Improved Dwarf Mongoose Optimization for Constrained Engineering Design Problems

Abstract

This paper proposes a modified version of the Dwarf Mongoose Optimization Algorithm (IDMO) for constrained engineering design problems. This optimization technique modifies the base algorithm (DMO) in three simple but effective ways. First, the alpha selection in IDMO differs from the DMO, where evaluating the probability value of each fitness is just a computational overhead and contributes nothing to the quality of the alpha or other group members. The fittest dwarf mongoose is selected as the alpha, and a new operator ω is introduced, which controls the alpha movement, thereby enhancing the exploration ability and exploitability of the IDMO. Second, the scout group movements are modified by randomization to introduce diversity in the search process and explore unvisited areas. Finally, the babysitter's exchange criterium is modified such that once the criterium is met, the babysitters that are exchanged interact with the dwarf mongoose exchanging them to gain information about food sources and sleeping mounds, which could result in better-fitted mongooses instead of initializing them afresh as done in DMO, then the counter is reset to zero. The proposed IDMO was used to solve the classical and CEC 2020 benchmark functions and 12 continuous/discrete engineering optimization problems. The performance of the IDMO, using different performance metrics and statistical analysis, is compared with the DMO and eight other existing algorithms. In most cases, the results show that solutions achieved by the IDMO are better than those obtained by the existing algorithms.

Introduction

The drive to better each circumstance or situation in human life is high. Optimization tries to find the best solution amongst a pool of other solutions. As such, optimization occurs naturally in many human endeavors and is deeply rooted in science, businesses, ecology, and manufacturing [1]. Basically, optimization is accomplished using either the mathematical or metaheuristic approach. The mathematical methods suffer from being gradient-dependent, time or computationally complex, and problem-dependent [2]. On the other hand, the metaheuristic approach does not guarantee that a better or optimal solution will be found; however, a near-optimal solution will be found. The trade-off of speed with the quality of the solution and associated drawbacks of the mathematical approach could be attributed to the surge in the rate at which researchers are proposing nature-inspired algorithms. Also, some attribute this surge to the ease of mimicking nature’s way of solving problems [3].

Another reason for this surge is the emergence of new application areas in various optimization domains and their attendant technology. These emerging real-life application areas require optimization of various components like speed, profit, risk, efficiency, or cost, often nonlinear and complex, requiring a robust and efficient metaheuristic optimization technique. Though the metaheuristic methods do not guarantee that it would find the optimal solution, its stochastic nature ensures at least a near-optimal solution in the shortest possible time. Also, the “No Free Lunch Theorem (NFLT)” stipulates that no one algorithm can optimally solve all optimization problems. No matter how robust an algorithm is, it can only solve that problem or problems for which its robustness is established. Therefore, there is a need to develop new algorithms, hybridize two or more existing algorithms, or improve existing algorithms to solve emerging optimization problems.

Many aspects of natural problem-solving techniques have been used for developing metaheuristic algorithms. One of the first approaches that use nature as a source of inspiration is the Genetic Algorithm (GA), whose source of inspiration is natural selection in the theory of evolution [4]. Another popular approach is Particle Swarm Optimization (PSO) which is inspired by the intelligent way birds flock together [5]. They have been used to solve problems in domains such as the traveling salesman problem [6], optimal control [7], and many more. These algorithms have recorded significant success in their respective areas of applications. However, some authors have criticized the over-reliance on the metaphor-based paradigm by the nature-inspired metaheuristic algorithms [8–10].

Nature-inspired metaheuristic algorithms imitate some aspects of the problem-solving technique in nature. They are stochastic and gradient-independent, and their parameters can be tuned depending on the problem to be solved [2]. Each nature-inspired metaheuristic algorithm has its unique way of searching the problem space for the optimal solution. Its robustness and efficiency are measured by how effective the search mechanism is under different problem landscapes and parameter uncertainty. Other desired features of these algorithms include implementation simplicity, flexibility, and robustness. The research efforts on nature-inspired metaheuristic algorithms are enormous, and impossible to review all the contributions except in a review study [11]. However, some selected nature-inspired metaheuristic algorithms proposed between 2019 and 2022 are summarized in Table 1

Table 1.

Some nature-inspired metaheuristic algorithms with their source of inspiration (2019–2021)

Acronym	Name	References
BO	Butterfly Optimization Algorithm	[47]
SSA	Squirrel Search Algorithm	[48]
DHOA	Deer Hunting Optimization Algorithm	[49]
EPC	Emperor Penguins Colony	[50]
HHO	Harris Hawks Optimization	[51]
SOA	Seagull Optimization Algorithm	[52]
ASO	Atom Search Optimization	[53]
NRO	Nuclear Reaction Optimization	[54]
HGSO	Henry Gas Solubility Optimization	[55]
BWO	Black Window Optimization	[56]
ChOA	Chimp Optimization Algorithm	[57]
MPA	Marine Predators Algorithm	[58]
MRFO	Manta Ray Foraging Optimization	[59]
AVOA	African Vultures Optimization Algorithm	[60]
GTO	Artificial Gorilla Troop Optimizer	[61]
RFO	Red Fox Optimization Algorithm	[62]
CHIO	Coronavirus Herd Immunity Optimization	[63]
RCM	Red Colobuses Monkey	[64]
GOA	Gazelle Optimization Algorithm	[65]
RHSO	Red Hyraxes Swarm Optimizer	[66]
PDO	Prairie Dog Optimization	[67]
EOSA	Ebola Optimization Search Algorithm	[68]
PPO	Predator–Prey Optimization	[69]

Aside from developing new nature-inspired algorithms, developers have also hybridized existing metaheuristic algorithms. The bibliographic diversity of hybridized approaches is enormous; therefore, a few examples of this approach involving the DMO are mentioned in this study. The DMO was hybridized with a Multi-Hop Routing Scheme (DMOSC-MHRS) and applied to solve the clustering problem [12]. The binary variant of DMO, the BDMO, was hybridized with a local search algorithm simulated annealing to tackle feature selection challenges of varying dimensional problems (BDMSAO) [13]. Moreover, in [14], the study proposed a Binary Dwarf Mongoose Optimizer (BDMO) and was applied to solve the multiclass high-dimensional feature selection problem. The success history-based adaptive differential evolution (SHADE) was hybridized with a modified Whale Optimization Algorithm (WOA) such that the two algorithms work stand-alone and only share information about the best-found solution [15].

These inspirational sources can be categorized into swarm-based, evolutionary-based, human, and physics-based, as in [16], and some further categorized them as system-based and bio-based, as can be found in [13]. The swarm-based category, also known as Swarm Intelligent (SI), derives their inspirations from the behavior or social interaction of animals, fish, birds and so on. This group has attracted much attention in the past decades as more methods were developed based on this inspirational point. The evolutionary-based are nature-based that commence their process by random generation of solutions’ population. The human-based category is based on the activities that humans perform. For example, the teaching–learning process involving students and teachers was used to form a popular algorithm in this group. Similarly, the physics-based is inspired by physics laws in nature. For example, nuclear reaction, atom search, and others.

The authors in [17] proposed a novel self-adaptive beneficial factor-based improved SOS (SaISOS), which improved the Symbiosis Organism Search (SOS) algorithm. Also, a modified Whale Optimization Algorithm (WOA) was developed for prediction using COVID-19 chest X-ray pictures [18]. The Quasi-Oppositional Based Learning (QOBL) strategy and Symbiosis Organisms Search (SOS) were incorporated to form a Quasi-Oppositional Symbiosis Organisms Search (QOSOS) algorithms for solving unconstrained global optimization problems [19]. The mLBOA, a new BOA variant, was proposed to solve the IEEE CEC 2017 benchmark suite [20]. The ensemble of Butterfly Optimization Algorithm (BOA) and Symbiosis Organism Search (SOS) called (h-BOASOS) was used to optimize the weight and cost of the cantilever retaining wall [21]. The authors [22] proposed a modified WOA called m-SDWOA by combining the WOA with the modified Symbiotic Organisms Search (SOS).

The DMO, a recently developed swarm-based metaheuristic algorithm, has received some attention as researchers have improved, modified, or hybridized it to solve various optimization problems. Shortly after its development, the DMO was employed to modify long short-time memory (LSTM) in predicting the effect of ${\mathrm{AI}}_2{\mathrm{O}}_3$ content of nanoparticle on the thermal expansion’s coefficient in $\mathrm{Cu}-\mathrm{A}{\mathrm{I}}_2{\mathrm{O}}_3$ nanocomposites which uses the in situ chemical method in its preparation [23]. Alumni nitrate was used to prepare the nanocomposite, which was added to a copper nitrate solution. After which the ${\mathrm{AI}}_2{\mathrm{O}}_3$ and $\mathrm{CuO}$ in powdered form were obtained, and removal of the leftover liquid was done by thermal treatment for 1 h at 850° C. The study investigated the impact of ${\mathrm{AI}}_2{\mathrm{O}}_3$ content of the $\mathrm{Cu}-\mathrm{A}{\mathrm{I}}_2{\mathrm{O}}_3$ nanocomposite. The machine learning model proposed could predict the thermal expansion coefficient (TEC), which was evaluated using various temperature rates, achieving an accuracy of up to 99%, as reported by the researchers.

The DMO was also applied to the clustering problem in [12], where a novel DMO-secure-based clustering combined with a multi-hop scheme of routing (DMOSC-MHRS) was developed in the internet of drones arena. Moreover, in [14] the study proposed a binary dwarf mongoose optimizer (BDMO) and was applied to solve the multiclass high-dimensional feature selection problem. The proposed method used 18 high-dimensional feature selection datasets and was compared against 10 state-of-the-art feature selection techniques. The method produced higher validation accuracy on 15 of the 18 datasets utilized. Afterward, the BDMO was hybridized with a local search Algorithm Simulated Annealing (SA) known as BDMSAO [13] to tackle feature selection challenges of varying dimensional problems. The proposed hybrid method yielded the highest classification accuracy obtainable on 50% of the 18 datasets employed for validation. It was compared with other popular methods in the literature with promising results.

This algorithm has been gaining ground in other application areas. For example, in [47] work, a parameter estimation of an autoregressive exogenous (ARX) model was developed based on the DMO. In [24], a new method was developed that incorporated the Dwarf Mongoose Optimization Algorithm (DMOA), generalized normal distribution (GNF), and opposition-based learning strategy (OBL), which is abbreviated as GNDDMOA. The proposed method was evaluated using 8 data clustering and 23 test functions and was compared with other popular techniques. This proposed GNDDMOA yielded the best results when employed to solve data clustering applications. Finally, the DMO was modeled with machine learning-driven ransomware detection called DWOML-RWD [25]. The method was majorly proposed to recognize and classify ransomware. This study first carried out the pre-processing stage of feature selection with a Krill Herd Optimization (EKHO) using dynamic oppositional-based learning (QOBL).

The proposed optimization technique (IDMO) modifies the base algorithm (DMO) in three simple but effective ways. First, the alpha selection in IDMO differs from the DMO, where evaluating the probability value of each fitness is just a computational overhead and contributes nothing to the quality of the alpha or other group members. The fittest dwarf mongoose is selected as the alpha, and a new operator ω is introduced, which controls the alpha movement, thereby enhancing the exploration ability and exploitability of the IDMO. Second, the scout group movements are modified by randomization to introduce diversity in the search process and explore unvisited areas. Finally, the babysitter's exchange criterium is modified such that once the criterium is met, the babysitters that are exchanged interact with the dwarf mongoose exchanging them to gain information about food sources and sleeping mounds, which could result in better-fitted mongooses instead of initializing them afresh as done in DMO, then the counter is reset to zero. The proposed improved algorithm is used to solve benchmark test functions and twelve (12) different optimization problems in the engineering domain. The major contributions of this study can be summarized as follows:

• The alpha selection in IDMO differs from the DMO, where evaluating the probability value of each fitness is just a computational overhead and contributes nothing to the quality of the alpha or other group members. The fittest dwarf mongoose is selected as the alpha.

• A new operator ω is introduced, which controls the alpha movement, thereby enhancing the exploration ability and exploitability of the IDMO.

• The scout group movements are modified by randomization to introduce diversity in the search process and explore unvisited areas.

• The babysitter's exchange criterium is modified such that once the criterium is met, the babysitters that are exchanged interact with the dwarf mongoose exchanging them to gain information about food sources and sleeping mounds, which could result in better-fitted mongooses instead of initializing them afresh as done in DMO, then the counter is reset to zero.

The rest of the paper is organized as follows: Sect. 2 presents the dwarf mongoose optimization algorithm (DMO). Section 3 presents the improved dwarf mongoose optimization algorithm (IDMO). The experimental setup, results, and detailed discussion are presented in Sect. 4. Finally, the conclusion and future work is presented in Sect. 5.

The Dwarf Mongoose Optimization Algorithm (DMO)

This section presents the base algorithm, namely the dwarf mongoose optimization (DM) algorithm. First, the nature-derived inspiration which motivated the design of the novel DMO is described to allow for the understanding of the conceptualization of the flow of the algorithm. This study justified and provided a linkage of the real-life activities or phenomena in the domain of consideration to further buttress the optimization process inherent among dwarf mongoose creatures. The second sub-section is focused on presenting a summary of the mathematical and procedural models of the DMO algorithm. Then the section is concluded by drawing up the gap in the existing implementation of DMO with respect to the natural phenomenon in the domain. This is aimed at providing a ground for a new variant.

Inspiration

The behavioral pattern of an organism is an embodiment of its natural disposition to the demand of survival and evolvement strategy. This strategy allows the organism to adapt to its environment, evolve into a bigger specie, develop a fitness mechanism to wade off predators, source food, and build a social system with related and non-related organisms. When the population of such an organism is considered, their combined behavioral pattern demonstrates a great system with natural phenomena typical of providing computational solution patterns. A careful understudy of the dwarf mongoose population revealed that an interesting and useful natural phenomenon is inherent in their existential strategy. One of the fundamental attributes of the dwarf mongoose, peculiar to the Helogale specie, is the ability to cohabit in groups—a natural disposition that reflects a complete optimization process.

A group of dwarf mongoose lives in a territory that is demarcated using anal secretion, a behavioral approach for wading off competing groups or predators, and for keeping group members within the territory size. The secretion is made on horizontal objects within the territory using secretion from anal or checks glands. The smell of the secretion, which may remain detectable between 20 and 25 days, serves two purposes: reassures group members of safety within the territory and puts predators to flight [26] [27]. Unlike other organisms, which expand their population size for the benefit of resource enrichment, the dwarf mongoose prefers to shrink its population size to allow for the sustainability of available resources for the group. This resource control strategy is one of the group survival strategies in addition to the anti-predation strategy. Whereas the former supports foraging and nutrition needs, the latter ensures that the group members are kept from an intruder's reach.

In fact, compared to secretion measure for wading off an enemy, the dwarf mongoose has an aggressive skull-crush attack, a bite aimed at their prey's eyes—an approach that allows for adaptation and disallows intruding dwarf mongoose from depleting their group resources. The prey hunted down serves as a food source, which any individual extensively and intensively searches for within the group to have a full meal. This food search has characterized the dwarf mongoose with a seminomadic lifestyle, allowing a group to forage for food intensively from the current location and extensively over a long distance. This foraging behavior promotes relocation of the group's territory, referred to as a mound, to allow for searching for a new mound where the groups pitch their territory and sleep. This sleeping mound is reported to change almost every night [28]. Meanwhile, while the group moves about during the foraging activity, cohesion of the entire group members is achieved using a peep vocalization to alert members of the presence of a predator or intruder [29].

The group of dwarf mongooses maintains a caste structure that delineates members into subgroups. These subgroups include the alpha (male and female), juvenile, scout, and babysitters. The task of vocalization associated with foraging activity is reserved for the alpha female. Meanwhile, even the call for the group members to move out of their mound for foraging activity, the direction of the journey, and the distance covered, are imitated through the vocalization of this same alpha female. Furthermore, the alpha female is the only member who can and should birth, rare, and raise young dwarf mongooses. An attempt by other female subgroup members will be fought harshly and considered an act of insubordination [30], which group members will aggressively fight and most likely lead to the offender's departure from the group or such young, birthed ones being killed. In fact, to prevent such an extreme reaction, the alpha female prevents the subordinate males from mating with subordinate females.

Meanwhile, babysitters are elected from among the subordinates to keep watch on the babies of the alpha females since such young ones are prevented from foraging with the group. However, their mothers may carry them during mound relocation. This carrying capacity of the young dwarf mongoose often limits the distance covered when relocating to another mound. This presents an advantage to the young ones since the dwarf mongooses attack all enemies as a closed group [31]. This attack on an enemy—which could also mean another group of dwarf mongoose—is led by the alpha male, and the alpha female comes in the rear, considering they might be carrying the young ones. The numerical strength of a group determines if it will win the fight. Unfortunately, while the group may win against a ground predator, they are known to be weak in dealing with predator launching from an aerial position. Group territoriality determines the optimization of group size, which optimizes an individual’s fitness and impacts the cost/benefit relationship in the group. Considering the rich foraging, anti-predation, and group territory sustenance strategy of the dwarf mongoose, the DMO algorithm was designed and implemented to solve real-life optimization problems. In the next subsection, we present the algorithmic and mathematical models of the DMO.

The DMO Model

Motivated by the natural phenomena of the dwarf mongoose, the design of the DMO [32] algorithm is discussed in the following paragraph. The authors model all subgroups identified with the dwarf mongoose population by incorporating the alpha (male and female), scouts, and babysitter subgroups. The adaptive nature of the animals in their environment concerning predation and foraging was also simulated and applied in the design phase. In Fig. 1, the complete optimization process of the dwarf mongoose is represented. First, the scouting group moves out to source food while those designated for babysitting are allowed to remain in the mound. It is assumed that the procedure for looking out for food, known as foraging, demonstrates the exploration phase of the optimization process. Moreover, since the location of a new food source allows for a group of dwarf mongoose to settle down in a mound, the DMO models that as the exploitation or intensification phase. Meanwhile, the algorithm also simulated the discovery of new mounds resulting from the exploration process. The figure also showed that the design for the exchange of babysitters among the scout group is provisioned.

Fig. 1.

The optimization procedures of the DMO

The representation of a group of dwarf mongooses, which also represents the entire population, is modeled using Eq. (1). Since the population of the dwarf mongoose is often composed of the alpha groups, juvenile group, and scout (including babysitters), we derive the alpha group from the population to allow for allocating the remaining individuals in the population for the other two subgroups. Meanwhile, considering the role of the alpha female ($\alpha )$ in the population, they applied Eq. (2) to compute this special subgroup of alphas. This is made possible by first evaluating the fitness $\mathrm{fit}$ of the entire group and as well an individual fitness to know what individuals are characterized and suitable for the alpha female ($\alpha )$.

$$\mathit{X}=\left[\begin{array}{c}\begin{array}{ccc}{x}_{1,1}& \begin{array}{cc}{x}_{1,2}& \cdots \end{array}& \begin{array}{cc}{x}_{1,d-1}& x_{1,d}\end{array}\end{array}\\ \begin{array}{ccc}{x}_{2,1}& \begin{array}{cc}{x}_{2,2}& \cdots \end{array}& \begin{array}{cc}{x}_{2,d-1}& x_{2,d}\end{array}\end{array}\\ \begin{array}{c}\begin{array}{ccc}⋮& \begin{array}{cc}⋮& x_{i,j}\end{array}& \begin{array}{cc}⋮& ⋮\end{array}\end{array}\\ \begin{array}{ccc}{x}_{n,1}& \begin{array}{cc}{x}_{n,2}& \cdots \end{array}& \begin{array}{cc}{x}_{n,d-1}& x_{n,d}\end{array}\end{array}\end{array}\end{array}\right],$$ 1

$$\alpha =\frac{{\mathit{fit}}_i}{{\sum }_{i=1}^n{\mathit{fit}}_i}.$$ 2

Similarly, the scout groups, which represent the workforce of the dwarf mongoose population, are computed and extracted from the entire population. To achieve this, they developed Eq. (3) to represent this derivation process, allowing for the foraging activity and searching for new sleeping mounds simultaneously [31].

$${\mathit{X}}_{\mathit{i}+1}=\left\{\begin{array}{c}{\mathit{X}}_{\mathit{i}}-CF\ast phi\ast rand\ast {[\mathit{X}}_{\mathit{i}}-\overrightarrow{\mathit{M}}]if{\phi }_{i+1}>{\phi }_i\\ {\mathit{X}}_i+CF\ast phi\ast rand\ast {[\mathit{X}}_i-\overrightarrow{\mathit{M}}]else\end{array},,\right)$$ 3

where $\mathrm{rand}$ is a random number between $\left[0,1\right]$ and $\mathrm{CF}={\left(1,-,\frac{\mathrm{iter}}{{\mathrm{Max}}_{\mathrm{iter}}}\right)}^{\left(2,\frac{\mathrm{iter}}{{\mathrm{Max}}_{\mathrm{iter}}}\right)}$ is the collective-volatile movement control parameter and $\overrightarrow{\mathit{M}}={\sum }_{i=1}^n\frac{X_i\times {\mathrm{sm}}_i}{X_i}$ determines the movement of the mongoose to the new sleeping mound, and $\phi =\frac{{\sum }_{i=1}^n{\mathrm{sm}}_i}{n}$.

The predatory and foraging activities within a mound of dwarf mongoose require that individuals move from one location to another. This will often require that an update mechanism be applied for computing each group member's position. They leverage the vocalization $\mathrm{peep}$ facility of the alpha female ($\alpha )$ in addition to a randomly generated variable $\mathit{phi}$, in the range of [−1,1] to compute the current position of an individual. Using Eq. (4), the position update model is described.

$${\mathit{X}}_{\mathit{i}+1}={\mathit{X}}_{\mathit{i}}+\mathrm{phi}\ast \mathrm{peep}.$$ 4

Another important milestone characterized by predatory and foraging activities is the discovery of a new sleeping mound. This new mound is assumed to change during the iterative process of optimization. To model the discovery of the sleeping mound ($\mathrm{sm}$), Eq. (5) is applied for computing this, and Eq. (6) allows for averaging $\phi$ the sleeping mound values.

$${\mathrm{sm}}_i=\frac{{\mathrm{fit}}_{i+1}-{\mathrm{fit}}_i}{\max \{\left|{\mathrm{fit}}_{i+1},,,{\mathrm{fit}}_i\right|\}}.$$ 5

The procedure for the application of mathematical models described in this section is encoded in the following listing:

1. Initialize all control parameters

2. The population of the dwarf mongoose is first generated

3. Subgrouping the entire population into alpha (male and female), scouts, and babysitters

4. Determine the available number of search agents by subtracting the babysitters from the entire population

5. Set the rate of exchange for babysitting tasks as L

6. While the termination condition is not satisfied, do the following:

   • i.
     Compute the fitness of the mongoose population/group
   • ii.
     Activate and set the time counter
   • iii.
     Apply Eq. (2) to deduce the size of the alpha female
   • iv.
     Update the position of a potential food source using Eq. (4)
   • v.
     Iterate over each individual and compute the fitness of $X_i$
   • vi.
     Derive the sleeping mound for the population using Eq. (5)
   • vii.
     Determine the movement vector $\overrightarrow{M}$
   • viii.
     Exchange the babysitters
   • ix.
     Compute the scout group using Eq. (3)
   • x.
     Update the solution so far

7. Return the best solution

The procedure described above was translated into a pseudocode, and then implemented as the DMO algorithm. Although the DMO demonstrates some measure of novelty, a further study of the domain revealed that some important component of the optimization process derivable from the natural existence and co-existence among the dwarf mongoose presents an opportunity for advancing the algorithm. The inclusion of these phenomena is necessitated by enhancing performance and balancing the exploration and exploitation stages. The next section presents a detailed design and discussion of the improved dwarf mongoose optimization algorithm.

The pseudocode for the algorithm is given in algorithm listing 1.

The Improved Dwarf Mongoose Optimization Algorithm (IDMO) Model

The section presents the improved dwarf mongoose optimization algorithm (IDMO).

The IDMO Model

The IDMO is proposed to enhance the exploration and exploitation of the DMO. This limitation is evident in the solution returned by DMO for F9, F15, and F17, which shows how DMO could not find the optimal solution. This optimization technique modifies the base algorithm (DMO) in three simple but effective ways. First, the alpha selection in IDMO differs from the DMO, where evaluating the probability value of each fitness is just a computational overhead and contributes nothing to the quality of the alpha or other group members. The fittest dwarf mongoose is selected as the alpha, and a new operator ω is introduced, which controls the alpha movement, thereby enhancing the exploration ability and exploitability of the IDMO. Second, the scout group movements are modified by randomization to introduce diversity in the search process and explore unvisited areas. Finally, the babysitter's exchange criterium is modified such that once the criterium is met, the babysitters that are exchanged interact with the dwarf mongoose exchanging them to gain information about food sources and sleeping mounds, which could result in better-fitted mongooses instead of initializing them afresh as done in DMO, then the counter is reset to zero.

The proposed IDMO achieves optimization in three phases, as shown in Fig. 2. This model shows how the scouting activity is separated from the foraging, which is not the case in DMO, where the scouting and foraging are the same activity. The individual dwarf mongooses are the search agents as modeled as a $n\times d$ matrix shown in Eq. 6. At the exploration phase, the modified alpha (Eq. 8) leads the group to uncharted territories using steps modeled in Eq. 9. Equation 10 models a new operator ω, which controls the alpha movement, thereby enhancing the exploration ability and exploitability of the IDMO. As shown in Eq. 11, the scout group movements are modified by randomization to introduce diversity in the search process and explore unvisited areas. The exploitation is achieved after the babysitter exchange criterium is met and babysitters are exchanged, as shown in Eq. 12. This phase refines the obtained solution toward the optimal solution.

Fig. 2.

The model of the proposed IDMO

Population Initialization

The IDMO population is initialized stochastically as a matrix of candidate dwarf mongooses (X), as shown in Eq. (6). The population vector ranges between the upper bound (U) and lower bound (L) of the optimization problem

$$\mathit{X}=\left[\begin{array}{c}\begin{array}{ccc}{x}_{1,1}& \begin{array}{cc}{x}_{1,2}& \cdots \end{array}& \begin{array}{cc}{x}_{1,d-1}& x_{1,d}\end{array}\end{array}\\ \begin{array}{ccc}{x}_{2,1}& \begin{array}{cc}{x}_{2,2}& \cdots \end{array}& \begin{array}{cc}{x}_{2,d-1}& x_{2,d}\end{array}\end{array}\\ \begin{array}{c}\begin{array}{ccc}⋮& \begin{array}{cc}⋮& x_{i,j}\end{array}& \begin{array}{cc}⋮& ⋮\end{array}\end{array}\\ \begin{array}{ccc}{x}_{n,1}& \begin{array}{cc}{x}_{n,2}& \cdots \end{array}& \begin{array}{cc}{x}_{n,d-1}& x_{n,d}\end{array}\end{array}\end{array}\end{array}\right],$$ 6

where $n$ is the number of dwarf mongoose in a mound, $x_{i,j}$ denotes the position of the jth dimension of the ith population and each $x_{i,j}$ is defined in Eq. (7).

$$x_{i,j}=\mathrm{rand}\times \left(U,-,L\right)+L.$$ 7

Alpha Group

The population size of this group is modeled by subtracting the number of babysitters from the total number of dwarf mongooses. The alpha female $(\alpha )$ leads this group, and the fittest dwarf mongoose is selected as the alpha female, as given in Eq. 8. The alpha selection in IDMO differs from the DMO, where evaluating the probability value of each fitness is just a computational overhead and contributes nothing to the quality of the alpha or other members.

$$\alpha =\min ({\mathrm{fit}}_1,{\mathrm{fit}}_2,\cdots ,{\mathrm{fit}}_n).$$ 8

The alpha female keeps the group together using vocalization modeled by $\mathrm{peep}.$

The IDMO searches the problem space moving around as defined in Eq. 9. Initially defined as the fittest dwarf mongoose, it drags the other family members toward a potential food source. This scenario is a clear departure from the DMO, where only the alpha’s vocalization is used to influence the position of the other dwarf mongoose. In IDMO, the position of the alpha is used to set the position of the other mongoose and a new operator $\omega$, defined in Eq. 10, which controls the alpha movement, thereby enhancing the exploration ability and exploitability of the IDMO

$${\mathit{X}}_{\mathit{i}+1}=\alpha +\mathrm{phi}\ast \mathrm{rand}\ast {(\mathit{X}}_{\mathit{i}}-{\mathit{X}}_{\mathit{k}}),$$ 9

$${\omega =e}^{-4\ast {{(C}_{\mathrm{iter}}/{\mathrm{Max}}_{\mathrm{iter}})}^2},$$ 10

where $\mathrm{phi}=\left(\frac{\mathrm{peep}}{2}\right)\ast \mathrm{rand}\ast \omega$, ${\mathit{X}}_i$ is the previous dwarf position, $\mathit{rand}$ is a uniformly distributed random number [−1,1]. ${\mathit{X}}_{\mathit{k}}$ is a randomly selected dwarf mongoose.

Scout Group

The responsibility of the scouts is to look for a suitable sleeping mound since the dwarf mongooses are known to be seminomadic, never returning to the previous sleeping mound. The IDMO models the scout group scouting for the next sleeping mound after foraging activities. The scouts’ fitness is considered a potential sleeping mound since the dwarf mongooses are known to stay around abundant food sources, and the fittest scout is considered the selected sleeping mound. The scouts are modeled as given in Eq. 11.

$${\mathit{X}}_{\mathit{i}+1}=\alpha +\mathrm{phi}\ast \mathrm{rand}\ast {(\mathit{X}}_{\mathit{k}}-{\mathit{X}}_{\mathit{h}})/2,$$ 11

where $\mathrm{rand}$ is a random number between$\left[0,1\right]$, and ${\mathit{X}}_{\mathit{k}}{,\mathit{X}}_{\mathit{h}}$ are randomly selected dwarf mongooses.

The Babysitters

The babysitter’s exchange criterium is given in Eq. 12. Once the criterium is met, the exchanged babysitters interact with the dwarf mongooses to gain information about food sources and the next sleeping mound, which could result in better-fitted mongooses instead of initializing them afresh as done in DMO, then the counter is reset to zero. This improvement is modeled in Eq. 13, where the dwarf mongooses to replace the babysitters are randomly selected, and their information is passed to the babysitters as shown. If L gets to zero, it is reset by multiplying it with the current iteration and CF.

$$L=\left\{\begin{array}{c}Roundup\left(0.6,,\ast ,n,,\ast ,\dim ,,\ast ,\left(\frac{1}{C_{\mathrm{iter}}}\right)\right),\\ L\ast {\mathrm{C}}_{\mathrm{iter}}\ast CFwhenL<0\end{array}\right)$$ 12

$${\mathit{X}}_{\mathit{i}+1}={(\mathit{X}}_{\mathit{j}}+\mathrm{rand}\ast \left({\alpha -(\mathit{X}}_k,+,{\mathit{X}}_{\mathit{h}},)/2\ast \mathrm{br}\right),$$ 13

where $\mathrm{CF}={\left(1,-,\frac{C_{\mathrm{iter}}}{{\mathrm{Max}}_{\mathrm{iter}}}\right)}^{\left(2,\frac{C_{\mathrm{iter}}}{Ma{x}_{\mathit{iter}}}\right)}$ controls the collective-volitive movement of the dwarf mongooses, ${\mathit{X}}_{\mathit{j}}{,\mathit{X}}_{\mathit{k}}{,\mathit{X}}_{\mathit{h}}$ are randomly selected dwarf mongooses to replace the babysitters, and $\mathrm{br}$ is the birthrate.

The computation complexity of the IDMO is significantly reduced because it simplifies the overhead of alpha selection. The intuitive and detailed process of IDMO is shown in Fig. 3. The optimization process starts when the dwarf mongooses forage, led by the alpha female. A select few are left behind, called the babysitters, to tend the nest. The search for abundant food sources simulates the exploration phase of the IDMO. At midday, the babysitters are exchanged so they can feed since the dwarf mongooses are not known to bring food for the young or others. When this change occurs, the return to already known food sources to enable the exchanged babysitters to feed quickly before new food sources or sleeping mounds are found. This scenario simulates the exploitation phase of the IDMO. The scouting for a sleeping mound at the end of the day further explores and exploits the search space. Like the DMO, the IDMO algorithm has only one specific parameter to be fine-tuned (the number of babysitters.

Fig. 3.

The flowchart of the proposed IDMO

Conceptual Advantage of the IDMO

The superiority of the proposed IDMO can be theoretically attributed to the fact that the IDMO processes are stochastic in all ramifications. Starting with the population generation and the updating steps of the alpha and scout groups. The selection of the babysitters and dwarf mongooses to exchange them is entirely stochastic, which improves these solutions using enhanced exploratory and exploitation. The IDMO has only one parameter that can be tuned, and its implementation is simple and flexible.

As listed in Algorithm 2, the following algorithm reflects the mathematical model and procedural listing for the IDMO model.

Results and Discussion

The proposed improvements of the IDMO were tested to establish performance using 31 benchmark functions (classical and CEC2020 benchmark functions [33, 34], twelve (12) engineering benchmark problems, and real-world feature selection problems. The results of IDMO for benchmark functions were compared with that of DMO and eight existing population-based metaheuristic algorithms, namely differential evolution (DE), arithmetic optimization algorithm (AOA), particle swarm optimization (PSO), constriction-coefficient-based (PSO) and GSA (CPSOGSA), salp swarm algorithm (SSA), grey wolf optimizer (GWO), biogeography-based optimization (BBO), sine cosine algorithm (SCA). All the algorithms and optimization problems considered were implemented using MATLAB R2020b, and Table 2 presents the different algorithm control parameters used for the experiments. The population size and the maximum number of iterations used for all algorithms are 50 and 1000, respectively. Windows 10 OS environment, Intel Core i7-7700@3.60 GHz CPU, and 16G RAM were used to conduct the experiments. The results of 30 independent runs of each algorithm are collated using the “best, worst, average, and SD” performance indicators. Further statistical analysis was carried out using mean, standard deviation, and Friedman and Wilcoxon tests.

Table 2.

Algorithm control parameters

Algorithm	References	Name of the parameter	Value of the parameter
AOA	[70]	$\alpha$	5
		$\mu$	0.05
PSO	[5]	C1, C2	2
		Wmax	0.9
		Wmin	0.2
CPSOGSA	[71]	${\phi }_1,{\phi }_2$	2.05
GWO	[72]	A	[0,2]
		r1f r2	[0,1]
SCA	[73]	a	2
SSA	[74]	c2, c3	[0,1]
BBO	[75]	nKeep	0.2
		Pmutation	0.9
DE	[76]	Lower bound of scaling factor	0.2
		Upper bound of scaling factor	0.8
		PCR	0.8

Benchmark Test Function

The results of all the algorithms used in this study are presented in Table 3. The exploitation ability of the IDMO was tested using the unimodal, separable, and non-separable benchmark functions (F1-F9). Clearly, the IDMO, AOA, and GWO found the global minimum solution for F1 and F2. However, the DMO, CPSOGSA, PSO, DE, SSA, and SCA found near-optimal solutions, and the BBO could not find the optimal solution. All the algorithms failed to find the global minimum for F5 and showed promising results for F6-F9. In most cases, the IDMO outperformed the DMO and performed competitively with GWO. The result confirms the exploitative capability of IDMO.

Table 3.

Result of classical benchmark functions

Function	Dim	Global	Value	IDMO	DMO	AOA	CPSOGSA	PSO	BBO	DE	SSA	SCA	GWO
F1	50	0	Best	0.00E + 00	2.00E−07	0.00E + 00	0.00E + 00	6.49E−05	1.07E + 01	2.36E−05	2.88E−08	3.83E−02	0.00E + 00
			Worst	0.00E + 00	6.81E−07	0.00E + 00	1.00E + 04	4.52E−03	1.78E + 01	6.72E−05	5.86E−08	2.12E + 02	0.00E + 00
			Average	0.00E + 00	3.87E−07	0.00E + 00	5.00E + 02	8.00E−04	1.38E + 01	3.89E−05	4.30E−08	4.52E + 01	0.00E + 00
			SD	0.00E + 00	1.13E−07	0.00E + 00	2.24E + 03	1.15E−03	1.68E + 00	1.00E−05	7.40E−09	7.45E + 01	0.00E + 00
F2	50	0	Best	0.00E + 00	1.64E−04	0.00E + 00	1.41E-08	1.59E−03	1.36E + 00	4.43E−04	7.24E−02	9.04E−06	0.00E + 00
			Worst	0.00E + 00	4.28E−04	0.00E + 00	7.31E + 01	3.59E + 01	1.81E + 00	8.23E−04	5.28E + 00	3.36E−02	0.00E + 00
			Average	0.00E + 00	2.80E−04	0.00E + 00	5.87E + 00	2.00E + 00	1.57E + 00	5.88E−04	2.04E + 00	5.38E−03	0.00E + 00
			SD	0.00E + 00	8.60E−05	0.00E + 00	1.60E + 01	7.99E + 00	1.15E−01	1.04E−04	1.33E + 00	7.76E−03	0.00E + 00
F3	50	0	Best	0.00E + 00	1.23E−01	0.00E + 00	5.46E + 03	2.17E + 03	8.61E + 02	7.18E + 04	4.84E + 02	9.10E + 03	0.00E + 00
			Worst	6.95E + 00	8.64E−01	3.10E−01	2.33E + 04	5.45E + 03	1.63E + 03	1.07E + 05	7.73E + 03	5.56E + 04	3.45E−08
			Average	3.55E−01	4.14E−01	5.15E−02	1.33E + 04	4.04E + 03	1.20E + 03	9.15E + 04	2.05E + 03	2.46E + 04	3.31E−09
			SD	1.55E + 00	2.02E-01	7.07E−02	5.91E + 03	9.77E + 02	2.11E + 02	9.99E + 03	1.67E + 03	1.07E + 04	8.82E−09
F4	50	0	Best	0.00E + 00	8.53E−01	3.85E−02	3.83E + 01	7.20E + 00	1.77E + 00	1.94E + 01	7.62E + 00	3.50E + 01	0.00E + 00
			Worst	1.06E−03	9.38E−01	7.60E−02	9.35E + 01	1.42E + 01	2.33E + 00	2.60E + 01	1.77E + 01	7.23E + 01	0.00E + 00
			Average	5.32E−05	9.03E−01	4.92E−02	7.52E + 01	9.84E + 00	2.10E + 00	2.23E + 01	1.29E + 01	5.61E + 01	0.00E + 00
			SD	2.38E−04	1.92E−02	9.13E−03	1.98E + 01	1.70E + 00	1.36E−01	2.19E + 00	2.33E + 00	8.88E + 00	0.00E + 00
F5	50	0	Best	4.36E + 01	1.08E + 00	4.77E + 01	4.46E + 01	4.64E + 01	2.44E + 02	7.59E + 01	4.55E + 01	3.99E + 02	4.59E + 01
			Worst	4.59E + 01	6.12E + 00	4.89E + 01	3.05E + 02	2.07E + 02	1.37E + 03	2.76E + 02	8.13E + 02	1.26E + 06	4.84E + 01
			Average	4.45E + 01	3.40E + 00	4.85E + 01	8.00E + 01	1.31E + 02	3.98E + 02	1.82E + 02	1.55E + 02	3.05E + 05	4.68E + 01
			SD	5.60E−01	1.56E + 00	3.01E−01	7.06E + 01	3.95E + 01	2.44E + 02	7.13E + 01	2.23E + 02	4.11E + 05	6.95E−01
F6	50	0	Best	1.70E−02	2.17E−07	5.39E + 00	0.00E + 00	1.90E−05	1.03E + 01	1.50E−05	3.48E−08	8.35E + 00	7.11E−01
			Worst	7.79E−01	6.96E−07	6.88E + 00	7.89E−01	4.33E−03	1.87E + 01	7.00E−05	6.35E−08	6.41E + 02	2.50E + 00
			Average	4.76E−01	4.05E−07	6.23E + 00	4.81E−02	7.00E−04	1.26E + 01	3.89E−05	4.37E−08	7.24E + 01	1.51E + 00
			SD	2.30E−01	1.47E−07	4.01E−01	1.76E−01	1.01E−03	1.93E + 00	1.36E−05	7.17E−09	1.43E + 02	4.46E−01
F7	50	0	Best	4.14E−04	3.43E−02	1.04E−06	8.22E−02	3.01E−02	5.71E−03	5.92E−02	5.10E−02	2.19E−02	1.32E−04
			Worst	5.58E−03	8.82E−02	5.14E−05	2.10E−01	7.77E−02	1.42E−02	9.32E−02	2.70E−01	2.50E + 00	2.20E−03
			Average	1.83E−03	5.76E−02	1.76E−05	1.31E−01	4.80E−02	9.38E−03	7.47E−02	1.76E−01	3.65E−01	7.48E−04
			SD	1.48E−03	1.25E−02	1.42E−05	3.79E−02	1.36E−02	2.96E−03	1.02E−02	5.41E−02	5.58E−01	4.68E−04
F8	50	−20,949	Best	−1.40E + 04	−6.60E + 03	−8.83E + 03	−1.40E + 04	−1.44E + 04	−1.42E + 04	−1.40E + 04	−1.46E + 04	−5.94E + 03	−1.12E + 04
			Worst	−1.02E + 04	−5.38E + 03	−7.19E + 03	−1.07E + 04	−1.12E + 04	−1.15E + 04	−1.27E + 04	−1.05E + 04	−4.80E + 03	−5.71E + 03
			Average	−1.16E + 04	−6.00E + 03	−7.93E + 03	−1.22E + 04	−1.27E + 04	−1.27E + 04	−1.33E + 04	−1.23E + 04	−5.27E + 03	−9.07E + 03
			SD	1.06E + 03	3.30E + 02	5.25E + 02	7.85E + 02	7.06E + 02	6.93E + 02	3.75E + 02	1.07E + 03	3.40E + 02	1.19E + 03
F9	50	0	Best	0.00E + 00	8.28E−01	0.00E + 00	1.23E + 02	1.62E + 02	5.63E + 01	1.81E + 02	4.28E + 01	8.46E−01	0.00E + 00
			Worst	1.87E + 01	1.13E + 00	0.00E + 00	3.15E + 02	2.96E + 02	1.16E + 02	2.24E + 02	1.41E + 02	2.07E + 02	1.02E + 01
			Average	1.61E + 00	1.01E + 00	0.00E + 00	2.05E + 02	2.14E + 02	7.33E + 01	2.00E + 02	8.02E + 01	7.51E + 01	6.85E−01
			SD	5.03E + 00	7.39E−02	0.00E + 00	4.96E + 01	3.64E + 01	1.69E + 01	1.01E + 01	2.43E + 01	6.17E + 01	2.38E + 00
F10	50	0	Best	0.00E + 00	4.32E−04	0.00E + 00	0.00E + 00	1.68E−03	9.22E−01	8.21E−04	1.27E + 00	8.37E−01	0.00E + 00
			Worst	0.00E + 00	1.54E−03	0.00E + 00	1.91E + 01	1.47E + 00	1.49E + 00	1.74E−03	3.67E + 00	2.05E + 01	0.00E + 00
			Average	0.00E + 00	7.31E−04	0.00E + 00	7.64E + 00	4.29E−01	1.18E + 00	1.34E−03	2.46E + 00	1.84E + 01	0.00E + 00
			SD	0.00E + 00	2.56E−04	0.00E + 00	8.72E + 00	5.63E−01	1.18E−01	2.29E−04	6.08E−01	5.08E + 00	0.00E + 00
F11	50	0	Best	0.00E + 00	3.59E−06	9.76E−02	1.27E + 00	9.93E−05	1.08E + 00	2.35E−05	9.60E−07	1.50E−01	0.00E + 00
			Worst	0.00E + 00	6.24E−03	7.64E−01	9.22E + 01	5.88E−02	1.17E + 00	2.93E−04	3.94E−02	3.47E + 00	1.88E−02
			Average	0.00E + 00	8.27E−04	3.89E−01	2.17E + 01	1.06E−02	1.12E + 00	9.31E−05	7.31E−03	1.29E + 00	1.63E−03
			SD	0.00E + 00	1.46E−03	1.79E−01	3.58E + 01	1.65E−02	2.21E−02	6.69E−05	9.82E−03	7.62E−01	5.08E−03
F12	50	0	Best	3.25E−05	1.03E + 00	5.01E−01	4.77E + 00	2.61E−06	1.36E−02	7.08E−06	3.07E + 00	2.21E + 00	2.29E−02
			Worst	1.83E−02	4.44E + 00	6.59E−01	1.10E + 01	7.58E−01	2.87E−02	4.50E−05	1.35E + 01	6.61E + 06	8.12E−02
			Average	7.47E−03	2.48E + 00	5.96E−01	7.54E + 00	1.44E−01	2.11E−02	2.12E−05	7.48E + 00	6.31E + 05	4.71E−02
			SD	5.59E−03	1.04E + 00	3.35E−02	2.07E + 00	2.24E−01	4.27E−03	9.94E−06	2.25E + 00	1.54E + 06	1.70E−02
F13	50	0	Best	3.89E−03	6.80E−01	4.60E + 00	1.10E−02	4.25E−04	4.55E−01	3.83E−05	1.10E−02	7.71E + 00	1.03E + 00
			Worst	5.78E−01	3.83E + 00	4.95E + 00	4.95E + 01	2.34E−01	7.56E−01	1.79E−04	7.93E + 01	5.49E + 07	2.19E + 00
			Average	3.27E−01	1.50E + 00	4.82E + 00	2.31E + 01	5.43E-02	5.85E−01	1.04E−04	3.58E + 01	4.42E + 06	1.55E + 00
			SD	1.51E−01	7.69E−01	1.04E−01	1.38E + 01	7.41E−02	7.84E−02	3.44E−05	2.67E + 01	1.23E + 07	2.98E−01
F14	2	1	Best	9.98E−01	4.00E−03	9.98E−01	9.98E−01	9.98E−01	9.98E−01	9.98E−01	9.98E−01	9.98E−01	9.98E−01
			Worst	9.98E−01	1.94E−02	1.27E + 01	1.27E + 01	9.98E−01	1.55E + 01	9.98E−01	9.98E−01	1.00E + 00	1.27E + 01
			Average	9.98E−01	7.61E−03	8.22E + 00	2.97E + 00	9.98E−01	4.58E + 00	9.98E−01	9.98E−01	9.98E−01	2.76E + 00
			SD	5.09E−17	4.11E−03	4.68E + 00	2.90E + 00	0.00E + 00	3.68E + 00	0.00E + 00	1.53E−16	1.17E−03	3.51E + 00
F15	4	0.003	Best	3.07E−04	1.94E−06	3.86E−04	3.07E−04	3.07E−04	5.27E−04	4.37E−04	3.74E−04	3.96E−04	3.07E−04
			Worst	3.07E−04	6.35E−04	8.00E−02	2.04E−02	2.04E−02	1.38E−03	7.63E−04	1.22E−03	1.35E−03	3.08E−04
			Average	3.07E−04	3.61E−04	1.08E−02	4.62E−03	1.82E−03	6.64E−04	6.11E−04	7.26E−04	8.55E−04	3.07E−04
			SD	2.76E−13	1.75E−04	1.84E−02	8.08E−03	4.39E−03	1.78E−04	1.01E−04	2.54E−04	3.46E−04	7.80E−09
F16	2	3	Best	3.00E + 00	4.86E−04	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00
			Worst	3.00E + 00	5.55E−02	3.00E + 01	3.00E + 00	3.00E + 00	3.00E + 01	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00
			Average	3.00E + 00	1.24E−02	4.35E + 00	3.00E + 00	3.00E + 00	4.35E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00
			SD	9.34E−16	1.41E−02	6.04E + 00	1.25E−15	2.70E−16	6.04E + 00	6.52E−16	6.18E−14	1.42E−05	3.45E−06
F17	3	−3.86	Best	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00
			Worst	−3.86E + 00	−3.86E + 00	−3.85E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	−3.85E + 00	−3.85E + 00
			Average	−3.86E + 00	−3.86E + 00	−3.85E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00	-3.86E + 00	−3.86E + 00	−3.86E + 00	−3.86E + 00
			SD	2.28E−15	2.28E−15	3.96E−03	2.05E−15	2.28E−15	1.86E−15	2.28E−15	1.11E−14	2.50E−03	2.83E−03
F18	6	−3.32	Best	−3.32E + 00	−3.32E + 00	−3.23E + 00	−3.32E + 00	−3.32E + 00	−3.32E + 00	−3.32E + 00	−3.32E + 00	−3.20E + 00	−3.32E + 00
			Worst	−3.32E + 00	−3.32E + 00	−2.93E + 00	−3.20E + 00	−3.20E + 00	−3.20E + 00	−3.32E + 00	−3.20E + 00	−2.59E + 00	−3.08E + 00
			Average	−3.32E + 00	−3.32E + 00	−3.12E + 00	−3.22E + 00	−3.27E + 00	−3.28E + 00	−3.32E + 00	−3.23E + 00	−3.02E + 00	−3.26E + 00
			SD	5.61E−12	4.89E−16	5.86E−02	4.36E−02	6.07E−02	5.82E−02	4.05E−05	4.90E−02	1.55E−01	7.27E−02
F19	4	−10.1532	Best	−1.02E + 01	−1.02E + 01	−6.59E + 00	−1.02E + 01	−1.02E + 01	−1.02E + 01	−1.02E + 01	−1.02E + 01	−6.47E + 00	−1.02E + 01
			Worst	−1.02E + 01	-1.02E + 01	−2.77E + 00	−2.63E + 00	−2.63E + 00	−2.63E + 00	−1.02E + 01	−2.63E + 00	−4.97E−01	−5.06E + 00
			Average	−1.02E + 01	−1.02E + 01	−4.45E + 00	−5.89E + 00	−7.89E + 00	−7.14E + 00	-1.02E + 01	−8.90E + 00	−4.06E + 00	−9.90E + 00
			SD	6.17E−15	1.03E−11	9.60E−01	3.36E + 00	3.25E + 00	3.50E + 00	2.21E−05	2.64E + 00	1.85E + 00	1.14E + 00
F20	4	−10.4028	Best	−1.04E + 01	−1.04E + 01	−8.22E + 00	−1.04E + 01	−1.04E + 01	−1.04E + 01	−1.04E + 01	−1.04E + 01	−8.09E + 00	−1.04E + 01
			Worst	−1.04E + 01	−1.04E + 01	−2.79E + 00	−2.75E + 00	−2.77E + 00	−1.84E + 00	−1.04E + 01	−5.09E + 00	−9.07E−01	−1.04E + 01
			Average	−1.04E + 01	−1.04E + 01	−4.74E + 00	−8.44E + 00	−8.19E + 00	−5.87E + 00	−1.04E + 01	−9.87E + 00	−4.15E + 00	−1.04E + 01
			SD	3.67E−15	3.42E−03	1.38E + 00	3.12E + 00	3.17E + 00	3.50E + 00	5.81E−05	1.63E + 00	2.28E + 00	1.61E−04
F21	4	−10.5363	Best	−1.05E + 01	−1.05E + 01	−6.02E + 00	−1.05E + 01	−1.05E + 01	−1.05E + 01	−1.05E + 01	−1.05E + 01	−8.34E + 00	−1.05E + 01
			Worst	−1.05E + 01	−1.05E + 01	−2.17E + 00	−2.42E + 00	−2.42E + 00	−2.42E + 00	−1.05E + 01	−2.43E + 00	−9.47E−01	−1.05E + 01
			Average	−1.05E + 01	−1.05E + 01	−4.30E + 00	−8.06E + 00	−9.72E + 00	−7.16E + 00	−1.05E + 01	−8.56E + 00	−4.96E + 00	−1.05E + 01
			SD	5.04E−15	1.06E−04	1.01E + 00	3.53E + 00	2.50E + 00	3.85E + 00	4.99E−13	3.18E + 00	1.66E + 00	1.65E−04
Mean rank				2.88	3.64	6.17	7.67	5.64	6.21	4.29	5.98	8.76	3.76
Rank				1	2	7	9	5	8	4	6	10	3

The study used the multimodal, separable, and non-separable benchmark functions (F10-F21) to test the exploration ability of the IDMO. The algorithms seem to perform relatively well, finding optimal or near-optimal solutions. The IDMO, however, returned the best solution for most of the functions and performed competitively for others. The null hypothesis of related-samples Friedman's test assumed that the distributions of IDMO, DMO, CPSOGSA, AOA, GWO, DE, BBO, SCA, SSA, and PSO are the same. The test results are summarized in Table 4; a p value of 0.000 is returned by the test, which is far less than the significant tolerance level of 0.05, so the hypothesis is rejected. The lowest mean rank is returned by the IDMO, as shown in Fig. 4, which means it ranked first for all the 21 benchmark functions.

Table 4.

Summary of Friedman’s test

Total N	21
Test statistic	75.228
Degree of freedom	9
Asymptotic Sig. (two-sided test)	0.000

Fig. 4.

Mean ranking of the 10 algorithms used

The Wilcoxon Signed Ranks Test on classical benchmark test functions shown in Table 5 confirmed the superiority of the optimization capability of IDMO. The IDMO has higher R + values in 7 out of 9 cases, and the number of R − and ties combined is more for the remaining two cases. This outcome implies that IDMO significantly outperforms DMO, AOA, CPSOGSA, PSO, BBO, DE, SSA, SCA, and GWO for all of the classical benchmark test functions. Their respective p-values returned by Wilcoxon's test at tolerance level, α = 0.05, show a significant difference in the performance of the algorithms in 7 out of 9 cases, which means that IDMO significantly outperforms 7 out of 9 algorithms on all classical test functions.

Table 5.

Wilcoxon signed ranks test for classical benchmark functions

Algorithms		N	Mean rank	Sum of ranks	Z	Asymp. Sig. (two-tailed)
DMO–IDMO	Negative ranks	5	12.40	62.00	−0.686b	0.492
	Positive ranks	12	7.58	91.00
	Ties	4
	Total	21
AOA–IDMO	Negative ranks	3	5.67	17.00	−2.983b	0.003
	Positive ranks	15	10.27	154.00
	Ties	3
	Total	21
CPSOGSA–IDMO	Negative ranks	2	11.00	22.00	−2.938b	.003
	Positive ranks	17	9.88	168.00
	Ties	2
	Total	21
PSO–IDMO	Negative ranks	3	11.00	33.00	−2.286b	.022
	Positive ranks	15	9.20	138.00
	Ties	3
	Total	21
BBO–IDMO	Negative ranks	1	19.00	19.00	−3.211b	.001
	Positive ranks	19	10.05	191.00
	Ties	1
	Total	21
DE–IDMO	Negative ranks	4	9.00	36.00	−1.036b	.300
	Positive ranks	10	6.90	69.00
	Ties	7
	Total	21
SSA–IDMO	Negative ranks	2	11.50	23.00	−2.722b	.006
	Positive ranks	16	9.25	148.00
	Ties	3
	Total	21
SCA–IDMO	Negative ranks	0	.00	.00	−3.920b	< ,001
	Positive ranks	20	10.50	210.00
	Ties	1
	Total	21
GWO–IDMO	Negative ranks	4	5.00	20.00	−2.040b	.041
	Positive ranks	10	8.50	85.00
	Ties	7
	Total	21

^bBased on negative ranks

Also, the IDMO returned smaller values for the standard deviation than the other exiting algorithms, which translates to IDMO being a more stable algorithm. The convergence rate comparison in Fig. 5 confirms this assertion and reveals that IDMO converges toward the best solution early in the iteration process. This convergence can be attributed to the alpha effectively pulling the other population members toward the optimal solution early in the iteration. In some cases (F2-F4), this effect is seen in the middle of the iteration. In the case of F8, IDMO converged toward the optimal solution late in the iteration process.

Fig. 5.

Convergence rate comparison for classical and CEC2020 benchmark functions

The CEC2020 Test Functions

The robustness and stability of the proposed IDMO were further tested using all the functions found in CEC 2020 benchmark test suite. The results of these tests are presented in Table 6. Clearly, the IDMO returned better solutions than other algorithms because it could find the global optimum for F22, F23, F25, F27, and F29. The IDMO performed competitively for the rest of the functions, returning solutions very close to optimal ones. The solutions returned by the AOA, GWO, and CPSOGSA are also competitive. Friedman’s test was used to carry out a non-parametric test on the results, as presented in Table 7. The IDMO also returned the lowest mean ranks of these comparisons, ranking first amongst all the algorithms compared, as shown in Fig. 6. The convergence rate comparison for the CEC 2020 is also shown in Fig. 5. The IDMO, closely followed by AOA, GWO, and CPSOGSA, provided the best convergence.

Table 6.

Results of CEC2020 test functions

Function	Global Opt	Value	IDMO	DMO	AOA	CPSOGSA	PSO	BBO	DE	SSA	SCA	GWO
F22	100	Best	100.07	409.33	1.31E + 09	103.32	107.58	101.45	326.91	18,406,000	2.06E + 08	4132.3
		Worst	101.43	22,626	1.34E + 10	12,616	5618	9479.3	5794.6	12,320	1.38E + 09	4.81E + 08
		Average	100.33	5538.9	5.06E + 09	4335.2	1676.6	2057.6	1992.1	2825.4	6.75E + 08	1.96E + 07
		SD	0.27695	6041.8	2.40E + 09	3972	1566.3	2482.6	1501.2	2797.2	2.97E + 08	8.75E + 07
F23	1100	Best	1108.1	1499.5	1506.9	1347.8	1115.1	1250.5	1308.2	1482.1	1738.8	1116.8
		Worst	1507	2439.7	2621.8	2859.8	1905.4	2390.9	1754.7	2526.7	2785.6	1876.5
		Average	1242.2	2064.3	2018.7	2188.3	1466.3	1783.9	1552.7	1916.5	2301.3	1524.2
		SD	97.941	206.86	310.98	358.44	187.56	263.22	111.6	300.71	250.68	210.7
F24	700	Best	713.47	724.85	766.97	729.55	712.76	714.08	716.08	733.85	762.11	717.17
		Worst	727.11	744.13	828.09	821.18	732.39	729.92	727.08	751.65	796.07	746.98
		Average	718.36	733.3	801	759.03	721.39	720.9	722.16	731.28	775.54	730.58
		SD	3.2551	4.6801	15.2	23.211	4.9189	4.36	2.245	10.17	9.1564	10.56
F25	1900	Best	1900.6	1900.5	3854.5	1900.7	1900.5	1900.1	1900.8	1903	1909.1	1900.5
		Worst	1901.8	1902.7	186,740	1936.5	1901.9	1904.6	1901.9	1903.6	2082.2	1903.5
		Average	1901.3	1901.3	62,921	1902.8	1901.1	1901.7	1901.6	1901.7	1923.9	1901.9
		SD	0.30019	0.56783	52,820	6.4186	0.37742	0.94168	0.28284	0.7055	30.854	0.79346
F26	1700	Best	1719.8	2689.5	34,413	2180.1	2261.1	2208	4677.1	2951.6	5195.9	3109.4
		Worst	1814.9	9413.5	369,980	61,662	8108.7	121,810	98,655	17,343	69,371	352,510
		Average	1759.8	4343.8	170,260	12,593	4208	25,290	27,794	5886.3	19,164	65,047
		SD	21.325	1491.5	92,783	14,552	1651.5	33,507	19,248	3862.2	14,917	128,330
F27	1600	Best	1600.5	1600.6	1600.5	1600.5	1600.5	1600.5	1600.5	1600.7	1601	1600.5
		Worst	1600.5	1601.1	1630.8	1660.7	1659	1619.3	1600.9	1603.1	1602.4	1622.2
		Average	1600.5	1600.9	1615.2	1616.6	1612.9	1603.3	1600.7	1601.1	1601.5	1603.2
		SD	0.00086963	0.12188	8.8082	22.669	11.984	5.9223	0.083246	0.49274	0.29381	5.9343
F28	2100	Best	2103.6	2360	2707.9	2527.5	2134.6	2125.9	2239.1	3520	3206.3	2533
		Worst	2125.1	2968.9	57,694	23,244	4599.7	22,806	5027.4	19,475	26,612	17,583
		Average	2109.2	2579.1	8043.3	8128.1	2701.9	8606.8	3169.1	6999.2	10,094	8628.2
		SD	4.4189	134.97	9733.1	6196.7	537.8	7009.3	790.72	4990.2	5331.4	4655.6
F29	2200	Best	2200.1	2239.3	2417.6	2246.9	2200	2300.6	2274.2	2311.5	2273.1	2301.5
		Worst	2303.2	2303.5	3346.4	3376.6	2304.8	2309.7	2301.6	2309.3	2424.6	2321.4
		Average	2279.3	2287.8	2791.4	2336.5	2298.7	2302.6	2299.9	2296	2368.4	2307.9
		SD	42.131	19.046	254.97	196.71	18.67	1.6989	4.9212	22.805	31.759	4.9224
F30	2400	Best	2416.3	2500	2688.9	2500	2500	2500	2560.5	2503.2	2534.2	2729.5
		Worst	2736.9	2614.1	2975.6	2824.4	2764.2	2767.4	2753.8	2768.8	2801.5	2771
		Average	2539.7	2522.1	2827.1	2758.7	2723.9	2715.7	2725.6	2747.5	2768.2	2748.3
		SD	65.097	29.925	70.872	72.626	55.138	86.578	44.927	8.3183	55.568	12.484
F31	2500	Best	2601.8	2898	2950.8	2898.1	2600.1	2897.8	2899.9	2904.7	2923.3	2900.2
		Worst	2898.1	2940.7	3445.1	3024.2	2948	2949.4	2947.5	2950.6	2971.2	2949.9
		Average	2887.9	2903.9	3149.9	2940.4	2907.5	2932.1	2913.9	2925.7	2956.8	2934.3
		SD	54.03	9.8305	106.65	37.844	62.419	22.393	13.589	23.926	14.119	17.203
Mean rank			1.25	3.75	9.2	7.70	3.20	5.35	4.30	4.95	8.50	5.80
Rank			1	3	10	8	2	6	4	5	9	7

Table 7.

Summary of Friedman’s test

Total N	10
Test statistic	62.694
Degree of freedom	9
Asymptotic sig. (2−sided test)	.000

Fig. 6.

Mean ranking for CEC functions

A further Wilcoxon Signed Ranks Test for CEC 2020 test functions is presented in Table 8. A p value less than the 0.05 significance level indicates that the obtained results for IDMO significantly outperform the other algorithms considered in this study. Also, this assertion is further confirmed by the IDMO obtaining higher R + values than R − or ties in all the cases. The implication is that the IDMO significantly outperforms all the algorithms used in the comparison for all the CEC 2020 test functions. Thus, it can be concluded that the efficiency, searchability, and robustness of IDMO are better than the other state-of-the-art algorithms used in this study.

Table 8.

Wilcoxon signed ranks test for CEC 2020 test functions

Algorithm		N	Mean rank	Sum of ranks	Z	Asymp. sig. (two-tailed)
DMO—IDMO	Negative ranks	1	5.00	5.00	−2.073b	0.038
	Positive ranks	8	5.00	40.00
	Ties	1
	Total	10
AOA—IDMO	Negative ranks	0	0.00	0.00	−2.803b	0.005
	Positive ranks	10	5.50	55.00
	Ties	0
	Total	10
CPSOGSA—IDMO	Negative ranks	0	0.00	0.00	−2.803b	0.005
	Positive ranks	10	5.50	55.00
	Ties	0
	Total	10
PSO—IDMO	Negative ranks	1	1.00	1.00	−2.701b	0.007
	Positive ranks	9	6.00	54.00
	Ties	0
	Total	10
BBO—IDMO	Negative ranks	0	0.00	0.00	−2.803b	0.005
	Positive ranks	10	5.50	55.00
	Ties	0
	Total	10
DE—IDMO	Negative ranks	0	0.00	0.00	−2.803b	0.005
	Positive ranks	10	5.50	55.00
	Ties	0
	Total	10
SSA—IDMO	Negative ranks	0	0.00	0.00	−2.803b	0.005
	Positive ranks	10	5.50	55.00
	Ties	0
	Total	10
SCA—IDMO	Negative ranks	0	0.00	0.00	−2.803b	0.005
	Positive ranks	10	5.50	55.00
	Ties	0
	Total	10
GWO—IDMO	Negative ranks	0	0.00	0.00	−2.803b	0.005
	Positive ranks	10	5.50	55.00
	Ties	0
	Total	10

^bBased on negative ranks

Engineering Problems

This section presents the results of experiments conducted using the IDMO to optimize twelve (12) benchmark problems in different engineering domains. The population size and the maximum number of iterations used for IDMO, DMO, and AOA are 50 and 1000, respectively. The results for the remaining comparative algorithms are obtained from their respective literature. The number of function evaluations (FEs) is set at $\mathrm{population}\mathrm{size}\times \mathrm{maximum}\mathrm{iterations}$. Windows 10 OS environment, Intel Core i7-7700@3.60 GHz CPU, and 16G RAM were used to conduct the experiments. The results of 30 independent runs of each algorithm are collated using the “best, worst, average, and SD” performance indicators. Further statistical analysis was carried out using mean, standard deviation, and Friedman and Wilcoxon tests. The details about the 12 engineering problems can be found at the following:

• The welded beam design problem [35, 36]

• The compression spring design problem (CSD) [37]

• The pressure vessel design problem (PVD) [38]

• The speed reducer design problem (SRD) [39]

• The three-bar truss design problem (3-BTD) [40]

• The gear train design problem (GTD) [41]

• The cantilever beam design problem (CBD) [42]

• The optimal design of I-shaped beam (IBD) [43]

• The tubular column design (TCD) [44]

• The piston lever design problem (PLD) [43]

• The corrugated bulkhead design problem (Cbhd) [45]

• The reinforced concrete beam design problem (RCB) [46]

Results of the Welded Beam Design Problem (WBD)

The results and statistical analysis of optimizing the WBD using IDMO and other existing results in the literature are presented in Table 9. The IDMO found the least minimum cost, and Fig. 7 shows that this value was found at the 400th iteration. This means the results were achieved within 20,000 function evaluations (FEs). The least minimum cost for IDMO was achieved within the desired combination of the four (4) problem design variables, as shown in Table 9. The IDMO also returned the minor standard deviation and average values, thereby confirming the stability and robustness of the IDMOin solving the WBD problem.

Table 9.

The comparative results of WBD

Algorithm	h	l	t	b	Best	Worst	Average	SD	FEs
IDMO	0.200952683	3.353311664	9.036343673	0.20574711	1.6952	1.781	1.7192	0.021464	20,000
DMO [32]	0.205234326	3.26225324	9.037040824	0.205730354	1.6953	1.6991	1.6961	0.00097687	35,000
AOA [70]	0.247498553	2.536812819	10	0.250284392	1.831	2.928	2.3579	0.27284	40,000
BBO [77]	0.185486	4.3129	8.439903	0.235902	1.918055	3.606933	2.630412	4.11E—01	50,000
PSO [77]	0.219292	3.430416	8.433559	0.236204	1.85272	3.841845	2.613785	4.71E—01	50,000
ICA [77]	0.205799	3.469634	9.03495	0.205806	1.725135	2.237755	1.79433	1.10E—01	50,000
WCA [78]	0.205728	3.470522	9.03662	0.205729	1.724856	1.744697	1.726427	4.29E—03	46,500
ABC [79]	0.20573	3.470489	9.036624	0.20573	1.724852	NA	1.741913	3.10E—02	30,000
EO [80]	0.2057	3.4705	9.0366	0.2057	1.724853	1.736725	1.726482	3.26E—03	15,000
TEO [81]	0.205681	3.472305	9.035133	0.205796	1.725284	1.931161	1.76804	5.82E—02	NA
SSA [74]	0.2057	3.4714	9.0366	0.2057	2.246638	1.725886	1.823426	1.28E—01	50,000
WSA [77]	0.20573	3.470489	9.036624	0.20573	1.724852	1.725068	1.724908	4.15E—05	50,000
GWO [77]	0.205677	3.470894	9.038558	0.205739	1.728487	1.725232	1.72631	7.71E—04	50,000
SHO [82]	0.205563	3.474846	9.035799	0.205811	1.725661	1.726064	1.725828	2.87E—04	NA
WOA [83]	0.205396	3.484293	9.037426	0.206276	1.730499	NA	1.732	0.0226	9900
SNS [84]	0.2057296	3.4704887	9.0366239	0.2057296	1.724852	1.725051	1.72488	5.18E—05	9000

Fig. 7.

Convergence rate graph for WBD

Results of the Compression Spring Design Problem (CSD)

The results of optimizing the CSD with IDMO and other existing results are presented in Table 10. The IDMO failed to return the least cost of the objective function. However, its result is better than that of the DMO and AOA. The IDMO needed 20 iterations, corresponding to 1,000 FEs, as shown in Fig. 8.

Table 10.

Summary of comparative results for CSD

Algorithm	d	D	N	Best	Worst	Mean	SD	NFEs
IDMO	0.13914	1.3	11.88923	3.6619	3.6923	3.6632	0.005662	1,000
DMO [32]	0.13915	1.3	11.89243	3.6619	3.6619	3.6619	1.57E−15	20,000
AOA [70]	0.148317	1.3	15	4.3133	6.585	6.0337	0.52677	50,000
CPSO [85]	0.051728	0.357644	11.244543	0.0126747	0.012924	0.01273	5.20E—05	200,000
HPSO [86]	NA	NA	NA	0.0126652	0.0127191	0.0127072	1.58E—05	81,000
CDE [87]	0.051689	0.356718	11.288968	0.0126702	0.01279	0.012703	2.70E—05	204,800
PSO [77]	0.05169	0.356737	11.28885	0.012857	0.071802	0.019555	1.17E—02	20,000
QPSO [77]	0.0518	0.359	11.279	0.012669	0.018127	0.013854	1.34E—03	20,000
G−QPSO [77]	NA	NA	NA	0.012666	0.015869	0.012996	6.28E—04	20,000
WCA [88]	0.05168	0.3565	11.3004	0.012665	0.012952	0.012746	8.06E—05	11,750
ABC [79]	0.051749	0.358179	11.203763	0.012665	NA	0.012709	1.28E—02	30,000
APSO [89]	0.052588	0.378343	10.138862	0.0127	0.014937	0.013297	6.85E—04	120,000
IAPSO [89]	0.051685	0.356629	11.294175	0.01266523	0.01782864	0.01367653	1.57E—03	20,000
WOA [83]	0.0512	0.3452	12.004	0.0126763	NA	0.0127	3.00E—04	4410
MCEO [90]	0.051994	0.364109	10.868421	0.01266051	0.01350901	0.0127196	3.79E—05	2000
EO [80]	0.051207	0.345215	12.004032	0.012666	0.013997	0.013017	3.91E—04	15,000
SNS [84]	0.051587	0.354268	11.434058	0.01266525	0.01276587	0.01268472	2.39E—05	9000

Fig. 8.

Convergence rate graph for CSD

Result of the Pressure Vessel Design Problem (PVD)

Similarly, Table 11 presents the comparative results and statistical analysis of the IDMO and several other existing optimization algorithms. The IDMO returned the least cost of the design problem, achieved at the 20th iteration, equivalent to 1,000 FEs, as shown in Fig. 9. Similarly, the stability of the IDMO is confirmed by returning the least value of the average and standard deviation performance indicators.

Table 11.

Summary of the comparative results for PVD

Algorithm	Ts	Th	R	L	Best	Worst	Average	SD	FEs
IDMO	0.5479	0.2469	43.4967	160.0482	4527.2	5106.1	4533.2	279.92	1000
DMO [32]	0.4611	0.2401	40.3196	200	4527.3	4527.3	4527.3	2.96E−11	5000
AOA [70]	0.3764	0.3945	40.6441	200	4739.3	6763.9	5592.3	549.35	45,000
SAP [91]	0.8125	0.4375	40.3239	200	6288.8	6308.2	6293.8	7.41E + 00	3000
HPSO [85]	0.8125	0.4375	42.0984	176.6366	6059.7	6288.7	6099.9	8.62E + 01	81,000
CDE [87]	0.8125	0.4375	42.0984	176.6376	6371.1	6059.7	6085.2	4.30E + 01	204,800
CPSO [85]	0.8125	0.4375	42.0913	176.7465	6061.1	6363.8	6147.1	8.65E + 01	200,000
PSO [92]	0.8125	0.4375	42.0984	176.6366	6693.7	14,076.3	8756.7	1.49E + 03	8000
QPSO [92]	0.8125	0.4375	42.0984	176.6374	6059.7	8017.3	6839.9	4.79E + 02	8000
G−QPSO [92]	0.8125	0.4375	42.0984	176.6372	6059.7	7544.5	6440.4	4.48E + 02	8000
ABC [79]	0.8125	0.4375	42.0985	176.6366	6059.7	NA	6245.3	2.05E + 02	30,000
CS [93]	0.8125	0.4375	42.0985	176.6366	6059.7	6495.4	6447.7	5.03E + 02	15,000
WOA [83]	0.8125	0.4375	42.0983	176.639	6059.7	NA	6068.05	6.57E + 01	6300
APSO [89]	0.8125	0.4375	42.0984	176.6374	6059.7	7544.5	6470.7	3.27E + 02	200,000
EO [80]	0.8125	0.4375	42.0985	176.6366	6059.7	7544.5	6668.1	5.66E + 02	15,000
CGO [94]	0.8125	0.4345	42.0892	176.7587	6247.7	6331	6251	1.07E + 01	100,000
SNS [84]	0.8125	0.4375	42.0985	176.6366	6059.7	6410.1	6097.1	9.28E + 01	6000

Fig. 9.

Convergence rate graph for PVD

Results of the Speed Reducer Design Problem (SRD)

The best and statistical results of optimizing the SRD with IDMO compared with other existing methods are presented in Table 12. It can be seen that the IDMO returned the least cost of the objective function. The “average and standard deviation” values returned by the IDMO confirm its stability and robustness. The required number of function evaluations (FEs) for the IDMO algorithm is 500, which is equivalent to the 10th iteration, as shown in Fig. 10, which is much lower than that of other algorithms.

Table 12.

Summary of comparative results for SRD

Algorithm	× 1	× 2	× 3	× 4	× 5	× 6	× 7	Best	Worst	Average	SD	FEs
IDMO	3.5	0.7	17	7.3	7.8	3.4	5.3	2993.7	3007.3	2998.2	3.2165	500
DMO [32]	3.5	0.7	17	7.3	7.7	3.4	5.3	2993.6	2993.6	2993.6	4.63E−13	10,000
AOA [32]	3.6	0.7	17	7.5	8.3	3.6	5.3	3059.2	3222.5	3119.1	32.434	40,000
CS [93]	3.5	0.7	17	7.6	7.8	3.3	5.3	3001	3009	3007.2	4.96E + 00	250,000
ABC [79]	3.5	0.7	17	7.3	7.8	3.4	5.3	2997.1	NA	2997.1	0.00E + 00	30,000
WCA [78]	3.5	0.7	17	7.3	7.7	3.4	5.3	2994.5	2994.5	2994.5	7.40E—03	15,150
APSO [89]	3.5	0.7	18	8.1	8.0	3.4	5.3	3187.6	4443.0	3822.6	3.66E + 02	30,000
SHO [82]	3.5	0.7	17	7.3	7.8	3.4	5.3	2998.6	3003.9	2999.6	1.93E + 00	NA
SSA [74]	3.5	0.7	17	7.3	7.7	3.4	5.3	2996.0	3015.7	3005.6	4.63E + 00	NA
WOA [83]	NA	NA	NA	NA	NA	NA	NA	2996.6	3233.6	3042.9	4.08E + 01	NA
CSS [93]	NA	NA	NA	NA	NA	NA	NA	2996.5	3106.2	3005.7	4.86E + 00	NA
CGO [94]	NA	NA	NA	NA	NA	NA	NA	2994.4	2995.5	2994.5	0.110282	100,000
FACSS [95]	NA	NA	NA	NA	NA	NA	NA	2996.4	3006.4	2999.4	4.82E + 00	NA
SNS [84]	3.5	0.7	17	7.3	7.7	3.4	5.3	2994.5	2994.5	2994.5	7.00E—06	3750

Fig. 10.

Convergence rate graph for SRD

Results of the Three-Bar Truss Design Problem (3-BTD)

The best results and statical analysis of optimizing the 3-BTD with IDMO and several other existing algorithms are presented in Table 13. The best objective value returned by the IDMO is the least and better than that of other algorithms. This was achieved within the 5th iteration (250 FEs), as shown in Fig. 11.

Table 13.

Summary of comparative results for 3−BTD

Algorithm	× 1	× 2	Best	Worst	Average	SD	FEs
IDMO	0.219498	0.188424	106.93	106.93	106.93	4.85E−05	250
DMO [32]	0.219515	0.188373	106.93	106.93	106.93	3.01E−14	500
AOA [32]	0.224256	0.192452	106.93	107.16	106.97	0.045939	45,000
GA [77]	0.788915	0.407569	263.89589	264.82081	263.96804	1.66862E—01	50,000
PSO [77]	0.788669	0.408265	263.89584	264.5849	263.95741	1.36897E—01	50,000
ICA [77]	0.788625	0.408389	263.89585	263.91413	263.89933	4.11693E—03	50,000
CS [93]	0.78867	0.40902	263.97156	NA	264.0669	9.00000E—05	15,000
WCA [78]	0.788651	0.408316	263.89584	263.8962	263.8959	8.71000E—05	5250
GWO [77]	0.788648	0.408325	263.89601	263.90422	263.89796	1.61422E—03	50,000
WSA [77]	0.788683	0.408227	263.89584	263.89743	263.89607	3.11960E—04	50,000
CGO [94]	NA	NA	263.89584	263.89601	263.89585	2.51E—05	100,000
AOS [96]	NA	NA	263.89584	263.89585	263.89584	8.26E—09	100,000
SNS [84]	0.7886847	0.4082211	263.89584	263.89586	263.89585	3.31056E—06	4800

Fig. 11.

Convergence rate graph for 3-BTD

Results of the Gear Train Design Problem (GTD)

The best results and statical analysis of optimizing the GTD with IDMO and several other existing algorithms are presented in Table 14. Though all the algorithms found the global optimum solution as the ‘best’ value, the IDMO returned the least value for the standard deviation and average values, indicating a more stable and robust optimization. Also, this was achieved within the minimum number of 5,000 function evaluations, as shown in Fig. 12.

Table 14.

Comparative result for GTD

Algorithm	× 1	× 2	× 3	× 4	Best	Worst	Average	SD	FEs
IDMO	48	17	22	54	2.70E−12	2.36E−09	1.22E−09	6.16E−10	5,000
DMO [32]	49	19	16	43	2.70E−12	2.31E−11	8.81E−12	9.50E−12	20,000
AOA [32]	55	14	34	60	2.31E−11	2.73E−08	1.28E−08	1.14E−08	45,000
GA [77]	49	19	16	43	2.70E−12	1.5247E−08	1.6212E−09	3.2174E−09	50,000
PSO [77]	34	13	20	53	2.31E−11	1.0222E−06	7.9383E−08	1.8147E−07	50,000
ICA [77]	43	16	19	49	2.70E−12	2.3576E−09	8.0417E−10	7.7862E−10	50,000
CS [93]	43	16	19	49	2.70E−12	6.51E−09	9.6633E−10	6.4529E−10	50,000
ABC [79]	49	16	19	43	2.70E−12	1.3616E−09	1.6800E−10	4.5748E−09	50,000
MSFWA [97]	49	19	16	43	2.70E−12	1.36165E−09	1.68012E−10	7.2953E−08	50,000
SNS [84]	43	19	16	49	2.70E−12	1.36165E−09	1.68012E−10	3.74894E−10	25,000

Fig. 12.

Convergence rate for GTD

Results of the Cantilever Beam Design Problem (CBD)

The best results and statical analysis of optimizing the CBD with IDMO and several other existing algorithms are presented in Table 15. It can be observed that the best solution for solving the CBD problem obtained is IDMO, which returned the optimal values for the five (5) design variables within the 80th iteration or 4000 FEs as shown in Fig. 13. Furthermore, the superiority and robustness of IDMO are shown by the least “average and standard deviation” values it returned.

Table 15.

Comparative result for CBD

Algorithm	× 1	× 2	× 3	× 4	× 5	Best	Worst	Average	SD	FEs
IDMO	5.689588	5.020755	4.261692	3.312994	2.040863	1.3004	1.3005	1.3004	1.31E−05	4,000
DMO [32]	5.694297	5.025255	4.253986	3.314226	2.037547	1.3004	1.3004	1.3004	2.26E−16	15,000
AOA [32]	6.322849	4.41889	4.119345	7.754706	1.46564	1.3847	3.2824	1.9122	0.52199	45,000
SOS [98]	6.01878	5.30344	4.49587	3.49896	2.15564	1.33996	NA	1.33997	1.1E—5	15,000
CGO [94]	6.01513	5.3093	4.495	3.50142	2.15278	1.33997	1.340602	1.340052	1.23E—04	100,000
AOS [96]	NA	NA	NA	NA	NA	1.339957	1.491711	1.351954	0.0249974	100,000
MGA [99]	NA	NA	NA	NA	NA	1.3399756	1.3402011	1.3400526	6.99E—05	100,000
SNS [84]	6.01545	5.31066	4.488	3.50528	2.15428	1.3399576	1.3399576	1.3399576	1.1102E—15	12,000

Fig. 13.

Convergence rate for CBD

Results of the Optimal Design of I-Shaped Beam (IBD)

The best results and statical analysis of optimizing the IBD with IDMO and several other existing algorithms are presented in Table 16. All the algorithms could find optimal solutions, and the least standard deviation was returned by the IDMO. This result confirms the IDMO's stability and superiority. Also, the results were achieved within the smallest possible function evaluations (150), as can be seen in Fig. 14.

Table 16.

Summary of comparative results for IBD

Algorithm	× 1	× 2	× 3	× 4	Best	Worst	Average	SD	FEs
IDMO	80	50	0.9	2.321795	0.013074	0.013078	0.013075	8.8592E−07	150
DMO [32]	80	50	0.9	2.321793	0.013074	0.013074	0.013074	1.523E−13	2,500
AOA [32]	80	50	0.9	2.321547	0.013074	0.013161	0.013089	1.6536E−05	4,000
GWO [100]	80	50	0.9	2.3217	0.0131	NA	NA	NA	NA
EMGO-FCR [100]	80	50	0.9	2.32	0.0131	NA	NA	NA	NA
CS [93]	80	50	0.9	2.3216	0.0130747	0.01353646	0.0132165	0.0001345	5000
SOS [98]	80	50	0.9	2.3217	0.0130741	NA	0.0130884	4.0E−5	5000
AOS [96]	NA	NA	NA	NA	0.0130741	0.013814	0.0131788	1.555E−04	100,000
SNS [84]	80	50	0.9	2.3217	0.0130741	0.0130764	0.0130743	4.313E−07	3600

Fig. 14.

Convergence rate for IBD

Results of the Tubular Column Design (TCD)

The TCD problem has been previously solved using various optimizers, and some of the best results obtained by these optimizers and IDMO are presented in Table 17. The number of function evaluations needed to obtain the results and statistical analysis of some optimizers is reported. It can be seen from Fig. 15 that the IDMO obtained this result in the 10th iteration or 500 FEs.

Table 17.

Summary of comparative results for TCD

Algorithm	× 1	× 2	Best	Worst	Average	SD	FEs
IDMO	6.182678144	0.2	24.615	2.46E + 01	24.615	1.95E−06	500
DMO [32]	6.182683216	0.2	24.615	2.46E + 01	24.615	1.81E−14	4,000
AOA [32]	6.179782123	0.2	24.615	2.47E + 01	24.631	0.021115	25,000
ISA [101]	5.45115623	0.29196547		2.65E + 01	26.531	NA	3000
CS [93]	5.45139	0.29196	26.53217	26.53972	26.53504	1.93E—03	15,000
FA [102]	NA	NA	26.52	NA	28.74	2.08	3000
AOS [96]	NA	NA	26.531378	26.608314	26.531614	1.0300E—03	100,000
SNS [84]	5.45115623	0.29196547	26.486361	26.486371	26.486362	2.2160E—06	1250

Fig. 15.

Convergence rate for TCD

The Piston Lever Design Problem (PLD)

The best results and statical analysis of optimizing the PLD with IDMO and several other existing algorithms are presented in Table 18. It can be seen that the results returned by the IDMO, DMO, and AOA are relatively small compared to those of CS, ISA, CGO, AOS, MGA, and SNS. Figure 16 shows that the result for IDMO was obtained at the 60th iteration or 3000 FEs.

Table 18.

Comparative result for PLD

Algorithm	× 1	× 2	× 3	× 4	Best	Worst	Average	SD	FEs
IDMO	0.05	0.144897318	4.11572157	120	4.695	167.87	10.136	29.791	3000
DMO [32]	0.05	0.125073578	4.116042166	120	4.6949	4.7006	4.6987	0.0027386	5000
AOA [32]	500	500	2.578147082	120	7.532	488.42	320.17	115.54	35,000
PSO [103]	133.3	2.44	117.14	4.75	122	294	166	51.7	50,000
DE [103]	129.4	2.43	119.8	4.75	159	199	187	14.2	50,000
GA [103]	250	3.96	60.03	5.91	161	216	185	18.2	50,000
HPSO [103]	135.5	2.48	116.62	4.75	162	197	187	13.4	50,000
HPSO with Q-learning [103]	NA	NA	NA	NA	129	168	151	13.4	50,000
CS [93]	0.05	2.043	120	4.085	8.4271	168.592	40.2319	59.0552	50,000
ISA [101]	NA	NA	NA	NA	8.4	610.6	226.5	111.2	12,500
CGO [94]	NA	NA	NA	NA	8.41281381	167.472809	45.04866	67.24763	100,000
AOS [96]	NA	NA	NA	NA	8.41914274	167.664986	33.741276	93.46674724	100,000
MGA [ [99]	NA	NA	NA	NA	8.41340665	167.473213	32.468893	29.96370439	100,000
SNS [84]	0.05	2.042	120	4.083	8.41269835	167.472775	24.318974	47.71792646	5000

Fig. 16.

Convergence rate for PLD

Results of the Corrugated Bulkhead Design Problem (CBHD)

The best results and statical analysis of optimizing the CBhD with IDMO and several other existing algorithms are presented in Table 19. The results showed the IDMO returned the least value for the objective function within the least possible function evaluations, as shown in Fig. 17.

Table 19.

Summary of comparative result for CBhD)

	× 1	× 2	× 3	× 4	Best	Worst	Average	SD	FEs
IDMO	48.31	54.78	61.93	0.43	4.6972	1.548	4.6693	4.3569	2500
DMO [32]	48.14	5.45E + 01	62.04	0.43	4.8201	4.6699	4.5407	0.70271	20,000
AOA [32]	43.46	74.92	100	0.39	5.002	5.9617	5.031	0.99549	40,000
FA [102]	37.12	33.04	37.19	0.73	7.21	NA	10.23	1.95	12,000
LF−FA [102]	57.69	34.15	57.69	1.05	6.95	NA	8.83	1.26	12,000
LS−LF−FA [102]	57.69	34.13	57.55	1.05	6.86	NA	7.44	0.67	12,000
AD−IFA [102]	NA	NA	NA	NA	6.84	NA	7.21	0.58	12,000
AOS [96]	NA	NA	NA	NA	6.843	7.0669	7.0608	6.49E – 04	100,000
SNS [84]	57.69	34.15	57.69	1.05	6.843	6.8431	6.843	2.09E—05	3125

Fig. 17.

Convergence rate for CBhD

Results of the Reinforced Concrete Beam Design Problem (RCB)

The best results and statical analysis of optimizing the IBD with IDMO and several other existing algorithms are presented in Table 20. It can be seen that values returned by the IDMO, DMO, and AOA are relatively small compared to FA, CS, AOS, and SNS. There is a significant improvement in the results available in the literature. It is important to note that this improvement was achieved within the least possible function evaluations (100) or 2nd iteration, as shown in Fig. 18.

Table 20.

Comparative result for RCB

Algorithm	× 1	× 2	× 3	Best	Worst	Average	SD	Fes
IDMO	6.16	28	5	357.5	357.6	357.55	1.12E−06	100
DMO [32]	6	33	5	357.6	357.6	357.6	1.17E−13	150
AOA [32]	8.4	28	5	357.6	357.6	357.6	0	200
FA [104]	6.32	34	8.5	359.208	669.15	460.706	80.7387	25,000
CS [93]	6.32	34	8.5	359.208	NA	NA	NA	5000
AOS [96]	6.32	34	8.5	359.208	362.2535	359.3306872	0.596149	100,000
SNS [84]	6.32	34	8.5	359.208	362.634	359.3222001	0.6149858	1000

Fig. 18.

Convergence rate for RCB

Summary of Results

This study proposes a modified version of the dwarf mongoose optimization algorithm (IDMO) for constrained engineering design problems. This optimization technique modifies the base algorithm (DMO) in three simple but effective ways. The proposed method solved 21 classical, 10 CEC2020, and 12 constrained benchmark engineering optimization problems at a relatively low computational cost. This section summarizes the obtained results.

The unimodal test functions provide a good tool to test the exploitation capabilities of IDMO. The results of these experiments show that IDMO effectively exploited the search regions and returned the optimal or near-optimal solution compared to the DMO and eight other algorithms. This scenario can be attributed to the newly introduced operator ω, which controls the alpha movement ensuring continuous neighborhood searching during exploitation and stops when the next phase is activated. Similarly, the results of IDMO on multimodal test functions, which are excellent tools for measuring the exploration ability of algorithms, were superior compared to the other algorithms used. Hence, it can be concluded that IDMO has very good exploration capabilities.

The IDMO demonstrated high optimization capabilities while solving the 12 engineering benchmark problems. It returned superior results for almost all the 12 engineering problems, as seen from the results. Also, the convergence analysis for the benchmark and engineering problems further confirms the superiority of the IDMO. As seen from the convergence figure, the IDMO found quality solutions early in the iteration process and converged steadily toward the optimal solution. Further statistical analysis of the obtained results using Friedman’s test showed that IDMO ranked first in the final ranking of all algorithms based on their general performance. A conclusion can be made that the proposed algorithm could perform real-world optimization tasks efficiently.

Conclusion and Future Work

This study proposes a modified version of the dwarf mongoose optimization algorithm (IDMO) for constrained engineering design problems. This optimization technique modifies the base algorithm (DMO) in three simple but effective ways. Firstly, the alpha selection in IDMO differs from the DMO, where evaluating the probability value of each fitness is just a computational overhead and contributes nothing to the quality of the alpha or other group members. The fittest dwarf mongoose is selected as the alpha, and a new operator ω is introduced, which controls the alpha movement, thereby enhancing the exploration ability and exploitability of the IDMO. Secondly, the scout group movements are modified by randomization to introduce diversity in the search process and explore unvisited areas. Finally, the babysitter's exchange criterium is modified such that once the criterium is met, the babysitters that are exchanged interact with the dwarf mongoose exchanging them to gain information about food sources and sleeping mound, which could result in better-fitted mongooses instead of initializing them afresh as done in DMO, then the counter is reset to zero.

The proposed method solved 21 classical, 10 CEC2020, and 12 constrained benchmark engineering optimization problems at a relatively low computational cost. The results and further statistical analysis showed that the IDMO is an effective tool for optimizing the selected optimization problems. A comparison with DMO and other state-of-the-art algorithms further solidifies the superiority of the IDMO. The IDMO achieved more balanced exploitation and exploration due to introducing a new operator ω, which controls the alpha movement. Also, the randomization of the scout group and babysitters' updating steps greatly influenced the performance of the IDMO.

While this study greatly improved the exploration and exploitation capabilities of the DMO, the overall effect or influence of the alpha female persists. Though the randomization of the updating steps prevents the IDMO from being trapped in local minima, the alpha size could still prevent effective exploitation. In the future, efforts could be made to address the issues mentioned above. Also, the capability of the IDMO could be tested on CEC 2011, CEC 2014, and CEC 2017 benchmark functions and other complex real-world optimization problems such as feature selection and classification using deep learning models.

Data availability statement

All data generated or analyzed during this study are included in this article.

Declarations

Conflict of interest

The authors declare that there is no conflict of interest with regard to the publication of this paper.