Simple multi-objective examples
This tutorial contains a number of examples of multi-objective programs from the literature.
Required packages
This tutorial requires the following packages:
using JuMP
import HiGHS
import MultiObjectiveAlgorithms as MOA
Bi-objective linear problem
This is example is taken from Example 6.3 (from Steuer, 1985), page 154 of Multicriteria Optimization (2nd ed), M. Ehrgott, Springer 2005. The code was adapted from an example in vOptGeneric by @xgandibleux.
model = Model()
set_silent(model)
@variable(model, x1 >= 0)
@variable(model, 0 <= x2 <= 3)
@objective(model, Min, [3x1 + x2, -x1 - 2x2])
@constraint(model, 3x1 - x2 <= 6)
set_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))
set_attribute(
model,
MOA.Algorithm(),
MOA.Lexicographic(; all_permutations = true),
)
optimize!(model)
solution_summary(model)
* Solver : MOA[algorithm=MultiObjectiveAlgorithms.Lexicographic, optimizer=HiGHS]
* Status
Result count : 2
Termination status : OPTIMAL
Message from the solver:
"Solve complete. Found 2 solution(s)"
* Candidate solution (result #1)
Primal status : FEASIBLE_POINT
Dual status : NO_SOLUTION
Objective value : [0.00000e+00,0.00000e+00]
Objective bound : [0.00000e+00,-9.00000e+00]
Relative gap : Inf
Dual objective value : -9.00000e+00
* Work counters
Solve time (sec) : 1.06978e-03
Simplex iterations : 1
Barrier iterations : 0
Node count : -1
for i in 1:result_count(model)
print(i, ": z = ", round.(Int, objective_value(model; result = i)), " | ")
println("x = ", value.([x1, x2]; result = i))
end
1: z = [0, 0] | x = [0.0, -0.0]
2: z = [12, -9] | x = [3.0, 3.0]
Bi-objective linear assignment problem
This is example is taken from Example 9.38 (from Ulungu and Teghem, 1994), page 255 of Multicriteria Optimization (2nd ed), M. Ehrgott, Springer 2005. The code was adapted from an example in vOptGeneric by @xgandibleux.
C1 = [5 1 4 7; 6 2 2 6; 2 8 4 4; 3 5 7 1]
C2 = [3 6 4 2; 1 3 8 3; 5 2 2 3; 4 2 3 5]
n = size(C2, 1)
model = Model()
set_silent(model)
@variable(model, x[1:n, 1:n], Bin)
@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])
@constraint(model, [i = 1:n], sum(x[i, :]) == 1)
@constraint(model, [j = 1:n], sum(x[:, j]) == 1)
set_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))
set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
optimize!(model)
solution_summary(model)
* Solver : MOA[algorithm=MultiObjectiveAlgorithms.EpsilonConstraint, optimizer=HiGHS]
* Status
Result count : 6
Termination status : OPTIMAL
Message from the solver:
"Solve complete. Found 6 solution(s)"
* Candidate solution (result #1)
Primal status : FEASIBLE_POINT
Dual status : NO_SOLUTION
Objective value : [6.00000e+00,2.40000e+01]
Objective bound : [6.00000e+00,7.00000e+00]
Relative gap : 0.00000e+00
* Work counters
Solve time (sec) : 1.00813e-02
Simplex iterations : -1
Barrier iterations : -1
Node count : 1
for i in 1:result_count(model)
print(i, ": z = ", round.(Int, objective_value(model; result = i)), " | ")
println("x = ", round.(Int, value.(x; result = i)))
end
1: z = [6, 24] | x = [0 1 0 0; 0 0 1 0; 1 0 0 0; 0 0 0 1]
2: z = [9, 17] | x = [0 0 1 0; 0 1 0 0; 1 0 0 0; 0 0 0 1]
3: z = [12, 13] | x = [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]
4: z = [16, 11] | x = [0 0 0 1; 0 1 0 0; 0 0 1 0; 1 0 0 0]
5: z = [19, 10] | x = [0 0 1 0; 1 0 0 0; 0 0 0 1; 0 1 0 0]
6: z = [22, 7] | x = [0 0 0 1; 1 0 0 0; 0 0 1 0; 0 1 0 0]
Bi-objective shortest path problem
This is example is taken from Exercise 9.5 page 269 of Multicriteria Optimization (2nd edition), M. Ehrgott, Springer 2005. The code was adapted from an example in vOptGeneric by @xgandibleux.
M = 50
C1 = [
M 4 5 M M M
M M 2 1 2 7
M M M 5 2 M
M M 5 M M 3
M M M M M 4
M M M M M M
]
C2 = [
M 3 1 M M M
M M 1 4 2 2
M M M 1 7 M
M M 1 M M 2
M M M M M 2
M M M M M M
]
n = size(C2, 1)
model = Model()
set_silent(model)
@variable(model, x[1:n, 1:n], Bin)
@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])
@constraint(model, sum(x[1, :]) == 1)
@constraint(model, sum(x[:, n]) == 1)
@constraint(model, [i = 2:n-1], sum(x[i, :]) - sum(x[:, i]) == 0)
set_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))
set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
optimize!(model)
solution_summary(model)
* Solver : MOA[algorithm=MultiObjectiveAlgorithms.EpsilonConstraint, optimizer=HiGHS]
* Status
Result count : 4
Termination status : OPTIMAL
Message from the solver:
"Solve complete. Found 4 solution(s)"
* Candidate solution (result #1)
Primal status : FEASIBLE_POINT
Dual status : NO_SOLUTION
Objective value : [8.00000e+00,9.00000e+00]
Objective bound : [8.00000e+00,4.00000e+00]
Relative gap : 0.00000e+00
* Work counters
Solve time (sec) : 5.61881e-03
Simplex iterations : -1
Barrier iterations : -1
Node count : 1
for i in 1:result_count(model)
print(i, ": z = ", round.(Int, objective_value(model; result = i)), " | ")
X = round.(Int, value.(x; result = i))
print("Path:")
for ind in findall(val -> val ≈ 1, X)
i, j = ind.I
print(" $i->$j")
end
println()
end
1: z = [8, 9] | Path: 1->2 2->4 4->6
2: z = [10, 7] | Path: 1->2 2->5 5->6
3: z = [11, 5] | Path: 1->2 2->6
4: z = [13, 4] | Path: 1->3 3->4 4->6
This tutorial was generated using Literate.jl. View the source .jl
file on GitHub.