## 4 Multi-Objective Hybrid Resource Consolidation Algorithm
### Hybrid Flower Pollination Resource Consolidation Algorithm
#### 4.2.1 Flower Pollination Algorithm  
One of the fascinating processes of reproduction in the natural world is flower pollination, and its evolutionary features are employed in the design of a unique optimization algorithm by [37]. It is a global optimization algorithm that is centered on population. The following steps summarize the flower pollination procedures. _Procedure 1_: cross-pollination, which is termed as biotic, is considered global pollination carried out by pollinators called pollen vectors. The pollen vectors could be bats, birds, and bees. The pollen vector that carries pollen grain performs Levy flights. _Procedure 2_: self-pollination is considered abiotic, which is a local pollination process. This pollination occurs on the same flower or nearby flowers. _Procedure 3_: flower constancy is passive in terms of the reproduction chances, and is directly proportional to the resemblance of two flowers involved. _Procedure 4_: the switch probability \(p\in[0,1]\) controls both the global pollination and local pollination. Considering the real-life proximity and other factors such as wind, a significant fraction \(p\) in the whole pollination process can be achieved by local pollination. Considering the above procedures, the FPA comprises local and global pollination operators. From the FPA, each pollen grain is handled as a solution, \(S_{i}X_{i}\), and the solutions are set with non-uniform vectors in the viable search space. The preliminary formula is expressed in Equation (7) follows:  
\[S_{i}X_{i}=lower+RD_{v}(upper-lower) \tag{7}\]  
Such that \(i\in\{1,\ldots,NP\}\), the population size, is represented as NP, and \(RD_{v}\) is the random vector between \([0,1]^{D}\) with D-dimension. The lower boundary of the search space is \(lower=l_{1},\ldots,l_{d},\ldots,l_{D}\) and the upper boundary is \(upper=u_{1},\ldots,u_{d},\ldots,u_{D}\). Considering the global pollination process, pollen vectors such as birds have a comparatively large movement coverage and could carry pollens over a lengthy distance. Hence, procedures 1 and 3 are expressed as in Equation (8). \[S_{i}X_{i}^{t+1}=S_{i}X_{i}^{t}+\propto L(\beta)\big{(}gbest-S_{i}X_{i}^{t} \big{)} \tag{8}\]  
where \(S_{i}X_{i}^{t}\) is the solution \(i\) at the cycle of execution \(t\), the present global best solution is termed as gbest, \(\beta\) represents the step factor, the flight characteristics of a bird could be numerically mimicked by a levy distribution symbolized by \(L(\beta)\) in Equation (3), and it can be perceived as a varying step factor for measuring the intensity of pollination. The levy distribution when \(L>0\) can be expressed as in Equation (9). \[L\,\sim\,\frac{\beta\varnothing(\beta)\sin(\,\pi\beta/2)}{\pi}\cdot\frac{1}{S^{ 1+\beta}} \tag{9}\]  
\[s\gg s_{0}>0,\;s_{0}=0.1\;,\]  
where \(\varnothing(\beta)\) represents the standard gamma function with \(\beta=1.5\). In addition, \(s\) is obtained with 2 Gaussian distribution \(V\) and \(W\) as follows in Equation (10). \[s=\frac{V}{|V|^{\frac{1}{\beta}}}\,,\;V\,\sim\,N\Big{(}0,\mu^{2}\Big{)},\;W\, \sim\,N(0,1),\;\mu^{2}=\bigg{\{}\frac{\varnothing(1+\beta)}{\beta\varnothing [(1+\beta)/2]}\cdot\frac{\sin(\pi\beta/2)}{2^{(\beta-1)/2}}\bigg{\}}^{1/\beta} \tag{10}\]  
where \(N\big{(}0,\mu^{2}\big{)}\) represents the normal distribution attached with the mean score 0 and variance \(\mu^{2}\); \(N(0,\;1)\) represents the standard normal distribution. In the case where the pollination process involves local pollination, pollen grains are spread to a local neighbor, and the mathematical model can be formulated considering procedures 2 and 3 as in Equation (11). \[S_{i}X_{i}^{t+1}=S_{i}X_{i}^{t}+\vartheta\Big{(}S_{i}X_{j}^{t}-S_{i}X_{k}^{t} \Big{)} \tag{11}\]  
where \(S_{i}X_{j}^{t}\) and \(S_{i}X_{k}^{t}\) are the pollen grain, which are non-uniformly selected from different flowers in the same plant, such that \(j\)_and_\(k\,\in\,\{1,\;\ldots,\;NP\}\) and \(\vartheta\) represent D-dimensional non-uniform vector in \([0,1]^{D}\).