Quokka swarm optimization: A new nature-inspired metaheuristic optimization algorithm

1 Introduction

The primary aim of optimization is to identify the most suitable solution while considering the constraints and requirements of the problem at hand. Multiple solutions may exist for a given problem, and the objective function serves as a means to compare them and select the best one. The nature of the problem dictates the choice of function to be utilized. In transportation network optimization, e.g., a common objective is the reduction of trip time or costs. However, a critical aspect of the optimization process lies in selecting the appropriate objective function. Multi-objective problems involve the optimization of multiple objectives simultaneously, employing various objective functions [1]. In recent years, bio-inspired optimization algorithms have gained popularity due to their ease of use, robustness, and ability to perform parallel computation [2]. Many real-world problems are highly complex, featuring a diverse array of potential solutions, nonlinear constraints, and interconnected variables. This complexity underscores the need for a method capable of addressing challenging optimization problems in real-time.

The use of metaheuristic algorithms is one such method [3]. In order to be successful, a search algorithm must establish an effective equilibrium between exploration and exploitation. In this respect, Evolutionary algorithms (EAs) [4] like Genetic algorithms (GAs) [5], Evolutionary strategies (ES) [6], Evolutionary programming (EP) [7], and Genetic programming [8] are examples of additional well-known instance. Metaheuristics are broadly classified into three main classes: EAs, physics based (PB), and Swarm intelligence (SI) algorithms. EAs, which constitute the first class, get their inspiration from the genetic and evolutionary characteristics of living things. The GA is the most widely used algorithm in this field. In an effort to imitate the ideas of Darwinian evolution, Holland created GA in 1975 [9]. The second class is PB. Typically, these optimization methods imitate physical rules. The algorithm most widely favored in this category is Gravitational search algorithm (GSA) based algorithmic rule, in which its fundamental physical theory is inspired from Newton’s law of universal gravitation [10]. In SI, techniques simulate the intelligence of swarms, flocks, schools, or groups of animals in their natural habitat. The most popular algorithms in this class are Ant colony optimization [11], Particle swarm optimization (PSO) [12], and Artificial bee colony (ABC) algorithm [13]. Metaheuristics frequently possess the capability to partition the search process into distinct exploration and exploitation phases [14], notwithstanding the variations among them. The exploration phase entails thoroughly investigating the search space, which necessitates the utilization of stochastic operators for widespread, random search across the entire space. Conversely, the term “exploitation” signifies the local search proficiency focused on the previously explored regions, derived from the initial exploration phase. Because of the stochastically based nature of metaheuristics, it is thought to be difficult to achieve the proper balance between these stages [15]. The behavior of Quokka swarms served as inspiration for the novel SI method that is presented in this study. The suggested algorithm can strike a harmonious balance between the exploration and exploitation phases and has a small number of parameters, which makes it simple to build and suitable for addressing numerous optimization problems. The constraints posed by current algorithms, encompass concerns regarding convergence speed, scalability, and the ability to handle complex optimization tasks effectively. These challenges emphasize the need for fresh perspectives in swarm optimization. Our proposed algorithm endeavors to address these obstacles by integrating pioneering functionalities aimed at boosting performance and surmounting the constraints present in previous methodologies.

The existing algorithms include some limitation related with scalability, convergence speed, and the capacity to manage intricate optimization jobs efficiently. These difficulties highlight the need for new ideas in swarm optimization. The suggested algorithm attempts to overcome these challenges by including innovative features designed to increase efficiency and overcome the limitations of earlier approaches.

The main contributions of this work are:

1. The main aim of this research is to design and develop a new algorithm called Quokka Swarm Optimization (QSO) as a novel metaheuristic algorithm tailored for solving complex optimization problems.

2. The efficiency of the QSO algorithm in solving both single and multi-objective optimization problems, which means that it efficiently explores trade-off solutions, providing decision-makers with a diverse set of best solutions to choose from.

3. This algorithm has an adaptive parameter control, where it continuously adjusts parameters such as acceleration coefficients and swarm size. This adaptability improves convergence speed and solution quality, making it highly effective across a wide range of optimization problems.

The remainder of this work is structured as follows: related works are displayed in Section 2. The mathematical model and inspiration for the QSO algorithm are presented in Section 3. Results and a description of test functions are presented in Section 4. Section 5 concludes the work and makes a number of recommendations for additional research trajectories.

2 Related works

SI-inspired optimization techniques have gained popularity in recent years. Operating in a decentralized manner, these techniques mimic the behavior of swarms of social insects, bird flocks, or schooling fish. Such strategies offer advantages of robustness and adaptability compared to conventional methods. SI serves as a valuable project paradigm for algorithms addressing complex issues due to these inherent properties. This field of artificial intelligence (AI) is centered on developing intelligent multi-agent systems, drawing inspiration from the cooperative behaviors observed in social insects such as ants, termites, bees, and wasps, as well as other animal species like birds or fish [16].

In 2013, Sahil and Parikshit explored the effectiveness of ABC algorithms for tackling the Traveling Salesman Problem (TSP). The fundamental purpose of TSP is for a salesman to visit a number of destinations and return to the initial city using the shortest route possible. Because domain experts are difficult to find and knowledge extraction from experts is a challenging operation, data-driven modeling becomes important. To develop rule-based form data, one must use soft computing techniques. Among the methodologies are neural networks, GAs, and PSO. The fundamental ABC algorithm offers notable strengths including robustness, rapid convergence, high adaptability, and minimal parameter configuration requirements. However, it can suffer from premature convergence during the later stages of search and occasional deviations from the required optimal value accuracy [17].

In 2014, Seyedali et al. proposed the Gray wolf optimizer (GWO), a metaheuristic influenced by gray wolves (Canis lupus). The GWO algorithm mimics the gray wolves’ innate leadership structure and hunting techniques. Alpha, beta, delta, and omega are the four different species of gray wolves used to represent the leadership hierarchy. Additionally, the three fundamental hunting techniques – looking for prey, encircling prey, and attacking prey – are used. A comparison study using PSO, Gravitational search technique (GSA), Differential evolution (DE), EP, and Evolution strategy (ES) is used to validate the results after the technique is tested on 29 well-known test functions. The outcomes imply that the GWO algorithm can perform better than a number of well-known metaheuristics. The study also addresses how to solve three common engineering design issues (tension/compression spring, welded beam, and pressure vessel designs), and it illustrates how the suggested approach is really used in optical engineering. Results from conventional engineering design concerns and real-world applications show that the suggested approach is appropriate for challenging problems with unknown search spaces. Compared with traditional optimization algorithms such as PSO and GA, GWO has the advantages of fewer parameters, simple principles, and easy implementation. However, GWO has the disadvantages of slow convergence speed, low solution accuracy, and easy to fall into the local optimum [18].

In 2015 Seyedali proposed Ant lion optimizer (ALO) algorithm. This study introduces ALO, an innovative algorithm inspired by nature. The ALO algorithm mimics the natural hunting behavior of ant lions, encompassing five key processes: the random movement of ants, building traps, capturing ants, collecting prey, and reconstructing traps. The proposed algorithm is evaluated in three stages. The first step is to test the different ALO properties using a set of 19 mathematical functions. The construction of three-bar trusses, cantilever beams, and gear trains are three classic engineering challenges that ALO resolves. Finally, ALO makes minor real-world adjustments to the designs of two ship propellers. In the first and second test phases, a variety of techniques from the literature are compared to the ALO algorithm. Results from the test function demonstrate that the proposed approach can produce highly competitive results in terms of better exploration, local optima avoidance, exploitation, and convergence. The majority of classical engineering problems are solved using the ALO method, which shows that it is effective in solving constrained problems with various search horizons. The most effective designs for ship propellers discovered show how useful the suggested technique is for solving real-world issues with unknowable search areas. ALO is a recent metaheuristic algorithm that simulates antlion’s foraging performance in finding and attacking ants. However, the ALO algorithm suffers from local optima stagnation and slow convergence speed for some optimization problems [19].

In 2016, Mohammed and Ramzy introduced a Novel optimization algorithm inspired by the traveling behavior of camels. Camels, renowned for their remarkable adaptations enabling survival in harsh desert conditions, served as the inspiration for this innovative algorithm. The Camel algorithm was employed to uncover optimal solutions across a spectrum of benchmark test functions. Through experimental comparison with GA and PSO, it became evident that the Camel algorithm exhibits a robust and practical search capability, consistently yielding superior results for a diverse range of test functions [20].

In 2018, Belal and Amal presented an algorithm designed to tackle the practical challenge of university examination timetabling, known as Meerkat swarm optimization (MSO). The MSO approach represents an innovative and intelligent solution inspired by the cooperative behaviors observed in meerkat swarms, such as foraging and nurturing. Within the MSO algorithm, meerkats are divided into two sub-groups, one dedicated to foraging and the other to caring for pups in burrows, each led by their respective alpha leaders. This strategic partitioning promotes solution diversity, enhancing users’ ability to explore and exploit the search space effectively. The effectiveness of this approach is evaluated across seven benchmark examination scheduling scenarios. Comparative analysis with Bee colony optimization (BCO) and BCO with the tournament selection method reveals that the proposed MSO algorithm offers promising solutions for various examination timetabling instances [21].

In 2019, Maha and Belal proposed a Blue monkey swarm optimization (BMSO), this algorithm is inspired from the blue monkey swarms in nature. The BMSO method determines the number of males in a given group. Outside of the breeding season, blue monkey groups, like other forest guenons, usually have only one adult male. Erythrocebus patas is a species of monkey related to the Patas monkey. This algorithm is tested by using 43 well-known optimization test functions as benchmarks. A performance comparison of BM’s results with those of ABC, GSA, Biogeography-based optimizer (BBO), and PSO also serves to validate the method. The results obtained demonstrated the competitiveness of the BMSO algorithm when pitted against the selected metaheuristic algorithms; furthermore, the BMSO algorithm may converge to the global optimal via optimization problems. Additionally, this method handles real-world issues with limitations and ambiguous search spaces quite effectively. It should be highlighted that the BMSO technique incorporates a number of variables and can outperform other algorithms in a number of test functions [22].

In 2020, Belal Al-Khateeb et al. proposed a Rock hyraxes swarm optimization (RHSO) takes inspiration from the collective behavior of rock hyrax swarms in their natural habitat. RHSO algorithm replicates the foraging techniques and unique perspective employed by rock hyraxes in their search for food. Rock hyraxes are social animals that live in colonies or groups. To maintain the colony’s safety, a dominant male keeps a constant check on it. The RHSO method is evaluated using 48 test functions (22 unimodal and 26 multimodal), which are often utilized in the optimization field. A comparison efficiency analysis compares RHSO to PSO, ABC, GSA, and GWO. The collected results confirmed the RHSO algorithm’s superiority over the chosen algorithms, and the data showed the RHSO’s capacity to optimize in the direction of the global optimal as it does well in both exploitation and exploration tests. With constraints and a new search space, RHSO is also quite effective at solving problems in the real world. It is worth noting that the RHSO method has few variables and can outperform the selected algorithms in many test functions [23].

Finally, in 2022, Zuyan et al., proposed an Egret swarm optimization algorithm (ESOA). The Great Egret and the Snowy Egret, two species of egrets, were the inspiration for the ESOA. A sit-and-wait technique, an aggressive strategy, and discriminatory criteria make up the three primary parts of ESOA. The egret employs a trainable sit-and-wait strategy, guiding it to pinpoint the most probable solutions using a pseudo-gradient estimator. In contrast, an aggressive technique involves random wandering and encirclement processes to facilitate an optimal search for solutions. By leveraging the discriminant model, these two approaches are harmonized. This proposed method not only presents a parallel framework but also a dependable mechanism for learning parameters from historical data, applicable across a wide range of scenarios. The performance of the ESOA is compared against PSO, GA, and DE on 36 benchmark functions and three engineering challenges [24].

When studying the literature research, it was found that the QSO is similar to other algorithms in some respects, such as being often inspired by natural phenomena or observed collective behaviors. However, it differs in the design of the mathematical model and the implementation approach. Additionally, the QSO algorithm stands out from other algorithms by having fewer variables, which makes it easier to understand and use. Table 1 explains the summary of the related works.

3 Quokka animals

In this section, the motivation of the proposed algorithm and the mathematical model are discussed.

3.2 Mathematical model and algorithm

In this section, we introduce the QSO algorithm along with its corresponding mathematical model.

3.2.1 Position update

The update of each quokka’s position within the group is influenced by the best quokka position within that group, as described by the following equations:

(1) D new = ( T + H ) ( 0.8 × D old ) + ∆ w × rand × ∆ X ,

(2) X new = X old + D new × N ,

(Where D ^old represents the Drought and its value between [0,1]), T represents the ratio of the temperature, which is between 0.2 and 0.44, and H represents the ratio of the humidity and it is between 0.3 and 0.65. The reason for choosing these values is that quokka animals are able to tolerate temperatures and humidity between these ratios. ∆w represents the differences of weight between the leader and the quokka i, rand represents the random number and its value is between 0 and 1, ∆X represents the differences of position between the leader and quokka i, quokka’s new position is represented by X new , while the old position is represented by X old , and N represents the ratio of Nitrogen where its number is between 0 and 1, this ratio was chosen because quokkas need this amount of nitrogen. When the value is closer to 0, this will have a negative effect on the quokka because it will increase the dehydration rate, whereas when the value is closer to 1, it will be better for the quokka. The number 0.8 in the first equation indicates that the sum of temperature and humidity should not exceed this number because the quokka animal does not tolerate high temperature and humidity.

3.2.2 Quokka optimization algorithm

The behavior of quokka animals is simulated by the QSO algorithmic program. In this section, we will elucidate the pseudo-code of the proposed QSO algorithm. In the exploratory mode, the QSO initially generates random solutions and evaluates their fitness. It then initializes values for temperature, humidity, and nitrogen. When the global optimum may be in proximity, it transitions from exploration to local exploitation mode, concentrating on promising locations and designating the quokka with the highest fitness as the leader. The leader represents the best solution to optimization problems thereafter. The search agents recommence their exploratory movements, transitioning into a new phase of exploitation. Humidity and position are updated for each quokka according to equations (1) and (2). The fitness of the leader is evaluated, and subsequently, the fitness for every quokka is updated. Upon reaching the stopping condition, the procedure kept going and returned the leader as the closest approximation to the best solution to the optimization problem. Figure 1 shows the flowchart of the QSO algorithm.

1 – Initialize the quokka population b i (i = 1…n).

2 – Initialize the temperature T, humidity H, and nitrogen N, where (T ∊ [0.2, 0.44]), (H ∊ [0.3,0.65]), (N ∊ [0,1]).

3 – Compute the fitness for each quokka.

4 – Start loop.

5 – Select the best quokka to be the leader.

6 – Update drought (D) and position (X) for each Quokka by using equations (1) and (2).

7 – Find the fitness of the leader.

8 – Update the fitness for every quokka.

9 – If not stop condition return to step 5.

10 – End loop.

11 – Return the best solution.

Figure 1

Flowchart of QSO algorithm.

4 Results and discussion

In this section, we scrutinize and assess the performance of the QSO algorithm using a set of 43 benchmark functions, including some commonly used ones in the field. We selected these functions to contrast the outcomes of our algorithm with those of several well-known metaheuristic algorithms. Tables 2–4 illustrate the 43 test functions chosen, where D denotes the function’s dimension, Lb and Ub represent the function’s lower and upper bounds in the search space, respectively, and Opt denotes the optimal value. Typically, the test functions employed are optimization functions designed for minimization, and they can belong to either unimodal or multimodal benchmark categories. Unimodal test functions can be used to assess the convergence and application of an algorithm because they have a single ideal value. Despite having several optimum values, multimodal test functions are more challenging to execute than unimodal test functions. To reach and approximate the global optimum, an algorithm should steer clear of local optima; as a result, multimodal test functions can be used to benchmark algorithm exploration and local optima avoidance. We use these benchmark functions to validate the algorithm, which leads to identifying potential weaknesses or errors that need to be addressed. Furthermore, it helps us to compare the performance of the algorithm with other algorithms.

The QSO algorithm and the comparison algorithms are tested for each benchmark function under identical conditions (1,000 iterations), separate runs for 30 times, and a population size of 30. Tables 5–10 display the statistical findings (average and standard deviation). The QSO method is compared against BMSO [22], ABC [28], PSO [29], GSA [30], GWO [31], and BBO [32] for the purpose of confirming the results.

5 Exploitation analysis

The outcomes in Table 4 showed that for all unimodal test functions, QSO is superior to the chosen methods. Employing unimodal functions to evaluate the algorithm’s exploitation capabilities, the findings collected demonstrated QSO’s supremacy in utilizing the best value, demonstrating QSO’s superior exploitation capabilities. Figure 2 displays the average values of QSO, ABC, BMSO, GSA, BBO, GWO, and PSO for the unimodal benchmark functions. According to the findings of the unimodal test functions provided in Tables 5 and 6, QSO gets the optimal value in seven of them (F6, F7, F10, F12, F14, F15, and F16). On the other hand, QSO converges to optimal values in four test functions (F2, F3, F17, and F18). Furthermore, in the remaining seven test functions (F1, F4, F5, F8, F9, F11, and F13), QSO can come close to the best values of the specified algorithms. It should be noted that these values outperformed those achieved by other selected algorithms when tested with the same functions. Compared to the methods mentioned earlier, the findings of QSO demonstrate its effectiveness and competitiveness. Furthermore, it surpasses them in the majority of tests, with the results clearly demonstrating its superior performance. Specifically, when compared to unimodal performance, QSO consistently yields excellent results in identifying the best solution value for unimodal functions, highlighting its superior exploitation search capabilities.

Figure 2

Unimodal benchmark function for QSO and other algorithms.

6 Exploration analysis

The multimodal functions are used to evaluate an algorithm’s capacity for exploration. The multimodal results shown in Tables 7 and 8 for fixed and Tables 9 and 10 for not fixed illustrate how the suggested algorithm was explored. These outcomes demonstrate QSO’s effectiveness and strength in comparison to the other algorithms used for comparison. Using 25 multimodal test functions (17 fixed and 8 non-fixed), QSO is found to be superior and more effective than the other methods. Figures 3 and 4 display the average values of QSO, BMSO, GSA, ABC, BBO, GWO, and PSO for the multimodal benchmark functions. According to the findings of the multimodal (fixed) test functions provided in Tables 7 and 8, QSO gets the optimal value in seven of them (F19, F20, F21, F22, F23, F24, and F25). It, on the other hand, converges to optimal values in one test function (F26). Based on the results in Tables 9 and 10 for multimodal (not fixed) test functions, QSO gets the optimal value in eight of them (F30, F36, F37, F38, F39, F40, F41, and F43), it converges to optimal values in three functions (F29, F34, and F42). Additionally, within the remaining six test functions (F27, F28, F31, F32, F33, and F35), QSO closely approaches the best values attained by the specified algorithms. It is noteworthy that these values outperformed those achieved by other selected algorithms when subjected to the same test functions. In comparison to the methods stated above, QSO findings demonstrate that it is effective and competitive. Furthermore, it excels in the majority of tests, underscoring the accuracy, versatility, and efficiency of QSO’s exploration techniques. The results indicate that, when compared to multimodal performance, QSO consistently achieves strong results in identifying the best solution values for multimodal functions, reinforcing its superiority in exploration search.

Figure 3

Multimodal benchmark function (fixed) for QSO and other algorithms.

Figure 4

Multimodal benchmark function (not fixed) for QSO and other algorithms.

The scalability of the QSO algorithm may necessitate modifications to parameter settings or adaptation mechanisms to better handle large-scale optimization problems. One limitation of scalability is premature convergence, which may occur as the algorithm struggles to maintain diversity and exploration with increasing problem size. Furthermore, the performance of swarm optimization algorithms can be sensitive to parameter settings such as swarm size and termination criteria. Tuning these parameters becomes more challenging as problem size increases, and finding the best parameter values may require extensive experimentation

7 Conclusion and future work

This study suggests a novel optimization technique based on the quokka animal’s behavior, QSO, as an alternate strategy for tackling optimization problems. The suggested QSO algorithm updates location, causing solutions to move towards or away from the target, ensuring the exploration and utilization of search space. QSO’s performance in terms of exploitation and exploration is evaluated using 43 test functions. The acquired findings showed that QSO outperformed ABC, PSO, GSA, BMSO, BBO, and GWO. The results obtained from the unimodal test function demonstrate the QSO algorithm’s excellence in exploitation. The resulting multimodal test function result then demonstrates QSO’s exploring capacity. The QSO algorithm stands out from other algorithms by having fewer variables, which makes it easier to understand and use. The QSO algorithm can be applied to solve various optimization problems, including function optimization and parameter tuning, because it has the ability to explore large solution spaces and find the best solutions for these problems. Future research avenues could encompass diverse areas, including addressing various optimization challenges and evolving the QSO methodology into a multi-objective optimization algorithm, given its current single-objective nature. Additionally, hybridizing the QSO algorithm with different optimization algorithms could further improve its performance and versatility across a wide range of problem domains. The limitation of the algorithm lies in its single-objective nature; however, it can evolve into a multi-objective optimization algorithm, as discussed in this section.