## 3 Methodology--Mathematical Modelling and Heuristic Optimization Method
#### 3.2.1 Black Hole Optimization-Based Clustering
The event horizon and the photon sphere can be calculated as follows:  
\[r_{eh}=\frac{2 gM}{c^{2}},\text{ and }\mathrm{r}_{ps}=\frac{3}{2}r_{s} \tag{30}\]  
where \(g\) is [PERSON]'s gravitational constant, \(M\) is the mass, and \(c\) is the speed of light. Black hole algorithm belongs to the swarm intelligence algorithm, which are inspired either by living bodies, like ants [88], bees [89], fishes [90], bats [91], krill herds [92], fireflies [93], fruit flies [94], bacteria's [95], or by other natural phenomena, like gravitation [96], big-bang [97], or intelligent water drop [98]. Black hole optimization is used in a wide range of NP-hard optimization problems, like investigating the critical slip surface of soil slope [99], solving the non-unicost set covering problem [100], optimization of consignment-store-based supply chain [101], thermodynamic optimization of a Penrose process [102], power flow optimization [103], and design of electromagnetic devices [104], but one of its most important application fields is the clustering. The black hole algorithms have six phases as follows:  
Figure 11: Impact of photon sphere and event horizon on moving particles and photons. * _big-bang phase_: this phase is the initialization of the position and velocity of stars in the multidimensional search space. The stars represent potential solutions of the optimization problem, where the coordinates of the stars in the search space are the values of the decision variables. Stars can be initialized only inside the search space. \[\overrightarrow{p_{i}}=(p_{i,1},p_{i,2}\cdots p_{i,n})\in\left[p_{i,j}^{min} \cdots p_{i,j}^{max}\right]\] (31)
* _evaluation phase_: this phase includes the calculation of the objective function based on the parameters represented by the coordinates of the star. \[\overrightarrow{v_{i}}=f(p_{i,1},p_{i,2}\cdots p_{i,n})\] (32)
* _selection of black hole_: within the frame of this phase a new black hole is defined as having the highest value of objective function. This star has the highest weight (represented by the value of objective function) and therefore it has the highest force of gravity and it is the center of movement of stars in the next movement phase. \[\overline{\mathrm{P}}_{BH}=\overline{\mathrm{P}}_{i}\to v_{BH}=\max_{i}(v_{i})\] (33)
* _moving of stars_: in this phase of the algorithm, a new position of stars is calculated. The movement of the stars can be influenced only by the black hole, but it is also possible to take into consideration the gravity force of the other stars. \[p_{i,j}(t+\Delta t)=p_{i,j}(t)+Rnd\big{|}p_{BH}(t)-p_{i,j}(t)\big{|}\] (34)
* _decreasing the event horizon and the photon sphere_: in this phase, the size of the event horizon and the photon sphere is decreased based on the Hawking radiation, which describes the lost weight process of black holes. This phase makes it possible to prevent the absorption of stars representing solutions of the optimization problem near the optimum: \[r_{eh}=\frac{v_{BH}}{\varpi\cdot\sum_{i=1}^{m}v_{i}}\text{ and }r_{ps}=\frac{3}{2}\frac{v_{BH}}{\varpi\cdot\sum_{i=1}^{m}v_{i}},\] (35) where \(\varpi\) is the number of the current iteration step. * _shift the position of the black hole_: in this phase of the optimization we use the idea of Hawking radiation. Particles can escape and the black hole's mass reduces because if a particle-antiparticle pair is created beyond the event horizon, it is possible to have one drawn into the black hole itself while the other is ejected [105]. The position of the black hole is shifted using the following calculation: \[\overline{\mathrm{P}}_{BH}(t+\Delta t)=\mathrm{p}_{BH}(t+\Delta t)+\frac{v_{ BH}}{\varpi\cdot\mu\cdot\sum_{i=1}^{m}v_{i}}\] (36) where \(\mu\) is the shift-factor. We demonstrate the clustering with a short example shown in Tables 2 and 3. Table 2 shows the parameters of the clustering problem.