Примеры простых многоцелевых задач

using JuMP
import HiGHS
import MultiObjectiveAlgorithms as MOA

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())
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       : 0.00000e+00
  Dual objective value : -1.50000e+01

* Work counters
  Solve time (sec)   : 7.48412e-01
  Simplex iterations : 0
  Barrier iterations : 0
  Node count         : 0

for i in 1:result_count(model)
    @assert is_solved_and_feasible(model; result = i)
    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]

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)   : 7.36149e-01
  Simplex iterations : 0
  Barrier iterations : 0
  Node count         : 0

for i in 1:result_count(model)
    @assert is_solved_and_feasible(model; result = i)
    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]

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)   : 7.49016e-03
  Simplex iterations : 0
  Barrier iterations : 0
  Node count         : 0

for i in 1:result_count(model)
    @assert is_solved_and_feasible(model; result = i)
    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