# On Approximation of Any Ordered Fuzzy Number by A Trapezoidal Ordered Fuzzy Number

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Ordered Fuzzy Numbers—Basic Facts

**Theorem**

**1.**

**Definition**

**1.**

**Remark:**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- the symbol $\overline{)+}$ denotes such addition operator on $\mathbb{K},$ which is an extension of the addition operator $\u201c+\u201d$ on $\mathbb{R}$;
- the symbol $\u2609$ denotes such dot product operator on $\mathbb{K}$, which is an extension of the dot product operator $\u201c\xb7\u201d$ on $\mathbb{R}$.

**Example**

**1.**

**Example**

**2.**

**Definition**

**2.**

**Definition**

**3.**

**Example**

**3.**

**Example**

**4.**

## 3. Evaluation of Imprecision

**Example**

**5.**

**Example**

**6.**

**Lemma**

**1.**

**Proof.**

## 4. Approximation Problem

#### 4.1. Unconditional Approximation (UC-Approximation)

**Example**

**7.**

#### 4.2. Approximation under Criterion of Constant Ambiguity (CA-Approximation)

**Example**

**8.**

#### 4.3. Approximation under Criterion of Constant Imprecision (CI-Approximation)

**Example**

**9.**

#### 4.4. Approximation under Invariant Criterion (IC-Approximation)

#### 4.5. Approximation under Invariant Criterion with Constant Ambiguity (ICCA-Approximation)

**Example**

**10.**

#### 4.6. Approximation under Invariant Criterion with Constant Imprecision (ICCI-Approximation)

## 5. Recommendations

- If the researcher accepts the phenomenon of zero-error propagation, then one should choose CI- or ICCA-approximation method if applicable. Otherwise, one should choose UC-approximation.
- If, in a considered case, the approximation method provides no solution, then the researcher chooses the second of CI- or ICCA-approximation method,
- If, in the considered case, CI- and ICCA-approximation methods do not have solutions, then the researcher should choose the CA-approximation method,
- If, in the considered case, the CA-approximation method does not have a solution, then the researcher should choose UC-approximation.

## 6. Conclusions

- the impact of approximation on the ordering of OFNs,
- the impact of the approximation on a solution to equations determined by OFNs,
- the impact of approximation on formal modelling of real objects.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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$\overleftrightarrow{\mathit{\mathcal{L}}}$ | $\mathit{a}(\overleftrightarrow{\mathit{\mathcal{L}}})$ | $\mathit{a}({\overleftrightarrow{\mathit{T}\mathit{r}}}_{\mathit{U}\mathit{C}}(\overleftrightarrow{\mathit{\mathcal{L}}}))$ | $\mathit{e}(\overleftrightarrow{\mathit{\mathcal{L}}})$ | $\mathit{e}({\overleftrightarrow{\mathit{T}\mathit{r}}}_{\mathit{U}\mathit{C}}(\overleftrightarrow{\mathit{\mathcal{L}}}))$ |
---|---|---|---|---|

$\overleftrightarrow{\mathcal{U}}$ | $6.3614$ | $6.0000$ | $0.1365$ | $0.1429$ |

$\overleftrightarrow{\mathcal{V}}$ | $-7.8362$ | $-7.500$ | $0.1396$ | $0.1429$ |

$\overleftrightarrow{\mathcal{Z}}$ | $-2.0414$ | $-2.000$ | $0.2041$ | $0.2000$ |

$\overleftrightarrow{\mathit{\mathcal{L}}}$ | $\mathit{e}(\overleftrightarrow{\mathit{\mathcal{L}}})$ | $\mathit{e}({\overleftrightarrow{\mathit{T}\mathit{r}}}_{\mathit{C}\mathit{A}}(\overleftrightarrow{\mathit{\mathcal{L}}}))$ | $\mathit{e}({\overleftrightarrow{\mathit{T}\mathit{r}}}_{\mathit{U}\mathit{C}}(\overleftrightarrow{\mathit{\mathcal{L}}}))$ |
---|---|---|---|

$\overleftrightarrow{\mathcal{U}}$ | $0.1365$ | $0.1358$ | $0.1429$ |

$\overleftrightarrow{\mathcal{V}}$ | $0.1396$ | $0.1375$ | $0.1429$ |

$\overleftrightarrow{\mathcal{Z}}$ | $0.2041$ | $0.1967$ | $0.2000$ |

$\overleftrightarrow{\mathit{\mathcal{L}}}$ | $\mathit{e}(\overleftrightarrow{\mathit{\mathcal{L}}})$ | $\mathit{e}({\overleftrightarrow{\mathit{T}\mathit{r}}}_{\mathit{I}\mathit{C}\mathit{C}\mathit{A}}(\overleftrightarrow{\mathit{\mathcal{L}}}))$ | $\mathit{e}({\overleftrightarrow{\mathit{T}\mathit{r}}}_{\mathit{C}\mathit{A}}(\overleftrightarrow{\mathit{\mathcal{L}}}))$ | $\mathit{e}({\overleftrightarrow{\mathit{T}\mathit{r}}}_{\mathit{U}\mathit{C}}(\overleftrightarrow{\mathit{\mathcal{L}}}))$ |
---|---|---|---|---|

$\overleftrightarrow{\mathcal{U}}$ | $0.1365$ | 0.1141 | $0.1358$ | $0.1429$ |

$\overleftrightarrow{\mathcal{V}}$ | $0.1396$ | 0.1213 | $0.1375$ | $0.1429$ |

$\overleftrightarrow{\mathcal{Z}}$ | $0.2041$ | - | $0.1967$ | $0.2000$ |

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**MDPI and ACS Style**

Piasecki, K.; Łyczkowska-Hanćkowiak, A.
On Approximation of Any Ordered Fuzzy Number by A Trapezoidal Ordered Fuzzy Number. *Symmetry* **2018**, *10*, 526.
https://doi.org/10.3390/sym10100526

**AMA Style**

Piasecki K, Łyczkowska-Hanćkowiak A.
On Approximation of Any Ordered Fuzzy Number by A Trapezoidal Ordered Fuzzy Number. *Symmetry*. 2018; 10(10):526.
https://doi.org/10.3390/sym10100526

**Chicago/Turabian Style**

Piasecki, Krzysztof, and Anna Łyczkowska-Hanćkowiak.
2018. "On Approximation of Any Ordered Fuzzy Number by A Trapezoidal Ordered Fuzzy Number" *Symmetry* 10, no. 10: 526.
https://doi.org/10.3390/sym10100526