add_min_set_objective {oppr} | R Documentation |
Add minimum set objective
Description
Set the objective of a project prioritization problem()
to
minimize the cost of the solution whilst ensuring that all targets are met.
This objective is conceptually similar to that used in Marxan
(Ball, Possingham & Watts 2009).
Usage
add_min_set_objective(x)
Arguments
x |
ProjectProblem object. |
Details
A problem objective is used to specify the overall goal of the
project prioritization problem.
Here, the minimum set objective seeks to find the set of actions that
minimizes the overall cost of the prioritization, while ensuring that the
funded projects meet a set of persistence targets for the conservation
features (e.g. populations, species, ecosystems). Let represent
the set of conservation actions (indexed by
). Let
denote
the cost for funding action
. Also, let
represent each
feature (indexed by
),
represent the persistence target
for feature
, and
denote the probability that each
feature will go extinct given the funded conservation projects.
To guide the prioritization, the conservation actions are organized into
conservation projects. Let denote the set of conservation projects
(indexed by
), and let
denote which actions
comprise each conservation project
using zeros and ones. Next, let
represent
the probability of project
being successful if it is funded. Also,
let
denote the enhanced probability that each feature
associated with the project
will persist if all of the actions that comprise project
are funded
and that project is allocated to feature
.
For convenience,
let
denote the actual probability that each
associated with the project
is expected to persist if the project is funded. If the argument
to
adjust_for_baseline
in the problem
function was set to
TRUE
, and this is the default behavior, then
, where
n
corresponds to the
baseline "do nothing" project. This means that the probability
of a feature persisting if a project is allocated to a feature
depends on (i) the probability of the project succeeding, (ii) the
probability of the feature persisting if the project does not fail,
and (iii) the probability of the feature persisting even if the project
fails. Otherwise, if the argument is set to FALSE
, then
.
The binary control variables in this problem indicate whether
each project
is funded or not. The decision
variables in this problem are the
,
, and
variables.
Specifically, the binary
variables indicate if project
is funded or not based on which actions are funded; the binary
variables indicate if project
is used to manage
feature
or not; and the semi-continuous
variables
denote the probability that feature
will go extinct.
Now that we have defined all the data and variables, we can formulate
the problem. For convenience, let the symbol used to denote each set also
represent its cardinality (e.g. if there are ten features, let
represent the set of ten features and also the number ten).
The objective (eqn 1a) is to minimize the cost of the funded actions.
Constraints (eqn 1b) ensure that the persistence targets are met.
Constraints (eqn 1c) calculate the probability that each feature
will go extinct according to their allocated project.
Constraints (eqn 1d) ensure that feature can only be allocated to projects
that have all of their actions funded. Constraints (eqn 1e) state that each
feature can only be allocated to a single project. Constraints (eqn 1f)
ensure that a project cannot be funded unless all of its actions are
funded. Constraints (eqns 1g) ensure that the probability variables
() are bounded between zero and one. Constraints (eqns 1h) ensure
that the action funding (
), project funding (
), and
project allocation (
) variables are binary.
Value
ProjectProblem object with the objective added to it.
References
Ball IR, Possingham HP & Watts M (2009) Marxan and relatives: software for spatial conservation prioritisation. Spatial conservation prioritisation: Quantitative methods and computational tools, 185-195.
See Also
Examples
# load the ggplot2 R package to customize plot
library(ggplot2)
# load data
data(sim_projects, sim_features, sim_actions)
# build problem with minimum set objective and targets that require each
# feature to have a 30% chance of persisting into the future
p <- problem(sim_projects, sim_actions, sim_features,
"name", "success", "name", "cost", "name") %>%
add_min_set_objective() %>%
add_absolute_targets(0.3) %>%
add_binary_decisions()
## Not run:
# solve problem
s <- solve(p)
# print solution
print(s)
# plot solution, and add a dashed line to indicate the feature targets
plot(p, s) +
geom_hline(yintercept = 0.3, linetype = "dashed")
## End(Not run)