Simple multi-objective examples

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This tutorial was generated using Literate.jl. Download the source as a .jl file.

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 example is taken from Example 6.3 (from Steuer, 1985), page 154 of Ehrgott, M. (2005). Multicriteria Optimization. Springer, Berlin. 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())
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)   : 1.26982e-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)), " | ")
    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 example is taken from Example 9.38 (from Ulungu and Teghem, 1994), page 255 of Ehrgott, M. (2005). Multicriteria Optimization. Springer, Berlin. 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)   : 9.46593e-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)), " | ")
    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 example is taken from Exercise 9.5 page 269 of Ehrgott, M. (2005). Multicriteria Optimization. Springer, Berlin. 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)   : 7.05886e-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