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. 2019 Dec 23;28:105046. doi: 10.1016/j.dib.2019.105046

Datasets on statistical analysis and performance evaluation of backtracking search optimisation algorithm compared with its counterpart algorithms

Bryar A Hassan a,c,∗, Tarik A Rashid b
PMCID: PMC6948123  PMID: 31921951

Abstract

In this data article, we present the data used to evaluate the statistical success of the backtracking search optimisation algorithm (BSA) in comparison with the other four evolutionary optimisation algorithms. The data presented in this data article is related to the research article entitles ‘Operational Framework for Recent Advances in Backtracking Search Optimisation Algorithm: A Systematic Review and Performance Evaluation’ [1]. Three statistical tests conducted on BSA compared to differential evolution algorithm (DE), particle swarm optimisation (PSO), artificial bee colony (ABC), and firefly algorithm (FF). The tests are used to evaluate these mentioned algorithms and to determine which one could solve a specific optimisation problem concerning the statistical success of 16 benchmark problems taking several criteria into account. The criteria are initializing control parameters, dimension of the problems, their search space, and number of iterations needed to minimise a problem, the performance of the computer used to code the algorithms and their programming style, getting a balance on the effect of randomization, and the use of different type of optimisation problem in terms of hardness and their cohort. In addition, all the three tests include necessary statistical measures (Mean: mean-solution, S.D.: standard-deviation of mean-solution, Best: the best solution, Worst: the worst solution, Exec. Time: mean runtime in seconds, No. of succeeds: number of successful minimisation, and No. of Failure: number of failed minimisation).

Keywords: Backtracking search optimisation algorithm, Statistical analysis, BSA experimental data, BSA performance evaluation

Specifications table

Subject Theoretical Computer Science
Specific subject area Evolutionary computation
Type of data Table
Graph
How data were acquired Instruments: software, program
Data format Raw
Parameters for data collection Basic statistical measures (Mean: mean-solution, S.D.: standard-deviation of mean-solution, Best: the best solution, Worst: the worst solution, Exec. Time: mean runtime in seconds, No. of succeeds: number of successful minimisation, and No. of Failure: number of failed minimisation) of the 30-solutions obtained by BSA, DE, PSO, ABC, and FF algorithms in three tests.
Description of data collection The data was collected from the MATLAB simulations of running five optimisation algorithms for minimising sixteen benchmark problems. The algorithms are BSA, DE, PSO, ABC, and FF. Each of these algorithms was run thirty times on each of the sixteen benchmarking problems to evaluate the performance of each of them. The collected data are presented on Mendeley Data: Hassan, Bryar; Rashid, Tarik (2019), “Data for statistical analysis and performance evaluation of backtracking search optimisation algorithm compared with its competitive algorithms”, Mendeley Data, v3 https://doi.org/10.17632/hx8xbyjmf5.3
Data source location Institution: Kurdistan Institution for Strategic Studies and Scientific Research
City/Town/Region: Sulaimani
Country: Iraq
Latitude and longitude (and GPS coordinates) for collected samples/data: 35.521700, 45.466605
Data accessibility Raw/primary data are available on Mendeley Data: Hassan, Bryar; Rashid, Tarik (2019), “Data for: statistical analysis and performance evaluation of backtracking search optimisation algorithm compared with its competitive algorithms”, Mendeley Data, v3 https://doi.org/10.17632/hx8xbyjmf5.3
Related research article B.A. Hassan, T.A. Rashid, Operational framework for recent advances in backtracking search optimisation algorithm: A systematic review and performance evaluation, Appl. Math. Comput. (2019) 124919., DOI: https://doi.org/10.1016/j.amc.2019.124919 [1]
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Value of the Data
  • •

    The statistical analysis data on of backtracking search optimisation algorithm with its competitive algorithms may be useful for the researcher to better adapt these techniques for particular problems.

  • •

    This dataset shows the best parameter settings used in the tests, which aid scholars to enhance reanalysis or/and reproducibility.

  • •

    This dataset helps better understanding the BSA algorithm in depth by providing experiments and additional information.

  • •

    This data provided gives an insight into BSA statistical performance compared with its competitors to encourage further researches on BSA in real-world applications.

  • •

    Three tests on sixteen benchmark functions are used to statistically evaluate the performance of BSA compared to DE, PSO, ABS, and FF.

  • •

    Necessary statistical measures are obtained by BSA, DE, PSO, ABC, and FF algorithms in three tests for sixteen benchmark functions.

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1. Data

The dataset contains the simulation result of performance evaluation of BSA compared to its counterpart algorithms (DE, PSO, ABS, and FF). The performance of these algorithms is evaluated on minimising sixteen benchmark functions. This evaluation is conducted by three tests. The optimisation benchmark problems used in these tests are presented in Table 1. The data files (xlsx format) are the raw data results from Tests 1, 2, and 3. Additionally, information about the Tests 1, 2, and 3 are depicted in the tables as follows: (i) Table 2, Table 3, Table 4 present basic statistics of 30-solutions obtained by the algorithms in Test 1. (ii) Table 5, Table 6, Table 7 present basic statistics of 30-solutions obtained by the algorithms in Test 2. (iii) Table 8, Table 9 presents the ratio of successful minimisation of the functions for Tests 1 and 2.

Table 1.

The sixteen benchmark problems.

Name Success rate (%)
Ackley (F1) 48.25
Alpine01 (F2) 65.17
Bird (F3) 59.00
Leon (F4) 41.17
CrossInTray (F5) 74.08
Easom (F6) 26.08
Whitley (F7) 4.92
EggCrate (F8) 64.92
Griewank (F9) 6.08
HolderTable (F10) 80.08
Rastrigin (F11) 39.50
Rosenbrock (F12) 44.17
Salomon (F13) 10.33
Sphere (F14) 82.75
StyblinskiTang (F15) 70.50
Schwefel26 (F16) 62.67
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Table 2.

Basic statistics of the 30-solutions obtained by BSA, DE, PSO, ABC, and FF in Test 1 (default search space with Nvar1 dimensions).

Functions Statistics BSA DE PSO ABC FF
F1 Mean 688.3666667 209.8 NC NC 323.3333333
S.D. 61.19808614 5.761944116 NC NC 7.716231586
Best 584 193 NC NC 301
Worst 830 220 NC NC 334
Exec. time 0.3129113 0.767031967 NC NC 8.876311833
No. of Succeeds 30 30 0 0 30
No. of Failures 0 0 30 30 0
F2 Mean 78.96666667 305.1666667 NC NC 93.43333333
S.D. 11.21723838 22.1672068 NC NC 78.09882481
Best 63 245 NC NC 12
Worst 109 348 NC NC 248
Exec. time 0.079013967 0.613814533 NC NC 1.1518529
No. of Succeeds 30 30 0 0 30
No. of Failures 0 0 30 30 0
F3 Mean 288.2 128.7666667 54.59259259 35.33333333 32.5
S.D. 57.68487521 40.0548618 4.901014489 7.169539654 14.28708122
Best 193 72 44 23 6
Worst 387 201 62 53 63
Exec. time 0.121238167 0.392393267 0.097268 0.087277867 0.656926867
No. of Succeeds 30 30 27 30 30
No. of Failures 0 0 3 0 0
F4 Mean 387.7 3960.133333 857 128.3333333 1480.666667
S.D. 127.0707514 244.6563717 675.7609249 136.9530756 34.20963096
Best 47 3546 5 26 1388
Worst 588 4472 1747 641 1520
Exec. time 0.142008467 10.51661843 1.519789933 0.217805708 22.88727067
No. of Succeeds 30 30 30 24 30
No. of Failures 0 0 0 6 0
F5 Mean 121.1666667 38.26666667 27.23333333 14.76666667 3.766666667
S.D. 34.72362143 8.337424283 9.814427569 4.031627834 0.678910554
Best 45 23 6 7 2
Worst 175 52 40 24 5
Exec. time 0.027849683 0.095730833 0.033622767 0.042479367 0.057955067
No. of Succeeds 30 30 30 30 30
No. of Failures 0 0 0 0 0
F6 Mean 760.4333333 208.5 63.83333333 NC 128.1666667
S.D. 63.97665702 33.34847357 3.939922398 NC 20.77977022
Best 617 162 54 NC 91
Worst 880 326 71 NC 166
Exec. time 0.251127833 0.5434189 0.094766533 NC 1.785906667
No. of Succeeds 30 30 30 0 30
No. of Failures 0 0 0 30 0
F7 Mean 972 4809.6 NC NC NC
S.D. 349.46797 1851.094463 NC NC NC
Best 555 1616 NC NC NC
Worst 1926 8339 NC NC NC
Exec. time 0.505736867 8.1352712 NC NC NC
No. of Succeeds 30 10 0 0 0
No. of Failures 0 20 30 30 30
F8 Mean 119.1 2897.133333 637.5666667 15.875 5.071428571
S.D. 30.35121994 22.93278534 4.636313438 2.997281377 1.741190981
Best 63 2855 627 11 3
Worst 169 2944 650 21 10
Exec. time 0.044835 4.779958067 1.223019467 0.029835208 0.066557357
No. of Succeeds 30 30 30 24 28
No. of Failures 0 0 0 6 2
F9 Mean 373.1 805.2666667 NC NC NC
S.D. 82.92345868 100.8679346 NC NC NC
Best 285 631 NC NC NC
Worst 611 1081 NC NC NC
Exec. time 0.172796067 1.5184459 NC NC NC
No. of Succeeds 30 30 0 0 0
No. of Failures 0 0 30 30 30
F10 Mean 314.1333333 72.83333333 54.26666667 NC 20.83333333
S.D. 61.64119387 14.48681525 7.315233672 NC 11.21293383
Best 209 43 40 NC 5
Worst 418 111 65 NC 44
Exec. time 0.1276912 0.181376167 0.088567367 NC 0.2915058
No. of Succeeds 30 30 30 0 30
No. of Failures 0 0 0 30 0
F11 Mean 1038.933333 562.3666667 NC NC NC
S.D. 128.4778079 28.48046316 NC NC NC
Best 811 508 NC NC NC
Worst 1293 634 NC NC NC
Exec. time 0.379444367 0.744079967 NC NC NC
No. of Succeeds 30 30 0 0 0
No. of Failures 0 0 30 30 30
F12 Mean 1119.566667 NC NC NC NC
S.D. 307.9534989 NC NC NC NC
Best 578 NC NC NC NC
Worst 2001 NC NC NC NC
Exec. time 0.3752452 NC NC NC NC
No. of Succeeds 30 0 0 0 0
No. of Failures 0 30 30 30 30
F13 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F14 Mean 151.9 154.6 1967.9 NC 12.53333333
S.D. 31.17840765 4.123941871 517.0026579 NC 1.008013866
Best 107 148 1782 NC 11
Worst 208 162 4691 NC 15
Exec. time 0.092171037 0.463599767 3.315528667 NC 0.156159333
No. of Succeeds 30 30 30 0 30
No. of Failures 0 0 0 30 0
F15 Mean 490.7666667 NC 100 659.6666667 245.8461538
S.D. 56.80073053 NC 5.253570215 108.7443485 4.017589531
Best 345 NC 94 516 240
Worst 598 NC 109 949 253
Exec. time 0.2717145 NC 0.188506 1.422079033 3.067869077
No. of Succeeds 30 0 11 30 13
No. of Failures 0 30 19 0 17
F16 Mean NC 254.9615385 NC NC NC
S.D. NC 14.32614608 NC NC NC
Best NC 231 NC NC NC
Worst NC 289 NC NC NC
Exec. time NC 0.572146308 NC NC NC
No. of Succeeds 0 26 0 0 0
No. of Failures 30 4 30 30 30
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Table 3.

Basic statistics of the 30-solutions obtained by BSA, DE, PSO, ABC, and FF in Test 1 (default search space with Nvar2 dimensions).

Functions Statistics BSA DE PSO ABC FF
F1 Mean 1391.866667 600.8 NC NC 373.0333333
S.D. 240.1736058 11.12747454 NC NC 4.31903033
Best 949 572 NC NC 362
Worst 1899 626 NC NC 381
Exec. time 0.546482767 2.1215337 NC NC 9.566863367
No. of Succeeds 30 30 0 0 30
No. of Failures 0 0 30 30 0
F2 Mean 467.4333333 2653.133333 NC NC 346.0333333
S.D. 54.7487574 311.8812477 NC NC 34.27121375
Best 397 2136 NC NC 227
Worst 601 3367 NC NC 408
Exec. time 0.2806565 6.5514224 NC NC 6.9452777
No. of Succeeds 30 30 0 0 30
No. of Failures 0 0 30 30 0
F3 Mean 346.6333333 170.8666667 60.79310345 35.6 32.7
S.D. 84.07528783 38.91346131 28.86592073 6.003447286 16.26748396
Best 182 99 44 26 5
Worst 505 236 209 45 67
Exec. time 0.190157233 0.5601642 0.115230897 0.0631293 0.644730733
No. of Succeeds 30 30 29 30 30
No. of Failures 0 0 1 0 0
F4 Mean 471.6333333 5530.1 1152.866667 91.71428571 1492
S.D. 139.2421566 266.6691544 496.0978583 41.26953602 38.38821658
Best 180 4762 4 23 1323
Worst 862 6071 1644 160 1533
Exec. time 0.193297133 14.77711263 2.1152108 0.253110071 22.1456451
No. of Succeeds 30 30 30 28 30
No. of Failures 0 0 0 2 0
F5 Mean 169.1 41.76666667 29.86666667 13.8 3.733333333
S.D. 49.42800407 11.49717607 11.64335931 4.604345773 0.868344971
Best 70 14 2 2 2
Worst 248 62 54 22 5
Exec. time 0.044610567 0.106829767 0.046657733 0.044076667 0.051693733
No. of Succeeds 30 30 30 30 30
No. of Failures 0 0 0 0 0
F6 Mean 912.9 267.5666667 64.8 NC 121.7333333
S.D. 97.77713928 41.86487568 4.574290976 NC 34.65985786
Best 678 194 55 NC 19
Worst 1045 386 75 NC 173
Exec. time 0.318349367 0.724312633 0.127859867 NC 1.7146897
No. of Succeeds 30 30 30 0 30
No. of Failures 0 0 0 30 0
F7 Mean 4764.566667 NC NC NC NC
S.D. 624.9610572 NC NC NC NC
Best 3887 NC NC NC NC
Worst 6810 NC NC NC NC
Exec. time 3.934701977 NC NC NC NC
No. of Succeeds 30 0 0 0 0
No. of Failures 0 30 30 30 30
F8 Mean 133.1666667 3542.366667 637.3333333 15.55172414 4.666666667
S.D. 35.99337423 39.65409344 5.441433212 2.599260979 1.109400392
Best 35 3436 625 9 3
Worst 203 3610 651 21 8
Exec. time 0.063654233 6.444122133 1.30194 0.030338172 0.077771407
No. of Succeeds 30 30 30 29 27
No. of Failures 0 0 0 1 3
F9 Mean 587.0333333 1297 NC NC NC
S.D. 80.61123535 45.5290737 NC NC NC
Best 469 1234 NC NC NC
Worst 841 1423 NC NC NC
Exec. time 0.295489567 2.409856833 NC NC NC
No. of Succeeds 30 30 0 0 0
No. of Failures 0 0 30 30 30
F10 Mean 366.8666667 91.63333333 199.1666667 NC 19.8
S.D. 42.6790212 14.87994098 554.8413929 NC 10.03579799
Best 301 63 36 NC 5
Worst 457 125 2450 NC 45
Exec. time 0.187796133 0.2636237 0.357409 NC 0.275910433
No. of Succeeds 30 30 30 0 30
No. of Failures 0 0 0 30 0
F11 Mean 3761.9 4211.533333 NC NC NC
S.D. 503.0939687 835.0827642 NC NC NC
Best 2890 3339 NC NC NC
Worst 4679 7001 NC NC NC
Exec. time 1.841760067 6.437364733 NC NC NC
No. of Succeeds 30 30 0 0 0
No. of Failures 0 0 30 30 30
F12 Mean 1518.7 NC NC NC NC
S.D. 520.5891987 NC NC NC NC
Best 871 NC NC NC NC
Worst 2901 NC NC NC NC
Exec. time 0.457080213 NC NC NC NC
No. of Succeeds 30 0 0 0 0
No. of Failures 0 30 30 30 30
F13 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F14 Mean 417 484.4333333 NC 714.3 54.36666667
S.D. 56.51914537 7.219147592 NC 77.84739714 2.85854235
Best 303 469 NC 568 50
Worst 531 497 NC 834 63
Exec. time 0.2572861 0.841805833 NC 2.2013179 0.5315005
No. of Succeeds 30 30 0 30 30
No. of Failures 0 0 30 0 0
F15 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F16 Mean NC 991 NC NC NC
S.D. NC 1078.5 NC NC NC
Best NC 52.59882474 NC NC NC
Worst NC 991 NC NC NC
Exec. time NC 1200 NC NC NC
No. of Succeeds 0 12 0 0 0
No. of Failures 30 18 30 30 30
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Table 4.

Basic statistics of the 30-solutions obtained by BSA, DE, PSO, ABC, and FF in Test 1 (default search space with Nvar3 dimensions).

Functions Statistics BSA DE PSO ABC FF
F1 Mean 2212.033333 1265.4 NC NC 409.8
S.D. 285.8103402 17.29380994 NC NC 2.578425074
Best 1789 1231 NC NC 404
Worst 2705 1297 NC NC 414
Exec. time 1.2222052 4.161145333 NC NC 6.678406533
No. of Succeeds 30 30 0 0 30
No. of Failures 0 0 30 30 0
F2 Mean 1437.033333 1233.333333 NC NC 453.4666667
S.D. 192.1665017 188.0521449 NC NC 26.07381211
Best 1129 907 NC NC 411
Worst 1982 1619 NC NC 542
Exec. time 0.946699633 2.589517933 NC NC 9.553941067
No. of Succeeds 30 30 0 0 30
No. of Failures 0 0 30 30 0
F3 Mean 364.7666667 186.8666667 53.80769231 38.33333333 28.26666667
S.D. 73.52535726 51.15475731 5.592990118 8.326663998 19.5711493
Best 242 124 39 21 4
Worst 525 284 61 56 81
Exec. time 0.217162567 0.4063774 0.105839692 0.0705598 0.603975867
No. of Succeeds 30 30 26 30 30
No. of Failures 0 0 4 0 0
F4 Mean 525.7 6225.6 909.4 84.7037037 1489.633333
S.D. 218.5027657 461.136198 641.2112245 43.79481425 32.82606548
Best 284 5442 6 23 1409
Worst 1269 7563 1837 186 1531
Exec. time 0.2490285 14.56781393 2.5651712 0.266280556 20.34113953
No. of Succeeds 30 30 30 27 30
No. of Failures 0 0 0 3 0
F5 Mean 173.5333333 45.3 24.2 14.53333333 3.666666667
S.D. 39.28299916 10.16807038 10.69772776 4.904138529 0.884086645
Best 82 25 6 2 2
Worst 234 64 41 23 5
Exec. time 0.109724933 0.132266967 0.043759533 0.045434933 0.051644167
No. of Succeeds 30 30 30 30 30
No. of Failures 0 0 0 0 0
F6 Mean 996.1 296.5666667 64.16666667 NC 113.7333333
S.D. 88.20757259 47.62691454 4.609460536 NC 26.92509832
Best 861 209 55 NC 56
Worst 1197 408 76 NC 168
Exec. time 0.4946741 0.879912567 0.118392067 NC 1.790483167
No. of Succeeds 30 30 30 0 30
No. of Failures 0 0 0 30 0
F7 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F8 Mean 118.2 3750.9 637.0333333 15.03571429 4.5
S.D. 27.9425519 44.63055239 6.392254652 2.486737307 0.989949494
Best 78 3643 624 10 2
Worst 190 3811 649 20 6
Exec. time 0.169004103 6.867129 1.4425986 0.029832 0.073269769
No. of Succeeds 30 30 30 28 26
No. of Failures 0 0 0 2 4
F9 Mean 965.9666667 2588.033333 NC NC NC
S.D. 76.88974791 72.99715787 NC NC NC
Best 855 2514 NC NC NC
Worst 1150 2898 NC NC NC
Exec. time 0.582512467 5.4244718 NC NC NC
No. of Succeeds 30 30 0 0 0
No. of Failures 0 0 30 30 30
F10 Mean 397.2333333 92.83333333 56.66666667 NC 20.83333333
S.D. 43.98172295 15.20681184 5.168427583 NC 10.19493902
Best 309 41 49 NC 7
Worst 471 114 67 NC 38
Exec. time 9565.148219 0.241654933 0.099469833 NC 0.284334333
No. of Succeeds 30 30 30 0 30
No. of Failures 0 0 0 30 0
F11 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F12 Mean 16054.2 NC NC NC NC
S.D. 22085.31046 NC NC NC NC
Best 9065 NC NC NC NC
Worst 132801 NC NC NC NC
Exec. time 6.478484033 NC NC NC NC
No. of Succeeds 30 0 0 0 0
No. of Failures 0 30 30 30 30
F13 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F14 Mean 943.9 1060 NC 6468.933333 113.1333333
S.D. 84.9061754 15.59840841 NC 311.951912 3.329422994
Best 798 1034 NC 5878 106
Worst 1109 1097 NC 7116 120
Exec. time 0.621603966 3.001068333 NC 19.45626283 1.025718967
No. of Succeeds 30 30 0 30 30
No. of Failures 0 0 30 0 0
F15 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F16 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
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Table 5.

Basic statistics of the 30-solutions obtained by BSA, DE, PSO, ABC, and FF in Test 2 (two-variable dimensions with R1).

Functions Statistics BSA DE PSO ABC FF
F1 Mean 72.33333333 NC 0 19.33333333 60.36666667
S.D. 16.00287331 NC NC 2.368374 23.48217715
Best 45 NC NC 14 5
Worst 101 NC NC 23 95
Exec. time 0.028775833 NC NC 0.159118567 1.193958333
No. of Succeeds 30 0 NC 30 30
No. of Failures 0 30 0 0 0
F2 Mean 39.5 15.26666667 794.25 9.133333333 4.033333333
S.D. 11.70838191 2.935318033 379.0831654 1.907034769 0.808716878
Best 22 10 154 4 2
Worst 76 22 1146 13 5
Exec. time 0.00711624 0.045706 1.445961536 0.0771137 0.064563133
No. of Succeeds 30 30 28 30 30
No. of Failures 0 0 2 0 0
F3 Mean 139.5 53.33333333 53.75 NC 31.63333333
S.D. 28.07717198 14.21347911 9.777619948 NC 12.64224809
Best 73 32 40 NC 9
Worst 206 79 97 NC 59
Exec. time 0.076975133 0.084473933 0.092407821 NC 0.493045367
No. of Succeeds 30 30 28 0 30
No. of Failures 0 0 2 30 0
F4 Mean 377.1333333 89.69230769 1213.866667 16 9.269230769
S.D. 68.4693404 33.18405549 159.6013856 47.125 3.053623321
Best 286 28 926 33.18369492 5
Worst 649 191 1550 6 19
Exec. time 0.200669233 0.293420846 2.0382361 147 0.126300154
No. of Succeeds 30 26 30 0.281062708 26
No. of Failures 0 4 0 24 4
F5 Mean 45.56666667 15.03333333 21.8 7.433333333 3.133333333
S.D. 17.02469471 5.18275212 9.459168149 3.339247627 0.730296743
Best 13 5 3 2 2
Worst 76 24 37 13 4
Exec. time 0.019755067 0.032688133 0.039887567 0.077219 0.0421937
No. of Succeeds 30 30 30 30 30
No. of Failures 0 0 0 0 0
F6 Mean 248.2666667 30.86666667 794.25 20.3 11.96666667
S.D. 31.22325982 4.108303898 379.0831654 1.600646421 7.716901886
Best 202 25 154 17 4
Worst 311 45 1146 24 34
Exec. time 0.0888098 0.051433367 0.0724958 0.214975967 0.094171033
No. of Succeeds 30 30 28 30 30
No. of Failures 0 0 2 0 0
F7 Mean 64.36666667 139.5714286 NC 27.6 249.9473684
S.D. 12.86396553 16.33997274 NC 22.41568999 26.50885321
Best 41 103 NC 9 193
Worst 91 170 NC 89 282
Exec. time 0.038450533 0.427799464 NC 0.25814615 3.344931158
No. of Succeeds 30 28 0 20 19
No. of Failures 0 2 30 10 11
F8 Mean 248.5666667 18.48148148 636.9 12.24 4.851851852
S.D. 23.91894839 2.359438825 5.984750737 2.241279397 0.948833442
Best 206 15 627 8 4
Worst 300 23 651 16 7
Exec. time 0.042607433 0.032870519 0.713972767 0.09773984 0.068419926
No. of Succeeds 30 27 30 25 27
No. of Failures 0 3 0 5 3
F9 Mean 25.23333333 166.5769231 100.7142857 9.076923077 539.3793103
S.D. 7.833100999 110.8112532 5.13552591 5.483962632 45.94671898
Best 14 89 92 2 413
Worst 47 691 109 19 597
Exec. time 0.016005467 0.554092731 0.162314571 0.046853615 6.89237431
No. of Succeeds 30 26 14 26 29
No. of Failures 0 4 16 4 1
F10 Mean 36.9 60.15 NC NC 5.333333333
S.D. 17.7906407 179.1662783 NC NC 1.688364508
Best 16 14 NC NC 3
Worst 79 821 NC NC 10
Exec. time 0.029989533 0.17157865 NC NC 0.084701533
No. of Succeeds 30 20 0 0 30
No. of Failures 0 10 30 30 0
F11 Mean 111.3666667 81.86666667 105.5 29.22222222 9.96
S.D. 16.08486831 3.559671947 4.289120157 6.405126152 3.813135193
Best 79 76 93 16 4
Worst 147 89 117 45 20
Exec. time 0.034564033 0.277437967 0.1759911 0.152298889 0.14458784
No. of Succeeds 30 30 30 27 25
No. of Failures 0 0 0 3 5
F12 Mean 157.9333333 60.77272727 722.8 35.88888889 8.083333333
S.D. 35.35332334 24.92069673 104.1181027 24.95585847 2.019829237
Best 98 8 414 5 5
Worst 236 103 857 94 13
Exec. time 0.145922333 0.145608045 1.219939367 0.322523741 0.111710708
No. of Succeeds 30 22 30 27 24
No. of Failures 0 8 0 3 6
F13 Mean 271.8 147.137931 646.36 133.7666667 10.92592593
S.D. 40.4086027 52.0286488 5.321967055 100.097774 2.840990157
Best 201 57 634 27 6
Worst 344 260 656 349 17
Exec. time 0.051252967 0.229098862 1.19053768 1.2913116 0.168615333
No. of Succeeds 30 29 25 30 27
No. of Failures 0 1 5 0 3
F14 Mean 232.7666667 10.2 637.5 6 2.68
S.D. 20.00979645 2.483630619 5.544490897 1.560378995 0.476095229
Best 190 6 626 3 2
Worst 275 14 654 9 3
Exec. time 0.077608533 0.034451 1.2173071 0.05095925 0.033408
No. of Succeeds 30 20 30 24 25
No. of Failures 0 10 0 6 5
F15 Mean 52.06666667 22.43333333 44.16666667 19.1 11.7
S.D. 21.2244708 2.51455533 6.454527058 3.262641198 5.408486307
Best 21 18 27 10 3
Worst 90 27 52 25 23
Exec. time 0.031657633 0.0592526 0.0887543 0.138674967 0.151984633
No. of Succeeds 30 30 30 30 30
No. of Failures 0 0 0 0 0
F16 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
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Table 6.

Basic statistics of the 30-solutions obtained by BSA, DE, PSO, ABC, and FF in Test 2 (two variable dimensions with R2).

Functions Statistics BSA DE PSO ABC FF
F1 Mean 425.7333333 NC NC NC 249.4137931
S.D. 65.63111822 NC NC NC 24.0735736
Best 338 NC NC NC 192
Worst 535 NC NC NC 295
Exec. time 0.129019233 NC NC NC 5.009911103
No. of Succeeds 30 0 0 0 29
No. of Failures 0 30 30 30 1
F2 Mean 156.0333333 39.10344828 732.0740741 38.33333333 61
S.D. 16.09558374 5.1084302 407.9909431 6.199740447 22.28486896
Best 126 26 166 27 11
Worst 188 49 1158 49 93
Exec. time 0.039772367 0.112101655 1.457569407 0.197538433 0.884010333
No. of Succeeds 30 29 27 30 27
No. of Failures 0 1 3 0 3
F3 Mean 1087.666667 233.3888889 81.73333333 NC 208.0333333
S.D. 391.2479885 71.84144215 6.356822744 NC 33.0146084
Best 293 103 72 NC 117
Worst 1719 376 97 NC 250
Exec. time 0.1936562 0.3856845 0.1286414 NC 3.080032433
No. of Succeeds 30 18 30 0 30
No. of Failures 0 12 0 30 0
F4 Mean 419.7666667 194.0714286 1253.52381 219.6896552 93.8
S.D. 191.3771657 65.9005584 154.2182282 132.8401573 24.72515588
Best 189 64 981 33 47
Worst 898 367 1517 636 145
Exec. time 0.263658433 0.674849321 2.085545905 1.637981655 1.34841468
No. of Succeeds 30 28 21 29 25
No. of Failures 0 2 9 1 5
F5 Mean 99.93333333 29 54 NC 40.3
S.D. 22.2539161 3.695290572 10.22168081 NC 26.1786304
Best 53 22 28 NC 3
Worst 143 36 68 NC 85
Exec. time 0.043615233 0.0518579 0.0923664 NC 0.349824067
No. of Succeeds 30 30 30 0 30
No. of Failures 0 0 0 30 0
F6 Mean 671.8 230.1333333 75.13333333 NC NC
S.D. 107.4107167 48.5107406 5.888231005 NC NC
Best 424 162 61 NC NC
Worst 883 341 91 NC NC
Exec. time 0.229059867 0.646945533 0.120696033 NC NC
No. of Succeeds 30 30 30 0 0
No. of Failures 0 0 0 30 30
F7 Mean 112.6333333 157.6551724 NC 30.61111111 445.8666667
S.D. 30.11527661 16.00061575 NC 9.798792776 33.75033418
Best 84 125 NC 16 343
Worst 232 186 NC 58 499
Exec. time 0.0441853 0.534595448 NC 0.272917111 5.854646267
No. of Succeeds 30 29 0 18 30
No. of Failures 0 1 30 12 0
F8 Mean 328.6666667 32.08 644.2333333 23.76 110.7666667
S.D. 31.07425293 3.414674216 5.992428172 3.3326666 24.39712156
Best 261 24 633 17 53
Worst 419 37 661 29 151
Exec. time 0.0566179 0.05600776 1.142361633 0.18997352 1.5272697
No. of Succeeds 30 25 30 25 30
No. of Failures 0 5 0 5 0
F9 Mean 181.5333333 175.8 794.25 201.5769231 742.3103448
S.D. 44.83743305 19.91256751 379.0831654 144.6715378 47.85178266
Best 104 145 154 32 547
Worst 262 213 1146 488 807
Exec. time 0.1036287 0.459873733 0.419778435 0.98108872 9.115128828
No. of Succeeds 30 30 28 26 29
No. of Failures 0 0 2 4 1
F10 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F11 Mean 138.1666667 96.83333333 122.4827586 43 149.8148148
S.D. 28.06652606 4.177801215 5.925855512 7.541883054 38.67421824
Best 94 85 114 31 50
Worst 201 103 142 55 210
Exec. time 0.059349433 0.282379867 0.199407138 0.424502846 2.34202537
No. of Succeeds 30 30 29 26 27
No. of Failures 0 0 1 4 3
F12 Mean 572.8333333 368.2222222 936.1 409.5882353 80.37037037
S.D. 122.3739558 99.16549234 243.0311957 225.0005719 27.80999841
Best 411 210 605 55 19
Worst 934 565 1858 944 120
Exec. time 0.310757733 1.056444556 1.5107343 3.453786647 1.234736259
No. of Succeeds 30 27 30 17 27
No. of Failures 0 3 0 13 3
F13 Mean 427.8666667 179.8333333 764.962963 204.9259259 118.4333333
S.D. 68.73172902 39.81689412 229.5504774 118.616034 25.96374572
Best 311 80 647 23 56
Worst 601 270 1392 457 163
Exec. time 0.123285933 0.2651661 1.240302815 1.956210444 1.893014833
No. of Succeeds 30 30 27 27 30
No. of Failures 0 0 3 3 0
F14 Mean 269.0666667 23.85185185 642.8333333 17.16666667 41.35714286
S.D. 29.88706713 3.134224278 6.772578246 1.340560125 15.61626475
Best 206 18 630 15 14
Worst 339 29 658 20 74
Exec. time 0.0919827 0.066911815 1.159890367 0.1416615 0.626989893
No. of Succeeds 30 27 30 24 28
No. of Failures 0 3 0 6 2
F15 Mean 135.0333333 36.86666667 68.33333333 31.33333333 163.8
S.D. 27.06758251 3.702127378 3.950847428 3.283536081 25.20180618
Best 85 27 58 24 87
Worst 233 44 74 38 207
Exec. time 0.033026633 0.087536133 0.1189461 0.236676 1.600936967
No. of Succeeds 30 30 30 30 30
No. of Failures 0 0 0 0 0
F16 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
Open in a new tab

Table 7.

Basic statistics of the 30-solutions obtained by BSA, DE, PSO, ABC, and FF in Test 2 (two-variable dimensions with R3).

Functions Statistics BSA DE PSO ABC FF
F1 Mean 556.6333333 NC NC NC 279.75
S.D. 112.7296073 NC NC NC 33.31962437
Best 396 NC NC NC 171
Worst 898 NC NC NC 315
Exec. time 0.189253867 NC NC NC 5.1484892
No. of Succeeds 30 0 0 0 20
No. of Failures 0 30 30 30 10
F2 Mean 154.6666667 46.16666667 847.8571429 47.46666667 88.46666667
S.D. 24.94730078 6.411645296 367.6544991 12.19308264 26.17017694
Best 113 27 165 32 36
Worst 201 63 1152 95 141
Exec. time 0.067592267 0.139866267 1.5282565 0.340741933 1.4476881
No. of Succeeds 30 30 28 30 30
No. of Failures 0 0 2 0 0
F3 Mean 1909.433333 182.75 84.46666667 NC 249.6
S.D. 591.0772564 82.99949799 10.78547341 NC 29.66897834
Best 1402 112 67 NC 159
Worst 3725 302 118 NC 294
Exec. time 0.3167525 0.139866267 0.127532267 NC 3.4745561
No. of Succeeds 30 4 30 0 30
No. of Failures 0 26 0 30 0
F4 Mean 444.0666667 220.0357143 1482.153846 238.2592593 118.6153846
S.D. 81.262 62.51487654 824.6896479 120.881249 31.97258441
Best 309 100 955 45 51
Worst 659 365 5332 415 180
Exec. time 0.179457933 0.7116585 1.445961536 2.140455593 1.692470923
No. of Succeeds 30 28 26 27 26
No. of Failures 0 2 4 3 4
F5 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F6 Mean 1096.466667 570.2666667 82.76666667 NC NC
S.D. 263.5093054 179.2411078 8.071590594 NC NC
Best 691 295 66 NC NC
Worst 1590 984 100 NC NC
Exec. time 0.3102513 1.701697967 0.111516034 NC NC
No. of Succeeds 30 30 30 0 0
No. of Failures 0 0 0 30 30
F7 Mean 165.1333333 158.3103448 NC 36.83333333 474.3
S.D. 44.85281165 18.3655412 NC 20.98809186 31.4896261
Best 98 117 NC 22 409
Worst 292 188 NC 116 528
Exec. time 0.107162467 0.423307759 NC 0.336140889 6.128933267
No. of Succeeds 30 29 0 18 30
No. of Failures 0 1 30 12 0
F8 Mean 319.0666667 35.30769231 643.8333333 27.2 145.3043478
S.D. 38.74935669 2.412786452 6.198349799 2.645751311 19.00104012
Best 271 31 634 22 115
Worst 401 40 657 31 182
Exec. time 0.0543112 0.060988577 1.248304667 0.21520208 2.085555391
No. of Succeeds 30 26 30 25 23
No. of Failures 0 4 0 5 7
F9 Mean 145.1666667 176.3 370.7083333 179.9615385 789.7666667
S.D. 36.97257449 17.78831465 400.0615691 118.135001 27.78945885
Best 99 147 118 29 725
Worst 254 207 1277 460 842
Exec. time 0.08441 0.555940233 0.588364917 0.943245115 8.9030898
No. of Succeeds 30 30 24 26 30
No. of Failures 0 0 6 4 0
F10 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
F11 Mean 143.9333333 98.93333333 129.9333333 45.42857143 195.5517241
S.D. 38.00901285 4.555658349 19.80061534 7.593508785 30.62513698
Best 89 88 115 25 91
Worst 211 108 227 60 242
Exec. time 0.1755195 0.305618167 0.2071013 0.423829429 3.17409831
No. of Succeeds 30 30 30 28 29
No. of Failures 0 0 0 2 1
F12 Mean 1379.166667 427.9230769 898.9 569.5714286 116.5
S.D. 156.3588012 117.8514058 254.7484696 246.4879018 38.30820432
Best 1101 263 585 342 24
Worst 1879 667 1753 932 181
Exec. time 0.4534084 0.7652505 1.454673867 4.816096714 1.725299179
No. of Succeeds 30 26 30 7 28
No. of Failures 0 4 0 23 2
F13 Mean 381.5666667 171.5333333 700.1785714 224.8076923 154.1
S.D. 102.9180066 53.29147783 134.0732397 135.1523642 29.25553293
Best 205 75 645 46 76
Worst 756 276 1374 456 199
Exec. time 0.1637319 0.519127567 1.213026786 2.209036731 2.474901667
No. of Succeeds 30 30 28 26 30
No. of Failures 0 0 2 4 0
F14 Mean 280.0333333 26.35714286 644.2333333 18.06896552 66.78571429
S.D. 26.29210362 3.291001038 6.371722732 2.750727624 25.65635744
Best 235 17 633 10 22
Worst 344 31 657 21 105
Exec. time 0.0942649 0.082852964 1.2564309 0.157990172 1.06789675
No. of Succeeds 30 28 30 29 28
No. of Failures 0 2 0 1 2
F15 Mean NC 39.46666667 70.66666667 33.3 185.5666667
S.D. NC 2.885556581 4.309839051 3.052980454 33.59359887
Best NC 34 62 26 111
Worst NC 44 77 39 241
Exec. time NC 0.1049036 0.106110233 0.2379349 1.4060248
No. of Succeeds 0 30 30 30 30
No. of Failures 30 0 0 0 0
F16 Mean NC NC NC NC NC
S.D. NC NC NC NC NC
Best NC NC NC NC NC
Worst NC NC NC NC NC
Exec. time NC NC NC NC NC
No. of Succeeds 0 0 0 0 0
No. of Failures 30 30 30 30 30
Open in a new tab

Table 8.

The success and failure ratio for minimising the sixteen benchmark functions in Test 1.

Variable dimensions BSA
 DE
 PSO
 ABC
 FF
Success Failure Success Failure Success Failure Success Failure Success Failure
Nvar1: 10 11 2 13 3 9 7 6 10 10 6
Nvar2: 30 13 3 12 4 6 10 5 11 9 7
Nvar3: 60 11 5 10 6 6 10 5 11 9 7
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Table 9.

The success and failure ratio for minimising the sixteen benchmark functions in Test 2.

Search space BSA
 DE
 PSO
 ABC
 FF
Success Failure Success Failure Success Failure Success Failure Success Failure
R1: [-5, 5] 15 1 14 2 12 4 13 3 15 1
R2: [-250, 205] 14 2 13 3 12 4 12 4 13 3
R3: [-500, 500] 12 4 12 4 10 6 10 6 12 4
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2. Experimental design, materials, and methods

The data presented in this data article is collected from the simulation results of BSA and the other four competitive algorithms (DE, PSO, ABS, and FF) applied to minimise sixteen benchmark functions are presented in three tests [[2], [3], [4], [5]] as follows:

  • 1.

    Several iterations are needed to minimise a specific function with Nvar variables with the default search space for the population size of 30. Nvars take values of 10, 30, and 60. For each benchmark function, each algorithm is run for 30 times with 2000 iterations for Nvar values of 10, 30, and 60.

  • 2.
    Several iterations are needed to minimise the functions with two variables for three different sized solution spaces for the population size of 30. For each benchmark function, each algorithm is run for 30 times with 2000 iterations for three different ranges (R1, R2, and R3), where
    • (a)
      R1: [-5, 5]
    • (b)
      R2: [-250, 250]
    • (c)
      R3: [-500, 500]
  • 3.

    Determining the ratio of successful minimisation of the functions for Tests 1 and 2 is needed to compare the successful rate of BSA with its competitive algorithms.

Table 2, Table 3, Table 4 present basic statistics (Mean: mean-solution, S.D.: standard-deviation of mean-solution, Best: the best solution, Worst: the worst solution, Exec. Time: mean runtime in seconds, No. of succeeds: number of successful minimisation, and No. of Failure: number of failed minimisation) of 30-solutions obtained by BSA, DE, PSO, ABC, and FF in Test 1 to minimise F1–F16 functions with default search space with number of variables 10, 30, and 60 respectively.

In addition, Table 5, Table 6, Table 7 present basic statistics (Mean: mean-solution, S.D.: standard-deviation of mean-solution, Best: the best solution, Worst: the worst solution, Exec. Time: mean runtime in seconds, No. of succeeds: number of successful minimisation, and No. of Failure: number of failed minimisation) of 30-solutions obtained by BSA, DE, PSO, ABC, and FF in Test 2 to minimise F1–F16 functions with default number of variables in three different search spaces R1, R2, and R3 respectively.

In Test 3, the ratio of successful minimisation of the functions for Tests 1 and 2 are presented in Table 8, Table 9.

Funding

Funding information is not applicable/No funding was received.

Acknowledgements

The authors would like to thank the referees for their remarkable suggestions. This paper's technical content has significantly improved based on their suggestions.

Footnotes

Appendix A

Supplementary data to this article can be found online at https://doi.org/10.1016/j.dib.2019.105046.

Conflict of Interest

The authors declare that they have no conflict of interest.

Appendix A. Supplementary data

The following is the Supplementary data to this article:

Multimedia component 1
mmc1.zip (90.7KB, zip)

References

  • 1.Hassan B.A., Rashid T.A. Operational framework for recent advances in backtracking search optimisation algorithm: a systematic review and performance evaluation. Appl. Math. Comput. 2019:124919. [Google Scholar]
  • 2.Ali M.M., Khompatraporn C., Zabinsky Z.B. A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J. Glob. Optim. 2005;31:635–672. [Google Scholar]
  • 3.Global Optimization Benchmarks and AMPGO. http://infinity77.net/global_optimization
  • 4.Jamil M., Yang X.-S. A literature survey of benchmark functions for global optimization problems. ArXiv Prepr. ArXiv1308.4008. 2013 [Google Scholar]
  • 5.Civicioglu P. Backtracking search optimization algorithm for numerical optimization problems. Appl. Math. Comput. 2013;219:8121–8144. [Google Scholar]

Associated Data

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Supplementary Materials

Multimedia component 1
mmc1.zip (90.7KB, zip)

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