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Continuous Time Chaotic Systems for Whale Optimization Algorithm.

I. INTRODUCTION

Meta-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
Publication:Advances in Electrical and Computer Engineering
Article Type:Technical report
Date:Nov 1, 2018
Words:5692
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