K. Deb et al. (Eds.): SEAL 2010, LNCS 6457, pp. 177–186, 2010. © Springer-Verlag Berlin Heidelberg 2010

A Comparative Study of Different Variants of Genetic Algorithms for Constrained Optimization

Saber M. Elsayed, Ruhul A. Sarker, and Daryl L. Essam

School of Engineering and Information Technology, University of New South Wales at ADFA (UNSW@ADFA), Canberra 2600, Australia saber.elsayed@student.adfa.edu.au, {r.sarker,d.essam}@adfa.edu.au

Abstract. Over the last few decades, many different variants of Genetic Algorithms (GAs) have been introduced for solving Constrained Optimization Problems (COPs). However, a comparative study of their performances is rare. In this paper, our objective is to analyze different variants of GA and compare their performances by solving the 36 CEC benchmark problems by using, a new scoring scheme introduced in this paper and, a nonparametric test procedure. The insights gain in this study will help researchers and practitioners to decide which variant to use for their problems.

Keywords: Constrained Optimization, genetic algorithm, a non-parametric test.

1 Introduction

Over the last few decades, many different variants of GA have been introduced for solving COPs. These variants differ mainly in their use of crossover and mutation operators. Due to the variability of function properties in practical COPs, one such variant, even if it works well for one problem or a class of problems, does not guaran-tee that it will work well for another class of problems or a range of problems. As a result, finding a suitable variant is still a headache for researchers and practitioners. Keeping this in mind, in this research, we have implemented ten variants of GA using five different specialized crossover and two mutation operators. The crossovers consi-dered in this paper are widely used in practice, such as the blend crossover (BLX-α) [7], the simulated binary crossover (SBX) [2], the simplex crossover (SPX) [13], the parent centric crossover (PCX) [3] and the triangular crossover (TC) [6]. Each of these operators has its own positives and negatives when applied to evolutionary problem solving. Interestingly, none of these crossovers is suitable for all possible scenarios of the constrained problems. The mutations considered are non-uniform and polynomial mutations.

The variants are compared based on their performances in solving a set of 36 prob-lem instances introduced in CEC2010 (18 test problems, 2 instances each [9]). In this paper, we have introduced a new scoring scheme to compare the performance of dif-ferent algorithms which is very simple to use, and easy to judge the performances of different algorithms. The scheme produces a single score for each test problem. To judge the performance over a set of problems, an overall score can be generated by

178 S.M. Elsayed, R.A. Sarker, and D.L. Essam

simply adding the individual scores. The decision produced by the proposed scheme is consistent with a nonparametric test procedure. The analysis shows that no single operator of GA is able to reach the high quality solutions for all the test problems. In addition, no one is clearly the winner. These insights of algorithms’ performances are analyzed and discussed. We believe the insights will help researchers and practition-ers to decide the appropriate variant for their problems.

This paper is organized as follows. After the introduction, the search operators and the constraint handling technique used are discussed in section 2. Section 3 presents the new comparison technique. Section 4 provides the computational results and ana-lyzes the performances. Finally the conclusions are drawn.

2 Search Operators and Constraint Handling

In this section, the search operators and constraint handling technique used in this research are briefly discussed.

As indicated earlier, we have implemented five different crossovers with two muta-tions. These crossover and mutation operators are briefly reviewed below.

Blend Crossover (BLX-α) has an advantage of generating diverse offspring [7], that allows GA to converge, diverge, or adapt to changing fitness landscapes without incurring extra parameters or mechanisms [8]. However, it has a disadvantage in the consideration of the epistasis problems, in which a piece of information of one variable is very dependent to another one [8]. Also, BLX-α works well for separable functions, but it does not perform well in solving optimization problems with non-separable functions [12].

Simulated Binary Crossover (SBX) is widely used in practice. The SBX operator has been found to work well in many test problems having a continuous search space when compared to other real-coded crossover implementations. The SBX operator can restrict offspring solutions to any arbitrary closeness to the parent solutions, the-reby not requiring any separate mating restriction scheme for better performance. SBX is also useful in problems where the bounds of the optimum point are not known a priori and where there are multiple optima [2].

Simplex Crossover (SPX) is a multi-parent recombination operator for real-coded GAs. The SPX uses the property of a simplex in the search space. The simplex cros-sover works well on functions having multimodality and/or epistasis with a medium number of parents: three parents on a low dimensional function and four parents on a high dimensional function [13]. However, SPX fails on functions that consist of tightly linked sub-functions [13].

Parent Centric Crossover (PCX) allows a large probability of creating a solution near to each parent, rather than near the centroid of the parents [3]. PCX is a self adaptive type approach that has shown excellent performance in solving some test problems when implemented with a real coded GA [3]. However, GA with PCX has a difficulty in separable multimodal problems compared to other EAs such as DE [11].

Triangular Crossover (TC) is a three-parent crossover approach that concentrates on the boundary of a feasible region. TC works well where the optimal solution lies on the boundary of the feasible region of a problem, where the problem also has a single bounded feasible region in the continuous domain [6].

A Comparative Study of Different Variants of GAs for Constrained Optimization 179

In Non-uniform Mutation, the step size is decreased as the generations increase, thus making a uniform search in the initial stage and very little at the later stages [10]. In contrast, in Polynomial Mutation, the probability of mutating a solution near to the parent is higher than the probability of mutating one distant from it. The shape of the probability distribution is directly controlled by an external parameter and the distri-bution remains unchanged throughout the entire evolution process [4]. Hence non-uniform mutation is often better at refining a solution, while polynomial mutation maintains diversity throughout a run.

In this paper, we measure the superiority of feasible points (during a tournament [5]) as follows: i) between two feasible solutions, the fittest one (according to the fit-ness function) is better, ii) a feasible solution is always better than an infeasible one, iii) between two infeasible solutions, the one having the smaller sum of its constraint violation is preferred. The equality constraints are transformed to inequalities of the form: 0, 1, … , , where is a small number.

3 A New Comparison Technique

In this section, we propose a new technique for comparing different variants of GA. It can also be used to compare any other stochastic algorithms. To judge the quality of any variant, we assign a score of ‘1.0’ if a variant obtains the best fitness value for a given test instance and ‘0.0’ if a variant fails to achieve any feasible solution. If a va-riant achieves a feasible solution, but not the best fitness, it will receive a fractional score (between 0.0 and 1.0) as discussed below. We assume that all of the problem instances have a minimization objective function.

For a variant i and test instance j, and a total number of test problems , we define as the actual fitness, and , as the overall best

and worst fitness value for a test instance j, respectively. The score of a variant i for instance j is then:

1 | | | | , if is feasible 0, (1)

where 1 and 1. A value 1 will differentiate between the worst feasible and any infeasible solution by having a small positive value for Sij. A higher value of p will put a higher emphasis on good solutions. In this study, we use a = 1.1 and p = 2. In a similar way we can also calculate scores for averages. In that case, the final score for a variant i can be calculated as follows: ∑ 1 ∑ (2)

where, is the final score of variant i for test problem j, is the feasibility ratio of

variant i for test problem j, is the score based on the best solutions, is the score based on the average values, and is a constant 0, 1 . A higher value of (1 or close to 1), will put a higher emphasis on the best solutions, which is ap-propriate when we are interested in only the best fitness value, while a lower value of (0 or close to 0) will put a higher emphasis on the average solutions, which is

180 S.M. Elsayed, R.A. Sarker, and D.L. Essam

appropriate when we are interested in a number good alternative solutions. In this study, we use 0.5 to make a balance between the best and the average results. The overall score ( for each variant i can then be calculated using the following equation. ∑ (3)

If a variant finds the best fitness value for all test instances, then OSi = J. For the worst fitness value for all test instances, OSi = 0. In statistical significance testing, if a variant is significantly better in some instances and significantly worse in some other instances, it is not easy to decide the best performing variant, as the precise difference in magnitudes is not reflected in the test results. In that case, the proposed scoring scheme would provide better insights in such comparisons, as the precise difference in magnitudes is taken into account. In addition, this is not a pair-wise comparison, but rather a simultaneous comparison for any number of algorithms /samples.

4 Experimental Results and Analysis

As discussed earlier, the variants are designed using one of five crossover operators with one of two mutation operators. The crossover operators are TC, SBX, PCX, SPX and BLX-α, and the mutation operators are non-uniform (NU) and polynomial (P) mutations. The test problems details are presented in Table 1. The parameters settings are: initial population size 30, crossover rate (CR) = 100%, mutation rate (MR) = 10%, tournament size (TS) = 3, α = 0.366 according to [12], the index parameter 3, 0.01 according to [3]. As suggested for the number of parents in SPX, three parents are used on a low 10D function and four parents on a high 20D function [13], we have used 5 parents for 30D. For the non-uniform muta-tion, b 5, and for the polynomial external parameter 10, finally ε 0.0001. The detailed results showing best fitness (b), mean (M), standard deviation (Sd) and the average feasibility ratio (Avg. Fr) are presented in Table 2 for 10D, and in Table 3 for 30D. All results are out of 25 independent runs. Due to space and formatting limi-tation, we only provide the results of the best variants. However the analysis is performed using all ten variants.

Firstly, based on the feasibility ratio for 10D and 30D, we found that TC-NU is in the 1st position followed by, BLX-α-NU, PCX-NU, SBX-NU, SPX-NU, PCX-P, SBX-P, TC-P, BLX-α-P and SPX-P, with total averages of FR 84%, 83.5%,83.5%, 83%, 78.5%, 78%, 75%, 72%, 69% and 61.5%, respectively. So, it is clear that the non-uniform mutation is superior to polynomial in regards of the feasibility ratio. The number of best solutions found by each variant, TC-NU, TC-P, SBX-NU, SBX-P, SPX-NU, SPX-P, PCX-NU, PCX-P, BLX-α-NU, and BLX-α-P are 3, 0, 4, 2, 3, 0, 3, 4, 3 and 0, for 10D, respectively. Note that all variants with the non-uniform muta-tion were able to reach the same solution for C16, while TC-NU, SBX-NU, SPX-NU, PCX-NU, PCX-P and BLX-α-NU are able to solve C03. Further only SBX-NU, SPX-NU, PCX-NU, and BLX- α-NU are able to solve C04, while all variants except TC-NU cannot solve C11. For 30D, the numbers of best solutions found by each variant (in the same order) are: 3, 2, 2, 2, 1, 0, 1, 0, 4 and 0. No variant can solve C03, C04 and C11, and the variants with only the non-uniform mutation are able to solve C12.

A Comparative Study of Different Variants of GAs for Constrained Optimization 181

Table 1. Properties of the CEC2010 test problems. D is the number of decision variables, | |/| | is the estimated ratio between the feasible region and the search space, I is the number of inequality constraints, E is the number of equality constraints

Prob Search Range Objective Type Number of constraints Feasibility Region E I 10D 30D

C01 [0, 10] Non Separable 02 Non

Separable0.997689 1.000000

C02 [ 5.12, 5.12] Separable 1 Separable2 Non

Separable0.000000 0.000000

C03 [ 1000, 1000] Non Separable 1 Separable 0 0.000000 0.000000

C04 [ 50, 50] Separable2 Non Separable,

2 Separable0 0.000000 0.000000

C05 [ 600, 600] Separable 2 Separable 0 0.000000 0.000000C06 [ 600, 600] Separable 2 Rotated 0 0.000000 0.000000C07 [ 140, 140] Non Separable 0 1 Separable 0.505123 0.503725C08 [ 140, 140] Non Separable 0 1 Rotated 0.379512 0.375278C09 [ 500, 500] Non Separable 1 Separable 0 0.000000 0.000000C10 [ 500, 500] Non Separable 1 Rotated 0 0.000000 0.000000C11 [ 100, 100] Rotated 1 Non Separable 0 0.000000 0.000000C12 [ 1000, 1000] Separable 1 Non Separable 1 Separable 0.000000 0.000000

C13 [ 500, 500] Separable 02 Separable, 1 Non Separable

0.000000 0.000000

C14 [ 1000, 1000] Non Separable 0 3 Separable 0.003112 0.006123C15 [ 1000, 1000] Non Separable 0 3 Rotated 0.003210 0.006023

C16 [ 10, 10] Non Separable 2 Separable1 Separable, 1 Non Separable

0.000000 0.000000

C17 [ 10, 10] Non Separable 1 Separable2 Non

Separable0.000000 0.000000

C18 [ 50, 50] Non Separable 1 Separable 1 Separable 0.000010 0.000000 According to the problem properties, it can be seen, that TC with the non-uniform

mutation is the only variant that is able to reach a feasible solution for 10D rotated objective functions (e.g., C11). In contrast, PCX with the polynomial mutation is pre-ferred for either separable or non-separable objective function, with only rotated equality constraints, (e.g. C06 and C10). Also, PCX with the polynomial mutation is the best for the separable objective function, with separable equality constraints and non-separable inequality constraints, (e.g. C02). For the non-separable objective functions with only inequality constraints (rotated or separable), it is preferable to use the non-uniform mutation with BLX-α, or TC (e.g. C07, C08, C14 and C15). For the non-separable objective function with both equality constraints (separable) and in-equality constraints (only non-separable or only separable), SBX and PCX with the polynomial mutation reached more robust solutions (e.g. C17 and C18). In test prob-lems where the objective function is non-separable/or separable, and does have a mix of non-separable and separable inequality constraints, the non-uniform mutation is preferred with SBX (e.g. C13). Finally, for the separable objective function with only equality constraints (separable), it is preferable to use the non-uniform mutation with SBX or PCX (e.g. C05). The non-uniform mutation is preferred in the test problems where the objective function is non-separable with only inequality constraints (non-separable) (e.g. C01). Also, the same mutation performs well for test problems with a separable objective function with equality constraints (only non-separable) and in-equality constraints (only separable), (e.g. C03 and C09).To compare the variants, we have performed a non-parametric test (Wilcoxon Signed Rank Test) [1] that allows us

182 S.M. Elsayed, R.A. Sarker, and D.L. Essam

Table 2. Function values out of different GA variants, for 10D test problems

Pr. TC-NU SBX-NU SBX-P SPX-NU PCX-NU PCX-P BLX-α-NU1 b -0.74726 -0.74728 -0.74641 -0.74731 -0.74729 -0.74687 -0.74729

M -0.72839 -0.72992 -0.69813 -0.68265 -0.71796 -0.70001 -0.71900Sd 0.02172 0.02331 0.04945 0.06857 0.02762 0.05356 0.04048

2 b -2.20403 -2.20140 -2.25830 -1.60231 -2.24070 -2.27097 -2.18671M -0.07300 -1.18436 -1.69085 1.53784 -0.90938 -1.73062 0.75009Sd 1.85484 1.31417 0.72668 1.49232 1.40541 0.65883 1.94768

3 b 1.1E+10 10028669 - 3.44E+09 40707467 3.516912 3.99E+08M 2.8E+13 7.54E+13 - 1.01E+14 6.49E+13 1.39E+14 5.13E+13Sd 3.7E+13 2.22E+14 - 2.14E+14 1.01E+14 2.83E+14 1.03E+14

4 b - 0.004377 - 0.00045 0.014563 - 14.02194M - - - 4.387388 8.102383 - 15.10635Sd - - - 7.87455 11.4379 - 1.53359

5 b -231.962 -473.5781 -237.2348 -464.6771 -400.9999 -427.6626 -265.2265M -106.338 -204.7016 -105.0468 -50.0089 -244.1168 -160.4987 -150.5770Sd 77.2131 93.7422 105.7225 241.3993 59.3223 130.6904 100.1925

6 b -575.789 -575.7960 -576.2588 -576.1843 -576.7053 -572.6023 -574.6536M -83.7290 -160.4517 -345.7701 -68.3604 -144.7258 -87.8436 -189.3297Sd 321.1026 483.5311 334.8700 379.0355 392.5743 328.1714 380.5476

7 b 3.35E-21 1.22E-20 6.70E-11 9.32E-17 1.38E-21 2.97E-12 4.64E-22M 5.40E-19 1.70E-18 5.72E-09 1.90E-15 3.02E-20 1.88E-11 1.47E-20Sd 1.10E-18 2.34E-18 1.92E-08 2.38E-15 2.56E-20 2.01E-11 2.44E-20

8 b 1.20E-20 5.87E-20 3.52E-10 1.76E-16 4.95E-21 3.25E-11 1.93E-22M 2.17E-19 1.31E-18 4.62E-08 0.5147464 6.29E-20 1.80E-08 1.10E-20Sd 2.33E-19 1.93E-18 5.72E-08 0.6465979 6.96E-20 5.34E-08 1.76E-20

9 b 207.7978 375.54613 24.32602211 4.30E+08 218.49958 0.0920886 374.7197M 2.6E+07 4.58E+08 3.74E+09 4.88E+12 1.04E+06 5.47E+09 1.89E+09Sd 1.1E+08 2.24E+09 17348769574 4.77E+12 2209044.7 2.71E+10 7.33E+09

10

b 0.248143 60.033514 6.61064209 1.34E+10 69.104363 7.73E-06 75.291298M 9.6E+06 5.93E+09 2.50E+09 5.02E+12 2.97E+08 3.31E+08 5.49E+09Sd 3.0E+07 2.44E+10 1.05E+10 4.24E+12 1.21E+09 1.27E+09 1.16E+10

11

b -0.00106 - - - - - -M - - - - - - -Sd - - - - - - -

12

b -115.445 -374.4726 -304.7074 -161.8349 -304.6433 -260.0960 -303.0400M 44.3670 20.1254 8.7836 25.7515 5.2216 -27.8116 -26.1526Sd 126.6006 384.7512 241.1720 135.0566 197.5852 71.4188 112.6932

13

b -67.3679 -68.2885 -68.1437 -68.0511 -67.3388 -68.2287 -67.8926M -61.2049 -63.2528 -64.9840 -61.7082 -63.3479 -65.4860 -63.7720Sd 3.4104 1.9835 2.8576 3.5873 1.9832 1.9890 2.2564

14

b 0.1758 0.0910 0.2467 0.0882 0.0818 0.0000 0.2258M 4783.407 7931.4724 2335.7943 4197.8557 5130.0824 8377.1144 10656.307Sd 18106.75 16877.661 11231.1540 18074.121 18711.9980 17430.0030 23138.143

15

b 4.3E+07 3.42E+11 4.0787E+10 5.84E+10 1.09E+11 5.33E+08 9E+08M 2.4E+12 7.057E+13 4.0229E+13 1.39E+13 3.2E+13 4.56E+12 2.03E+13Sd 5.3E+12 6.744E+13 5.2273E+13 1.65E+13 4.14E+13 6.48E+12 5.72E+13

16

b 0 0 4.78173E-13 1.11E-16 0 1.33E-14 0M 0.153187 0.0119858 0.1556867 0.633979 0.19949 0.245698 0.280637Sd 0.221928 0.0293591 0.15681833 0.317308 0.255503 0.248255 0.488704

17

b 0.111628 0.0035024 0.000678985 15.44855 0.001073 0.000924 0.106927M 6.632889 6.594218 3.80935626 198.1898 9.641727 3.198288 19.50653Sd 7.659975 8.162406 5.49315642 181.9092 11.98974 4.413743 32.84651

18

b 0.353893 2.488E-10 0.082019072 1.97547 0.007973 0.003178 0.075067M 1107.45 762.80204 104.199786 6695.769 834.5436 26.33942 1004.014Sd 1545.649 1441.3715 178.745095 4695.514 1723.946 51.25978 1384.054

Fr 89% 88% 81% 83% 88% 86% 89%

A Comparative Study of Different Variants of GAs for Constrained Optimization 183

Table 3. Function values out of different GA variants, for 30D test problems

Pr. TC-NU SBX-NU SBX-P SPX-NU PCX-NU PCX-P BLX-α-NU1 b -0.81714 -0.821671 -0.815216 -0.815679 -0.817997 -0.817050 -0.817855

M -0.79892 -0.781180 -0.794330 -0.764440 -0.799509 -0.792670 -0.783985Sd 0.014196 0.031671 0.014201 0.030688 0.012201 0.019100 0.029011

2 b -1.25658 -1.126926 -1.652844 3.390203 -1.789475 -1.422927 -0.640858M -0.13683 1.289422 -0.525870 4.545690 0.996184 -0.296748 1.283116Sd 0.84786 1.361109 0.640276 0.636066 1.383242 0.921409 0.915661

3 b - - - - - - -M - - - - - - -Sd - - - - - - -

4 b - - - - - - -M - - - - - - -Sd - - - - - - -

5 b -457.322 -439.8795 -429.2540 215.0637 -431.9744 -378.6791 -435.1072M -145.951 -274.4893 -162.1598 481.4305 -335.2496 -33.7101 -111.7567Sd 102.1364 173.4741 214.5627 97.3591 83.9306 386.8334 181.6593

6 b -252.319 -517.8180 -436.9482 45.3549 -519.5224 -310.4290 -521.3494M -147.094 -352.6657 -271.6617 339.0001 -123.7365 -214.1711 -409.0158Sd 57.6696 228.0035 104.7799 194.7921 231.2404 97.0451 175.9162

7 b 7.12E-19 4.43E-17 0.111546449 5.9393812 1.96E-16 0.063957987 4.70E-20M 0.197979 3.08E-16 1.034270613 9.5030099 5.78E-16 1.972052263 6.12E-19Sd 0.494948 1.60E-16 0.847198591 2.4415226 2.97E-16 0.98146394 3.65E-19

8 b 4.50E-19 6.38E-17 0.429089011 4.9494844 1.30E-16 3.072100581 1.63E-19M 1.629232 0.435555 3.794085142 11.129745 0.5939709 5.021856481 0.5939381Sd 1.643779 0.6440672 1.473428046 2.9225308 0.857331 1.392382668 1.0303173

9 b 1.8E+09 2.29E+12 7.86E+10 2.52E+13 3.97E+12 2.19E+11 4.43E+12M 1.4E+11 1.94E+13 2.03E+12 3.33E+13 1.84E+13 3.36E+12 1.41E+13Sd 1.6E+11 1.17E+13 1.86E+12 8.78E+12 1.05E+13 2.85E+12 5.73E+12

10 b 4.3E+11 2.19E+12 1.36E+11 9.26E+12 7.41E+12 4.31E+11 5.86E+10M 2.9E+12 1.74E+13 3.35E+12 2.87E+13 2.00E+13 3.56E+12 1.05E+13Sd 2.3E+12 1.09E+13 3.67E+12 1.53E+13 8.17E+12 2.02E+12 5.43E+12

11 b - - - - - - -M - - - - - - -Sd - - - - - - -

12 b -0.13045 -0.190433 - -0.192083 -0.117789 - -0.086306M 0.003453 3.155255 - -0.054965 0.998620 - 8.751375Sd 0.148807 4.731518 - 0.130925 1.993884 - 12.498368

13 b -60.5334 -65.79880 -62.000733 -63.25430 -64.448800 -64.213130 -64.56370M -55.6431 -61.68457 -59.837132 -58.94818 -61.069500 -60.510728 -59.50287Sd 2.509570 2.120910 1.756991 3.094427 1.768428 2.112393 1.987751

14 b 8.47152 4.81551 161.87403 22.70745 10.92657 85.12049 0.12090M 2.7E+03 7.08E+03 4.09E+05 1.53E+05 6.56E+03 4.55E+03 4.14E+03Sd 6.6E+03 1.30E+04 1.83E+06 7.44E+05 1.83E+04 9.17E+03 1.24E+04

15 b 52.11062 6017.2691 3.43E+10 7.26E+12 225712214 6.88E+10 0.7314271M 9.8E+03 3.13E+12 6.76E+11 2.84E+13 3.42E+12 2.01E+12 2.61E+12Sd 3.8E+04 6.67E+12 6.99E+11 2.51E+13 6.67E+12 2.12E+12 5.23E+12

16 b 0.749937 0.290163 0.000841 1.017013 0.258111 0.001214 0.824704M 1.017990 0.958698 0.852487 1.080749 1.004627 0.777738 1.027524Sd 0.103935 0.188695 0.336429 0.044407 0.238504 0.397705 0.054663

17 b 0.749937 60.024590 3.598506 182.72766 49.991087 3.623220 90.666040M 124.8363 258.79140 93.818115 1016.4893 265.721340 86.013982 420.60479Sd 45.59627 155.21920 61.064014 503.07045 214.048620 72.386892 180.46487

18 b 6.2E+03 2.63E-01 5.57E+01 2.54E+03 2.44E+02 9.42E+01 1.63E+03M 1.1E+04 3.14E+03 1.72E+03 1.57E+04 4.70E+03 1.55E+03 5.70E+03Sd 2.9E+03 3.99E+03 1.94E+03 5.57E+03 7.16E+03 1.28E+03 2.06E+03

Fr 79% 78% 69% 74% 79% 70% 78%

184 S.M. Elsayed, R.A. Sarker, and D.L. Essam

Table 4. The Wilcoxon non-parametric test for ten variants of GA for both 10D and 30D, based on the best fitness values obtained

b a

10D 30D

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9

Table 5. The Wilcoxon non-parametric test for ten variants of GA for both 10D and 30D, based on the average fitness values obtained

b a

10D 30D

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9

to judge the difference between paired scores when it cannot make the assumption required by the paired-samples test, such as that the populations should be normally distributed. The results based on the best and average fitness values are presented in Table 4 and 5, respectively. As a null hypothesis, it is assumed that there is no signifi-cant difference between the best/or mean values of two samples, whereas the alterna-tive hypothesis is that there is a significant difference of the two samples at the 5% significance level. Based on the test results /rankings, we assign one of three signs (+, , and ) for the comparison of any two algorithms (shown in the last column), where the “ ” sign means the first algorithm (a) is significantly better than the second (b), “ ” sign means that the first algorithm significantly worse, and “ ” sign means that there is no significant difference between the two algorithms. From the results it can be seen that there is no significant difference between most of the algo-rithms, with the exception of SPX with both non-uniform and polynomial in 30D, but SPX with non-uniform is a little bit competitive to the other variants in the 10D test problems. Note that the numbers in the second row of Tables 4 and 5 represent the algorithms TC-NU, TC-P, SBX- NU, SBX-P, SPX-NU, SPX-P, PCX- NU, PCX-P, BLX-α-NU and BLX-α-P, respectively.

A Comparative Study of Different Variants of GAs for Constrained Optimization 185

Fig. 1. The effect of changing on the overall scoring over 36 test problems

Fig. 2. The convergence pattern for 10 variants of GA, for one problem “C01”. Note that, the x-axis is in the log scale.

Based on our proposed scoring scheme, when considered all 36 test problems, the overall scores for the variants SBX-NU, PCX-NU, PCX-P, TC-NU, BLX-α-NU, SBX-P, TC-P, BLX-α-P, SPX-NU and SPX-P are 20.62, 20.09, 19.12, 18.98, 18.66, 17.65, 15.2, 14.45, 10.98, and 2.48, respectively. Note that, the maximum possible score is 36 when a variant dominates all other variants. That means SBX-NU is the best performing variant for the 36 test problems considered in this paper. However it is not easy to make a similar decision with the Wilcoxon Signed Rank Test results.

To see the effect of the constant on the overall scorings, we have plotted versus the overall scores for all the variants in Fig. 1, which shows that TC-NU is the best when is less than 0.2, and SBX-NU is the best when is above 0.2. However, PCX-NU is always in the second place irrespective of the value of .

186 S.M. Elsayed, R.A. Sarker, and D.L. Essam

Finally, a sample convergence pattern for problem “C01” for all variants is shown in Fig. 2.

5 Conclusions

In this paper, ten different variants of GA were implemented and extensively tested on 36 well-known benchmark problems. The results obtained showed that there is no single variant that is able to reach the best results for all test problems. However, the non-uniform mutation is suitable in most of the test problems in both 10D and 30D, but there are also some test problems in which the polynomial mutation works better. The analysis and insights provided in this paper will help researchers and practitioners to decide the variant suitable for their problems.

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