Multi-Objective Optimization

OptimShortestPaths provides comprehensive support for multi-objective optimization through Pareto front computation.

Overview

When you have multiple conflicting objectives (e.g., minimize cost AND minimize time), there's no single "best" solution. Instead, you need to find the Pareto front - the set of solutions where improving one objective requires sacrificing another.

Computing the Pareto Front

Basic Usage

using OptimShortestPaths
using OptimShortestPaths.MultiObjective

# Create multi-objective graph
edges = [
    MultiObjectiveEdge(1, 2, [1.0, 5.0], 1),  # [cost, time] for edge 1->2
    MultiObjectiveEdge(2, 3, [2.0, 1.0], 2)   # [cost, time] for edge 2->3
]

# Build adjacency list
adjacency = [Int[] for _ in 1:3]
for (idx, edge) in enumerate(edges)
    push!(adjacency[edge.source], idx)
end

graph = MultiObjectiveGraph(
    3,                      # n_vertices
    edges,                  # edges with weights
    2,                      # n_objectives
    adjacency,              # adjacency list
    ["Cost", "Time"]        # objective names
)

source = 1
target = 3

# Compute Pareto front
pareto_solutions = compute_pareto_front(graph, source, target; max_solutions=1000)

# Each solution has:
for sol in pareto_solutions
    println("Objectives: ", sol.objectives)  # [total_cost, total_time]
    println("Path: ", sol.path)              # Vertex sequence
end

All upcoming examples repeat this lightweight setup so each block can be copied into a fresh Julia REPL and run on its own. If you are working through the page sequentially, feel free to skip the duplicated using and graph-construction code and reuse the graph, source, and target variables already in scope.

Bounded Pareto Computation

To prevent exponential growth of the Pareto set:

using OptimShortestPaths
using OptimShortestPaths.MultiObjective

edges = [
    MultiObjectiveEdge(1, 2, [1.0, 5.0], 1),
    MultiObjectiveEdge(2, 3, [2.0, 1.0], 2)
]

adjacency = [Int[] for _ in 1:3]
for (idx, edge) in enumerate(edges)
    push!(adjacency[edge.source], idx)
end

graph = MultiObjectiveGraph(3, edges, 2, adjacency, ["Cost", "Time"])
source = 1
target = 3

pareto_solutions = compute_pareto_front(
    graph, source, target;
    max_solutions=1000  # Stop after 1000 solutions
)

Scalarization Methods

Each helper in this section returns a ParetoSolution, giving you direct access to the objectives vector and reconstructed path.

Weighted Sum Approach

Convert multiple objectives into a single weighted sum:

using OptimShortestPaths
using OptimShortestPaths.MultiObjective

edges = [
    MultiObjectiveEdge(1, 2, [1.0, 5.0], 1),
    MultiObjectiveEdge(2, 3, [2.0, 1.0], 2)
]

adjacency = [Int[] for _ in 1:3]
for (idx, edge) in enumerate(edges)
    push!(adjacency[edge.source], idx)
end

graph = MultiObjectiveGraph(3, edges, 2, adjacency, ["Cost", "Time"])
source = 1
target = 3

weights = [0.7, 0.3]  # 70% cost, 30% time
solution = weighted_sum_approach(graph, source, target, weights)
println("Objectives: ", solution.objectives)
println("Path: ", solution.path)

Minimization Only

weighted_sum_approach currently requires all objectives to be minimization (:min). Transform maximization objectives first.

Epsilon-Constraint Method

Optimize one objective while constraining others:

using OptimShortestPaths
using OptimShortestPaths.MultiObjective

edges = [
    MultiObjectiveEdge(1, 2, [1.0, 5.0], 1),
    MultiObjectiveEdge(2, 3, [2.0, 1.0], 2)
]

adjacency = [Int[] for _ in 1:3]
for (idx, edge) in enumerate(edges)
    push!(adjacency[edge.source], idx)
end

graph = MultiObjectiveGraph(3, edges, 2, adjacency, ["Cost", "Time"])
source = 1
target = 3

# Minimize cost subject to: time ≤ 10.0
solution = epsilon_constraint_approach(
    graph, source, target,
    1,              # Objective index to minimize (cost)
    [Inf, 10.0]     # Constraints on objectives [cost, time]
)
println("Objectives: ", solution.objectives)
println("Path: ", solution.path)

Lexicographic Optimization

Optimize objectives in priority order:

using OptimShortestPaths
using OptimShortestPaths.MultiObjective

edges = [
    MultiObjectiveEdge(1, 2, [1.0, 5.0], 1),
    MultiObjectiveEdge(2, 3, [2.0, 1.0], 2)
]

adjacency = [Int[] for _ in 1:3]
for (idx, edge) in enumerate(edges)
    push!(adjacency[edge.source], idx)
end

graph = MultiObjectiveGraph(3, edges, 2, adjacency, ["Cost", "Time"])
source = 1
target = 3

priorities = [1, 2]  # First minimize obj 1 (cost), then obj 2 (time)
solution = lexicographic_approach(graph, source, target, priorities)
println("Objectives: ", solution.objectives)
println("Path: ", solution.path)

Decision Support

Finding the Knee Point

The "knee point" offers the best trade-off between objectives:

using OptimShortestPaths
using OptimShortestPaths.MultiObjective

# Setup from previous examples
edges = [
    MultiObjectiveEdge(1, 2, [1.0, 5.0], 1),
    MultiObjectiveEdge(2, 3, [2.0, 1.0], 2)
]

adjacency = [Int[] for _ in 1:3]
for (idx, edge) in enumerate(edges)
    push!(adjacency[edge.source], idx)
end

graph = MultiObjectiveGraph(3, edges, 2, adjacency, ["Cost", "Time"])
source = 1
target = 3

pareto_solutions = compute_pareto_front(graph, source, target)

# Find solution with best trade-off
best_solution = get_knee_point(pareto_solutions)

println("Best trade-off: ", best_solution.objectives)
println("Path: ", best_solution.path)

The knee point maximizes the angle between solutions, representing the steepest change in the Pareto curve.

Working with Objective Senses

Minimization and Maximization

using OptimShortestPaths
using OptimShortestPaths.MultiObjective

# Define mixed objectives
edges = [MultiObjectiveEdge(1, 2, [5.0, 8.0], 1)]  # [cost_to_minimize, profit_to_maximize]

# Build adjacency list
adjacency = [Int[] for _ in 1:2]
push!(adjacency[1], 1)

# Specify senses
graph = MultiObjectiveGraph(
    2,                               # n_vertices
    edges,                           # edges
    2,                               # n_objectives
    adjacency,                       # adjacency list
    ["Cost", "Profit"],              # objective names
    objective_sense = [:min, :max]   # Minimize cost, maximize profit
)

# Pareto front respects both senses
pareto_front = compute_pareto_front(graph, 1, 2)

Converting Maximization to Minimization

For scalarization methods that require :min:

# Original: maximize profit
# Transform: minimize negative profit

original_profit = 100.0
minimization_objective = -original_profit

# Or subtract from baseline
baseline = 1000.0
minimization_objective = baseline - original_profit

Example: Cost-Time Trade-off

using OptimShortestPaths
using OptimShortestPaths.MultiObjective

# Supply chain network: minimize cost AND time
edges = [
    MultiObjectiveEdge(1, 2, [10.0, 1.0], 1),  # Cheap but slow
    MultiObjectiveEdge(1, 3, [30.0, 0.5], 2),  # Expensive but fast
    MultiObjectiveEdge(2, 4, [5.0, 2.0], 3),   # Cheap and slow
    MultiObjectiveEdge(3, 4, [15.0, 1.0], 4)   # Moderate
]

# Build adjacency list
adjacency = [Int[] for _ in 1:4]
for (idx, edge) in enumerate(edges)
    push!(adjacency[edge.source], idx)
end

graph = MultiObjectiveGraph(
    4,                      # n_vertices
    edges,                  # edges
    2,                      # n_objectives (cost, time)
    adjacency,              # adjacency list
    ["Cost", "Time"]        # objective names
)

# Find all Pareto-optimal paths
pareto_front = compute_pareto_front(graph, 1, 4)

println("Found ", length(pareto_front), " Pareto-optimal solutions:")
for (i, sol) in enumerate(pareto_front)
    println("  $i. Cost: $(sol.objectives[1]), Time: $(sol.objectives[2])")
end

# Select best trade-off
best = get_knee_point(pareto_front)
println("\nBest trade-off: Cost=$(best.objectives[1]), Time=$(best.objectives[2])")

Performance Considerations

Pareto set size: Can grow exponentially; use max_solutions to bound it
Number of objectives: 2-3 objectives typical; 4+ can be slow
Graph size: Pareto computation is slower than single-objective
Dominated solutions: Automatically filtered during computation