A Radial Memetic Algorithm to Resolve the No-Wait Job-Shop Scheduling Problem

Abstract

A new radial memetic algorithm is proposed to resolve the no-wait job-shop scheduling problem. Basically, each sequencing solution is factorized as a distance-based ranking model, i.e., each solution is decomposed in n − 1 terms, where n is the number of jobs to be sequenced. After that, a cumulative radial distribution of hydrogen is considered to produce new factorizations using the offspring information (genes). Such radial distribution is applied in the local optimization procedure of the memetic algorithm. A benchmarking dataset is used to show the performance of this new experimental technique, as well as other current procedures. Statistical tests were implemented to confirm the performance of the proposed scheme.

1. Introduction and Related Work

In the no-wait job-shop scheduling problem (NWJSP), the operations of each job must be executed in sequence. At any particular moment, each machine can execute no more than one operation, and more than one operation of a given job cannot be carried out at the particular moment. In addition, waiting time is not permitted for any consecutive operation of each job. Interruptions are not allowed during the production process once any job has started. These characteristics widely exist in real-world manufacturing or production processes. For example, in computer systems, semiconductors testing, food, pharmaceutical, chemical, concrete, and steel, among others.

Some contributions in the real-world no-wait job-shop environments can be found in diverse fields, such as the steel-making industry [1,2], concrete manufacturing [3,4], the chemical and pharmaceutical industries [5], and the food industry [6]. Such examples are evidence that show how the NWJSP has gained increasing attention from researchers. Lately, heuristics and meta-heuristics have been used by researchers to resolve the NWJSP. It is suitable to identify new strategies for building competitive algorithms because the NWJSP’s complexity is NP-hard even for two machine cases [7].

In many previous studies, the NWJSP is normally sectioned into two sub-problems: sequencing and timetabling. The sequencing sub-problem aims to identify the best job processing sequence. The timetabling sub-problem aims to obtain the best start time for each job, according to a previously defined processing sequence that yields the minimum makespan. Significant studies have been published to resolve the NWJSP using the aforementioned sub-problems. Among those studies, a fast deterministic variable neighborhood search (VNS) is proposed by [8]. A VNS searches the optima results iteratively in a $k$-insertion neighborhood until no better solution can be found for the sequencing sub-problem. A VNS employs the non-delay approach to tackle the timetabling sub-problem. Another relevant method is a non-deterministic hybrid algorithm (GASA), which is also proposed by [8]. A GASA uses features of genetic algorithms and simulated annealing to identify the best sequence of jobs in the sequencing sub-problem, whereas the non-delay algorithm is also considered for the timetabling sub-problem. A complete local search with memory (CLM) is discussed by [9]. In the CLM, all the solutions are recorded to eliminate re-calculating. The solutions are produced by 1-insertion for the sequencing sub-problem. CLM utilizes the non-delay timetabling approach. In addition, the inverse timetabling procedure is used in this research, i.e., the non-delay procedure is executed on reversed jobs (the processing route of every job is reversed). A modified complete local search with memory (MCLM) is presented by [10]. Normally, in the CLM, VNS and GASA, both sub-problems, the sequencing and the timetabling, are separately resolved. However, in this research, both sub-problems are comprehensively resolved. A complete local search is embedded into an enhanced timetabling procedure to solve the sequencing problem. The enhanced timetabling procedure is built using the non-delay approach and the inverse non-delay approach.

Three different sequencing algorithms, i.e., a tabu search (TS), a hybrid of tabu search with variable neighborhood search (TSVNS), and a hybrid of tabu search with particle swarm optimization (TSPSO) are shown in [11]. Afterwards, the sequencing algorithms are combined with four different timetabling methods, among them, the non-delay procedure. A multi-start simulated annealing with bi-directional shift timetabling algorithm (MSA-BST) is detailed in [12]. To enhance the quality of the solution, a bi-directional shift timetabling procedure that allows delay timetabling, and, consequently, enlarges the search space, is embedded into the multi-start simulated annealing algorithm. The performance of the proposed MSA-BST is compared to the hybrid artificial bee colony algorithm (HABC), as detailed by [13].

Based on previous literature reviews and to the best of our knowledge, there are no proposed algorithms that integrate distance-based ranking models and gas chemistry concepts, such as hydrogen, as methods of improvement in the construction of solutions for the NWJSP. A distance-based ranking model is suitable when there is a need to rank a set of $k$ items according to some judgment [14]. In this paper, the aforementioned items are jobs to sequence in the NWSJP. Each sequence of jobs in the sequencing sub-problem produces a ranking. A simple ranking model can work to rank the jobs, i.e., the most preferred job is elected first, the best of the remaining jobs is elected in second place, and so on. In general, probability models have been used to produce rankings. For example, [15] detailed a distance-based ranking model, i.e., an exponential model for rankings. The authors show how the ranking process is split into $k-1$ stages. Their exponential model considers that the accuracy of the choice made at any stage is independent of the accuracies in the other stages. The accuracy is assessed with respect to a ‘central ranking’, i.e., the probability of observing any ranking $\pi$ decreases as the distance between $\pi$ and the central ranking increases.

Unlike previous studies, we used the radial distribution of hydrogen as part of a distance-based ranking model. A radial distribution of hydrogen can be built to represent the distance between an electron and the core for a hydrogen atom. A radial distribution of hydrogen can be interpreted as the probability of finding an electron at a certain distance from the core. The integration of both features, i.e., the radial distribution of hydrogen to generate offspring according to a distance-based ranking model, has not been reported as a proposal for improvement in a memetic algorithm.

The non-delay procedure’ results are considered in all the comparisons among methods in this research for the timetabling sub-problem. The main goal of this paper is to evaluate the performance of the proposed radial memetic algorithm (RADMEME) to tackle the sequencing sub-problem. In summary,

Table 1. Differences between the RADMEME approach and others.

Research	Technique/Method
Schuster and Framinan [8]	Variable neighborhood search
Schuster and Framinan [8]	Genetic algorithms and simulated annealing
Framinan and Schuster [9]	Complete local search with memory
Zhu et al. [10]	Modified complete local search with memory
Samarghandi et al. [11]	Tabu search
Samarghandi et al. [11]	Tabu search with variable neighborhood search
Samarghandi et al. [11]	Tabu search with particle swarm optimization
Ying and Lin [12]	Multi-start simulated annealing
Sundar et al. [13]	Hybrid artificial bee colony algorithm
RADMEME approach	Radial memetic algorithm

shows the approaches between the RADMEME and other current algorithms for the NWJSP.

Table 2. Most common local search techniques and the proposed radial distribution.

Technique	Approach
Hill climbing	Hill climbing is a straightforward local search algorithm that starts with an initial solution and iteratively moves to the best neighboring solution that improves the objective function.
Local beam search	Local beam search represents a parallelized adaptation of hill climbing, designed specifically to counteract the challenge of becoming ensnared in local optima. Instead of starting with a single initial solution, local beam search begins with multiple solutions, maintaining a fixed number (the “beam width”) simultaneously.
Simulated annealing	Simulated annealing is a probabilistic local search algorithm inspired by the annealing process in metallurgy. It allows the algorithm to accept worse solutions with a certain probability, which decreases over time.
* Radial distribution of the hydrogen	A cumulative radial distribution of hydrogen can be built using information from a solution. Thus, it is possible to sample a new solution of such distribution, and to try to improve the objective function.

* The contribution of this paper.

depicts the most common local search techniques used in evolutionary algorithms.

The research goals and expected outcomes are listed below.

• To resolve the NWJSP thru the RADMEME
• To evaluate the performance of the proposed RADMEME
• To present a different technique for local optimization
• To offer a competitive-novel memetic algorithm
• To show a different experimental technique for the NWJSP

2. Problem Statement

The most important assumptions of the NWJSP are detailed in [12,16], i.e., each job has its own operation sequence, which indicates its own processing route. No operation can be interrupted once started. No job can be processed on more than one machine simultaneously. No machine can process more than one job simultaneously. No job is allowed to re-enter previous machines. No waiting time is allowed between two consecutive operations of the same job. No failures are considered throughout the production plan. All the processing times are known in advance. The objective is to minimize the makespan by searching for the best timetabling. Based on the MILP model shown in [17], the NWJSP considered herein can be formally specified as follows

• J a set of independent jobs
• M a set of machines
• O a set of operations

Each job $j (j\in J)$ has its own processing order $\{O_{j1},O_{j2},\dots , O_{j{n}_j} \}$, where $n_j$ is the total number of operations in job $j$. Each operation must be processed on a specified machine. Operation $O_{jk}(j\in J;k=\mathrm{1,2},\dots ,n_j)$ of job $j$ requires the exclusive use of machine $M_i(M_i\in M)$, on which it must be processed without interruption for $P_{jk}$ time units.

• ${\mathit{C}}_j$ denotes the completion time of the final operation of job $j$
• ${\mathit{S}}_{jk}$ denotes the starting time of operation $O_{jk}$
• $\mathit{T}$ denotes a sufficiently large positive number

  $$\mathrm{minimize} C_{max}=\underset{j\in J}{\max }\{S_{j{n}_j}+P_{j{n}_j}\}$$

  (1)

  subject to

  $$S_{jk}+P_{jk}=S_{jk+1}, \forall k=\mathrm{1,2},\dots ,n_j-1,j\in J$$

  (2)

  $$S_{ji}+P_{ji}=S_{j’i}+Ty, \forall j, j´\in J, i\in M$$

  (3)

  $$S_{j´i}+P_{j´i}=S_{ji}+T\left(1-y\right), \forall j, j´\in J, i\in M$$

  (4)

  $$y\in \left\{\mathrm{0,1}\right\}; S_{jk}\ge 0, k=\mathrm{1,2},\dots ,n_j,j\in J$$

  (5)

  Equation (1) defines the makespan of the schedule. Constraint set Equation (2) is no-wait constraints on consecutive operations. Constraint sets Equations (3) and (4) guarantee that every job is assigned to one position in the sequence of jobs. Constraint set Equation (5) defines the domain of variable $y$ and the ranges of the starting times of all operations.

3. The RADMEME Approach

As any memetic algorithm, the RADMEME contains the same basic procedures, i.e., solution representation, initial population, selection, cross, mutation, local optimization, and replacement. However, the main difference is located in the local optimization, where the radial distribution of hydrogen is executed. All these elements are detailed below.

3.1. Solution Representation and Initial Population

In the NWJSP is very suitable to represent all the solutions as permutations of jobs $\pi =({\pi }_{\left[0\right]}, {\pi }_{\left[1\right]}, \dots , {\pi }_{\left[n\right]})$. Each member of the population is a sequence of jobs to be processed. Each member of the initial population is randomly built.

3.2. Fitness Evaluation

The non-delay procedure is used to obtain the makespan, i.e., the fitness for each member of the population. The non-delay timetabling procedure starts each job as early as possible without violating ordinal constraint, i.e., ordinal constraint is $S_{{\pi }_{\left[0\right]}}\le \dots \le S_{{\pi }_{[n-1]}}$ for processing sequence $({\pi }_{\left[0\right]}, \dots , {\pi }_{[n-1]})$.

3.3. Selection Process

The bubble sort method is used to select the best candidates from the population. The selected candidates are used in order to create new offspring.

3.4. Cross Operator

The partially mapped crossover (PMX) operator is considered in this research to create new offspring. This operator utilizes the genetic material of two randomly selected parent solutions to propose a new offspring. Basically, we randomly select two cut points on the first parent and the second parent. After, we construct a mapping table for the jobs between the two cut points in the first parent and the second parent. Then, we copy the jobs between the two cut points from the second parent to the new offspring by keeping the jobs’ position from the second parent. Now, we copy the jobs except between two cut points from the first parent to the new offspring by keeping jobs’ position from the first parent. Finally, keep the jobs between the two cut points unchanged.

To provide a clear insight into how PMX operator works, consider that we have the following two parent solutions:

$${\pi }_1 [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]$$

$${ \pi }_2 [ 5, 4, 6, 9, 2, 1, 7, 8, 3 ]$$

We select points of crossover at 3 to 6 positions as follows:

$${\pi }_1 \left[ 1, 2 \right| 3, 4, 5, 6 | 7, 8, 9 ]$$

$${\pi }_2 \left[ 5, 4 \right| 6, 9, 2, 1 | 7, 8, 3 ]$$

By copying the jobs between the two cut points from the second parent, the offspring is partially built as follows:

$$\sigma \left[ , \right| 6, 9, 2, 1 | , , ]$$

By copying the remaining jobs from the first parent, the offspring takes the next shape:

$$\sigma \left[ 3 , 4 \right| 6, 9, 2, 1 | 5 , 7 , 8 ]$$

3.5. Mutation Operator

The swap mutation is also used in permutation encoding. To perform the swap mutation, we select two alleles at random and swap their positions. It preserves most of the adjacency information, but links broken disrupts order more. As an example, let $\sigma \left[ 3 , 4 \right| 6, 9, 2, 1 | 5 , 7 , 8 ]$ as an offspring. If we randomly select position 4 and 5, the results would be $\sigma \left[ 3 , 4 \right| 6, 2, 9, 1 | 5 , 7 , 8 ]$.

3.6. Local Optimization

Based on [14], any permutation, i.e., any ranking, can be decomposed in $n-1$ terms. Let $\pi (5, 4, 3, 2, 1)$ be a permutation, i.e., as an offspring with five jobs, $n=5$. If we want to know how many positions job 5 can be in the permutation (from where it is and to the right of where it is), then we can easily realize that it can be in the first position, the second position, and so on. Thus, the number of moves to place job 5 in each position are: zero moves for the first position, one move for the second position, two moves for the third position, three moves for the fourth position, and four moves for the fifth position. The possible values would be either zero or one or two or three or four, i.e., $V_1 \left\{\begin{array}{c}0\\ 1\\ 2\\ 3\\ 4\end{array}\right.$, where $V_1$ represents the first term of the aforementioned decomposition.

Now, if we want to know how many positions the job 4 can be in the permutation (from where it is and to the right of where it is), then easily we can realize that it can be in the second position, the third position, and so on. The procedure is the same as that conducted with job 5. Thus, the possible values would be $V_2\left\{\begin{array}{c}0\\ 1\\ 2\\ 3\end{array}\right.$ where $V_2$ represents the second term of the aforementioned decomposition. Note that there is only $n-1$ terms of such decomposition. It is due to the last job cannot be located anywhere else to its right.

Therefore, each term of such decomposition, can take a value between $0$ and $n-j$, where $j$ is a position in the permutation. The possible values of such decomposition, i.e., each $V_j$ works as input parameter in the radial distribution of hydrogen.

The radial distribution function, for the hydrogen, is detailed as follows:

$$P\left(r\right)=4{\left({\displaystyle \frac{Z}{a_0}}\right)}^3{e}^{-\left({\displaystyle \frac{2Zr}{a_0}}\right)}{r}^2$$

(6)

where $Z$ represents the atomic number of the element, i.e., the hydrogen $Z=1$. $a_0$ indicates the Bohr radius, and $r$ is the distance (radius), in picometers ($pm$) of electron to the core.

The Bohr radius is defined as $a_0={\displaystyle \frac{h^2}{4{\pi }^{2}me}}=52.9 \mathrm{p}\mathrm{m}$, $h$ is the Planck constant, $m$ is the mass of the electron, and $e$ its charge.

With this radial distribution function, a cumulative distribution should be built. By computing the cumulative radial distribution of hydrogen, we can use it to generate a new decomposition.

The process to obtain a decomposition is computed value per value. Firstly, a random value should be generated for each position. Then, each random value, is interpolated in the cumulative probability distribution, to identify which value is elected.

Figure 1 depicts an example of such construction with five jobs, i.e., $n=5$, for the first position $(j=1)$. The abscissa axis, represents all the possible distances between the electron and the core in picometers. However, all the possible distances are proportionally assigned and distributed to the possible values for job 5, i.e., $V_1 \left\{\begin{array}{c}0\\ 1\\ 2\\ 3\\ 4\end{array}\right.$. This procedure is repeated for each term of decomposition, and all the possible distances, are proportionally assigned and distributed to the possible values from each $V_j$. In this way, a decomposition vector is built.

Once the decomposition is obtained, it is transformed into a permutation by the procedure depicted in [18]. The resulting permutation can be considered as a “neighbour” from the offspring.

When evaluating this permutation, if its fitness is better than the offspring, then the offspring is replaced.

3.7. Replacement

The replacement is executed using the bubble method between the parents and the offspring. The RADMEME framework is depicted below.

$P_0\leftarrow \mathrm{G}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e} \mathrm{M} \mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}$

$F_0\leftarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{i}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h} \mathrm{m}\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} P_0$

$\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{t}\leftarrow \mathrm{S}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} P_0$

$\mathrm{t}≔1$

$\mathrm{D}\mathrm{o}$

$S_t\leftarrow \mathrm{S}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t} \mathrm{o}\mathrm{n}\mathrm{e}\mathrm{s} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} D_{t-1}$

$O_t\leftarrow \mathrm{E}\mathrm{x}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{P}\mathrm{M}\mathrm{X} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r} S_t$

${ O\prime }_t\leftarrow \mathrm{E}\mathrm{x}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{w}\mathrm{a}\mathrm{p} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r} O_t$

$F_t\leftarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{i}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} {O\prime }_t$

$R\leftarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{c}\mathrm{u}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{l} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}.$

${ L}_t\leftarrow \mathrm{B}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{d} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s} \mathrm{u}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g} R$

${ T}_t\leftarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{d} \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} L_t$

${ TF}_t\leftarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{i}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} T_t$

${ O\prime }_t\leftarrow \mathrm{U}\mathrm{p}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e} {O\prime }_t \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} T_t$

$\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{t}\leftarrow \mathrm{i}\mathrm{f} \mathrm{a}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{y}, \mathrm{u}\mathrm{p}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} {O\prime }_t$

${ D}_t\leftarrow \mathrm{R}\mathrm{e}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{o}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} {O\prime }_t \& D_{t-1}$

$\mathrm{t}≔\mathrm{t}+1$

$\mathrm{U}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{l} (\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{p}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{i}\mathrm{s} \mathrm{m}\mathrm{e}\mathrm{t})$

$\mathrm{O}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}:\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{t}$

4. Results and Comparison

The main standard benchmarking datasets for evaluating job-shop scheduling problems are used as input data for the comparison. The aforementioned datasets used in this research are the [19] instances; the [20] instances; the [21] instances; the [22] instances; the [23] instances; finally, the [24] instances. More than 80 instances were used, and all the instances contain necessary and detailed information such as the set of jobs, the feasible machines per each operation, and the processing times.

In order to validate the scientific relevance of this paper, some algorithms are proposed as a benchmark for comparison with the RADMEME scheme. The algorithms are as follows:

The VNS proposed by [8].

The GASA also shown by [8].

The CLM discussed by [9].

The MCLM presented by [10].

The HABC detailed by [13].

The MSA-BST depicted by [12].

The TS, the TSVNS, and the TSPSO, shown by [11].

The average relative percentage deviation (ARPD) is computed in order to compare the performance of each algorithm. The results of each aforementioned algorithm were directly obtained from the literature.

The distribution of the experimental results, for the [19] instances, is detailed in Figure 2. The most of the RADMEME results are concentrated between 0 and 1.5, and the most MSA-BST results are located between 0.25 and 3.5. This means that in most runs, the RADMEME is closer to the best reported results than the MSA-BST.

The distribution of the experimental results for the [20] instances is shown in Figure 3. Again, most of the RADMEME results are concentrated between 0 and 1.5, and the other algorithms’ results are located extremely close to zero. This means that in most runs, the other algorithms are closer to the best reported results than the RADMEME.

The distribution of the experimental results for the [21] instances is detailed in Figure 4. Most of the RADMEME results are concentrated between 0 and 3.5, and most of the remaining algorithms’ results are located far from the RADMEME results. This means that in most runs, the RADMEME is closer to the best reported results than the rest of the algorithms used in the comparison.

The distribution of the experimental results for the [22] instances is presented in Figure 5. It is clear from the figure, that the RADMEME obtains better results than TS, TSVNS, and TSPSO algorithms. However, based on the results, the HABC, and MSA-BST algorithms are closer to zero.

The distribution of the experimental results for the [23] instances is detailed in Figure 6. The most of the RADMEME results are concentrated between 0 and 2.5, i.e., almost practically the same as the HABC and the MSA-BST. In addition, the RADMEME outperforms the TS, TSVNS, and TSPSO performances. The latter are found far from the RADMEME results.

The distribution of the experimental results for the [24] instances is detailed in Figure 7. Most of the RADMEME results are concentrated between 0.07 and 1.6, and most MSA-BST results are located between 2.8 and 3.5. This means that in most runs, the RADMEME is closer to the best reported results than the MSA-BST.

Finally, a Dunnett statistical test is detailed in order to show the RADMEME performance. Figure 8 depicts a Dunnett test for the [19] instances. There is no statistically significant difference between the algorithms.

Figure 9 details a Dunnett test, for the [20] instances. There is no statistically significant difference between the algorithms.

Figure 10 presents a Dunnett test, for the [21] instances. There is no statistically significant difference between the HABC, MCLM, MSA-BST, and RADMEME. In addition, RADMEME outperforms the CLM, GASA, TS, TSPSO, TSVNS, and VNS algorithms.

Figure 11 depicts a Dunnett test, for the [22] instances. There is no statistically significant difference between the algorithms.

Figure 12 details a Dunnett test for [23] instances. There is no statistically significant difference between the HABC, MSA-BST, and RADMEME. Furthermore, RADMEME shows a better performance than the TS, TSPSO, and TSVNS algorithms.

Figure 13 presents a Dunnett test, for the [24] instances. There is a statistically significant difference between the MSA-BST, and RADMEME.

5. Conclusions

The RADMEME scheme detailed above is suitable to tackle the NWJSP. It is a well-known NP-hard issue.

Based on the results shown in Section 4, the RADMEME scheme is competitive. The RADMEME can be extended to other types of scheduling problems because the solution representation for other scheduling issues might be the same, i.e., by permutations. Other scheduling issues can be considered such as flow-shop, open-shop, etc. The RADMEME also can be applied for other combinatorial problems such as routing issues, i.e., vehicle routing situations. Therefore, other optimization problems should be resolved by the RADMEME scheme, in order to confirm its performance.

The set of instances used in the comparison are considered benchmarking. Therefore, the use of the Dunnett test is clearly justified and forceful. The performance of the RADMEME scheme should be taken into account in the literature.

The proposal of the RADMEME, i.e., using the radial distribution of hydrogen to enhance the performance’s offspring is substantial.

For future work, other greedy procedures should be implemented to help the RADMEME find more suitable sequencing solutions. In addition, other radial distributions should be considered to obtain new offspring, to enhance the performance of the RADMEME. These insights could be useful for distributions related to other chemical elements.

Finally, application tools for users should be implemented in practice and real-life situations using this approach.