# Indistinguishability Operators via Yager t-norms and Their Applications to Swarm Multi-Agent Task Allocation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries on Response Threshold Methods and Indistinguishability Operators

#### 2.1. Response Threshold Methods

#### 2.2. Fuzzy Markov Chains

**Theorem**

**1.**

#### 2.3. RTM via Indistinguishability Operators

**(i)**- $E(x,x)=1$; (Reflexivity)
**(ii)**- $E(x,y)=E(y,x)$; (Symmetry)
**(iii)**- $E(x,z)\ge T\left(E\right(x,y),E(y,z\left)\right)$. (Transitivity)

## 3. Yager Possibilitic Response Function as an Indistinguishability Operator

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

## 4. Experimental Results

#### 4.1. Task Description

#### 4.2. Experimental Framework

#### 4.3. Experiments with Randomly Placed Tasks

#### 4.4. Experiments with Tasks Arranged in Clusters

#### 4.4.1. Convergence Analysis

#### 4.4.2. Probabilistic vs. Possibilistic

#### 4.4.3. Impact of the Parameter $nTH$

#### 4.4.4. Impact of the Number of Clusters

#### 4.5. Summary and Discussion of the Experimental Results

## 5. Conclusions and Future Work

## Author Contributions

## Funding

_{−}Proyecto PGC2018-095709-B-C21. This work is also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project PROCOE/4/2017 (Direcció General d’Innovació i Recerca, Govern de les Illes Balears) and by projects ROBINS and BUGWRIGHT2. These two latest projects have received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreements No 779776 and No 871260, respectively. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Environments with 120 tasks used for the experiments. Blue dots represent the position of the tasks or objects.

**Figure 2.**Number of iterations to converge to a stable possibility distribution with 120 tasks and clustered environments.

**Figure 3.**Possibilistic transition matrix induced by the OPRF with $nTH=2$ for the 120 tasks arranged into 8 clusters.

**Figure 4.**Possibilistic transition matrix induced by the YPRF with $nTH=2$ for the 120 tasks arranged into 8 clusters.

**Figure 5.**Probabilistic transition matrix induced by the OPRF with $nTH=2$ for the 120 tasks arranged into 8 clusters.

**Figure 6.**Probabilistic transition matrix induced by the YPRF with $nTH=2$ for the 120 tasks arranged into 8 clusters.

**Figure 7.**Probabilistic transition matrix induced by the OPRF with $nTH=8$ for the 120 tasks arranged into 8 clusters.

**Figure 8.**Possibilistic transition matrix, induced by the OPRF, powered after reaching a stationary possibilistic distribution (${P}^{\tau}$) for the 120 tasks arranged into 8 clusters and different values of $nTH$.

**Figure 9.**Possibilistic transition matrix, induced by the YPRF, powered after reaching a stationary possibilistic distribution (${P}^{\tau}$) for the 120 tasks arranged into 8 clusters and different values of $nTH$.

**Figure 10.**Probabilistic transition matrix, induced by the OPRF, powered after 500 iterations (${P}^{500}$) for the 120 tasks arranged into 8 clusters and different values of $nTH$.

**Figure 11.**Probabilistic transition matrix, induced by the YPRF, powered after 500 iterations (${P}^{500}$) for the 120 tasks arranged into 8 clusters and different values of $nTH$.

**Figure 12.**Possibilistic transition matrix, induced by the OPRF, powered after reaching a stationary possibilistic distribution (${P}^{\tau}$) for the 120 tasks, $nTH=4$ and different number of clusters of tasks.

**Figure 13.**Possibilistic transition matrix, induced by the YPRF, powered after reaching a stationary possibilistic distribution (${P}^{\tau}$) for the 120 tasks, $nTH=4$ and different number of clusters of tasks.

**Table 1.**Steps required to converge for probabilistic/possibilistic Markov chains with random environments.

Function | nTH | % Prob. | Steps Prob. | $\mathit{\sigma}$ Prob. | % Fuzzy | Steps Fuzzy | $\mathit{\sigma}$ Fuzzy |
---|---|---|---|---|---|---|---|

ORPF | 1 | 41.00% | 159.81 | 81.53 | 100% | 23.28 | 3.57 |

2 | 41.40% | 172.17 | 92.00 | 100% | 23.28 | 3.57 | |

4 | 38.00% | 193.28 | 81.87 | 100% | 23.28 | 3.57 | |

8 | 36.60% | 232,21 | 85.00 | 100% | 23.28 | 3.57 | |

YPRF | 1 | 41.40% | 160.00 | 81.84 | 100% | 23.29 | 3.57 |

2 | 36.00% | 204.30 | 78.97 | 100% | 23.29 | 3.57 | |

4 | 12.80% | 435.42 | 41.29 | 100% | 23.29 | 3.57 | |

8 | 0.00% | - | - | 100% | 23.29 | 3.57 |

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**MDPI and ACS Style**

Bibiloni-Femenias, M.-d.-M.; Guerrero, J.; Miñana, J.-J.; Valero, O.
Indistinguishability Operators via Yager *t*-norms and Their Applications to Swarm Multi-Agent Task Allocation. *Mathematics* **2021**, *9*, 190.
https://doi.org/10.3390/math9020190

**AMA Style**

Bibiloni-Femenias M-d-M, Guerrero J, Miñana J-J, Valero O.
Indistinguishability Operators via Yager *t*-norms and Their Applications to Swarm Multi-Agent Task Allocation. *Mathematics*. 2021; 9(2):190.
https://doi.org/10.3390/math9020190

**Chicago/Turabian Style**

Bibiloni-Femenias, Maria-del-Mar, José Guerrero, Juan-José Miñana, and Oscar Valero.
2021. "Indistinguishability Operators via Yager *t*-norms and Their Applications to Swarm Multi-Agent Task Allocation" *Mathematics* 9, no. 2: 190.
https://doi.org/10.3390/math9020190