## 3 Optimization Methodology  
### Flower Pollination Algorithm (FPA)  
The river pollination method of flowering plants served as the basis for the creation of the FPA ([PERSON] et al. 2012 [46; 47]). Through the process of pollination, flowers of a plant are crucial to reproduction. Pollinators including insects, birds, and other animals assist the spread of pollen during pollination. Self-pollination or cross-pollination are terms used to describe pollination. Cross-pollination is the term used to describe the pollination process that occurs when pollen from a plant's flower is used. Self-pollination, on the other hand, is the act of fertilizing a flower of a plant with pollen from the same flower or from many flowers of the same plant. In the absence of a dependable pollinator, self-pollination frequently takes place. Cross-pollination typically occurs over great distances, and pollinators like bees, bats, birds, and flies are able to travel great distances. Cross-pollination is hence comparable to global pollination. Additionally, a Levy distribution governs how bees and birds fly. Further, the consistency of two blooms can be used as an increment step by comparing or contrasting them. The traditional FPA mostly relies on flower constancy behavior, which is explained in the following manner. Elite flower species can be visited by pollinators, while other flower species can be avoided. Because it facilitates the transmission of pollen to the same plants' flowers, such flower constancy has the benefit of promoting the reproduction of the same flower species. The following is a list of the FPA's primary guidelines. Rule 1: Cross-pollination is a form of global pollination, and pollinators follow the Levy distribution when they fly. Rule 2: The process of self-pollination is regarded as local pollination. Rule 3: The likelihood of reproduction is obtained by pollinators like insects and is equivalent to flowering persistence. Based on how similar the two flowers are, this probability is determined. Rule 4: A switching probability \(\mathrm{p}\) equal to either 0 or 1 that is slightly skewed in favor of local pollination can be used to control the transition between local and global pollination. The update equations for the FPA are derived by incorporating the aforementioned rules. Mathematically, the first and third principles can be written as Equation (25). \[u_{n}^{i+1}=u_{n}^{i}+\delta L(x)\Big{(}u_{best}-u_{n}^{i}\Big{)} \tag{25}\]  
where \(L(x)\) is the Levy-based step size showing the amount of pollination, \(\delta\) is a scaling factor to control the step size, and \(u_{n}^{i}\) represents the \(n\)-th pollen or solution \(u_{n}\) at iteration \(u_{best}\) is the best solution among all the solutions thus far. As a result, \(L(x)\) is as in Equation (26). \[L\approx\frac{x\Gamma(\gamma)\text{sin}\big{(}\frac{\pi x}{2}\big{)}}{\pi}\frac{1 }{s^{1}+x},\hskip 14.226378 pts>>s_{0}>0 \tag{26}\]  
where the standard gamma function is represented by \(\Gamma(\gamma)\), and this distribution is acceptable for big steps \(s>0\). Despite \(s\gg 0\) being necessary, \(s_{0}\) can actually be as low as 0.1. It is challenging to produce pseudorandom step sizes that accurately match the Levy distribution though [48]. As a result, the Mantegna method [49], a useful algorithm that has been documented in the literature, is utilized in the FPA to generate these random values. Two Gaussian distributions, \(G_{u}\) and \(G_{v}\), can be used to derive the step size s in Equation (27). \[s\frac{G_{u}}{|G_{v}|^{\frac{1}{4}}},G_{u}\,\sim N\Big{(}0,var^{2}\Big{)},G_{v }\,\sim N(0,1) \tag{27}\]  
The expression \(G_{u}\,\sim N\big{(}0,var^{2}\big{)}\) denotes that the samples were taken from a Gaussian distribution with a mean and variance of 0 and \(var^{2}\), respectively. Equation (28) is used to obtain the variance, or \(var^{2}\).