Red Panda Optimization Algorithm: An Effective Bio-Inspired Metaheuristic Algorithm for Solving Engineering Optimization Problems

This paper presents a new bio-inspired metaheuristic algorithm called Red Panda Optimization (RPO) that imitates the natural behaviors of red pandas in nature. The main design idea of RPO is derived from two characteristic natural behaviors of red pandas: (i) foraging strategy, and (ii) climbing trees to rest. The proposed RPO approach is mathematically modeled in two phases of exploration based on the simulation of red pandas’ foraging strategy and exploitation based on the simulation of red pandas’ movement in climbing trees. The main advantage of the proposed approach is that there is no control parameter in its mathematical modeling, and for this reason, it does not need a parameter adjustment process. The performance of RPO is evaluated on fifty-two standard benchmark functions including unimodal, high-dimensional multimodal, and fixed-dimensional multimodal types as well as CEC 2017 test suite. The optimization results obtained by the proposed RPO approach are compared with the performance of twelve well-known metaheuristic algorithms. The simulation results show that RPO, by maintaining the balance between exploration and exploitation, is effective in solving optimization problems and its performance is superior over competitor algorithms. Based on the analysis of the optimization results, RPO has provided more successful performance compared to the competitor algorithms in 100% of unimodal functions, 100% of high-dimensional multimodal functions, 100% of fixed-dimensional multimodal functions, and 86.2% of CEC 2017 test suite benchmark functions. Also, the statistical analysis of the Wilcoxon rank sum test shows that the superiority of RPO in the competition with the compared algorithms is significant from a statistical point of view. In addition, the results of implementing RPO on four engineering design problems confirms the ability of the proposed approach to handle real-world optimization applications.


I. INTRODUCTION
Optimization problems are a type of problems that have more than one feasible solution. According to this definition, optimization is the process of finding the best feasible solution among the available solutions for a problem [1]. From a mathematical point of view, an optimization problem can be modeled considering three main parts: decision variables, The associate editor coordinating the review of this manuscript and approving it for publication was Xiong Luo . constraints, and objective function. The main goal in optimization is to set values for decision variables such that the objective function is optimized according to the constraints of the problem [2]. There are numerous optimization problems in science that have become more complex with the advancement of technology, and this is the reason for the need to powerful tools for solving optimization problems [3].
Problem solving methods in optimization studies are classified into two groups: deterministic and stochastic approaches [4]. Deterministic approaches are effective tools for solving linear, convex, continuous, differentiable, and low-dimensional problems [5]. However, in case of more complex optimization problems, deterministic approaches lose their efficiency due to getting stuck in local optima. This is while many of today's optimization problems and real-world applications are non-linear, non-convex, discontinuous, non-differentiable, and high-dimensional [6]. These disadvantages and the inability of deterministic approaches to solve complex optimization problems have prompted researchers to develop stochastic approaches. Stochastic approaches, without the need for derivative information from the objective function and problem constraints, are able to provide suitable solutions for optimization problems based on the random search process in the problem-solving space [7]. Metaheuristic algorithms are one of the most effective stochastic approaches in solving optimization problems. Advantages such as simplicity of concepts, convenient implementation, no dependence on the type of problem, no need for derivative information, efficient performance in solving nonlinear, non-convex, high-dimensional, and NPhard problems, as well as desirable efficiency in nonlinear and unknown search spaces are the main reasons for the popularity of metaheuristic algorithms [8].
The nature of random search in metaheuristic algorithms means that there is no guarantee of achieving the global optimal solution with these approaches. However, since the solutions obtained by metaheuristic algorithms are close to the global optima, they are acceptable and known as quasi-optimal solutions [9]. In order to organize an effective search process, metaheuristic algorithms must be able to scan the problem-solving space appropriately at both global and local levels. Global search with the concept of exploration leads to the ability of the algorithm to comprehensively search the problem-solving space with the aim of discovering the main optimal area and preventing the algorithm from getting stuck in local optima. Local search with the concept of exploitation leads to the ability of the algorithm to achieve possible better solutions near the discovered solutions [10]. In addition to exploration and exploitation abilities, what leads to the success of metaheuristic algorithms in the optimization process is the balancing of exploration and exploitation during the search process [11]. The efforts of researchers to achieve more effective solutions for optimization problems have led to the design of numerous metaheuristic algorithms [12]. These algorithms are employed in various optimization applications in science, such as energy [13], [14], [15], [16], protection [17], energy carriers [18], [19], and electrical engineering [20], [21], [22], [23], [24], [25].
The main research question in the study of metaheuristic algorithms is that considering various algorithms presented so far, is there still a need to design newer metaheuristic algorithms? In response to this question, No Free Lunch (NFL) theorem [26] explains that there is no specific metaheuristic algorithm to be considered as the best optimizer for all optimization problems. In fact, the optimal performance of an algorithm in solving a set of optimization problems is not a guarantee for the similar performance of that algorithm in solving other optimization problems. According to NFL theorem, a successful algorithm in solving an optimization problem may even fail in solving another problem. Therefore, there is no guarantee for the success or failure of implementing an algorithm on a problem. By keeping open the field of metaheuristic algorithms study, NFL theorem encourages researchers to provide more effective solutions to optimization problems by designing newer metaheuristic algorithms.
The innovation and novelty of this paper is the introduction and design of a new metaheuristic algorithm called Red Panda Optimization (RPO) to solve optimization problems. The main contributions of this paper are as follows: • The proposed RPO approach is based on the simulation of red panda behaviors in nature.
• The fundamental inspiration for RPO design is the foraging strategy and tree climbing ability of red pandas.
• The mathematical model of RPO is presented in two phases of exploration and exploitation.
• The efficiency of RPO in optimization has been evaluated on fifty-two benchmark functions consisting of unimodal, high-dimensional multimodal, and fixed-dimensional multimodal types, as well as CEC 2017 test suite.
• The performance of RPO is compared with twelve well-known metaheuristic algorithms.
• The effectiveness of RPO in handling real-world applications is examined on four engineering design problems.
The rest of this article is organized in this way, first the literature review is presented in section II. Then, the proposed Red Panda Optimization (RPO) algorithm is introduced and mathematically modeled in section III. Simulation studies and results are presented in section IV. The performance of the proposed RPO in solving real-world applications is evaluated in section V. Finally, conclusions and several proposals for future studies are provided in section VI.

II. LECTURE REVIEW
Metaheuristic algorithms have been developed with inspiration from various natural phenomena, animal life in nature, biological sciences, physical laws and phenomena, rules of games, human interactions, and other evolutionary processes. Based on the idea used in the design, metaheuristic algorithms can be broadly classified into five groups: swarmbased, evolutionary-based, physics-based, human-based, and game-based approaches [27].
Swarm-based metaheuristic algorithms are developed based on simulating the natural swarm behavior of animals, birds, aquatic animals, insects, plants, and other living organisms in nature. Among the well-known algorithms of this group, one can mention Particle Swarm Optimization (PSO) [28], Ant Colony Optimization (ACO) [29], Artificial Bee Colony (ABC) [30], and Firefly Algorithm (FA) [31]. ACO was proposed based on modeling the 57204 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. ability of ant swarm to identify the shortest communication path between nests and food sources. ABC was designed based on simulating interactions and natural behaviors of colony bees in obtaining food resources. FA was inspired by the communication feature of flashing light in the firefly's swarm. Finding food resources, migration, and chasing are common natural behaviors among animals, whose simulation has inspired researchers to design several swarm-based algorithms such as: Coati Optimization Algorithm (COA) [32], Golden Jackal Optimization (GJO) [33], White Shark Optimizer (WSO) [34], Marine Predator Algorithm (MPA) [35], African Vultures Optimization Algorithm (AVOA) [36], Pelican Optimization Algorithm (POA) [37], Tunicate Swarm Algorithm (TSA) [38], Honey Badger Algorithm (HBA) [39], Whale Optimization Algorithm (WOA) [40], Reptile Search Algorithm (RSA) [41], Green Anaconda Optimization (GAO) [42], Cuckoo Search Algorithm (CSA) [43], and Grey Wolf Optimizer (GWO) [44].
Evolutionary-based metaheuristic algorithms are introduced based on the concepts of genetics, biology, natural selection, and survival of the fittest. Genetic Algorithm (GA) [45] and Differential Evolution (DE) [46] are widely used approaches in this group. GA and DE were developed based on reproductive process modeling, biology concepts and stochastic operators such as selection, crossover, and mutation. Artificial Immune Systems (AISs) is another evolutionary approach that has been introduced based on the ability of the human body's defense system against diseases and microbes [47]. Some other evolutionary-based metaheuristic algorithms are: Evolution Strategy (ES) [48], Genetic programming (GP) [49], and Cultural Algorithm (CA) [50].
Physics-based metaheuristic algorithms are designed based on simulating concepts, phenomena, and laws in physics. Simulated Annealing (SA) [51] is one of the most widely used physics-based approaches. SA was developed based on the modeling of metal annealing phenomenon in physics, where metals are melted under heat and then cooled in order to achieve ideal crystal. The modeling of physical forces has been the starting point for the introduction of several physics-based algorithms, such as: Spring Search Algorithm (SSA) [52], Momentum Search Algorithm (MSA) [53], and Gravitational Search Algorithm (GSA) [54]. SSA was introduced based on the simulation of Hooke's law, spring elastic force, and Newton's laws of motion in a system consisting of weights connected by springs. MSA was proposed based on the modeling of the force resulting from the momentum between the bullets. GSA was designed based on simulating the gravitational force that masses exert on each other at different distances. The physical phenomenon of matter state transitions for water was employed in design of the Water Cycle Algorithm (WCA) [55]. Some other physics-based metaheuristic algorithms are: Nuclear Reaction Optimization (NRO) [56], Lichtenberg Algorithm (LA) [57], Archimedes Optimization Algorithm (AOA) [58], Equilibrium Optimizer (EO) [59], Multi-Verse Optimizer (MVO) [60], and Electro-Magnetism Optimization (EMO) [61].
Human-based metaheuristic algorithms have been developed based on the simulation of human interactions, communication, thinking, and decision-making in social and individual lives. Teaching-Learning Based Optimization (TLBO) [62] is one of the most widely used human-based algorithms. The basic inspiration in its design was modelling the educational interactions of students and teachers in the classroom. The economic activities of both the poor and the rich sections of the society, who are trying to improve their economic conditions, have been a source of inspiration in the design of Poor and Rich Optimization (PRO) [63]. Therapeutic interactions between patients and physicians were employed in the design of Doctor and Patients Optimization (DPO) [64]. The cooperation between the members of a team who are trying to achieve the team goal was the basic idea in the design of Teamwork Optimization Algorithm (TOA) [65]. Some other human-based metaheuristic algorithms are: Ali Baba and the Forty Thieves (AFT) [66], Skill Optimization Algorithm (SOA) [67], Language Education Optimization (LEO) [68], Coronavirus Herd Immunity Optimizer (CHIO) [69], War Strategy Optimization (WSO) [70], and Driving Training-Based Optimization (DTBO) [71].
Game-based metaheuristic algorithms have been introduced based on modeling the rules of various individual and group games, the strategy of players, coaches, referees, and other influential persons of the games. Football Game Based Optimization (FGBO) [72] and Volleyball Premier League (VPL) [73] are two game-based approaches, which were designed based on the simulation of competitions between clubs in soccer and volleyball leagues. The skill of the players in putting together the pieces of the puzzle has been a source of inspiration in the design of Puzzle Optimization Algorithm (POA) [74]. The strategy of players in throwing darts and collecting points in the darts game was employed in the design of Darts Game Optimizer (DGO) [75]. Some other game-based metaheuristic algorithms are: Tug of War Optimization (TWO) [76], Billiards Optimization Algorithm (BOA) [77], Dice Game Optimization (DGO) [78], Ring Toss Game-Based Optimization (RTGBO) [79], Orientation Search Algorithm (OSA) [80], and Archery Algorithm (AA) [81].
Based on the best knowledge obtained from the literature review, no metaheuristic algorithm has been designed based on simulating the natural behavior of the red panda. Meanwhile, the behavior of foraging and resting on trees among red pandas are intelligent activities that has the potential to design a metaheuristic algorithm. In order to address this research gap in the studies of metaheuristic algorithms, in this paper, a new metaheuristic algorithm is introduced based on the mathematical modeling of the natural behavior of the red panda, which is discussed in the next section. VOLUME 11, 2023 57205 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

Start RPO.
Input: The problem information (variables, objective function, and constraints). Set RPO population size (N ) and the total number of iterations (T ). Generate the initial population matrix at random using (1) and (2). Evaluate the objective function by (3). For t = 1 to T For i = 1 to N Phase 1: : The strategy of red pandas in foraging Update food positions set for the ith RPO member using (4).
Determine the selected food by the ith red panda at random. Calculate new position of the ith RPO member based on the 1 st phase of RPO using (5).
Update the ith RPO member using (6).

Phase 2: Skill in climbing and resting on the tree
Calculate new position of the ith RPO member based on the 2 nd phase of RPO using (7).
Update the ith RPO member using (8).
Save the best candidate solution found so far. end Output: The best solution obtained by RPO. End RPO.

III. RED PANDA OPTIMIZATION ALGORITHM
In this section, the proposed Red Panda Optimization (RPO) algorithm is introduced, then its mathematical modeling is presented.

A. INSPIRATION OF RPO
The red panda is a small endemic animal of the southern China and eastern Himalayas. It has dense reddish-brown hair on its body and legs, a black belly and legs, white-lined ears, a mainly white muzzle, and a ringed tail. It has a headto-body length of 51-63.5 cm, a tail length of 28-48.5 cm, and weighs between 3.2 and 15 kg. Because of its flexible joints and curved semi-retractile claws, it is well adapted to climbing [82]. The red panda inhabits temperate broadleaf and mixed forests as well as coniferous forests, favoring steep slopes with dense bamboo cover close to water sources. It is largely arboreal and solitary [83]. A picture of the red panda is shown in Figure 1.
Red panda is largely herbivorous and eats mainly bamboo leaves and shoots, as well as blooms and fruits. The red panda has a good sense of sight, smell, and hearing and uses its long white whiskers to search for food at night [84]. According to observations, the red panda is a nocturnal animal. Due to its high ability to climb, it sleeps and rests in high places, especially trees during the day [85].
Among the natural behaviors of the red panda, its foraging strategy based on its high ability of hearing, sight, and smell, as well as the high skill of this animal in climbing trees, is much more impressive. Mathematical modeling of these natural behaviors of the red panda is the basis for the design of the proposed RPO approach, which is explained below.

B. MATHEMATICAL MODELLING
In this subsection, first the initialization of the proposed RPO approach is described, then based on the simulation of the natural behaviors of the red panda, the mathematical model of updating the candidate solutions in two phases of exploration and exploitation is presented.

1) INITIALIZATION
The proposed RPO approach is a population-based metaheuristic algorithm, whose members consist of red pandas. In RPO design, each red panda is a candidate solution to the problem, which suggests certain values for the problem variables based on its position in the search space. Therefore, from a mathematical point of view, each red panda (i.e., candidate solution) is modeled using a vector. Together, the red pandas of the algorithm population can be mathematically modeled using a matrix according to (1). Each row of this matrix represents a red panda (i.e., candidate 57206 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   using (2).
where, X is the population matrix of red pandas' locations, X i is the ith red panda (i.e., candidate solution), x i,j is its jth dimension (problem variable), N is the number of red pandas, m indicates the number of problem variables, r i,j are random numbers in the interval [0, 1], lb j , and ub j are the lower bound and upper bound of the jth problem variable, respectively. Considering that the position of each red panda is a candidate solution for the problem, the objective function of the problem corresponding to each of these candidate solutions can be evaluated. The set of evaluated values for the objective function can be represented using a matrix according to (3).
where F is the objective function values vector and F i denotes the value of the objective function obtained by the ith red panda.
The evaluated values for the objective function of the problem are the main criterion in determining the quality of the candidate solutions. In other words, the best obtained value for the objective function corresponds to the best candidate solution and similarly the worst value obtained for the objective function corresponds to the worst candidate solution.
Since the candidate solutions are updated in each iteration, the best and worst candidate solutions must also be updated in each iteration. After the implementation of the algorithm, the best candidate solution obtained during the iterations of the algorithm is presented as a solution to the problem. The process of updating candidate solutions in the proposed RPO consists of two phases of exploration and exploitation, which are described as follows.

2) PHASE 1: THE STRATEGY OF RED PANDAS IN FORAGING (EXPLORATION)
The position of red pandas in the first phase of RPO is modeled based on their movement in order to forage in the wild. Red pandas are highly skilled in identifying and moving towards the location of food sources using their high abilities VOLUME 11, 2023 57209 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. in smell, hearing, and vision. In RPO design, for each red panda, the location of other red pandas that lead to better objective function values is considered as the location of food resources. The set of proposed food resource positions for each red panda based on the comparison of the objective function values is modeled using (4). Among these proposed positions, one position is randomly determined as the food position selected by the corresponding red panda.
where PFS i is the set of proposed food sources for ith red panda and X best is the location of the red panda with best value for the objective function (best candidate solution).
Moving towards the food source leads to big changes in the position of red pandas, which improves the capability of the proposed algorithm in exploration and global search in the problem-solving space. In order to model the behavior of red pandas during foraging, first a new position is calculated for each red panda based on movement towards the location of food source (the best candidate solution) using (5). Then, if the value of the objective function is improved in the new position, the position of the red panda is updated to the position calculated in the exploration phase using (6).
57210 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where, X P1 i is the new position of the ith red panda based on the first phase of RPO, x P1 i,j is its jth dimension, F P1 i represents its objective function value, SFS i is the selected food source for ith red panda, SFS i,j denotes its jth dimension, r is a random number in the interval [0, 1], and I is a random number selected from the set {1, 2} randomly.

3) PHASE 2: SKILL IN CLIMBING AND RESTING ON THE TREE (EXPLOITATION)
The position of red pandas in the second phase of the RPO is modeled based on the skill of this animal in climbing trees and resting on them. Red pandas spend most of their time resting on trees. After foraging on the ground, this animal climbs the nearby trees. Moving towards the tree and climbing it leads to small changes in the position of red pandas, which increases the capability of the proposed RPO algorithm in exploitation and local search in promising areas. In order to mathematically model the natural behavior of red pandas in climbing trees, first a new position is calculated for each red panda using (7). Then, if the value of the objective function is improved, this new position replaces the previous position of the corresponding red panda using (8).
x P2 i,j = x i,j + lb j + r · ub j − lb j t , i = 1, 2, . . . , N , j = 1, 2, . . . , m, and t = 1, 2, . . . , T , where X P2 i is the new position of the ith red panda based on the second phase of RPO, x P2 i,j is its jth dimension, F P2 i indicates its objective function value, r is a random number in the interval [0, 1], t represents the iteration counter of the algorithm, and T is the maximum number of iterations.

C. REPETITIONS PROCESS, FLOWCHART, AND PSEUDO-CODE OF RPO
The proposed RPO approach is an iteration-based metaheuristic algorithm. After updating the position of all red pandas based on the exploration and exploitation phases, the first iteration of the RPO is completed. Then, based on the new values, the algorithm enters the next iteration and the process of updating the position of the red pandas is repeated using (4) to (8) until the last iteration of the algorithm. After completion of RPO implementation, the position of the best red panda, which results in the best value for the objective function, is presented as the solution of the problem. The implementation steps of RPO are presented in the form of a flowchart in Figure 2 and its pseudocode is given in Algorithm 1.

D. COMPUTATIONAL COMPLEXITY
In this subsection, the computational complexity analysis of the proposed RPO approach is discussed. RPO initialization has a computational complexity equal to O(Nm), where N is the number of red pandas and m denotes the number of problem variables. In each iteration of the algorithm, the position of red pandas is updated in the two phases of exploration and exploitation. Therefore, the red pandas update process has a computational complexity equal to O(2NmT), where T is the maximum number of the algorithm iterations. Therefore, the total computational complexity of RPO is equal to O (Nm(1 + 2T).

IV. SIMULATION STUDIES AND DISCUSSION
In this section, simulation studies on the performance of the proposed RPO in solving optimization problems are presented. For this purpose, fifty-two standard benchmark functions consisting of unimodal, high-dimensional multimodal, and fixed-dimensional multimodal types as well as CEC 2017 test suite [86] are employed. Also, in order to analyze the quality of RPO in providing appropriate solutions, the results obtained from the proposed approach are compared with the performance of twelve well-known metaheuristic algorithms including: GA, PSO, GSA, TLBO, GWO, MVO,  VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. WOA, TSA, MPA, RSA, WSO, and AVOA. The values of the control parameters for these competitor algorithms are specified in Table 1. Simulation results are reported using six statistical indicators: mean, best, worst, standard deviation (std), median, and rank. The ranking criterion for metaheuristic algorithms in solving each benchmark function is to provide a better value for the mean index.

A. EVALUATION OF UNIMODAL TEST FUNCTIONS
Seven benchmark functions F1 to F7 are selected from the unimodal type. Because these functions have no local optima, they are suitable options for evaluating the exploitation power of metaheuristic algorithms. The optimization results for the functions F1 to F7 using RPO and the competitor algorithms are presented in Table 2. Based on the optimization results, RPO with high exploitation ability has converged to the global optima in solving functions F1, F2, F3, F4, F5, and F6. In solving the function F7, RPO is the first-best optimizer. The analysis of the simulation results shows that the proposed RPO approach has provided better results and in total by winning the first rank, it has achieved a superior performance in the optimization of unimodal benchmark functions compared to the competitor algorithms.

B. EVALUATION OF HIGH-DIMENSIONAL MULTIMODAL TEST FUNCTIONS
Six benchmark functions F8 to F13 are selected from high-dimensional multimodal type. In addition to the global optima, these functions have a large number of local optima, and for this reason, they are suitable options for evaluating the exploration power of metaheuristic algorithms. The implementation results of RPO and the competitor algorithms for the functions F8 to F13 are reported in Table 3. The optimization results show that RPO with high exploration ability has converged to the global optima in the optimization of F9 and F11 functions in addition to identifying the main optimal area in the search space. In solving the functions F8, F10, F12, and F13, RPO has provided suitable solutions with high exploration ability and is the first-best optimizer for these functions. The comparison of the simulation results indicates that the proposed RPO approach, with a high exploration ability in the case of high-dimensional multimodal functions, has obtained superior performance over the competitor algorithms.

C. EVALUATION OF FIXED-DIMENSIONAL MULTIMODAL TEST FUNCTIONS
Ten benchmark functions F14 to F23 have been selected from the fixed-dimensional multimodal type. These functions, compared to functions F8 to F13, have a lower number of local optima. Functions F14 to F23 are suitable options for evaluating the ability of metaheuristic algorithms in balancing exploration and exploitation features during the search process. The results of using RPO and the competitor algorithms for optimizing the functions F14 to F23 are presented in Table 4. Based on the optimization results, RPO is the first-best optimizer for the functions F14, F15, F21, F22, and F23. In solving the functions F16, F17, F18, F19, and F20, RPO has the same conditions as some of the competing algorithms considering the mean index criterion. However, RPO has provided a more effective performance in handling these functions by providing better results from the std index viewpoint. What is evident from the analysis of simulation results, RPO has achieved better results in solving fixed-dimensional multimodal functions with an appropriate ability to balance exploration and exploitation, and compared to the competitor algorithms, it has provided superior performance in optimizing these functions.
The performance of RPO and the competitor algorithms in solving benchmark functions F1 to F23 is illustrated in the form of convergence curves in Figure 3.

D. EVALUATION OF CEC 2017 TEST SUITE
RPO's performance in solving optimization problems is evaluated on CEC 2017 test suite. This test suite has thirty benchmark functions consisting of three unimodal functions of C17-F1 to C17-F3, seven multimodal functions of C17-F4 to C17-F10, ten hybrid functions of C17-F11 to C17-F20, and ten composition functions of C17-F21 to C17-F30. The C17-F2 function has been excluded from the simulation studies due to its unstable behavior. The full description of CEC 2017 test suite is provided in [86]. The implementation results of RPO and the competitor algorithms on CEC 2017 test suite are reported in Table 5. Based on the optimization result, RPO is the first best optimizer for functions C17-F1, C17-F4 to C17-F8, C17-F10 to C17-F21, C17-F23, C17-F24, and C17-F26 to C17-F30. The performance of RPO and the competitor algorithms in solving the CEC 2017 test suite is drawn as boxplot diagrams in Figure 4. Analysis of the simulation results shows that RPO has provided better results for most of the benchmark functions of CEC 2017 test suite and overall, by winning the first rank, it has provided superior performance over the competitor algorithms in solving CEC 2017 test suite.

E. STATISTICAL ANALYSIS
In this subsection, statistical analysis is presented on the performance of RPO and the competitor algorithms to determine VOLUME 11, 2023  whether RPO has a significant statistical superiority or not. For this purpose, Wilcoxon rank sum test [87] is employed, which is a non-parametric statistical test to determine the significant difference between the average of two data samples. In Wilcoxon rank sum test, an index called p-value is utilized to evaluate whether RPO is significantly superior over any of the competitor algorithms from a statistical point of view.
The results of the statistical analysis on the performance of RPO and the competitor algorithms are reported in Table 6. Based on the results, in cases where the p-value is less than 0.05, RPO has a significant statistical superiority over the corresponding competitor algorithm.

V. RPO FOR REAL-WORLD APPLICATION
In this section, the effectiveness of the proposed RPO approach for solving optimization problems in real-world applications is tested. In this regard, RPO is implemented on four engineering design problems.

A. PRESSURE VESSEL DESIGN PROBLEM
Pressure vessel design is an engineering minimization problem with the aim of reducing design cost. The schematic of this design is presented in Figure 5. Pressure vessel design mathematical model is as follows [88]: Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  The results of implementing RPO and the competitor algorithms for solving pressure vessel design problem are reported in Tables 7 and 8. Based on the obtained results, RPO has provided the optimal solution of this design with the design values equal to (0.778027, 0.384579, 40.31228, 200) and the corresponding objective function value is equal to (5882.895). The convergence curve of RPO while achieving the solution for pressure vessel design is drawn in Figure 6. Based on the simulation results, RPO has provided superior performance in pressure vessel design optimization compared to the competitor algorithms.

B. SPEED REDUCER DESIGN PROBLEM
Speed reducer design is an engineering minimization problem with the aim of reducing the weight of the speed reducer. The schematic of this design is illustrated in Figure 7. The mathematical model of speed reducer design is as follows [89], [90]: Schematic view of speed reducer design problem.
Speed reducer design optimization results using RPO and the competitor algorithms are reported in Tables 9 and 10. Based on the obtained results, RPO has provided the optimal solution of this design with the design values equal to (3.5, 0.7, 17, 7.3, 7.8, 3.350215, 5.286683) and the corresponding   objective function value is equal to (2996.348). The RPO convergence curve during solving the speed reducer design is drawn in Figure 8. Analysis of the simulation results shows that RPO is superior over the competitor algorithms by providing better results in the optimization of speed reducer design.

C. WELDED BEAM DESIGN PROBLEM
Welded beam design is a real-world application with the aim of minimizing the fabrication cost of the welded beam. The schematic of this design is shown in Figure 9. The mathematical model of the welded beam design is as follows [40]:    The results of employing RPO and the competitor algorithms in handling the welded beam design problem are presented in Tables 11 and 12. Based on the obtained results, RPO has provided the optimal solution of this design with the design values equal to (0.20573, 3.470489, 9.036624, 0.20573) and the corresponding objective function value is equal to (1.72468). The convergence curve of RPO while reaching the solution for welded beam design is drawn in Figure 10. What is evident from the simulation results, RPO has a higher ability compared to the competitor algorithms in dealing with the welded beam design problem.

D. TENSION/COMPRESSION SPRING DESIGN PROBLEM
Tension/compression spring design is a real-world application aimed at minimizing the weight of tension/compression spring. The schematic of this design is presented in Figure 11. The mathematical model of tension/compression spring design is as follows [40]: VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
The implementation results of RPO and the competitor algorithms for the tension/compression spring design problem are reported in Tables 13 and 14. Based on the obtained VOLUME 11, 2023 57223 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.    results, RPO has provided the optimal solution of this design with the design values equal to (0.051689, 0.356718, 11.28897) and the corresponding objective function value is equal to (0.012602). The convergence curve of RPO while achieving the optimal solution for tension/compression spring design is drawn in Figure 12. What can be concluded from the simulation results, RPO has provided a more effective performance compared to the competitor algorithms in solving the tension/compression spring design problem.

VI. CONCLUSION AND FUTURE WORKS
In this paper, a new bio-inspired metaheuristic algorithm called Red Panda Optimization (RPO) was introduced, which can be applied to solve optimization problems. The fundamental inspiration of RPO is simulation of the behavior of red pandas when foraging and their ability to climb trees to rest. The implementation steps of RPO were described and mathematically modeled in two phases of exploration and exploitation. The effectiveness of RPO in solving optimization problems was evaluated considering fifty-two benchmark functions consisting of unimodal, highdimensional multimodal, fixed-dimensional multimodal, and CEC 2017 test suite. The optimization results of unimodal functions indicated the high ability of RPO in local search and exploitation. The optimization results of multimodal functions showed that RPO has a high ability in global search and exploration. Also, the optimization results of CEC 2017 test suite showed the high capability of the proposed RPO approach in providing simultaneous exploration and exploitation during the search process. The results obtained from the implementation of RPO were compared with the performance of twelve well-known metaheuristic algorithms. The simulation results showed that the proposed RPO approach by balancing exploration and exploitation features during the search process, has provided superior performance over the competitor algorithms. Based on the simulation results, the proposed RPO approach provided better results compared to the competitor algorithms in 100% of unimodal functions, 100% of high-dimensional multimodal functions, 100% of fixed-dimensional multimodal functions, and 86.2% of CEC 2017 test suite benchmark functions. In addition, the implementation of RPO on four engineering design problems showed that the proposed algorithm has a high ability to solve optimization problems in real-world applications.
Following the introduction of the proposed RPO approach, several research paths are activated for further studies. The development of binary and multi-objective versions of RPO is one of the most significant research potentials in this regard. The use of RPO for solving optimization problems in various fields of science as well as optimization tasks in real-world applications is one of the other suggestions of this paper for future investigations.