## 3 Methodology--Mathematical Modelling and Heuristic Optimization Method
#### 3.2.1 Black Hole Optimization-Based Clustering  
There is a wide range of clustering algorithms in the literature, which is suitable for the solution of clustering problems without complex restrictions. K-means clustering, mean-shift clustering, density-based spatial clustering, and agglomerative hierarchical clustering algorithms belong to the most well-known general clustering algorithms. However, their implementation codes are quite simple, but they have disadvantages:  
* K-mean: the classes must be defined [84];
* Mean-shift: the size of the sliding window must be defined [85];
* Density-based spatial clustering: its performance is low in the case of varying density of points [86];
* Agglomerative hierarchical clustering: its complexity is \(O\big{(}n^{3}\big{)}\) while K-means is linear [87]. The constraints and the complexity of the multi-dimensional search space make the clustering problem NP-hard, which means that heuristic clustering methods are suitable to solve the clustering problem of the extended logistics resource optimization in matrix production. The idea of black holes was first suggested by [PERSON] and [PERSON]. They proposed the existence of \"invisible stars\". They calculated its mass and size, which is the so-called event horizon in today's science. Later, in 1916 [PERSON] predicted the existence of black holes \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Evaluation Function** & \begin{tabular}{c} **Black Hole** \\ **Optimization** \\ \end{tabular} & \begin{tabular}{c} **Genetic** \\ **Algorithm** \\ \end{tabular} & \
\begin{tabular}{c} **Harmony** \\ **Search** \\ \end{tabular} &
\begin{tabular}{c} **Flower** \\ **Pollination** \\ \end{tabular} \\ \hline Ackley & \(3.66\times 10^{-7}\) & \(4.67\times 10^{-6}\) & \(1.28\times 10^{-7}\) & \(3.45\times 10^{-7}\) \\ Bukin & \(2.45\times 10^{-6}\) & \(5.45\times 10^{-7}\) & \(9.08\times 10^{-7}\) & \(5.61\times 10^{-8}\) \\ Cross-in-tray & \(8.55\times 10^{-9}\) & \(7.32\times 10^{-9}\) & \(6.98\times 10^{-8}\) & \(6.12\times 10^{-7}\) \\ Easom & \(1.18\times 10^{-5}\) & \(2.09\times 10^{-4}\) & \(8.18\times 10^{-9}\) & \(4.02\times 10^{-8}\) \\ Eggholder & \(5.50\times 10^{-7}\) & \(3.12\times 10^{-7}\) & \(1.98\times 10^{-8}\) & \(1.39\times 10^{-8}\) \\ Three hump camel back & \(1.51\times 10^{-6}\) & \(4.17\times 10^{-8}\) & \(7.79\times 10^{-10}\) & \(6.60\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance analysis of various heuristic algorithms: error values in the case of six benchmark functions after 50 iteration steps. with his general relativity theory. The first black hole called Cygnus X-1 was recognized by [PERSON] in 1971. Black holes are strange and fascinating places in space where the gravitation forces are so high that they can trap not only particles, planets, and stars but also light. Black holes are born when stars die. There are three types of black holes depending on their size and weight: supermassive black holes, stellar black holes, and miniature black holes. The black hole optimization is based on this phenomenon of black holes. Black holes have four layers: space outside the photonsphere, space between photonsphere and event horizon, space inside the event horizon, and the singularity, where the mass of the black hole in concentrated in one single point. The photon sphere is a spherical region of non-spinning black holes; photons reaching the photon sphere are not captured but they are forced to travel in orbits. The distance between particles and the black hole has a great impact on the behavior of the particles. If the distance between a particle is much higher than the Schwarzschild radius, then the particle can move in any direction. If this distance is larger than the Schwarzschild radius, but this difference is not too much, the space-time is deformed, and more particles are moving towards the center of the black hole than in other directions (Figure 11).