This is a preference structure for Modeling Preferences in MCDA that includes $J$ - the Incomparability relation.

Assume:

- there are different experts $\{1, 2, 3\}$
- they evaluate 4 projects $a, b, c, d$
- investment $a$ is preferred to investment $b$ if estimates from $a$ are higher than from $b$ (or $a$ dominates $b$)
- i.e. there is Unanimity between the experts

$a$ | $b$ | $c$ | $d$ | |
---|---|---|---|---|

1 | 10 | 8 | 7 | 6 |

2 | 9 | 7 | 5 | 6 |

3 | 12 | 8 | 9 | 4 |

We can infer the following relations:

- $a \ P \ b$ because all three experts agree
- but $b \ J \ c$:
- 1st expert say $b \ P \ c$
- but 3rd say $c \ P \ b$
- therefore we cannot compare $a$ and $b$

so we have partial order:

- $P$ is transitive
- and $J$ is not empty