# Application of Game Method for Modelling and Temporal Intuitionistic Fuzzy Pairs to the Forest Fire Spread in the Presence of Strong Wind

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## Abstract

**:**

## 1. Introduction

## 2. Wildfire in the Kresna Gorge, Bulgaria

## 3. Definition and Properties of Temporal Intuitionistic Fuzzy Pairs

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Theorem**

**3.**

## 4. Game Method for Modelling

#### 4.1. GMM Basics

- [A.1.]
- $R\to R$;
- [A.2.]
- $L\to L$;
- [A.3.]
- $S\to S$;
- [A.4.]
- $n\to n-1,$ for $n\in [1,9]$.

#### 4.2. Application of GMM to Forest Fire Spread in the Presence of Wind

- (1)
- The way of defining the new borders of the fire spread zone is visualized in Figure 5 for the three idealized cases of: (a) no fire, (b) mild wind, or (c) strong wind. (Nota bene: Here we visualize the three cases specifically for northwest wind).In particular, for every currently affected cell at this step, its own zone of subsequent fire spread is expanded:
- (a)
- with its four neighbouring cells, as shown in Figure 5a;
- (b)
- with its four neighbouring cells and the additional three cells in the direction of the mild wind as shown in Figure 5b, i.e., each currently burning cell affects seven other cells at the subsequent iteration;
- (c)
- with its four neighbouring cells and the additional eight cells in the direction of the strong wind as shown in Figure 5c, i.e., each currently burning cell affects 12 other cells at the subsequent iteration.

When these zones of subsequent fire spread are defined for all the currently affected (burning) cells at this step, the cumulative area, obtained as a union of these zones, defines the complete zone that at the next step will be affected (burning). - (2)
- Burning is represented by:
- −
- either leaving them as unchanged (according to rules [A.1.] to [A.3.] from Section 4.1) for unaffectable cells such as $R,L$ or S,
- −
- or decrementing them by 1 (according to rule [A.4.]) for the cells that are affectable, i.e., represent combustible forest mass labeled with a number in the $[1,9]$ interval with respect to the density of the “fuel”;

- (3)
- The algorithm terminates when all the cells in the grid reach the value of 0, meaning that all cells containing any flammable material have already burnt out, and/or the remaining cells are ones that may not be affected by the firespread.

## 5. Model and Simulation

## 6. Results and Discussion

## 7. Conclusions and Directions for Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**The representation of element $x\left(t\right)$ in time-moments ${t}_{1},{t}_{2},{t}_{3},\dots \in T$.

**Figure 5.**A step of the wildfire development under three different scenarios for the wind intensity (no wind, mild wind, strong wind). Reflecting the real-life scenario, we consider it for the case of wind from northwest to southeast direction.

**Figure 6.**Investigated area at the ignition point (Iteration 1): TIFP $\langle \mu \left(1\right),\nu \left(1\right)\rangle =\langle 0,\phantom{\rule{3.33333pt}{0ex}}0.998\rangle $.

**Figure 7.**The investigated area at Iteration 2: TIFP $\langle \mu \left(2\right),\nu \left(2\right)\rangle =\langle 0,\phantom{\rule{3.33333pt}{0ex}}0.976\rangle $.

**Figure 8.**The investigated area at Iteration 3: TIFP $\langle \mu \left(3\right),\nu \left(3\right)\rangle =\langle 0,0.920\rangle $.

**Figure 9.**The investigated area at Iteration 4: TIFP $\langle \mu \left(4\right),\nu \left(4\right)\rangle =\langle 0.002,0.844\rangle $.

**Figure 10.**The investigated area at Iteration 6: TIFP $\langle \mu \left(6\right),\nu \left(6\right)\rangle =\langle 0.052,0.676\rangle $.

**Figure 11.**The investigated area at Iteration 7: TIFP $\langle \mu \left(7\right),\nu \left(7\right)\rangle =\langle 0.146,0.602\rangle $.

**Figure 12.**The investigated area at Iteration 17: TIFP $\langle \mu \left(17\right),\nu \left(17\right)\rangle =\langle 0.907,0.002\rangle $.

**Figure 13.**The investigated area at Iteration 21: TIFP $\langle \mu \left(21\right),\nu \left(21\right)\rangle =\langle 0.998,0.000\rangle $.

**Figure 14.**GMM simulation of the 2017 Kresna Gorge wildfire: Standard linear graphical interpretation of TIFPs.

**Figure 15.**GMM simulation of the 2017 Kresna Gorge wildfire: Modified linear graphical interpretation of TIFPs.

**Figure 16.**GMM simulation of the 2017 Kresna Gorge wildfire: Triangular graphical interpretation of TIFPs.

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**MDPI and ACS Style**

Mavrov, D.; Atanassova, V.; Bureva, V.; Roeva, O.; Vassilev, P.; Tsvetkov, R.; Zoteva, D.; Sotirova, E.; Atanassov, K.; Alexandrov, A.;
et al. Application of Game Method for Modelling and Temporal Intuitionistic Fuzzy Pairs to the Forest Fire Spread in the Presence of Strong Wind. *Mathematics* **2022**, *10*, 1280.
https://doi.org/10.3390/math10081280

**AMA Style**

Mavrov D, Atanassova V, Bureva V, Roeva O, Vassilev P, Tsvetkov R, Zoteva D, Sotirova E, Atanassov K, Alexandrov A,
et al. Application of Game Method for Modelling and Temporal Intuitionistic Fuzzy Pairs to the Forest Fire Spread in the Presence of Strong Wind. *Mathematics*. 2022; 10(8):1280.
https://doi.org/10.3390/math10081280

**Chicago/Turabian Style**

Mavrov, Deyan, Vassia Atanassova, Veselina Bureva, Olympia Roeva, Peter Vassilev, Radoslav Tsvetkov, Dafina Zoteva, Evdokia Sotirova, Krassimir Atanassov, Alexander Alexandrov,
and et al. 2022. "Application of Game Method for Modelling and Temporal Intuitionistic Fuzzy Pairs to the Forest Fire Spread in the Presence of Strong Wind" *Mathematics* 10, no. 8: 1280.
https://doi.org/10.3390/math10081280