A Self-Adaptive Cuckoo Search Algorithm Using a Machine Learning Technique

Abstract

Metaheuristics are intelligent problem-solvers that have been very efficient in solving huge optimization problems for more than two decades. However, the main drawback of these solvers is the need for problem-dependent and complex parameter setting in order to reach good results. This paper presents a new cuckoo search algorithm able to self-adapt its configuration, particularly its population and the abandon probability. The self-tuning process is governed by using machine learning, where cluster analysis is employed to autonomously and properly compute the number of agents needed at each step of the solving process. The goal is to efficiently explore the space of possible solutions while alleviating human effort in parameter configuration. We illustrate interesting experimental results on the well-known set covering problem, where the proposed approach is able to compete against various state-of-the-art algorithms, achieving better results in one single run versus 20 different configurations. In addition, the result obtained is compared with similar hybrid bio-inspired algorithms illustrating interesting results for this proposal.

1. Introduction

Recent studies about bio-inspired procedures to solve complex optimization problems have demonstrated that finding good results and the best performance are laborious tasks, so it is necessary to apply an off-line parameter adjustment on metaheuristics [1,2,3,4,5]. This adjustment is considered an optimization problem itself, and several studies are proposing some solutions to solve that, but it always depends on his static therms [6]. Many of these studies use mathematical ways to change the values of one of each parameter during their execution. In this context, parameters like the population size of metaheuristics are initially set without considering their variation or behaviors. We consider using a machine learning (ML) technique that lets us analyze the population and determine the values of the number of solutions and the abandon probability.

As we can see in [7], machine learning and optimization are two topics of artificial intelligence that are rapidly expanding, having a wide range of computer science applications. Due to the rapid progress in the performance of computing and communication techniques, these two research areas have proliferated and drawn widespread attention in a wide variety of applications [8]. For example, in [9], the authors present different clustering techniques to evaluate the credit risk in a determinate population of Europe. Although both fields belong to different communities, they are fundamentally based on artificial intelligence, and the techniques from ML and optimization frequently interact with each other and themselves to improve their learning and/or search capabilities.

On the other hand, advances in operations research and computer science have brought forward new solution approaches in optimization theory, such as heuristics and metaheuristics. While the former are experience-based procedures, which usually provide good solutions in short computing times, metaheuristics are general templates that can easily be tailored to address a wide range of problems. They have been shown to provide near-optimal solutions in reasonable computing times to problems for which traditional methods are not applicable [10]. Moreover, as we can see in [11], the tendency to use a hybrid method to solve some type of recent problem, such as those that COVID-19 has brought with it, has recently proven its effectiveness. In one of the important studies of these two main topics, we are interested in exploring the integration to use ML into metaheuristics, in terms to enhance any of the characteristics or attributes of those algorithms: the solutions, performance, or time to get the results. In this way, we propose to use an unsupervised machine learning technique that let us learning in the search space of the metaheuristic, exploiting the characteristic of their attributes that let us use to enhance the metaheuristic parameters in an online way: spatial clustering based on noise application density (DBSCAN) is one of those techniques that gathers those characteristics. We propose to use noise clustering to associate with the abandon probability and the solutions clustering result for determining the number of nests of the cuckoo search algorithm (CSA). They are studies that include some hybrid propose to enhance the CSA whit an ML technique, as in [12], when the authors present a Kmean technique to determinate the discrete parameters to improve the CSA. In [13], the contribution is a hybrid method with a sin-cousin algorithm to enhance the search space to be used on the CSA. Other different scenarios, but similar cases are in [14], they present a CuckooVina, a combination of cuckoo search and differential evolution in a conformational search method. Thus, some studies propose a hybridization to enhance CSA, but they do not respond to our search to find some ML technique that allows enriching the self-adaptation capabilities of CSA in the way we present.

To illustrate and prove our approach, we apply an improvement of the cuckoo search algorithm with a self-adaptive capability (SACSADBSCAN). This approach was tested on the set covering problem, whose goal is to cover a range of needs at the lowest cost and is a widely used problem to demonstrate research.

The remainder of the paper is structured as follows: in Section 2, we introduce the theoretical background on the optimization and clustering algorithms that have been integrated into the proposed approach, then, in Section 3, we present the original CSA, and his parameters that need to be set to run, moreover we introduce to the DBSCAN algorithm, how is the characteristic operation of this algorithm, and how we propose to exploit them in the metaheuristics. Following this, in Section 4 we present the integration of DBSCAN on CSA, how parameter values are configured in the execution of our approach. Then, in Section 5 experimental results and discussions are shown. Finally, we conclude and suggest some lines of future research in Section 6.

2. Related Work

As previously mentioned, parameter adjustment of metaheuristics is a complex task and is considered an optimization problem itself, in many cases, this depends on the try-error test to find a good combination of parameters. In this scenario, some studies use different ways, many of those with a mathematical formula to vary the values of the parameters of the CSA.

For example, to determine the step size $\alpha$ and $Pa$ parameter of the Cuckoo Search, Ref. [15] uses a mathematical formula that depends on the range of those minimal and maximal values of those parameters and the number of iterations of the algorithm. With that idea, the target is to use a bigger target area, also those neighboring areas. In [16], authors propose a self-adaptive step size in his studies to vary the $\alpha$ parameter according to the fitness in each iteration, applying a mathematical formula to get the value to set. In addition, Ref. [17] vary the $\alpha$ and $Pa$ values according to their propose, $\alpha$ changes his value according to the algorithm iterations, meanwhile $Pa$ vary randomly depending on the dimension map of the problem. Similar studies are in [18] where Q-Learning is used to set the step size value of the cuckoo search algorithm in a self-adaptive way.

Another study case of adaptive CSA is [19], where the author varies the step size and the abandon probability considering the historical values of them, including their best and worst values to determine the new one. Following with the vary in population, in [20] the authors divide the cuckoo population between making an analysis to them and improve the results.

In another case, such as in [21] use a hybrid algorithm to set the values of their parameters and reduce the population, when the diversity of population decreases, the population is reduced, allowing the algorithm to diversify in subsequent iterations.

As we can see, the machine learning techniques that we have cited up to this point are used to determine a static value in the metaheuristic parameters, which is used as it is the initial value. However, it is used to vary the value of the metaheuristic parameters. During its execution, it is seen minimally in the literature.

In other terms, machine learning has also been applied to the metaheuristic to compact the heuristic space through a clustering procedure [22] and artificial networks [23]. These manuscripts use forecasting and classification. In deep, machine learning integrates with metaheuristics is proposed to forecast different classes of problems, such as electric load [24], economic recessions [25], optimise the low-carbon flexible job shop scheduling problem [26] and other industrial problems [27]. As we mentioned, several dataset groups can be classified by employing machine learning. In this context, we can found image classification problem [28], electronic noise classification of bacterial food bones pathogens [29], urban management [30] and to mention some recently studies. Finally, in [31] the authors use the DBSCAN algorithm to make a binarization strategy and transform the continuous search space to a binary one.

Although machine learning has a big presence in the metaheuristic discipline, its use to set values is quite limited. The use to vary, in an online way, the value of the parameters of the metaheuristic is an area to dig deeper. In this way, we believe that the self-adaptive with DBSCAN, beyond the enhancement of metaheuristics, there seems to be a lot of scope for exploration, using a machine learning technique like DBSCAN with his characteristic we can make an analysis, and thanks to that, change the values of the parameters, is the aim pursued in this work.

3. Theoretical Background

3.1. Cuckoo Search Algorithm

Metaheuristics belong to a class of approximation methods. These are types of higher-level general-purpose algorithms that can be used for a wide range of different types of problems. They have been widely used to solve large-scale combinatorial optimization problems within acceptable computational time, but they do not guarantee optimum solutions [32].

There are a lot of metaheuristics. A full set of all nature-inspired algorithms can be found in [33], and one of them is CSA, which has several study cases. CSA [34] is inspired by the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other bird species. The steps of CSA are described below:

• Each cuckoo lays an egg at a time and drops it into a randomly selected nest.
• The best nests with high-quality eggs will be carried over to the next generations.
• The number of available host nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probability $Pa\in [0,1]$. In this case, a new random solution is generated.

Every new generation is determinate by Lévy flight [34], that is given by the Equation (2)

$$\begin{array}{cc}\hfill & x_i^d(t+1)=x_i^d+\alpha \otimes Levy\left(\beta \right)\hfill \\ \hfill & \forall i\in \{1,...,n\}\vee \forall d\in \{1,...,m\}\hfill \end{array}$$

(1)

where $x_i^d$ is the element d of a solution i at iteration t. $x_i^d(t+1)$ is a solution in the iteration $t+1$. $\alpha >0$ is the step size which should be related to the scales of the problem of interest, the upper ($Ub$) and lower bounds ($Lb$) that the problem need to be determinated, in this scenario values between 0 and 1.

$$\begin{array}{cc}\hfill & Levy\sim u=t^{\beta },(0<\beta <3)\hfill \end{array}$$

(2)

The Lévy flight represents a random walk while the random step length is drawn from a Lévy distribution which has an infinite variance with an infinite mean.

3.2. Density Based Spatial Clustering Application with Noise

Density-Based Spatial Clustering of Applications with Noise [35] is a popular data grouping algorithm that uses automated analysis techniques to similar group information together. It can be used to find clusters of any shape in a data collection with noise and outliers. Clusters are dense sections of data space separated by regions with a lower density of points.

The goal is to define dense areas that can be determined by the number of things in close proximity to a place. It is crucial to understand that DBSCAN has two important parameters that are required for work.

• Epsilon ($ϵ$): Determines how close points in a cluster can be seen to each other.
• Minimum points (MinPts): The minimal amount of points required to produce a concentrated area.

The underlying premise is that there must be at least a certain number of points in the vicinity of a particular radius for each given group. The $ϵ$ parameter determines the surrounding radius around a given point; every point x in the data set is tagged as a central point with a neighbor above or equal to MinPts; x is a limit point if the number of neighbors is less than MinPts. Finally, if a point is neither a center nor a border, it is referred to as a point of noise or an outlying point.

When users do not know much about the data being modified, this technique proves helpful in metaheuristics. See pseudo-code in Algorithms 1 and 2 to understand how this strategy works.

The computed groups are shown on the right side of Figure 1. Each iteration of the process adjusts the positions until the algorithm converges. It is important to mention that the noise points are those that do not remain in any cluster at the time of iterating the algorithm and we can visualize them as the white points around the clusters identified in the graph (Red, Yellow, Blue and Green), all these noise points are grouped into a single cluster, and in this way all the points will belong to a specific cluster.

Algorithm 1: Cuckoo Search pseudo-code.

Algorithm 2: DBSCAN pseudo-code.

4. Proposed Approach: Integrating DBSCAN in CSA

In this section, we describe how DBSCAN was integrated on CSA, in addition to all the keys elements for this technique to work. We add a shortcode at the end of each iteration to decide whether intervention is appropriate to implement parameter control intervention and use the DBSCAN algorithm to calculate the values of those parameters (see Algorithm 3).

Algorithm 3: Integration CSA with DBSCAN.

In the following sections, we explain some relevant topics that are important to mention to understand our approach. Section 4.1 explains under which criteria the DBSCAN intervening in the metaheuristic to make an analysis on the search space and make the clustering on the solutions. Then, Section 4.2 indicates how the noise cluster is fundamental to determinate the abandon probabilities.

4.1. Free Execution Parameter

We decide to include a variable to control the moment to intervene the parameter values of the CSA to make the metaheuristic maintain its independence of executions and its specific behavior, in this case, we consider it prudent to set it on one hundred iterations of a free run. If these execution values are reached, then the algorithm performs a procedure to update the CSA parameter values, as is described in line 39 of the Algorithm 3.

4.2. Online Parameter Setting

As we mention in the previous section, the update of the values of the parameters in the CSA occurs when the value of the free execution parameter has reached its limit, then the DBSCAN algorithm can be run and used the generated clusters to infer in the parameter values of the metaheuristic. How to associate the parameters of the $Pa$ and $Nest$ is detailed in the following sections.

4.2.1. Probability Abandon Nest

To set the value of the probability of nest abandon, we use the value of the noise point obtained from the DCSCAN execution. That value indicates the points that are excluded from any cluster, so we can associate the number of noise points to make the metaheuristic consider those probabilities to explore new points of the search space. To make this possible, we associate the percentage number of noise points to set the abandon probabilities value. To let the metaheuristic keep the normal execution we use bounds between 10 to 40 percentage. That indicates, that if the noise points are more than 40% then, the $Pa$ values are set to 0.4, in the same way, if the noise point is less than 10%, the $Pa$ values are set to 0.1. In other cases, the values are set to the percentage values of noise points.

4.2.2. Number of Nests

To determine when varying the number of nests on the CSA, our approach considers keeping in memory the last best fitness value to compare with the new candidate solutions, if the best value of fitness does not vary on the fourth intervention of the DBSCAN, we consider that it is necessary to increase the number of nests, to amplify the search space. So, in this case, the number of nests increases in five on five every time this scenario occurs. On the other hand, if the global best value improves four times consecutively, then the number of nest decrease, eliminating the worst five.

4.3. Exploration Influence

During the tests, we realize that the CSA solution space converges to a single large cluster that considers all possible solutions while evaluating the executions, in some instances, certain clusters appear to have values compressed very close to each other. We deem it entirely appropriate in this scenario to have CSA explore the search space. For this, half of the cluster points are renovated to new random ones, allowing the metaheuristic to diversify the search space. The replacement criteria are according to the following Formula (3)

$$ExpInfl=\frac{{\displaystyle \sum _{ClusterSol=1}^{n}Fitnes{s}_{ClusterSol}}}{n}$$

(3)

where $ClusterSol$ corresponding to all solutions in the current cluster evaluates. $ExplInfl$ is evaluated with the best possible solution of the global population, if the absolute value of the difference between both is larger than one, then we renew a half-point on those clusters.

4.4. Set Covering Problem

Many studies use the Set Covering Problem (SCP) to represent real scenarios into the studies, as we can see in the area of management of crew on airlines [36], in the optimization of the location of the emergency buildings [37], manufacturing [38] and production [39]. As we can see, SCP is a classical combinatorial optimization problem. It is belong to the NP-hard class [40] and is formally defined as well: let $A=\left(a_{ij}\right)$ be a binary matrix with M-rows ($\forall i\in I=\{1,\dots ,M\}$) and N-columns ($\forall j\in J=\{1,\dots ,N\}$), and let $C=\left(c_j\right)$ be a vector representing the cost of each column j, assuming that $c_j>0,\forall j=\{1,...,N\}$. Then, it observes that a column j covers a row i if $a_{ij}=1$. Therefore, it has:

$$a_{ij}=\left\{\begin{array}{c}1,\mathrm{if}\mathrm{row}i\mathrm{can}\mathrm{be}\mathrm{covered}\mathrm{by}\mathrm{column}j\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.$$

The SCP entails identifying a group of materials that can be used to address a lot of purposes for the least amount of money. A feasible solution corresponds to a subset of columns in its matrix form, and the demands are associated with rows and regarded as constraints. The goal of the challenge is to find the columns that best cover all of the rows.

The Set Covering Problem identifies a low-cost subset S of columns that covers each row with at least one column from S. The SCP can be expressed using integer programming as follows:

$$\begin{array}{c}{\displaystyle \mathtt{minimize}\sum _{j=1}^n{c}_j{x}_j}\\ \\ \mathrm{subject}\mathrm{to}:\\ \\ {\displaystyle \sum _{j=1}^n{a}_{ij}{x}_j\ge 1\forall i\in I}\\ x_j\in \{0,1\}\forall j\in J\end{array}$$

(4)

Instances: We use 65 instances from Beasley’s OR-library, which are arranged into 11 sets, to evaluate the algorithm’s performance when solving the SCP. To represent the instances, we present on

Table 1. Instances taken from the Beasley’s OR-Library.

Instance Group	M	N	Cost Range	Density (%)	Best Known
4	200	1000	[1, 100]	2	Known
5	200	2000	[1, 100]	2	Known
6	200	1000	[1, 100]	5	Known
A	300	3000	[1, 100]	2	Known
B	300	3000	[1, 100]	5	Known
C	400	4000	[1, 100]	2	Known
D	400	4000	[1, 100]	5	Known
NRE	500	5000	[1, 100]	10	Unknown (except NRE.1)
NRF	500	5000	[1, 100]	20	Unknown (except NRF.1)
NRG	1000	10,000	[1, 100]	2	Unknown (except NRG.1)
NRH	1000	10,000	[1, 100]	5	Unknown

the following details: instance group, the number of rows M, number of columns N, the cost range, density (percentage of non-zeroes in the matrix).

Reducing the instance size of SCP: In [41] different pre-processing approaches have been proposed in particular to reduce the size of the SCP, with Column Domination and Column Inclusion being the most effective. These methods are used to accelerate the processing of the algorithm.

Column Domination is the process of removing unnecessary columns from a problem in such a way that the final solution is unaffected.

Steps:

• All the columns are ordered according to their cost in ascending order.
• If there are equal cost columns, these are sorted in descending order by the number of rows that the column j covers.
• Verify if the column j whose rows can be covered by a set of other columns with a cost less than $c_j$ (cost of the column j).
• It is said that column j is dominated and can be eliminated from the problem.

Column Inclusion: when the domination process has terminated, the process of inclusion is performed, which means if a row is covered only by one column, means that there is no best column to cover those rows, which implies, that column must be included in the optimal solution. All of this process will be included to data of instances and let the new solutions satisfying the constraints.

5. Experimental Results

To evaluate the performance of our proposal, we test the instances of the SCP [42], comparing the original CS with different configurations, versus the SACSDBCAN.

5.1. Methodology

To adequately evaluate the performance of metaheuristics, a performance analysis is required [43]. For this work, we compare the supplied best solution of the CSA to the best-known result of the benchmark retrieved from the OR-Library [44]. Figure 2 depicts the procedures involved in doing a thorough examination of the enhanced metaheuristic. We create objectives and recommendations for the experimental design to show that the proposed approach is a viable alternative for determining metaheuristic parameters. Then, as a vital indicator for assessing future results, we evaluate the best value. We use ordinal analysis and statistical testing to evaluate whether a strategy is significantly better in this circumstance. Lastly, we detail the hardware and software aspects that were used to replicate computational experiments, and we present all of the results in tables and graphs.

As a result, we conduct a contrast statistical test for each case, using the Kolmogorov–Smirnov–Lilliefors process [45] to measure sample autonomy and the Mann–Whitney–Wilcoxon [46] test to statistically evaluate the data, in Figure 3 we describe and determinate the organization.

The Kolmogorov-Smirnov-Lilliefors test allows us to assess sample independence by calculating the $Z_{MIN}$ or $Z_{MAX}$ (depending on whether the task is minimization or maximization) obtained from each instance’s 31 executions.

The relative percentage deviation is used to assess the results (RPD). The RPD value computes the difference between the objective value $Z_{m}in$ and the minimal best-known value $Z_{o}pt$ for each instance in our experiment, and it is determined as follows:

$$RPD=\left({\displaystyle \frac{Z_{min}-Z_{opt}}{Z_{opt}}}\right)$$

5.2. Set Covering Problem Results

Infrastructure: Java 1.8 was used to implement SACSDBSCAN. The Personal Computer (PC) has the common attributes: MacOS with a 2.7 GHz Intel Core i7 CPU and 16 GB of RAM.

Setup variables: The configuration for our suggested approach is shown in

Table 2. SACSDBSCAN Parameters for SCP.

Population Initial	Abandon Probability $\mathit{Pa}$	$\mathit{\alpha }$	Max Iterations	${\mathit{L}}_{\mathit{b}}$ and ${\mathit{U}}_{\mathit{b}}$
10	0.1–0.4	0.01	5000	0 and 1

below.

Sixty-five SCP instances were considered, each of which was run 31 times; the results are presented in

Table 3. Results of SACSDBSCAN for instance 4, 5, 6 and A.

Instance	SACSDBSCAN
	Nest = 10 to 50
	Pa = 0.1 to 0.45
	BKS	Fit	RDP	AVG	MIN	MAX	STD
4.1	429	429	0.00	429	429	430	0.352
4.2	512	512	0.00	513	512	518	1.552
4.3	514	516	0.00	517	516	520	1.407
4.4	494	494	0.00	495	494	497	0.775
4.5	512	512	0.00	512	512	514	0.828
4.6	560	560	0.00	560	560	564	1.121
4.7	430	430	0.00	430	430	433	0.799
4.8	492	492	0.00	495	492	499	2.640
4.9	641	641	0.00	646	641	653	3.680
4.10	514	514	0.00	514	514	516	0.743
5.1	253	253	0.00	253	253	254	0.516
5.2	302	303	0.00	306	303	312	3.355
5.3	226	226	0.00	227	226	229	1.345
5.4	242	242	0.00	243	242	245	1.234
5.5	211	211	0.00	211	211	212	0.458
5.6	213	213	0.00	213	213	213	0.000
5.7	293	293	0.00	293	293	296	0.816
5.8	288	288	0.00	288	288	289	0.258
5.9	279	279	0.00	280	279	281	0.632
5.10	265	265	0.00	266	265	268	1.047
6.1	138	138	0.00	141	138	144	1.807
6.2	146	146	0.00	148	146	150	1.598
6.3	145	145	0.00	148	145	148	1.060
6.4	131	131	0.00	131	131	133	0.737
6.5	161	161	0.00	163	161	167	1.751
A.1	253	254	0.00	256	254	259	1.234
A.2	252	254	0.01	256	254	259	1.502
A.3	232	232	0.00	234	232	236	1.060
A.4	234	235	0.00	238	235	242	2.586
A.5	236	236	0.00	237	236	238	0.640

and

Table 4. Results of SACSDBSCAN for instance B, C, D, NRE, NRF, NRG and NRH.

Instance	SACSDBSCAN
	Nest = 10 to 50
	Pa = 0.1 to 0.45
	BKS	Fit	RDP	AVG	MIN	MAX	STD
B.1	69	73	0.05	76	73	79	2.610
B.2	76	76	0.00	80	76	87	3.218
B.3	80	80	0.00	84	80	88	2.673
B.4	79	79	0.00	82	79	88	2.560
B.5	72	72	0.00	73	72	78	1.710
C.1	227	227	0.00	229	227	234	1.993
C.2	219	219	0.00	221	219	225	1.685
C.3	243	243	0.00	246	243	250	2.131
C.4	219	219	0.00	221	219	224	1.309
C.5	215	215	0.00	216	215	217	0.799
D.1	60	61	0.02	64	61	66	1.414
D.2	66	70	0.06	70	70	70	0.000
D.3	72	76	0.05	79	76	82	1.859
D.4	62	63	0.02	66	63	67	1.486
D.5	61	63	0.03	64	63	66	0.910
NRE.1	29	29	0.00	30	29	30	0.516
NRE.2	30	31	0.03	32	31	34	0.862
NRE.3	27	28	0.04	29	28	31	0.976
NRE.4	28	30	0.07	31	30	33	0.834
NRE.5	28	29	0.03	29	29	30	0.516
NRF.1	14	15	0.07	15	15	16	0.458
NRF.2	15	15	0.00	16	15	17	0.458
NRF.3	14	16	0.13	17	16	17	0.507
NRF.4	14	15	0.07	15	15	16	0.389
NRF.5	15	15	0.00	15	15	16	0.258
NRG.1	176	188	0.06	194	188	197	2.789
NRG.2	154	160	0.04	165	160	167	2.282
NRG.3	166	180	0.08	182	180	183	0.862
NRG.4	168	177	0.05	183	177	186	2.728
NRG.5	168	181	0.07	183	181	184	0.799
NRH.1	63	69	0.09	71	69	72	0.976
NRH.2	63	66	0.05	67	66	67	0.414
NRH.3	59	66	0.11	67	66	68	0.738
NRH.4	63	63	0.00	64	63	66	1.528
NRH.5	55	59	0.07	60	59	61	0.707

.

Overview: The algorithms are ranked in order of $Z_{min}$ achieved. The instances that obtained $Z_{min}$ are also displayed.

• Score
  • CS got 38/65 $Z_{min}$.
  • SACSDBSCAN got 10/65 $Z_{min}$.
• The two algorithm got $Z_{min}$ in the same instances: 5.4–6.4–A.4–B.2–B.3–B.4–B.5–D.3–D.5–NRE.1–NRE.2–NRE.3–NRF.1–NRF.3–NRF.4–NRF.5–NRH.5

As can be seen in the results of the algorithms that solved SCP, we compare the distribution of the samples of each instance using a violin plot, which allows us to observe the entire distribution of the data. We provide and discuss the most difficult instances of each group to create a resume of all the instances below (4.10, 5.10, 6.5, A.5, B.5, C.5, D.5, NRE.5, NRF.5, NRG.5 and NRH.5):

In [47], the authors display the results obtained, the information is detailed, and the configuration that they use. The information is organized as follows: MIN: the minimum value reached, MAX: the maximum value reached, AVG: the average value, BKS: the best-known solution, RPD is determined by Equation (43), and lastly the average of the fitness obtained.

The first method was the standard cuckoo search algorithm with various settings, and the second was SACSDBSCAN, as previously indicated.

Table 5. Comparison results for set of groups 4, 5, 6, A. SACSDBSCAN v/s CS Nest = 20, 30.

Instance	BKS	SACSDBSCAN			CS															CS
		Nest = 10 to 50			Nest = 20															Nest = 30
		Pa = 0.1 to 0.45			Pa = 0.1			Pa = 0.15			Pa = 0.25			Pa = 0.35			Pa = 0.45			Pa = 0.1			Pa = 0.15			Pa = 0.25			Pa = 0.35			Pa = 0.45
		Fit	RPD	AVG	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD
4.1	429	429	0	429	468	476	9.09	471	476	9.79	471	476	9.79	468	476	9.09	468	475	9.09	470	476	9.56	470	476	9.56	470	476	9.56	474	477	10.49	471	476	9.79
4.2	512	512	0	513	604	612	17.97	608	613	18.75	597	614	16.60	597	613	16.60	597	613	16.60	597	613	16.60	606	612	18.36	597	613	16.60	597	612	16.60	597	613	16.60
4.3	516	516	0	517	578	586	12.45	575	587	11.87	578	585	12.45	575	586	11.87	575	585	11.87	578	585	12.45	575	585	11.87	568	584	10.51	575	586	11.87	575	584	11.87
4.4	494	494	0	495	568	581	14.98	560	581	13.36	571	581	15.59	571	581	15.59	560	580	13.36	560	579	13.36	560	579	13.36	568	580	14.98	560	581	13.36	568	579	14.98
4.5	512	512	0	512	575	615	12.30	592	618	15.63	601	619	17.38	592	617	15.63	594	616	16.02	591	613	15.43	594	614	16.02	596	617	16.41	592	615	15.63	598	620	16.80
4.6	560	560	0	560	643	653	14.82	637	651	13.75	637	652	13.75	637	652	13.75	643	652	14.82	637	651	13.75	637	650	13.75	643	652	14.82	637	651	13.75	637	651	13.75
4.7	430	430	0	430	518	527	20.47	518	527	20.47	518	527	20.47	518	526	20.47	518	526	20.47	518	524	20.47	518	526	20.47	514	526	19.53	515	525	19.77	518	526	20.47
4.8	492	492	0	495	552	558	12.20	546	557	10.98	547	557	11.18	552	558	12.20	552	558	12.20	546	555	10.98	546	555	10.98	544	557	10.57	546	556	10.98	550	556	11.79
4.9	641	641	0	646	763	788	19.03	763	787	19.03	763	788	19.03	763	785	19.03	763	788	19.03	762	785	18.88	752	782	17.32	763	786	19.03	763	787	19.03	763	787	19.03
4.10	514	514	0	514	588	608	14.40	588	609	14.40	588	608	14.40	588	607	14.40	588	608	14.40	588	609	14.40	588	609	14.40	588	607	14.40	588	607	14.40	595	608	15.76
5.1	253	253	0	253	303	311	19.76	305	310	20.55	303	310	19.76	303	311	19.76	305	311	20.55	300	309	18.58	302	310	19.37	303	310	19.76	303	309	19.76	303	310	19.76
5.2	302	303	0.003	306	374	382	23.84	374	383	23.84	374	382	23.84	370	383	22.52	374	381	23.84	369	383	22.19	370	382	22.52	374	382	23.84	368	383	21.85	374	382	23.84
5.3	226	226	0	227	257	262	13.72	259	263	14.60	257	262	13.72	257	262	13.72	257	263	13.72	257	262	13.72	259	263	14.60	257	262	13.72	257	263	13.72	257	262	13.72
5.4	242	242	0	243	271	275	11.98	269	275	11.16	268	274	10.74	271	274	11.98	271	275	11.98	270	274	11.57	268	274	10.74	271	275	11.98	271	274	11.98	268	274	10.74
5.5	211	211	0	211	254	258	20.38	257	260	21.80	254	258	20.38	256	259	21.33	252	259	19.43	252	258	19.43	252	258	19.43	254	258	20.38	251	259	18.96	252	258	19.43
5.6	213	213	0	213	256	268	20.19	260	269	22.07	259	268	21.60	260	268	22.07	256	268	20.19	256	267	20.19	256	267	20.19	265	269	24.41	256	267	20.19	260	269	22.07
5.7	293	293	0	293	344	352	17.41	345	352	17.75	341	352	16.38	344	352	17.41	344	352	17.41	341	350	16.38	342	351	16.72	343	351	17.06	343	351	17.06	341	351	16.38
5.8	288	288	0	288	355	365	23.26	359	365	24.65	355	364	23.26	357	365	23.96	359	365	24.65	355	364	23.26	359	365	24.65	355	363	23.26	355	363	23.26	355	364	23.26
5.9	279	279	0	280	338	356	21.15	346	356	24.01	340	356	21.86	338	357	21.15	343	356	22.94	338	355	21.15	338	355	21.15	338	355	21.15	343	356	22.94	338	356	21.15
5.10	265	265	0	266	304	310	14.72	306	310	15.47	307	310	15.85	306	310	15.47	308	310	16.23	305	310	15.09	305	310	15.09	306	310	15.47	305	310	15.09	306	310	15.47
6.1	138	138	0	141	169	176	22.46	173	177	25.36	169	176	22.46	169	176	22.46	169	176	22.46	169	176	22.46	169	176	22.46	169	177	22.46	169	175	22.46	169	176	22.46
6.2	146	146	0	148	168	175	15.07	168	174	15.07	168	176	15.07	168	175	15.07	168	175	15.07	168	174	15.07	168	173	15.07	168	174	15.07	168	174	15.07	168	174	15.07
6.3	145	145	0	148	166	175	14.48	169	176	16.55	169	176	16.55	171	176	17.93	169	176	16.55	166	175	14.48	169	174	16.55	169	175	16.55	166	174	14.48	166	174	14.48
6.4	131	131	0	131	142	149	8.40	142	149	8.40	147	149	12.21	147	149	12.21	142	149	8.40	142	149	8.40	145	149	10.69	146	149	11.45	142	149	8.40	142	149	8.40
6.5	161	161	0	163	196	205	21.74	195	205	21.12	202	206	25.47	196	205	21.74	196	205	21.74	200	205	24.22	199	204	23.60	196	204	21.74	196	204	21.74	198	204	22.98
A.1	253	254	0.004	256	277	279	9.49	274	279	8.30	274	279	8.30	274	279	8.30	277	279	9.49	274	278	8.30	277	279	9.49	274	278	8.30	274	278	8.30	274	279	8.30
A.2	252	254	0.01	256	315	318	25.00	310	317	23.02	310	318	23.02	310	317	23.02	313	317	24.21	314	317	24.60	310	316	23.02	314	317	24.60	312	318	23.81	311	317	23.41
A.3	232	232	0	234	263	267	13.36	256	266	10.34	263	266	13.36	258	266	11.21	263	267	13.36	258	266	11.21	261	266	12.50	258	266	11.21	263	266	13.36	262	266	12.93
A.4	234	235	0.004	238	274	277	17.09	273	278	16.67	272	277	16.24	274	277	17.09	271	277	15.81	271	276	15.81	273	277	16.67	275	277	17.52	271	277	15.81	271	276	15.81
A.5	236	236	0	237	271	276	14.83	271	276	14.83	266	276	12.71	269	276	13.98	271	276	14.83	267	275	13.14	266	274	12.71	271	276	14.83	267	275	13.14	271	275	14.83

,

Table 6. Comparison results for set of groups B, C, D, NRE, NRF, NRG, NRH. SACSDBSCAN v/s CS Nest = 20, 30.

Instance	BKS	SACSDBSCAN			CS															CS
		Nest = 10 to 50			Nest = 20															Nest = 30
		Pa = 0.1 to 0.45			Pa = 0.1			Pa = 0.15			Pa = 0.25			Pa = 0.35			Pa = 0.45			Pa = 0.1			Pa = 0.15			Pa = 0.25			Pa = 0.35			Pa = 0.45
		Fit	RPD	AVG	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD
B.1	69	73	0.04	76	88	91	27.54	88	91	27.54	89	91	28.99	87	91	26.09	88	92	27.54	88	91	27.54	89	91	28.99	88	91	27.54	89	92	28.99	87	92	26.09
B.2	76	76	0	80	96	98	26.32	95	99	25.00	96	99	26.32	94	99	23.68	96	99	26.32	94	98	23.68	95	98	25.00	94	99	23.68	95	99	25.00	95	99	25.00
B.3	80	80	0	84	91	93	13.75	91	93	13.75	91	93	13.75	91	93	13.75	91	93	13.75	91	92	13.75	90	92	12.50	91	93	13.75	91	93	13.75	91	93	13.75
B.4	79	79	0	82	102	107	29.11	102	107	29.11	101	107	27.85	104	108	31.65	104	108	31.65	100	106	26.58	101	107	27.85	100	107	26.58	102	107	29.11	102	106	29.11
B.5	72	72	0	73	86	89	19.44	87	90	20.83	86	90	19.44	87	89	20.83	85	90	18.06	86	89	19.44	86	89	19.44	88	89	22.22	85	89	18.06	83	89	15.28
C.1	227	227	0	229	267	270	17.62	264	269	16.30	267	270	17.62	267	270	17.62	265	269	16.74	265	269	16.74	264	268	16.30	267	270	17.62	264	269	16.30	265	269	16.74
C.2	219	219	0	221	266	273	21.46	268	273	22.37	267	273	21.92	267	274	21.92	266	272	21.46	268	272	22.37	266	272	21.46	267	272	21.92	266	271	21.46	269	272	22.83
C.3	243	243	0	246	308	313	26.75	307	313	26.34	308	314	26.75	306	313	25.93	309	314	27.16	306	312	25.93	309	313	27.16	306	313	25.93	306	313	25.93	305	313	25.51
C.4	219	219	0	221	263	267	20.09	264	266	20.55	262	267	19.63	262	266	19.63	263	267	20.09	262	266	19.63	262	266	19.63	263	266	20.09	263	266	20.09	263	267	20.09
C.5	215	215	0	216	248	256	15.35	248	256	15.35	250	256	16.28	248	257	15.35	251	256	16.74	250	256	16.28	251	255	16.74	248	255	15.35	248	255	15.35	251	256	16.74
D.1	60	61	0.02	64	70	72	16.67	70	72	16.67	70	73	16.67	70	73	16.67	71	73	18.33	71	72	18.33	70	72	16.67	70	72	16.67	70	72	16.67	70	72	16.67
D.2	66	70	0.06	70	79	80	19.70	79	80	19.70	78	80	18.18	79	81	19.70	79	81	19.70	78	80	18.18	77	80	16.67	79	80	19.70	79	80	19.70	79	80	19.70
D.3	72	76	0.05	79	86	88	19.44	87	89	20.83	87	89	20.83	87	89	20.83	85	88	18.06	86	88	19.44	86	88	19.44	86	88	19.44	84	88	16.67	87	89	20.83
D.4	62	63	0.02	66	72	73	16.13	71	72	14.52	70	72	12.90	69	73	11.29	71	73	14.52	69	72	11.29	70	72	12.90	71	72	14.52	70	72	12.90	70	73	12.90
D.5	61	63	0.03	64	70	71	14.75	70	71	14.75	71	72	16.39	71	72	16.39	70	72	14.75	70	71	14.75	70	71	14.75	70	71	14.75	70	71	14.75	70	72	14.75
NRE.1	29	29	0	30	31	33	6.90	32	33	10.34	31	33	6.90	32	33	10.34	32	33	10.34	32	33	10.34	32	33	10.34	31	32	6.90	32	33	10.34	32	33	10.34
NRE.2	30	31	0.03	32	37	38	23.33	35	38	16.67	36	38	20	37	38	23.33	36	38	20	36	37	20	36	38	20	37	38	23.33	36	38	20	36	38	20
NRE.3	27	28	0.04	29	35	36	29.63	35	36	29.63	34	36	25.93	34	36	25.93	34	36	25.93	35	36	29.63	34	36	25.93	35	36	29.63	35	36	29.63	35	36	29.63
NRE.4	28	30	0.07	31	34	35	21.43	34	35	21.43	35	35	25.00	34	35	21.43	35	36	25.00	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43
NRE.5	28	29	0.03	29	33	35	17.86	33	35	17.86	34	35	21.43	33	35	17.86	33	35	17.86	32	35	14.29	32	35	14.29	33	35	17.86	34	35	21.43	32	35	14.29
NRF.1	14	15	0.07	15	18	18	28.57	17	18	21.43	18	18	28.57	18	18	28.57	18	18	28.57	18	18	28.57	18	18	28.57	18	18	28.57	18	18	28.57	17	18	21.43
NRF.2	15	15	0	16	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20
NRF.3	14	16	0.13	17	20	21	42.86	19	21	35.71	20	21	42.86	20	21	42.86	20	21	42.86	19	21	35.71	20	21	42.86	19	21	35.71	19	21	35.71	19	21	35.71
NRF.4	14	15	0.07	15	18	19	28.57	19	20	35.71	19	20	35.71	19	19	35.71	19	20	35.71	18	19	28.57	18	19	28.57	18	19	28.57	18	19	28.57	18	19	28.57
NRF.5	15	15	0	15	16	17	6.67	16	17	6.67	16	17	6.67	16	17	6.67	16	17	6.67	16	16	6.67	16	17	6.67	16	16	6.67	16	17	6.67	16	17	6.67
NRG.1	176	188	0.06	194	230	235	30.68	228	235	29.55	230	235	30.68	232	235	31.82	230	236	30.68	229	234	30.11	228	236	29.55	229	235	30.11	228	235	29.55	229	235	30.11
NRG.2	154	160	0.04	165	187	192	21.43	189	192	22.73	187	192	21.43	187	192	21.43	188	193	22.08	187	191	21.43	185	191	20.13	188	191	22.08	187	192	21.43	188	192	22.08
NRG.3	166	180	0.08	182	198	200	19.28	198	200	19.28	198	200	19.28	197	200	18.67	196	199	18.07	198	200	19.28	198	199	19.28	196	199	18.07	198	200	19.28	197	199	18.67
NRG.4	168	177	0.05	183	216	220	28.57	213	221	26.79	217	221	29.17	216	221	28.57	215	222	27.98	216	220	28.57	214	219	27.38	216	220	28.57	214	219	27.38	217	221	29.17
NRG.5	168	181	0.07	183	219	223	30.36	218	224	29.76	218	223	29.76	220	224	30.95	212	224	26.19	219	223	30.36	218	223	29.76	221	224	31.55	220	224	30.95	216	224	28.57
NRH.1	63	69	0.09	71	82	84	30.16	82	84	30.16	83	85	31.75	82	85	30.16	83	85	31.75	82	85	30.16	82	85	30.16	83	84	31.75	82	85	30.16	82	85	30.16
NRH.2	63	66	0.05	67	79	82	25.40	79	82	25.40	77	82	22.22	80	82	26.98	79	82	25.40	78	82	23.81	79	81	25.40	78	82	23.81	79	82	25.40	80	82	26.98
NRH.3	59	66	0.11	67	74	77	25.42	76	77	28.81	76	77	28.81	76	77	28.81	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12
NRH.4	63	63	0	64	74	76	17.46	74	76	17.46	75	76	19.05	72	76	14.29	74	76	17.46	74	76	17.46	72	75	14.29	74	76	17.46	74	76	17.46	74	76	17.46
NRH.5	55	59	0.07	60	67	69	21.82	67	69	21.82	67	69	21.82	67	69	21.82	67	69	21.82	67	69	21.82	67	69	21.82	67	69	21.82	66	69	20	67	69	21.82

,

Table 7. Comparison results for set of groups 4, 5, 6, A. SACSDBSCAN v/s CS Nest = 40, 50.

Instance	BKS	SACSDBSCAN			CS															CS
		Nest = 10 to 50			Nest = 40															Nest = 50
		Pa = 0.1 to 0.45			Pa = 0.1			Pa = 0.15			Pa = 0.25			Pa = 0.35			Pa = 0.45			Pa = 0.1			Pa = 0.15			Pa = 0.25			Pa = 0.35			Pa = 0.45
		Fit	RPD	AVG	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD
4.1	429	429	0	429	470	475	9.56	471	476	9.79	467	476	8.86	471	476	9.79	469	475	9.32	470	475	9.56	470	475	9.56	470	475	9.56	471	476	9.79	470	476	9.56
4.2	512	512	0	513	597	611	16.60	597	611	16.60	597	611	16.60	597	611	16.60	597	612	16.60	602	611	17.58	597	611	16.60	597	612	16.60	606	613	18.36	597	611	16.60
4.3	516	516	0	517	578	585	12.45	575	585	11.87	575	585	11.87	575	584	11.87	568	585	10.51	575	584	11.87	574	584	11.67	575	584	11.87	575	583	11.87	565	584	9.92
4.4	494	494	0	495	567	579	14.78	568	579	14.98	568	580	14.98	560	581	13.36	569	579	15.18	560	577	13.36	571	582	15.59	560	578	13.36	568	579	14.98	560	578	13.36
4.5	512	512	0	512	575	616	12.30	575	612	12.30	589	616	15.04	589	615	15.04	594	616	16.02	592	614	15.63	598	617	16.80	592	615	15.63	592	616	15.63	592	619	15.63
4.6	560	560	0	560	639	651	14.11	639	651	14.11	629	649	12.32	639	652	14.11	637	650	13.75	637	650	13.75	637	649	13.75	639	651	14.11	637	650	13.75	637	650	13.75
4.7	430	430	0	430	517	524	20.23	515	525	19.77	515	524	19.77	509	525	18.37	518	525	20.47	518	524	20.47	516	525	20	517	523	20.23	518	525	20.47	511	524	18.84
4.8	492	492	0	495	543	555	10.37	552	556	12.20	547	556	11.18	546	555	10.98	539	556	9.55	546	556	10.98	546	556	10.98	546	555	10.98	546	555	10.98	546	555	10.98
4.9	641	641	0	646	763	784	19.03	763	785	19.03	763	786	19.03	763	784	19.03	763	780	19.03	760	781	18.56	763	787	19.03	753	785	17.47	760	785	18.56	761	784	18.72
4.10	514	514	0	514	588	608	14.40	588	608	14.40	596	606	15.95	588	608	14.40	588	605	14.40	588	607	14.40	595	607	15.76	588	605	14.40	588	607	14.40	588	605	14.40
5.1	253	253	0	253	303	310	19.76	305	310	20.55	303	309	19.76	303	310	19.76	303	309	19.76	301	309	18.97	299	309	18.18	303	310	19.76	303	309	19.76	299	309	18.18
5.2	302	303	0.003	306	374	381	23.84	374	381	23.84	373	383	23.51	371	381	22.85	371	381	22.85	369	381	22.19	369	381	22.19	370	382	22.52	373	380	23.51	371	381	22.85
5.3	226	226	0	227	257	262	13.72	257	262	13.72	257	262	13.72	257	262	13.72	257	262	13.72	257	262	13.72	257	262	13.72	257	261	13.72	256	262	13.27	259	262	14.60
5.4	242	242	0	243	270	274	11.57	268	274	10.74	268	274	10.74	268	274	10.74	268	273	10.74	271	274	11.98	268	273	10.74	270	275	11.57	270	274	11.57	270	274	11.57
5.5	211	211	0	211	254	259	20.38	252	258	19.43	252	259	19.43	255	259	20.85	255	259	20.85	252	258	19.43	255	259	20.85	252	258	19.43	253	257	19.91	252	258	19.43
5.6	213	213	0	213	265	269	24.41	260	268	22.07	256	268	20.19	265	268	24.41	256	268	20.19	256	267	20.19	260	268	22.07	260	269	22.07	259	268	21.60	256	268	20.19
5.7	293	293	0	293	341	351	16.38	341	350	16.38	341	350	16.38	341	351	16.38	341	351	16.38	341	349	16.38	346	352	18.09	341	351	16.38	343	350	17.06	341	350	16.38
5.8	288	288	0	288	355	363	23.26	355	361	23.26	356	364	23.61	355	364	23.26	357	364	23.96	355	363	23.26	355	362	23.26	354	363	22.92	352	362	22.22	359	363	24.65
5.9	279	279	0	280	338	354	21.15	341	355	22.22	338	354	21.15	343	354	22.94	326	356	16.85	346	354	24.01	337	356	20.79	338	355	21.15	343	356	22.94	338	355	21.15
5.10	265	265	0	266	305	309	15.09	306	310	15.47	306	309	15.47	305	310	15.09	306	310	15.47	303	309	14.34	302	309	13.96	305	309	15.09	302	310	13.96	305	309	15.09
6.1	138	138	0	141	165	176	19.57	168	174	21.74	165	174	19.57	167	175	21.01	169	175	22.46	169	176	22.46	162	175	17.39	169	175	22.46	169	175	22.46	167	174	21.01
6.2	146	146	0	148	168	175	15.07	168	174	15.07	168	174	15.07	168	173	15.07	168	175	15.07	168	172	15.07	168	172	15.07	168	174	15.07	168	172	15.07	168	172	15.07
6.3	145	145	0	148	166	174	14.48	165	174	13.79	169	175	16.55	169	175	16.55	171	175	17.93	168	173	15.86	166	173	14.48	169	174	16.55	168	173	15.86	169	174	16.55
6.4	131	131	0	131	142	148	8.40	142	149	8.40	145	149	10.69	142	149	8.40	146	149	11.45	142	148	8.40	142	149	8.40	148	149	12.98	140	148	6.87	147	149	12.21
6.5	161	161	0	163	196	203	21.74	196	204	21.74	199	204	23.60	196	204	21.74	196	205	21.74	198	204	22.98	196	204	21.74	196	203	21.74	199	204	23.60	196	204	21.74
A.1	253	254	0.004	256	273	278	7.91	274	278	8.30	278	279	9.88	273	278	7.91	273	278	7.91	277	279	9.49	274	278	8.30	273	278	7.91	274	278	8.30	274	278	8.30
A.2	252	254	0.01	256	308	317	22.22	307	316	21.83	313	317	24.21	310	316	23.02	312	317	23.81	310	316	23.02	311	316	23.41	310	316	23.02	310	316	23.02	310	316	23.02
A.3	232	232	0	234	263	266	13.36	264	266	13.79	258	266	11.21	258	266	11.21	258	265	11.21	258	266	11.21	261	266	12.50	264	266	13.79	262	266	12.93	258	266	11.21
A.4	234	235	0.004	238	272	276	16.24	271	276	15.81	274	277	17.09	273	276	16.67	273	277	16.67	271	276	15.81	271	276	15.81	274	277	17.09	272	276	16.24	271	276	15.81
A.5	236	236	0	237	267	275	13.14	269	275	13.98	270	274	14.41	271	274	14.83	270	275	14.41	271	274	14.83	269	274	13.98	271	275	14.83	267	274	13.14	266	274	12.71

and

Table 8. Comparison results for set of groups B, C, D, NRE, NRF, NRG, NRH. SACSDBSCAN v/s CS Nest = 40, 50.

Instance	BKS	SACSDBSCAN			CS															CS
		Nest = 10 to 50			Nest = 40															Nest = 50
		Pa = 0.1 to 0.45			Pa = 0.1			Pa = 0.15			Pa = 0.25			Pa = 0.35			Pa = 0.45			Pa = 0.1			Pa = 0.15			Pa = 0.25			Pa = 0.35			Pa = 0.45
		Fit	RPD	AVG	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD	Fit	AVG	RPD
B.1	69	73	0.04	76	88	91	27.54	88	91	27.54	87	91	26.09	88	91	27.54	87	90	26.09	87	90	26.09	87	90	26.09	88	91	27.54	87	91	26.09	87	91	26.09
B.2	76	76	0	80	94	98	23.68	94	98	23.68	94	98	23.68	94	98	23.68	95	98	25.00	96	98	26.32	94	97	23.68	94	98	23.68	94	98	23.68	94	98	23.68
B.3	80	80	05	84	90	92	12.50	91	92	13.75	89	92	11.25	91	92	13.75	91	93	13.75	91	92	13.75	90	92	12.50	91	92	13.75	90	92	12.50	91	93	13.75
B.4	79	79	0	82	102	106	29.11	102	107	29.11	103	107	30.38	100	106	26.58	101	106	27.85	102	106	29.11	100	106	26.58	101	106	27.85	101	106	27.85	101	106	27.85
B.5	72	72	0	73	87	89	20.83	85	89	18.06	85	89	18.06	85	89	18.06	86	89	19.44	85	89	18.06	86	89	19.44	86	89	19.44	86	89	19.44	86	89	19.44
C.1	227	227	0	229	267	269	17.62	263	269	15.86	264	269	16.30	265	269	16.74	264	269	16.30	265	269	16.74	265	268	16.74	264	269	16.30	267	269	17.62	263	268	15.86
C.2	219	219	0	221	268	272	22.37	264	272	20.55	265	271	21.00	266	271	21.46	260	272	18.72	268	272	22.37	262	271	19.63	266	271	21.46	267	271	21.92	267	272	21.92
C.3	243	243	0	246	308	312	26.75	307	312	26.34	306	312	25.93	299	312	23.05	309	313	27.16	308	313	26.75	302	312	24.28	306	312	25.93	302	311	24.28	304	312	25.10
C.4	219	219	0	221	263	266	20.09	263	266	20.09	263	266	20.09	263	266	20.09	264	267	20.55	262	266	19.63	262	266	19.63	262	265	19.63	263	265	20.09	263	266	20.09
C.5	215	215	0	216	247	254	14.88	248	255	15.35	247	253	14.88	249	255	15.81	251	256	16.74	250	254	16.28	248	254	15.35	248	255	15.35	248	254	15.35	248	255	15.35
D.1	60	61	0.02	64	70	72	16.67	70	72	16.67	71	72	18.33	70	72	16.67	69	72	15.00	70	72	16.67	70	72	16.67	70	72	16.67	70	72	16.67	70	72	16.67
D.2	66	70	0.06	70	78	80	18.18	78	80	18.18	78	80	18.18	78	80	18.18	78	80	18.18	78	80	18.18	78	80	18.18	78	80	18.18	78	80	18.18	78	80	18.18
D.3	72	76	0.05	79	86	88	19.44	86	88	19.44	86	88	19.44	85	88	18.06	86	88	19.44	85	88	18.06	86	88	19.44	87	88	20.83	86	88	19.44	84	88	16.67
D.4	62	63	0.02	66	71	72	14.52	71	72	14.52	71	72	14.52	70	72	12.90	71	72	14.52	70	72	12.90	70	72	12.90	71	72	14.52	71	72	14.52	71	72	14.52
D.5	61	63	0.03	64	70	71	14.75	70	72	14.75	70	71	14.75	70	71	14.75	70	71	14.75	70	71	14.75	69	71	13.11	70	71	14.75	70	71	14.75	70	71	14.75
NRE.1	29	29	0	30	32	32	10.34	32	33	10.34	32	33	10.34	31	33	6.90	31	33	6.90	32	32	10.34	32	33	10.34	32	32	10.34	31	33	6.90	32	32	10.34
NRE.2	30	31	0.03	32	36	38	20	36	38	20	36	38	20	36	38	20	36	38	20	37	38	23.33	36	38	20	36	38	20	36	38	20	37	38	23.33
NRE.3	27	28	0.04	29	34	36	25.93	35	36	29.63	34	36	25.93	35	36	29.63	34	36	25.93	34	36	25.93	34	36	25.93	34	36	25.93	34	36	25.93	34	36	25.93
NRE.4	28	30	0.07	31	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43	34	35	21.43
NRE.5	28	29	0.03	29	32	34	14.29	32	35	14.29	33	35	17.86	32	35	14.29	33	35	17.86	33	35	17.86	33	34	17.86	33	34	17.86	32	34	14.29	33	34	17.86
NRF.1	14	15	0.07	15	17	18	21.43	18	18	28.57	18	18	28.57	18	18	28.57	17	18	21.43	17	18	21.43	18	18	28.57	18	18	28.57	18	18	28.57	17	18	21.43
NRF.2	15	15	0	16	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20	18	19	20
NRF.3	14	16	0.13	17	19	21	35.71	19	21	35.71	19	21	35.71	20	21	42.86	20	21	42.86	20	21	42.86	20	21	42.86	20	21	42.86	19	21	35.71	20	21	42.86
NRF.4	14	15	0.07	15	18	19	28.57	18	19	28.57	18	19	28.57	18	19	28.57	18	19	28.57	18	19	28.57	18	19	28.57	19	19	35.71	18	19	28.57	18	19	28.57
NRF.5	15	15	0	15	16	16	6.67	16	16	6.67	16	16	6.67	16	16	6.67	16	17	6.67	16	16	6.67	16	16	6.67	16	16	6.67	16	16	6.67	16	16	6.67
NRG.1	176	188	0.06	194	228	234	29.55	229	234	30.11	227	234	28.98	229	234	30.11	230	234	30.68	232	234	31.82	229	234	30.11	230	234	30.68	227	234	28.98	231	235	31.25
NRG.2	154	160	0.04	165	186	191	20.78	181	190	17.53	188	191	22.08	187	191	21.43	187	191	21.43	187	191	21.43	187	191	21.43	186	191	20.78	187	191	21.43	185	190	20.13
NRG.3	166	180	0.08	182	198	199	19.28	197	199	18.67	195	199	17.47	197	199	18.67	195	199	17.47	197	199	18.67	197	199	18.67	197	199	18.67	196	199	18.07	197	200	18.67
NRG.4	168	177	0.05	183	215	219	27.98	213	219	26.79	214	220	27.38	214	221	27.38	216	221	28.57	215	219	27.98	213	220	26.79	212	219	26.19	215	219	27.98	217	220	29.17
NRG.5	168	181	0.07	183	220	223	30.95	215	222	27.98	220	223	30.95	218	223	29.76	218	223	29.76	218	222	29.76	218	222	29.76	220	223	30.95	219	222	30.36	219	223	30.36
NRH.1	63	69	0.09	71	82	84	30.16	81	84	28.57	82	84	30.16	83	84	31.75	83	84	31.75	80	84	26.98	81	84	28.57	83	84	31.75	82	84	30.16	82	84	30.16
NRH.2	63	66	0.05	67	79	82	25.40	78	81	23.81	79	81	25.40	80	82	26.98	80	82	26.98	78	81	23.81	77	81	22.22	79	81	25.40	79	81	25.40	79	81	25.40
NRH.3	59	66	0.11	67	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12	75	77	27.12
NRH.4	63	63	0	64	73	75	15.87	73	75	15.87	74	76	17.46	74	75	17.46	73	75	15.87	73	75	15.87	73	75	15.87	74	75	17.46	73	75	15.87	74	75	17.46
NRH.5	55	59	0.07	60	67	68	21.82	67	68	21.82	67	68	21.82	67	68	21.82	67	69	21.82	67	68	21.82	67	68	21.82	67	68	21.82	67	68	21.82	65	68	18.18

show the behavior of our proposed algorithm versus the original algorithm. The best results are highlighted with underline and maroon color. For example, in the instance 4.1, the best reached solution by our proposal overcomes than the classical CS algorithm. The same strategy is used in all comparisons.

The distribution of the data in all instances (Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14), shows that the performance of our proposal is better than the traditional the cuckoo search optimizer. Concentrating the largest distribution of results in the optimal values, while in original CS they are visibly distant. For example, in the instance B.5, we can see that the distribution is better in our proposal, but at the same time, reflex the behavior to move the result on the best values in his executions, getting the center of the distribution in the best value quartile. Other instance that shows this behaviour can be observed in C.5. Here, our proposal generates again a large number of optimum results.

The scenario in D.5 and NRE.5, is similar in all of them, the behavior of the ASCSDBSCAN show that reach best results compare to CS. In instance NRF.5 we can see that the behavior of both algorithms obtains a similar figure, reflexing the results obtained from the nature of the sample data. In this scenario, we can rescue results in our proposal, obtains betters solutions. The instance NRG.5 is the only scenario where the results of this instance are at six point of distance to obtain an $Z_{opt}$ in SACSDBSCAN. Finally, for the instance NRH.5, the behavior of the ASCSDBSCAN is again superior than CS.

5.3. Statistical Test

As previously stated, we offer the following hypotheses in order to determine independence:

-

$H_0$: states that $Z_{min}$/$Z_{max}$ follows a normal distribution.

-

$H_1$: states the opposite.

The test performed has yielded $p\_value$ lower than $0.05$; therefore, $H_0$ cannot be assumed. Now that we know that the samples are independent, and it cannot be assumed that they follow a normal distribution, it is not feasible to use the central limit theorem. Therefore, for evaluating the heterogeneity of samples we use a non-parametric evaluation called Mann–Whitney–Wilcoxon test to compare all the results of the hardest instances we propose the following hypotheses:

-

$H_0$: CS is better than SACSDBSCAN.

-

$H_1$: states the opposite.

Finally, the statistical contrast test reveals which technique is considerably superior.

The Wilcoxon signed rank test was used to compare SCP on the algorithms techniques for the hardest instances (

Table 9. p-values for instance 4.10.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$1.711\times {10}^{-12}$	Not applicable

,

Table 10. p-values for instance 5.10.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$2.228\times {10}^{-12}$	Not applicable

,

Table 11. p-values for instance 6.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$1.711\times {10}^{-12}$	Not applicable

,

Table 12. p-values for instance A.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$3.055\times {10}^{-12}$	Not applicable

,

Table 13. p-values for instance b.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$3.417\times {10}^{-12}$	Not applicable

,

Table 14. p-values for instance c.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$4.100\times {10}^{-12}$	Not applicable

,

Table 15. p-values for instance d.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$1.288\times {10}^{-12}$	Not applicable

,

Table 16. p-values for instance NRE.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$1.915\times {10}^{-12}$	Not applicable

,

Table 17. p-values for instance NRF.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$2.406\times {10}^{-12}$	Not applicable

,

Table 18. p-values for instance NRG.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$3.296\times {10}^{-12}$	Not applicable

and

Table 19. p-values for instance NRH.5.

	CS	SACSDBSCAN
CS	Not applicable	SWS
SACSDBSCAN	$2.482\times {10}^{-12}$	Not applicable

). Smaller p-values than $0.05$ define that $H_0$ cannot be assumed because the significance level is also set to $0.05$.

To conduct the test run that supports the study, we use a method from the PISA system. We specify all data distributions (each in a file and each data in a line) in this procedure, and the algorithm returns a p-value for the hypotheses.

The following tables show the result of the Mann–Whitney–Wilcoxon test. To understand them, it is necessary to know the following acronyms:

• SWS = Statistically without significance.

In all the cases, as mentioned above, the p-values reported are less than 0.05, and SWS suggests that they have no statistical significance. So, with this knowledge, in each instance mentioned, we can see the SACSDBSCAN algorithm was better than the original CS.

If we focus on the instances where our proposal improves the result obtained in comparison to the original CS algorithm, we can infer that the solutions achieved are distributed in a centered way on their optimal value, which reflects that the behavior of this algorithm is very positive. This is reflected in the violin Figure 8 or Figure 10 and Figure 14.

5.4. Comparison Results in Similar Hybrid Algorithms

Within the literature, recent studies can be found that use hybrid algorithms that solve the coverage problem [5,48,49,50]. However, to compare a hybrid algorithm that resembles our proposal, we have considered making a comparison of results with hybrid algorithms that work with bio-inspired metaheuristics, improved by ML, and they solve the set covering problem. In this scheme, we have three algorithms. The first one is the crow search algorithm boosted by the DBSCAN method (called CSADBSCAN). The second studied approach was the integration between the crow search algorithm and the Kmean method (CSAKmean). Both hybridizations were proposed by Valdivia et al. in [51]. Finally, we employ an improved version of the cuckoo search algorithm with the Kmean transition algorithm (KMTA), recently proposed by García et al. in [52].

Table 20. Best values reached by an improved bio-inspired algorithm with ML that solve the set covering problem Instances 4, 5, 6 and A.

Instance	BKS	SACSDBSCAN		CSADBSCAN		CSAKmean		KMTA
		Best	RDP	Best	RDP	Best	RDP	Best	RDP
4.1	429	429	0.000	429	0.000	429	0.000	430	0.002
4.2	512	512	0.000	512	0.000	513	0.002	N/R	N/R
4.3	516	516	0.000	516	0.000	516	0.000	N/R	N/R
4.4	494	494	0.000	494	0.000	495	0.002	N/R	N/R
4.5	512	512	0.000	512	0.000	514	0.004	N/R	N/R
4.6	560	560	0.000	560	0.000	560	0.000	N/R	N/R
4.7	430	430	0.000	430	0.000	430	0.000	N/R	N/R
4.8	492	492	0.000	492	0.000	493	0.002	N/R	N/R
4.9	641	641	0.000	641	0.000	645	0.006	N/R	N/R
4.10	514	514	0.000	514	0.000	513	0.002	N/R	N/R
5.1	253	253	0.000	253	0.000	253	0.000	253	0.000
5.2	302	303	0.003	302	0.000	308	0.020	N/R	N/R
5.3	226	226	0.000	226	0.000	228	0.009	N/R	N/R
5.4	242	242	0.000	242	0.000	242	0.000	N/R	N/R
5.5	211	211	0.000	211	0.000	211	0.000	N/R	N/R
5.6	213	213	0.000	213	0.000	213	0.000	N/R	N/R
5.7	293	293	0.000	293	0.000	293	0.000	N/R	N/R
5.8	288	288	0.000	288	0.000	288	0.000	N/R	N/R
5.9	279	279	0.000	279	0.000	279	0.000	N/R	N/R
5.10	265	265	0.000	265	0.000	267	0.008	N/R	N/R
6.1	138	138	0.000	140	0.014	140	0.014	138	0.000
6.2	146	146	0.000	146	0.000	146	0.000	N/R	N/R
6.3	145	145	0.000	145	0.000	147	0.014	N/R	N/R
6.4	131	131	0.000	131	0.000	131	0.000	N/R	N/R
6.5	161	161	0.000	162	0.006	163	0.012	N/R	N/R
A.1	253	254	0.004	254	0.004	254	0.004	254	0.004
A.2	252	254	0.008	256	0.016	257	0.020	N/R	N/R
A.3	232	232	0.000	233	0.004	235	0.013	N/R	N/R
A.4	234	235	0.004	236	0.009	235	0.004	N/R	N/R
A.5	236	236	0.000	236	0.000	236	0.000	N/R	N/R
AVG	–	–	0.001	–	0.002	–	0.005	–	0.002

and

Table 21. Best values reached by an improved bio-inspired algorithm with ML that solve the set covering problem Instances B, C, D, NRE, NRF, NRG, and NRH.

Instance	BKS	SACSDBSCAN		CSADBSCAN		CSAKmean		KMTA
		Best	RDP	Best	RDP	Best	RDP	Best	RDP
B.1	69	73	0.058	69	0.000	74	0.072	69	0.000
B.2	76	76	0.000	81	0.066	83	0.092	N/R	N/R
B.3	80	80	0.000	82	0.025	84	0.050	N/R	N/R
B.4	79	79	0.000	83	0.051	84	0.063	N/R	N/R
B.5	72	72	0.000	78	0.083	72	0.000	N/R	N/R
C.1	227	227	0.000	233	0.026	228	0.004	229	0.009
C.2	219	219	0.000	226	0.032	226	0.032	N/R	N/R
C.3	243	243	0.000	253	0.041	254	0.045	N/R	N/R
C.4	219	219	0.000	224	0.023	225	0.027	N/R	N/R
C.5	215	215	0.000	222	0.033	215	0.000	N/R	N/R
D.1	60	61	0.017	68	0.133	66	0.100	60	0.000
D.2	66	70	0.061	73	0.106	71	0.076	N/R	N/R
D.3	72	76	0.056	82	0.139	82	0.139	N/R	N/R
D.4	62	63	0.016	70	0.129	67	0.081	N/R	N/R
D.5	61	63	0.033	72	0.180	66	0.082	N/R	N/R
NRE.1	29	29	0.000	70	1.414	30	0.034	29	0.000
NRE.2	30	31	0.033	83	1.767	34	0.133	N/R	N/R
NRE.3	27	28	0.037	74	1.741	34	0.259	N/R	N/R
NRE.4	28	30	0.071	83	1.964	33	0.179	N/R	N/R
NRE.5	28	29	0.036	92	2.286	30	0.071	N/R	N/R
NRF.1	14	15	0.071	372	25.571	17	0.214	14	0.000
NRF.2	15	15	0.000	74	3.933	18	0.200	N/R	N/R
NRF.3	14	16	0.143	335	22.929	19	0.357	N/R	N/R
NRF.4	14	15	0.071	52	2.714	18	0.286	N/R	N/R
NRF.5	15	15	0.000	65	3.333	16	0.067	N/R	N/R
NRG.1	176	188	0.068	287	0.631	197	0.119	176	0.000
NRG.2	154	160	0.039	208	0.351	168	0.091	N/R	N/R
NRG.3	166	180	0.084	211	0.271	183	0.102	N/R	N/R
NRG.4	168	177	0.054	250	0.488	186	0.107	N/R	N/R
NRG.5	168	181	0.077	230	0.369	183	0.089	N/R	N/R
NRH.1	63	69	0.095	N/R	N/R	71	0.127	64	0.016
NRH.2	63	66	0.048	N/R	N/R	71	0.127	N/R	N/R
NRH.3	59	66	0.119	N/R	N/R	69	0.169	N/R	N/R
NRH.4	63	63	0.000	N/R	N/R	68	0.079	N/R	N/R
NRH.5	55	59	0.073	N/R	N/R	61	0.109	N/R	N/R
AVG	–	–	0.021	–	1.162	–	0.059	–	0.003

present best values reached in CSADBSCAN, CSAKmean and KMTA. Those algorithms implement different strategies to improve metaheuristics with ML. To resume and centering the results of the best values obtained, we add the AVG measure in the final row of each table. Unfortunately, KMTA only reports results of the first of each family instance, so N/R means Not Reported.

Table 20 shows how SACSDBSCAN obtain better results average in comparison with the other algorithms for the instances 4, 5, 6 y A. Table 21 shows how SACSDBSCAN obtain better results average in comparison with CSADBSCAN and CSAKmean in a very wide value to CSADBSCAN, a difference to 1.141 and 0.038 distance with CSAKmean. Only KMTA obtain the best AVG value with the reported best values with 0.018 difference.

6. Conclusions

In this paper, we can conclude that the use of the machine learning technique to make a metaheuristic autonomous parameters setting has the 38/65 min values fitness on the test that we make, in one single configuration over the 20 different configurations, that demonstrates that with our proposed algorithm, it is not necessary make the complex task to find the best parameter setting of the metaheuristic CS, that is, in most of the time, in a try-and-error way. The result of the experiment demonstrates that it is positive to use the DBSCAN algorithm to infer his result and use that information to let us make changes to the values of the parameters. The comparison results with other bio-inspired hybrid algorithms applied on the set covering problem demonstrate that the use of DBSCAN obtains better results on average fitness values in comparison with studies that report all the best results values. The exploration criteria that we use can let the algorithm vary the search space to find another best candidate, as we saw in the box graphs and the distribution of the instances. In addition, the use of the noise points associate with the $Pa$, lets the SACSDBSCAN keep the variety on his behavior and not forget the stochastic factor that characterizes a metaheuristic.

The free execution parameter allows the metaheuristic to maintain its natural behavior across executions. At the time of reaching the freedom parameter, we can analyze the results of the metaheuristics and classify its results space to be able to make its classification and corresponding intervention of the possible candidate solutions, eliminating the worst for possible new better solutions.

As future work, we consider implementing an improvement to the criterion of population increase and decrease using clusterization strategies. In another line of work, we want to implement different machine learning techniques to be able to perform algorithms that allow effective use of the population increase/decrease in metaheuristics, and thus be able to deliver tools that make these algorithms more efficient. In addition, we are considering contributing new works that provide comparisons of this algorithm with other hybrid variants of cuckoo search, and other types of metaheuristics that use ML to improve their performance, different from those presented in this work, used to solve continuous problems.