1 Introduction

Intelligent swarm optimization algorithms have become increasingly common due to their success in solving real-world problems. Intelligent swarm algorithms are created by imitating social behavior in the flock. Particularly in the exploitation and exploration movements of the continuous search space, these behavior styles are imitated. Many intelligent swarm algorithms have been proposed in the literature in recent years. For example, Dwarf Mongoose Optimization (DMO) algorithm is a newly proposed intelligent swarm optimization algorithm in recent years [1]. It is a heuristic algorithm created by imitating dwarf mongooses lifeforms. The Coati Optimization Algorithm (COA) has been proposed by imitating the hunting behavior of coatis in nature [2]. Snake Optimizer (SO) is an intelligent swarm algorithm created by imitating the mating behavior of snakes [3]. The Mountain Gazelle Optimization (MGO) algorithm is inspired by the social behavior of mountain gazelles in natural life [4]. Aquila Optimizer has been created by imitating the behavior of the aquila creature in natural life [5]. These examples are swarm-based algorithms selected from the literature and newly proposed in the last few years. It is possible to find many examples of swarm-based heuristic algorithms like these in the literature.

When heuristic algorithms are first proposed, they are mostly recommended for continuous optimization problems. Their success is made on continuous optimization problems. But real-world problems are not always problems with continuous variables. Problems whose search space consists of discrete variables are called discrete optimization problems. Binary optimization is a special form of discrete optimization. In binary optimization, search space variable values consist of binary values. Many real-world problems are solved more efficiently by using these binary variables (knapsack problem [6], feature selection problem [7], uncapacitated facility location problem [8], etc.). A proposed heuristic algorithm for continuous optimization can be re-updated to solve binary optimization problems. In this study, Dwarf Mongoose Optimization (DMO), which is a newly proposed intelligent swarm-based heuristic algorithm in recent years, was chosen. When the literature is examined, it is seen that there is no transfer function-based binary versions of DMO to solve binary optimization problems. Taking advantage of this literature gap, a Binary DMO is proposed for binary optimization problems. In DMO, transfer functions are used while converting continuous search space to binary search space. The success of a transfer function directly affects the binary algorithm. Therefore, many transfer functions have been proposed in the literature (S-shaped [6, 7], V-shaped [6, 7], Z-shaped [9], U-shaped [10], Taper-shaped [11], X-shaped [12], etc.). The success of a transfer function depends on its exploration and exploitation capabilities in binary search space. To maximize the success of a binary optimization algorithm, the right transfer function must be chosen. In this study, twelve transfer functions that have been newly proposed in recent years have been selected (four Z-shaped, four U-shaped, and four Taper-shaped transfer functions). The success of BinDMO is examined in detail for each transfer function in this study. The achievements of twelve different BinDMO variants were tested on thirteen different unimodal and multimodal classical benchmark functions. Binary values obtained with BinDMO were converted back to continuous values with a special transformation benchmark equation, and calculations were carried out [8, 13]. The effectiveness of population sizes on the effectiveness of BinDMO was also investigated. The most successful variation of BinDMO and the heuristic algorithms (Snake Optimizer (SO) [3], Prairie Dog Optimization Algorithm (PDO) [28], and Ali Baba and the forty thieves (AFT) [29]) selected from the literature were run at similar parameter settings on CEC-2017 benchmark functions for a fair comparison, and comparisons were made. Table 1 shows studies on other binary optimization using binary DMO and contributions of the current study.

Table 1 Summary of related works
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The motivations and contributions of this study can be summarized as follows:

  • This study proposed the first transfer function-based binary DMO (BinDMO) algorithm and tested its effectiveness on thirteen unimodal and multimodal classical benchmark functions and the CEC-2017 test functions. When Table 1 is examined, other binary versions of DMO have been proposed for the feature selection problem.

  • In this study, the effect of different transfer functions on the success of BinDMO was examined in detail. The transfer functions selected in this study have been newly proposed in recent years (four Z-shaped, four U-shaped, and four Taper-shaped). The results obtained constitute a source for literature comparisons. When Table 1 is examined, in other binary DMO versions, transfer functions are not used when converting the continuous search space to binary search space.

  • The success of BinDMO has been compared with various heuristic algorithms (SO, PDO, and AFT) selected from the literature and proposed in recent years. BinDMO managed to enter the top in the CEC-2017 test functions.

The remainder of this work is structured as follows: DMO, BinDMO, and transfer functions are explained in detail in Sect. 2. In Sect. 3, the performance tests of BinDMO on classical thirteen unimodal and multimodal test functions for twelve different transfer functions are performed and compared with each other. The most successful BinDMO variation was determined, and this variation was compared with Binary SO, Binary PDO, and Binary AFT heuristic algorithms in CEC-2017 benchmarks. Convergence graphs were drawn. In Sect. 4, obtained results were interpreted and discussed.

1.1 Related works

Aldosari et al. proposed a normal distributed dwarf mongoose optimization algorithm for global optimization and data clustering applications [14]. They tested the success of their proposed algorithm on twenty-three test functions and eight data clustering tasks. Sadoun et al. proposed a modification of long short-term memory (LSTM) using dwarf mongoose optimization (DMO) [15]. Akinola et al. proposed a new hybrid method called the BDMSAO which combines the binary variants of the DMO (or BDMO) and simulated annealing (SA) algorithm [16]. In hybrid method, the BDMO is employed and used as the global search method and the simulated annealing (SA) as the local search component to enhance the limited exploitative mechanism of the BDMO [16]. The new method was tested on eighteen (18) UCI machine learning datasets of low and medium dimensions [16]. It was also tested using three high-dimensional medical datasets to assess its robustness [16]. Agushaka et al. proposed improved dwarf mongoose optimization for constrained engineering design problems [17]. Alissa et al. proposed a new model of dwarf mongoose optimization with machine learning-driven ransomware detection (DWOML-RWD) [18]. The presented DWOML-RWD model was mainly developed for the recognition and classification of goodware/ransomware. Mehmood et al. proposed dwarf mongoose optimization metaheuristics for autoregressive exogenous model identification [19]. Agushaka et al. proposed an advanced dwarf mongoose optimization for solving CEC 2011 and CEC 2017 benchmark problems [20]. Alrayes et al. proposed dwarf mongoose optimization-based secure clustering with routing technique in internet of drones [21]. Balasubramaniam et al. enabled deep learning for heart disease detection with feature selection and dwarf mongoose optimization [22]. They proposed DMOA-SqueezeNet method for feature selection problem [22]. The heart disease was predicted using SqueezeNet adjusted by the dwarf mongoose optimization algorithm (DMOA) [22]. Dora et al. proposed a solution of reactive power dispatch problems using enhanced dwarf mongoose optimization algorithm [23]. Akinola et al. proposed a binary version of the dwarf mongoose optimization (BDMO) to solve 8 high-dimensional feature selection problems [30]. The feature selection problem is to select appropriate feature subsets [30]. Its main goal is to eliminate noisy, irrelevant, and redundant feature subsets that can negatively impact the accuracy of the learning model [30]. Thus, classification performance is improved without loss of information [30]. For this reason, binary algorithms are frequently used to determine optimal subsets [30]. The effectiveness of BDMO was tested using 18 high-dimensional datasets [30]. Elaziz et al. proposed a DMOAQ [31]. DMOAQ is a new feature selection technique [31]. They modified the performance of the dwarf mongoose optimization (DMO) algorithm using quantum-based optimization (QBO) [31]. It is tested with well-known high-dimensional datasets [31]. Al-Shourbaji et al. proposed an AEO-DMOA for feature selection (FS) problem [32]. It consists of a combination of Artificial Ecosystem-based Optimization (AEO) and dwarf mongoose optimization algorithm (DMOA) [32]. The performance of the AEO-DMOA is investigated on seven datasets and a eighteen CEC2017, and ten CEC2019 benchmark functions [32].

2 Materials and methods

2.1 Basic dwarf mongoose optimization algorithm (DMO)

The dwarf mongoose is found in the semi-desert and savanna scrub regions of Africa. The dwarf mongoose is the smallest African carnivore, with an average total body length of 47 cm and an adult weight of about 400 gr. [24]. Dwarf mongooses have certain territories. They mark vertical or horizontal objects in their area. Thus, dwarf mongooses feel more secure. All family members contribute to the marking of the territory. They contribute to the marking depending on the priority order in the group [25]. The hunting mode of dwarf mongoose is the mode in which small prey is caught [1]. Some of the mongoose herd, consisting of male and female individuals, serve as babysitters. Dwarf mongooses do not build nests for their young. Dwarf mongooses cannot catch prey large enough to feed the entire group. They do not have a deadly bite. This led the dwarf mongoose to adopt a semi-nomadic life. This lifestyle avoids overuse of a particular area [1].

2.2 Mathematical Model of DMO

The DMO algorithm was created by imitating the lifestyles of dwarf mongooses. DMO optimization begins with the initialization of the candidate population of mongoose as shown in Eqs. 1–2. The population is randomly generated between the lower (VarMin) and upper (VarMax) bounds of the search space.

$$X = \left[ {\begin{array}{*{20}c} {x_{1,1 } x_{1,2} \ldots x_{{1,{\text{Dim}} - 1}} x_{{1,{\text{Dim}}}} } \\ {x_{2,1 } x_{2,2} \ldots x_{{2,{\text{Dim}} - 1}} x_{{2,{\text{Dim}}}} } \\ \vdots \\ {x_{i,j} } \\ \vdots \\ {x_{{{\text{Pop}},1 }} x_{{{\text{Pop}},2}} \ldots x_{{{\text{Pop,Dim}} - 1}} x_{{\text{Pop,Dim}}} } \\ \end{array} } \right]$$
(1)
$$x_{i,j} = {\text{unifrnd}}\left( {{\text{VarMin}},{\text{VarMax}},{\text{Dim}}} \right)$$
(2)

where \(X\) is current population and \(x_{i,j}\) is the position of the jth dimension of the ith population. Pop is population size, and Dim is dimension size [1].

The dwarf mongoose population is divided into three groups. These are alpha group, scouts, and babysitters.

Alpha group: After the initial random population is created, the fitness value of each individual in the population is calculated. By using these fitness values, the alpha female (\(\alpha\)) value is calculated according to Eq. 3 [1].

$$\alpha = \frac{{fit_{i} }}{{\sum\nolimits_{{i = 1}}^{{Pop}} {fit_{i} } }}$$
(3)

The number of mongooses in the alpha group corresponds to Pop–Babysitter (bs). Babysitter is the number of babysitters. The vocalization of the alpha female, which keeps the family on a path, is indicated by peep. The number of babysitters was chosen as 3, and the peep value was chosen as 2 [1]. Initially, the first sleep mound (sm) is set to ∅ and each mongoose sleeps in it. The candidate uses Eq. 4 to construct a food position [1].

$$X_{i + 1} = X_{i} + {\text{phi}}*{\text{peep}}$$
(4)

where \({\text{phi}}\) is a uniformly distributed random number [− 1, 1]. After each iteration, the sm value is updated again according to Eqs. 5–6. \(\varphi\) is the average value of the sleeping mound [1].

$$sm_{i} = \frac{{fit_{{i + 1}} + fit_{i} }}{{\max \left\{ {\left| {fit_{{i + 1}} + fit_{i} } \right|} \right\}}}$$
(5)
$$\varphi = \frac{{\mathop \sum \nolimits_{i = 1}^{Pop} sm_{i} }}{n}$$
(6)

Scout group: The scout group search for a new sleeping mound. Mongooses do not reuse their previous sleeping mound. Scouts also perform in foraging. If scouts get far enough in search space, they discover a new sleeping mound. Scout group movement is shown in Eqs. 7–9 [1].

$$X_{i + 1} = \left\{ {\begin{array}{*{20}c} {X_{i} - {\text{CF}}*{\text{phi}}*{\text{rand}}*\left[ {X_{i} - \vec{M}} \right] \;\;\;\;\; {\text{if }}\varphi_{i + 1} > \varphi_{i} } \\ {X_{i} + {\text{CF}}*{\text{phi}}*{\text{rand}}*\left[ {X_{i} - \vec{M}} \right] \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{else}}} \\ \end{array} } \right.$$
(7)
$${\text{CF}} = \left( {1 - \frac{{{\text{iter}}}}{{{\text{Max}}_{{{\text{iter}}}} }}} \right)^{{\left( {2*\frac{{{\text{iter}}}}{{{\text{Max}}_{{{\text{iter}}}} }}} \right)}}$$
(8)
$$\vec{M} = \mathop \sum \limits_{i = 1}^{Pop} \frac{{X_{i} \times sm_{j} }}{{X_{i} }}$$
(9)

where \({\text{rand}}\) is a random number between [0, 1]. \({\text{iter}}\) is the current number of iterations, and \({\text{Max}}_{{{\text{iter}}}}\) is the maximum number of iterations.

The babysitters: The number of babysitters is calculated based on population size, and they reduce the overall population size. The babysitters fitness weight is set to zero. This causes the average weight of the alpha group to decrease in the next iteration. This means that group movement is blocked [1].

The DMO algorithm is shown in Algorithm 1.

figure a

2.3 Definitions of binary DMO (BinDMO) and transfer functions

Dwarf mongoose optimization (DMO) algorithm is a swarm-based heuristic algorithm that has been proposed for continuous optimization problems in recent years. In this study, DMO algorithm binary version (BinDMO) was obtained with some updates. The reason for choosing DMO algorithm in this study is that its transfer function-based binary version has not been proposed in the literature. In this study, the continuous search space of DMO has been converted to binary space by using twelve newly proposed transfer functions in recent years. The details of the transfer functions selected in this study are given in Table 2. In BinDMO, twelve transfer functions map the continuous search space to the binary search space. Transfer functions directly affect the success of a binary algorithm. That's why choosing the right transfer function in binary optimization is very important. In this study, the effect of each transfer function on BinDMO was analyzed in detail.

Table 2 Transfer functions (Z-shaped, U-shaped, Tapper-shaped) (A = [−5,5]) [6, 7, 9,10,11,12]
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Initially, BinDMO starts working as a continuous heuristic algorithm. The working steps of BinDMO are the same as DMO until the fitness function is calculated. Before calculating the fitness value of a population individual, it is first converted to binary variable space with the help of transfer functions. Then, the fitness value is calculated by converting it back to the continuous search space with Eq. 10 [13]. The comparison and displacement of new candidate solutions within the population with existing individuals is the same as in DMO.

$$C_{i,j} = {\text{ VarMin}}_{i,j} + \frac{{\left( {{\text{Var}}\max i_{i,j} - {\text{VarMin}}_{i,j} } \right) \times {\text{DecValue}}_{i,j} }}{{{\text{MaxValue}}}}$$
(10)

where \(C_{i,j}\) denotes the continuous value of the jth dimension of the ith dwarf mongoose. The \({\text{VarMin}}_{i,j}\) and \({\text{VarMax}}_{i,j}\) denote lower and upper bounds of the jth dimension of the ith dwarf mongoose. \({\text{DecValue}}_{i,j}\) denotes the integer value in the decimal number system of the binary number in the jth dimension of the ith dwarf mongoose. \({\text{MaxValue}}\) denotes the maximum decimal integer value that the binary number can take, according to the bit length determined for each dimension [13].

The BinDMO algorithm is shown in Algorithm 2.

figure b

3 Conclusions and discussion

All the applications in this section are coded on the MATLAB program, and a PC with a Corel i5 processor and 12 GB ram has been chosen for their performance. The properties of the classical functions are given in Table 3. Eight of the functions are unimodal, and the five of the functions are multimodal. In addition, all of the functions are variable dimension. These benchmark test functions are available at https://www.sfu.ca/~ssurjano/optimization.html.

Table 3 Classical benchmark functions
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3.1 Parameter analyzes

In this subsection, the impact of population sizes on the effectiveness of BinDMO is analyzed. Five different population values (10, 20, 30, 40, and 50) were chosen randomly. In the tests performed, the maximum iteration value was taken as 300, the problem dimension was taken as 10, and the transfer function was taken as T1 (Z1-shaped). nBabysitter number has set as 3 and peep value has set as 2. The peep and nBabysitter parameter values are used as in the code shared by the Agushaka et al. (https://www.mathworks.com/matlabcentral/fileexchange/105125-dwarf-mongoose-optimization-algorithm) [1]. Thirteen different unimodal and multimodal classical benchmark test functions have been selected. Each test function was run 20 times independently, and the average (Ave) and standard deviation values (Std) of the obtained results were calculated. The best results are marked in bold in the result tables. The results of the average and standard deviations are shown in Table 4. When Table 4 is examined, the success of the algorithm increases as the population size increases. The most successful results were obtained when the population size was 50.

Table 4 The average (Ave) and standard deviation (Std) results of the BinDMO for different population sizes (Pop = population size) (Transfer function = T1)
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3.2 Comparison of BinDMO for Z-shaped, U-shaped, and taper-shaped transfer functions on classical benchmark functions

In this subsection, BinDMO was run on thirteen unimodal and multimodal benchmark functions for twelve different Z-shaped, U-shaped, and Taper-shaped transfer functions and the most successful transfer function was determined. The parameter settings set for BinDMO are shown in Table 5. BinDMO was run 20 times for each benchmark function. The results are shown in Tables 6 and 7. Calculations were made on the results obtained according to five different comparison criteria (best, worst, average (Ave), standard deviation (Std), and time). Figures 1, 2, 3, 4, 5 and 6 show the results of the total average, total standard deviation, and total time of transfer functions and groups of transfer functions. According to the total average results, the most successful BinDMO variation was the one using the T1 transfer function and the least successful BinDMO variation was the one using the T5 transfer function. When evaluated in terms of transfer function groups, the most successful transfer function group is Z-shaped transfer functions (according to total average results). According to the total standard deviation results, the most successful BinDMO variation was the one using the T1 transfer function. When evaluated in terms of transfer function groups, the most successful transfer function group is Z-shaped transfer functions (according to total standard deviation results). According to the total time results, the most successful BinDMO variation was the one using the T10 transfer function. When evaluated in terms of transfer function groups, the most successful transfer function group is Taper-shaped transfer functions (according to total time results). Figure 7 shows convergence graphs for the variants of the BinDMO on thirteen different unimodal and multimodal classic benchmark functions (Dimension = 10).

Table 5 Parameter settings
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Table 6 BinDMO results for different transfer functions on classical benchmark functions (Dim = 10)
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Table 7 BinDMO results for different transfer functions on classical benchmark functions (Dim = 10)
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Fig. 1
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The total average result graph for the different transfer functions (Dim = 10)

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Fig. 2
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The total average result graph for groups of the different transfer functions (Dim = 10)

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Fig. 3
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The total standard deviation result graph for the different transfer functions (Dim = 10)

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Fig. 4
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The total standard deviation result graph for groups of the different transfer functions (Dim = 10)

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Fig. 5
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The total time result graph for the different transfer functions (Dim = 10)

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Fig. 6
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The total Time result graph for groups of the different transfer functions (Dim = 10)

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Fig. 7
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Convergence graphs for the variants of the BinDMO on classic benchmark functions (Dim = 10)

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Table 8 shows the Wilcoxon signed test results between the T1 transfer function and other transfer functions. Wilcoxon signed test was performed to find out whether there is a semantic difference between two different transfer function result datasets [7, 26]. Thus, the similarities between the results of the BinDMO variations were examined. ( +) indicates that there is a semantic difference, and (−) indicates that there is no semantic difference. According to Table 10, Z-shaped transfer function results obtained different results from U-shaped and Taper-shaped transfer function results.

Table 8 The Wilcoxon signed test results for the best variation of the BinDMO with other variations of the BinDMO on (Dim = 10)
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Table 9 shows the average, standard deviation, and time results of the best variation of the BinDMO (with T1) for different dimensions (10, 20, and 30) on classical benchmark functions.

Table 9 The average (Ave), standard deviation (Std), and time results of the best variation of the BinDMO for different dimensions on classical benchmark functions (Dim = {10, 20, and 30})
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3.3 Comparison of BinDMO with other current state-of-the-art algorithms on CEC-2017 benchmark functions

BinDMO and some current heuristic algorithms selected from the literature were compared in twenty-nine CEC-2017 benchmark functions in terms of four different comparison criteria (average (Ave), standard deviation (Std), time, and best). CEC-2017 benchmark functions details are shown in Table 10 [27]. Comparison algorithms Snake Optimizer (SO) [3], Prairie Dog Optimization Algorithm (PDO) [28], and Ali Baba and the forty thieves (AFT) [29]. All algorithms were run 20 times for each benchmark function. The parameter settings used in the comparisons are shown in Table 11. Comparison results are shown in Tables 12 and 13. Table 14 shows the one-way ANOVA test results among the BinDMO, Binary SO, Binary PDO, and Binary AFT on CEC-2017 benchmark functions. Table 15 shows the Wilcoxon signed test results between the BinDMO with other current state-of-the-art algorithms on CEC-2017 benchmark functions. Figure 8 shows the convergence graphs for the BinDMO with other current state-of-the-art algorithms on five different CEC-2017 benchmark functions (F1, F5, F10, F15, and F25). Figure 9 shows boxplot graphs for the BinDMO with other current state-of-the-art algorithms on CEC-2017 benchmark functions. In order to make a fair comparison, Z1-shaped transfer function is used in all heuristic algorithms selected for comparison.

Table 10 CEC-2017 test functions [27]
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Table 11 Parameter settings
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Table 12 Comparison results of BinDMO with other current state-of-the-art algorithms on CEC-2017 benchmark functions (Dim = 10)
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Table 13 Comparison results of BinDMO with other current state-of-the-art algorithms on CEC-2017 benchmark functions (Dim = 10)
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Table 14 Analysis of one-way ANOVA test results range BinDMO, Binary SO, Binary PDO, and Binary AFT
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Table 15 The Wilcoxon signed test results for the BinDMO with other current state-of-the-art algorithms on CEC-2017 benchmark functions (Dim = 10)
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Fig. 8
figure 8figure 8

Convergence graphs for the BinDMO with other current state-of-the-art algorithms on five different CEC-2017 benchmark functions (F1, F5, F10, F15, and F25) (Dim = 10)

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Fig. 9
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Boxplot graphs for the BinDMO, Binary SO, Binary PDO, Binary AFT on CEC-2017 benchmark functions (Dim = 10)

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BinDMO was the most successful heuristic algorithm according to average result ranks. After BinDMO, Binary SO is in the second place. The most unsuccessful heuristic was Binary AFT. The fastest working heuristic algorithm according to time result ranks is Binary AFT. It is followed by Binary SO and BinDMO, respectively.

In addition to statistical tests such as best, average (Ave), and standard deviation (Std), one-way ANOVA test was also applied on comparison algorithms. ANOVA test is used to determine whether the proposed algorithm is statistically significantly different from other comparison algorithms [33]. The difference between the groups is determined with one-way ANOVA [34]. There must be at least three comparison groups for one-way ANOVA test [33, 34]. A p value obtained as a result of the test indicates whether the given algorithm is statistically significant or not. If the resulting p value is less than 0.05, the relevant algorithm is statistically significant. Table 14 shows the one-way ANOVA test results between BinDMO and other comparison algorithms. According to the results, there is a semantic difference between BinDMO results and other comparison algorithms in all CEC-2017 benchmark functions except F3, F11, F12, F13, F14, F15, F19, F29, and F30 benchmark functions.

Wilcoxon signed test is applied to find out whether there is a semantic difference between two different groups [7, 26]. Wilcoxon sign test was applied between BinDMO and each comparison algorithm. Thus, it was examined whether there was a semantic difference between BinDMO and each comparison algorithm. Table 15 shows the Wilcoxon signed test results between the BinDMO with other current state-of-the-art algorithms on CEC-2017 benchmark functions. According to Table 15, there is a semantic difference between BinDMO results and Binary PDO, Binary SO, and Binary AFT results. According to the Wilcoxon test results, the h value was true in most cases.

According to Fig. 8, BinDMO showed rapid convergence at F1, F15, and F25 benchmark functions. In F5 and F10 benchmark functions, Binary SO converged quickly.

Figure 9 shows the data distributions of BinDMO and other current state-of-the-art algorithms on CEC-2017 benchmark functions. Each boxplot shows five characteristics of each algorithm: minimum value, first (25%) quartile, median, third (75%) quartile, and maximum value. According to Fig. 9, BinDMO data are sparsely distributed in benchmark functions such as F3, F4, F9, F11, F12, F13, F14, F15, and F17. In F21, the BinDMO box plot is better distributed. In addition, the fact that the median values of BinDMO in the boxplots are outside the box plots of other comparison algorithms in benchmarks such as F1, F5, F6, F7, F8, F10, F16, and F21 shows that there is a semantic difference between BinDMO with other comparison algorithms.

3.4 Discussion on the results

In this study, a binary version of the DMO algorithm based on transfer functions is proposed. In recent years, twelve newly proposed transfer functions have been used (four Z-shaped (T1, T2, T3, and T4), four U-shaped (T5, T6, T7, and T8), and four Taper-shaped (T9, T10, T11, and T12)). In the literature review, it was noticed that there were no versions of Binary DMO (BinDMO) based on transfer functions and this study was proposed to close this gap. In the study, twelve different BinDMO variations were obtained according to twelve transfer functions (T1, T2, …., T12). A population size analysis was conducted with BinDMO1 (with T1 transfer function). According to the results of this analysis, as the population size increased, the success of the proposed algorithm also increased.

Variations of BinDMO have been tested on thirteen unimode and multimode classical benchmark functions, and the most successful BinDMO variation has been identified. According to the average and standard deviation results, the most successful BinDMO variation was BinDMO1. In other words, the Z1-shaped transfer function (T1) increased the success of Binary DMO more than other transfer functions. When the total average and standard deviation results were compared according to different transfer function groups, it was seen that the Z-shaped transfer function group was more successful than the other transfer function groups (U-shaped and Taper-shaped). According to the time results, the most successful BinDMO variation was BinDMO10. When the total time results were compared according to different transfer function groups, it was seen that the group of the Taper-shaped transfer functions worked in less time than the other transfer function groups (Z-shaped and U-shaped). Since the most important indicator of the success of an algorithm is its average results, BinDMO1 has been determined as the most successful BinDMO variation.

The most successful BinDMO version (BinDMO1) was compared with various heuristic algorithms selected from the literature on CEC-2017, which consists of twenty-nine functions. These heuristic algorithms are Snake Optimizer (SO), Prairie Dog Optimization Algorithm (PDO), and Ali Baba and the forty thieves (AFT). In order to make a fair comparison for each algorithm, the Z1-shaped transfer function was chosen as the transfer function while making equal parameter settings. BinDMO was the most successful heuristic algorithm according to average result ranks. After BinDMO, Binary SO is in the second place. The most unsuccessful heuristic was Binary AFT. The fastest working heuristic algorithm according to time result ranks is Binary AFT. It is followed by Binary SO and BinDMO, respectively. According to comparisons of average results, BinDMO has been the most successful heuristic. This shows that BinDMO can compete with literature algorithms. BinDMO’s running time is significantly worse than compared algorithms. While the convergence speed of BinDMO was quite fast in the F1, F15, and F25 benchmark functions, it fell behind Binary SO in the F5 and F10 benchmark functions.

The summaries of the discussion are shown in Tables 16, 17 and 18. While determining the success of BinDMO variations, transfer function groups, and comparison algorithms, they were evaluated in terms of total average results, total average standard deviation results, total time results, and convergence speeds. Since the total average results are the most important factor affecting the success of BinDMO variations, transfer function groups, and comparison algorithms, they are ranked according to the total average results. According to Tables 16, 17 and 18, the most successful BinDMO variation, the most successful transfer function group, and the most successful comparison algorithm were BinDMO1, the group of the Z-shaped transfer functions, and BinDMO, respectively.

Table 16 Summary of discussion of the variations results of the BinDMO (sorted according to total average results)
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Table 17 Summary of discussion of the groups results of the transfer functions (sorted according to total average results)
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Table 18 Summary of discussion of the results of the BinDMO and other current state-of-the-art algorithms (sorted according to total average results)
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4 Conclusion

Dwarf mongoose optimization (DMO) algorithm is a swarm-based heuristic algorithm that has been proposed for continuous optimization problems in recent years. DMO is an algorithm created by imitating the dwarf mongoose life style. When the literature is reviewed, binary versions of DMO are rarely seen. It was noticed that transfer function-based binary DMO analysis was not performed. In this study, continuous search space in DMO is mapped to binary search space with twelve different transfer functions (four Z-shaped (T1, T2, T3, and T4), four U-shaped (T5, T6, T7, and T8), and four Taper-shaped (T9, T10, T11, and T12)). Selected transfer functions have been newly proposed in the literature in recent years. DMO has been updated as Binary DMO (BinDMO) according to transfer functions. Twelve different variations of BinDMO were thus obtained (BinDMO1 (with T1), BinDMO2 (with T2), …, BinDMO12 (with T12)). In Table 4, five different population sizes (10, 20, 30, 40, and 50) were analyzed with BinDMO1 (with T1) and the most successful population size was determined as 50. It has been observed that there is a directly proportional relationship between the increase in the population size and the success of BinDMO. The proposed BinDMO variations are analyzed in detail on thirteen classical unimodal and multimodal benchmark functions. Results from each BinDMO variation were evaluated in terms of best, worst, average (Ave), standard deviation (Std), and time. According to Tables 6 and 7, the most successful BinDMO variation in terms of average and standard deviation results has been determined to be BinDMO1 (with T1 (Z1-shaped transfer function)) and the least successful BinDMO variation in terms of average and standard deviation results has been determined to be BinDMO5 (with T5). According to Fig. 1, the minimum total average result belongs to T1 (BinDMO1), and according to Fig. 3, the minimum total standard deviation result belongs to T1 (BinDMO1). According to the total mean and total standard deviation results (Figs. 2 and 4), the most successful transfer function group is the Z-shaped transfer function group. The results showed that Z-shaped transfer functions proved to be more successful than other function groups (U-shaped and Taper-shaped). According to Fig. 5, the minimum total time result belongs to T10 (BinDMO10) (Taper10-shaped transfer function). According to Fig. 6, Taper-shaped transfer functions in terms of total time results worked in a shorter time than other transfer functions groups (U-shaped and Taper-shaped). According to the total time results, the most successful transfer function group is the Taper-shaped transfer function group. Although taper-shaped transfer functions work in a shorter time, according to total average and standard deviation results, the most successful transfer function group is Z-shaped transfer functions. Since mean and standard deviation results are more effective on the success of an algorithm than time, the most successful BinDMO version was evaluated based on mean and standard deviation results. When the average and standard deviation results were examined, the most successful BinDMO version was determined to be BinDMO1. Whether there was a semantic difference in the results between BinDMO1 and other BinDMO versions was determined by Wilcoxon statistical test. Wilcoxon sign test results are shown in Table 8. According to Table 8, while there is no a semantic difference in most functions between BinDMO1 and BinDMO2, BinDMO3, BinDMO4, there is semantic difference in most functions between BinDMO1 and BinDMO5-BinDMO12. This shows that BinDMO1 results are similar to BinDMO5-BinDMO12 results. After determining the most successful BinDMO variation, the success of BinDMO1 was presented in three different dimensions (10, 20, and 30) on classical benchmarks in Table 9.

The results of BinDMO1 were compared with three different heuristic algorithms selected from the literature on CEC-2017, which consists of twenty-nine benchmark functions. These heuristic algorithms are Snake Optimizer (SO), Prairie Dog Optimization Algorithm (PDO), and Ali Baba and the forty thieves (AFT). They are newly proposed algorithms in recent years. Parameter settings were chosen similar to ensure that all comparisons were fair. According to Table 12, the most successful comparison algorithm in terms of total average and standard deviation results was BinDMO, while the least successful comparison algorithm in terms of total average and standard deviation results was Binary AFT. Additionally, Binary SO in terms of total average and standard deviation results took the second place. When the total time results are evaluated according to Table 13, BinDMO is the second slowest running algorithm. The fastest working heuristic algorithm according to time result ranks is Binary AFT. It is followed by Binary SO and BinDMO, respectively. When the total best results are evaluated according to Table 13, BinDMO is the second best algorithm. The best heuristic algorithm according to best result ranks is Binary SO. It is followed by BinDMO and Binary AFT, respectively. One-way ANOVA test was applied on comparison algorithms. ANOVA test is used to determine whether BinDMO is statistically significantly different from other comparison algorithms. According to Table 14, there is a semantic difference between BinDMO results and other comparison algorithms in all CEC-2017 benchmark functions except F3, F11-F15, F19, F29, and F30 benchmark functions. Additionally, Wilcoxon sign test was applied between BinDMO and each comparison algorithm. Wilcoxon statistical test results are shown in Table 15. According to Table 15, there is a semantic difference between the results of BinDMO and other comparison algorithms in most CEC-2017 benchmark functions. According to Fig. 8, BinDMO showed rapid convergence at F1, F15, and F25 benchmark functions. In F5 and F10 benchmark functions, Binary SO converged quickly. Binary PDO and Binary AFT showed slower convergence to the optimum result than BinDMO and Binary SO. According to Fig. 9, while the distribution of BinDMO results (minimum value, first quartile, median, third quartile, and maximum value) is closer to each other in most functions, the distribution of Binary AFT and Binary PDO results (minimum value, first quartile, median, third quartile, and maximum value) is further away from each other. When the results are evaluated in general, it is seen that BinDMO can be used in binary optimization problems.

In the future, it is considered that the binary form of DMO will be tested on a different binary optimization problem, such as the feature selection problem and knapsack problem. In addition, the success of DMO on large-sized data sets will be demonstrated, and if necessary, it is planned to create a hybrid structure with a different heuristic algorithm. Different transfer functions used in the literature will be investigated, and their effects on BinDMO will be analyzed.