Spontaneous Process Algorithm

Spontaneous Process Algorithm (SPA) a physics-based algorithm which has been inspired by nuclear reaction. In the projected algorithm complete exploration space is in a potted container [14-17]. In the projected algorithm, exploration space symbolizes a nucleus which possesses mass number, charge, location, potential & kinetic energy. Sequence of operators will direct the nucleus which has the quality of dissimilar condition of reaction and used to avoid to get trapped in local optimum.
Every nucleus symbolizes the variables and potential solution in exploration space, then the population i-th nucleus is initialized by,
\(Y_{i,d}={\text{lower}\ \text{bound}}_{d}+\text{random}\bullet\left({\text{upper}\ \text{bound}}_{d}-\text{lowerbound}_{d}\right)\)(10)
Due to nuclear fission and fusion cycle, in two dissimilar arbitrary nuclei the heated neutrons will be engendered by the nuclear fusion,
\(i-th\ \text{heated}\ \text{neutron}\ \left({he}_{i}\right)=\frac{\left(i-th\ \text{nucleus}(Y_{i})+j-th\ \text{nucleus}(Y_{j})\right)}{2}\)(11)
Secondary Fission [14-17] espouses the variation between the present solution and heated neutron to make use of equivalent range. Once\(\text{random}\leq\text{probability}\ \text{of}\ \beta\text{\ decay}\ \)then the formation procedure of Secondary fission,
\(Y_{i}^{\text{fission}}=\text{Gaussian}\left(Y_{\text{best}},\sigma_{1}\right)+\left(\text{random}\ n\bullet Y_{\text{best}}-\text{Mutation}\ \text{factor}\bullet{he}_{i}\right)\)(12)
\(\sigma_{1}=\left(\frac{\log(\text{current}\ \text{generation})}{\text{currenet}\ \text{generation}}\right)\bullet\left|Y_{i}-Y_{\text{best}}\right|\)(13)
\(\text{mutation}\ \text{factor}=\text{round}\left(\text{random}+1\right)\)(14)
Primary fission is created by,
\(Y_{i}^{\text{fission}}=\text{Gaussian}\left(Y_{i},\sigma_{2}\right)+\left(\text{random}\ n\bullet Y_{\text{best}}-\text{Mutation}\ \text{factor}\bullet{he}_{i}\right)\)(15)
\(\sigma_{2}=\left(\frac{\log(\text{current}\ \text{generation})}{\text{currenet}\ \text{generation}}\right)\bullet\left|Y_{\text{random}}-Y_{\text{best}}\right|\)(16)
\(\text{mutation}\ \text{factor}=\text{round}\left(\text{random}+2\right)\)(17)
Present nucleus information will be preserved then twig from the Gaussian walk is defined by,
\(Y_{i}^{\text{fission}}=\text{gaussian}\ \left(Y_{i}\bullet\sigma_{2}\right)\)(18)
In ionization phase all nucleus are categorized based on the fitness values [14-17]. Then the i-th nucleus can be described by a probability value which wholly abide by uniform distribution,
\(\text{Probability}_{i}=\frac{\text{rank}\left(\text{fitness}{\ Y}_{i}^{\text{fitness}}\right)}{\text{Total}\ \text{number}\ \text{of}\ \text{nuclei}}\)(19)
\({\text{ionization}\text{\ \ }Y}_{i}^{\text{fitness}}\) d-th variable is defined by,
\(Y_{i,d}^{\text{ion}}=Y_{r1,d}^{\text{fitness}}+\ \text{random}\bullet\left(Y_{r2,d}^{\text{fitness}}-Y_{i,d}^{\text{fitness}}\right),\text{random}\leq 0.5\)(20)
\(Y_{i,d}^{\text{ion}}=Y_{r1,d}^{\text{fitness}}-\ \text{random}\bullet\left(Y_{r2,d}^{\text{fitness}}-Y_{i,d}^{\text{fitness}}\right),\text{random}>0.5\)(21)
Exploitation will be improved, by adding a slim disturbance in\({\ Y}_{r1,d}^{\text{fitness}}\) as follows:
\(Y_{i,d}^{\text{ion}}=Y_{i,\ d}^{\text{fitness}}+\text{round}\ (\text{random})\bullet\left(Y_{\text{worst},d}^{\text{fitness}}-Y_{\text{best},d}^{\text{fitness}}\right)\)(22)
Value of i-th ion probability is given by,
\(\text{probability}_{i}=\frac{\text{rank}\left(\text{fitness}Y_{i}^{\text{ion}}\right)}{\text{Total}\ \text{number}\ \text{of}\ \text{nuclei}}\)(23)
By applying different plans of dissimilar operators superiority of exploration is enhanced which also used to imitate the collision and fusion;
\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}+\ \text{random}\bullet\left(Y_{r1}^{\text{ion}}-Y_{\text{best}}^{\text{ion}}\right)+\ \text{random}\bullet\left(Y_{r2}^{\text{ion}}-Y_{\text{best}}^{\text{ion}}\right)-e^{-\text{norm}\left(Y_{r1}^{\text{ion}}-Y_{r2}^{\text{ion}}\right)}.\left(Y_{r1}^{\text{ion}}-Y_{r2}^{\text{ion}}\right)\)(24)
When fusion does not happening Coulomb force [14-17] will decrease the forthcoming velocity and it described as,
\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}-0.5\bullet\left(\sin\left(2\pi.\text{frequency}\bullet g+\pi\right)\bullet\frac{\text{maximum}\ \text{generation}-\text{current}\ \text{generation}}{\text{maximum}\ \text{generation}}+1\right).\left(Y_{r1}^{\text{ion}}-Y_{r2}^{\text{ion}}\right)\ \text{random}>0.5\)(25)
\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}-0.5\bullet\left(\sin\left(2\pi.\text{frequency}\bullet g+\pi\right)\bullet\frac{\text{current}\ \text{generation}}{\text{maximum}\ \text{generation}}+1\right).\left(Y_{r1}^{\text{ion}}-Y_{r2}^{\text{ion}}\right)\ \text{random}\leq 0.5\)
(26)
Levy flight is a rank of non-Gaussian random procedure whose capricious walks are haggard from Levy stable distribution has been applied to avoid local optimum [14-17]. Allocation by L(s) ~ |s|-1-β where 0 < ß < 2 is an index. Scientifically defined as,
\(L\left(s,\gamma,\mu\right)=\left\{\par \begin{matrix}\sqrt{\frac{\gamma}{2\pi}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\\ 0\ \ \ \ \text{if}\ s\leq 0\\ \end{matrix}\right.\ \exp\left[-\frac{\gamma}{2\left(s-\mu\right)}\right]\frac{1}{\left(s-\mu\right)^{\frac{3}{2}}}\text{\ \ \ \ }\text{if}\ 0\ \ <\mu<s<\infty\text{\ \ \ \ \ \ \ \ \ \ }\)(27)
\(F\left(k\right)=\exp\left[-\alpha\left|k\right|^{\beta}\right],\ 0<\beta\leq 2,\)(28)
\(s=\frac{u}{\left|v\right|^{\frac{1}{\beta}}}\) (29)
\(Y^{t+1}=Y^{t}+\text{random}\ \left(\text{size}\left(D\right)\right)\oplus Levy\left(\beta\right)\sim 0.01\frac{u}{\left|v\right|^{\frac{1}{\beta}}}\left(y_{j}^{t}-\text{gb}\right)\)(30)
\(u\sim N\left(0,\sigma_{u}^{2}\right)\text{\ \ \ }v\sim N\left(0,\sigma_{v}^{2}\right)\)(31)
With
\(\sigma_{u}=\left\{\frac{G\left(1+\beta\right)\sin\left(\text{πβ}/2\right)}{G\left[\left(1+\beta\right)/2\right]{\beta 2}^{\left(\beta-1\right)/2}}\right\}^{\frac{1}{\beta}}\ ,\sigma_{v}=1\)(32)
Then,
\(\text{Levy}\left(y\right)=0.01\times\frac{u\times\sigma}{\left|v\right|^{\frac{1}{\beta}}}\)(33)
\(Y_{i,d}^{\text{ion}}=Y_{i,d}^{\text{Fission}}+\left(\alpha\oplus Levy\ (\beta)\right)_{d}\bullet\left(Y_{i,d}^{\text{fission}}-Y_{\text{best},d}^{\text{fission}}\right)\)(34)
\(Y_{i,d}^{\text{ion}}=Y_{i,d}^{\text{Fission}}+\left(\alpha\oplus Levy\ (\beta)\right)_{d}\bullet\left({\text{upper}\ \text{bound}}_{d}-\text{lowerbound}_{d}\right)\)(35)
\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}+\left(\alpha\oplus Levy\ (\beta)\right)\oplus\bullet\left(Y_{i}^{\text{ion}}-Y_{\text{best}}^{\text{ion}}\right)\)(36)
Boundary control has been done by,
\(Y_{i,d}=\text{lowerbound}_{d}+\text{random}\ \bullet\left({\text{upper}\ \text{bound}}_{d}-\text{lowerbound}_{d}\right)\)(37)
  1. Start
  2. Population initiated by\(Y_{i,d}={\text{lower}\ \text{bound}}_{d}+\text{random}\bullet\left({\text{upper}\ \text{bound}}_{d}-\text{lowerbound}_{d}\right)\)
  3. Compute the fitness function of the population
  4. While\(\left(\text{current}\ \text{generation}<\text{maximum}\ \text{iteration}\right)\ \text{do}\)
  5. Current generation = current generation+1
  6. \(\text{For}\ i=1\ \text{to}\ N\ \text{do}\)
  7. Heated neutron computed by\(i-th\ h\text{eated}\ \text{neutron}\ \left({he}_{i}\right)=\frac{\left(i-th\ \text{nucleus}(Y_{i})+j-th\ \text{nucleus}(Y_{j})\right)}{2}\)
  8. Modernize the population by\({\ \sigma}_{1}=\left(\frac{\log(\text{current}\ \text{generation})}{\text{currenet}\ \text{generation}}\right)\bullet\left|Y_{i}-Y_{\text{best}}\right|\)
  9. For \(Y_{i}^{\text{fitness}}\) compute the fitness value and boundary conditions are checked
  10. Modernize the \(Y_{i}^{\text{fitness}},\ Y_{i}^{\text{generation}}\)
  11. Check the probability by using\(\text{Probability}_{i}=\frac{\text{rank}\left(\text{fitness}{\ Y}_{i}^{\text{fitness}}\right)}{\text{Total}\ \text{number}\ \text{of}\ \text{nuclei}}\)
  12. \(\text{For}\ i=1\ \text{to}\ N\ \text{do}\)
  13. \(\text{For}\ d=1\ \text{to}\ N\ \text{do}\)
  14. Apply the levy flight by,
\(Y_{i,d}^{\text{ion}}=Y_{i,d}^{\text{Fission}}+\left(\alpha\oplus Levy\ (\beta)\right)_{d}\bullet\left(Y_{i,d}^{\text{fission}}-Y_{\text{best},d}^{\text{fission}}\right)\)\(Y_{i,d}^{\text{ion}}=Y_{i,d}^{\text{Fission}}+\left(\alpha\oplus Levy\ (\beta)\right)_{d}\bullet\left({\text{upper}\ \text{bound}}_{d}-\text{lowerbound}_{d}\right)\)\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}+\left(\alpha\oplus Levy\ (\beta)\right)\oplus\bullet\left(Y_{i}^{\text{ion}}-Y_{\text{best}}^{\text{ion}}\right)\)
Ion states are modernized by
\(Y_{i,d}^{\text{ion}}=Y_{r1,d}^{\text{fitness}}+\ \text{random}\bullet\left(Y_{r2,d}^{\text{fitness}}-Y_{i,d}^{\text{fitness}}\right),\text{random}\leq 0.5\)\(Y_{i,d}^{\text{ion}}=Y_{r1,d}^{\text{fitness}}-\ \text{random}\bullet\left(Y_{r2,d}^{\text{fitness}}-Y_{i,d}^{\text{fitness}}\right),\text{random}>0.5\)\(Y_{i,d}^{\text{ion}}=Y_{i,\ d}^{\text{fitness}}+\text{round}\ (\text{random})\bullet\left(Y_{\text{worst},d}^{\text{fitness}}-Y_{\text{best},d}^{\text{fitness}}\right)\)
  1. End for
  2. For \(Y_{i}^{\text{ion}}\) compute the fitness value and boundary conditions are checked
  3. Modernize the \(Y_{i}^{\text{fitness}},\ Y_{i}^{\text{ion}}\)
  4. End for
  5. Compute the probability by\(\text{probability}_{i}=\frac{\text{rank}\left(\text{fitness}Y_{i}^{\text{ion}}\right)}{\text{Total}\ \text{number}\ \text{of}\ \text{nuclei}}\)
  6. \(\text{For}\ i=1\ \text{to}\ N\ \text{do}\)
  7. Apply the levy flight through\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}+\left(\alpha\oplus Levy\ (\beta)\right)\oplus\bullet\left(Y_{i}^{\text{ion}}-Y_{\text{best}}^{\text{ion}}\right)\)
  8. Fusion population modernized by,
\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}+\ \text{random}\bullet\left(Y_{r1}^{\text{ion}}-Y_{\text{best}}^{\text{ion}}\right)+\ \text{random}\bullet\left(Y_{r2}^{\text{ion}}-Y_{\text{best}}^{\text{ion}}\right)-e^{-\text{norm}\left(Y_{r1}^{\text{ion}}-Y_{r2}^{\text{ion}}\right)}.\left(Y_{r1}^{\text{ion}}-Y_{r2}^{\text{ion}}\right)\)\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}-0.5\bullet\left(\sin\left(2\pi.\text{frequency}\bullet g+\pi\right)\bullet\frac{\text{maximum}\ \text{generation}-\text{current}\ \text{generation}}{\text{maximum}\ \text{generation}}+1\right).\left(Y_{r1}^{\text{ion}}-Y_{r2}^{\text{ion}}\right)\ \text{random}>0.5\)\(Y_{i}^{\text{fusion}}=Y_{i}^{\text{ion}}-0.5\bullet\left(\sin\left(2\pi.\text{frequency}\bullet g+\pi\right)\bullet\frac{\text{current}\ \text{generation}}{\text{maximum}\ \text{generation}}+1\right).\left(Y_{r1}^{\text{ion}}-Y_{r2}^{\text{ion}}\right)\ \text{random}\leq 0.5\)
  1. For \(Y_{i}^{\text{fusion}}\) compute the fitness value and boundary conditions are checked
  2. Modernize the \(Y_{i}^{\text{fusion}},\ Y_{i}^{\text{ion}}\)
  3. End for
  4. End while
  5. Output; Best solution