Trade-off between exploration and exploitation with genetic algorithm using a novel selection operator
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Abstract
As an intelligent search optimization technique, genetic algorithm (GA) is an important approach for non-deterministic polynomial (NP-hard) and complex nature optimization problems. GA has some internal weakness such as premature convergence and low computation efficiency, etc. Improving the performance of GA is a vital topic for complex nature optimization problems. The selection operator is a crucial strategy in GA, because it has a vital role in exploring the new areas of the search space and converges the algorithm, as well. The fitness proportional selection scheme has essence exploitation and the linear rank selection is influenced by exploration. In this article, we proposed a new selection scheme which is the optimal combination of exploration and exploitation. This eliminates the fitness scaling issue and adjusts the selection pressure throughout the selection phase. The \(\chi ^2\) goodness-of-fit test is used to measure the average accuracy, i.e., mean difference between the actual and expected number of offspring. A comparison of the performance of the proposed scheme along with some conventional selection procedures was made using TSPLIB instances. The application of this new operator gives much more effective results regarding the average and standard deviation values. In addition, a two-tailed t test is established and its values showed the significantly improved performance by the proposed scheme. Thus, the new operator is suitable and comparable to established selection for the problems related to traveling salesman problem using GA.
Keywords
Genetic algorithm Selection pressure Selection operators Statistical analysis Traveling salesman problemIntroduction
Several modern meta-heuristic algorithms have been developed during the last five decades for solving the non-deterministic polynomial (NP-hard) and complex nature optimization problems. According to some specified criteria, these algorithms can be divided into different groups such as stochastic, deterministic, population, and iterative-based, etc. If an algorithm is trying to improve the solution according to the probabilistic rules, it is called stochastic algorithm. If an algorithm is trying to increase the solution quality with a set of solutions, it is called population-based and to seek the better solution to using multiple iterations called an iterative approach. The two important classifications of population-based algorithms are swarm intelligence and evolutionary approaches which depend on simulation theory with natural phenomenon.
A very common issue about GA is premature convergence to find the optimal solution of a problem. This is strongly linked to the loss of population diversity. If it is very low then a very quick convergence will be observed by GA; otherwise, time-consuming and may cause wastage of computational resources. Hence, there is essential to find a trade-off between exploration (i.e., exploring the new areas of search space) and exploitation (i.e., using already detected points to search the optimum). Therefore, the performance of the GA highly depends on its genetic operators, in general. The first operator is selection being used to choose the set of chromosomes for mating process, the crossover is the second one and used to create new individuals, and the last one is the mutation used for random changes. The balance between exploration and exploitation can be adjusted either by selection pressure in a selection approach or by the recombination operators with adjustment of their probabilities.
The selection scheme is the procedure to choose a sub-population (set of individuals) from the current population that will form the next population. GA is one of those algorithms whose performance is highly affected by the choice of selection operator. Without this mechanism, GA is only simple random sampling giving different results in each generation. Hence, we can say that the selection operator is the backbone of the GA process. Usually, the choice of the selection mechanism depends on the complexity of the problem. A hard approach combined with a conservative replacement mechanism and soft one manipulate an algorithm without sufficient exploring capability which may cause to stuck off on local optima.
There are several selection operators used and reviewed in the literature. A study about various selection approaches and results showed that different schemes perform well in different problems [5]. Thus, the most suitable selection approach has to be chosen in relation to a specific problem to enhance the optimality of desired result. Goldberg and Deb [6] did a comprehensive study of some traditional selection methods through the solutions of differential equations. Another popular study to adjust the probabilistic noise level throughout the mating pool to regulate the selection pressure [7]. Abd-Rahman et al. [8] established a hybrid roulette-tournament selection operator for solving a real-valued shrimp diet formulation problem which can also be generalized to evolutionary algorithm-related problems. A detailed study is about the selection process in GA and examined some common issues in various selection operators in Ref. [5].
The main objective of this study is to present the performance of selection operators that have a major impact on the GAs process. In this way, a new selection operator is proposed that intended to enhance the average quality of the population and gives a better trade-off between exploration and exploitation.
The rest of this article is presented as follows: in “Background” we present the background of selection schemes. The proposed selection operator is presented in “Proposed selection operator” with the statistical properties of a sampling algorithm. The traveling salesman problem (TSP) is discussed and reviewed in “Test problem (traveling salesman problem)”. Performance evaluation of the proposed scheme and conclusions are given in “Performance evaluation” and “Conclusions”, respectively.
Background
The FPS has been widely used selection scheme in various fields such as spanning tree [9], scheduling [10, 11], sources allocation problem [12], menu planning [13], and traveling salesman problem [14]. Throughout the selection procedure, there is no change in the segment size and selection probability. It is easy to implement and gives a high probability for the best individual; these aspects are the main strengths [15]. Another advantage of this approach is that it provides no bias with unlimited spread [16]. However, the difficulty is encountered when a significant difference appears in the fitness values [14, 17, 18]. The scaling problem which is the major drawback of this scheme was first pointed out by Grefenstette [19]. It has happened when population evolves, the ratio between the variance and the fitness average becomes increasingly small. The selection pressure, therefore, drops as the population converges [7]. On the other hand, high selection pressure may lead to premature convergence to a sub-optimal solution.
Proposed selection operator
Motivation
Split rank selection
In this research, we propose an alternative selection scheme [split rank selection (SRS)] that maintains a fine balance between exploration and exploitation. This approach not only eliminates the fitness scaling problem, but also provides an adequate selection pressure throughout the selection phase. In this scheme, all individuals are sorted from worst to best according to their fitness values. All the individuals get a different rank even if they have the same fitness value. Consider a population, a combination of K individuals, i.e., population size (usually it is even). Now, we divide the K individuals into two equal portions.
The sampling procedure
In a two-step selection procedures, i.e., FPS, LRS, ERS, and SRS, etc., a sampling mechanism is required to choose the individuals for mating process. That sampling mechanism fills the mating pool with the individual’s copies of the given population, while respecting the selection probabilities \(p_i\), such that the expected and observed number of individuals are equals. Among the broad variety of sampling mechanisms, we used roulette wheel sampling scheme (or Monte Carlo sampling) for testing the accuracy of the proposed SRS operator.
The \(\chi ^2\) goodness-of-fit measure
Classes \(C_j\) and overall expectation \(\xi _j\) for SRS
j | \(C_j\) | \(\xi _j\) |
---|---|---|
1 | 1–43 | 14.9368 |
2 | 44–61 | 14.9211 |
3 | 62–75 | 15.1421 |
4 | 76–90 | 15.4248 |
5 | 91–103 | 15.6230 |
6 | 104–114 | 14.8549 |
7 | 115–124 | 14.8053 |
8 | 125–133 | 14.3841 |
9 | 134–142 | 15.3876 |
10 | 143–150 | 14.5203 |
Test problem (traveling salesman problem)
The traveling salesman problem (TSP) is one of the most famous benchmark, significant, and historic hard combinatorial optimization problems. The main objective of TSP is to find the shortest Hamiltonian tour in a complete graph with n nodes. It was documented by Euler in 1759 (his interest was how to get rid of the knight’s tour problem) [28]. It is a fundamental problem in the fields of computer science, engineering, operations research, discrete mathematics, and graph theory. In this problem, a salesman visits all cities (nodes) exactly once (the constraint) and then returns to the initial point to complete a tour. It has many applications such as a variety of vehicle routing [29], scheduling [30], and bioinformatics [31] which can easily be transformed into the TSP.
Performance evaluation
In this section, we evaluate the performance of the SRS in comparison to other selection schemes. At first, we present basic information about benchmarks and the parameters setting for GA in “Computational testing methodology”. Second, MATLAB software (version R2015a) was used to compare the simulation study among selection operators, and a detailed discussion on results is given in “Simulation results and discussion”.
Computational testing methodology
The benchmark problems
Problem name | No. of cities | Optimal tour length |
---|---|---|
ftv33 | 34 | 1286 |
berlin52 | 52 | 7542 |
ft70 | 70 | 38,673 |
kroA100 | 100 | 21,282 |
ftv170 | 171 | 2755 |
brg180 | 180 | 1950 |
pr226 | 226 | 80,369 |
rbg323 | 323 | 1326 |
rbg403 | 403 | 2465 |
pa561 | 561 | 2763 |
Parametric configuration for GA
Parameter | Setting |
---|---|
Representation | Permutation |
Population size | 150 |
Crossover criteria | PMX, OX, and CX |
Crossover rate | 80% |
Mutation method | EM |
Mutation rate | 5% |
Maximum generation | 5000 |
Number of trails | 30 |
Replacement in GA | Steady-state GA |
Results of different selection strategies with PMX (crossover) and EM (mutation) operators
Instance | Optimal | Selection scheme | Average | Improvement in SRS (%) | SD | t test |
---|---|---|---|---|---|---|
ftv33 | 1286 | FPS | 1480 | 5.95 | 117 | \(-\) 3.29 |
LRS | 1503 | 7.39 | 138 | \(-\) 3.71 | ||
ERS | 1588 | 12.34 | 223 | \(-\) 4.48 | ||
BTS | 1461 | 4.72 | 110 | \(-\) 2.68 | ||
PTS | 1551 | 10.25 | 186 | \(-\) 4.23 | ||
SRS | 1392 | – | 88 | – | ||
berlin52 | 7542 | FPS | 7703 | 1.17 | 187 | \(-\) 2.28 |
LRS | 7717 | 1.35 | 173 | \(-\) 2.79 | ||
ERS | 7903 | 3.67 | 235 | \(-\) 6.13 | ||
BTS | 7695 | 1.07 | 148 | \(-\) 2.44 | ||
PTS | 7812 | 2.55 | 199 | \(-\) 4.80 | ||
SRS | 7613 | – | 109 | – | ||
ft70 | 38,673 | FPS | 40,692 | 2.04 | 1333 | \(-\) 2.97 |
LRS | 40,174 | 0.77 | 1086 | \(-\) 1.29 | ||
ERS | 42,239 | 5.63 | 1457 | \(-\) 7.95 | ||
BTS | 39,954 | 0.22 | 958 | \(-\) 0.41 | ||
PTS | 40,578 | 1.76 | 1223 | \(-\) 2.73 | ||
SRS | 39,863 | – | 747 | – | ||
kroA100 | 21,282 | FPS | 21,883 | 1.43 | 418 | \(-\) 2.98 |
LRS | 21,962 | 1.78 | 535 | \(-\) 3.23 | ||
ERS | 22,808 | 5.42 | 876 | \(-\) 7.06 | ||
BTS | 21,806 | 1.08 | 506 | \(-\) 2.01 | ||
PTS | 21,980 | 1.90 | 435 | \(-\) 3.83 | ||
SRS | 21,571 | – | 392 | – | ||
ftv170 | 2755 | FPS | 3086 | 4.18 | 249 | \(-\) 2.40 |
LRS | 3129 | 5.50 | 281 | \(-\) 2.93 | ||
ERS | 3266 | 9.46 | 301 | \(-\) 4.96 | ||
BTS | 3163 | 6.51 | 266 | \(-\) 3.65 | ||
PTS | 3178 | 6.95 | 283 | \(-\) 3.74 | ||
SRS | 2957 | – | 157 | – | ||
brg180 | 1950 | FPS | 2199 | 6.18 | 217 | \(-\) 2.74 |
LRS | 2241 | 7.94 | 239 | \(-\) 3.36 | ||
ERS | 2189 | 5.76 | 226 | \(-\) 2.47 | ||
BTS | 2118 | 2.60 | 211 | \(-\) 1.13 | ||
PTS | 2254 | 8.47 | 261 | \(-\) 3.39 | ||
SRS | 2063 | – | 164 | – | ||
pr226 | 80,369 | FPS | 82,180 | 0.73 | 1392 | \(-\) 2.02 |
LRS | 82,321 | 0.93 | 1444 | \(-\) 2.42 | ||
ERS | 83,233 | 1.99 | 1554 | \(-\) 5.10 | ||
BTS | 82,115 | 0.66 | 1101 | \(-\) 2.11 | ||
PTS | 82,821 | 1.50 | 1327 | \(-\) 4.30 | ||
SRS | 81,577 | – | 863 | – | ||
rbg323 | 1326 | FPS | 1631 | 7.48 | 218 | \(-\) 2.24 |
LRS | 1694 | 10.92 | 262 | \(-\) 3.06 | ||
ERS | 1659 | 9.04 | 236 | \(-\) 2.64 | ||
BTS | 1621 | 6.91 | 209 | \(-\) 2.11 | ||
PTS | 1712 | 11.86 | 241 | \(-\) 3.53 | ||
SRS | 1509 | – | 203 | – | ||
rbg403 | 2465 | FPS | 2844 | 3.97 | 247 | \(-\) 2.01 |
LRS | 2803 | 2.57 | 275 | \(-\) 1.19 | ||
ERS | 2897 | 5.73 | 271 | \(-\) 2.77 | ||
BTS | 2788 | 2.04 | 234 | \(-\) 1.05 | ||
PTS | 2820 | 3.16 | 260 | \(-\) 1.52 | ||
SRS | 2731 | – | 185 | – | ||
pa561 | 2763 | FPS | 2979 | 3.52 | 191 | \(-\) 2.66 |
LRS | 2958 | 2.84 | 165 | \(-\) 2.37 | ||
ERS | 3024 | 4.96 | 186 | \(-\) 3.87 | ||
BTS | 2911 | 1.27 | 128 | \(-\) 1.24 | ||
PTS | 2993 | 3.98 | 179 | \(-\) 3.16 | ||
SRS | 2874 | – | 102 | – |
Results of different selection strategies with OX (crossover) and EM (mutation) operators
Instance | Optimal | Selection scheme | Average | Improvement in SRS (%) | SD | t test |
---|---|---|---|---|---|---|
ftv33 | 1286 | FPS | 1498 | 8.74 | 183 | \(-\) 3.50 |
LRS | 1471 | 7.07 | 141 | \(-\) 3.39 | ||
ERS | 1532 | 10.77 | 196 | \(-\) 4.18 | ||
BTS | 1424 | 4.00 | 122 | \(-\) 2.05 | ||
PTS | 1510 | 9.47 | 157 | \(-\) 4.32 | ||
SRS | 1367 | – | 91 | – | ||
berlin52 | 7542 | FPS | 7671 | 1.15 | 172 | \(-\) 2.20 |
LRS | 7618 | 0.46 | 155 | \(-\) 0.93 | ||
ERS | 7809 | 2.89 | 201 | \(-\) 5.11 | ||
BTS | 7607 | 0.32 | 158 | \(-\) 0.63 | ||
PTS | 7732 | 1.93 | 167 | \(-\) 3.80 | ||
SRS | 7583 | – | 135 | – | ||
ft70 | 38,673 | FPS | 40,468 | 2.60 | 1106 | \(-\) 3.92 |
LRS | 39,993 | 1.44 | 1215 | \(-\) 2.03 | ||
ERS | 40,897 | 3.62 | 1414 | \(-\) 4.73 | ||
BTS | 39,641 | 0.57 | 1223 | \(-\) 0.79 | ||
PTS | 40,376 | 2.38 | 1288 | \(-\) 3.26 | ||
SRS | 39,417 | – | 968 | – | ||
kroA100 | 21,282 | FPS | 21,598 | 1.26 | 392 | \(-\) 3.43 |
LRS | 21,593 | 1.24 | 409 | \(-\) 3.25 | ||
ERS | 21,879 | 2.53 | 513 | \(-\) 5.55 | ||
BTS | 21,487 | 0.75 | 388 | \(-\) 2.05 | ||
PTS | 22,036 | 3.22 | 469 | \(-\) 7.71 | ||
SRS | 21,326 | – | 186 | – | ||
ftv170 | 2755 | FPS | 3164 | 5.91 | 251 | \(-\) 3.52 |
LRS | 3107 | 4.18 | 196 | \(-\) 2.90 | ||
ERS | 3223 | 7.63 | 277 | \(-\) 4.29 | ||
BTS | 3083 | 3.44 | 156 | \(-\) 2.70 | ||
PTS | 3198 | 6.91 | 188 | \(-\) 5.06 | ||
SRS | 2977 | – | 148 | – | ||
brg180 | 1950 | FPS | 2140 | 5.79 | 229 | \(-\) 2.44 |
LRS | 2177 | 7.40 | 212 | \(-\) 3.34 | ||
ERS | 2246 | 10.24 | 258 | \(-\) 4.16 | ||
BTS | 2053 | 1.80 | 193 | \(-\) 0.81 | ||
PTS | 2217 | 9.07 | 207 | \(-\) 4.23 | ||
SRS | 2016 | – | 158 | – | ||
pr226 | 80,369 | FPS | 81,715 | 0.74 | 1012 | \(-\) 2.62 |
LRS | 81,967 | 1.05 | 1378 | \(-\) 2.99 | ||
ERS | 83,030 | 2.31 | 1554 | \(-\) 6.09 | ||
BTS | 81,878 | 0.94 | 1229 | \(-\) 2.92 | ||
PTS | 82,372 | 1.53 | 1331 | \(-\) 4.52 | ||
SRS | 81,110 | – | 756 | – | ||
rbg323 | 1326 | FPS | 1625 | 8.43 | 206 | \(-\) 2.66 |
LRS | 1666 | 10.68 | 215 | \(-\) 3.37 | ||
ERS | 1764 | 15.65 | 249 | \(-\) 4.80 | ||
BTS | 1597 | 6.83 | 223 | \(-\) 2.02 | ||
PTS | 1659 | 10.31 | 216 | \(-\) 3.23 | ||
SRS | 1488 | – | 193 | – | ||
rbg403 | 2465 | FPS | 2859 | 4.72 | 311 | \(-\) 2.00 |
LRS | 2776 | 1.87 | 268 | \(-\) 0.85 | ||
ERS | 2864 | 4.89 | 318 | \(-\) 2.04 | ||
BTS | 2727 | 0.11 | 254 | \(-\) 0.05 | ||
PTS | 2863 | 4.86 | 301 | \(-\) 2.10 | ||
SRS | 2724 | – | 201 | – | ||
pa561 | 2763 | FPS | 2955 | 3.25 | 114 | \(-\) 3.51 |
LRS | 2981 | 4.09 | 143 | \(-\) 3.87 | ||
ERS | 3107 | 7.98 | 168 | \(-\) 7.00 | ||
BTS | 2890 | 1.07 | 138 | \(-\) 1.01 | ||
PTS | 3036 | 5.83 | 155 | \(-\) 5.30 | ||
SRS | 2859 | – | 97 | – |
Results of different selection strategies with CX (crossover) and EM (mutation) operators
Instance | Optimal | Selection scheme | Average | Improvement in SRS (%) | SD | t test |
---|---|---|---|---|---|---|
ftv33 | 1286 | FPS | 1547 | 8.40 | 209 | \(-\) 2.62 |
LRS | 1562 | 9.28 | 218 | \(-\) 2.85 | ||
ERS | 1643 | 13.76 | 241 | \(-\) 4.17 | ||
BTS | 1509 | 6.10 | 225 | \(-\) 1.78 | ||
PTS | 1596 | 11.22 | 257 | \(-\) 3.16 | ||
SRS | 1417 | – | 173 | – | ||
berlin52 | 7542 | FPS | 7723 | 1.19 | 199 | \(-\) 2.02 |
LRS | 7739 | 1.38 | 164 | \(-\) 2.67 | ||
ERS | 7951 | 4.01 | 243 | \(-\) 6.16 | ||
BTS | 7694 | 0.81 | 171 | \(-\) 1.51 | ||
PTS | 7804 | 2.20 | 187 | \(-\) 3.97 | ||
SRS | 7632 | – | 146 | – | ||
ft70 | 38,673 | FPS | 40,731 | 2.07 | 1325 | \(-\) 2.69 |
LRS | 40,232 | 0.86 | 1277 | \(-\) 1.12 | ||
ERS | 41,146 | 3.06 | 1362 | \(-\) 3.95 | ||
BTS | 40,128 | 0.60 | 1264 | \(-\) 0.79 | ||
PTS | 41,089 | 2.93 | 1147 | \(-\) 4.16 | ||
SRS | 39,887 | – | 1093 | – | ||
kroA100 | 21,282 | FPS | 21,975 | 2.68 | 512 | \(-\) 3.65 |
LRS | 22,032 | 2.93 | 638 | \(-\) 3.50 | ||
ERS | 23,143 | 6.72 | 756 | \(-\) 10.59 | ||
BTS | 21,876 | 2.68 | 407 | \(-\) 3.21 | ||
PTS | 22,435 | 3.78 | 564 | \(-\) 7.40 | ||
SRS | 21,588 | – | 274 | – | ||
ftv170 | 2755 | FPS | 3190 | 5.80 | 276 | \(-\) 3.16 |
LRS | 3156 | 4.78 | 219 | \(-\) 3.02 | ||
ERS | 3256 | 7.71 | 325 | \(-\) 3.78 | ||
BTS | 3108 | 3.31 | 222 | \(-\) 2.04 | ||
PTS | 3269 | 8.08 | 297 | \(-\) 4.26 | ||
SRS | 3005 | – | 164 | – | ||
brg180 | 1950 | FPS | 2220 | 7.48 | 228 | \(-\) 3.14 |
LRS | 2215 | 7.27 | 201 | \(-\) 3.28 | ||
ERS | 2285 | 10.11 | 246 | \(-\) 4.16 | ||
BTS | 2188 | 6.12 | 225 | \(-\) 2.55 | ||
PTS | 2243 | 8.43 | 218 | \(-\) 3.67 | ||
SRS | 2054 | – | 179 | – | ||
pr226 | 80,369 | FPS | 82,997 | 1.60 | 1367 | \(-\) 4.37 |
LRS | 82,442 | 0.93 | 1056 | \(-\) 2.98 | ||
ERS | 83,564 | 2.26 | 1431 | \(-\) 6.05 | ||
BTS | 82,592 | 1.11 | 1108 | \(-\) 3.46 | ||
PTS | 83,670 | 2.39 | 1042 | \(-\) 7.79 | ||
SRS | 81,673 | – | 941 | – | ||
rbg323 | 1326 | FPS | 1723 | 10.50 | 226 | \(-\) 3.23 |
LRS | 1706 | 9.61 | 252 | \(-\) 2.75 | ||
ERS | 1859 | 17.05 | 320 | \(-\) 4.55 | ||
BTS | 1684 | 8.43 | 231 | \(-\) 2.50 | ||
PTS | 1744 | 11.58 | 228 | \(-\) 3.58 | ||
SRS | 1542 | – | 208 | – | ||
rbg403 | 2465 | FPS | 2911 | 5.50 | 339 | \(-\) 2.20 |
LRS | 2827 | 2.69 | 297 | \(-\) 1.15 | ||
ERS | 2943 | 6.52 | 364 | \(-\) 2.51 | ||
BTS | 2784 | 1.19 | 288 | \(-\) 0.51 | ||
PTS | 2832 | 2.86 | 303 | \(-\) 1.21 | ||
SRS | 2751 | – | 209 | – | ||
pa561 | 2763 | FPS | 3015 | 4.28 | 192 | \(-\) 3.17 |
LRS | 2997 | 3.70 | 178 | \(-\) 2.88 | ||
ERS | 3159 | 8.64 | 247 | \(-\) 5.51 | ||
BTS | 2955 | 2.33 | 164 | \(-\) 1.90 | ||
PTS | 3068 | 5.93 | 156 | \(-\) 5.18 | ||
SRS | 2886 | – | 113 | – |
Simulation results and discussion
Table 4 summarizes the results of six competing selection schemes with PMX and EM as crossover and mutation operators respectively. Results compare on the basis of average, SD, and improved performance of the SRS in percentage (%) values. The significant improvements in the results of SRS with respect to each other approach are indicated through t values. The proposed operator is indicated less average values for all ten benchmarks with stable results (low SD), as well. According to the critical value (\(t = -\ 2.00\)), all computed t scores are less than − 2.00 for all ten benchmark instances and bold t test values have shown the significantly improved performance by the proposed operator. The other t test values which are not bold (non-significant), but negative values indicates a slightly improved performance with respect to an average by the proposed operator. In other words, the simulation results found by the SRS are statistically significant and better than the other five selection approaches (i.e., FPS, LRS, ERS, BTS, and PTS).
We continue our simulation study to check the performance of the proposed operator along with other selection methods and different techniques of crossover and mutation operators. Likewise, in Table 6, we tested the performance of SRS with the pair of CX (crossover operator) and EM (mutation operator). The simulation results indicate the lower average and SD values for all benchmarks by the SRS. Based on statistical perspectives, the SRS outperforms (bold t test values) all the other selection methods for all ten benchmark instances (\(t \le - \ 2.00\)), but, in some cases, only BTS and LRS give non-significant results with the proposed operator. The non-bold t test values are all negative which means that the proposed operator is not worse than any other competing selection operators used in this study. Besides, we can clearly see from Figs. 3, 4 and 5 and analyses performed on the ‘rbg403’ instance that SRS produces lower average results using three different crossover and one mutation operators. We also observe that FPS and BTS produced faster results in early stages, but lead to premature convergence because of high selection pressure. On the other hand, the proposed operator work efficiently throughout the generations taking care of selection pressure and population diversity.
Conclusions
Exploration and exploitation are the two main techniques which employed normally to all the optimization methods. The fitness proportional selection approach has essence exploitation and linear rank approach is influenced by exploration. This article presented a new split ranked selection operator which is a great trade-off between exploration and exploitation. In the proposed procedure, the individuals are ranked according to their fitness scores from worst to best, thus overcoming the fitness scaling issue. After this, split the whole population into two portions and assigning them probabilities for selection based on their ranks. The \(\chi ^2\) goodness-of-fit test confirms that there is insignificant difference between the expected and the actual number of offspring. To evaluate the performance of the proposed operator, we conducted a series of simulation study along with some conventional operators. Computational results proved the superior performance of the new selection scheme in comparison with the traditional GA approaches. The significance of such improvement is also validated through two-tailed t test. Hence, the proposed operator might be a good candidate to get optimum or near to optimum results. Moreover, researchers might be more confident to apply it for any problems related to evolutionary algorithms.
Notes
References
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