| add_max_phylo_div_objective {oppr} | R Documentation |
Add maximum phylogenetic diversity objective
Description
Set the objective of a project prioritization problem() to
maximize the phylogenetic diversity that is expected to persist into the
future, whilst ensuring that the cost of the solution is within a
pre-specified budget (Bennett et al. 2014, Faith 2008).
Usage
add_max_phylo_div_objective(x, budget, tree)
Arguments
x |
ProjectProblem object. |
budget |
|
tree |
|
Details
A problem objective is used to specify the overall goal of the
project prioritization problem.
Here, the maximum phylogenetic diversity objective seeks to find the set
of actions that maximizes the expected amount of evolutionary history
that is expected to persist into the future given the evolutionary
relationships between the features (e.g. populations, species).
Let I represent the set of conservation actions (indexed by
i). Let C_i denote the cost for funding action i, and
let m denote the maximum expenditure (i.e. the budget). Also,
let F represent each feature (indexed by f), W_f
represent the weight for each feature f (defaults to zero for
each feature unless specified otherwise), and E_f
denote the probability that each feature will go extinct given the funded
conservation projects.
To describe the
evolutionary relationships between the features f \in F,
consider a phylogenetic tree that contains features f \in F
with branches of known lengths. This tree can be described using
mathematical notation by letting B represent the branches (indexed by
b) with lengths L_b and letting T_{bf} indicate which
features f \in F are associated with which phylogenetic
branches b \in B using zeros and ones. Ideally, the set of
features F would contain all of the species in the study
area—including non-threatened species—to fully account for the benefits
for funding different actions.
To guide the prioritization, the conservation actions are organized into
conservation projects. Let J denote the set of conservation projects
(indexed by j), and let A_{ij} denote which actions
i \in I comprise each conservation project
j \in J using zeros and ones. Next, let P_j represent
the probability of project j being successful if it is funded. Also,
let B_{fj} denote the enhanced probability that each feature
f \in F associated with the project j \in J
will persist if all of the actions that comprise project j are funded
and that project is allocated to feature f.
For convenience,
let Q_{fj} denote the actual probability that each
f \in F associated with the project j \in J
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
Q_{fj} = (P_{j} \times B_{fj}) + \bigg(\big(1 - (P_{j} B_{fj})\big)
\times (P_{n} \times B_{fn})\bigg), 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
Q_{fj} = P_{j} \times B_{fj}.
The binary control variables X_i in this problem indicate whether
each project i \in I is funded or not. The decision
variables in this problem are the Y_{j}, Z_{fj}, E_f,
and R_b variables.
Specifically, the binary Y_{j} variables indicate if project j
is funded or not based on which actions are funded; the binary
Z_{fj} variables indicate if project j is used to manage
feature f or not; the semi-continuous E_f variables
denote the probability that feature f will go extinct; and
the semi-continuous R_b variables denote the probability that
phylogenetic branch b will remain in the future.
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 F
represent the set of ten features and also the number ten).
\mathrm{Maximize} \space (\sum_{b = 0}^{B} L_b R_b) + \sum_{f}^{F}
(1 - E_f) W_f \space \mathrm{(eqn \space 1a)} \\
\mathrm{Subject \space to} \space
\sum_{i = 0}^{I} C_i \leq m \space \mathrm{(eqn \space 1b)} \\
R_b = 1 - \prod_{f = 0}^{F} ifelse(T_{bf} == 1, \space E_f, \space
1) \space \forall \space b \in B \space \mathrm{(eqn \space 1c)} \\
E_f = 1 - \sum_{j = 0}^{J} Z_{fj} Q_{fj} \space \forall \space f \in F
\space \mathrm{(eqn \space 1d)} \\
Z_{fj} \leq Y_{j} \space \forall \space j \in J \space \mathrm{(eqn \space
1e)} \\
\sum_{j = 0}^{J} Z_{fj} \times \mathrm{ceil}(Q_{fj}) = 1 \space \forall
\space f \in F \space \mathrm{(eqn \space 1f)} \\
A_{ij} Y_{j} \leq X_{i} \space \forall \space i \in I, j \in J \space
\mathrm{(eqn \space 1g)} \\
E_{f}, R_{b} \geq 0, E_{f}, R_{b} \leq 1 \space \forall \space b \in B
\space f \in F \space \mathrm{(eqn \space 1h)} \\
X_{i}, Y_{j}, Z_{fj} \in [0, 1] \space \forall \space i \in I, j \in J, f
\in F \space \mathrm{(eqn \space 1i)}
The objective (eqn 1a) is to maximize the expected phylogenetic diversity
(Faith 2008) plus the probability each feature will remain multiplied
by their weights (noting that the feature weights default to zero).
Constraint (eqn 1b) limits the maximum expenditure (i.e.
ensures that the cost of the funded actions do not exceed the budget).
Constraints (eqn 1c) calculate the probability that each branch
(including tips that correspond to a single feature) will go extinct
according to the probability that the features which share a given
branch will go extinct.
Constraints (eqn 1d) calculate the probability that each feature
will go extinct according to their allocated project.
Constraints (eqn 1e) ensure that feature can only be allocated to projects
that have all of their actions funded. Constraints (eqn 1f) state that each
feature can only be allocated to a single project. Constraints (eqn 1g)
ensure that a project cannot be funded unless all of its actions are funded.
Constraints (eqns 1h) ensure that the probability variables
(E_f) are bounded between zero and one. Constraints (eqns 1i) ensure
that the action funding (X_i), project funding (Y_j), and
project allocation (Z_{fj}) variables are binary.
Although this formulation is a mixed integer quadratically constrained programming problem (due to eqn 1c), it can be approximated using linear terms and then solved using commercial mixed integer programming solvers. This can be achieved by substituting the product of the feature extinction probabilities (eqn 1c) with the sum of the log feature extinction probabilities and using piecewise linear approximations (described in Hillier & Price 2005 pp. 390–392) to approximate the exponent of this term.
Value
ProjectProblem object with the objective added to it.
References
Bennett JR, Elliott G, Mellish B, Joseph LN, Tulloch AI, Probert WJ, Di Fonzo MMI, Monks JM, Possingham HP & Maloney R (2014) Balancing phylogenetic diversity and species numbers in conservation prioritization, using a case study of threatened species in New Zealand. Biological Conservation, 174: 47–54.
Faith DP (2008) Threatened species and the potential loss of phylogenetic diversity: conservation scenarios based on estimated extinction probabilities and phylogenetic risk analysis. Conservation Biology, 22: 1461–1470.
Hillier FS & Price CC (2005) International series in operations research & management science. Springer.
See Also
Examples
# load data
data(sim_projects, sim_features, sim_actions, sim_tree)
# plot tree
plot(sim_tree)
# build problem with maximum phylogenetic diversity objective and $200 budget
p1 <- problem(sim_projects, sim_actions, sim_features,
"name", "success", "name", "cost", "name") %>%
add_max_phylo_div_objective(budget = 200, tree = sim_tree) %>%
add_binary_decisions()
## Not run:
# solve problem
s1 <- solve(p1)
# print solution
print(s1)
# plot solution
plot(p1, s1)
# build another problem that includes feature weights
p2 <- p1 %>%
add_feature_weights("weight")
# solve problem with feature weights
s2 <- solve(p2)
# print solution based on feature weights
print(s2)
# plot solution based on feature weights
plot(p2, s2)
## End(Not run)