Kookaburra Optimization Algorithm: A New Bio-Inspired Metaheuristic Algorithm for Solving Optimization Problems

In this paper, a new bio-inspired metaheuristic algorithm named the Kookaburra Optimization Algorithm (KOA) is introduced, which imitates the natural behavior of kookaburras in nature. The fundamental inspiration of KOA is the strategy of kookaburras when hunting and killing prey. The KOA theory is stated, and its mathematical modeling is presented in the following two phases: (i) exploration based on the simulation of prey hunting and (ii) exploitation based on the simulation of kookaburras’ behavior in ensuring that their prey is killed. The performance of KOA has been evaluated on 29 standard benchmark functions from the CEC 2017 test suite for the different problem dimensions of 10, 30, 50, and 100. The optimization results show that the proposed KOA approach, by establishing a balance between exploration and exploitation, has good efficiency in managing the effective search process and providing suitable solutions for optimization problems. The results obtained using KOA have been compared with the performance of 12 well-known metaheuristic algorithms. The analysis of the simulation results shows that KOA, by providing better results in most of the benchmark functions, has provided superior performance in competition with the compared algorithms. In addition, the implementation of KOA on 22 constrained optimization problems from the CEC 2011 test suite, as well as 4 engineering design problems, shows that the proposed approach has acceptable and superior performance compared to competitor algorithms in handling real-world applications.


Introduction
There are many problems in science, engineering, mathematics, and real-world applications that have more than one feasible solution. These multi-solution problems are known as optimization problems. According to this definition, the process of determining the best feasible solution among all the available solutions for this type of problem is called optimization [1]. Optimization problems are mathematically modeled using three main parts: decision variables, constraints, and objective functions. The goal of optimization is to determine the optimal values for the decision variables in such a way that, respecting metaheuristic algorithms active, the NFL theorem encourages and motivates researchers to be able to provide more effective solutions for optimization problems by introducing newer metaheuristic algorithms.
The novelty and innovation of this article is in the design of a new metaheuristic algorithm called the Kookaburra Optimization Algorithm (KOA), which is used in solving optimization problems. The scientific contributions of this study are as follows: • KOA is designed based on mimicking the natural behavior of kookaburras in the wild; • The fundamental inspiration of KOA is derived from (i) the kookaburras' strategy during hunting and (ii) the behavior of kookaburras when they slam their prey into a tree to ensure that the prey is killed; • The implementation steps of KOA are described and mathematically modeled in two phases of exploration and exploitation based on simulating the behavior of kookaburras in nature; • The effectiveness of KOA in solving optimization problems has been evaluated in the CEC 2017 test suite; • The performance of KOA in handling real-world applications has been tested on 22 constrained optimization problems from the CEC 2011 test suite as well as 4 engineering design problems; • The results of KOA have been compared with the performance of 12 well-known metaheuristic algorithms.
The structure of the paper is as follows: the literature review is presented in Section 2. Then the proposed Kookaburra Optimization Algorithm (KOA) is introduced and modeled in Section 3. The simulation studies and results are presented in Section 4. The effectiveness of KOA in solving real-world applications is investigated in Section 5. The conclusions and suggestions for future research are provided in Section 6.

Kookaburra Optimization Algorithm
In this section, the inspiration source and theory of the proposed Kookaburra Optimization Algorithm (KOA) approach are stated, then its implementation steps are mathematically modeled in order to be used in solving optimization problems.

Inspiration of KOA
The Kookaburra of the Dacelo genus is a bird from the group of terrestrial tree kingfishers that lives on land, is carnivorous, and belongs to the Coraciiformes and Alcedininae families. This bird lives in the native habitats of New Guinea and Australia. They are found in habitats ranging from arid savannah to humid forest, as well as near running water or in suburban areas with tall trees. The sound of this bird is similar to human laughter, and with this sound, the bird basically warns its enemies not to approach its territory [88].
Kookaburras can be found in different colors such as blue, brown, and white, and behind the eyes of this bird there is a dark brown spot, which gives the bird an angry awe along with the special shape of the feathers on its head. Kookaburra is between 28 and 47 cm long, and its weight is about 300 g [89]. A picture of a kookaburra is shown in Figure 1.
Kookaburras are carnivorous birds that feed on mice, insects, snakes, frogs, small reptiles, and small birds. The beak of the kookaburra is suitable for diving and hunting. The bird dives towards the prey with an open beak, and after hunting, it returns to the branch of the tree from which it flew from and knocks the prey against the tree several times to make sure it is dead. Then he holds the prey tightly between his claws, crushes it, and eats it [90].
Among the natural behaviors of the kookaburra in the wild, the strategy of this animal in hunting and knocking the prey against the tree in order to ensure that the prey is killed is much more significant. These natural kookaburra behaviors are the intelligent processes employed in the design of proposed KOA approach. Kookaburras are carnivorous birds that feed on mice, insects, snakes, frogs, small reptiles, and small birds. The beak of the kookaburra is suitable for diving and hunting. The bird dives towards the prey with an open beak, and after hunting, it returns to the branch of the tree from which it flew from and knocks the prey against the tree several times to make sure it is dead. Then he holds the prey tightly between his claws, crushes it, and eats it [90].
Among the natural behaviors of the kookaburra in the wild, the strategy of this animal in hunting and knocking the prey against the tree in order to ensure that the prey is killed is much more significant. These natural kookaburra behaviors are the intelligent processes employed in the design of proposed KOA approach.

Algorithm Initialization
The proposed KOA approach is a population-based optimizer that is able to provide suitable solutions for optimization problems in an iterative-based process based on a random search in the problem-solving space. The KOA population consists of kookaburras that are placed in the problem-solving space so that each kookaburra determines values for the decision variables based on its position in the problem-solving space; therefore, each kookaburra is a candidate solution to the problem that can be modeled using a vector. Kookaburras together form the KOA population matrix, which can be modeled using a matrix according to Equation (1). The position of the kookaburras at the beginning of KOA implementation is randomly initialized using Equation (2).

Algorithm Initialization
The proposed KOA approach is a population-based optimizer that is able to provide suitable solutions for optimization problems in an iterative-based process based on a random search in the problem-solving space. The KOA population consists of kookaburras that are placed in the problem-solving space so that each kookaburra determines values for the decision variables based on its position in the problem-solving space; therefore, each kookaburra is a candidate solution to the problem that can be modeled using a vector. Kookaburras together form the KOA population matrix, which can be modeled using a matrix according to Equation (1). The position of the kookaburras at the beginning of KOA implementation is randomly initialized using Equation (2).
x 1,1 · · · x 1,d · · · x 1,m . . . . . . . . . . . . . . . (1) Here, X is the KOA population matrix, X i is the ith kookaburra (candidate solution), x i,d is its dth dimension in search space (decision variable), N is the number of kookaburras, m is the number of decision variables, r is a random number in interval [0, 1], lb d and ub d are the lower bound and upper bound of the dth. decision variable, respectively.
Considering that the position of each kookaburra in the problem-solving space is a candidate solution for the problem corresponding to each kookaburra, the objective function of the problem can be evaluated. The set of evaluated values for the objective function of the problem can be represented using a vector according to Equation (3).
Here, F is the vector of evaluated objective function and F i is the evaluated objective function based on the ith kookaburra.
The evaluated values for the objective function are a suitable criterion for measuring the quality of candidate solutions and population members. The best evaluated value for the objective function corresponds to the best member and the worst evaluated value for the objective function corresponds to the worst member. Considering that in each iteration, the position of the kookaburras in the problem-solving space is updated, the objective function of the problem is reevaluated and based on the comparison of the new values, the best member of the population is also updated.

Mathematical Modelling of KOA
The proposed KOA approach updates the position of kookaburras in the following two phases: exploration and exploitation, in an iterative-based process in order to improve candidate solutions based on the simulation of natural kookaburra behaviors in the wild. Next, the process of updating the KOA population in the search space is presented.

Phase 1: Hunting Strategy (Exploration)
The kookaburra is a carnivorous bird that feeds on other small birds, reptiles, insects, mice, frogs, etc. Although this bird has weak legs, they have a very strong neck that helps them in hunting. The strategy of kookaburras in selecting prey and attacking it leads to large displacement in their position. This process represents the global search with the concept of exploration, which refers to the detailed scanning of the problem-solving space with the aim of avoiding getting stuck in the local optimal in order to discover the main optimal area.
In order to simulate the hunting strategy of kookaburras, the position of other kookaburras, which have a better objective function value, is considered as the prey location in KOA design for each kookaburra. Therefore, based on the comparison of the objective function values, the available prey set for each kookaburra is determined using Equation (4).
CP i = {X k : F k < F i and k = i}, where i = 1, 2, . . . , N and k ∈ {1, 2, . . . , N} Here, CP i is the set of candidate prey for ith kookaburra, X k is the kookaburra with a better objective function value than the ith kookaburra, and F k is the objective function value.
In the KOA design, it is assumed that each kookaburra randomly selects a prey and attacks it. Based on the simulation of the movement of the kookaburra towards the prey in the hunting strategy, a new position for the kookaburra is calculated using Equation (5). In this case, if the value of the objective function is improved in the new position, this new position will replace the previous position of the corresponding kookaburra according to Equation (6).
x P1 i,d = x i,d + r·(SCP i,d − I·x i,d ), i = 1, 2, . . . , N, and d = 1, 2, . . . , m Here, X P1 i is the new suggested position of the ith kookaburra based on first phase of KOA, x P1 i,d is its dth dimension, F P1 i is its objective function value, r is a random number with a normal distribution in the range of [0, 1], SCP i,d is the dth dimension of selected prey for ith kookaburra, I is a random number from set {1, 2}, N is the number of kookaburra, and m is the number of decision variables.

Phase 2: Ensuring That the Prey Is Killed (Exploitation)
The second characteristic behavior of kookaburras is that after attacking the prey, the kookaburra carries the prey with itself and makes sure that the prey is killed by repeatedly hitting it against the tree. The kookaburra then holds the prey tightly between its claws and crushes and eats it. This behavior of kookaburras, which happens near the hunting ground, leads to small changes in their position. This process, which represents the local search with the concept of exploitation, refers to the ability of the algorithm to achieve better solutions near the obtained solutions and promising areas.
In the KOA design, in order to simulate this behavior of kookaburras based on their movement near the hunting place, a random position is calculated using Equation (7). In fact, it is assumed that this displacement occurs randomly in a neighborhood to the center of each kookaburra with a radius equal to (ub . The radius of this neighborhood is first set to the maximum value; then, during successive iterations, this radius becomes smaller so that the local search with the aim of converging towards better solutions can be performed more accurately. The new position calculated for each kookaburra replaces its previous position if it improves the value of the objective function according to Equation (8).
. . , N, d = 1, 2, . . . , m, and t = 1, 2, . . . , T Here, X P2 i is the new suggested position of the ith kookaburra based on the second phase of KOA, x P2 i,d is its dth dimension, F P2 i is its objective function value, t is the iteration counter of the algorithm, and T is the maximum number of algorithm iterations.

Repetition Process, Pseudocode, and Flowchart of KOA
The first iteration of KOA is completed after updating the location of all kookaburras based on the first and second phases. At the end of each iteration, the best solution obtained until that iteration is updated and saved. Then, based on the updated positions and the new evaluated values for the objective function, the algorithm enters the next iteration. The process of updating the position of kookaburras continues until the last iteration of the algorithm based on Equations (4)- (8). In the end, the best candidate solution obtained during the iterations of the algorithm is presented as the proposed solution by KOA for the problem. The steps of KOA implementation are presented as a flowchart in Figure 2, and its pseudo code is presented in Algorithm 1.

Computational Complexity of KOA
In this subsection, the analysis of the computational complexity of KOA is discussed. The KOA initialization steps have a complexity equal to O(Nm), where N is the number of kookaburras and m is the number of decision variables of the problem. In each iteration of KOA, the position of each kookaburra in the problem-solving space is updated in the two phases of exploration and exploitation. Therefore, the process of updating kookaburras has a complexity equal to O(2NmT), where T is the maximum number of iterations of the algorithm. Therefore, the total computational complexity of the proposed KOA approach is equal to O(Nm(1 + 2T)).  Set KOA population size (N) and iterations (T).

3.
Generate the initial population matrix at random using Equation (2). , ← Evaluate the objective function.  Set KOA population size (N) and iterations (T).

3.
Generate the initial population matrix at random using Equation (2).
Evaluate the objective function. 5.
For t = 1 to T 6.
For i = 1 to N 7.
Determine the candidate preys set using Equation (4).
Choose the prey for the ith KOA member at random.

10.
Calculate new position of ith KOA member using Equation (5).
Phase 2: Ensuring that the prey is killed (exploitation) 13.
Calculate new position of ith KOA member using Equation (7).

Simulation Studies and Results
In this section, simulation studies are presented on the performance of KOA in dealing with optimization scenarios. The performance of KOA has been evaluated on 29 standard benchmark functions from the Competitions on Evolutionary Computation (CEC) 2017 test suite for problem dimensions equal to 10, 30, 50, and 100. In order to measure the performance quality of KOA, the obtained results have been compared with the performance of the following 12 well-known metaheuristic algorithms: GA [43], PSO [26], GSA [52], TLBO [63], MVO [55], GWO [34], WOA [38], MPA [36], TSA [40], RSA [42], AVOA [31], and WSO [32]. The control parameters of metaheuristic algorithms are specified in Table 1. The optimization results are reported using the following six statistical indicators: mean, best, worst, standard deviation (std), median, and rank. The ranking criterion of metaheuristic algorithms for each of the benchmark functions is the value of the mean index.

Statistical Analysis
In this subsection, using statistical analysis, it has been checked whether the superiority of KOA against competitor algorithms is significant from a statistical point of view or not. For this purpose, the Wilcoxon rank sum test [92] is used, which is a nonparametric test and is used to determine the significant difference between the average of two data samples. In this test, based on the values calculated for the p-value index, it is determined whether there is a statistically significant difference between the performance of the two algorithms or not.
The results of statistical analysis on the performance of KOA and each of the competitor algorithms in handling the CEC 2017 test suite in different dimensions of the The optimization results show that the proposed KOA approach, with a high ability in both exploration and exploitation and the ability to balance them during the search process, has been able to provide suitable results for the benchmark functions. The analysis of the simulation results indicates that KOA, by providing better results and obtaining the rank of the first best optimizer in most of the benchmark functions, has provided a superior performance compared to competitor algorithms in addressing the CEC 2017 test suite in different dimensions of the problem equal to 10, 30, 50, and 100.

Statistical Analysis
In this subsection, using statistical analysis, it has been checked whether the superiority of KOA against competitor algorithms is significant from a statistical point of view or not. For this purpose, the Wilcoxon rank sum test [92] is used, which is a non-parametric test and is used to determine the significant difference between the average of two data samples. In this test, based on the values calculated for the p-value index, it is determined whether there is a statistically significant difference between the performance of the two algorithms or not.
The results of statistical analysis on the performance of KOA and each of the competitor algorithms in handling the CEC 2017 test suite in different dimensions of the problem are reported in Table 6. Based on the results obtained from the Wilcoxon rank sum test, in cases where the p-value is less than 0.05, the proposed KOA approach has a statistically significant superiority in competition with the corresponding metaheuristic algorithm.

KOA for Real-World Applications
In this section, the efficiency of KOA in addressing real-world applications is challenged. For this purpose, 22 real-world constrained optimization problems from the CEC 2011 test suite as well as 4 classical engineering design problems are employed.

Evaluation CEC 2011 Test Suite
In this subsection, the ability of KOA and competitor algorithms in handling the CEC 2011 test suite is evaluated. This test suite consists of 22 constrained optimization problems from real-world applications. The full description and details of CEC 2011 test suite are provided in [93]. The optimization results of CEC 2011 test suite using KOA and competitor algorithms are reported in Table 7. Also, the boxplot diagrams obtained from the performance of metaheuristic algorithms in solving optimization problems C11-F1 to C11-F22 are drawn in Figure 7.

Pressure Vessel Design Problem
Pressure vessel design is a real-world optimization application aimed at minimizing construction cost. Pressure vessel design schematic is shown in Figure 8 and its The optimization results show that KOA, with its high ability to balance exploration and exploitation, has been able to provide suitable results for optimization problems in real-world applications. Based on the simulation results, KOA is the first best optimizer for C11-F1 to C11-F22. What is concluded from the analysis of the simulation results is that KOA has provided better results in most of the optimization problems compared to the competitor algorithms in handling the CEC 2011 test suite. Also, based on the statistical analysis and the results obtained from the Wilcoxon rank sum test, the superiority of KOA compared to competitor algorithms is significant from a statistical point of view.

Pressure Vessel Design Problem
Pressure vessel design is a real-world optimization application aimed at minimizing construction cost. Pressure vessel design schematic is shown in Figure 8 and its mathematical model is as follows [94]:

Speed Reducer Design Problem
Speed reducer design is an engineering challenge with the aim of minimizing the weight of the speed reducer. The schematic of speed reducer design is shown in Figure 10 and its mathematical model is as follows [95,96]: Subject to: Minimize: f (x) = 0.6224x 1 x 3 x 4 + 1.778x 2 x 2 3 + 3.1661x 2 1 x 4 + 19.84x 2 1 x 3 . Subject to: The pressure vessel design optimization results using KOA and competitor algorithms are reported in Tables 8 and 9. Based on the results, KOA has provided the optimal design with the values of design variables equal to (0.7780271, 0.3845792, 40.312284, 200) and the corresponding objective function value equal to (5882.8955). The convergence curve of KOA during the pressure vessel design optimization is drawn in Figure 9. Based on the comparison of optimization results, it is evident that KOA has provided superior performance in the pressure vessel design optimization compared to the competitor algorithms.

Speed Reducer Design Problem
Speed reducer design is an engineering challenge with the aim of minimizing the weight of the speed reducer. The schematic of speed reducer design is shown in Figure 10 and its mathematical model is as follows [ Subject to: Figure 9. KOA's performance convergence curve on pressure vessel design.

Speed Reducer Design Problem
Speed reducer design is an engineering challenge with the aim of minimizing the weight of the speed reducer. The schematic of speed reducer design is shown in Figure 10 and its mathematical model is as follows [95,96]:   Consider: 9 ≤ x 6 ≤ 3.9, and 5 ≤ x 7 ≤ 5.5.
The implementation results of KOA and competitor algorithms on the speed reducer design are presented in Tables 10 and 11. Based on the results, KOA has provided the optimal design with the values of design variables equal to (3.5, 0.7, 17, 7.3, 7.8, 3.3502147, 5.2866832) and the corresponding objective function value equal to (2996.3482). The convergence curve of KOA while achieving the optimal design for the speed reducer design problem is drawn in Figure 11. The analysis of the simulation results shows the superiority of KOA performance compared to the competitor algorithms in order to handle the speed reducer design.

Welded Beam Design
Welded beam design is a real-world application in engineering with the aim of minimizing the fabrication cost of the welded beam. The schematic of welded beam design is shown in Figure 12 and its mathematical model is as follows [38] 14 3.561 × 10 13 5.7774686 11 Figure 12. Schematic of welded beam design. Figure 12. Schematic of welded beam design. with 0.1 ≤ x 1 , x 4 ≤ 2 and 0.1 ≤ x 2 , x 3 ≤ 10.
The results of employing KOA and competitor algorithms in handling the welded beam design problem are reported in Tables 12 and 13. Based on the results, KOA has provided the optimal design with the values of design variables equal to (0.2057296, 3.4704887, 9.0366239, 0.2057296) and the corresponding objective function value equal to (1.7246798). The convergence process of KOA towards the optimal solution for the welded beam design problem is drawn in Figure 13. What is clear from the analysis of the optimization results is that KOA has provided a more effective performance compared to the competitor algorithms in the optimization of the welded beam design.

Tension/Compression Spring Design
Tension/compression spring design is an engineering subject of real-world applications with the aim of minimizing the weight of tension/compression spring. The schematic of welded beam design is shown in Figure 14 and its mathematical model is as follows [38]: Consider: = , , = , , .
Subject to: The optimization results of tension/compression spring design using KOA and competitor algorithms are reported in Tables 14 and 15. Based on the results, KOA has provided the optimal design with the values of design variables equal to (0.0516891, 0.3567177, 11.288966) and the corresponding objective function value equal to (0.0126019). The convergence curve of KOA to the optimal solution for the tension/compression spring design problem is plotted in Figure 15. The analysis of the simulation results shows that KOA has provided superior performance in dealing with tension/compression spring design by providing better results compared to the competitor algorithms.

Tension/Compression Spring Design
Tension/compression spring design is an engineering subject of real-world applications with the aim of minimizing the weight of tension/compression spring. The schematic of welded beam design is shown in Figure 14 and its mathematical model is as follows [   Minimize: f (x) = (x 3 + 2)x 2 x 2 1 . Subject to: with 0.05 ≤ x 1 ≤ 2, 0.25 ≤ x 2 ≤ 1.3 and 2 ≤ x 3 ≤ 15.
The optimization results of tension/compression spring design using KOA and competitor algorithms are reported in Tables 14 and 15. Based on the results, KOA has provided the optimal design with the values of design variables equal to (0.0516891, 0.3567177, 11.288966) and the corresponding objective function value equal to (0.0126019). The convergence curve of KOA to the optimal solution for the tension/compression spring design problem is plotted in Figure 15. The analysis of the simulation results shows that KOA has provided superior performance in dealing with tension/compression spring design by providing better results compared to the competitor algorithms.

Conclusions and Future Works
In this paper, a new metaheuristic algorithm named the Kookaburra Optimization Algorithm (KOA) was introduced, which has applications in dealing with optimization issues. The fundamental inspiration for KOA is derived from the strategy of kookaburras when hunting and their behavior to ensure that the prey is killed. The theory of KOA was stated and mathematically modeled in the two phases of exploration and exploitation, which are based on simulating the natural behaviors of kookaburras. The effectiveness of

Conclusions and Future Works
In this paper, a new metaheuristic algorithm named the Kookaburra Optimization Algorithm (KOA) was introduced, which has applications in dealing with optimization issues. The fundamental inspiration for KOA is derived from the strategy of kookaburras when hunting and their behavior to ensure that the prey is killed. The theory of KOA was stated and mathematically modeled in the two phases of exploration and exploitation, which are based on simulating the natural behaviors of kookaburras. The effectiveness of the proposed KOA approach in handling optimization tasks was evaluated on the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100. The optimization results showed that KOA, with its high ability in exploration, exploitation, and establishing a balance between them, has been able to provide suitable solutions for the benchmark functions. The quality of KOA in the optimization process was compared with the performance of the 12 well-known metaheuristic algorithms. Based on the simulation results, by achieving better results for most benchmark functions, KOA provided superior performance compared to competitor algorithms in the handling of the CEC 2017 test suite. Also, the implementation of KOA on 22 constrained optimization problems from the CEC 2011 test suite, as well as 4 engineering design problems, indicated the capability of the proposed approach in addressing real-world applications.
By introducing the proposed approach of KOA, several research tasks are proposed for further studies. Designing binary and multi-purpose versions of KOA is one of the most special research proposals of this study. Also, using KOA to solve optimization problems in different sciences and real-world applications are other research proposals for future studies.  Acknowledgments: The financial support of NSERC Canada through a research grant is acknowledged.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The information of CEC 2017 test suite is determined in Table A1.