	0.00	0.50	0.00	250 × 8 × 2	1.00	0.00	0.00	0.00	0.00
30 × 6 × 2	1.00	0.00	0.00	0.00	0.00	250 × 8 × 3	1.00	0.00	0.00	0.00	0.00
30 × 8 × 1	1.00	0.00	0.00	0.00	0.00	350 × 2 × 1	1.00	0.00	0.00	0.00	0.00
30 × 8 × 2	1.00	0.00	0.00	0.00	0.00	350 × 2 × 2	1.00	0.00	0.00	0.00	0.00
50 × 2 × 1	1.00	0.00	0.00	0.00	0.00	350 × 2 × 3	1.00	0.00	0.00	0.00	0.00
50 × 2 × 2	1.00	0.00	0.00	0.00	0.00	350 × 4 × 1	1.00	0.00	0.00	0.00	0.00
50 × 4 × 1	1.00	0.00	0.00	0.00	0.00	350 × 4 × 2	1.00	0.00	0.00	0.00	0.00
50 × 4 × 2	1.00	0.00	0.00	0.00	0.00	350 × 4 × 3	1.00	0.00	0.00	0.00	0.00
50 × 6 × 1	1.00	0.00	0.00	0.00	0.00	350 × 6 × 1	1.00	0.00	0.00	0.00	0.00
50 × 6 × 2	1.00	0.00	0.00	0.00	0.00	350 × 6 × 2	1.00	0.00	0.00	0.00	0.00
50 × 8 × 1	1.00	0.00	0.00	0.00	0.00	350 × 6 × 3	1.00	0.00	0.00	0.00	0.00
50 × 8 × 2	1.00	0.00	0.00	0.00	0.00	350 × 8 × 1	1.00	0.00	0.00	0.00	0.00
150 × 2 × 1	1.00	0.00	0.00	0.00	0.00	350 × 8 × 2	1.00	0.00	0.00	0.00	0.00
150 × 2 × 2	1.00	0.00	0.00	0.00	0.00	350 × 8 × 3	1.00	0.00	0.00	0.00	0.00

describe the corresponding results of the five algorithms. A, I, T, H, TL denote ATLBO, IMOFA, TICA, HMOTLBO, and TLBO. Figure 4 shows the convergence curves of ATLBO, IMOFA, TICA, HMOTLBO, and TLBO on $30\times 2$ and $50\times 2$ instances. Figure 5 shows box plots of the three metrics based on Table 3, Table 4 and Table 5.

As shown in Table 3, Table 4 and Table 5, for eight small-scale instances, the three metrics of ATLBO are greater than TLBO. ATLBO obtains smaller $D{I}_R$ than TLBO and $\mathcal{C}(A,TL)$ is greater than $\mathcal{C}(TL,A)$ for 22 of the 24 medium-scale instances. For the large-scale instances, ATLBO obtains smaller $D{I}_R$ than TLBO; moreover, $\mathcal{C}(A,TL)$ and $\rho$ are 1, that is, all the solutions of TLBO are dominated by non-dominated solutions produced by ATLBO. There are notable performance differences between ATLBO and TLBO. This conclusion can also be seen from Figure 4 and Figure 5. Box plots for all the algorithms illustrate the data in the Table 3, Table 4 and Table 5. The performance differences between ATLBO and TLBO show that the new strategies, such as an adaptive learner phase, have positive impacts on the performances of ATLBO, so the new strategies are effective.

As shown in Table 3, Table 4 and Table 5, ATLBO obtains smaller $\mathcal{C}(I,A)$ than $\mathcal{C}(A,I)$ on 68 instances and has $\mathcal{C}(A,I)$ of 1 on at least 64 instances; that is, all solutions of IMOFA are dominated by non-dominated solutions of ATLBO. ATLBO converges significantly better than IMOFA. It can be found from Table 5 that $\rho$ of ATLBO is better than that of IMOFA on 68 instances and $\rho$ of IMOFA is 0 on 65 instances; that is, IMOFA cannot generate any members for reference set ${\Omega }^{*}$. Table 4 shows that ATLBO obtains smaller $D{I}_R$ than IMOFA on 64 of 68 instances. Significant performance differences between ATLBO and IMOFA can also be seen from Figure 4 and Figure 5.

When ATLBO is compared with TICA, it is found that ATLBO performs better than TICA for most instances. Table 4 shows that ATLBO produces smaller $\mathcal{C}(T,A)$ than $\mathcal{C}(A,T)$ on all instances and obtains $\mathcal{C}(A,T)$ of 1 on 56 instances. As shown in Table 5, ATLBO produces a larger $\rho$ than TICA on 58 instances and $\rho$ of ATLBO is 1 on 54 instances. ATLBO also performs better than TICA on $D{I}_R$ because ATLBO obtains better $D{I}_R$ than TICA on 65 instances. The above analyses show that ATLBO can provide better results than TICA. This conclusion can also be drawn from Figure 4 and Figure 5. They depict the notable differences between all the algorithms.

It can also be concluded from Table 3, Table 4 and Table 5 that ATLBO outperforms HMOTLBO on the three metrics. ATLBO has smaller $\mathcal{C}(H,A)$ than $\mathcal{C}(A,H)$ on all instances, obains larger $\rho$ than HMOTLBO on all instances and better $D{I}_R$ than HMOTLBO on 67 instances. The performance differences between ATLBO and HMOTLBO can also be seen in Figure 4 and Figure 5. Figure 5 shows the significant difference of ATLBO. It can be concluded that ATLBO has promising search abilities for solving EHFSPs with additional resources.

In ATLBO, the teacher phase consists of self-learning and teaching for each teacher, good solutions are used fully, the global ability is intensified, and an adaptive learner phase is applied. As a result, mutual learning on a low quality class is avoided and a reinforcement search of some better solutions is conducted to produce high quality solutions. These new strategies lead to a good balance between exploitation and exploration and better results than for the comparative algorithms; thus, ATLBO is a very competitive algorithm for EHFSPs with additional resources.

5. Conclusions

In this study, an EHFSP with additional resources is studied and a new algorithm called ATLBO, which has multiple teachers, is proposed to minimize the makespan and the total energy consumption. After two classes are formed, a teacher phase is first executed, which consists of the teacher’s self-learning and the teacher’s training. Then, an adaptive learner phase is presented, in which the quality of the two classes is used to decide adaptively the learner phase or the reinforcement search of the temporary solution set, and an adaptive formation of classes is also applied. The computational results obtained demonstrate that new strategies are effective and that ATLBO is a very competitive algorithm for EHFSPs with additional resources.

EHFSPs with various constraints have attracted much attention. In the future, we will consider EHFSPs with a batch processing machine in a casting process by applying new optimization strategies, such as meta-heuristic learning. TLBO with new optimization mechanisms and its application to production scheduling are also topics that will be considered in the future.