# Computational Optimized Monitoring Methodology to Avoid Crowd Crush Accidents with Scattered Data

## Abstract

**:**

## 1. Introduction

## 2. Optimization Methodology

**Lemma**

**1.**

**Proof.**

- (i)
- First, let us consider the points ${u}^{\prime}={u}_{\pm}$. Observe that$${\psi}_{i}\left({u}_{\pm}\right){\psi}_{j}^{\u2033}\left({u}_{\pm}\right)=-\frac{{w}_{j}^{2}\left(\right|a|-|{x}_{i}{\left|\right)}^{2}{(a\pm {x}_{i})}^{2}}{|a{x}_{i}|}<0.$$This means that if ${\psi}_{i}\left({u}_{\pm}\right)$ is greater (or less) than zero, then ${\psi}_{i}^{\u2033}\left({u}_{\pm}\right)$ is less (or greater) than zero, and thus the function ${\psi}_{i}\left(u\right)$ has a local maximum (or local minimum) at ${u}_{\pm}$. Since ${\varphi}_{i}\left(u\right)=\left|{\psi}_{i}\left(u\right)\right|$, it follows that the function ${\varphi}_{i}\left(u\right)$ has a local maximum at ${u}_{\pm}$.
- (ii)
- Next, consider the point ${u}^{\prime}={u}_{0}.$ Since the function $\varphi \left(u\right)$ has a local minimum at ${u}_{0}$ and $\varphi \left(u\right)={\varphi}_{i}\left(u\right)$ on $({u}^{\prime}-\u03f5,{u}^{\prime}+\u03f5)$, it is trivial that $\varphi \left({u}_{0}\right)={\varphi}_{i}\left({u}_{0}\right)=0$.

**Theorem**

**2.**

**Proof.**

- Step1.
- Compute the distance $2a$ between ${\mathbf{F}}_{1}$ and ${\mathbf{F}}_{2}$; choose the coordinates system such that ${\mathbf{F}}_{1}(a,0)$ and ${\mathbf{F}}_{2}(-a,0)$; by a rigid-motion transform to ${\mathbf{P}}_{j}$ associate all the coordinates $({x}_{j},{y}_{j})$ for $j=1,\cdots ,n$;
- Step2.
- If ${\mathbf{F}}_{1}$, ${\mathbf{F}}_{2}$ and ${\mathbf{P}}_{j},j=1,\cdots ,n$ are on one specific circle, then it is done.
- Step3.
- Find all the intersection points ${u}_{j,k}$’s of the graphs $y={\varphi}_{j}\left(u\right)$ and $y={\varphi}_{k}\left(u\right)$ for all $j,k;$
- Step4.
- For all such the intersection points ${u}_{j,k}$’s, evaluate ${\varphi}_{i}\left({u}_{jk}\right)$ for all $i=1,\cdots ,n$; then compute $\varphi \left({u}_{j,k}\right)={max}_{i}{\varphi}_{i}\left({u}_{j,k}\right)$; and find the minimum $\varphi \left({u}^{*}\right)={min}_{j,k}\varphi \left({u}_{j,k}\right)$.
- Step5.
- If $\varphi \left({u}^{*}\right)\le {max}_{j}{w}_{j}\left|{y}_{j}\right|$, then $\varphi \left({u}^{*}\right)$ is the global minimum. If $\varphi \left({u}^{*}\right)>{max}_{j}{w}_{j}\left|{y}_{j}\right|$, the global minimum of the function $\varphi \left(u\right)$ does not exist.

## 3. Numerical Results

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The constrained optimization model of monitoring a crowd crush risk by using the position of cell phones. The constraint is to pass through ${\mathbf{F}}_{1}$ and ${\mathbf{F}}_{2}$. The blue dot is the position of the cell phone. ${\mathbf{F}}_{1}$ and ${\mathbf{F}}_{2}$ are constrained points. u is the optimal solution of minimax problem with radius r.

**Figure 2.**The validation figure of the constrained optimization problem of finding a circle in 2D that is closest to all four points, (3.58, 1.78), (−3.00, 2.61), (2.12, −3.45), (−3.26, −2.34) among all the circles that are constrained to pass through (−4, −4) and (4, 4). The solution is the center (0.034375, 0.034375) with radius $1.7081$.

**Figure 3.**The constrained optimization model (the specific time t) of finding a circle in 2D that is closest to all cell phone 20 position points among all the circles that are constrained to pass through (5, 8) and (−5, 2). The solution is the center (−13.4130, 27.355) with radius $20.582$.

**Figure 4.**The constrained optimization model (the specific time t) of finding a circle in 2D that is closest to all cell phone 20 position points.

**Figure 5.**The constrained optimization model (the specific time t) of finding a circle in 2D that is closest to all cell phone 20 position points.

**Figure 6.**The constrained optimization model (the specific time t) of finding a circle in 2D that is closest to all cell phone 20 position points, which depend on the time t. Earlier time: solid dots; the minimax black circle. Middle time: asterisk; the minimax wavy black circle. Later time: upper-oriented triangle; the minimax wavy gray circle.

**Figure 7.**The closed dots are the 20 cell phone position data at the earlier time. The open dots are the 20 cell phone position data at the later time. From the suggested model, I analyze the multiperiod optimized crowd crush monotiring at those times. The solid line circle with all 20 cell phone position data at the earlier time (closed dots) has the center (−13.413, 27.355) with the minimax radius $20.582$, and the wavy line circle with 20 points (open dots) at the later time has the center (−10.222, 22.038) with the minimax radius $18.830$.

**Table 1.**${\mathbf{F}}_{\mathbf{1}}$ = (5.00, 8.00) and ${\mathbf{F}}_{\mathbf{2}}$ = (−5.00, 2.00) are given with the equal weighted 20 points (Value is 1). The minimax value is $20.582$.

${\mathit{P}}_{\mathit{j}}$ | ${\mathit{x}}_{\mathit{j}}$ | ${\mathit{y}}_{\mathit{j}}$ |
---|---|---|

${P}_{1}$ | 2.700 | 4.900 |

${P}_{2}$ | −0.200 | 3.53 |

${P}_{3}$ | 4.97 | 1.40 |

${P}_{4}$ | −2.30 | 8.63 |

${P}_{5}$ | 8.70 | 13.9 |

${P}_{6}$ | −8.20 | 3.53 |

${P}_{7}$ | 19.00 | 6.90 |

${P}_{8}$ | −12.3 | 8.63 |

${P}_{9}$ | 33.7 | 27.9 |

${P}_{10}$ | −25.3 | 22.6 |

${P}_{11}$ | 6.70 | 4.90 |

${P}_{12}$ | −6.20 | 3.53 |

${P}_{13}$ | 2.7 | 1.92 |

${P}_{14}$ | −26.3 | 18.6 |

${P}_{15}$ | 8.37 | 15.9 |

${P}_{16}$ | −8.62 | 23.5 |

${P}_{17}$ | 9.97 | 9.90 |

${P}_{18}$ | −16.3 | 2.63 |

${P}_{19}$ | 21.7 | 2.90 |

${P}_{20}$ | −15.3 | 9.83 |

Solution | −13.413 | 27.355 |

**Table 2.**${\mathbf{F}}_{\mathbf{1}}$ = (6.00, 6.00) and ${\mathbf{F}}_{\mathbf{2}}$ = (−8.00, 8.00) are given with the equal weighted 20 points (Value is 1). The minimax value is $13.117$.

${\mathit{P}}_{\mathit{j}}$ | ${\mathit{x}}_{\mathit{j}}$ | ${\mathit{y}}_{\mathit{j}}$ |
---|---|---|

${P}_{1}$ | 2.700 | 14.900 |

${P}_{2}$ | −8.7200 | 13.50 |

${P}_{3}$ | 6.97 | 21.40 |

${P}_{4}$ | −12.30 | 18.63 |

${P}_{5}$ | 28.20 | 13.9 |

${P}_{6}$ | −8.20 | 3.53 |

${P}_{7}$ | 19.0 | 6.90 |

${P}_{8}$ | −12.2 | 28.60 |

${P}_{9}$ | 33.7 | 27.9 |

${P}_{10}$ | −25.3 | 22.6 |

${P}_{11}$ | 16.7 | 4.90 |

${P}_{12}$ | −6.42 | 13.50 |

${P}_{13}$ | 12.7 | 6.92 |

${P}_{14}$ | −26.7 | 18.6 |

${P}_{15}$ | 8.70 | 5.90 |

${P}_{16}$ | −8.62 | 23.5 |

${P}_{17}$ | 9.70 | 9.90 |

${P}_{18}$ | −16.3 | 12.6 |

${P}_{19}$ | 21.7 | 2.90 |

${P}_{20}$ | −2.83 | 12.5 |

Solution | 1.5994 | 25.196 |

**Table 3.**${\mathbf{F}}_{\mathbf{1}}$ = (25.00, 18.00) and ${\mathbf{F}}_{\mathbf{2}}$ = (−25.00, 12.00) are given with the equal weighted 20 points (Value is 1). The minimax value is $15.5322$.

${\mathit{P}}_{\mathit{j}}$ | ${\mathit{x}}_{\mathit{j}}$ | ${\mathit{y}}_{\mathit{j}}$ |
---|---|---|

${P}_{j}$ | ${x}_{j}$ | ${y}_{j}$ |

${P}_{1}$ | −16.20 | 23.50 |

${P}_{2}$ | 12.700 | 21.90 |

${P}_{3}$ | −26.30 | 28.60 |

${P}_{4}$ | 18.40 | 15.90 |

${P}_{5}$ | −8.62 | 23.50 |

${P}_{6}$ | 20.00 | 9.90 |

${P}_{7}$ | −16.30 | 2.63 |

${P}_{8}$ | 21.70 | 22.90 |

${P}_{9}$ | −15.30 | 9.83 |

${P}_{10}$ | 12.70 | 4.90 |

${P}_{11}$ | −0.20 | 3.53 |

${P}_{12}$ | 15.00 | 21.40 |

${P}_{13}$ | −2.30 | 28.60 |

${P}_{14}$ | 8.70 | 13.90 |

${P}_{15}$ | −8.20 | 23.50 |

${P}_{16}$ | 19.00 | 6.90 |

${P}_{17}$ | −12.30 | 28.60 |

${P}_{18}$ | 33.70 | 27.90 |

${P}_{19}$ | −25.3 | 22.60 |

${P}_{20}$ | 26.7 | 4.90 |

Solution | −0.32476 | 17.706 |

$\mathit{K}\mathit{I}\mathit{M}$ | $\mathit{D}\mathit{T}$ | $\mathit{k}-\mathit{N}\mathit{N}$ | $\mathit{L}\mathit{o}\mathit{g}\mathit{R}$ |
---|---|---|---|

$1.0000$ | 0.7941 | 0.7683 | 0.6328 |

$NB$ | $C4.5$ | $SVM$ | $LC$ |

0.7126 | 0.7452 | 0.7448 | 0.6408 |

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**MDPI and ACS Style**

Kim, D.
Computational Optimized Monitoring Methodology to Avoid Crowd Crush Accidents with Scattered Data. *AppliedMath* **2022**, *2*, 711-720.
https://doi.org/10.3390/appliedmath2040042

**AMA Style**

Kim D.
Computational Optimized Monitoring Methodology to Avoid Crowd Crush Accidents with Scattered Data. *AppliedMath*. 2022; 2(4):711-720.
https://doi.org/10.3390/appliedmath2040042

**Chicago/Turabian Style**

Kim, Dongyung.
2022. "Computational Optimized Monitoring Methodology to Avoid Crowd Crush Accidents with Scattered Data" *AppliedMath* 2, no. 4: 711-720.
https://doi.org/10.3390/appliedmath2040042