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# Dynamic Multi-Criteria Decision Analysis

## Site Selection in ADRIA
At each time step where an intervention is performed at a site, the sites for intervention can be selected either randomly or by using a site selection algorithm. When an algorithm is used, site selection is performed by iteratively updating how each site/reef in the domain performs against a range of criteria. Currently, ADRIA allows the following criteria to be used in site selection:

* Connectivity, calculated as ```centrality*(site coral cover)*(site area)```.
* Wave damage probability.
* Heat stress probability.
* Priority predecessors, or sites with the strongest connectivity to "priority sites" (defined by the user in the ```prioritysites``` parameter).
* Priority zones, such as GBRMPA management zones (defined by the user in the ```priorityzones``` parameter in order of preference).
* Coral cover relative to the site carrying capacity.
* Site depth, for which median site depth is used.

In the case of site selection for seeding, the coral cover criterium is defined as the area not occupied by coral relative to the site carrying capacity. For shading, the coral cover criterium is defined as the area already covered with coral relative to the carrying capacity. All other criteria are identical for seeding and shading, so that sites selected for these two interventions tend to be similar. The priority predecessors and priority zones criteria allow the user to incorporate spatial objectives, such as through GBRMPA management zone layers or data layers from other spatial management software such as MARXAN.

These criteria are used within methods based on multi-criteria decision analysis (MCDA) to rank sites from most to least suitable for intervention at each intervention time step when ADRIA is run. The criteria are weighted according to the user's preference and the weights can be defined in ADRIA through the input scenario parameters, or inline, as in the following example:

```julia
@info "Loading data package"
here = @__DIR__
dom = ADRIA.load_domain(joinpath(here, "Example_domain"))

@info "Creating 5 scenarios based on parameter bounds using the Sobol' method"
scens = ADRIA.sample(dom, 5)

@info "Adjusting heat_stress and wave_stress weightings"
scens.wave_stress .= [0.7,0.6,0.1,0.4]
scens.heat_stress .= [0.0,0.2,0.1,0.7]

```
The criteria weights and parameters which the user defines in ADRIA are described in the table below.

|Parameter |Description |
|-----------------------------|-----------------------------------------------------------------------------------|
|```heat_stress``` |Weight for heat stress when selecting seeding or shading sites. |
|```wave_stress``` |Weight for wave stress when selecting seeding or shading sites. |
|```shade_connectivity``` |Weight for larval connectivity when selecting shading sites. |
|```in_seed_connectivity``` |Weight for in-degree larval connectivity when selecting seeding sites. |
|```out_seed_connectivity``` |Weight for out-degree larval connectivity when selecting seeding sites. |
|```coral_cover_high``` |Weight of coral cover area covered when selecting shading sites. |
|```coral_cover_low``` |Weight of coral cover area available for coral gorwth when selecting seeding sites.|
|```seed_priority``` |Weight of being a priority predecessor site to a priority site when seeding. |
|```shade_priority``` |Weight of being a priority predecessor site to a priority site when shading. |
|```zone_seed``` |Weight of being in/connected to a site in a zone when seeding. |
|```zone_shade``` |Weight of being in/connected to a site in a zone when seeding. |
|```coral_cover_tol``` |% of area of corals to be seeded allowed on a site when selecting sites for seeding|
|```deployed_coral_risk_tol```|Risk tolerance for heat and wave risk when seeding or shading corals. |
|```depth_min``` |Minimum depth of sites when selecting for seeding or shading corals. |
|```depth_offset``` |depth_min+depth_offset is maximum depth of sites when seeding or shading corals. |

## MCDA methods
Multi-criteria decision analysis encompasses a series of methods for conducting decision making in a way that is formalised, structured and transparent. It evaluates the decision objective according to numerical measures of decision criteria assembled by the decision maker. Constructing explicit criteria over which to evaluate alternatives allows a transperant evaluation of benefits, negatives and trade-offs in coming to a decision solution.

Commonly, comparison of criteria in MCDA is acheived through the construction of a “values matrix” or “decision matrix”, which contains measures of value for each alternative against an array of criteria. The measure of value, for example, could be a relative ranking of worth, probability of success, or risk of failure. Formally, these values form an $N$ by $M$ matrix where entry $x_{i,j}$ is the value of the $i$th alternative as evaluated against the $j$th criterium. The decision matrix is generally normalised, so that
Multi-criteria decision analysis encompasses a series of methods for conducting decision making in a way that
is formalised, structured and transparent. It evaluates the decision objective according to numerical measures
of decision criteria assembled by the decision maker. Constructing explicit criteria over which to evaluate
alternatives allows a transperant evaluation of benefits, negatives and trade-offs in coming to a decision
solution.

Commonly, comparison of criteria in MCDA is acheived through the construction of a ''values matrix'' or ''decision
matrix'', which contains measures of value for each alternative against an array of criteria.
The measure of value, for example, could be a relative ranking of worth, probability of success, or risk
of failure. Formally, these values form an $N$ by $M$ matrix where entry $x_{i,j}$ is the value of the
$i$th alternative as evaluated against the $j$th criterium. The decision matrix is generally normalised,
so that

$$\sum_{i=1}^N\sqrt{x_{i,j}^2}=1 \forall j.$$

The user also weights each criterium with a weighting representing it's relative importance in the decison problem (which could, for example, be decided through stakeholder engagement or expert opinion). These weightings $w_j, j =1,...,M$, are also normalised so that
The user also weights each criterium with a weighting representing it's relative importance in the decison
problem (which could, for example, be decided through stakeholder engagement or expert opinion). These
weightings $w_j, j =1,...,M$, are also normalised so that
$$\sum_{j=1}^M w_j=1.$$

The final decision matrix used in the criteria ranking algorithms in ADRIA has the form,
$$X_{i,j} = w_j x_{i,j}$$

The particular MCDA technique chosen prescribes a strategy for aggregating criteria in the decision matrix and forming rankings of alternatives. ADRIA allows the user to select from 3 main aggregation methods, which are set through the ```guided``` parameter:

1. Order-ranking : ```guided = 1```
2. TOPSIS : ```guided = 2```
3. VIKOR : ```guided = 3```

The ```guided``` parameter can be set in the input parameter DataFrame or inline, as in the example below.
The particular MCDA technique chosen prescribes a strategy for aggregating criteria in the decision matrix and
forming rankings of alternatives. ADRIA allows the user to select from 3 main aggregation methods, which are set
through the ```guided``` parameter, which can be set in the criteria DataFrame. This is a regular DataFrame, with
values which can be set as in the example below.

```julia
@info "Loading data package"
here = @__DIR__
dom = ADRIA.load_domain(joinpath(here, "Example_domain"))
# Loading data package.
dom = ADRIA.load_domain("path to a domain data package")

@info "Creating 5 scenarios based on parameter bounds using the Sobol' method"
# Creating 5 scenarios based on parameter bounds using the Sobol' method
scens = ADRIA.sample(dom, 5)

@info "Setting the guided parameter for each run"
# Setting the guided parameter for each run
scens.guided[1] = -1.0 # counterfactual run
scens.guided[2] = 0.0 # unguided selection (randomised)
scens.guided[3] = 1.0 # order ranking
Expand Down Expand Up @@ -124,7 +84,9 @@ $$S^- = \textbf{min}_{i}(S_i),S^+ = \textbf{max}_{i}(S_i)$$
$$R^- = \textbf{min}_{i}(R_i),R^+ = \textbf{max}_{i}(R_i)$$
$$Q_i = \delta \frac{S_i-S^-}{S^+-S^-} + (1-\delta)\frac{R_i-R^-}{R^+-R^-}$$

Here, $\delta$ is a weighting which balances strategy between maximising all criteria and regret due to the worst performing criteria. Additional steps to the algorithm give guidelines for choosing between a single best alternative and n (n>1) best alternatives.
Here, $\delta$ is a weighting which balances strategy between maximising all criteria and regret due to the worst
performing criteria. Additional steps to the algorithm give guidelines for choosing between a single best
alternative and n (n>1) best alternatives.

## Post-ranking distance sorting
If highly ranked sites happen to be very close together in a particular domain, this can represent an issue for site selection. This is because the site selection algorithm will seed or shade in a very small region, representing a significant loss of resources if a high damage, localised event, such as a ship grounding, were to occur. To represent an insurance policy against this possibility, the user can also specify distance thresholds for selected sites. If sites are selected which are too close, sites will be replaced by the next highest ranked sites which are further apart until the distance threshold is satisfied. The distance sorting parameter inputs are described in the table below:
Expand Down

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