# Evaluation of Uncertainties in the Design Process of Complex Mechanical Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Complexity

## 3. Uncertainty

## 4. Fair Distribution of Resources

- If in the probability space $\left(\mathsf{\Omega},A,P\right)$
- $\mathsf{\Omega}$ is a finite non-empty set;
- A is an $\sigma -algebra$ off the subset in $\mathsf{\Omega}$ of the events;
- P is a probability measure on A,

- One thus obtains the data, $\overline{\Delta}\text{}\mathrm{and}\text{}\underset{\_}{\Delta}$, for the distribution from the subset $\underset{\_}{\mathrm{S}}\text{}\mathrm{and}\text{}\overline{\mathrm{S}}$, while the sub-rules are:
- If $\overline{\Delta}\text{}0$ then $\underset{\_}{\mathrm{v}}=0,\text{}\mathrm{i}\in \left\{m+1,\dots ,u\right\}.$If $\underset{\_}{\Delta}\text{}0$ then the distribution is consistent.
- If $\overline{\Delta}>0$ and $\underset{\_}{\Delta}>0$ then the value of the upper set $\underset{\_}{\Delta}$ determines the lower distributions and the one of the lower set $\underset{\_}{\Delta}$ determines the higher distribution (Table 1).

## 5. Measure of the Information of a Distribution of Imprecise Data

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

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Well-Known Values | Values Included in Intervals | Values of the Laplace-Bernuille Distribution |
---|---|---|

${p}_{1}={v}_{1}={\underset{\_}{v}}_{1}={\overline{v}}_{1}$ | ${p}_{k+1}\in \left[{\underset{\_}{v}}_{m},{\overline{v}}_{m}\right]$ | ${p}_{m+1}\in \left[\overline{\Delta}/u,\underset{\_}{\Delta}/u\right]$ |

${p}_{k}={v}_{k}={\underset{\_}{v}}_{k}={\overline{v}}_{k}$ | ${p}_{m}\in \left[{\underset{\_}{v}}_{m},{\overline{v}}_{m}\right]$ | ${p}_{u}\in \left[\overline{\Delta}/u,\underset{\_}{\Delta}/u\right]$ |

${\mathit{v}}_{\mathit{i}}$ | ${\underset{\_}{\mathit{v}}}_{\mathit{i}}$ | ${\overline{\mathit{v}}}_{\mathit{i}}$ | $\left({\overline{\mathit{v}}}_{\mathit{i}}-{\underset{\_}{\mathit{v}}}_{\mathit{i}}\right)$ | - | ${\mathit{v}}_{\mathit{i}}$ | ${\underset{\_}{\mathit{v}}}_{\mathit{i}}$ | ${\overline{\mathit{v}}}_{\mathit{i}}$ | $\left({\overline{\mathit{v}}}_{\mathit{i}}-{\underset{\_}{\mathit{v}}}_{\mathit{i}}\right)$ | res |
---|---|---|---|---|---|---|---|---|---|

1 | 0.1 | 0.1 | 0 | crisp | 1 | 0.1 | 0.1 | 0 | 0.1 |

2 | 0.1 | 0.3 | 0.2 | unce | 2 | 0.1 | 0.3 | 0.2 | 0.2 |

3 | 0.2 | 0.4 | 0.2 | unce | 3 | 0.2 | 0.4 | 0.2 | 0.3 |

4 | - | - | - | unkn | 4 | $\overline{\Delta}/2=0.1$ | $\underset{\_}{\Delta}/2=0.3$ | 0.2 | 0.2 |

5 | - | - | - | unkn | 5 | $\overline{\Delta}/2=0.1$ | $\underset{\_}{\Delta}/2=0.3$ | 0.2 | 0.2 |

- | ${\displaystyle \sum}_{1}^{3}}{\underset{\_}{v}}_{i}=0.4$ | ${\displaystyle \sum}_{1}^{3}}{\overline{v}}_{i}=0.4$ | - | - | - | ${\displaystyle \sum}_{1}^{5}}{\underset{\_}{v}}_{i}=0.6$ | ${\displaystyle \sum}_{1}^{5}}{\overline{v}}_{i}=1.4$ | ${{\displaystyle \sum}}^{\text{}}=0.8$ | ${{\displaystyle \sum}}^{\text{}}=1$ |

- | $\overline{\Delta}=1-{\displaystyle {\displaystyle \sum}_{1}^{3}}{\overline{v}}_{i}=0.2$ | - | - | - | $1-{\displaystyle {\displaystyle \sum}_{1}^{5}}{v}_{i}=0.4$ | - | - | ||

- | $\underset{\_}{\Delta}=1-{\displaystyle {\displaystyle \sum}_{1}^{3}}{\underset{\_}{v}}_{i}=0.6$ | - | - | - | ${\displaystyle \sum}_{1}^{5}}\left({\overline{v}}_{i}-{\underset{\_}{v}}_{i}\right)=0.8$ | - | - |

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Villecco, F.; Pellegrino, A.
Evaluation of Uncertainties in the Design Process of Complex Mechanical Systems. *Entropy* **2017**, *19*, 475.
https://doi.org/10.3390/e19090475

**AMA Style**

Villecco F, Pellegrino A.
Evaluation of Uncertainties in the Design Process of Complex Mechanical Systems. *Entropy*. 2017; 19(9):475.
https://doi.org/10.3390/e19090475

**Chicago/Turabian Style**

Villecco, Francesco, and Arcangelo Pellegrino.
2017. "Evaluation of Uncertainties in the Design Process of Complex Mechanical Systems" *Entropy* 19, no. 9: 475.
https://doi.org/10.3390/e19090475