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Tom-van-Woudenberg committed Apr 30, 2024
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- Constraints and bounds as for single-objective optimization problems.

## Model
Three different ways of solving multi-objective optimization problems were introduced:
Three different ways of solving multi-objective optimization problems were introduced, which all effectively convert the problem to a single-objective optimization problem:

1. Weighted objective function: setting pre-determined weight on the two objectives, effectively converting the problem to a single-objective optimization problem. In general this requires the two objectives to have a comparable unit:
1. Weighted objective function: setting pre-determined weight on the two objectives. In general this requires the two objectives to have a comparable unit:

```{math}
:label: multi_objective_optimization_weighted
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\mathop {\min }\limits_x \left( {{\delta }_{i}} \cdot f_{1,\text{normalized}}\left( x \right) + \delta_j \cdot f_{2,\text{normalized}}\left( x \right) \right)
```

All of these methods could also be applied to problems which include more than two goals.

### Normalize objective functions
Normalizing the objectives functions can be done by setting the domain of every goal $f$ between $0$ and $1$ by finding (or estimating) the lower and upper bounds for these objective functions within the domain:

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\end{align}
```

:::{card} Test yourself
<iframe src="https://tudelft.h5p.com/content/1292259181041373527/embed" aria-label="MOO models" width="1088" height="637" frameborder="0" allowfullscreen="allowfullscreen" allow="autoplay *; geolocation *; microphone *; camera *; midi *; encrypted-media *"></iframe><script src="https://tudelft.h5p.com/js/h5p-resizer.js" charset="UTF-8"></script>
:::

## Method

Because the models are all single-objective, we can use our earlier methods to solve these problems.

## Questions, discussions and comments
<script src="https://utteranc.es/client.js"
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