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Article

Binary Secretary Bird Optimization Algorithm for the Set Covering Problem

by
Broderick Crawford
1,*,
Felipe Cisternas-Caneo
1,
Ricardo Soto
1,
Claudio Patricio Toledo Mac-lean
1,
José Lara Arce
1,
Fabián Solís-Piñones
1,
Gino Astorga
2 and
Giovanni Giachetti
3
1
Escuela de Ingeniería Informática, Pontificia Universidad Católica de Valparaíso, Avenida Brasil 2241, Valparaíso 2362807, Chile
2
Escuela de Negocios Internacionales, Universidad de Valparaíso, Alcalde Prieto Nieto 452, Viña del Mar 2572048, Chile
3
Facultad de Ingeniería, Universidad Andres Bello, Antonio Varas 880, Providencia, Santiago 7591538, Chile
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(15), 2482; https://doi.org/10.3390/math13152482 (registering DOI)
Submission received: 9 June 2025 / Revised: 23 July 2025 / Accepted: 24 July 2025 / Published: 1 August 2025

Abstract

The Set Coverage Problem (SCP) is an important combinatorial optimization problem known to be NP-complete. The use of metaheuristics to solve the SCP includes different algorithms. In particular, binarization techniques have been explored to adapt metaheuristics designed for continuous optimization problems to the binary domain of the SCP. In this work, we present a new approach to solve the SCP based on the Secretary Bird Optimization Algorithm (SBOA). This algorithm is inspired by the natural behavior of the secretary bird, known for its ability to hunt prey and evade predators in its environment. Since the SBOA was originally designed for optimization problems in continuous space and the SCP is a binary problem, this paper proposes the implementation of several binarization techniques to adapt the algorithm to the discrete domain. These techniques include eight transfer functions and five different discretization methods. Taken together, these combinations create multiple SBOA adaptations that effectively balance exploration and exploitation, promoting an adequate distribution in the search space. Experimental results applied to the SCP together with its variant Unicost SCP and compared to Grey Wolf Optimizer and Particle Swarm Optimization suggest that the binary version of SBOA is a robust algorithm capable of producing high quality solutions with low computational cost. Given the promising results obtained, it is proposed as future work to focus on complex and large-scale problems as well as to optimize their performance in terms of time and accuracy.

1. Introduction

The use of optimization and, in particular, metaheuristics, is gaining more and more attention in modern industry as it allows to tackle problems that, until some time ago, were not possible to address. Today, it is possible to tackle complex and large-scale problems, with metaheuristics allowing us to obtain good quality solutions in a reasonable processing time [1]. Metaheuristics, in general, allow us to address different types of problems and can also adapt to changes in the environment, being able to work in different domains. All these characteristics make them attractive for solving current real-world problems in a variety of areas, such as logistics [2,3], manufacturing industry [4,5], transport [6,7], health [8,9], and mining [10,11], among others.
The Set Covering Problem (SCP) is a combinatorial optimization problem known to be NP-hard that is ever-present in various industries [12] and one of the 21 problems present in Karp [13]. It involves finding a subset of columns from a binary matrix A of size m × n that covers all rows at the lowest possible cost. This problem has significant applications in various fields, such as emergency service location [14], crew scheduling in mass transportation systems [15], and other related problems discussed in [16]. There is a special case of SCP where all sets have the same cost in this case the task is to minimize the selection of sets to cover the given universe. Practical applications in the industry are engineering design [17] and vehicle path planning [18], among others.
The use of metaheuristics to solve the SCP includes algorithms such as Genetic Algorithms [19], Particle Swarm Optimization [20], Ant Colony Optimization [21], Tabu Search [22], Electromagnetism metaheuristic [23], Artificial Bee Colony [24], and GRASP [25], among others. In particular, binarization techniques have been explored to adapt metaheuristics designed for continuous optimization problems to the binary domain of the SCP.
In this work, we present a new approach to solving the SCP based on the Secretary Bird Optimization Algorithm (SBOA) [26]. The SBOA was selected for this research for several compelling reasons, drawn directly from its original design and validated performance. First, it is a recent metaheuristic whose authors specifically engineered it to address common shortcomings of existing algorithms, aiming to improve convergence speed, enhance optimization accuracy, and effectively avoid local optima. These characteristics are highly desirable for tackling NP-hard problems like the SCP. Second, the SBOA achieves a robust balance between its exploration and exploitation phases by uniquely combining three different search strategies: differential evolution, Brownian motion, and Lévy flights. Finally, its effectiveness is not merely theoretical; the SBOA has demonstrated superior performance against numerous advanced algorithms on standard benchmark suites (CEC-2017 and CEC-2022) and has been successfully applied to solve twelve different constrained engineering problems.
Given these strengths, the SBOA emerges as a strong and novel candidate for adaptation to the binary domain to solve the SCP. The relevance of finding new and efficient solutions for this problem is underscored by its wide range of applications, which have become even more critical in modern industry. For example, in the post-pandemic era, supply chains and logistics face continuous challenges where the SCP is present [27], including the efficient allocation of resources and crews [28,29], balancing assembly lines, and optimizing the location of distribution centers to meet growing demand.
Since the SBOA was originally designed for optimization problems in continuous spaces and the SCP is a binary problem, this work proposes the implementation of several binarization techniques to adapt the algorithm to the discrete domain. These techniques include eight transfer functions with “S”-shaped and “V”-shaped variants and five different discretization methods. Together, these combinations create multiple adaptations of the SBOA that effectively balance exploration and exploitation, promoting an adequate distribution in the search space. The results suggest that the binary version of the SBOA (BSBOA) is a robust algorithm capable of producing high-quality solutions with low computational cost.
This article is organized as follows: We present a brief description of the Set Coverage Problem in Section 2, how to resolve SCP with continuos metaheuristics in Section 3, and an outline of the Secretary Bird Optimitation Algorithm in Section 4. We explain how and make the binary version of the SBOA in Section 5. Finally, we present our results, discussions, conclusions, and possible future lines of research in Section 6, Section 7 and Section 8.

2. Set Covering Problem

The Set Covering Problem (SCP) is a combinatorial optimization problem categorized as NP-hard. Its main objective is to find a minimal subset of elements that completely covers a universal set of requirements while minimizing the associated cost.

2.1. Formal Mathematical Formulation

Formally, let U = { u 1 , u 2 , , u m } be a universal set composed of m elements, and let S = { S 1 , S 2 , , S n } be a collection of n subsets, where each subset S j has an associated cost c j . The problem consists of identifying an optimal subset S S that covers all elements in U at the minimum possible cost.
To model this problem and solve it computationally, the formal set-based definition is translated into a matrix formulation. A binary matrix represents the problem A with m rows and n columns. Each of the m rows corresponds to an element u i of the universal set U and each of the n columns represents an available subset S j . The value in the matrix, a i , j , is 1 if subset j covers element i (that is, u i S j ), and 0 otherwise.
A = a 1 , 1 a 1 , 2 a 1 , n a 2 , 1 a 2 , 2 a 2 , n a m , 1 a m , 2 a m , n
To decide which columns (subsets) are chosen, the binary decision variable x j is used, which takes the value of one if column j is selected for the solution, and zero if it is not.
Thus, the main objective is to minimize the total cost of the solution, summing the costs of only the selected columns, as indicated by Equation (1).
min j = 1 n c j x j
The main constraint of this problem is the coverage constraint, which ensures that each element (row) is covered by at least one selected column. This is achieved with the following equation, which must hold for every row i:
j = 1 n a i , j x j 1
In addition, we must ensure that the decision variable is binary; therefore, x j { 0 , 1 } .

2.2. SCP Practical Example

Let us assume a company needs to form an oversight committee for a new technology project. For the project to be successful, the committee must cover five areas of expertise: financial, legal, technical, marketing, and logistics. The company has identified six external consultants, each with a different set of skills and an associated cost (fee).
The goal is to apply the Set Covering Problem (SCP) to select the committee with the lowest total cost that guarantees coverage of all required areas of expertise.

2.2.1. Problem Definition

1.
Elements to be Covered: The five required areas of expertise.
  • R 1 : Financial Expertise.
  • R 2 : Legal Expertise.
  • R 3 : Technical Expertise.
  • R 4 : Marketing Expertise.
  • R 5 : Logistics Expertise.
2.
Available Sets: The six candidates, each with their cost and skills.
  • C 1 : Anne (Cost: $4)—Skills: Financial, Legal.
  • C 2 : Ben (Cost: $3)—Skills: Technical, Marketing.
  • C 3 : Carla (Cost: $6)—Skills: Legal, Technical, Logistics.
  • C 4 : David (Cost: $4)—Skills: Marketing, Logistics.
  • C 5 : Elena (Cost: $5)—Skills: Financial, Technical.
  • C 6 : Frank (Cost: $4)—Skills: Legal, Marketing.
Thus, Table 1 shows the relationship between both previous lists in the so-called Incidence Matrix. This matrix reflects the coverage (columns) of each element (rows) and allows us to validate whether the requirements (problem constraints) are met.

2.2.2. Mathematical Formulation

Let x j = 1 if consultant j is selected and 0 otherwise. The problem is formulated to minimize the total cost, subject to the constraint that each area of expertise is covered.
  • Objective Function (minimize cost):
    min ( Z ) = 4 x 1 + 3 x 2 + 6 x 3 + 4 x 4 + 5 x 5 + 4 x 6
  • Constraints (cover each expertise):
    x 1 + x 5 1 ( Financial ) x 1 + x 3 + x 6 1 ( Legal ) x 2 + x 3 + x 5 1 ( Technical ) x 2 + x 4 + x 6 1 ( Marketing ) x 3 + x 4 1 ( Logistics )
  • Decision Variables:
    x j { 0 , 1 } j { 1 , , 6 }

2.2.3. Optimal Solution

The optimal solution for this problem is to select Anne ( C 1 ), Ben ( C 2 ), and David ( C 4 ).
  • Selected Consultants: {Anne, Ben, David}
  • Coverage:
    -
    Anne covers the Financial and Legal areas.
    -
    Ben covers the Technical and Marketing areas.
    -
    David covers the Marketing and Logistics areas.
  • Minimum Total Cost: $ 4 ( Anne ) + $ 3 ( Ben ) + $ 4 ( David ) = $ 11 .
This combination fulfills all expertise requirements at the lowest possible cost.

2.3. Unicost Set Covering Problem (Unicost SCP)

The unicost problem is a specific variant of the Set Covering Problem (SCP), in which all column costs are equal ( c j = 1 for all j J ). The primary objective in this case is to minimize the number of selected columns while ensuring that each row is covered by at least one of them. This variant simplifies the general SCP model by focusing exclusively on the minimum number of columns required to cover all rows, emphasizing structural optimization without considering cost variations.
Mathematically, the model can be formulated as follows:
Minimize j J x j
subject to
j J a i j x j 1 for all i I ,
j { 0 , 1 } for all j J ,
where a i j is a coefficient indicating whether column j covers row i, x j is a binary variable that takes the value 1 if column j is selected and 0 otherwise, I is the set of rows, and J is the set of columns.
The unicost problem, like the general SCP, is known to be NP-hard and has been applied in various fields, such as scheduling, logistics, and resource optimization.

3. Continuous Metaheuristics Solving Set Covering Problem

To apply the Secretary Bird Optimization Algorithm (SBOA) to the Set Covering Problem (SCP), it is necessary to adapt the original search approach, designed for a continuous space, to a binary environment. As discussed in Section 1, various metaheuristics have been successfully applied to solve the SCP, including Genetic Algorithms [30], Ant Colony Optimization [31], Particle Swarm Optimization [32], and others. All these metaheuristics have one characteristic in common: they are metaheuristics designed to solve continuous optimization problems that were modified to resolve binary problems like the SCP.
This adaptation is achieved through a Two-Step Technique that converts continuous solutions into binary solutions, ensuring compliance with the SCP requirements. This technique is fundamental for applying bioinspired algorithms to binary combinatorial optimization problems, such as the SCP.

Two-Step Technique

In the literature, there are different ways to binarize continuous metaheuristics [33], but the most widely used is the Two-Step Technique [34]. As its name suggests, the binarization process is performed in two stages:
  • Application of a transfer function, which transforms the continuous value into a value within the range [ 0 , 1 ] .
  • Application of a binarization rule, which determines the assignment of a 1 or a 0.
In the literature [33], a variety of transfer functions are proposed for this purpose. They are generally categorized into two main families based on their shape and behavior: S-shaped and V-shaped functions. These two families represent the most common and well-established approaches for mapping continuous search spaces to binary ones.
  • S-Shaped Functions: These sigmoidal functions produce a value in the range [ 0 , 1 ] that represents the probability of a solution’s component becoming ‘1’. An input value close to zero yields a probability near 0.5, while large positive or negative values push the probability towards 1 or 0, respectively. This behavior models a form of probabilistic switch.
  • V-Shaped Functions: In contrast, these functions relate the probability of change to the magnitude of the continuous value, rather than its sign. A small step (a value close to zero) results in a low probability of changing the bit, while a large step increases this probability. This is conceptually linked to the notion of velocity or momentum in swarm algorithms, where a larger “move” is more likely to alter the solution’s state.
Table 2 and Figure 1 present the standard functions from both families that are commonly used in this research area. The notation d j i observed in the table corresponds to the continuous value of the j-th dimension of the i-th individual, resulting from the perturbation performed by the continuous metaheuristic.
Additionally, in the literature [33], we can find five different binarization rules, of which we can highlight the following:
  • Standard (STD): If the condition is satisfied, the standard binarization rule returns 1; otherwise, it returns 0. Mathematically, it is defined as follows:
    X new j = 1 if rand T ( d i j ) , 0 else .
  • Elitist (ELIT): The best value is assigned if a random value is within the probability; otherwise, a zero value is assigned. Mathematically, it is defined as follows:
    X new j = X Best j if rand < T ( d i j ) , 0 else .
  • Complement (COM): If the condition is satisfied, the second step operator returns the complement of the actual value.
    x new j = complement x w j if rand T d w j , 0 else .

4. Secretary Bird Optimization Algorithm

The Secretary Bird Optimization Algorithm (SBOA) is a metaheuristic inspired by the natural behavior of the secretary bird, particularly its unique strategies for hunting and evading predators. It was proposed in 2024 by Youfa Fu, Dan Liu, Jiadui Chen, and Ling He [26]. The algorithm was originally designed to solve continuous optimization problems, leveraging its exploration and exploitation phases to efficiently find optimal solutions.
The operation of the SBOA is divided into two main phases that mimic the bird’s survival instincts: exploration, which models its hunting strategy, and exploitation, which models its escape strategy.

4.1. Exploration Phase (Hunting Strategy)

This phase simulates how the secretary bird hunts its prey, like snakes, and corresponds to a global search in the solution space. The process is modeled in three stages that reflect the bird’s hunting tactics:
  • Searching for Prey: The bird begins by exploring the terrain to locate hidden prey. The algorithm mimics this by creating a new solution based on the difference between two randomly selected solutions from the population. This enhances diversity and allows the algorithm to scan new, unexplored areas of the search space. This movement is modeled by Equation (11).
    x i j new , P 1 = x i j + ( x random 1 x random 2 ) · R 1 ,
  • Wearing Down the Prey: Once prey is found, the bird does not attack immediately but circles and provokes it to deplete its energy. The algorithm models this by moving towards the best solution found so far ( x best ), simulating how the bird focuses on its target, while introducing a random component to avoid premature convergence. This is shown in Equation (12).
    x i j new , P 1 = x best + exp t T 4 · ( R B 0.5 ) · ( x best x i j ) ,
  • Attacking the Prey: When the prey is exhausted, the bird executes a swift, lethal attack. To simulate this decisive action, the algorithm performs a “jump” toward the best solution using a Lévy flight (RL). This feature models the bird’s powerful strike, combining frequent small steps with occasional long jumps to accelerate convergence towards the global optimum. The movement is described in Equation (13).
    x i j new , P 1 = x best + 1 t T 2 × t T · x i j · R L ,

4.2. Exploitation Phase (Escape Strategy)

This phase models how the secretary bird evades predators, which corresponds to a refined local search for a better solution. The bird chooses one of two strategies with equal probability:
  • Camouflage ( C 1 ): The bird may hide in its environment to avoid being detected. The algorithm simulates this by making small, subtle adjustments to its current position, moving locally around the best-known solution to refine it.
  • Fleeing ( C 2 ): If camouflage is not an option, the bird runs or flies to escape danger. The algorithm models this with a larger, more random movement that allows it to jump to other regions of the search space, effectively avoiding stagnation.
Both escape strategies are modeled using Equation (14).
x i j new , evasion = C 1 : x best + ( 2 · R B 1 ) · 1 t T 2 · x i j , if r < 0.5 , C 2 : x i j + R 2 · ( x random K · x i j ) , else .

4.3. Solution Selection

A key aspect of the SBOA is how it determines if a new solution is superior to the current one. The algorithm employs a greedy selection mechanism for this purpose.
The quality of each solution is measured by an objective function (F), which, in the context of the Set Covering Problem, is the total cost to be minimized. After generating a new candidate position in either the exploration or exploitation phase, the algorithm compares the fitness of the new solution ( F i new , P 1 ) with that of the current one ( F i ).
The new solution is accepted only if it offers a better fitness value (i.e., a lower cost). Otherwise, the current solution is kept. This process, shown in Equation (15), ensures that the population’s quality either improves or remains the same in each iteration, guiding the search effectively towards the optimal solution.
X i = X i new , P 1 if F i new , P 1 < F i X i else
The pseudo-code of the SBOA is detailed in the pseudo-code of Algorithm 1.
Algorithm 1 Secretary Bird Optimization Algorithm.
Input: Problem Setting ( D i m , u b , l b , P o p _ s i z e ( N ) ) , M a x _ I t e r ( T ) , C u r r _ I t e r ( t )
Output: Best solution
1:
Initialize the population randomly.
2:
for  t = 1 T   do
3:
    Update Secretary Bird x best .
4:
    for  i = 1 N  do
5:
        Exploration Phase:
6:
        if  t 1 3 T  then
7:
           Calculate new status using Searching Stage Equation (11).
8:
           Update the i-th Secretary Bird using Update Equation (15).
9:
        else if  1 3 T < t < 2 3 T  then
10:
           Calculate new status using Consuming Stage Equation (12).
11:
           Update the i-th Secretary Bird using Update Equation (15).
12:
        else if  1 3 T < t < 2 3 T  then
13:
           Calculate new status using Attacking Stage Equation (13).
14:
           Update the i-th Secretary Bird using Update Equation (15).
15:
        end if
16:
        Exploitation Phase:
17:
        if  r < 0.5  then
18:
           Calculate new status using C1 in Exploitation Phase Equation (14).
19:
        else
20:
           Calculate new status using C 2 in Exploitation Phase Equation (14).
21:
        end if
22:
        Update the i-th Secretary Bird using Update Equation (15).
23:
    end for
24:
    Output the best solution obtained by SBOA for the given optimization problem.
25:
    Save the best candidate solution so far.
26:
end for
27:
Output the best solution obtained by SBOA for the given optimization problem.

5. Binary Secretary Bird Optimization Algorithm

As explained in Section 4, the Algorithm Optimization based on the Behavior of the Secretary Bird (SBOA) is a metaheuristic designed to solve continuous optimization problems. To address problems such as feature selection, it is necessary to transform the solutions into the binary domain.
Furthermore, in Section 3, it was highlighted that the Two-Step Technique is one of the most widely used approaches for binarizing continuous metaheuristics. In [33,34], eight different transfer functions and five binarization rules are described, which can be applied in this context.
In this work, the binarization scheme was chosen based on established findings in the literature that analyze the relationship between different transfer functions, binarization rules, and the exploration–exploitation balance. For the experiments, we use the V3 transfer function and the Elitist discretization method, considering the work carried out by Lanza-Gutierrez et al. in [35], where they recommend the use of V3 to solve small and medium-sized problems and for a better exploration–exploitation relation recommend Elitist as a discretization technique.
Additionally, we selected a function from the V-shaped family because its behavior is particularly well-suited for swarm intelligence algorithms. Unlike S-shaped functions, V-shaped functions relate the probability of a bit changing to the magnitude (or “velocity”) of the agent’s move, not its direction. A larger step size corresponds to a higher probability of flipping a bit. Among these, the V3 transfer function, shown in Equation (16), was chosen due to recommendations in the literature for its robust performance on small and medium-sized problems.
T ( d j i ) = d j i 1 + ( d j i ) 2
To complement the transfer function, the elitist binarization rule was selected, as shown in Equation (17). This rule introduces a strong exploitation pressure by giving a chance to preserve bits from the best-found solution in the population ( X Best ). This element of elitism is intended to balance the highly explorative nature of the SBOA’s hunting phases, thereby creating a more focused and effective search process.
X new k = X Best k if rand < T ( d i k ) , 0 else .
In this way, the Binary Secretary Bird Optimization Algorithm (BSBOA) is constructed. The process begins with the initialization of the solutions in the binary domain. In each iteration, the binary solutions are modified using Equations (11)–(14), which represent the movement equations specific to the SBOA. Once the solutions are perturbed, they exit the binary domain, and a binarization process is applied using Equations (16) and (17). This cycle is repeated until the defined number of iterations is completed.
Algorithm 2 presents the binary version of the SBOA, where the key section is the binarization that occurs in line 23.
Algorithm 2 Binary Secretary Bird Optimization Algorithm.
Input: Problem Setting ( D i m , u b , l b , P o p _ s i z e ( N ) ) , M a x _ I t e r ( T ) , C u r r _ I t e r ( t )
Output: Best solution
1:
Initialize the population randomly.
2:
for  t = 1 T   do
3:
    Update Secretary Bird x best .
4:
    for  i = 1 N  do
5:
        Exploration Phase:
6:
        if  t 1 3 T  then
7:
           Calculate new status using Searching Stage Equation (11).
8:
           Update the i-th Secretary Bird using Update Equation (15).
9:
        else if  1 3 T < t < 2 3 T  then
10:
           Calculate new status using Consuming Stage Equation (12).
11:
           Update the i-th Secretary Bird using Update Equation (15).
12:
        else if  1 3 T < t < 2 3 T  then
13:
           Calculate new status using Attacking Stage Equation (13).
14:
           Update the i-th Secretary Bird using Update Equation (15).
15:
        end if
16:
        Exploitation Phase:
17:
        if  r < 0.5  then
18:
           Calculate new status using C1 in Exploitation Phase Equation (14).
19:
        else
20:
           Calculate new status using C 2 in Exploitation Phase Equation (14).
21:
        end if
22:
        Update the i-th Secretary Bird using Update Equation (15).
23:
        Binarization of population X
24:
    end for
25:
    Save the best candidate solution so far.
26:
end for
27:
Output the best solution obtained by SBOA for the given optimization problem.

Algorithm Complexity Analysis

Each algorithm requires a certain amount of time to perform its optimization tasks, and these can vary for the same problem. Evaluating algorithmic complexity is an effective way to demonstrate performance in terms of runtime. Big O notation is one of the most widely used tools for complexity analysis [36], and we will use it in this paper to analyze the complexity of BSBOA. Let N be the population size, D i m the number of decision variables, and T the maximum number of iterations. Thus, for the initialization process of random solutions, we have a complexity of O ( N ) . During the optimization process, the complexity is O(T × N) + 2 · O(T × N × Dim), which includes the search for the best positions per iteration (O(T × N)), the updating of the positions of all solutions per iteration (O(T × N × Dim)), and the binarization of all solutions per iteration (O(T × N × Dim)). Thus, the algorithmic complexity of our proposal is O(N × (T × Dim + 1)).

6. Experimental Result

To develop the validation of our proposal, we have used the instances offered in OR-library [37] for both the pesos version (SCP) [35] and for the Unicot version (USCP) [17]. Our proposal was compared to two continuous metaheuristics of great relevance, such as Binary Particle Swarm Optimization [38] and Binary Grey Wolf Optimizer [39].

6.1. Parameter Setting

Before performing experimentation, we perform internal tests for parameter configuration. Specifically, we have carried out experimentation with the population size and the number of iterations. For the population size, we tested from [10, 100] in increments of 10, and the population sizes we tested were [20, 50, 70, 100, 150, 200, 300, 400, 500, 600, 700, 800, 900, 1000].
Table 3 shows the subset of instances used for parameter configuration. We have used these instances because we consider that they are representative of the entire portfolio of existing instances of OR-library. The table contains the following details: The first column shows the name of the instance, the second column shows the sample to the type of problem that the instance (SCP or USCP) belongs, the third column refers to the amount of restrictions that the instance possesses, the fourth column refers to the amount of decision variables to optimize, the fifth column refers to the density of some that the matrix has in view of Section 3, and the last column refers to the optimal value of the instance. This experiment was conducted in a team using a Windows 10 operating system, an Intel Core i9-10900 K 3.70 GHz Processor, and 64 GB of RAM.
Table 4 shows the best configurations for each instance executed, considering computing time and the fitness reached. Thus, it is observed that the ideal population size for this experiment is 10. For the number of iterations, the value changes for each instance; therefore, we have determined that the average iterations will be used in experimentation. Thus, the number of iterations is 600.
Thus, Table 5 shows the final configuration of the experimentation carried out in this work. The global configuration used by all the metaheuristics (All MH) and the parameters of each metaheuristic are highlighted.

6.2. SCP and USCP Instances Resolved

Table 6 and Table 7 show the instances used to solve the SCP and USCP, respectively. Each table shows the instance name (column Instance), the number of constraints (column M), the number of decision variables (column N), the density of ones in the matrix mentioned in Section 2 (column Density %), and the instance’s optimum. It should be noted that the underlined and bold optima are not global optima but the best results reported in the literature. Thus, for the present work, we solved 22 instances for the SCP and 17 instances for the USCP.

6.3. Results of SCP

This subsection shows the results of running the SBOA with the previously established parameters and compared with the GWO and PSO algorithms. In Table 8 and Table 9, the results are shown when running the different instances of OR-library. For each algorithm and instance, the best known optimal value, the best value achieved, the worst value achieved, the average, and the standard deviation of the solutions obtained of the 30 executions are shown in the tables [40].
A convergence graph represents how the metaheuristic finds progressively better and better solutions as iterations increase. In order to find good solutions in a reasonable time to provide answers to real-world problems, it is expected that the number of iterations will not be excessive in order to avoid the excessive use of computational resources. As described in Crawford et al. [41] and Lemus-Romani et al. [42], graphs were used to document the optimization process while considering the fitness achieved as the iterations progressed. This is shown in Figure 2, Figure 3, Figure 4 and Figure 5, where the relationship between the number of iterations and the fitness achieved is shown, corresponding to the x and y axes, respectively, where it can be seen that there is good convergence without the algorithm being trapped in a local optimum [43].
On the other hand, in Table 10 and Table 11, the execution of the different instances for the three Algorithms, the SBOA, GWO and PSO, are shown, considering their execution time.
Figure 6, Figure 7, Figure 8 and Figure 9 show graphs with the performance of the three metaheuristics (SBOA, PSO and GWO), where it can be seen that although the times of the SBOA in relation to the iterations were not better than PSO and GWO, they were promising.

6.4. Results of USCP

Table 12 shows the results obtained when running the different instances of the USCP for the three metaheuristics investigated in this work (SBOA, GWO and PSO).
Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 show the convergence plots of the unicost instances, where it can be observed that, in general, the SBOA achieves good performance with few iterations without being trapped in a local optimum.
Table 13 and Table 14 show the times used for the different USCP instances for the three algorithms studied.
Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 show graphs with the performance of the three metaheuristics (SBOA, PSO and GWO), enabling us to compare it with the times achieved by the SBOA.

Statistical Tests

To validate our work, Table 15, Table 16, Table 17 and Table 18 show the statistical significance tests for the algorithms worked on: SBP, PSO and GWO. This table shows the p-values that act as indicators of statistical significance. We use the nonparametric Wilcoxon–Mann–Whitney test to perform this validation [44] since we have two independent samples and we cannot assume normality for at least one of them. The hypotheses considered are the following:
H 0 : μ A μ B ,
H 1 : μ A < μ B
where μ A and μ B represent the average value delivered by algorithms A and B. We consider that if the p-value is less than 0.05, the hypothesis H 0 will be rejected, with hypothesis H 1 being accepted and these cases are highlighted in bold and underlined in the tables.
The BSBOA demonstrated remarkable ability in solving the Set Covering Problem (SCP) and its unicost variant (USCP) using the V3 transfer function and elitist rule. For the SCP, it stood out by generating high-quality solutions with lower costs compared to metaheuristics, like GWO and PSO; in the USCP, where costs are homogeneous, it maintained competitive performance with less pronounced differences. Additionally, the BSBOA achieved favorable execution times, establishing itself as an efficient and versatile option for combinatorial optimization problems.

7. Discussion

The experimental outcomes consistently show BSBOA’s ability to yield high-quality solutions, often surpassing GWO and PSO in solution value and robustness (lower standard deviations) for the SCP, while maintaining competitive performance for the USCP. The computational analysis confirms BSBOA’s operational efficiency with a generally low computational burden.
BSBOA’s effectiveness stems from SBOA’s core design, inspired by the secretary bird’s natural behaviors of hunting and evasion, fostering a well-balanced interplay between exploration and exploitation crucial for NP-hard problems. Its systematic adaptation from continuous to binary domains, involving a rigorous exploration of eight transfer functions and five discretization methods, was pivotal. The selection of the V3 transfer function and the Elitist rule, based on prior research for optimal exploration–exploitation balance in medium-sized problems, ensures this effective translation and harmonious interaction with the discrete solution space.
However, it is crucial to acknowledge certain trade-offs and constraints. The iterative nature of the binarization process, applying transfer functions and rules in each cycle, introduces a computational overhead that could be significant for extremely large-scale problems or real-time systems, despite current observed efficiencies. Moreover, while our systematic validation identified an optimal configuration (V3 and Elitist) for the tested instances, the generalizability of this specific setup across all possible SCP and USCP variations, particularly those with distinct structural properties or much higher dimensions, requires further extensive validation. The current reliance on standard benchmark instances, while a common practice, also represents a limitation in fully assessing its performance in the intricate, diverse conditions of real-world applications.
These considerations underscore the ongoing need for refinement and broader validation. Such endeavors are vital to fully realize BSBOA’s potential for industrial and engineering challenges.

8. Conclusions

In this work, a solution to the SCP was provided using the SBOA, which is inspired by the natural behavior of the secretary bird. On the one hand, the SBOA was originally designed to work on continuous space optimization problems; on the other hand, the SCP is a binary problem. Several binarization techniques were proposed to adapt the algorithm to the discrete domain. Combinations of eight transfer functions were used, along with five discretization methods. Subsequently, the binary version of the SBOA was compared with the GWO and PSO algorithms, providing a solution to the SCP and its USCP variant. The BSBOA is positioned as a robust and effective tool for resolving both the SCP and USCP, excelling in solution quality and execution efficiency compared to GWO and PSO. Its ability to handle different levels of complexity and balance exploration and exploitation makes it ideal for practical optimization applications. Future work is to extend its applicability to more complex and large-scale problems, as well as to optimize its performance in terms of time and accuracy.

Author Contributions

Conceptualization, B.C., F.C.-C. and R.S.; methodology, B.C., F.C.-C., C.P.T.M.-l., J.L.A. and F.S.-P.; software, F.C.-C., C.P.T.M.-l., J.L.A. and F.S.-P.; validation, B.C., F.C.-C., R.S., G.A. and G.G.; formal analysis, C.P.T.M.-l., J.L.A. and F.S.-P.; investigation, B.C., F.C.-C., R.S., C.P.T.M.-l., J.L.A., F.S.-P., G.A. and G.G.; resources, C.P.T.M.-l., J.L.A., F.S.-P. and G.A.; writing—original draft preparation, C.P.T.M.-l., J.L.A., F.S.-P. and G.A.; writing—review and editing, B.C., F.C.-C., R.S. and G.G.; supervision, B.C., F.C.-C. and R.S.; funding acquisition, B.C. and R.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding authors.

Acknowledgments

Felipe Cisternas-Caneo was supported by the National Agency for Research and Development ANID BECAS/DOCTORADO NACIONAL 21230203.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:
SBOASecretary Bird Optimization Algorithm
BSBOABinary Secretary Bird Optimization Algorithm
PSOParticle Swarm Optimization
GWOGrey Wolf Optimizer
ACOAnt Colony Optimization
GAGenetic Algorithm
BCSOBinary Cat Swarm Optimization
BBHABinary Black Hole Algorithm
CSCuckoo Search
TSTabu Search
SCPSet Covering Problem
USCPUnicost Set Covering Problem

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Figure 1. S-shaped and V-shaped transfer functions.
Figure 1. S-shaped and V-shaped transfer functions.
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Figure 2. Convergence analysis of the instances Scp 41, Scp 51 and Scp 61.
Figure 2. Convergence analysis of the instances Scp 41, Scp 51 and Scp 61.
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Figure 3. Convergence analysis of the instances Scp a1, Scp b1 and Scp c1.
Figure 3. Convergence analysis of the instances Scp a1, Scp b1 and Scp c1.
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Figure 4. Convergence analysis of the instances Scp d1, Scp nre1 and Scp nrf1.
Figure 4. Convergence analysis of the instances Scp d1, Scp nre1 and Scp nrf1.
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Figure 5. Convergence analysis of the instances Scp nrg1 and Scp nrh1.
Figure 5. Convergence analysis of the instances Scp nrg1 and Scp nrh1.
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Figure 6. Time analysis of the instances Scp 41, Scp 51 and Scp 61.
Figure 6. Time analysis of the instances Scp 41, Scp 51 and Scp 61.
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Figure 7. Time analysis of the instances Scp a1, Scp b1 and Scp c1.
Figure 7. Time analysis of the instances Scp a1, Scp b1 and Scp c1.
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Figure 8. Time analysis of the instances Scp d1, Scp nre1 and Scp nrf1.
Figure 8. Time analysis of the instances Scp d1, Scp nre1 and Scp nrf1.
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Figure 9. Time analysis of the instances Scp nrg1, Scp nre1 and Scp nrh1.
Figure 9. Time analysis of the instances Scp nrg1, Scp nre1 and Scp nrh1.
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Figure 10. Convergence analysis of the instances UScp 41, UScp 51 and UScp 61.
Figure 10. Convergence analysis of the instances UScp 41, UScp 51 and UScp 61.
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Figure 11. Convergence analysis of the instances UScp a1, UScp b1 and UScp c1.
Figure 11. Convergence analysis of the instances UScp a1, UScp b1 and UScp c1.
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Figure 12. Convergence analysis of the instances UScp d1, UScp unre1 and UScp unrg1.
Figure 12. Convergence analysis of the instances UScp d1, UScp unre1 and UScp unrg1.
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Figure 13. Convergence analysis of the instances UScp unrh1, UScp clr10 and UScp clr11.
Figure 13. Convergence analysis of the instances UScp unrh1, UScp clr10 and UScp clr11.
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Figure 14. Convergence analysis of the instances UScp clr12, UScp cyc06 and UScp cyc07.
Figure 14. Convergence analysis of the instances UScp clr12, UScp cyc06 and UScp cyc07.
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Figure 15. Convergence analysis of the instance UScp cyc08.
Figure 15. Convergence analysis of the instance UScp cyc08.
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Figure 16. Time analysis of the instances UScp 41, UScp 51 and UScp 61.
Figure 16. Time analysis of the instances UScp 41, UScp 51 and UScp 61.
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Figure 17. Time analysis of the instances UScp a1, UScp b1 and UScp c1.
Figure 17. Time analysis of the instances UScp a1, UScp b1 and UScp c1.
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Figure 18. Time analysis of the instances UScp nre1, UScp nrf1 and UScp nrg1.
Figure 18. Time analysis of the instances UScp nre1, UScp nrf1 and UScp nrg1.
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Figure 19. Time analysis of the instances UScp nrh1, UScp clr10 and UScp clr11.
Figure 19. Time analysis of the instances UScp nrh1, UScp clr10 and UScp clr11.
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Figure 20. Time analysis of the instances UScp clr12, UScp cyc06 and UScp cyc07.
Figure 20. Time analysis of the instances UScp clr12, UScp cyc06 and UScp cyc07.
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Figure 21. Time analysis of the instance cyc08.
Figure 21. Time analysis of the instance cyc08.
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Table 1. Incidence Matrix (A), where a i j = 1 if consultant j covers requirement i.
Table 1. Incidence Matrix (A), where a i j = 1 if consultant j covers requirement i.
C1 (Anne)C2 (Ben)C3 (Carla)C4 (David)C5 (Elena)C6 (Frank)
R1 (Fin.)100010
R2 (Legal)101001
R3 (Tech.)011010
R4 (Mkt.)010101
R5 (Log.)001100
Table 2. S-shaped and V-shaped transfer functions.
Table 2. S-shaped and V-shaped transfer functions.
S-ShapedV-Shaped
NameEquationNameEquation
S1 T ( d j i ) = 1 1 + e 2 d j i V1 T ( d j i ) = erf π 2 d j i
S2 T ( d j i ) = 1 1 + e d j i V2 T ( d j i ) = tanh ( d j i )
S3 T ( d j i ) = 1 1 + e d j i / 2 V3 T ( d j i ) = d j i 1 + ( d j i ) 2
S4 T ( d j i ) = 1 1 + e d j i / 3 V4 T ( d j i ) = 2 π arctan π 2 d j i
Table 3. Instances used for parameter configuration.
Table 3. Instances used for parameter configuration.
InstanceType ProblemMNDensity (%)Optimum
41SCP20010002.00429
61SCP20020005.00138
b1SCP30030005.0069
d1SCP40040005.0060
clr10USCP51121012.3025
clr11USCP102333012.4023
cyc06USCP2401922.1060
cyc07USCP6724480.90144
Table 4. Parameter selection.
Table 4. Parameter selection.
PopIterInstanceBest FitnessWorst FitnessAverageTime SecondsTime Minutes
1050041433433433104.181.74
10706114114114110.320.17
10400b1696969233.973.9
10600clr10252725.4351.690.86
10900clr11232624.57185.93.1
10600cyc0660606052.070.87
10500cyc07144154149.71240.984.02
101000d1606160.431297.621.63
Table 5. Configuration of parameters.
Table 5. Configuration of parameters.
ParameterValue
All MHPopulation size10
Iterations600
Independent runs31
Transfer functionsV3
Method of discretizationElitist
PSO w m i n 0.1
w m a x 0.9
c 1 2
c 2 2
GWOalinearly decreases from 2 to 0
SBOACFpotentially decreases at 0
Table 6. Instances used for the SCP.
Table 6. Instances used for the SCP.
InstanceMNDensity (%)Optimum
4120010002.00429
4220010002.00512
5120020002.00253
5220020002.00302
6120010005.00138
6220010005.00146
A130030002.00253
A230030002.00252
B130030005.0069
B230030005.0076
C140040002.00227
C240040002.00219
D140040005.0060
D240040005.0066
NRE1500500010.0029
NRE2500500010.0030
NRF1500500020.0014
NRF2500500020.0015
NRG1100010,0002.00176
NRG2100010,0002.00154
NRH1100010,0005.0063
NRH2100010,0005.0063
Table 7. Instances used for the USCP.
Table 7. Instances used for the USCP.
InstanceMNDensity (%)Optimum
U4120010002.0038
U5120020002.0034
U6120010005.0021
UA130030002.0039
UB130030005.0022
UC140040002.0043
UD140040005.0024
UNRE1500500010.0017
UNRF1500500020.0010
UNRG1100010,0002.0061
UNRH1100010,0005.0034
CLR1051121012.3025
CLR11102333012.4023
CLR12204749512.5023
CYC062401922.1060
CYC076724480.90144
CYC08179210240.40344
Table 8. Fitness results per SCP instance.
Table 8. Fitness results per SCP instance.
MHInstanceOpt.BestWorstAvg. FitnessStd. Fitness
SBOA41429433.0433.0433.00.0
GWO433.0438.0433.4191.29
PSO433.0437.0433.2580.983
SBOA42512525.0527.0525.9680.999
GWO525.0527.0525.9350.982
PSO525.0527.0526.290.957
SBOA51253267.0269.0267.290.52
GWO267.0268.0267.1940.395
PSO267.0269.0267.3230.59
SBOA52302315.0322.0318.3871.979
GWO315.0323.0319.6772.74
PSO315.0323.0318.7422.17
SBOA61138141.0145.0141.9351.605
GWO141.0145.0142.9681.926
PSO141.0145.0141.8391.568
SBOA62146148.0154.0149.6772.277
GWO148.0154.0150.3232.085
PSO148.0154.0149.3872.058
SBOAA1253257.0257.0257.00.0
GWO257.0257.0257.00.0
PSO257.0257.0257.00.0
SBOAA2252258.0265.0260.3871.58
GWO256.0265.0260.4521.997
PSO258.0265.0261.3551.976
SBOAB16969.071.069.3870.79
GWO69.071.069.6450.9
PSO69.071.069.3870.79
SBOAB27676.077.076.1290.335
GWO76.082.076.4191.185
PSO76.077.076.1610.368
SBOAC1227231.0234.0233.0650.504
GWO231.0235.0232.8060.858
PSO231.0234.0232.9680.647
SBOAC2219221.0226.0222.3871.559
GWO221.0228.0222.5161.949
PSO221.0229.0222.6452.088
SBOAD16060.065.061.3871.006
GWO60.063.061.1610.919
PSO60.063.061.3231.028
SBOAD26667.069.067.710.727
GWO67.069.067.7740.791
PSO67.069.067.6770.69
SBOANRE12929.029.029.00.0
GWO29.030.029.0320.177
PSO29.029.029.00.0
SBOANRE23030.032.031.00.803
GWO30.033.031.0650.878
PSO30.032.030.710.632
SBOANRF11414.014.014.00.0
GWO14.015.014.0320.177
PSO14.014.014.00.0
SBOANRF21515.015.015.00.0
GWO15.016.015.0320.177
PSO15.015.015.00.0
Table 9. Fitness results per SCP instance.
Table 9. Fitness results per SCP instance.
MHInstanceOpt.BestWorstAvg. FitnessStd. Fitness
SBOANRG1176178.0184.0180.9681.492
GWO179.0185.0181.4841.72
PSO179.0184.0181.01.666
SBOANRG2154158.0161.0159.3550.598
GWO158.0162.0159.6770.929
PSO158.0162.0159.5161.043
SBOANRH16364.067.065.01.016
GWO64.067.065.5810.943
PSO64.067.065.4840.875
SBOANRH26364.067.065.0651.014
GWO64.067.065.1610.954
PSO64.066.065.1940.895
Table 10. Time per SCP instance.
Table 10. Time per SCP instance.
MHInstanceMin. Time (s)Max. Time (s)Avg. Time (s)Std. Time (s)
SBOA41157.704213.129183.11616.07
GWO153.028187.257173.5569.494
PSO154.711170.525165.334.352
SBOA42142.817218.333174.87519.872
GWO158.63182.134171.8146.122
PSO158.201167.463164.0512.481
SBOA51253.613374.187313.78730.033
GWO236.581282.194267.52914.559
PSO237.17271.079259.1718.931
SBOA52253.449379.583314.93234.542
GWO233.655283.615266.8813.466
PSO231.334271.309258.57710.615
SBOA6198.435157.754122.68113.947
GWO110.113123.04116.263.258
PSO102.526116.973109.842.943
SBOA6297.503142.421119.43513.442
GWO107.432121.659114.6063.808
PSO104.62115.671110.3253.257
SBOAA1776.617992.96886.2152.953
GWO548.174603.27583.82919.38
PSO547.932583.488564.8217.163
SBOAA2760.15978.413885.57558.469
GWO559.968613.854589.37714.932
PSO540.209587.692569.52311.293
SBOAB1492.081598.676549.8329.989
GWO340.665377.974355.2767.95
PSO331.606369.327355.7887.879
SBOAB2509.619609.138566.10727.889
GWO345.028399.848368.4759.286
PSO339.963369.457350.856.382
SBOAC11624.8842006.591800.822104.886
GWO1055.1711174.3781109.4226.687
PSO1057.7411215.1571083.76732.262
SBOAC21494.1191967.281785.954136.993
GWO1019.1271169.1761117.34230.362
PSO1034.2991112.9821075.74115.09
SBOAD1932.5271160.3151073.95565.504
GWO640.738702.894661.20113.484
PSO605.31667.894638.65613.116
SBOAD2943.9021147.3181060.20252.619
GWO608.824691.086646.15816.23
PSO594.241664.729628.35720.317
Table 11. Time per SCP instance.
Table 11. Time per SCP instance.
MHInstanceMin. Time (s)Max. Time (s)Avg. Time (s)Std. Time (s)
SBOANRE11163.3221343.9311301.28843.887
GWO709.169820.318756.53927.69
PSO681.815749.788714.20514.854
SBOANRE21111.7171419.1791314.76674.668
GWO683.767845.554754.17940.734
PSO669.604855.131758.74149.429
SBOANRF11032.8681327.7121225.84270.654
GWO565.83681.723619.61926.624
PSO546.366661.886604.70428.136
SBOANRF21099.4211391.4461272.06876.519
GWO573.592701.44633.94630.842
PSO588.031713.309635.21226.661
SBOANRG117,984.63754,355.9726,594.34512,842.352
GWO30,569.10862,689.42253,273.18611,121.198
PSO32,644.97361,491.4855,582.6786414.388
SBOANRG217,185.13152,717.74527,342.8289971.869
GWO25,719.30961,376.89153,300.76210,461.981
PSO31,367.54759,373.08754,333.0415643.738
SBOANRH110,512.10632,739.21916,587.8156416.691
GWO15,165.46134,183.71730,025.2514855.541
PSO21,114.65135,256.65431,207.5082997.95
SBOANRH210,276.13629,402.97215,774.8595539.228
GWO14,111.58234,673.10329,343.2995608.793
PSO19,751.68734,721.17430,023.6693445.378
Table 12. Fitness results per USCP instance.
Table 12. Fitness results per USCP instance.
MHInstanceOpt.BestWorstAvg. FitnessStd. Fitness
SBOAU413839.041.039.8390.627
GWO39.043.040.8711.07
PSO39.041.039.6450.598
SBOAU513435.037.035.5810.555
GWO35.038.036.290.681
PSO35.036.035.6130.487
SBOAU612121.022.021.290.454
GWO21.023.021.8390.514
PSO21.022.021.290.454
SBOAUA13939.041.040.3230.642
GWO40.043.041.3870.656
PSO40.041.040.4520.498
SBOAUB12222.024.022.8060.591
GWO22.024.023.3870.605
PSO22.023.022.7740.418
SBOAUC14343.045.044.3230.532
GWO44.048.046.3870.79
PSO44.046.044.7740.551
SBOAUD12425.026.025.2260.418
GWO25.027.026.1940.47
PSO25.026.025.0650.246
SBOAUNRE11717.018.017.5160.5
GWO17.018.017.7740.418
PSO17.017.017.00.0
SBOAUNRF11010.011.010.9030.296
GWO10.011.010.7740.418
PSO10.011.010.290.454
SBOAUNRG16162.063.062.4840.5
GWO63.068.066.1940.895
PSO62.064.063.290.579
SBOAUNRH13434.035.034.3230.467
GWO35.037.035.8390.514
PSO34.035.034.9350.246
SBOACLR102425.027.025.0970.39
GWO25.027.025.3870.748
PSO25.027.025.290.632
SBOACLR112323.023.023.00.0
GWO23.027.024.8391.919
PSO23.027.023.3871.183
SBOACLR122323.028.024.2581.586
GWO23.029.024.7742.121
PSO23.026.023.9681.257
SBOACYC066060.062.061.0970.817
GWO60.063.061.4520.978
PSO60.062.061.2260.831
SBOACYC07144147.0153.0150.8711.362
GWO147.0154.0151.9031.399
PSO144.0153.0150.4841.563
SBOACYC08344357.0365.0361.1292.324
GWO358.0372.0363.4192.721
PSO358.0367.0362.2581.883
Table 13. Time per USCP instance.
Table 13. Time per USCP instance.
MHInstanceMin. Time (s)Max. Time (s)Avg. Time (s)Std. Time (s)
SBOAU41127.133165.55144.0419.186
GWO122.253138.701128.1833.852
PSO117.762144.135128.7645.967
SBOAU51240.841298.561261.0114.843
GWO212.336229.391219.3894.053
PSO219.169248.563233.3248.576
SBOAU6187.424119.1498.9127.76
GWO68.66686.00477.1824.104
PSO62.44478.62573.1113.677
SBOAUA1596.009988.713718.13281.884
GWO271.85342.57299.67321.205
PSO155.901331.492281.7435.986
SBOAUB1261.626572.229354.13196.36
GWO163.543178.737169.9193.133
PSO163.947189.483177.0696.403
SBOAUC143.045.044.3230.532
GWO44.048.046.3870.79
PSO44.046.044.7740.551
SBOAUD1466.1611065.89691.041212.326
GWO280.281314.57292.7196.333
PSO280.569329.153302.32512.853
SBOAUNRE1863.6631732.2481337.399330.758
GWO410.564705.947434.39150.715
PSO423.553497.961460.51718.005
SBOAUNRF11070.3551650.3441551.32130.705
GWO424.606683.387589.4795.124
PSO449.775535.65485.25425.765
SBOAUNRG15414.28714,309.317441.4613000.263
GWO8737.81818,034.4615,638.1923071.343
PSO13,479.71921,729.84118,346.1112374.816
SBOAUNRH14168.71116,435.7247474.9383805.651
GWO4775.53611,231.7339150.0291921.657
PSO6517.54214,954.86911,776.281780.385
Table 14. Time per USCP instance.
Table 14. Time per USCP instance.
MetaheuristicInstanceMin. Time (s)Max. Time (s)Avg. Time (s)Std. Time (s)
SBOACLR1058.21286.82575.0666.622
GWO23.83129.50626.3741.434
PSO29.71136.12332.6891.824
SBOACLR11126.482196.981141.17611.534
GWO53.55869.38461.0664.716
PSO66.43885.25671.3014.728
SBOACLR12604.7511064.573801.739149.713
GWO338.341500.013407.82643.52
PSO501.697604.357550.12922.986
SBOACYC0653.30793.84668.638.254
GWO22.63826.96424.3171.085
PSO23.31228.31325.791.229
SBOACYC07282.102319.146306.656.968
GWO138.408163.015148.0456.521
PSO146.28175.9158.4265.986
SBOACYC08 5163.5094655.015226.937
GWO3080.4233591.2953391.635126.319
PSO3562.6423902.3473747.50584.342
Table 15. Average p-value of the SBOA compared with PSO and GWO for SCP41, ScP42, SCP51, SCP52, SCP61 and SCP62.
Table 15. Average p-value of the SBOA compared with PSO and GWO for SCP41, ScP42, SCP51, SCP52, SCP61 and SCP62.
4.14.25.15.26.16.2
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.050.03-≥0.05≥0.05
PSO≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-0.02≥0.05-≥0.05
GWO≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-0.030.02-≥0.05≥0.05-
Table 16. Average p-value of the SBOA compared with PSO and GWO for a1, a2, b1, b2, c1, c2, d1 and d2.
Table 16. Average p-value of the SBOA compared with PSO and GWO for a1, a2, b1, b2, c1, c2, d1 and d2.
a.1a.2b.1b.2
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-≥0.05≥0.05-0.05--≥0.05≥0.05-≥0.05≥0.05
PSO≥0.05-≥0.05≥0.05-≥0.051.0-≥0.05≥0.05-≥0.05
GWO≥0.05≥0.05--≥0.05-0.2≥0.05-≥0.05≥0.05-
c.1c.2d.1d.2
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05
PSO≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05
GWO≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-
Table 17. Average p-value of the SBOA compared with PSO and GWO for nre1, nre2, nrf1, nrf2, nrg1, nrg2, nrh1 and nrh2.
Table 17. Average p-value of the SBOA compared with PSO and GWO for nre1, nre2, nrf1, nrf2, nrg1, nrg2, nrh1 and nrh2.
nre1nre2nrf1nrf2
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05
PSO≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-≥0.05
GWO≥0.05≥0.05-≥0.05≥0.05-≥0.05≥0.05-0.33≥0.05-
nrg1nrg2nrh1nrh2
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-≥0.05≥0.05-≥0.05≥0.05-0.0030.02-≥0.05≥0.05
PSO≥0.05-≥0.05≥0.05-≥0.050.03-≥0.05≥0.05-≥0.05
GWO≥0.05≥0.05-≥0.05≥0.05-0.02≥0.05-≥0.05≥0.05-
Table 18. Average p-value of the SBOA compared with PSO and GWO for u41, u51, u61, ua1, ub1, uc1, uclr10, uclr12, ucyc06, ucyc07, ud1, unre1, unrf1, unre1 and unrh1.
Table 18. Average p-value of the SBOA compared with PSO and GWO for u41, u51, u61, ua1, ub1, uc1, uclr10, uclr12, ucyc06, ucyc07, ud1, unre1, unrf1, unre1 and unrh1.
u41u51u61ua1
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-≥0.050.0-≥0.050.0-≥0.050.0-≥0.050.0
PSO≥0.05-0.0≥0.05-0.0≥0.05-0.0≥0.05-0.0
GWO0.00.0-0.00.0-0.00.0-0.00.0-
ub1uc1uclr10uclr12
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-≥0.050.0-0.00.0-≥0.05≥0.05-0.5≥0.05
PSO≥0.05-0.00.0-0.0≥0.05-≥0.05≥0.05-≥0.05
GWO0.00.0-0.00.0-≥0.05≥0.05-≥0.05≥0.05-
ucyc06ucyc07ucyc07ud1
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-≥0.050.12-≥0.050.0-≥0.050.0-≥0.050.0
PSO≥0.05-≥0.05≥0.05-0.0≥0.05-0.0≥0.05-0.0
GWO≥0.05≥0.05-0.00.0-0.00.0-0.00.0-
unre1unrf1unre1unrh1
SBOPSOGWOSBOPSOGWOSBOPSOGWOSBOPSOGWO
SBO-0.00.04-0.0≥0.05-0.00.0-0.00.0
PSO0.0-0.00.0-0.00.0-0.00.0-0.0
GWO0.040.0-≥0.050.0-0.00.0-0.00.0-
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MDPI and ACS Style

Crawford, B.; Cisternas-Caneo, F.; Soto, R.; Mac-lean, C.P.T.; Lara Arce, J.; Solís-Piñones, F.; Astorga, G.; Giachetti, G. Binary Secretary Bird Optimization Algorithm for the Set Covering Problem. Mathematics 2025, 13, 2482. https://doi.org/10.3390/math13152482

AMA Style

Crawford B, Cisternas-Caneo F, Soto R, Mac-lean CPT, Lara Arce J, Solís-Piñones F, Astorga G, Giachetti G. Binary Secretary Bird Optimization Algorithm for the Set Covering Problem. Mathematics. 2025; 13(15):2482. https://doi.org/10.3390/math13152482

Chicago/Turabian Style

Crawford, Broderick, Felipe Cisternas-Caneo, Ricardo Soto, Claudio Patricio Toledo Mac-lean, José Lara Arce, Fabián Solís-Piñones, Gino Astorga, and Giovanni Giachetti. 2025. "Binary Secretary Bird Optimization Algorithm for the Set Covering Problem" Mathematics 13, no. 15: 2482. https://doi.org/10.3390/math13152482

APA Style

Crawford, B., Cisternas-Caneo, F., Soto, R., Mac-lean, C. P. T., Lara Arce, J., Solís-Piñones, F., Astorga, G., & Giachetti, G. (2025). Binary Secretary Bird Optimization Algorithm for the Set Covering Problem. Mathematics, 13(15), 2482. https://doi.org/10.3390/math13152482

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