abstract type AbstractNode end

Node type for directed, acyclic graph.

abstract type AbstractPathProbability end

Abstract path probability type.

abstract type AbstractPathUtility end

Abstract path utility type.

struct ChanceNode <: AbstractNode

A struct for chance nodes, includes the name, information set and states of the node

CompatiblePaths(diagram::InfluenceDiagram, Z::DecisionStrategy, fixed::FixedPath=Dict{Node, State}())

CompatiblePaths outer construction function. Interface for iterating over paths that are compatible and active given influence diagram and decision strategy.

1. Initialize path s of length n
2. Fill chance states s[C] by generating subpaths paths(C)
3. Fill decision states s[D] by decision strategy Z and path s

Examples

for s in CompatiblePaths(diagram, Z)
    ...
end

struct CompatiblePaths
    S::States
    C::Vector{Node}
    Z::DecisionStrategy
    fixed::FixedPath
end

CompatiblePaths type.

struct DecisionNode <: AbstractNode

A struct for decision nodes, includes the name, information set and states of the node

DecisionStrategy

Decision strategy type.

DecisionStrategy(z::DecisionVariables)

Extract values for decision variables from solved decision model.

Examples

Z = DecisionStrategy(z)

DecisionVariables(model::Model,  diagram::InfluenceDiagram; names::Bool=false, name::String="z")

Create decision variables and constraints.

Arguments

• model::Model: JuMP model into which variables are added.
• diagram::InfluenceDiagram: Influence diagram structure.
• names::Bool: Use names or have JuMP variables be anonymous.
• name::String: Prefix for predefined decision variable naming convention.

Examples

z = DecisionVariables(model, diagram)

struct DefaultPathProbability <: AbstractPathProbability

Path probability obtained as a product of the probability values corresponding to path s in each chance node.

Examples

julia> C = [2]
1-element Array{Int64,1}:
 2

julia> I_j = [[1]]
1-element Array{Array{Int64,1},1}:
 [1]

julia> X = [Probabilities(Node(2), [0.5 0.5; 0.2 0.8])]
1-element Array{Probabilities{2},1}:
 [0.5 0.5; 0.2 0.8]

julia> P = DefaultPathProbability(C, I_j, X)
DefaultPathProbability(Int16[2], Array{Int16,1}[[1]], Probabilities[[0.5 0.5; 0.2 0.8]])

julia> s = Path((1, 2))
(1, 2)

julia> P(s)
0.5

struct DefaultPathUtility <: AbstractPathUtility

Default path utility obtained as a sum of the utility values corresponding to path s in each value node.

Examples

julia> vals = Utility.([1.0 -2.0; 3.0 4.0])
2×2 Array{Float32,2}:
 1.0  -2.0
 3.0   4.0

julia> Y = [Utilities(Node(3), vals)]
1-element Array{Utilities{2},1}:
 [1.0 -2.0; 3.0 4.0]

julia> I_3 = [[1,2]]
1-element Array{Array{Int64,1},1}:
 [1, 2]

julia> U = DefaultPathUtility(I_3, Y)
DefaultPathUtility(Array{Int16,1}[[1, 2]], Utilities[[1.0 -2.0; 3.0 4.0]])

julia> s = Path((1, 2))
(1, 2)

julia> U(s)
-2.0f0

julia> t = Utility(-100.0)


julia> U(s, t)
-102.0f0

const FixedPath = Dict{Node, State}

FixedPath type.

Examples

julia> FixedPath(Dict(1=>1, 2=>3))
Dict{Int16,Int16} with 2 entries:
  2 => 3
  1 => 1

function FixedPath(diagram::InfluenceDiagram, fixed::Dict{Name, Name})

FixedPath outer construction function. Create FixedPath variable.

Arguments

• diagram::InfluenceDiagram: Influence diagram structure
• fixed::Dict{Name, Name}: Dictionary of nodes and their fixed states. Order is node=>state, and both are idefied with their names.

Example

julia> fixedpath = FixedPath(diagram, Dict("O" => "lemon"))
Dict{Int16,Int16} with 1 entry:
  1 => 1

julia> vec(collect(paths(states, fixedpath)))
3-element Array{Tuple{Int16,Int16},1}:
 (1, 1)
 (1, 2)
 (1, 3)

const ForbiddenPath = Tuple{Vector{Node}, Set{Path}}

ForbiddenPath type.

Examples

julia> ForbiddenPath(([1, 2], Set([(1, 2)])))
(Int16[1, 2], Set(Tuple{Vararg{Int16,N}} where N[(1, 2)])

julia> ForbiddenPath[
    ([1, 2], Set([(1, 2)])),
    ([3, 4, 5], Set([(1, 2, 3), (3, 4, 5)]))
]
2-element Array{Tuple{Array{Int16,1},Set{Tuple{Vararg{Int16,N}} where N}},1}:
 ([1, 2], Set([(1, 2)]))
 ([3, 4, 5], Set([(1, 2, 3), (3, 4, 5)]))

function ForbiddenPath(diagram::InfluenceDiagram, nodes::Vector{Name}, paths::Vector{NTuple{N, Name}}) where N

ForbiddenPath outer construction function. Create ForbiddenPath variable.

Arguments

• diagram::InfluenceDiagram: Influence diagram structure
• nodes::Vector{Name}: Vector of nodes involved in forbidden paths. Identified by their names.
• paths::Vector{NTuple{N, Name}}`: Vector of tuples defining the forbidden combinations of states. States identified by their names.

Example

ForbiddenPath(diagram, ["R1", "R2"], [("high", "low"), ("low", "high")])

mutable struct InfluenceDiagram
    Nodes::Vector{AbstractNode}
    Names::Vector{Name}
    I_j::Vector{Vector{Node}}
    States::Vector{Vector{Name}}
    S::States
    C::Vector{Node}
    D::Vector{Node}
    V::Vector{Node}
    X::Vector{Probabilities}
    Y::Vector{Utilities}
    P::AbstractPathProbability
    U::AbstractPathUtility
    translation::Utility
    function InfluenceDiagram()
        new(Vector{AbstractNode}())
    end
end

Hold all information related to the influence diagram.

Fields

• Nodes::Vector{AbstractNode}: Vector of added abstract nodes.
• Names::Vector{Name}: Names of nodes in order of their indices.
• I_j::Vector{Vector{Node}}: Information sets of nodes in order of their indices. Nodes of information sets identified by their indices.
• States::Vector{Vector{Name}}: States of each node in order of their indices.
• S::States: Vector showing the number of states each node has.
• C::Vector{Node}: Indices of chance nodes in ascending order.
• D::Vector{Node}: Indices of decision nodes in ascending order.
• V::Vector{Node}: Indices of value nodes in ascending order.
• X::Vector{Probabilities}: Probability matrices of chance nodes in order of chance nodes in C.
• Y::Vector{Utilities}: Utility matrices of value nodes in order of value nodes in V.
• P::AbstractPathProbability: Path probabilities.
• U::AbstractPathUtility: Path utilities.
• translation::Utility: Utility translation for storing the positive or negative utility translation.

Examples

diagram = InfluenceDiagram()

LocalDecisionStrategy{N} <: AbstractArray{Int, N}

Local decision strategy type.

function LocalDecisionStrategy(rng::AbstractRNG, diagram::InfluenceDiagram, d::Node)

Generate random decision strategy for decision node d.

Examples

rng = MersenneTwister(3)
diagram = InfluenceDiagram()
random_diagram!(rng, diagram, 5, 2, 3, 2, 2, rand(rng, [2,3], 5))
LocalDecisionStrategy(rng, diagram, diagram.D[1])

LocalDecisionStrategy(j::Node, z::Array{VariableRef})

Construct decision strategy from variable refs.

Name = String

Primitive type for node names. Alias for String.

Node = Int16

Primitive type for node index. Alias for Int16.

const Path{N} = NTuple{N, State} where N

Path type. Alias for NTuple{N, State} where N.

PathCompatibilityVariables(model::Model,
    diagram::InfluenceDiagram,
    z::DecisionVariables;
    names::Bool=false,
    name::String="x",
    forbidden_paths::Vector{ForbiddenPath}=ForbiddenPath[],
    fixed::FixedPath=Dict{Node, State}(),
    probability_cut::Bool=true,
    probability_scale_factor::Float64=1.0)

Create path compatibility variables and constraints.

Arguments

• model::Model: JuMP model into which variables are added.
• diagram::InfluenceDiagram: Influence diagram structure.
• z::DecisionVariables: Decision variables from DecisionVariables function.
• names::Bool: Use names or have JuMP variables be anonymous.
• name::String: Prefix for predefined decision variable naming convention.
• forbidden_paths::Vector{ForbiddenPath}: The forbidden subpath structures. Path compatibility variables will not be generated for paths that include forbidden subpaths.
• fixed::FixedPath: Path compatibility variable will not be generated for paths which do not include these fixed subpaths.
• probability_cut Includes probability cut constraint in the optimisation model.
• probability_scale_factor::Float64: Adjusts conditional value at risk model to be compatible with the expected value expression if the probabilities were scaled there.

Examples

x_s = PathCompatibilityVariables(model, diagram; probability_cut = false)

struct Probabilities{N} <: AbstractArray{Float64, N}

Construct and validate stage probabilities (probabilities for a single node).

Examples

julia> data = [0.5 0.5 ; 0.2 0.8]
2×2 Array{Float64,2}:
 0.5  0.5
 0.2  0.8

julia> X = Probabilities(Node(2), data)
2×2 Probabilities{2}:
 0.5  0.5
 0.2  0.8

julia> s = (1, 2)
(1, 2)

julia> X(s)
0.5

struct ProbabilityMatrix{N} <: AbstractArray{Float64, N}
    nodes::Vector{Name}
    indices::Vector{Dict{Name, Int}}
    matrix::Array{Float64, N}
end

Construct probability matrix.

function ProbabilityMatrix(diagram::InfluenceDiagram, node::Name)

Initialise a probability matrix for a given chance node. The matrix is initialised with zeros.

Examples

julia> X_O = ProbabilityMatrix(diagram, "O")
2-element ProbabilityMatrix{1}:
 0.0
 0.0

const State = Int

Primitive type for the number of states. Alias for Int16.

struct StateProbabilities
    probs::Dict{Node, Vector{Float64}}
    fixed::FixedPath
end

StateProbabilities type.

StateProbabilities(diagram::InfluenceDiagram, Z::DecisionStrategy, node::Node, state::State, prior_probabilities::StateProbabilities)

Associate each node with array of conditional probabilities for each of its states occuring in compatible paths given fixed states and prior probability. Fix node and state using their indices.

Examples

# Prior probabilities
prior_probabilities = StateProbabilities(diagram, Z)
StateProbabilities(diagram, Z, Node(2), State(1), prior_probabilities)

StateProbabilities(diagram::InfluenceDiagram, Z::DecisionStrategy, node::Name, state::Name, prior_probabilities::StateProbabilities)

Associate each node with array of conditional probabilities for each of its states occuring in compatible paths given fixed states and prior probability. Fix node and state using their names.

Examples

# Prior probabilities
prior_probabilities = StateProbabilities(diagram, Z)

# Select node and fix its state
node = "R"
state = "no test"
StateProbabilities(diagram, Z, node, state, prior_probabilities)

StateProbabilities(diagram::InfluenceDiagram, Z::DecisionStrategy)

Associate each node with array of probabilities for each of its states occuring in compatible paths.

Examples

StateProbabilities(diagram, Z)

struct States <: AbstractArray{State, 1}

States type. Works like Vector{State}.

Examples

julia> S = States(State.([2, 3, 2, 4]))
4-element States:
 2
 3
 2
 4

struct Utilities{N} <: AbstractArray{Utility, N}

State utilities.

Examples

julia> vals = Utility.([1.0 -2.0; 3.0 4.0])
2×2 Array{Float32,2}:
 1.0  -2.0
 3.0   4.0

julia> Y = Utilities(Node(3), vals)
2×2 Utilities{2}:
 1.0  -2.0
 3.0   4.0

julia> s = Path((1, 2))
 (1, 2)

julia> Y(s)
-2.0f0

const Utility = Float32

Primitive type for utility. Alias for Float32.

struct UtilityDistribution
    u::Vector{Float64}
    p::Vector{Float64}
end

UtilityDistribution type.

UtilityDistribution(diagram::InfluenceDiagram, Z::DecisionStrategy)

Construct the probability mass function for path utilities on paths that are compatible with given decision strategy.

Examples

UtilityDistribution(diagram, Z)

struct UtilityMatrix{N} <: AbstractArray{Utility, N}
    I_v::Vector{Name}
    indices::Vector{Dict{Name, Int}}
    matrix::Array{Utility, N}
end

Construct utility matrix.

function UtilityMatrix(diagram::InfluenceDiagram, node::Name)

Initialise a utility matrix for a value node. The matrix is initialised with Inf values.

Examples

julia> Y_V3 = UtilityMatrix(diagram, "V3")
2×3 UtilityMatrix{2}:
 Inf  Inf  Inf
 Inf  Inf  Inf

struct ValueNode <: AbstractNode

A struct for value nodes, includes the name and information set of the node

function add_node!(diagram::InfluenceDiagram, node::AbstractNode)

Add node to influence diagram structure.

Examples

julia> add_node!(diagram, ChanceNode("O", [], ["lemon", "peach"]))
1-element Array{AbstractNode,1}:
 ChanceNode("O", String[], ["lemon", "peach"])

function add_probabilities!(diagram::InfluenceDiagram, node::Name, probabilities::AbstractArray{Float64, N}) where N

Add probability matrix to influence diagram, specifically to its X vector.

Examples

julia> X_O = ProbabilityMatrix(diagram, "O")
2-element ProbabilityMatrix{1}:
 0.0
 0.0

julia> X_O["lemon"] = 0.2
0.2

julia> add_probabilities!(diagram, "O", X_O)
ERROR: DomainError with Probabilities should sum to one.:

julia> X_O["peach"] = 0.8
0.2

julia> add_probabilities!(diagram, "O", X_O)
1-element Array{Probabilities,1}:
 [0.2, 0.8]

function add_utilities!(diagram::InfluenceDiagram, node::Name, utilities::AbstractArray{T, N}) where {N,T<:Real}

Add utility matrix to influence diagram, specifically to its Y vector.

Examples

julia> Y_V3 = UtilityMatrix(diagram, "V3")
2×3 UtilityMatrix{2}:
 Inf  Inf  Inf
 Inf  Inf  Inf

julia> Y_V3["peach", :] = [-40, -20, 0]
3-element Array{Int64,1}:
 -40
 -20
   0

julia> Y_V3["lemon", :] = [-200, 0, 0]
3-element Array{Int64,1}:
 -200
    0
    0

julia> add_utilities!(diagram, "V3", Y_V3)
1-element Array{Utilities,1}:
 [-200.0 0.0 0.0; -40.0 -20.0 0.0]

julia> add_utilities!(diagram, "V1", [0, -25])
2-element Array{Utilities,1}:
 [-200.0 0.0 0.0; -40.0 -20.0 0.0]
 [0.0, -25.0]

conditional_value_at_risk(U_distribution::UtilityDistribution, α::Float64)

Calculate conditional value-at-risk.

conditional_value_at_risk(model::Model,
    diagram,
    x_s::PathCompatibilityVariables{N},
    α::Float64;
    probability_scale_factor::Float64=1.0) where N

Create a conditional value-at-risk (CVaR) objective.

Arguments

• model::Model: JuMP model into which variables are added.
• diagram::InfluenceDiagram: Influence diagram structure.
• x_s::PathCompatibilityVariables: Path compatibility variables.
• α::Float64: Probability level at which conditional value-at-risk is optimised.
• probability_scale_factor::Float64: Adjusts conditional value at risk model to be compatible with the expected value expression if the probabilities were scaled there.

Examples

α = 0.05  # Parameter such that 0 ≤ α ≤ 1
CVaR = conditional_value_at_risk(model, x_s, U, P, α)
CVaR = conditional_value_at_risk(model, x_s, U, P, α; probability_scale_factor = 10.0)

expected_value(model::Model,
    diagram::InfluenceDiagram,
    x_s::PathCompatibilityVariables)

Create an expected value objective.

Arguments

• model::Model: JuMP model into which variables are added.
• diagram::InfluenceDiagram: Influence diagram structure.
• x_s::PathCompatibilityVariables: Path compatibility variables.

Examples

EV = expected_value(model, diagram, x_s)

function generate_arcs!(diagram::InfluenceDiagram)

Generate arc structures using nodes added to influence diagram, by ordering nodes, giving them indices and generating correct values for the vectors Names, I_j, states, S, C, D, V in the influence digram. Abstraction is created and the names of the nodes and states are only used in the user interface from here on.

Examples

generate_arcs!(diagram)

function generate_diagram!(diagram::InfluenceDiagram;
default_probability::Bool=true,
default_utility::Bool=true,
positive_path_utility::Bool=false,
negative_path_utility::Bool=false)

Generate complete influence diagram with probabilities and utilities as well.

Arguments

• default_probability::Bool=true: Choice to use default path probabilities.
• default_utility::Bool=true: Choice to use default path utilities.
• positive_path_utility::Bool=false: Choice to use a positive path utility translation.
• negative_path_utility::Bool=false: Choice to use a negative path utility translation.

Examples

generate_diagram!(diagram)

function index_of(diagram::InfluenceDiagram, node::Name)

Get the index of a given node.

Example

julia> idx_O = index_of(diagram, "O")
1

function information_set(rng::AbstractRNG, j::Node, n_I::Int)

Generates random information sets for chance and decision nodes.

function information_set(rng::AbstractRNG, leaf_nodes::Vector{Node}, n::Int)

Generates random information sets for value nodes.

lazy_probability_cut(model::Model, diagram::InfluenceDiagram, x_s::PathCompatibilityVariables)

Add a probability cut to the model as a lazy constraint.

Examples

lazy_probability_cut(model, diagram, x_s)

function num_states(diagram::InfluenceDiagram, node::Name)

Get the number of states in a given node.

Example

julia> NS_O = num_states(diagram, "O")
2

function paths(states::AbstractVector{State}, fixed::FixedPath)

Iterate over paths with fixed states in lexicographical order.

Examples

julia> states = States(State.([2, 3]))
2-element States:
 2
 3

julia> vec(collect(paths(states, Dict(Node(1) => State(2)))))
3-element Array{Tuple{Int16,Int16},1}:
 (2, 1)
 (2, 2)
 (2, 3)

function paths(states::AbstractVector{State})

Iterate over paths in lexicographical order.

Examples

julia> states = States(State.([2, 3]))
2-element States:
 2
 3

julia> vec(collect(paths(states)))
6-element Array{Tuple{Int16,Int16},1}:
 (1, 1)
 (2, 1)
 (1, 2)
 (2, 2)
 (1, 3)
 (2, 3)

print_decision_strategy(diagram::InfluenceDiagram, Z::DecisionStrategy, state_probabilities::StateProbabilities; show_incompatible_states::Bool = false)

Print decision strategy.

Arguments

• diagram::InfluenceDiagram: Influence diagram structure.
• Z::DecisionStrategy: Decision strategy structure with optimal decision strategy.
• state_probabilities::StateProbabilities: State probabilities structure corresponding to optimal decision strategy.
• show_incompatible_states::Bool: Choice to print rows also for incompatible states.

Examples

print_decision_strategy(diagram, Z, S_probabilities)

print_risk_measures(U_distribution::UtilityDistribution, αs::Vector{Float64}; fmt = "%f")

Print risk measures.

print_state_probabilities(diagram::InfluenceDiagram, state_probabilities::StateProbabilities, nodes::Vector{Name}; prob_fmt="%f")

Print state probabilities with fixed states.

Examples

S_probabilities = StateProbabilities(diagram, Z)
print_state_probabilities(S_probabilities, ["R"])
print_state_probabilities(S_probabilities, ["A"])

print_statistics(U_distribution::UtilityDistribution; fmt = "%f")

Print statistics about utility distribution.

print_utility_distribution(U_distribution::UtilityDistribution; util_fmt="%f", prob_fmt="%f")

Print utility distribution.

Examples

U_distribution = UtilityDistribution(diagram, Z)
print_utility_distribution(U_distribution)

randomStrategy(diagram::InfluenceDiagram)

Generates a random decision strategy for the problem. Returns the strategy as well as the expected utility of the strategy and the paths that are compatible with the strategy.

Arguments

• diagram::InfluenceDiagram: Influence diagram structure.

Examples

objval, Z, S_active = randomStrategy(diagram)

random_diagram!(rng::AbstractRNG, diagram::InfluenceDiagram, n_C::Int, n_D::Int, n_V::Int, m_C::Int, m_D::Int, states::Vector{Int})

Generate random decision diagram with n_C chance nodes, n_D decision nodes, and n_V value nodes. Parameter m_C and m_D are the upper bounds for the size of the information set.

Arguments

• rng::AbstractRNG: Random number generator.
• diagram::InfluenceDiagram: The (empty) influence diagram structure that is filled by this function
• n_C::Int: Number of chance nodes.
• n_D::Int: Number of decision nodes.
• n_V::Int: Number of value nodes.
• m_C::Int: Upper bound for size of information set for chance nodes.
• m_D::Int: Upper bound for size of information set for decision nodes.
• states::Vector{State}: The number of states for each chance and decision node is randomly chosen from this set of numbers.

Examples

rng = MersenneTwister(3)
diagram = InfluenceDiagram()
random_diagram!(rng, diagram, 5, 2, 3, 2, 2, [2,3])

function random_probabilities!(rng::AbstractRNG, diagram::InfluenceDiagram, c::Node; n_inactive::Int=0)

Generate random probabilities for chance node c.

Examples

rng = MersenneTwister(3)
diagram = InfluenceDiagram()
random_diagram!(rng, diagram, 5, 2, 3, 2, 2, [2,3])
c = diagram.C[1]
random_probabilities!(rng, diagram, c)

function random_utilities!(rng::AbstractRNG, diagram::InfluenceDiagram, v::Node; low::Float64=-1.0, high::Float64=1.0)

Generate random utilities between low and high for value node v.

Examples

rng = MersenneTwister(3)
diagram = InfluenceDiagram()
random_diagram!(rng, diagram, 5, 2, 3, 2, 2, [2,3])
v = diagram.V[1]
random_utilities!(rng, diagram, v)

singlePolicyUpdate(diagram::InfluenceDiagram, model::Model)

Finds a feasible solution using single policy update and sets the model start values to that solution. Returns a vector of tuples consisting of the value of each improved solution starting from a random policy, time (in milliseconds) since the function call and the decision strategy that gave the improved value. The purpose of all this output is to allow us to examine how fast the method finds good solutions.

Arguments

• diagram::InfluenceDiagram: Influence diagram structure.
• model::Model: The decision model, modelled in JuMP
• z::DecisionVariables: The decision variables
• x_s::PathCompatibilityVariables: The path compatibility variables

Examples

solutionhistory = singlePolicyUpdate(diagram, model)

value_at_risk(U_distribution::UtilityDistribution, α::Float64)

Calculate value-at-risk.