diff --git a/README.md b/README.md
index d8adeb5..92db314 100644
--- a/README.md
+++ b/README.md
@@ -59,6 +59,7 @@ The value must be one of the algorithms supported by MOA:
  * `MOA.Hierarchical()`
  * `MOA.KirlikSayin()`
  * `MOA.Lexicographic()` [default]
+ * `MOA.RandomWeighting()`
  * `MOA.TambyVanderpooten()`
 
 Consult their docstrings for details.
diff --git a/src/algorithms/RandomWeighting.jl b/src/algorithms/RandomWeighting.jl
new file mode 100644
index 0000000..e163cd6
--- /dev/null
+++ b/src/algorithms/RandomWeighting.jl
@@ -0,0 +1,79 @@
+#  Copyright 2019, Oscar Dowson and contributors
+#  This Source Code Form is subject to the terms of the Mozilla Public License,
+#  v.2.0. If a copy of the MPL was not distributed with this file, You can
+#  obtain one at http://mozilla.org/MPL/2.0/.
+
+"""
+    RandomWeighting()
+
+A heuristic solver that works by repeatedly solving a weighted sum problem with
+random weights.
+
+## Supported optimizer attributes
+
+ * `MOI.TimeLimitSec()`: terminate if the time limit is exceeded and return the
+   list of current solutions.
+
+ * `MOA.SolutionLimit()`: terminate once this many solutions have been found.
+
+At least one of these two limits must be set.
+"""
+mutable struct RandomWeighting <: AbstractAlgorithm
+    solution_limit::Union{Nothing,Int}
+    RandomWeighting() = new(nothing)
+end
+
+MOI.supports(::RandomWeighting, ::SolutionLimit) = true
+
+function MOI.set(alg::RandomWeighting, ::SolutionLimit, value)
+    alg.solution_limit = value
+    return
+end
+
+function MOI.get(alg::RandomWeighting, attr::SolutionLimit)
+    return something(alg.solution_limit, default(alg, attr))
+end
+
+function optimize_multiobjective!(algorithm::RandomWeighting, model::Optimizer)
+    if MOI.get(model, MOI.TimeLimitSec()) === nothing &&
+       algorithm.solution_limit === nothing
+        error("At least `MOI.TimeLimitSec` or `MOA.SolutionLimit` must be set")
+    end
+    start_time = time()
+    solutions = SolutionPoint[]
+    sense = MOI.get(model, MOI.ObjectiveSense())
+    P = MOI.output_dimension(model.f)
+    variables = MOI.get(model.inner, MOI.ListOfVariableIndices())
+    f = _scalarise(model.f, ones(P))
+    MOI.set(model.inner, MOI.ObjectiveFunction{typeof(f)}(), f)
+    MOI.optimize!(model.inner)
+    status = MOI.get(model.inner, MOI.TerminationStatus())
+    if _is_scalar_status_optimal(status)
+        X, Y = _compute_point(model, variables, model.f)
+        push!(solutions, SolutionPoint(X, Y))
+    else
+        return status, nothing
+    end
+    # This double loop is a bit weird:
+    #   * the inner loop fills up SolutionLimit number of solutions. Then we cut
+    #     it back to nondominated.
+    #   * then the outer loop goes again
+    while length(solutions) < MOI.get(algorithm, SolutionLimit())
+        while length(solutions) < MOI.get(algorithm, SolutionLimit())
+            if _time_limit_exceeded(model, start_time)
+                return MOI.TIME_LIMIT, filter_nondominated(sense, solutions)
+            end
+            weights = rand(P)
+            f = _scalarise(model.f, weights)
+            MOI.set(model.inner, MOI.ObjectiveFunction{typeof(f)}(), f)
+            MOI.optimize!(model.inner)
+            status = MOI.get(model.inner, MOI.TerminationStatus())
+            if _is_scalar_status_optimal(status)
+                X, Y = _compute_point(model, variables, model.f)
+                push!(solutions, SolutionPoint(X, Y))
+            end
+        end
+        solutions = filter_nondominated(sense, solutions)
+    end
+    return MOI.OPTIMAL, filter_nondominated(sense, solutions)
+end
diff --git a/test/algorithms/RandomWeighting.jl b/test/algorithms/RandomWeighting.jl
new file mode 100644
index 0000000..a3239dd
--- /dev/null
+++ b/test/algorithms/RandomWeighting.jl
@@ -0,0 +1,192 @@
+#  Copyright 2019, Oscar Dowson and contributors
+#  This Source Code Form is subject to the terms of the Mozilla Public License,
+#  v.2.0. If a copy of the MPL was not distributed with this file, You can
+#  obtain one at http://mozilla.org/MPL/2.0/.
+
+module TestRandomWeighting
+
+using Test
+
+import HiGHS
+import MultiObjectiveAlgorithms as MOA
+import MultiObjectiveAlgorithms: MOI
+
+include(joinpath(dirname(@__DIR__), "mock_optimizer.jl"))
+
+function run_tests()
+    for name in names(@__MODULE__; all = true)
+        if startswith("$name", "test_")
+            @testset "$name" begin
+                getfield(@__MODULE__, name)()
+            end
+        end
+    end
+    return
+end
+
+function test_error_attribute()
+    model = MOA.Optimizer(HiGHS.Optimizer)
+    MOI.set(model, MOA.Algorithm(), MOA.RandomWeighting())
+    x = MOI.add_variables(model, 2)
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)
+    f = MOI.VectorOfVariables(x)
+    MOI.set(model, MOI.ObjectiveFunction{typeof(f)}(), f)
+    @test_throws(
+        ErrorException(
+            "At least `MOI.TimeLimitSec` or `MOA.SolutionLimit` must be set",
+        ),
+        MOI.optimize!(model),
+    )
+    return
+end
+
+function test_knapsack_min()
+    n = 10
+    W = 2137.0
+    C = Float64[
+        566 611 506 180 817 184 585 423 26 317
+        62 84 977 979 874 54 269 93 881 563
+    ]
+    w = Float64[557, 898, 148, 63, 78, 964, 246, 662, 386, 272]
+    model = MOA.Optimizer(HiGHS.Optimizer)
+    MOI.set(model, MOA.Algorithm(), MOA.RandomWeighting())
+    MOI.set(model, MOA.SolutionLimit(), 3)
+    MOI.set(model, MOI.Silent(), true)
+    x = MOI.add_variables(model, n)
+    MOI.add_constraint.(model, x, MOI.ZeroOne())
+    MOI.add_constraint(
+        model,
+        MOI.ScalarAffineFunction(
+            [MOI.ScalarAffineTerm(w[j], x[j]) for j in 1:n],
+            0.0,
+        ),
+        MOI.LessThan(W),
+    )
+    f = MOI.VectorAffineFunction(
+        [
+            MOI.VectorAffineTerm(i, MOI.ScalarAffineTerm(-C[i, j], x[j]))
+            for i in 1:2 for j in 1:n
+        ],
+        [0.0, 0.0],
+    )
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)
+    MOI.set(model, MOI.ObjectiveFunction{typeof(f)}(), f)
+    MOI.optimize!(model)
+    results = [
+        [1, 0, 1, 1, 1, 0, 1, 1, 0, 1] => [-3394, -3817],
+        [0, 1, 1, 1, 1, 0, 1, 0, 1, 1] => [-3042, -4627],
+        [0, 0, 1, 1, 1, 0, 1, 1, 1, 1] => [-2854, -4636],
+    ]
+    @test MOI.get(model, MOI.ResultCount()) == length(results)
+    for (i, (x_sol, y_sol)) in enumerate(results)
+        @test ≈(x_sol, MOI.get(model, MOI.VariablePrimal(i), x); atol = 1e-6)
+        @test ≈(y_sol, MOI.get(model, MOI.ObjectiveValue(i)); atol = 1e-6)
+    end
+    @test MOI.get(model, MOI.ObjectiveBound()) ≈ [-3394, -4636]
+    return
+end
+
+function test_knapsack_max()
+    n = 10
+    W = 2137.0
+    C = Float64[
+        566 611 506 180 817 184 585 423 26 317
+        62 84 977 979 874 54 269 93 881 563
+    ]
+    w = Float64[557, 898, 148, 63, 78, 964, 246, 662, 386, 272]
+    model = MOA.Optimizer(HiGHS.Optimizer)
+    MOI.set(model, MOA.Algorithm(), MOA.RandomWeighting())
+    MOI.set(model, MOA.SolutionLimit(), 3)
+    MOI.set(model, MOI.Silent(), true)
+    x = MOI.add_variables(model, n)
+    MOI.add_constraint.(model, x, MOI.ZeroOne())
+    MOI.add_constraint(
+        model,
+        MOI.ScalarAffineFunction(
+            [MOI.ScalarAffineTerm(w[j], x[j]) for j in 1:n],
+            0.0,
+        ),
+        MOI.LessThan(W),
+    )
+    f = MOI.VectorAffineFunction(
+        [
+            MOI.VectorAffineTerm(i, MOI.ScalarAffineTerm(C[i, j], x[j])) for
+            i in 1:2 for j in 1:n
+        ],
+        [1.0, 0.0],
+    )
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+    MOI.set(model, MOI.ObjectiveFunction{typeof(f)}(), f)
+    MOI.optimize!(model)
+    results = [
+        [0, 0, 1, 1, 1, 0, 1, 1, 1, 1] => [2855, 4636],
+        [0, 1, 1, 1, 1, 0, 1, 0, 1, 1] => [3043, 4627],
+        [1, 0, 1, 1, 1, 0, 1, 1, 0, 1] => [3395, 3817],
+    ]
+    @test MOI.get(model, MOI.ResultCount()) == length(results)
+    for (i, (x_sol, y_sol)) in enumerate(results)
+        @test ≈(x_sol, MOI.get(model, MOI.VariablePrimal(i), x); atol = 1e-6)
+        @test ≈(y_sol, MOI.get(model, MOI.ObjectiveValue(i)); atol = 1e-6)
+    end
+    @test MOI.get(model, MOI.ObjectiveBound()) ≈ [3395, 4636]
+    return
+end
+
+function test_time_limit()
+    n = 10
+    W = 2137.0
+    C = Float64[
+        566 611 506 180 817 184 585 423 26 317
+        62 84 977 979 874 54 269 93 881 563
+    ]
+    w = Float64[557, 898, 148, 63, 78, 964, 246, 662, 386, 272]
+    model = MOA.Optimizer(HiGHS.Optimizer)
+    MOI.set(model, MOA.Algorithm(), MOA.RandomWeighting())
+    MOI.set(model, MOI.Silent(), true)
+    MOI.set(model, MOI.TimeLimitSec(), 0.0)
+    x = MOI.add_variables(model, n)
+    MOI.add_constraint.(model, x, MOI.ZeroOne())
+    MOI.add_constraint(
+        model,
+        MOI.ScalarAffineFunction(
+            [MOI.ScalarAffineTerm(w[j], x[j]) for j in 1:n],
+            0.0,
+        ),
+        MOI.LessThan(W),
+    )
+    f = MOI.VectorAffineFunction(
+        [
+            MOI.VectorAffineTerm(i, MOI.ScalarAffineTerm(C[i, j], x[j])) for
+            i in 1:2 for j in 1:n
+        ],
+        [0.0, 0.0],
+    )
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+    MOI.set(model, MOI.ObjectiveFunction{typeof(f)}(), f)
+    MOI.optimize!(model)
+    @test MOI.get(model, MOI.TerminationStatus()) == MOI.TIME_LIMIT
+    @test MOI.get(model, MOI.ResultCount()) >= 1
+    return
+end
+
+function test_unbounded()
+    model = MOA.Optimizer(HiGHS.Optimizer)
+    MOI.set(model, MOA.Algorithm(), MOA.RandomWeighting())
+    MOI.set(model, MOI.Silent(), true)
+    @test MOI.supports(model, MOA.SolutionLimit())
+    MOI.set(model, MOA.SolutionLimit(), 10)
+    x = MOI.add_variables(model, 2)
+    MOI.add_constraint.(model, x, MOI.GreaterThan(0.0))
+    f = MOI.Utilities.operate(vcat, Float64, 1.0 .* x...)
+    MOI.set(model, MOI.ObjectiveFunction{typeof(f)}(), f)
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+    MOI.optimize!(model)
+    @test MOI.get(model, MOI.TerminationStatus()) == MOI.DUAL_INFEASIBLE
+    @test MOI.get(model, MOI.PrimalStatus()) == MOI.NO_SOLUTION
+    @test MOI.get(model, MOI.DualStatus()) == MOI.NO_SOLUTION
+    return
+end
+
+end  # module TestRandomWeighting
+
+TestRandomWeighting.run_tests()