Clouded Leopard Optimization: A New Nature-Inspired Optimization Algorithm

This paper proposes a new nature-inspired metaheuristic algorithm called Clouded Leopard Optimization (CLO), which mimics the natural behavior of clouded leopards in the wild. The fundamental inspiration of CLO is derived from two ways of natural behaviors of the clouded leopard, including hunting strategy and daily resting on trees. CLO is mathematically modeled in two phases of exploration and exploitation, based on the simulation of these two natural behaviors. CLO performance is evaluated in solving sixty-eight benchmark functions, including unimodal, multimodal, CEC 2015, and CEC 2017 types. The performance of CLO in solving optimization problems is compared with the performance of ten famous metaheuristic algorithms. The simulation results show that the proposed CLO approach with high ability in exploration, exploitation, and balancing between them has a high capability in optimization applications. Simulation results show that CLO performs better in most test functions than competitor algorithms. In addition, the implementation of CLO on four engineering design issues demonstrates the capability of the proposed approach in real-world applications.

problem is mathematically modeled using the three main 23 components of the decision variables, constraints, and the 24 objective function [2]. Optimization problem-solving meth-25 ods fall into two groups: deterministic methods and stochastic 26 methods. 27 Deterministic methods in both gradient-based and non- 28 gradient-based groups have good performance in dealing 29 with linear, convex, continuous, and differentiable optimiza-30 tion problems [3]. On the other hand, many real-world 31 The associate editor coordinating the review of this manuscript and approving it for publication was Sotirios Goudos . optimization problems have properties such as nonlinear, 32 non-convex, non-differentiable objective function, discrete 33 search space, and high dimensions. These features have led 34 to the failure of deterministic methods in handling such 35 optimization applications and thus led to the introduction of 36 another approach called stochastic methods [4]. 37 Stochastic methods in the optimization process use random 38 search in the problem-solving space based on search agents, 39 random operators, and trial and error processes. Metaheuris-40 tic algorithms are one of the most efficient stochastic methods 41 that, based on collective intelligence, provide effective perfor-42 mance in optimizing and providing solutions to optimization 43 problems [5]. 44 The optimization process in metaheuristic algorithms 45 begins with the random generation of a number of candidate 46 solutions. These candidate solutions are improved through 47 an iterative process based on different steps of the algo- 48 rithm. After the full implementation of the algorithm, the 49 best-improved candidate solution is presented as the solution 50 to the problem. However, the nature of random search in 51 • The CLO is mathematically modeled in two phases of 106 exploration and exploitation based on simulations of the 107 natural behaviors of clouded leopards.

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• CLO performance is evaluated on the optimization of 109 sixty-eight benchmark functions, including unimodal 110 functions, multimodal functions, CEC 2015 suite, and 111 CEC 2017 suite.

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• CLO's ability to handle optimization problems is com-113 pared with ten well-known metaheuristic algorithms.

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• CLO's ability to handle real-world applications is tested 115 on four engineering design challenges. 116 The rest of this paper is as follows: The literature review 117 is provided in Section II. The proposed CLO algorithm 118 is introduced in Section III. Then in Section IV, simula-119 tion studies are presented. The application of the proposed 120 CLO in solving real-world problems is studied in Section V. 121 Conclusions and suggestions for future studies are provided 122 in Section VI.

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Metaheuristic algorithms can be divided into nine 125 groups based on the main idea used in the design: 126 (i) swarm-based, (ii) biology-based, (iii) physics-based, 127 (iv) sport-based, (v) social-based, (vi) music-based, 128 (vii) chemistry-based, (viii) mathematical-based, and 129 (ix) hybrid algorithms [14], [15]. 130 Swarm-based algorithms have been developed based on 131 modeling natural swarm phenomena, the behavior of animals, 132 birds, aquatic animals, insects, and other living organisms. 133 PSO, ABC, and ACO are the most popular and widely used 134 swarm-based algorithms. The PSO is based on modeling the 135 collective intelligence behavior of birds and fish in search of 136 food sources. In the design of ABC, the behavior of bees in 137 a colony to find food sources is imitated. ACO is introduced 138 based on the ability of ants to find the shortest path between 139 the nest and food sources. Common behaviors in many living 140 things in nature, such as food search, hunting, and migration, 141 have been the main idea in the design of numerous swarm-142 based metaheuristic algorithms, such as Grey Wolf Optimiza-143 tion (GWO) [  Optimization (NRO) [33], Atom Search Optimization (ASO) 173 [34], and Multi-Verse Optimizer (MVO) [35]. players' efforts to win in the game of tug-of-war [38].  Atashpaz-Gargari et al. [38] proposed the first algorithm 185 of this type, as they introduced the Imperialist Competitive  [43], Election-Based Optimization Algorithm (EBOA) [44], 198 and Teamwork Optimization Algorithm (TOA) [45].    Among the mathematical-based algorithms, can be men-211 tioned to Base optimization algorithm (BOA) [50] and 212 Golden Sine Algorithm (GSA) [51] and Base optimization 213 algorithm (BOA) [50]. 214 Hybrid algorithms have been proposed based on com-215 bining two or more algorithms to increase their efficiency. 216 Among the hybrid algorithms can be mentioned Sine-cosine 217 and Spotted Hyena-based Chimp Optimization Algorithm 218 (SSC) [52] and Hybrid firefly algorithm with grouping attrac-219 tion (HFA-GA) [53]. 220 Plant intelligence has been the main idea used in the intro-221 duction of plant-based algorithms [54]. The simulation of 222 hydrological processes has been the main inspiration in the 223 design of water-based algorithms [55].

224
The clouded leopard is a wild cat that lives in dense forests 225 from the foothills of the Himalayas through the mainland of 226 Southeast Asia to southern China. The reason for naming 227 clouded leopards is its large dusky-gray blotches and irregular 228 spots and stripes that are very similar to clouds. Two charac-229 teristic natural behaviors of this animal are that it hunts by 230 night on the forest floor and rests in trees during the day.

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Based on the best knowledge gained from the literature 232 review, modeling the natural behaviors of clouded leopards 233 have not been employed in the design of any metaheuristic 234 algorithm so far. This is while the clouded leopard features 235 have the potential to be employed in designing a new opti-236 mizer. To address this research gap, the authors of this paper 237 have been motivated to develop a new metaheuristic algo-238 rithm based on mathematical modeling of clouded leopard 239 natural behaviors in the wild. 240 The metaheuristic algorithms that have been designed so 241 far have several advantages, such as simplicity of concepts, 242 easy implementation, efficiency in nonlinear, discontinuous, 243 non-convex, NP-hard problems, and efficiency in discrete and 244 unknown search spaces. However, many of the metaheuristic 245 algorithms designed in the literature also have disadvantages 246 that hinder their efficiency in handling some optimization 247 problems. These disadvantages are generally caused by the 248 weakness of metaheuristic algorithms in managing global 249 search and local search and balancing them during the opti-250 mization process.

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Suppose the metaheuristic algorithm does not have a 252 good exploration capability to provide a global search in 253 the problem-solving space. In that case, it will get stuck 254 in local optima and inappropriate solutions, especially in 255 solving complex optimization problems with a number of 256 local optima in their search space. One of the main reasons 257 that prevent the exploration of metaheuristic algorithms is 258 the excessive dependence of the algorithm population update 259 on specific population members, such as the population's 260 best members. This feature makes metaheuristic algorithms 261 unable to search effectively globally in the problem-solving 262 space and converge to local optima. In the design of the 263 proposed CLO approach, to increase the exploration ability 264 and provide an effective global search, the reliance of the 265 population update on specific members of the algorithm, 266 especially the best member of the population, has been 267 avoided. For this purpose, CLO population members are 268 moved to different areas of the search space in the explo-269 ration phase of CLO based on hunting strategy simulation. 270 In this phase, to increase the exploration capability of CLO,  stripe that continues down the foreleg and breaks up into 325 irregular spots. Dark dusky-grey irregular patches line the 326 flanks, which are framed behind by long, oblique, irregularly 327 curved, or looped stripes. Spots cover the underparts and 328 legs, and the tail has huge, uneven, paired spots. It has short, 329 strong legs and large paws [56]. Figure 1 shows an image of 330 a clouded leopard. Clouded leopards range in weight from 331 11.5 to 23 kg. Females range in length from 68.6 to 94 cm, 332 with a tail length of 61 to 82 cm. Males are bigger, measuring 333 81 to 108 cm in length with a tail of 74 to 91 cm in length. Its 334 shoulder height ranges between 50 and 55 cm [57].

335
The clouded leopard is a solitary and shy cat. Two natural 336 behaviors of clouded leopards have been recorded in the 337 observations, which is more significant than other behaviors 338 of this animal. These animals are resting on the trees during 339 the day. Their main activity is at night when they come down 340 from the trees to hunt on the ground [58]. Modeling these two 341 prominent behaviors in clouded leopards has been the major 342 inspiration in the proposed CLO design.  In the CLO design, each clouded leopard represents a member 348 of the population and represents a candidate solution to the 349 problem. The position of the clouded leopard in the search 350 space shows the values of the decision variables. Thus, from 351 a mathematical point of view, any clouded leopard can be 352 modeled using a vector, while the elements of this vector rep-353 resent decision variables. At the beginning of the algorithm, 354 the position of the clouded leopards in the search space is 355 randomly initialized using (1). tively, and '' · '' is the usual mathematical notation for scalar 365 multiplication.

366
Clouded leopards together form the CLO population, 367 which can be mathematically represented using a matrix 368 according to (2).
where X is the population matrix of CLO.   allows members of the population to scan different areas of 399 the search space globally and accurately. To model this activ-400 ity mathematically, for each clouded leopard, the position of 401 other population members in the search space is considered 402 as prey locations, one of which is randomly selected as the 403 target prey. Based on the position of the target prey, a new 404 situation is generated for the corresponding clouded leopard 405 using (4). If this new position has a better value for the 406 objective function, it replaces the previous position of the 407 corresponding clouded leopard according to (5).   Clouded leopards move to the trees to digest food and rest 422 after hunting and feeding prey. As a result, they spend most 423 of the day resting on trees. This behavior of clouded leopards 424 leads to a change in their position near where they are located. 425 This strategy represents exploitation and local search in meta-426 heuristic algorithms, which leads the algorithm to look for 427 better solutions around the discovered solutions. In order to 428 simulate this behavior of clouded leopards, a random position 429 near the location of each clouded leopard is generated using 430 (6). Then, if the value of the objective function improves in 431 the new position, it replaces the previous position according 432 to (7). 434 where X P 2 i is the new position suggested for the ith clouded 436 leopard (i.e., new candidate solution) based on the second 437 phase of CLO, x P 2 i,j is its jth dimension (i.e., decision variable), 438 F P 2 i is its objective function value, r i,j are randomly selected 439 numbers from the interval [0, 1].  For i = 1 : N 6.
Specify randomly the position of the target prey for the ith clouded leopard. 8.
Calculate the new status of the ith clouded leopard by (4). 9.
Calculate the new status of the ith clouded leopard using (6). 12.
Update the ith clouded leopard using (7). 13. end for 14. Save the best candidate solution so far. 15. end for 16. Output: The best solution obtained by CLO for the given optimization problem. End CLO.

461
In this section, the performance of the proposed CLO in  Table 1. 472 To make a fair comparison, the proposed CLO approach and 473 competitor algorithms in similar conditions are each imple-474 mented in benchmark functions in twenty independent execu-475 tions, while each execution contains 1000 iterations. Starting 476 search points of the optimization algorithms are randomly 477 initialized using (1). Simulations are implemented on the 478 software MATLAB R2022a using a 64-bit Core i7 processor 479 with 3.20 GHz and 16 GB main memory. The simulation 480 results are reported using seven indicators: mean, best, worst, 481 standard deviation (std), median, execution time (ET), and 482 rank.  Table 2. Based on the optimization 494 results, CLO has been able to provide the optimal global 495 solution for the F1, F2, F3, F4, F5, and F6 functions, which 496 demonstrates the high exploitation potential of the proposed 497 approach. In F7 optimization, CLO is the best optimizer. 498 Simulation results show that CLO provides better results, 499 is superior, and is more competitive than ten competitor 500 algorithms. High-dimensional multimodal functions are good options 504 for evaluating the exploration and global search capabili-505 ties of metaheuristic algorithms due to their multiple local 506 solutions in the search space. The optimization results of 507 high-dimensional multimodal functions F8 to F13 are pre-508 sented in Table 3. The optimization results show that CLO 509 has provided the optimal global solution for F9 and F11 510 functions. In addition, CLO is the best optimizer in solving 511 F8, F10, F12, and F13 functions. Based on the simulation 512 results, it is concluded that CLO has an acceptable explo-513 ration ability to identify the area containing the main opti-514 mization and also to prevent the algorithm from getting 515 stuck in local solutions, which has led to the superiority 516 of the proposed CLO approach compared to ten competitor 517 algorithms.   competitor algorithms in optimizing F1 to F23 functions are 536 shown in Figure 3.

538
In this subsection, a statistical analysis of the results 539 obtained from CLO versus the results of competitor algo-540 rithms is presented. The Wilcoxon sign-rank test [62] is 541 used for this purpose. The Wilcoxon sign-rank test is a 542 non-parametric test that shows whether there is a statistically 543 significant difference between the means of the two data 544 samples. Hence, the Wilcoxon sign-rank test is implemented 545 on CLO results and competitor algorithms to determine 546 whether the superiority of CLO over competitor algorithms is 547 significant.

548
The results of the Wilcoxon sign-rank test on the perfor-549 mance of CLO and competitor algorithms are presented in 550 Table 5. Based on the p-value index, in cases where the value 551 of this index is less than 0.05, the CLO approach has statis-552 tically significant superiority compared to the corresponding 553 competitor algorithm. Furthermore, the results of statistical 554 analysis show that in most cases, the p-value is less than 555 0.05, which indicates the statistically significant superiority 556 of CLO over competitor algorithms.

558
The proposed CLO approach is a population-based meta-559 heuristic algorithm that solves optimization problems in an 560 iterative process. Hence, the number of members of the 561 clouded leopard population (N ) and the number of algorithm 562 iterations (T ) affect the performance of the CLO. In this 563 regard, in this subsection, CLO sensitivity analysis to N and 564 T parameters is studied.

565
In order to analyze the sensitivity of CLO to the parameter 566 T , the proposed approach in independent executions for T 567 values of 200, 500, 800, and 1000 has been implemented on 568 F1 to F23 functions. The optimization results are presented in 569 Table 6, and the CLO convergence curves in Figure 4. What 570 is clear from the analysis of the simulation results is that as 571 the number of iterations of the algorithm increases, the CLO 572 converges to better solutions, and the values of the objective 573 function decrease.

574
The proposed approach in independent executions for N 575 values equal to 20, 30, 50, and 100 has been implemented on 576 the F1 to F23 functions to analyze the CLO sensitivity to the 577 parameter N . The results of the sensitivity analysis to param-578 eter N are presented in Table 7, and the CLO convergence 579 curves under this analysis are presented in Figure 5. Based 580 on the simulation results, it is observed that by increasing the 581 number of clouded leopards, better values are obtained for the 582 objective function. The optimization results of the CEC 2015 set, including 585 CEC1 to CEC15 using CLO and competitor algorithms, are 586 released in Table 8. The simulation results show that CLO 587 is the best optimizer for CEC1, CEC2, CEC4, CEC5, CEC6, 588 CEC7, CEC9, CEC10, CEC11, CEC12, CEC13, CEC14, and 589 CEC15 functions. In addition, in solving the CEC3 function, 590 VOLUME 10, 2022     presented in Table 9. What can be seen from the analysis of 597 the simulation results is that CLO is the first best optimizer

613
Unimodal functions have only one extremum in the 614 search space, which challenges the exploitation power of 615 metaheuristic algorithms in local search. To evaluate the 616 exploitation ability of metaheuristic algorithms, unimodal 617 test functions F1 to F7 have been selected. In optimizing 618 functions F1 to F4, the proposed CLO and RSA have similar 619 performance. They have converged to the global optimal with 620 clear superiority over other algorithms, with high exploitation 621 ability. In solving F5 and F6 test functions, the proposed CLO 622 approach has converged to the global optimum with obvious 623 superiority compared to competitor algorithms. In solving the 624 F7 test function, CLO has performed better than competitor 625 algorithms and has provided a better solution. The optimiza-626 tion results of unimodal functions show that the proposed 627 CLO approach has high exploitation and local search ability 628 and has achieved superior performance compared to com-629 petitor algorithms. Wilcoxon sign-rank test statistical anal-630 ysis shows that the superiority of CLO is significant from 631 a statistical point of view compared to any of the competi-632 tor algorithms. Therefore, the hypothesis of allocating the 633 exploitation update phase in CLO based on simulating the 634 natural behavior of clouded leopards in daily rest, which 635 aims to increase the ability of CLO in local search, has been 636     in the search space. In solving the F8 test function, the proposed CLO approach has provided better performance  approach has a more effective performance in solving this 694 function. In the F17 function, CLO, MPA, and GSA have 695 provided the same values for the ''mean'' index, but CLO has 696 performed better by providing a better value in the standard 697 deviation index. In functions F19, F20, and F23, the proposed 698 approach has similar conditions in the ''mean'' index with 699 some competitor algorithms. However, CLO has provided 700 more effective performance by providing better values in the 701 standard deviation index. The proposed CLO approach is the 702 first best optimizer for functions F15, F18, F21, and F22 703 by maintaining a balance between exploration and exploita-704 tion. The general comparison of the optimization results of 705 functions F14 to F23 shows that CLO has provided superior 706 performance by being ranked first compared to competitor 707 algorithms. Also, the Wilcoxon sign-rank test shows that 708 the superiority of the proposed approach is significant from 709 the point of view of statistical analysis. The analysis of the 710 results of fixed-dimensional multimodal test functions F14 711 to F23 concludes that from the point of view of balancing 712 exploration and exploitation, the proposed CLO approach is 713 superior compared to competitor algorithms.

714
In addition, to test functions F1 to F23, test sets of CEC 715 2015 and CEC 2017 have been selected to evaluate the perfor-716 mance of the proposed CLO approach. These test sets consist 717 of complex and standard test functions that challenge the 718 ability of metaheuristic algorithms to solve complex prob-719 lems. The optimization results of these functions show that 720 the proposed CLO approach with high power in exploration, 721 exploitation, and balancing global and local search has pro-722 vided superior performance in most test functions compared 723 to competitor algorithms.

724
The analysis of the simulation results shows that the pro-725 posed approach has a good and acceptable performance in 726 optimization applications with a high ability of exploration 727 in global search, high ability of exploitation in local search, 728 and high ability to create a balance between exploration and 729 exploitation.

730
The proposed CLO approach is a metaheuristic algorithm 731 from the stochastic-based approaches. The nature of CLO in 732 the optimization process is random search in the problem-733 solving space. These features create limitations in the pro-734 posed CLO approach. The first limitation of CLO is that 735 there is no claim that this method is the best optimizer for all 736 optimization problems. The second limitation of CLO is that 737 it is always possible to design newer metaheuristic algorithms 738 VOLUME 10, 2022       they are set at the border of the intervals. Also, to satisfy 752 the unequal and equal constraints, the penalty coefficient is 753 employed and added to the objective function value. In this 754    The welded beam is an engineering issue whose design 774 aims to minimize the fabrication cost of the welded beam. 775 A schematic of this problem is shown in Figure 8.     The speed reducer is a real-world application whose design 789 aims to minimize the weight of the speed reducer. The 790 schematic of this problem is shown in Figure 10.

791
The recruitment results of CLO and competitor algorithms 792 in determining the speed reducer design variables are reported 793 in Tables (14) and (15)     The pressure vessel is an engineering optimization challenge 805 whose design aims to minimize material and manufacturing 806 costs. The schematic of this problem is shown in Figure 12. 807 The results of optimizing the pressure vessel design prob-808 lem using CLO and competitor algorithms are released in 809  Tables 16 and 17. Based on the simulation results, CLO 810 has provided the optimal solution for this design with 811 proposed values for the design variables equal to (0.7782806 812 0.3847046, 40.32536, 199.819) and the corresponding objec-813 tive functions value 5883.3629. What can be seen from the 814 analysis of statistical results is that CLO has a better perfor-815 mance in solving the pressure vessel design problem com-816 pared to competitor algorithms. The CLO convergence curve 817 VOLUME 10, 2022  in determining the optimal solution for this design is shown 818 in Figure 13.  The optimization results showed that CLO with the abil-831 ity to strike the right balance between exploration and 832 exploitation was able to provide good results for optimiza-833 tion tasks. A comparison of CLO results with the perfor-834 mance of ten well-known metaheuristic algorithms showed 835 that CLO is superior in most cases and it provides better 836 results. In addition, the implementation of CLO on four 837 engineering design issues demonstrated the effectiveness of 838 the proposed approach in handling real-world optimization 839 applications.

840
Introducing the CLO activates several research tasks for the 841 future, the most specific of which are the design of binary and 842 multi-objective versions of the CLO. Moreover, the use of the 843 proposed CLO in optimization problems in various sciences 844