Fairness of allocation in times of scarcity

[SouthEndMusic]

Now that allocation is back to an LP solve per priority (#699), fairness in allocation has to be rethought.

On way to do it is to introduce a new variable $M$ to maximize, so the objective function becomes simply

$$ \max M $$

with either constraints

$$ M \le F_{\hat{i}\hat{j}} \quad \forall (\hat{i},\hat{j}) \in \hat{E}_S : \hat{j} \in \hat{U}_S, $$

so the objective is to maximize the minimum flow to a user, or

$$ M \le \frac{F_{\hat{i}\hat{j}}}{d^p_j(t)} \quad \forall (\hat{i},\hat{j}) \in \hat{E}_S : \hat{j} \in \hat{U}_S, $$

so the objective is to maximize the minimum demand fraction allocated to the users.

A problem with this is that this method doesn't 'care' about water that cannot be distributed fairly, so not all available water is distributed.

This isssue has been scheduled to be discussed with members of the RTC-Tools team.

[SouthEndMusic]

This issue has been discussed with the RTC-Tools team, concluding that the objective for allocation flows for a given priority $p$ can have the following forms.

1. All deviations from the demand flow have the same weight:

$$ \min \sum_{(\hat{i},\hat{j}) \in \hat{E}_S : \hat{j} \in \hat{U}_S} \left|F_{\hat{i}\hat{j}} - d^p_j(t) \right| $$

2. The same as 1 but quadratic:

$$ \min \sum_{(\hat{i},\hat{j}) \in \hat{E}_S : \hat{j} \in \hat{U}_S} \left(F_{\hat{i}\hat{j}} - d^p_j(t) \right)^2 $$

3. The same as 1 but demand fraction deviations are minimized:

$$ \min \sum_{(\hat{i},\hat{j}) \in \hat{E}_S : \hat{j} \in \hat{U}_S} \left|1 - \frac{F_{\hat{i}\hat{j}}}{d^p_j(t)} \right| $$

4. The same as 2 but scaled by the demands:

$$ \min \sum_{(\hat{i},\hat{j}) \in \hat{E}_S : \hat{j} \in \hat{U}_S} \frac{\left(F_{\hat{i}\hat{j}} - d^p_j(t) \right)^2}{d^p_j(t)} $$

If desired we could support custom weights for these terms as input from the modeller.