Polar Bear Optimization Algorithm: MetaHeuristic with Fast Population Movement and Dynamic Birth and Death Mechanism
Abstract
:1. Introduction
Related Works
2. Polar Bear Optimization Algorithm (PBO)
2.1. Basic Premise
Definition of the Optimization Problem
2.2. Global Move Using Ice Floes
2.3. Local Search While Hunting Seals
2.4. Dynamic Population Control by the Reproduction and Extinction by Starvation
Algorithm 1: Polar Bear Optimization Algorithm 

3. Experimental Results for Classic Test Functions
 average number of individuals in the population according to their fertility and mortality,
 average trajectory with respect to the ideal solution,
 average adaptation of the population,
 rate of convergence.
4. Application of PBO in Engineering Problems
4.1. Pressure Vessel Design Problem
 ${x}_{1}$—the coating thickness of cylinder,
 ${x}_{2}$—the coating thickness of hemispherical cylinders,
 ${x}_{3}$—the radius of the cylinder without it’s shell,
 ${x}_{4}$—the length of the cylinder.
4.2. Gear Train Problem
4.3. Welded Beam Design Problem
4.4. Compression Spring Design Problem
5. Discussion
6. Final Remarks
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Function Name  Function f  Range  ${\mathit{f}}_{\mathit{min}}$  Solution $\overline{\mathit{x}}$ 

DixonPrice  $f}_{1}\left(\overline{x}\right)={({x}_{1}1)}^{2}+\sum _{i=1}^{n}i{(2{x}_{i}^{2}{x}_{i1})}^{2$  $\langle 10,10\rangle $  0  $\left({2}^{\frac{{2}^{1}2}{{2}^{1}}},\dots ,{2}^{\frac{{2}^{n}2}{{2}^{n}}}\right)$ 
Griewank  ${f}_{2}\left(\overline{x}\right)=\sum _{i=1}^{n}\frac{{x}_{i}^{2}}{4000}\prod _{i=1}^{n}cos\left(\frac{{x}_{i}}{\sqrt{\left(i\right)}}\right)+1$  $\langle 10,10\rangle $  0  (0,…,0) 
Powell  $\begin{array}{ll}{f}_{3}\left(\overline{x}\right)=& {\displaystyle \sum _{i=1}^{n/4}}({({x}_{4i3}+10{x}_{4i2})}^{2}+5{({x}_{4i1}{x}_{4i})}^{2}\\ & +{({x}_{4i2}2{x}_{4i1})}^{4}+10{({x}_{4i3}{x}_{4i})}^{4})\end{array}$  $\langle 100,100\rangle $  0  (0,…,0) 
Rastragin  ${f}_{4}\left(\overline{x}\right)=10n+\sum _{i=1}^{n}[{x}_{i}^{2}10cos\left(2\pi {x}_{i}\right)]$  $\langle 10,10\rangle $  0  (0,…,0) 
Rosenbrock  ${f}_{5}\left(\overline{x}\right)=\sum _{i=1}^{n1}\left(100{({x}_{i+1}{x}_{i}^{2})}^{2}+{({x}_{i}1)}^{2}\right)$  $\langle 100,100\rangle $  0  (1,…,1) 
Hyper–Ellipsoid  $f}_{6}\left(\overline{x}\right)=\sum _{i=1}^{n}\sum _{j=1}^{i}{x}_{j}^{2$  $\langle 100,100\rangle $  0  (0,…,0) 
Schwefel  ${f}_{7}\left(\overline{x}\right)=418.9829n\sum _{i=1}^{n}{x}_{i}sin\left(\sqrt{{x}_{i}}\right)$  $\langle 500,500\rangle $  0  (420.97,…,420.97) 
Shubert  ${f}_{8}\left(\overline{x}\right)=\prod _{j=1}^{n}\left({\displaystyle \sum _{i=1}^{5}icos\left((i+1){x}_{j}\right)}\right)$  $\langle 10,10\rangle $  $186.7$  (0,…,0) 
Sphere  $f}_{9}\left(\overline{x}\right)=\sum _{i=1}^{n}{x}_{i}^{2$  $\langle 10,10\rangle $  0  (0,…,0) 
Sum squares  $f}_{10}\left(\overline{x}\right)=\sum _{i=1}^{n}i{x}_{i}^{2$  $\langle 10,10\rangle $  0  (0,…,0) 
StyblinskiTang  ${f}_{11}\left(\overline{x}\right)=\frac{1}{2}\sum _{i=1}^{n}\left({x}_{i}^{4}16{x}_{i}^{2}+5{x}_{i}\right)$  $\langle 10,10\rangle $  $39.2n$  (2.9,…,2.9) 
Weierstrass  $f}_{12}\left(\overline{x}\right)=\sum _{i=1}^{n}{\left([{x}_{i}+0.5]\right)}^{2$  $\langle 30,30\rangle $  0  $(\frac{1}{2},\dots ,\frac{1}{2})$ 
Zakharov  $f}_{13}\left(\overline{x}\right)=\sum _{i=1}^{n}{x}_{i}^{2}+{\left(0.5i{x}_{i}\right)}^{2}+{\left({\displaystyle \sum _{j=1}^{n}0.5j{x}_{j}}\right)}^{4$  $\langle 10,10\rangle $  0  (0,…,0) 
Function  BA  CSA  DA  FA  FPA  MFO  PBO  WWO  SA  GA  PSO  AACA 

${f}_{1}$  0.00832  0.007718  0.007801  0.009989  0.008947  0.007517  0.007512  0.009713  0.007113  0.007621  0.007501  0.007613 
${f}_{2}$  0.001837  0.000893  0.001092  0.002813  0.003815  0.000986  0.000052  0.000681  0.000102  0.000041  0.000052  0.000097 
${f}_{3}$  0.001241  0.000874  0.00983  0.000873  0.000781  0.000208  0.000189  0.000198  0.000174  0.000321  0.000193  0.000202 
${f}_{4}$  0.004602  0.004592  0.00513  0.005397  0.005284  0.004503  0.004483  0.003976  0.005612  0.004387  0.00404  0.005238 
${f}_{5}$  0.030147  0.030281  0.029999  0.37326  0.033059  0.029193  0.27936  0.009826  0.008427  0.04356  0.02134  0.03243 
${f}_{6}$  0.000981  0.000109  0.000087  0.000109  0.000054  0.000041  0.000039  0.007298  0.0002312  0.000074  0.000038  0.0003145 
${f}_{7}$  0.04618  0.045801  0.041982  0.04784  0.046871  0.039982  0.04352  0.043891  0.0446127  0.0424365  0.038932  0.041324 
${f}_{8}$  −187.387  −186.981  −186.789  −186.837  −186.811  −187.705  −187.702  −187.703  −186.942  −187.361  −186.924  −186.931 
${f}_{9}$  0.001026  0.000866  0.000986  0.000872  0.000201  0.000132  0.000064  0.000127  0.000243  0.001103  0.000059  0.000124 
${f}_{10}$  0.008765  0.002942  0.003856  0.004857  0.003927  0.000535  0.000289  0.008986  0.002476  0.003101  0.0004251  0.001621 
${f}_{11}$  −784.205  −783.804  −784.198  −784.301  −783.989  −784.098  −783.992  −784.102  −784.031  −783.994  −783.089  −783.995 
${f}_{12}$  −0.50968  −0.50991  −0.51012  −0.51830  −0.52380  −0.50910  −0.50901  −0.50992  −0.59851  −0.5312  −0.50941  −0.50993 
${f}_{13}$  0.004972  0.00478  0.005629  0.007615  0.002371  0.000689  0.000683  0.000983  0.004351  0.006254  0.0006831  0.0007452 
Function  BA  CSA  DA  FA  FPA  MFO  PBO  WWO  SA  GA  PSO  AACA 

${f}_{1}$  $0.400682$  $0.312220$  $0.280424$  $0.386053$  $0.189451$  $0.224049$  $0.216292$  $0.230397$  $0.233152$  $0.235467$  $0.253251$  $0.257632$ 
${f}_{2}$  $0.000442$  $0.000442$  $0.000403$  $0.000406$  $0.000567$  $0.000386$  $0.000382$  $0.000389$  $0.000371$  $0.000384$  $0.000392$  $0.000432$ 
${f}_{3}$  $0.299515$  $0.271036$  $0.214116$  $0.234468$  $0.247951$  $0.212719$  $0.211968$  $0.288523$  $0.217631$  $0.224521$  $0.235621$  $0.249893$ 
${f}_{4}$  $0.261550$  $0.348611$  $0.29821$  $0.342901$  $0.389093$  $0.24046$  $0.240397$  $0.240378$  $0.213452$  $0.224213$  $0.231273$  $0.245694$ 
${f}_{5}$  $0.304536$  $0.320787$  $0.342946$  $0.388872$  $0.324894$  $0.316903$  $0.311527$  $0.329623$  $0.317235$  $0.331231$  $0.334572$  $0.364121$ 
${f}_{6}$  $0.004838$  $0.003263$  $0.003032$  $0.004378$  $0.004008$  $0.003106$  $0.003240$  $0.003697$  $0.003123$  $0.003215$  $0.003521$  $0.004121$ 
${f}_{7}$  $0.290749$  $0.381693$  $0.341592$  $0.444689$  $0.374533$  $0.268699$  $0.295528$  $0.310741$  $0.283124$  $0.279523$  $0.283210$  $0.423187$ 
${f}_{8}$  $0.482759$  $0.455528$  $0.402704$  $0.44001$  $0.467686$  $0.413571$  $0.40463$  $0.485864$  $0.423981$  $0.445692$  $0.453982$  $0.473213$ 
${f}_{9}$  $0.002201$  $0.002041$  $0.002219$  $0.002166$  $0.002159$  $0.001971$  $0.001936$  $0.001825$  $0.002131$  $0.002201$  $0.002133$  $0.002213$ 
${f}_{10}$  $0.453699$  $0.485568$  $0.114836$  $0.540067$  $0.478353$  $0.442032$  $0.43376$  $0.440208$  $0.344561$  $0.413125$  $0.413241$  $0.512341$ 
${f}_{11}$  $0.00322$  $0.003358$  $0.003624$  $0.002953$  $0.002498$  $0.002730$  $0.002532$  $0.003324$  $0.002731$  $0.002745$  $0.002589$  $0.003421$ 
${f}_{12}$  $0.348204$  $0.391397$  $0.330937$  $0.420113$  $0.372893$  $0.313924$  $0.322731$  $0.357778$  $0.341831$  $0.328234$  $0.332198$  $0.375265$ 
${f}_{13}$  $0.004743$  $0.005072$  $0.005393$  $0.004565$  $0.003950$  $0.003657$  $0.003657$  $0.003771$  $0.003842$  $0.003642$  $0.004115$  $0.004932$ 
Function  BA  CSA  DA  FA  FPA  MFO  PBO  WWO  SA  GA  PSO  AACA 

${f}_{1}$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$ 
${f}_{2}$  $0.00172$  $0.00172$  $0.00172$  $0.00172$  $0.00172$  $0.00172$  $0.00172$  $0.00172$  $0.00172$  $0.00172$  $0.00172$  $0.00172$ 
${f}_{3}$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  N/A  N/A  N/A  $0.00286$  $0.00286$  $0.00286$  N/A  N/A 
${f}_{4}$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$ 
${f}_{5}$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$ 
${f}_{6}$  $0.00286$  $0.00286$  $0.00364$  $0.00286$  $0.00286$  N/A  N/A  $0.00341$  $0.00286$  $0.00286$  N/A  N/A 
${f}_{7}$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$ 
${f}_{8}$  $0.00345$  $0.00345$  $0.00331$  $0.00345$  $\underline{0.00501}$  $0.00339$  $\underline{0.00503}$  $0.00328$  $0.00341$  $0.00345$  $0.00340$  $0.00345$ 
${f}_{9}$  $0.00286$  $0.00286$  N/A  $0.00286$  $0.00286$  N/A  N/A  $0.00286$  $0.00286$  $0.00286$  N/A  N/A 
${f}_{10}$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$ 
${f}_{11}$  $0.0039$  $0.0039$  $0.004$  $0.0039$  $0.0039$  $0.0023$  $0.0031$  $0.0047$  $0.0039$  $0.0039$  $0.0031$  $0.0039$ 
${f}_{12}$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$ 
${f}_{13}$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $0.00286$  $N/A$  $N/A$  $0.00286$  $0.00286$  $0.00286$  N/A  N/A 
BA  CSA  DA  FA  FPA  MFO  PBO  WWO  SA  GA  PSO  AACA  

${x}_{1}$  0.81064  0.81958  0.81683  0.81082  0.82531  0.81738  0.81327  0.81038  0.81612  0.82034  0.81421  0.82313 
${x}_{2}$  0.44675  0.44468  0.44721  0.49577  0.44466  0.43488  0.43702  0.43629  0.44115  0.44014  0.43611  0.44537 
${x}_{3}$  42.15769  42.23808  42.14122  42.01106  42.15095  42.0162  42.04601  42.12014  42.10029  42.32500  42.13451  42.92300 
${x}_{4}$  176.7288  176.87528  177.12321  177.69137  178.23268  176.9014  176.75597  177.96564  176.72880  176.43206  176.65122  176.73561 
${g}_{1}\left(\overline{x}\right)$  6088.18314  6160.6034  6138.89315  6241.23766  6217.80235  6077.53166  6057.54657  6075.94324  6098.67000  6155.74048  6073.52853  6301.56641 
BA  CSA  DA  FA  FPA  MFO  PBO  WWO  SA  GA  PSO  AACA  

${x}_{1}$  57.4517  55.62869  52.44017  50.05234  51.15054  48.22301  50.04797  55.27473  51.34512  52.57125  51.32931  51.41824 
${x}_{2}$  19.48176  16.71727  17.00267  24.38656  22.45769  18.77492  23.32433  24.3159  21.32743  23.19834  21.02347  21.35921 
${x}_{3}$  18.58749  21.27387  22.99426  14.02798  17.97921  21.13716  14.82582  15.03104  14.98321  16.93254  14.79391  15.83209 
${x}_{4}$  43.68715  44.31047  51.67159  46.3547  55.90958  57.03833  47.88919  45.82914  47.43271  48.24512  47.82131  47.34128 
${g}_{2}\left(\overline{x}\right)$  1.53 × 10${}^{11}$  3.09 × 10${}^{13}$  3.02 × 10${}^{11}$  6.52 × 10${}^{13}$  4.83 × 10${}^{11}$  1.44 × 10${}^{14}$  1.37 × 10${}^{15}$  5.67 × 10${}^{12}$  1.71 × 10${}^{4}$  1.13 × 10${}^{4}$  3.08 × 10${}^{4}$  2.87 × 10${}^{5}$ 
BA  CSA  DA  FA  FPA  MFO  PBO  WWO  SA  GA  PSO  AACA  

${x}_{1}$  0.23848  0.22635  0.24685  0.25043  0.30209  0.21356  0.23258  0.25009  0.22413  0.22451  0.22131  0.22416 
${x}_{2}$  3.48082  3.47836  3.477  3.69551  3.48829  3.49944  3.49706  3.7379  3.51812  3.49359  3.47241  3.51512 
${x}_{3}$  8.71677  8.79874  8.56445  8.54958  8.98766  8.80996  8.91026  8.9735  8.64167  8.72150  8.65325  8.71412 
${x}_{4}$  0.22242  0.21859  0.24589  0.2787  0.24566  0.21283  0.21194  0.22149  0.25139  0.22451  0.21312  0.23151 
${g}_{3}\left(\overline{x}\right)$  1.84923  1.81413  2.00481  2.28456  2.20932  1.7549  1.79863  1.95437  2.02615  1.84247  1.73809  1.89509 
BA  CSA  DA  FA  FPA  MFO  PBO  WWO  SA  GA  PSO  AACA  

${x}_{1}$  $0.05205$  $0.05251$  $0.05241$  $0.05093$  $0.05193$  $0.0529$  $0.05102$  $0.05257$  0.05231  0.05242  0.05214  0.05253 
${x}_{2}$  $0.35145$  $0.35869$  $0.34079$  $0.35121$  $0.3487$  $0.35118$  $0.35756$  $0.33994$  0.34412  0.35325  0.34346  0.34512 
${x}_{3}$  $11.86055$  $10.9627$  $11.85907$  $11.58599$  $11.81457$  $11.74167$  $11.6994$  $11.37447$  11.62350  11.68242  11.62233  11.63542 
${f}_{v}\left(\overline{x}\right)$  $0.0132$  $0.01282$  $0.01297$  $0.01238$  $0.01299$  $0.0135$  $0.01275$  $0.01256$  0.01283  0.01328  0.01272  0.01299 
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Połap, D.; Woz´niak, M. Polar Bear Optimization Algorithm: MetaHeuristic with Fast Population Movement and Dynamic Birth and Death Mechanism. Symmetry 2017, 9, 203. https://doi.org/10.3390/sym9100203
Połap D, Woz´niak M. Polar Bear Optimization Algorithm: MetaHeuristic with Fast Population Movement and Dynamic Birth and Death Mechanism. Symmetry. 2017; 9(10):203. https://doi.org/10.3390/sym9100203
Chicago/Turabian StylePołap, Dawid, and Marcin Woz´niak. 2017. "Polar Bear Optimization Algorithm: MetaHeuristic with Fast Population Movement and Dynamic Birth and Death Mechanism" Symmetry 9, no. 10: 203. https://doi.org/10.3390/sym9100203