# Continuous Time Chaotic Systems for Whale Optimization Algorithm.

I. INTRODUCTIONMeta-heuristic algorithms are more effective than classical optimization algorithms; they can be easily transformed to apply to different problems, they do not require a derivative and they are independent of the problem [1]. Also, Yang and Deb mentioned that the two best features of meta-heuristic algorithms are concentration and diversification. Concentration is aimed at selecting the best candidates or solutions, whereas diversification allows the algorithm to discover the search field more efficiently [2]. There are many meta-heuristic algorithms proposed by different researchers in the literature. Some of the well-known meta-heuristic algorithms are Particle Swarm Optimization [3], Ant Colony Optimization [4], Genetic Algorithm [5], Artificial Bee Colony Algorithm [6], Golden Sine Algorithm [7] and Enhanced Grey Wolf Optimization Algorithm for Global Optimization [8]. WOA [9] developed by Mirjalili was used in this study. Because of it has few control parameters and is one of the actual meta-heuristic optimization algorithms. WOA is similar to the Grey wolf optimizer (GWO) [10] previously developed by the same author. The differences of BOA from GWO is searching optimum point with random or best agent, and the use of a spiral mechanism. GWO has been used to solve some problems in mathematics and physics [11, 12].

WOA requires random number arrays because of its stochastic structure. The fact that the random number sequences used do not have the spread spectrum or generate the same numbers may increase the risk of the algorithm entrapment in local optima and may reduce convergence speed. For this reason, chaotic maps should be used instead of random number sequences to improve the performance of the WOA. Chaotic maps are discrete-time systems exhibiting chaotic behavior. Theoretically proven that chaotic numbers, which generated by maps have unpredictable, spread spectrum characteristics and it is not periodic [1]. Various optimization algorithms with chaotic maps are available; Chaotic Firefly Algorithm Applied to Reliability-Redundancy Optimization [13], Chaos Embedded Particle Swarm Optimization Algorithms [14], Use of Chaotic Sequences in A Biologically Inspired Algorithm for Engineering Design Optimization [15] and Chaotic Maps Based On Binary Particle Swarm Optimization for Feature Selection [16] as examples.

In 2018 Sayed et al. [17], developed a chaotic version of WOA called Chaotic Whale Optimization Algorithm (CWOA) and used it for feature selecting problem. They did several experiments with 10 chaotic maps and showed that the algorithm using the circle map reached the best result. In 2017, Tanyildizi and Cigal [18] have proposed five different chaotic WOA versions using the chaotic map. The only similarity between the proposed method and that studies is the use of chaotic maps on WOA. The purpose of this work is to show that the use of continuous-time chaotic systems on WOA is useful in some cases instead of chaotic maps.

Ozkaynak published a new study titled "A novel method to improve the performance of chaos-based evolutionary algorithms" in 2015 and revealed a number of interesting results [19]. It has been shown by some experiments that continuous time chaotic systems give more successful results than chaotic maps.

Our Contribution:

* Ozkaynak has used histogram analysis and Nist statistical tests to confirm his claim in the study. However, these tests are necessary to demonstrate randomness, but not enough.

* Nist tests are hypothesis tests. Hypothesis testing has several problems.

* Experiments and simulations were carried out for both discrete and continuous time chaotic systems.

* The results obtained have provided a different approach to the previous studies in the field of optimization. It provides various implications on the success of the optimization problem, quality function and optimization parameters used.

* The study presented various results on chaos-based the Random Number Generation Approach (RNGA).

In this study, RNGA that is a continuous-time chaotic system which proposed in Ozkaynak's study was applied to WOA first time and compared with circle chaotic map. The outline of the study is as follows; In Section 2, we summarize the WOA. In Section 3, we summarize the continuous time chaotic systems proposed by Ozkaynak. In Section 4, whale optimization algorithms with chaotic maps are shown. In Section 5, numerical results are presented by mentioning experiments and simulations. Finally, Section 6 concludes the paper.

II. WHALE OPTIMIZATION ALGORITHM

This algorithm finds solutions for optimization problems by modeling hunting behaviors of humpback whales. Humpback whale is one of seven different whale species. These whales continue their lives by hunting small fish swarms. Their specialty is their own hunting methods called bubble-net strategy. In this strategy, whales find fish swarms close to the water surface and begin to create bubbles by circular motion. These bubbles, which are created deeper than prey, wrap around the prey and narrow its range of motion. Then the whale moves by narrowing the circle towards the water surface and it continues to create bubbles during this upward movement. As a result of this spiral movement, it is most likely to achieve its goal and catch the fish [20].

Mirjalili has created this algorithm by mathematically modeling the hunting method of humpback whales. The model consists of 3 sections; encircling prey, spiral movement and search for prey. The mathematical expression of encircling prey behavior is given in equation (1) and equation (2). Local search is being done with these equations.

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

In equations, indicates the current iteration, C and A are coefficient vectors, [X.sup.*] indicates best solution vector. D is the distance between the search agent's position and the best agent's positon. C and A is calculated as shown in equation (3) and equation (4).

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

In equation (4), r is a random vector in [0, 1] and a is linearly decreasing vector from two to zero.

The spiral motion behavior is modeled with the spiral equation shown in equation (5). In equation, b is logarithmic spiral constant, p is random number in [0, 1] and l is a random number in [-1,1]. As seen in the equation, spiral motion or encircling prey is applied at fifty percent probability. If p [greater than or equal to] 0.5, search agent updates position by spiral equation.

[mathematical expression not reproducible] (5)

Equation (6) and equation (7) are used for searching prey. These equations were created by using [X.sub.rand] which represents a random search agent instead of [X.sup.*](t) which represents the best search agent. Thus, global search is made.

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

A, C, P and I are critical variables affecting convergence speed and preventing trapped in local optima. Therefore, chaos has been employed on these variables to get maximum effect.

III. CHAOTIC WHALE OPTIMIZATION ALGORITHMS

In this section, circle map and lorenz system are integrated of WOA to show advantages of continuous-time chaotic system.

A. Chaotic Maps

Chaotic maps are discrete-time systems with chaotic behavior. It has been theoretically proven that the numbers produced by chaotic maps have unpredictable, spread spectral characteristics and are not periodic [21]. Because of that features of chaotic maps, they have been used in many applications as Artificial Neural Networks [22], Communication and Digital Simulation [23], Electronic Circuits [24, 25]. As mentioned in the introduction, one of these areas is also optimization. Many chaotic maps can be found in the literature [26]. Distributions of random number sequences obtained with chaotic maps can distributions can significantly affect the efficiency of global optimization algorithms [27]. Therefore, circle map, which is also successful in the studies of Sayed and ozkaynak is used in this study. Mathematical form of circle map with n dimension is given in equation (8).

[X.sub.n] + 1 = [X.sub.n] + b - (a / [2[pi]])sin([2[pi]][X.sub.n])mod(1) (8)

where a and b are control parameters. These were set to set to a = 0.5, b = 0.2 and the initial parameter was set to 0.1 in experiments.

B. Lorenz System

Ozkaynak has shown by experiments that many chaotic maps used in evolutionary algorithms do not distribute equally in statistical tests and that they fail. In that study, four different chaos-based random number generators have been proposed as a solution to this problem. It has been shown using experimental systems that continuous-time chaotic systems have better characteristics than chaotic maps. In this study, lorenz system which is a three dimensional continuous-time chaotic system is used. The lorenz system is a chaotic system that exhibits chaotic behavior within certain boundaries. This system was created in 1963 by meteorologist Ed Lorenz inspired by hydrodynamic systems [28]. Mathematical form of lorenz system with n dimension is given in equation (9).

[mathematical expression not reproducible] (9)

where a, b, c are control parameters and were set to a=10, b=28, c=8/3. x, y, x are initial values and were set to 0.1 in experiments.

C. Integration of Chaotic Systems into WOA

A parameter is for local search of algorithm and affects the distance traveled by the search agent, as mentioned before. In addition, algorithm performs local or global search according to the value of the A. Therefore A parameter is effective for convergence to optima. In this work, the random variable r where in equation (3) was substituted by the chaotic system function as shown in equation (10). r parameter sets values of A.

[??] = 2.[??].cs(t) - [??] (10)

IV. EXPERIMENTAL RESULTS

We have used three algorithms for comparison of chaotic systems in the experiments. First of them is standard WOA, second of them is WOA with circle map (CmWOA) and the other is WOA with Lorenz system (LsWOA). These algorithms were created by using the numbers obtained from chaotic systems instead of the r which controlling the A convergence vector shown in equation (3).

In the experiments, 23 benchmark functions were used. Benchmark functions can be used to measure and test the performance of optimization algorithms. Many benchmark functions are described in the literature [29]. Benchmark functions are divided into three categories as seven singlemode (F1-F7), six multi-mode (F8-13) and 10 fixed-size multi-mode (F14-F23). In addition, two engineering design problems called "pressure vessel design" and "spring tension/compression problem" were used in experiments. Engineering design problems are frequently used in the literature to test optimization algorithms because of their structures of nonlinear and complex [30].

Selected benchmark functions and engineering problems where cs(t) is random sequence obtained by chaotic system function. Flowchart of WOA and using of chaotic system functions is shown in Fig. 1.

are solved by simulating the WOA, CmWOA, and LsWOA. Since the use of continuous time chaotic systems instead of chaotic maps has been analyzed in experiments, the results with benchmark functions and engineering problems have not been compared with the literature.

Since WOA, CmWOA, and LsWOA are in a stochastic structure using nonparametric statistical tests to compare algorithms gives more reliable results. For this reason, Wilcoxon Signed Rank Test was also applied to the results of the experiments with significance value a = 0.05. Thus, it was statistically analyzed whether there was a significant difference in 95% probability between the algorithms.

A web page containing the simplified algorithm codes and dataset used was prepared for test [34].

A. Benchmark functions test results

In the tests, population sizes of the algorithms were set to 30 and the number of iterations was set to 500. Algorithms were run 30 times for each function. Statistical test results are shown in Table I.

In Table I, it is seen that CmWOA more converges to optimum in F1, F2, F10 functions according to average values compared to other algorithms. The WOA has given the best results in F8 function. All algorithms have reached the optimum value in F9 and F16 functions. In the F17 function, the average of the three algorithms is the same and closest to the optimum but according to best values both of the chaotic maps have reached the optimum point although WOA has got stuck on a local solution. In the other sixteen functions, LsWOA was more successful than other algorithms and it has improved the quality of the solution.

The convergence graphs of the algorithms for selected benchmark functions are shown in Fig. 2. The dotted line shows WOA, the solid line shows LsWOA and the dashed line shows CmWOA. According to table, convergence speeds of all algorithms are close to each other in F1 and F9 function. The chaotic versions in F3 function are almost the same speed and converge earlier than the WOA. LsWOA converges more slowly in F11 function. In F20 and F23 functions, LsWOA converges earlier.

In Table II, the Wilcoxon signed rank test results for the benchmark functions are shown separately. In the table [R.sup.+] and [R.sup.-] indicate the sum of the ranks. For F9, F11 and F16 functions, the Wilcoxon signed rank test was not applied because the algorithms resulted the same value in most of the 30 independent runs. These functions are shown in the table as N / A. When the 'WOA vs CmWOA' column is analyzed in table, it can be seen that there is a significant difference between the algorithms since p<0.05 in F1, F2, F3, F5, F10, F12 functions and CmWOA has improved performance of algorithm in this functions. Likewise, in F6 and F13 functions there is a significant difference between WOA and CmWOA but since [R.sup.+] > [R.sup.-] WOA is more successful in these functions. As a result, CmWOA has improved performance of algorithm in six functions and reduced performance in two functions. There was no significant difference in other functions. According to 'WOA vs LsWOA' column, there is a significant difference between WOA and LsWOA in F3, F5, F13, F15, F21, F22 functions. There was no significant difference in other functions. It can be said that lorenz system increases the search ability of the algorithm. The relationship between LsWOA and CmWOA, which is also the purpose of the study is shown in the column 'CmWOA vs LsWOA'. We mentioned that LsWOA gives better results according to Table I. Table II shows that there is a significant difference between the algorithms in F6, F12, F13, F14, F19 and F23 functions and that LsWOA is better than CmWOA. In F1 and F2 functions CmWOA is better than LsWOA since [R.sup.+] > [R.sup.-]. There is no statistically significant difference between these two algorithms in other functions.

B. Experiments with engineering design problems

To test algorithms on engineering problems population sizes was set to 50 and the number of iterations was set to 10000. The algorithms were run 30 times for each problem. 1) Pressure vessel design problem

As can be seen in Fig. 3, the pressure vessel is a cylinder closed with two semispherical caps. In this problem, the aim is to minimize the cost of material, welding and forming.

There are four design variables for this problem [31]. These are [T.sub.s] ([x.sub.1]) shell thickness, Th ([x.sub.2]) header thickness, R ([x.sub.3]) inner radius and L ([x.sub.4]) length of cylinder. The value ranges of the variables are defined as 0.0625 [less than or equal to] [x.sub.1], [x.sub.2] [less than or equal to] 99x0.0625 and 10 [less than or equal to] [x.sub.3], [x.sub.4] [less than or equal to] 200. Formulation and constraints of the problem are shown in equation (11).

min f([??]) = 0.6224[x.sub.1][x.sub.3][x.sub.4] + 1.7781[x.sub.2][x.sub.3.sup.2] + 3.1661[x.sub.1.sup.2][x.sub.4] +19.84[x.sub.1.sup.2][x.sub.3]

Constraints,

[mathematical expression not reproducible] (11)

Pressure vessel design problem is solved by simulating the WOA, CmWOA and LsWOA. The best solutions found are given in Table III and the statistical results obtained at the end of 30 runs are given in Table IV. The results show that WOA gives the minimum result but according to statistical results, LsWOA is more reliable because of lower std. value and lower average value.

2) Spring tension / compression problem

In this problem, the aim is to minimize the weight of a tension-compression spring depending on minimum deflection, shear stress, surge frequency and limits on outside diameter [32, 33]. The design is shown in Fig. 4 and the mathematical expression of the problem is given in equation (12). The value ranges of the variables are defined as 0.05 [less than or equal to] [x.sub.1] [less than or equal to] 2, 0.25 [less than or equal to] [x.sub.2] [less than or equal to] 1.3, 2 [less than or equal to] [x.sub.3] [less than or equal to] 15.

[mathematical expression not reproducible] (12)

Spring tension / compression problem is solved by simulating the WOA, CmWOA and LsWOA. The best solutions found are given in Table V and the statistics obtained at the end of 30 runs is given in Table VI. According to the results there is no difference in best values but LsWOA has low std. and low mean value.

Table VII shows Wilcoxon Signed Rank Test results for two engineering problems. According to the results, there is no significant difference between the algorithms for pressure vessel problem. In results of spring tension / compression problem, it can be said that there is a significant difference between the algorithms in comparison of WOA vs CmWOA and that WOA reduces the cost even more. There is no significant difference between WOA and LsWOA because of p > 0.05. According to 'CmWOA vs LsWOA' column there is a significant difference between the results and LsWOA reduces the cost more than CmWOA.

V. CONCLUSION

In this study, the results of using continuous time chaotic systems instead of one-dimensional chaotic maps are analyzed based on WOA. Two different chaotic versions of WOA were created using Circle chaotic map and Lorenz system. WOA and chaotic versions have been analyzed in 23 quality test functions and two engineering design problems. The chaotic patches used in the algorithm showed different effects on local and global searches. Performance of WOA can change according to the types of problems. For this reason, it is more convenient to evaluate the performance of the WOA according to the numerical examples of the test functions in different dimensions.

In experiments, entropy pools were created using the circle map and the Lorenz system to increase chaos and randomness. Each pool contains three million numbers. The random number sequences needed during the operation of the algorithms were randomized from these pools.

It has been observed that CmWOA is more successful in single-mode functions and LsWOA is more successful in multi-mode functions. As a result, it may be useful to use one-dimensional chaotic maps in one-dimensional problems and three-dimensional continuous time chaotic systems in multidimensional problems. This is a new study and detailed experiments can be done on different algorithms with different chaotic maps and continuous time chaotic systems. It is thought that this work could make progress in the literature.

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Erkan TANYILDIZI, Tuncay CIGAL

Department of Software Engineering, Firat University, 23119, Elazig, Turkey

etanyildizi@firat.edu.tr

Digital Object Identifier 10.4316/AECE.2018.04006

TABLE I. BENCHMARK FUNCTIONS TEST RESULTS (BEST VALUE (BEST), MEAN VALUE (MEAN), WORST VALUE (WORST), STANDARD DEVIATION (STD)) F Statistics WOA CmWOA LsWOA Opt. value mean 1,0591E-69 3,2261E-78 6,8636E-72 best 2,0176E-85 6,1191E-93 1,0062E-85 F1 worst 5,2955E-68 1,0983E-76 3,3533E-70 0 Std 7,4889E-69 1,6956E-77 4,7408E-71 mean 1,2756E-50 2,3701E-54 1,7471E-51 F2 best 2,3876E-58 3,9492E-62 2,6218E-58 0 worst 2,8602E-49 1,043E-52 2,9047E-50 Std 5,0675E-50 1,4762E-53 4,8935E-51 mean 49081,701 45686,1898 43373,2372 F3 best 20723,4497 18348,3312 6978,9602 0 worst 101365,6111 74376,2894 70567,2939 Std 16849,4697 13935,4029 14272,9904 mean 51,1163 46,6676 42,5246 F4 best 0,00451 0,089743 0,018835 0 worst 89,5215 88,4161 89,227 Std 28,9977 29,1535 26,4262 mean 28,1195 27,9382 27,915 F5 best 27,1397 27,1197 27,0772 0 worst 28,7798 28,7818 28,7409 Std 0,4912 0,45114 0,40563 mean 0,41129 0,59142 0,38125 F6 best 0,053468 0,16114 0,073151 0 worst 1,3303 1,4987 1,3921 Std 0,28541 0,27131 0,26472 mean 0,0038448 0,003864 0,0032432 F7 best 0,0002047 0,00012575 0,00010399 0 worst 0,034175 0,023973 0,015395 Std 0,0054157 0,0045304 0,0036228 mean -10374,7594 -10066,4633 -10249,052 F8 best -12569,4389 -12569,4742 -12568,8477 -12569,487 worst -7243,0724 -7924,5888 -6737,9118 Std 1819,3514 1740,2966 1779,2458 mean 2,2737E-15 0 0 F9 best 0 0 0 0 worst 5,6843E-14 0 0 Std 1,1252E-14 0 0 mean 4,7962E-15 4,0856E-15 4,583E-15 F10 best 8,8818E-16 8,8818E-16 8,8818E-16 0 worst 7,9936E-15 7,9936E-15 7,9936E-15 Std 2,5121E-15 2,6127E-15 2,5839E-15 mean 0,0032259 0,006995 2,2204E-18 F11 best 0 0 0 0 worst 0,16129 0,34975 1,1102E-16 Std 0,02281 0,049462 1,5701E-17 mean 0,048549 0,030741 0,025325 F12 best 0,0044129 0,01047 0,0052887 0 worst 1,3689 0,065711 0,10328 Std 0,191 0,013923 0,021673 mean 0,56822 0,70084 0,51035 F13 best 0,058122 0,17835 0,072139 0 worst 1,1896 1,4532 1,3026 Std 0,24793 0,31197 0,27627 mean 3,3877 3,7452 2,6895 F14 best 0,998 0,998 0,998 1 worst 10,7632 12,6705 10,7632 Std 3,7997 3,6395 2,8166 mean 0,00085463 0,0006957 0,0006879 F15 best 0,00030914 0,00030929 0,0003094 0,0003 worst 0,010872 0,00219 0,0022519 Std 0,0015656 0,0004103 0,00039466 mean -1,0316 -1,0316 -1,0316 F16 best -1,0316 -1,0316 -1,0316 -1,0316 worst -1,0316 -1,0316 -1,0316 Std 1,0295E-09 2,6294E-09 1,3688E-09 mean 0,3979 0,3979 0,3979 F17 best 0,39794 0,3980 0,3980 0,398 worst 0,39789 0,39789 0,39789 Std 0,000011845 0,000022079 0,0000 mean 3,0001 3,0196 3,0001 F18 best 3 3 3 3 worst 3,0012 3,9666 3,0029 Std 0,00030131 0,13665 0,00050483 mean -3,8561 -3,8509 -3,8579 F19 best -3,8628 -3,8628 -3,8628 -3,86 worst -3,8218 -3,7781 -3,8316 Std 0,0094423 0,018767 0,006703 mean -3,2297 -3,2335 -3,2402 F20 best -3,3219 -3,3215 -3,3219 -3,32 worst -2,8395 -2,9917 -3,0205 Std 0,10698 0,09838 0,091862 mean -8,0564 -8,5573 -9,013 F21 best -10,1525 -10,1526 -10,1532 -10,1532 worst -2,6286 -2,6277 -2,6284 Std 2,7328 2,4464 2,2874 mean -7,124 -7,2924 -8,0158 F22 best -10,4029 -10,4011 -10,401 -10,4028 worst -2,7638 -1,8331 -1,837 Std 2,8983 3,1954 2,9787 mean -6,9931 -6,6528 -7,4636 F23 best -10,5361 -10,5332 -10,5354 -10,5363 worst -1,6756 -1,6765 -1,676 Std 3,2634 3,5856 3,3214 TABLE II. THE RESULTS OF WILCOXON SIGNED RANK TEST FOR BENCHMARK FUNCTIONS F WOA vs CmWOA WOA vs LsWOA P value [R.sup.+] [R.sup.-] P value [R.sup.+] [R.sup.-] F1 1,47E-11 113 1162 0,3409 680 595 F2 1,63E-16 50 1225 0,49601 636 639 F3 0,04 460 815 0,01101 400 875 F4 0,261 571 704 0,06426 480 795 F5 0,04 464 811 0,02442 433 842 F6 0,0041 946 329 0,27093 701 574 F7 0,37 672 603 0,21186 555 720 F8 0,05 741 534 0,36317 601 674 F9 N/A N/A F10 0,00964 209 532 0,10749 352.5 550.5 F11 N/A N/A F12 0,00219 342 933 0,31561 588 687 F13 0,01831 854 421 0,04947 467 808 F14 0,20045 366 264 0,23885 288 378 F15 0,07186 824 451 0,04648 463 812 F16 N/A N/A F17 0,32636 112,5 140,5 0,10935 107 193 F18 0,40517 146 130 0,15151 114 186 F19 0,1423 720 505 0,35942 600 675 F20 0,40129 663,5 611,5 0,25785 570 705 F21 0,30854 585,5 689,5 0,03754 434 791 F22 0,48006 617,5 607,5 0,03216 446 829 F23 0,2327 686 539 0,28096 531 645 F CmWOA vs LsWOA P value [R.sup.+] [R.sup.-] F1 6,25538E-13 1150 125 F2 3,30755E-16 1212 63 F3 0,54235667 632 643 F4 0,46870962 557 718 F5 0,39743 611 664 F6 0,00008 245 1030 F7 0,20327 552 723 F8 0,31207 587 688 F9 N/A F10 0,18673 330,5 230,5 F11 N/A F12 0,03144 445 830 F13 0,00159 332 943 F14 0,02619 236,5 504,5 F15 0,30503 585 690 F16 N/A F17 N/A F18 0,19778241 90 120 F19 0,01743 382 794 F20 0,25463 569 706 F21 0,16109 535 740 F22 0,06944 484 791 F23 0,03144 426 799 TABLE III. BEST SOLUTIONS FOR PRESSURE VESSEL DESIGN PROBLEM Optimum variables Algorithm [T.sub.s] [x.sub.h] R L Opt. cost WOA 0,7801225 0,3864169 40,38487 199,0936 5895,8361 CmWOA 0,782327 0,3879404 40,47953 197,786 5903,5135 LsWOA 0,7791075 0,3874989 40,32804 199,8828 5899,2461 TABLE IV. PRESSURE VESSEL DESIGN PROBLEM STATISTICAL RESULTS Statistics WOA CmWOA LsWOA mean 6352,6508 6382,0097 6302,8902 best 5895,8361 5903,5135 5899,2461 worst 7438,7547 7509,5886 7388,0163 std 484,2424 474,8808 392,9247 TABLE V. BEST SOLUTIONS FOR SPRING TENSION / COMPRESSION PROBLEM Algorithm Optimum variables Optimum w d l cost WOA 0,0516384 0,3555 11,3607 0,012665 CmWOA 0,0519289 0,362515 10,957 0,012666 LsWOA 0,0517754 0,358798 11,168 0,012665 TABLE VI. SPRING TENSION / COMPRESSION PROBLEM STATISTICAL Statistics WOA CmWOA LsWOA mean 0,012967 0,013276 0,012888 best 0,012665 0,012666 0,012665 worst 0,014104 0,015865 0,013683 std 0,00035678 0,00067783 0,00022567 TABLE VII. THE RESULTS OF WILCOXON SIGNED RANK TEST BASED ON THE ENGINEERING PROBLEMS Problem WOA vs CmWOA P value [R.sup.+] [R.sup.-] pressure vessel design 0,40905 670 605 problem Spring tension / 0,00523 870 355 compression problem Problem WOA vs LsWOA P value [R.sup.+] [R.sup.-] pressure vessel design 0,32276 602 673 problem Spring tension / 0,2177 557 718 compression problem Problem CmWOA vs LsWOA P value [R.sup.+] [R.sup.-] pressure vessel design 0,21476 556 719 problem Spring tension / 0,00019 270 1005 compression problem

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Author: | Tanyildizi, Erkan; Cigal, Tuncay |
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Publication: | Advances in Electrical and Computer Engineering |

Article Type: | Technical report |

Date: | Nov 1, 2018 |

Words: | 5692 |

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