A New Adaptive Real-coded Memetic Algorithm

Hadi Nobahari Department of Aerospace Engineering

Sharif University of Technology Tehran, Iran

nobahari@sharif.edu

Davoud Darabi Department of Aerospace Engineering

Sharif University of Technology Tehran, Iran

ddarabi@ae.sharif.ir

Abstract— A new adaptive real-coded memetic algorithm has been developed for continuous optimization problems. The proposed algorithm utilizes an adaptive variant of Continuous Ant Colony System for local search. Here new adaptive strategies are utilized for online tuning of the number of local search steps and the width of the search interval over each dimension of the search space. A new crossover scheme is also developed and utilized. The new algorithm has been examined with CEC 2005 benchmarks and compared with three other state of the art works in the field. The comparisons have showed relatively better results.

Keywords- Adaptive Local Search, Real-coded, Memetic Algorithm, Steady State Genetic Algorithm, Continuous Ant Colony System.

I. INTRODUCTION It is established that pure Genetic Algorithms (GAs) can

be made more effective by hybridizing with other techniques [1-3]. GAs that have been combined with local search (LS) are called Memetic Algorithms (MAs) [1], [3], [4]. MAs are evolutionary algorithms that apply a separate LS process to refine new individuals. An important aspect concerning MAs is the trade-off between the exploration abilities of the GA, and the exploitation abilities of the LS that used [1], [3].

Under the initial formulation of GAs, the search space solutions are coded using the binary alphabet, however, other coding methods, such as real-coding, have also been taken into account to deal with the representation of the problem [1]. The use of real parameters makes it possible to use large domains for the variables, which is difficult to achieve in binary implementations. Another advantage when using real parameters is the ability to exploit the graduality of a continuous function. Moreover, using real coding the representation of the solutions is very close to the natural formulation of many problems. Therefore, the coding and decoding processes are avoided [5].

In Real-coded GA, decision variables are used directly (without coding) to form a chromosome-like structure. Each chromosome is a vector of floating point numbers the size of which is kept the same length of the vector, which is the solution of the problem. GAs and MAs that are based on real number representation are called Real-Coded Genetic Algorithms (RCGAs) and Real-Coded Memetic Algorithms (RCMAs), respectively [1, 3-5].

In the last years, there has been a growing interest in the analysis of experiments in the field of RCGA/RCMA

algorithms. Manuel Lozano et al. [3], proposed a real-coded memetic algorithm that applies a crossover hill-climbing to solutions produced by the genetic operators. Daniel Molina et al. [1], proposed a real-coded memetic algorithm that combines a high diversity global exploration with an adaptive LS method. They analyzed the performance of their proposal for the test problems proposed in the Special Session of the IEEE Congress on Evolutionary Computation in 2005 (CEC 2005). In their work, they introduced an adaptive LS method that adjusts the LS probability and the LS depth. They used the individual fitness to decide when LS can be applied (LS probability) and how many effort should be applied (the LS depth), where LS effort should be performed on the most promising regions. In the field of RCGA, there are more related activities [7-11].

In this study, a new Adaptive Real-Coded MA (ARCOMA) is proposed. The Steady State Genetic Algorithm (SSGA) is the base of proposed algorithm, as offered in [1, 12, 13]. In SSGA, usually only one or two offspring are produced in each generation. Although SSGAs are less common than generational GAs, different authors [3, 14] recommended their use for the design of steady state MAs (SSGA plus LS), because they may be more stable and allow the LS results to be maintained in the population [1].

Any optimization algorithm can be utilized for the LS. In this paper, the authors have adopted a simplified variant of their previously proposed metaheuristic, called Continuous Ant Colony System (CACS) [15], for this purpose.

The paper is organized as follows: In section 2, ARCOMA is described in details. In section 3, parameters tuning is addressed, the numerical results for CEC 2005 are presented and the comparison with other methods is carried out. Finally, the conclusion is made in section 4.

II. ADAPTIVE REAL-CODED MEMETIC ALGORITHM The new algorithm, called ARCOMA, is composed of

two main components: a real coded genetic algorithm to provide exploration within the whole solution space, and a continuous LS scheme to exploit the most promising subspaces. A detailed description on how these components work is presented in the following subsections.

To be able to solve various optimization problems, algorithms should have different setting or adaptive structures to meet problem requirements, especially, when dimensions of the problems are increased or in case the problem has several local minima. Thus, in the LS scheme,

2009 International Conference on Artificial Intelligence and Computational Intelligence

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DOI 10.1109/AICI.2009.259

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the number of search steps is adapted for each individual based on the distribution of the good solutions. In addition, in the new LS scheme the allowed search intervals are also adapted for each dimension of the problem. The authors have named the new scheme as Adaptive Real-coded Memetic Algorithm (ARCOMA) to differentiate it from the original RCMAs. The adaptation mechanisms, utilized here, can considerably match ARCOMA with a wide range of optimization problems.

A. General Setting Out of the Algorithm It is desired to find the global minimum of a

multivariable fitness function f(x), within a given search interval, in which the minimum occurs at a point xs. In general, f can be defined on a subset of Rn delimited by n intervals [ai,bi], i=1, …, n.

Figure 1 shows the general iterative structure of ARCOMA. A high level description of the sequential steps is shown in this figure. In the following subsections, these steps are discussed in details.

Figure 1. General flowchart of ARCOMA.

B. Initialization The parameters of ARCOMA, as defined in the next sub-

sections, must be set before the execution of the algorithm.

C. Creating the initial population The initial population is randomly distributed within the

whole solution space. The fitness function is evaluated for each individual. A local search is done and the process continues until Npop initial tuned individuals are generated.

D. Local Search A simplified variant of CACS [15] is utilized for LS.

Like in CACS, to provide a continuous pheromone model over the search space, the pheromone distribution is considered in the form of a normal Probability Distribution Function (PDF) and ants choose their next destinations using a random generator with the PDF. The fitness is calculated in the new point and some knowledge about the search space is acquired, used to update the pheromone distribution. In ARCOMA, the global exploration is carried out in the whole search space and ants search within limited search intervals. The major differences between CACS [15] and the variant proposed here is the use of only one ant and the use of two adaptation mechanisms. Here the history of the traveled points by the single ant is used to update pheromone.

The number of search steps for LS (NiLS) and the ratio of local to global search domains (DRL2GS) have been proposed by the authors as a new contribution. In other words, NiLS and DRL2GS are adapted based on the distribution of pheromone, when good solutions are closed to each other, pheromone distribution is confined and vice versa. When pheromone distribution is confined, NiLS is increased and DRL2GS is decreased to permit the more concentrated LS and vice versa. NiLS and DRL2GS are calculated as follows:

∑∑== −−

−=n

i ii

n

i ii

ii

yyyyxx

1*

1*

2*

)(1

)()(σ (1)

[ ]σ)(βN *LS iLS ii xx −= (2)

σLSL2GS αDR = (3)

where xi is the magnitude of each attributes in the i-th dimension of and yi is the corresponding fitness function. The * sign denotes the best solution found so far. Also βLS and αLS are algorithm parameters that should be tuned.

E. Genetic Operators The GA component of MAs should be more explorative,

because a part of the exploitation is made by LS. In this study a SSGA similar to that of [13] is utilized. In the following the genetic operators of ARCOMA are described.

1) Selection To generate each offspring, two parents must be selected

from the mating pool. The new offspring is then generated using crossover and mutation operators. The selection scheme adopted here is based on the Negative Assortative Mating (NAM) [16] as also proposed in [1]. In NAM a first parent is selected by the roulette wheel method and NNAM chromosomes are also selected with the same method. Then the similarity between each one of these chromosomes and the first parent is computed (similarity between two real-coded chromosomes is defined as the relative Euclidian distance). Then the one with less similarity is chosen to be the second parent.

2) Crossover A new crossover method is developed in this paper which

is based on BLX crossover method [19]. The new method

Initialization

Create the i-th individual

Local Search

i>Npop i=i+1

Select two parents to generate the k-th offspring

Crossover and Mutation

Local Search

k>Noffspring

Yes

No

Yes

No

Replacement

Stopping Condition Yes No

Stop

Take the j-th step

i>NLS

Yes

No

Update pheromone Distribution

j=1

k=k+1

369

carries out crossover within the direction of differences between two selected parents in the all search space. The authors have considered two states for the parents. Due to the limited number of pages, only the first state is presented here. Suppose the case when the position of the first parent (P1) in a specified dimension is smaller than that of the second parent (P2). The authors introduce two parameters α and β that show the maximum allowed expansions along the subtraction vector P2-P1 to put the new offspring within the search interval (Figure 2). The following relations are derived for αi and βi:

Figure 2. Parent2 (ith dimension) > Parent1 (ith dimension)

In this case, the following relations are used (Fig. 2):

)()()(a)(α

21

1

iPiPiiP

i −−= ,

)()()()(bβ

21

2

iPiPiPi

i −−= (4)

where i denotes the i-th dimension. Finally, utilizing the above equations for αi and βi, the

crossover equation is derived as follows:

( ) ( )( )randPPPPPP 12121 .βα1α −+++−−= (5)

where α and β are the permissible values of αi and βi over all dimensions.

3) Mutation The Nun-uniform mutation (Michalewics, 1992) has

been used in this paper [1,5]. 4) Replacement

The standard Replace Worst (RW) is applied. In RW, offspring replaces the worst individual only if the new one is better. This strategy is adequate as a combination with LS, because it is an elitist strategy, and it is recommended for MAs. Furthermore, it offers a high selective pressure, making it a good complement for NAM [1,2,18,19].

III. RESULTS AND DISCUSSION In this section, the results of ARCOMA are compared

with those of three others state of the art works using the benchmarks proposed in the CEC 2005 [20].

A. Parameter Tuning In order to tune the parameters of ARCOMA, the CEC

2005 benchmarks [20] have been used in this research. The parameters of ARCOMA that should be tuned are: number of population (Npop), mutation probability (Pmut), number of offsprings (NOffspring), number of Negative Assortative Mating (NNAM), LS iterations factor (αLS), and local to global search domain ratio factor (βLS). After doing trial and error by authors, tuned parameters have been obtained as follows: Npop=20, Pmut=0.12, NOffspring=2, NNAM=3, αLS =0.9, and βLS =11.0.

B. Comparison with another Algorithm Results In CEC 2005, 25 benchmarks were offered. The authors

examined ARCOMA for 1000 and 10000 fitness evaluations (FEs). All of the functions expressed in CEC 2005 benchmarks have been ran 25 times for 1000 and 10000 fitness evaluations (FEs). The ARCOMA results have been compared with those of Real-Coded Memetic Algorithm (RCMA) [1], Steady–State Population-Based Search Algorithm (SSPBSA) [9], and Estimation of Distribution Algorithm (EDA) [21] as presented in CEC 2005. Tables I to IV present the comparison between these algorithms with dimension D=10 and D=30 for 1000 and 10000 fitness evaluations respectively. It can easily be observed that ARCOMA has superior performance especially for 10000 function evaluations.

IV. CONCLUSION In this paper a new Adaptive Real-Coded Memetic

Algorithm, called ARCOMA was developed. This algorithm was compared with three other algorithms namely RCMA [1], SSPBSA [9] and EDA [21] based on the results reported for CEC 2005 benchmarks [20]. Results show that ARCOMA obtained acceptable results. Comparisons show that ARCOMA has superior results compared with other mentioned Real-coded algorithms. The most important feature of ARCOMA is that it utilizes adaptive local search operators. The adaptive features, embedded in ARCOMA provide the ability to solve a wide range of continuous optimization problems.

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Parameters for Real-coded Memetic Algorithm,” Proceeding of the 2005 IEEE congress on Evolutionary Computation, pages 888–895, 2005.

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[3] M. Lozano, F. Herrera, N. Krasnogor, D. Molina, “Real-Coded Memetic Algorithms with Crossover Hill-Climbing,” Evolutionary Computation Volume 12, Number 3, 2004.

[4] Z. Michalewicz, “Genetic Algorithms + Data Structures = Evolution Programs,” Springer-Verlag, 1996.

[5] F. Herrera, M. Lozano, J.L. Verdegay, “Tackling Real-coded Genetic Algorithms: Operators and Tools for Behavioural Analysis,” Kluwer Academic Publishers, 1998.

[6] M.M. Raghuwanshi, O.G. Kakde, “Survey on multi-objective evolutionary and real coded genetic algorithms,” The 8th Asia Pacific Symposium on Intelligent and Evolutionary Systems, Cairns, Australia, 6th - 7th December 2004.

[7] K. Deb, D. Joshi, A. Anand, “Real-Coded Evolutionary Algorithms with Parent-Centric Recombination,” IEEE, Vol.1, Pages 61-66, 2002.

[8] Kyoung-Jong Park, “The Application of Real-Coded Genetic Algorithm for Simulation Optimization,” First International Business Conference, Dearborn, Michigan, USA, August 7-9, 2008.

[9] A. Sinha, S. Tiwari, K. Deb, “A Population-Based, Steady-State Procedure for Real-Parameter Optimization,” Proceeding of the IEEE congress on Evolutionary Computation, 2005.

[10] S. García, D. Molina, M. Lozano, F. Herrera, “A study on the use of Non-parametric tests for analyzing the evolutionary algorithms’

β(P2-P1)

α(P2-P1) P2 P1 x1

x2

b1 P1(1) a1

P2-P1

P2(1)

370

behaviour: a case study on the CEC’2005 Special Session on Real Parameter Optimization,” J. Heuristics, April, 2008.

[11] H. Kita, “Comparison Study of Self-Adaptation in Evolution Strategies and Real-Coded Genetic Algorithms,” Evolutionary Computation, Vol. 9, Pages: 223 – 241, 2001.

[12] Alexandre C. M. DeOliviera, A. N. Lorena Luiz, “Real-Coded Evolutionary Approaches to Unconstrained Numerical Optimization,” In J. M. Abe and J. I. Silva Filho, editors, Advances in Logic, Artificial Intelligence and Robotics, volume 2. Congress of Logic Applied to Technology (LAPTEC2002), 3, Sao Paulo, Brazil, Pleiade, 2002.

[13] D. Whitley, “The genitor algorithm and selection pressure: why rank-based allocation of reproductive trials is best,” Proc. of the Third Int. Conf. on Genetic Algorithms, pages 116–121, 1989.

[14] M.W. Shannon Land, “Evolutionary Algorithms with Local Search for Combinational Optimization,” PhD thesis, Univ. California, San Diego, 1998.

[15] S. H. Pourtakdoust, H. Nobahari, “An Extension of Ant Colony System to Continuous Optimization Problems,” Lecture Notes in Computer Science, Vol. 3172. Springer-Verlag, Berlin Heidelberg Germany (2004) 294–301

[16] C. Fernandes, A. Rosa, “A Study of Non-random Matching and Varying Population Size in Genetic Algorithm Using a Royal Road Function,” Proceeding of the 2001 congress on Evolutionary Computation, (2001) 60–66

[17] L. J. Eshelman, D. J. Schaffer, “Real-coded Genetic Algorithms in Genetic Algorithms by preventing incest,” Foundation of Genetic Algorithms 2, (1993) 187–202

[18] W.E. Hart, “Adaptive Global Optimization with Local Search” PhD thesis, Univ. of California, San Diego, 1994.

[19] A. Sinha, “Designing efficient genetic and evolutionary algorithm hybrid,” PhD thesis, Indian Institute of Technology, Kharagpur, 1999.

[20] P.N. Suganthan, N. Hansen, J. J., Liang, and et al, “Problem definitions and evaluation criteria for the CEC 2005 Special Session on Real-Parameter Optimization,” Technical report, Nanyang Technological University, May 2005.

[21] Bo Yuan, M. Gallagher, “Experimental Results for the Special Session on Real-Parameter Optimization at CEC 2005: A Simple, Continuous EDA,” Proceeding of the IEEE congress on Evolutionary Computation, 2005.

TABLE I. COMPARISON OF ERROR VALUES ACHIEVED WHEN FES=1E3 AND DIMENSION =10

Algorithms ARCOMA RCMA EDA SSPBSA Functions Best Mean Best Mean Best Mean Best Mean

1 9.8396 106.813 12.6212 1844.514 744.29 1822.5 1.028e-7 2.9e-6 2 250.7585 1970.610 765.849 10556.41 1749.2 3461 0.1725 3.4961 3 147.5e3 4.252e6 1.105e6 2.76e7 4.391e6 1.2e7 1.07e5 1.2e6 4 1167.575 6541.1419 8912.104 22161.12 2079.1 4705.5 9724.3 18139 5 1210.15 2412.85 2624.7 1.148e4 3922.9 6853 7303.1 10731 6 16043.87 6.825e5 3991.61 7.982e7 2.278e7 1.345e8 9.9192 1674.5 7 1269.552 1274.6 4.4434 546.618 246.57 742.74 0.5738 0.9808 8 7.2474 29.7508 20.104 20.5497 20.427 20.724 20.356 20.696 9 9.322 41.7768 11.6317 35.4967 31.118 52.43 65.813 94.921 10 8.7425 10.9866 3.0367 50.9106 47.194 64.88 84.691 127.87 11 20.5973 20.7484 6.3678 8.7184 10.334 11.754 4.4173 7.297 12 1378.21 7509.08 861.52 3974.60 16150 59272 10.813 1495.6 13 1.462 3.3673 0.6776 6.8178 17.504 69.055 89.303 2376.7 14 3.6015 4.0149 3.9172 4.3313 3.7422 4.2331 4.126 4.3903 15 164.30 443.8394 305.32 544.36 578.89 671.87 737.83 841.46 16 130.83 216.2135 192.45 302.64 230.88 299.9 377.87 446.63 17 196.02 254.0757 266.06 423.95 234.69 339.74 406.58 527.16 18 835.67 1008.82 846.63 1052.3 848.53 1052.2 1092.4 1217.4 19 576.32 982.8 811.71 1015.4 899.52 1063.3 1114 1219.4 20 548.28 976.0723 882.32 1058.9 1003.9 1082.7 1062.9 1208.4 21 571.45 1164.09 510.64 1179.8 1099.1 1287 1220.2 1338.4 22 788.29 915.489 702.7463 981.1 885.35 945.61 936.98 1087.6 23 562.583 1171.8 1033.592 1316.684 1148.7 1287.8 1245.5 1396.6 24 213 554.8042 209.2101 1070.391 839.69 1040.6 485.67 763.14 25 209 815 446.438 687.9292 1025.7 1484.3 552.69 771.89

TABLE II. COMPARISON OF ERROR VALUES ACHIEVED WHEN FES=1E4 AND DIMENSION =10

Functions ARCOMA RCMA EDA SSPBSA Best Mean Best Mean Best Mean Best Mean 1 2.84e-12 75.95 1.315e-8 4.5979e-6 1.3e-7 1.3e-6 5.51e-9 8.7e-9 2 1.20e-11 830.7272 0.2317 3.659 7.1e-7 5.7e-6 9.20e-9 9.6e-9 3 0.0052 1627.9886 81420.14 515354.8 0.0051 215.3 2027.4 14100 4 1.279 4.377e6 101.9428 553.5116 2.0e-6 0.0001 141.97 407.66 5 .0005 1780.61 25.5889 125.3784 0.3996 1.8943 1413.4 2457 6 781.81 503761 5.5302 7.1079 6.4988 26.413 4.01e-9 0.4819 7 0.0014 1751.78 0.0859 0.4841 0.466 0.6961 0.0246 0.231 8 0.0079 2.853e5 20.104 20.247 20.354 20.525 20 20.023 9 10.9323 30.7655 2.9859 5.3094 6.0999 29.342 32.798 45.126

10 8.9546 37.5916 3.0367 8.8324 8.4075 29.426 41.234 52.479 11 6.7794 28.8574 4.9699 7.3346 2.8515 9.9626 4.1228 7.0579 12 4.3452 9.6431 17.8341 264.3701 0.0114 624.2 6.38e-9 149.26 13 2.3991 7105.36 0.6776 1.4253 1.6063 3.2076 2.5945 3.6308 14 1.3265 5549.52 2.914 3.514 3.4144 3.873 3.6667 4.2549 15 0.5126 5.3445 41.7191 305.706 344.46 479.14 636.75 724.6 16 0.3764 3.0328 102.9091 112.8299 144.92 176.23 173.17 195.27

371

TABLE II (CONTINUED). COMPARISON OF ERROR VALUES ACHIEVED WHEN FES=1E4 AND DIMENSION =10

Functions ARCOMA RCMA EDA SSPBSA Best Mean Best Mean Best Mean Best Mean

17 171.667 252.743 130.9578 156.6613 156.85 196.64 171.92 213.52 18 287.354 907.936 800.0159 806.7819 300 483.24 847.89 951.83 19 800.019 955.526 368.7447 772.0305 300 564.39 846.86 975.81 20 605.54 989.435 800.0156 800.1165 300 658.58 860.34 974.12 21 800.004 956.075 500 741.4137 300.01 540.97 941.88 1118.7 22 800.019 964.994 300.1545 721.0526 766.96 785.64 550.22 742.82 23 877.886 1159.77 559.4683 981.2801 559.47 651.66 1031.5 1164.1 24 613.459 1024.97 200 224.0172 200 223.95 408.23 411.19 25 770.014 906.416 201.6866 399.9965 377.38 386.15 409.6 411.27

TABLE III. COMPARISON OF ERROR VALUES ACHIEVED WHEN FES=1E3 AND DIMENSION =30

Functions ARCOMA RCMA EDA SSPBSA Best Mean Best Mean Best Mean Best Mean

1 5879.208 1.038e4 4785.153 2.075e4 6.476e4 8.35e4 2.0991 11.061 2 2.350e4 3.871e4 2.947e4 5.52e4 6.446e4 1.088e5 3249.3 9908 3 8.186e7 1.853e8 9.634e7 2.942e8 8.708e8 1.559e9 2.316e7 5.132e7 4 2.831e4 5.208e4 8.126e4 1.148e5 7.431e4 1.272e5 1.159e5 1.694e5 5 1.331e4 1.874e4 1.867e4 2.855e4 3.259e4 3.787e4 3.497e4 4.143e4 6 4.980e8 1.544e9 5.844e8 5.57e9 1.294e10 5.631e10 4.892e5 2.386e6 7 4863.592 5087.93 1103.454 3157.14 10214 10978 30.22 77.308 8 21.0844 21.1983 20.7311 20.9188 21.062 21.199 21.081 21.198 9 188.8456 261.802 182.1815 299.3973 432.67 480.69 443.13 503.16 10 288.2007 393.1644 333.253 516.9249 602.53 747.99 756.79 836.03 11 35.7921 43.5788 32.646 37.3589 40.776 45.612 27.683 33.04 12 1.767e5 4.049e5 2.738e5 4.155e5 1.396e6 1.763e6 3.84e4 1.047e5 13 19.1727 28.0269 408.69 3951.074 1.953e5 5.46e5 3.21e5 1.28e6 14 13.2752 13.7505 12.8606 13.867 13.86 14.211 13.99 14.358 15 536.3798 704.289 545.6772 761.5713 955.89 1147.2 936.96 1018.4 16 293.242 464.806 522.5558 643.0473 699.13 934.27 982.16 1198.2 17 351.0751 588.930 711.5639 1022.346 749.32 1033.6 1158.6 1358.9 18 1005.41 1067.76 959.7849 1090.253 1266.2 1331.1 1332.5 1421.9 19 988.810 1067.58 944.7712 1075.927 1269.6 1330 1332.4 1418.2 20 1000.92 1059.50 921.8594 1102.111 1208.2 1329 1332.4 1418.2 21 974.900 1251.55 1242.414 1317.16 1399.2 1471.1 1388.6 1562.1 22 1187.69 1292.75 1192.309 1333.117 1534.5 1695.1 1580.8 1913 23 1203.81 1260.27 1288.914 1368.447 1391.5 1480.5 1383.6 1562.2 24 1220.37 1350.97 1288.599 1446.678 1402.1 1502.4 1479.1 1589.5 25 1190.43 1348.79 1241.049 1462.123 1877 1920.5 1522.8 1690.5

TABLE IV. COMPARISON OF ERROR VALUES ACHIEVED WHEN FES=1E4 AND DIMENSION =30

Functions ARCOMA RCMA EDA SSPBSA Best Mean Best Mean Best Mean Best Mean

1 0.0518 0.2782 0.6236 3.4551 9667 13394 8.2e-9 9.47e-9 2 2946.3472 7818.5137 8489.698 1.408e4 1.730e4 2.278e4 1.5e-5 0.0002 3 1.083e7 2.109e7 1.806e7 4.143e7 5.989e7 1.057e8 3.692e5 8.557e5 4 2.571e4 4.827e4 2.148e4 2.865e4 1.873e4 2.786e4 1.159e5 1.694e5 5 6734.2926 9593.083 4801.445 6190.89 1.586e4 1.888e4 3.497e4 4.122e4 6 1083.021 5148.986 1149.044 6955.493 2.0813e9 3.947e9 17.453 244.45 7 4696.288 4696.288 8.5326 24.986 3601 4683.1 0.000 0.015 8 20.5376 20.9128 20.7054 20.8468 20.916 21.078 20.305 21.05 9 113.4322 161.2734 62.7457 105.1354 217.96 254.05 443.13 501.71 10 103.9435 252.8237 99.0068 139.014 266.02 300.88 756.79 836.03 11 27.0552 34.6952 32.3107 35.6986 40.776 43.037 24.257 30.079 12 1.0301e4 4.242e4 5.623e4 1.355e5 6.688e5 9.940e5 3.2737 2689 13 4.8545 11.9758 9.9731 15.1462 684.73 6289.5 3.210e5 1.285e6 14 12.1365 13.3036 12.1923 13.3974 13.563 13.895 13.99 14.323 15 342.6962 483.4107 371.6493 440.5251 532.82 569.12 882.01 979.4 16 142.0873 328.753 142.0165 355.7056 244.19 332.81 982.16 1192.5 17 257.285 502.8696 244.1006 362.3821 315.13 389.26 1158.6 1358.9 18 919.6125 983.5964 828.5554 891.5002 1033 1063 1256.5 1374.7 19 800.2645 948.6725 885.0832 894.7336 1023.5 1059.2 1277 1366.2 20 916.8904 957.7509 884.5718 896.0669 1028.4 1054.7 1217.7 1360.5 21 500.0316 1121.6439 501.0318 519.4592 1095 1189.2 1388.6 1562.1 22 927.57 1184.8001 967.4052 1011.22 1044.9 1104.9 1497.8 1861.6 23 646.3784 1178.1838 601.44 734.2925 1107.9 1188.5 1383.6 1562.2 24 244.604 1162.9731 229.4557 313.6706 1064.1 1167.3 1429 1571.4 25 200.2878 1196.9196 240.5694 408.7138 1303.5 1570.3 1095.3 1523.9

372