add_max_phylo_div_objective: Add maximum phylogenetic diversity objective

In oppr: Optimal Project Prioritization

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	numeric budget for funding actions.
tree	ape::phylo() phylogenetic tree describing the evolutionary relationships between the features. Note that the argument to tree must contain every feature, and only the features, present in the argument to x.

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 B_{fj}) + ((1 - (P_j B_{fj})) * (P_n \times B_{fn})), 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} * 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).

Maximize (sum_b^B L_b R_b) (eqn 1a); Subject to: sum_i^I C_i X_i <= m for all f in F (eqn 1b), R_b = 1 - prod_f^F ifelse(T_{bf} == 1, E_f, 1) for all b in B (eqn 1c), E_f = 1 - sum_j^J Y_{fj} Q_{fj} for all f in F (eqn 1d), Z_{fj} <= Y_j for all j in J (eqn 1e), sum_j^J Z_{fj} * ceil(Q_{fj}) = 1 for all f in F (eqn 1f), A_{ij} Y_{j} <= X_{i} for all i I, j in J (eqn 1g), E_f, R_b >= 0, E_f, R_b <= 1 for all b in B, f in F (eqn 1h), X_i, Y_j, Z_{fj} in [0, 1] for all i in I, j in J, f in F (eqn 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

objectives.

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)

oppr documentation built on Sept. 8, 2022, 5:07 p.m.