(PDF) Weighted differential evolution algorithm for numerical function optimization: a comparative study with cuckoo search, art

Neural Computing and Applications https://doi.org/10.1007/s00521-018-3822-5 (0123456789().,-volV)(0123456789().,-volV) ORIGINAL ARTICLE Weighted differential evolution algorithm for numerical function optimization: a comparative study with cuckoo search, artificial bee colony, adaptive differential evolution, and backtracking search optimization algorithms Pinar Civicioglu1 • Erkan Besdok2 • Mehmet Akif Gunen2 • Umit Haluk Atasever2 Received: 23 February 2018 / Accepted: 15 October 2018 The Natural Computing Applications Forum 2018 Abstract In this paper, weighted differential evolution algorithm (WDE) has been proposed for solving real-valued numerical optimization problems. When all parameters of WDE are determined randomly, in practice, WDE has no control parameter but the pattern size. WDE can solve unimodal, multimodal, separable, scalable, and hybrid problems. WDE has a very fast and quite simple structure, in addition, it can be parallelized due to its non-recursive nature. WDE has a strong exploration and exploitation capability. In this paper, WDE’s success in solving CEC’ 2013 problems was compared to 4 different EAs (i.e., CS, ABC, JADE, and BSA) statistically. One 3D geometric optimization problem (i.e., GPS network adjustment problem) and 4 constrained engineering design problems were used to examine the WDE’s ability to solve real-world problems. Results obtained from the performed tests showed that, in general, problem-solving success of WDE is sta- tistically better than the comparison algorithms that have been used in this paper. Keywords Cuckoo search algorithm Artificial bee colony algorithm Differential evolution algorithm Backtracking search optimization Particle swarm optimization 1 Introduction stochastic search mechanisms, searching for optimum solu- tions that belong to the related problem [8–11]. Random Evolutionary algorithms (EA) are commonly used for solving solution of a numerical optimization problem is demonstrated complex numerical optimization problems (i.e., multimodal, with a pattern containing individuals as the dimension of the non-differentiable, highly nonlinear, and constrained design problem [8, 12–15]. EAs can find the near optimum solution problems) [1–7]. EAs are population based, iterative, of a problem by using limited number of patterns. Patterns are kept in a pattern matrix having different patterns as the rows [8, 15]. Interaction models defined between the pattern matrix & Erkan Besdok elements bring in collective search capability to EAs. EAs can

be classified as swarm-inspired [1–4, 7–13], bio-inspired Pinar Civicioglu [16, 17], or nature-inspired [18, 19] algorithms. Bio-inspired

EAs, the most frequently used EAs, simulate various genetic Mehmet Akif Gunen processes such as selection, mutation, and crossover analog-

ically, with the help of the interaction models that they are Umit Haluk Atasever using. EAs use interaction models to generate trial patterns by

using the present patterns. If a trial pattern produces more 1 Department of Aircraft Electrics and Electronics, Faculty of feasible results than the pattern that it corresponds, then it is Aeronautics and Astronautics, Erciyes University, Kayseri, added to the pattern matrix instead of the related pattern at the Turkey next iteration. Problem-solving successes of EAs are gener- 2 Department of Geomatics Engineering, Engineering Faculty, ally sensitive to several parameters; i.e., structural properties Erciyes University, Kayseri, Turkey 123 Neural Computing and Applications of the problem, dimension of problem, initial form of the more feasible pattern, WDE produces a trial pattern by pattern matrix, random number generator that is being used, changing some randomly selected or all individuals of the total number of function evaluation value. related pattern with the related individuals of an interaction EAs have been used in the solution of various structural pattern. Though trial pattern generation strategy of WDE is engineering design problems [20, 21], such as communication very productive, there is a possibility of trapping to local applications [7], image processing applications [22], solution solution in hybrid problems. Every pattern matrix of WDE of some speech recognition problems [23], solution of sensor evolves into a randomly selected and permuted pattern deployment problems [24], various data mining applications matrix to provide swarming in every iteration. While WDE [25], design of IIR filters [26], video processing [27], and generates a trial pattern by allowing the changing of only solution of other several engineering problems [28–35]. one individual for the first pattern matrix, it produces a trial The nature of a numerical optimization problem defines pattern by allowing a randomly selected number of indi- whether it is a unimodal, multimodal, separable, non-sep- viduals for the second pattern matrix. WDE generates a arable, scalable or hybrid problem, or not [1, 10–12]. trial pattern by allowing the changing of all individuals for Unimodal problems have a single local solution, which is the third pattern matrix. the same with the global solution. Multimodal problems In this paper, numerical optimization problem-solving have several global solutions. Therefore, in the solution of capability of WDE is examined in detail by using CEC’ 2013 multimodal problems, EAs using interaction models that do [36] numerical functional optimization problems. Cuckoo not trap the local solutions should be used. In separable search algorithm (CS) [3, 8], artificial bee colony (ABC) [1], problems, since each variable is independent from the other adaptive differential evolution algorithm (JADE) [37], and variables, in the solution of this type of problems, each backtracking search optimization algorithm (BSA) [7] have variable can be optimized independently. In non-separable been successful in the solution of many different engineering problems, all variables have to be optimized together. In problems. Therefore, the success of WDE in the solution of scalable problems, computational complexity of problem functional optimization problems has been examined with changes with changing dimension of problem. Problem- statistical comparison with CS, ABC, JADE, and BSA. GPS- solving success of EAs is sensitive to whether they have Based Geodesic network adjustment problem has been solved trial pattern generation strategy appropriate to the nature of by using classical least square adjustment method and WDE. the related problem, or not. Engineering design problems have been solved by using EAs generally produce a trial pattern with the usage of a PSO2011 [9], CPI-JADE [38], A? [39], CS, ABC, JADE, crossover process defined between the patterns or a muta- BSA, and WDE. tion process based on a statistical distribution model. Pat- Differential evolutionary algorithm (DE) is a statistical, tern generation strategies used in EAs are frequently based powerful, and widely used evolutionary computation on interaction between patterns. (i.e., swarming). EAs algorithm developed to solve real-valued numerical opti- benefit from a trial pattern to evolve a pattern to a pattern mization problems. DE is robust, its implementation is that iteratively provides a better fitness value. In order to relatively easy, and its structure is simple. DE is used in the generate a trial pattern, patterns selected as the raw genetic solution of unimodal, multimodal, and hybrid problem material among the present patterns are mixed by using types. DE also has some disadvantages. The success of DE various pseudo-genetic operators. System equations defin- shows the sensitivity to the problem type. DE suffers from ing the trial pattern generation processes used in EAs premature convergence, especially when solving multi- generally show important similarities [8]. Besides, usage modal and hybrid problems. The global search capability of style of the related system equations shows significant DE is sufficient for many problem types. However, the differences in EAs. The strategy that an EA uses for trial local search ability of DE is weak. Also, DE suffers from pattern generation influences its problem-solving success low convergence speed. The standard DE tends to rapidly and speed. Researches continue to develop new EAs that reduce the diversity of the population. This restricts DE’s can solve complex engineering problems. search ability. The performance of DE generally decreases This paper introduces WDE that can solve different rapidly as the search space size increases. DE is sensitive to types of numerical problems (i.e., separable, non-separable, the selection of control parameters, and it is time-con- unimodal, multimodal, and hybrid problems). WDE is a suming and difficult to tune them for different problems. non-recursive, iterative algorithm; the swarming process of The success of DE depends on the crossover and mutation WDE is very efficient. Therefore, WDE is generally not strategy it uses at the moment. DE is not strong enough to trapped easily with local solution. This increases WDE’s determine efficient evolutionary direction and evolutionary success in solving numerical problems considerably. WDE step size [7, 8, 15]. uses a random crossover-based swarming strategy to gen- The need to develop a method that does not have the erate interaction pattern. While evolving a pattern into a inherent limitations of DE has motivated the development of 123 Neural Computing and Applications WDE. The problem-solving success of the WDE is, to a large unbounded search. The elitist EAs are generally successful extent, not dependent on the problem type, as opposed to the in solving the unimodal problems. However, the elitist DE. Also, WDE does not need to use a different mutation behavior that an EA exhibits may cause it to be trapped process like DE for each problem type. The crossover and with local solutions. Therefore, partially elitist functioning mutation process of WDE is different from the crossover and of an EA accelerates its convergence to solution and pre- mutation processes of DE; they are simpler and much more vents it from being trapped with local solutions. efficient. Compared to DE, WDE has the ability to determine The initialization process of WDE includes defining the very efficient evolutionary search direction and evolutionary initial population (i.e., pattern matrix; P). P is computed by step. In addition, WDE does not allow the diversity of the using Eq. 1; population to decline rapidly. For this reason, it may continue 0 1 to do efficient searches in progressive iterations, contrary to Pði0;j0Þ U lowðj0Þ ; upðj0Þ j @|{z} D A 2N ; |{z} sizeðPÞ DE. Because WDE does not have a control parameter in rows colums practice, the time-consuming and difficult parameter tuning ð1Þ process of DE is not available in WDE. This paper is organized as follows: In Sect. 2, nomen- Here, i0 ¼ ½1 : 2N; j0 ¼ ½1 : D; where i0; j0 2 Zþ : In clature is given. In Sects. 3 and 4, weighted differential Eq. 1, N is the pattern vector number, and D is the problem evolution algorithm (WDE) and experiments are presented, dimension. lowj0 ; upj0 are the lower and upper search limits respectively. In Sect. 5, conclusions and possible research of the j0th parameter. UðÞ denotes the continuous uniform directions are given. distribution. The objective function value of Pi0 is com- puted by using Eq. 2; 2 Nomenclature fitPði0Þ ¼ F Pði0Þ ð2Þ In Eq. 2, F denotes the objective function. The first selection process of WDE generates a sub-pattern matrix, Symbol Meaning/definition SubP, from P. WDE generates SubP by randomly selecting N pattern vectors from P in each iteration F Objective function 0 1 low, up Search Limits where@|{z} D A N ; |{z} sizeðSubPÞ. SubP is defined by N Size of population rows colums D Dimension of problem using Eq. 3; MaxCycle Maximum number of iterations gmin Global minimum SubP ¼ PðkÞ j f k ¼ jð1:NÞ j j ¼ permute ð1 : 2NÞ ð3Þ gbest Global minimizer jðÞ Uð0; 1Þ; jðÞ 6¼ 0 Uniform random number In Eq. 3, permuteðÞ denotes the permuting function. The objective function values, fitSubP, of the pattern vectors of kðÞ Nð0; 1Þ Normal random numbers SubP are defined in Eq. 4; a; b Uð0; 1Þ Uniform random numbers Pði0;j0Þ j Pði0;j0Þ Uðlowðj0Þ ; upðj0Þ Þ Patterns of pattern matrix fitSubP ¼ fitPðkÞ ð4Þ fitPði0Þ Fitness values of Pi0¼1:N The mutation process aims to generate new pattern vectors, permute() Permuting function Hadamart operator i.e., TempP. WDE regenerates TempPindex¼1:N ¼ 2 3 TempP1 4 . . . 5 in each iteration by using Eq. 5; TempPN X 3 Weighted differential evolution algorithm TempPðindexÞ ¼ ðw PðlÞ Þ j l ¼ jnk ðsee Eq:3 for j; kÞ (WDE) ð5Þ WDE is a bi-population based, iterative, evolutionary where index ¼ 1 : N and index 2 Zþ . In Eq.5, w ¼ search algorithm developed to solve real-valued numerical j3ðNÞ j ½N; 1 ¼ sizeðw Þ where w :¼ Pw w and w ¼ w optimization problems. WDE has been designed as a global D where D ¼ ½1ð1;DÞ . minimizer algorithm. WDE can perform bounded or 123 Neural Computing and Applications WDE updates the initial Mð1:N;1:DÞ ¼ 0 in each iteration ½gmin; gbest ¼ ½fitPðcÞ ; PðcÞ j fitPðcÞ ¼ minðfitPÞ ; c 2 i by using Eq. 6, and uses the updated M to control the ð15Þ mutation process; The pseudo-code of the WDE is given in Fig. 1. The values M ðindex;JÞ :=1 ð6Þ of the parameters used in WDE can be determined ran- In Eq. 6, J ¼ V ð1 : dK DeÞ j V ¼ permute ðj0Þ: K is domly. In that case, WDE theoretically has no control defined with the rule in Eq. 7; parameter. Since the values of the parameters used in WDE are determined randomly, WDE has no parameter tuning If a\b then K ¼ j3ð1Þ else K ¼ ð1 j3ð1Þ Þ ð7Þ process. Therefore, it is easy to use. In Eq. 7, a; b; j Uð0; 1Þ; ðÞ ¼ sizeðjð:Þ . The novelties of WDE introduced herein are as follows: In WDE, jðÞ generates ðÞ-sized real-valued uniform • The swarming process of WDE uses new stochastic random number, each time it is used. In WDE, the evolu- mutation control mechanism. tionary step size (i.e., scale factor), F, is computed by using • WDE’ s system equation is partially similar to the the rule given in Eq. 8; system equation of differential evolution algorithm if a\b thenF ¼ ½k3ðDÞ 0 j ½|{z} D ¼ sizeðFÞ 1 ; |{z} [15], but WDE’ s direction vector generation strategy is rows columns different. • The direction vectors generated in WDE are composed elseF ¼ ðk3ðNÞ DÞ j ½|{z} D ¼ sizeðFÞ N ; |{z} of the mixed vectors of different pattern vectors. rows columns • WDE uses a new method for boundary control. ð8Þ • The values of all parameters used in WDE are In WDE, trial vector, T, is generated by using Eq. 9; determined randomly. Therefore, WDE does not waste time to initial parameter tuning. T ¼ SubP þ F M TempP SubPðmÞ • Since it has a non-recursive structure, WDE can be ð9Þ jm ¼ permuteðiÞjm 6¼ ½1 : N parallelized easily. Therefore, it is rather fast. here, i ¼ 1 : N j i 2 Zþ . T 62 ½ low up values are updated Similarities and differences between WDE and comparison by using Eqs. 10 and 11; algorithms are listed below: • WDE does not partially or totally act elitist like ABC, if Tði;j0Þ \lowðj0Þ then 3 ð10Þ CS, and JADE. Tði;j0Þ ¼ lowðj0Þ þ jð1Þ upðj0Þ lowðj0Þ • WDE uses sub-populations for interacting pattern matrices. if Tði;j0Þ [ upðj0Þ ð11Þ • Mutation and crossover strategies of WDE are different then Tði;j0Þ ¼ upðj0Þ þ j3ð1Þ lowðj0Þ upðj0Þ than those of comparison algorithms. The objective function values of Ti¼1:N vectors are com- • WDE is a bijective search algorithm like BSA. puted by using Eq. 12; • WDE does not have any control parameters. • Boundary control mechanism of WDE is unique to fitT ¼ F ðTÞ ð12Þ itself. • Though basic system equations of WDE are somehow T and fitT are used in order to update SubP and fitSubP as similar to DE/rand/1/bin [15], BSA, ABC, and CS, per the greedy-selection rule. The relevant updating pro- usage style of these equations are quite different than cess of WDE is defined by using the rule given in Eq. 13; other algorithms. if fitTði Þ \fitSubPði Þ then ½SubPði Þ ; fitSubPði Þ : • WDE is structurally non-recursive as different from ¼ ½Tði Þ ; fitTði Þ j i 2 i DE/rand/1/bin [15], ABC, and JADE [37]. Therefore, it can be parallelized without being modified. ð13Þ • Functioning of WDE is analogically based on the The updated SubP and fitSubP are used in order to update cooperation of bio-interacting sub-populations. PðlÞ and fitPðlÞ values. The relevant updating process of • Functioning of WDE may be explained with the usage WDE is shown in Eq. 14; of bio-inspired evolutionary optimization processes (i.e., initialization, selection, mutation, and recombina- ½PðlÞ ; fitPðlÞ :¼ ½SubP; fitSubP ð14Þ tion/crossover) just like other EAs. Here, see Eq. 5 for l. WDE achieves the searched global solution by using Eq. 15; 123 Neural Computing and Applications Fig. 1 Pseudo-code of the WDE. The unoptimized MATLAB code of the WDE is publicly available at [40] 4 Experiments time. The same initial population has been used for each algorithm in the experiments. Dimension of pattern matrix Numerical optimization problem-solving capability of is 30, and stopping conditions used in the experiments are WDE that is introduced in this paper is examined by using given below: CEC’ 2013s benchmark problems (i.e., F1–F28) [36] that 1. Stop if the absolute value of the solution obtained for consist of very complex problems. the algorithm is smaller than 1016 . In this paper, in order to examine the success of WDE in 2. Stop if a better solution at the end of the last 200,000 the solution of real-world engineering problems, one geo- function evaluations has not been obtained. metric optimization problem (i.e., GPS network adjustment 3. Stop when the function evaluation number reaches to problem ), F29 [41–43], and 4 engineering design problems 2,000,000. have been used [2, 31]. Benchmark problems (i.e., F1–F28) have been solved In the tests performed in this paper, solutions obtained with for 50 trials by using a different initial population each WDE and comparison algorithms were pairwisely 123 Neural Computing and Applications compared by using two-tailed Wilcoxon signed-rank test benchmark function number that the related comparison [44]. For Wilcoxon signed-rank tests that have been carried algorithm obtains a statistically better result than WDE. out in this paper, H0 hypothesis is defined as ‘data come In the solution of CEC’ 2013 benchmark problems, from distributions with equal medians.’ ‘Significance level’ when results of WDE and comparison algorithms were is used as a ¼ 0:05. If a corrected p value lower than or examined in (þ; ¼; ) format, the following results are equal to a value is produced in a test, then H0 hypothesis is obtained as given in Table 3: CS (22,1,5), ABC (18,6,4), rejected for that test. Alternative hypothesis is determined JADE (22,4,2), BSA (13,8,7). Accordingly, WDE had as FA \FB . The validity of the alternative hypothesis is statistically better results (66.96%) than comparison algo- decided on by looking at whether Algorithm A provides a rithms in 75 out of a total of 112 piecewise comparisons. statistically better solution than Algorithm B or not, and the Successes of WDE and comparison algorithms are statis- sizes of ranks provided by Wilcoxon signed-rank test (i.e., tically similar in 19 (16.96%) comparisons. Comparison Rþ ; R as in [43]). algorithms achieved statistically better results than WDE Initial values of control parameters of the proposed only in 18 comparisons (16.07%). algorithm and the comparison algorithms used in this paper are given in Table 1. 4.2 GPS baseline network adjustment problem (F29) 4.1 Numerical function optimization problems: F1–F28 GPS network adjustment problem is a geometric opti- mization problem widely encountered in Geodesy [41–43]. In this section, success of WDE in numerical function In this paper, in order to review WDE’s capability of optimization problems has been examined with detailed solving the geometric optimization problems, the GPS applications. Basic statistical evaluations of the results (i.e., Network defined in [41] has been adjusted by using WDE . mean value of global minimum values (Ave), standard The relevant GPS network includes 6 geodesic points deviation value of global minimum values (Std), and run- defined as pointi ¼ hx; y; zi j i ¼ 1 : 6. point 1;2 are the time value in seconds (t (s)) obtained in the test performed fixed values in solution of the GPS network adjustment by using F1–F28 are given in Table 2. The best ‘Ave’ problem. points 1;2 are given in Table 4. The baseline values are marked with bold font in Table 2. values of the relevant GPS Network are given in Table 5. Mersenne Twister has been used in the tests as pseudo- The covariance values of the relevant observations are random number generator [45]. necessary for least squares adjustment (LSA) [41]. The Results that belong to the comparison of CEC’ 2013 related covariance values are given in page 323 of [41] for [36] benchmark problem (i.e., F1–F28)-solving successes F29. WDE does not need covariance values in order to of WDE and comparison algorithms by using Wilcoxon adjust the GPS network. On the relevant GPS Network, 13 signed-rank test [39] (p ¼ 0:05Þ are given in Table 3. baselines have been observed. Therefore, totally 13 3 ¼ On the last row of Table 3, results obtained from WDE 39 observation equations have been acquired. points 3:6 and comparison algorithms have been compared as have totally 4 3 ¼ 12 unknowns. Consequently, the rel- (þ; ¼; ) where (?) is the benchmark function number evant GPS network includes 39 12 ¼ 27 redundant that WDE obtains a statistically better result than the observations. Therefore, in this problem related comparison algorithm, (=) is the benchmark func- degree-of-freedom ¼ 27 [41]. In the adjustment process, tion number that the performances of WDE and the related the optimal values of totally 12 coordinate values of comparison algorithm are statistically equal and (-) is the point 3 : 6 have been searched. Therefore, the problem dimension of the GPS baseline network adjustment prob- lem is 12. According to definition, C1:6 ¼ ½ point 1 point 2 . . . point 6 T : In the objective Table 1 Initial values of control parameters of the proposed algorithm and the comparison algorithms function used for WDE, the baseline values have been computed by using # Algorithm Initial values of control parameters f ¼ Cfrom Cto j ffrom; tog 2 f1 : 6g. The relevant 1 ABC [1] Limit ¼ N D Sizeofempoyedbee ¼ Sizeofcolony=2 residual values, m, have been acquired by using Eq. 16; 2 CS [3, 8] b ¼ 1:50; p0 ¼ 0:25 v ¼ f ½DxDyDz 1 : 13 ð16Þ 3 JADE p1 ¼ p2 ¼ 0:30 jj j U(0,1) [37] In order to protect the centeroid of the GPS network during 4 BSA [7] mixrate ¼ 1:00 the search process, v :¼ v meanðvÞ update has been made. mean(v) denotes the mean values of the residuals. 123 Table 2 Results that belong to tests carried out by using F1–F28 F CS ABC JADE BSA WDE Ave Std t (s) Ave Std t (s) Ave Std t (s) Ave Std t (s) Ave Std t (s) F1 1400.000 8.81e-14 3.16 1400.000 0.00 4.99 1400.000 2.27e-13 8.75 1400.000 8.81e-14 2.26 1400.000 0.00 10.67 F2 9.38eþ05 4.71eþ05 51.97 2.56eþ06 7.87eþ05 36.68 2.18eþ05 1.38eþ05 413.83 6.84eþ03 5.51eþ03 54.30 21300.000 1.44e-13 65.86 Neural Computing and Applications F3 1.36eþ08 2.55eþ08 53.21 1.14eþ07 8.56eþ06 35.42 1.42eþ08 1.80eþ08 408.01 2.11e104 7.09eþ04 53.81 1.13eþ05 1.14eþ05 108.10 F4 219.507 1.47eþ03 48.45 38,231.404 4.22eþ03 30.67 -1073.517 4.59eþ01 649.37 1100.000 1.100e-06 51.98 1100.000 1.24e-13 68.67 F5 1000.000 9.51e-14 6.32 1000.000 1.19e-13 17.03 1000.000 1.22e-13 59.80 1000.000 1.14e-13 1.83 1000.000 0.00 17.46 F6 - 896.408 3.41 43.06 - 899.916 1.03e-01 25.88 - 887.471 2.09eþ01 573.27 900.000 1.08e-13 13.47 900.000 0.00 51.42 F7 - 728.928 1.18eþ01 37.96 - 750.511 1.16eþ01 32.56 - 756.180 1.92eþ01 548.48 786.287 7.36 62.24 - 783.078 5.03 120.76 F8 - 679.283 4.39e-02 5.57 - 679.273 5.19e-02 25.54 - 679.305 7.19e-02 24.01 - 679.287 9.17e-02 4.67 679.313 7.82e-02 16.14 F9 - 583.951 1.57 68.73 - 584.044 1.20 30.17 - 587.429 2.24 212.90 2 587.909 2.04 131.55 - 585.635 1.08 174.67 F10 - 499.995 5.73e-03 49.98 - 499.791 6.98e-02 49.21 - 497.559 2.04 35.04 - 499.948 5.57e-02 14.25 500.000 2.46e-09 108.93 F11 - 399.851 3.55e-01 15.50 400.000 4.58e-14 10.08 - 394.180 6.01 18.58 400.000 1.80e-14 3.71 400.000 3.22e-11 112.48 F12 - 231.668 2.47eþ01 60.74 - 211.882 1.98eþ01 33.90 - 240.141 1.62eþ01 233.69 - 268.261 6.68 16.17 283.416 3.37 116.99 F13 - 126.694 1.36eþ01 50.75 - 80.966 1.47eþ01 31.34 - 113.184 2.35eþ01 189.50 - 135.234 1.16eþ01 18.09 170.488 9.32 115.97 F14 - 56.783 4.90eþ01 53.47 99.887 4.11e-02 41.71 163.811 1.36eþ02 44.51 - 96.799 8.21 22.57 - 99.463 5.60e-01 112.44 F15 2241.617 3.71eþ02 37.39 1798.467 2.26eþ02 29.79 2145.751 3.74eþ02 199.12 1869.153 2.67eþ02 44.67 1705.872 2.27eþ02 114.41 F16 200.937 1.58e-01 15.55 200.798 9.43e-02 33.58 201.145 3.09e-01 114.86 200.587 1.13e-01 39.30 200.368 7.89e-02 138.06 F17 320.589 1.82 52.59 320.342 2.06e-03 41.09 320.436 1.16e-01 55.44 318.331 6.03 20.13 310.350 1.00eþ01 108.94 F18 475.076 1.28eþ01 41.07 526.096 1.43eþ01 30.32 492.532 1.89eþ01 179.49 444.290 6.49 59.99 431.403 2.45 111.91 F19 501.719 1.98e-01 46.49 500.076 3.49e-02 40.30 509.711 5.48 221.27 500.287 1.28e-01 50.64 500.450 9.12e-02 107.32 F20 607.551 6.84e-01 23.69 609.447 3.24e-01 25.07 607.080 1.52 103.74 606.164 8.00e-01 55.34 606.628 5.82e-01 114.02 F21 895.832 6.70eþ01 49.59 819.902 2.11eþ01 27.94 1040.000 8.00eþ01 19.53 1040.000 9.17eþ01 5.84 855.007 5.89eþ01 130.57 F22 1137.105 2.41eþ02 85.81 808.500 5.85 57.17 1046.627 1.21eþ02 144.97 800.000 1.08e-13 54.01 816.969 3.99 142.86 F23 3780.819 3.61eþ02 64.72 3499.165 3.10eþ02 33.36 3453.046 4.08eþ02 160.52 3144.507 2.23eþ02 78.81 2898.065 3.15eþ02 145.00 F24 1245.374 6.03 134.78 1231.720 1.43eþ01 37.04 1239.778 5.65 292.58 1236.889 5.49 144.50 1244.293 4.90 223.27 F25 1353.138 4.31 103.14 1357.813 4.74 32.37 1345.338 6.14 290.54 1344.466 4.66 146.44 1352.629 4.53 223.59 F26 1400.201 1.20e-01 209.75 1397.840 6.43 72.99 1420.577 4.89eþ01 300.96 1400.002 2.36e-04 195.24 1398.931 4.66 164.95 F27 2056.377 2.63eþ01 94.06 1700.000 7.49e-13 45.35 1909.972 6.30eþ01 299.81 1938.630 3.89eþ01 171.58 1959.547 1.12eþ02 242.51 F28 2696.670 2.81eþ02 105.38 2648.656 6.04eþ02 37.01 2870.237 2.72eþ02 236.45 2078.173 5.43eþ02 44.16 1712.140 3.72eþ02 154.15 123 Table 3 Comparison of CEC’ 2013 benchmark problem (i.e., F1–F28)-solving successes of WDE and comparison algorithms by using Wilcoxon signed-rank test (p ¼ 0:05) 123 Fnc CS ABC JADE BSA p value zval Ranksum Winner p value zval Ranksum Winner p value zval Ranksum Winner p value zval Ranksum Winner F1 4.02E-02 - 1.75 380 - 2.34E-10 - 6.23 210 ? 9.64E-06 - 4.27 280 ? 4.02E-02 - 1.75 380 - F2 1.98E-08 - 5.49 210 ? 1.98E-08 - 5.49 210 ? 1.98E-08 - 5.49 210 ? 1.98E-08 - 5.49 210 ? F3 7.88E-07 - 4.80 232 ? 3.40E-08 - 5.40 210 ? 9.59E-08 - 5.21 217 ? 1.00 4.56 578 = F4 1.65E-08 - 5.52 210 ? 1.65E-08 - 5.52 210 ? 1.65E-08 - 5.52 210 ? 1.65E-08 - 5.52 210 ? F5 1.86E-09 - 5.90 210 ? 1.63E-09 - 5.92 210 ? 1.57E-09 - 5.92 220 ? 2.34E-10 - 6.23 210 ? F6 1.50E-08 - 5.54 220 ? 4.00E-09 - 5.77 210 ? 4.00E-09 - 5.77 210 ? 2.73E-09 - 5.83 210 ? F7 3.40E-08 - 5.40 210 ? 3.95E-08 - 5.37 211 ? 2.27E-07 - 5.04 223 ? 9.55E-01 1.69 472 - F8 1.14E-01 - 1.20 365 - 5.99E-02 - 1.56 352 - 4.09E-01 - 0.23 401 - 7.39E-02 - 1.45 356 - F9 6.14E-04 - 3.23 290 ? 1.69E-04 - 3.58 277 ? 9.95E-01 2.58 505 = 1.00 3.83 551 = F10 2.19E-02 - 2.02 335 ? 3.40E-08 - 5.40 210 ? 3.40E-08 - 5.40 210 ? 3.39E-08 - 5.40 210 ? F11 1.00 3.89 550 = 1.00 5.53 610 = 7.84E-06 - 4.32 250 ? 1.00 5.69 610 = F12 3.40E-08 - 5.40 210 ? 3.40E-08 - 5.40 210 ? 3.40E-08 - 5.40 210 ? 1.11E-07 - 5.18 218 ? F13 4.59E-08 - 5.34 212 ? 3.40E-08 - 5.40 210 ? 3.40E-08 - 5.40 210 ? 6.17E-08 - 5.29 214 ? F14 3.40E-08 - 5.40 210 ? 1.00 5.40 609 = 3.40E-08 - 5.40 210 ? 9.80E-01 2.05 485 - F15 5.19E-05 - 3.88 266 ? 1.14E-01 - 1.20 365 - 1.37E-04 - 3.64 275 ? 2.83E-02 - 1.91 339 - F16 3.40E-08 - 5.40 210 ? 3.40E-08 - 5.40 210 ? 3.40E-08 - 5.40 210 ? 1.03E-06 - 4.75 234 ? F17 1.48E-07 - 5.13 220 ? 5.05E-E-01 0.01 410 - 8.12E-04 - 3.15 293 ? 6.35E-01 0.34 422 - F18 3.40E-08 - 5.40 210 ? 3.40E-08 - 5.40 210 ? 3.40E-08 - 5.40 210 ? 9.59E-08 - 5.21 217 ? F19 3.40E-08 - 5.40 210 ? 1.00 5.42 610 = 3.40E-08 - 5.40 210 ? 1.00 3.77 549 = F20 1.11E-04 - 3.69 273 ? 3.40E-08 - 5.40 210 ? 2.20E-01 - 0.77 381 - 9.40E-01 1.56 467 - F21 3.47E-01 - 0.39 395 - 7.95E-01 0.83 440 - 9.93E-06 - 4.27 254 ? 1.09E-05 - 4.25 254 ? F22 3.40E-08 - 5.40 210 ? 1.00 4.34 570 = 3.40E-08 - 5.40 210 ? 1.00 5.49 610 = F23 4.56E-07 - 4.91 228 ? 5.52E-06 - 4.40 247 ? 6.47E-05 - 3.83 268 ? 6.66E-03 - 2.48 318 ? F24 3.38E-01 - 0.42 394 - 9.99E-01 3.02 521 = 9.96E-01 2.66 508 = 1.00 3.96 556 = F25 4.30E-01 - 0.18 403 - 2.35E-03 - 2.83 305 ? 1.00 3.72 547 = 1.00 4.29 568 = F26 1.22E-08 - 5.58 210 ? 3.08E-04 - 3.42 287 ? 1.22E-08 - 5.58 210 ? 1.22E-08 - 5.58 210 ? F27 1.75E-06 - 4.64 238 ? 1.00 5.45 610 = 9.97E-01 2.77 512 = 9.97E-01 2.77 512 = F28 2.24E-07 - 5.05 223 ? 4.83E-06 - 4.42 246 ? 8.18E-08 - 5.24 216 ? 2.18E-02 - 2.02 335 ? Total ? 22 18 22 13 = 1 6 4 8 - 5 4 2 7 Neural Computing and Applications Neural Computing and Applications Table 4 Coordinates of fixed points; 1, 2 [41] for p value\0:05. In that case, the mean of errLSA and Points X (m) Y (m) Z (m) errWDE belongs to an equal population. Therefore, LSA and WDE have reached statistically the same success for the 1 402.35087 - 4,652,995.30109 4,349,760.77753 Derrloop values in solution of the GPS network adjustment 2 8086.03178 - 4,642,712.84739 4,360,439.08326 problem. The mean and standard deviation values of the residuals obtained after completion of the adjustment process are given in Table 8. The objective function used by WDE in order to adjust the After a review of Table 8, it is possible to say that the GPS Network is given in Eq. 17; GPS network adjustment results achieved by using WDE argmin vT v ð17Þ protect the centeroid of the relevant network better than point3 : 6 LSA. While solving the GPS network adjustment problem by The standard deviation values, m0 , of the residuals using WDE, the pattern matrix dimension has been selected computed for the baselines are given in Table 9. as N ¼ 20. The problem dimension is D ¼ 12. WDE does After a review of Table 9, it can be said that WDE has not use the search space limits while solving the GPS obtained a better m0 value compared to LSA. Therefore, network adjustment problem. WDE is successful in adjustment of the relevant GPS The initial pattern matrix has been generated by using network. Pði0;j0Þ Uð0; 1Þ. The global minimum value WDE has In solution of the GPS network adjustment problem, acquired in the final iteration is 0.0026484. WDE provides the following advantages in comparison to The coordinates of the adjusted network points obtained LSA: with solution of the GPS network adjustment problem and – There is no statistical difference between the results the relevant coordinate differences are given in Table 6. obtained in solution of the GPS network adjustment When the adjusted-coordinates given in Table 6 have problem by using WDE or LSA. However, in order to been compared by using Anova test, p value ¼ 0:00000 achieve the solution, LSA uses a mathematical model has been obtained for X, Y, and Z. Therefore, the means that is much more complex than the mathematical values of the relevant adjusted-coordinates obtained by model used by WDE. using LSA and WDE for p value \0:05 are statistically – In order to solve the relevant problem, LSA needs equal. The loop closure values, Derrloop , computed after the covariance values. However, WDE uses only the adjustment are given in Table 7. observed baseline parameters and fixed point parame- When the Derrloop values have been compared statisti- ters to solve the relevant problem. cally by using Anova test, p value ¼ 4:2603e08 has been – WDE protects the centeroid of the relevant network obtained. In that case, it can be said that the differences more accurately. between errLSA and errWDE are not statistically significant Table 5 The baseline values Observation Points Baseline components (m) used in GPS baseline network adjustment problem (i.e., F29) From To Dx Dy Dz [41] 1 1 3 11,644.22320 3601.21650 3399.25500 2 1 5 - 5321.71640 3634.07540 3173.66520 3 2 3 3960.54420 - 6681.24670 - 7279.01480 4 2 4 - 11,167.60760 - 394.52040 - 907.95930 5 4 3 15,128.16470 - 6286.70540 - 6371.05830 6 4 5 - 1837.74590 - 6253.85340 - 6596.66970 7 6 1 - 1116.45230 - 4596.16100 - 4355.89620 8 6 3 10,527.78520 - 994.93770 - 956.62460 9 6 5 - 6438.13640 - 962.06940 - 1182.23050 10 6 4 - 4600.37870 5291.77850 5414.43110 11 6 2 6567.23110 5686.29260 6322.39170 12 2 6 - 6567.23100 - 5686.30330 - 6322.38070 13 1 6 1116.45770 4596.15530 4355.91410 123 Neural Computing and Applications – WDE is capable of obtaining better m0 values than ZLSA ZWDE LSA. 0.00000 0.00000 0.00743 0.00324 0.00404 0.00257 4.3 Engineering design problems: F29–F32 YLSA YWDE F29–F32 are nonlinear constrained engineering optimiza- Coordinate differences (m) 0.00000 0.00000 0.00226 0.00181 0.00212 0.00120 tion problems, and they were widely analyzed in opti- mization literature [2, 31]. Therefore, F29–F32 also were solved in this study: XLSA XWDE – Pressure-vessel design problem, F29 [2, 31], Table 6 The coordinates of the adjusted network points obtained with solution of the GPS network adjustment problem and the coordinate differences – Speed-reducer design problem, F30 [2], 0.00000 0.00000 0.00238 0.00210 0.00568 0.00241 – Tension/compression string design problem, F31 [2, 31], – Welded-beam design problem, F32 [2, 31], 4,349,760.77,753 4,360,439.08326 4,353,160.05592 4,359,531.11878 4,352,934.44937 4,354,116.68679 In this paper, the constrained handling method [2, 45, 46] used in [2] is employed to solve F29–F32 problems. In the solution of F29–F32, problem-solving successes Z (m) of particle swarm optimization algorithm (PSO2011) [2, 13], cumulative-information-based differential evolu- Adjusted-coordinates by using WDE (m) tion algorithm (CPI-JDE) [38], Advanced Artificial - 4,642,712.84,739 - 4,652,995.30109 - 4,649,394.08482 - 4,643,107.37096 - 4,649,361.22199 - 4,648,399.14653 Cooperative Search Algorithm (A?) [39], CS, ABC, JADE, BSA, and WDE (proposed) were compared. CPI- JDE and JADE are relatively new, and they are very suc- cessful DE algorithms. Y (m) Statistical results that WDE and comparison algorithms have obtained for F29–F32 are given in Table 10. When Table 10 is examined, WDE is seen to have obtained better 402.35087 8086.03178 12,046.57838 - 3081.58523 - 4919.34476 1518.79878 results than PSO2011, CPI-JADE, ABC, and A?. Related problem-solving successes of WDE, JADE, BSA, and CS X (m) are statistically similar to a large extent. The detailed definitions of the mechanical-engineering- based nonlinear constrained optimization problems (i.e., 4,349,760.77753 4,360,439.08326 4,353,160.06335 4,359,531.12202 4,352,934.45341 4,354,116.68936 F29–F32) are given in Sects. 4.4, 4.5, 4.6, and 4.7. 4.4 Pressure-vessel design problem: F29 Z (m) This problem aims the design of a compressed air storage Adjusted-coordinates by using LSA (m) tank under a working pressure of 3000 psi and a minimum - 4,652,995.30109 - 4,642,712.84739 - 4,649,394.08256 - 4,643,107.36915 - 4,649,361.21987 - 4,648,399.14533 volume of 750 ft3 , the schematic structure of which is shown in Fig. 2. The mathematical model of this problem is defined in Eq. 18 [2, 31]. Y (m) argmin f ðxÞ ¼ 0:6224 x1 x3 x4 þ 19:84 x21 x3 x þ 1:7781 x2 x33 þ 3:1661 x21 x4 402.35087 8086.03178 12,046.58076 - 3081.58313 - 4919.33908 1518.80119 ð18Þ X (m) Subject to: Point# 1 2 3 4 5 6 123 Neural Computing and Applications Table 7 Post-adjustment loop Loop# Loop edges errLSA (1) errWDE Derrloop closure values (in m); Derrloop ¼ errLSA errWDE 1 ! ! ! 0.03770 0.04663 -0.00893 1; 3 ; 6; 3 ; 6; 1 2 ! ! ! ! 0.03210 0.03210 0.00000 1; 3 ; 2; 3 ; 6; 2 ; 6; 1 3 ! ! ! ! 0.03930 0.03930 0.00000 1; 3 ; 2; 3 ; 2; 6 ; 1; 6 4 ! ! ! 0.04900 0.05793 - 0.00893 1; 5 ; 6; 5 ; 6; 1 5 ! ! ! ! 0.06240 0.06240 0.00000 1; 5 ; 4; 5 ; 6; 4 ; 6; 1 6 ! ! ! ! ! 0.08030 0.08512 - 0.00482 1; 3 ; 2; 3 ; 2; 4 ; 4; 5 ; 1; 5 7 ! ! ! 0.02540 0.02595 - 0.00055 4; 5 ; 6; 4 ; 6; 5 8 ! ! ! ! 0.03260 0.03370 - 0.00110 4; 5 ; 2; 4 ; 6; 2 ; 6; 5 9 ! ! ! 0.00980 0.01173 - 0.00193 2; 4 ; 6; 2 ; 6; 4 10 ! ! ! 0.01620 0.01138 0.00482 2; 4 ; 2; 6 ; 6; 4 11 ! ! ! 0.03660 0.04142 - 0.00482 2; 4 ; 2; 3 ; 4; 3 12 ! ! ! ! 0.04350 0.04350 0.00000 1; 5 ; 6; 5 ; 6; 3 ; 1; 3 13 ! ! 0.03260 0.03370 - 0.00110 2; 6 ; 6; 2 14 ! ! ! ! 0.02240 0.02240 0.00000 2; 4 ; 2; 3 ; 6; 3 ; 6; 4 ; 15 ! ! ! ! ! 0.03680 0.04162 - 0.00482 6; 5 ; 4; 5 ; 2; 4 ; 2; 3 ; 6; 3 16 ! ! ! ! 0.03280 0.03280 0.00000 6; 5 ; 4; 5 ; 4; 3 ; 6; 3 17 ! ! 0.02900 0.00856 0.02044 1; 6 ; 6; 1 18 ! ! ! ! 0.01720 0.01720 0.00000 1; 6 ; 2; 6 ; 6; 2 ; 6; 1 Table 8 Mean and standard deviation values of residuals 0:0625 x1 ; x2 6:1875 ð20Þ Residuals LSA WDE and l (m) r l (m) r 10 x3 ; x4 200 ð21Þ vx - 0.0000669 0.0093955 0.0000000 0.0085625 vy 0.0017000 0.0051921 0.0000000 0.0051045 vz 0.0000746 0.0113934 0.0000000 0.0096674 4.5 Speed-reducer design problem: F30 This problem aims the design of the speed reducer shown in Fig. 3. The mathematical model of this problem is defined in Table 9 m0 values of the residual values obtained by using LSA and WDE Eq. 22 [2, 31]. Statistics Method argmin f ðxÞ ¼ 0:7854 x1 x22 ð3:3333 x23 þ 14:9334 x LSA (m) WDE (m) x3 43:0934Þ qffiffiffiffiffi vT v 0.00833 0.00735 1:508 x1 ðx26 þ x27 Þ þ 7:4777 ðx26 þ x27 Þ m0 ¼ 27 þ 0:78054 ðx4 x26 þ x5 x27 Þ ð22Þ c1 ðxÞ ¼ x1 þ 0:0193 x3 0 Subject to c2 ðxÞ ¼ x2 þ 0:0954 x3 0 4 ð19Þ c3 ðxÞ ¼ p x23 x4 p x33 þ 1; 296; 000 0 3 c4 ðxÞ ¼ x4 240 0 where the bounds are, 123 Neural Computing and Applications Table 10 The mean-solution (Ave) and standard deviation of mean- solution (Std) values obtained by the relevant algorithms for the F29– F32 Algorithm F Ave Std PSO2011 F29 6235.36961742 2.51e?02 F30 3108.37805916 1.53e?02 F31 0.01275660 1.12e-04 F32 1.86762054 1.16e-01 CPI-JDE F29 6310.75991379 2.23e?02 F30 3004.18039468 4.25e?01 F31 0.01368178 1.00e-03 F32 2.55281872 4.32e-01 A? F29 5885.33277360 1.70e-13 F30 2994.92524435 6.00e-13 Fig. 3 Speed-reducer problem F31 0.01267052 1.27e-05 F32 1.72485231 0.00e?00 27 CS F29 5910.20912757 1.45e?01 c1 ðxÞ ¼ 1 0 x1 x22 x3 F30 2994.92524435 4.90e-13 397:5 F31 0.01272432 2.68e-05 c2 ðxÞ ¼ 1 0 x1 x22 x23 F32 1.77712735 2.10e-02 ABC F29 6920.90919830 4.30e?02 1:93 x34 c3 ðxÞ ¼ 1 0 F30 3089.68804116 5.21e?01 x2 x3 x46 F31 0.01342275 4.39e-04 1:93 x35 c4 ðxÞ ¼ 1 0 F32 1.89243016 1.05e-01 x2 x3 x47 JADE F29 5885.33277360 0.00e?00 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 750:0 x4 2 F30 2994.92524435 4.60e-13 c5 ðxÞ ¼ þ16:9 106 1 0 F31 0.01266523 0.00e?00 110 x36 x2 x3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi F32 1.72485231 0.00e?00 1 750:0 x5 2 BSA F29 5885.33277360 0.00e?00 c6 ðxÞ ¼ þ157:5 106 1 0 85 x37 x2 x3 F30 2994.92524435 4.60e-13 x2 x3 F31 0.01266523 0.00e?00 c7 ðxÞ ¼ 1 0 40 F32 1.72485231 0.00e?00 5 x2 WDE (proposed) F29 5885.33277360 0.00e?00 c8 ðxÞ ¼ 1 0 x1 F30 2994.92524435 4.60e-13 x1 c9 ðxÞ ¼ 1 0 F31 0.01266523 0.00e?00 12 x2 F32 1.72485231 0.00e?00 1:5 x6 þ 1:9 c10 ðxÞ ¼ 1 0 x4 1:5 x7 þ 1:9 c11 ðxÞ ¼ 1 0 x5 ð23Þ where the bounds are, 2:6 x1 3:6 0:7 x2 0:8 17 x3 28 7:3 x4 8:3 ð24Þ 7:8 x5 8:3 2:9 x6 3:9 5:0 x7 5:5 Fig. 2 The schematic structure of pressure-vessel design problem 123 Neural Computing and Applications Fig. 4 Tension/compression spring problem 4.6 Tension/compression spring design problem: F31 Fig. 5 The schematic structure of welded-beam design problem The aim of this problem is to determine the values of the relevant parameters minimizing the weight of a ten- sion/compression spring under various constraints c1 ðxÞ ¼ tðxÞ tmax 0 [2, 31, 46, 47]. The schematic structure of a tension/com- c2 ðxÞ ¼ rðxÞ rmax 0 pression spring is shown in Fig. 4. c3 ðxÞ ¼ x1 x4 0 The mathematical model of this problem is defined in c4 ðxÞ ¼ 0:10471 x21 þ 0:04811 x3 x4 ð14 þ x2 Þ Eq. 25 [2, 32]. ð29Þ 5 0 argmin f ðxÞ ¼ ðx3 þ 2Þ x21 x2 ð25Þ c5 ðxÞ ¼ 0:125 x1 0 x c6 ðxÞ ¼ dðxÞ dmax 0 Subject to c7 ðxÞ ¼ P Pc ðxÞ 0 x32 x3 c1 ðxÞ ¼ 1 0 where 71785 x41 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4x22 x1 x2 1 2 t1 t2 x2 c2 ðxÞ ¼ þ 1 0 tðxÞ ¼ t12 þ þ t22 12566 x31 x2 x41 5108 x1 ð26Þ 2R 140:45 x1 P c3 ðxÞ ¼ 1 0 t1 ¼ pffiffiffi x22 x3 2 x1 x2 x1 þ x2 MR c4 ðxÞ ¼ 1 0 t2 ¼ 1:50 J x2 where the bounds are M ¼P Lþ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:05 x1 2:0 x22 x1 þ x3 2 R¼ þ 0:25 x2 1:3 ð27Þ 4 2 2 2:0 x3 15:0 pffiffiffi x2 x1 þ x3 2 J ¼2 2 x1 x2 þ 12 2 6PL 4.7 Welded-beam design problem: F32 rðxÞ ¼ x4 x23 4 P L3 This problem aims dimensioning of the welded steel beam dðxÞ ¼ E x4 x33 [2, 31], the structure of which is shown in Fig. 5, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi determining the welding length value. rffiffiffiffiffiffiffiffiffiffi! 4:013 E G x23 x64 =36 x3 E The mathematical model of this problem is given in Pc ðxÞ ¼ 1 L2 2L 4G Eq. 28; ð30Þ argmin f ðxÞ ¼ 1:10471 x21 x2 þ 0:04811 x3 x4 ð14 þ x2 Þ x The material properties and constraint values used above ð28Þ are given as follows: Subject to 123 Neural Computing and Applications P ¼ 6000 processing applications, and various industrial design L ¼ 14 problems. dmax ¼ 0:25 E ¼ 30 106 ð31Þ 6 Compliance with ethical standards G ¼ 12 10 tmax ¼ 13; 600 Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this paper. rmax ¼ 30; 000 Funding The authors would like to thank the editor and the referees where the bounds are, for their contribution in enhancing the technical quality of this paper. 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