In contrast to Uni-Objective Optimization problems, in Multi-Objective Optimization problems there are multiple

An usual model is:

- $\text{opt} f_1(x), ..., f_q(x), x \in A$
- but usually in this case there is no single
*optimal*solution - but a set of solutions where you cannot say which one is better

Example:

- suppose you want to buy a flat
- there are 2 criteria: cost and comfort
- you want to minimize the cost and maximize the comfort
- note that you cannot say that $d$ is better than $b$ or better than $a$
- but $c$ is clearly dominated by $b$: it's as comfortable as $c$, but cheaper
- the set of the best alternatives is called the
*Pareto-optimal*set of alternatives

We have $q$ criteria and $n$ items

Criterion 1 | Criterion 2 | ... | Criterion $q$ | |
---|---|---|---|---|

Item 1 | 100 | Medium | ... | 8 |

Item 2 | 100 | Medium | ... | 8 |

... | ... | ... | ... | ... |

Item $n$ | 55 | Very Bad | 8 |

Goal: to rank the items

- there are lots of conflicting criteria (like price and comfort)
- there are different units and scales
- the single optimal solution does not exist

Instead of "Item" it can be "Action", "Alternative", etc

Formally we can write it as:

- objective: $\text{opt} z(x)$
- $z: \mathbb{R}^n \to \mathbb{R}^m$
- constraints: $g_i(x) \geqslant 0, x \in \mathbb{R}^n$

But it is possible to draw a direct parallel with Voting Theory!

Voter 1 | Voter 2 | ... | Voter $q$ | |
---|---|---|---|---|

Candidate 1 | 100 | Medium | ... | 8 |

Candidate 2 | 100 | Medium | ... | 8 |

... | ... | ... | ... | ... |

Candidate $n$ | 55 | Very Bad | 8 |

So these two problems are similar:

- Each voter ranks all candidates (alternatives)
- We apply some voting mechanism and find the global preference (the "best" alternative)
- All properties of Voting Theory are still available:
- Unanimity,
- Monotonicity,
- Independence to Third Alternatives
- and others

However there are differences:

- Not all criteria have the same weight
- in Voting Theory all votes are equally important
- here some criteria may be more important than others

- We need more information than just ranking
- There are different scales
- Since the scales can be numerical, we can compare the intensity of preference

Suppose we have obtained the Pareto-optimal set of solutions. How do we choose the "best" solution?

There are several approaches:

Also MCDA is used for that:

- find the Pareto-optimal solutions
- apply MCDA to find the best one