# The urban planning problem

An "urban planning" problem based on an example from puzzlor.

using JuMP
import GLPK
import Test

function example_urban_plan()
model = Model(GLPK.Optimizer)
# x is indexed by row and column
@variable(model, 0 <= x[1:5, 1:5] <= 1, Int)
# y is indexed by R or C, the points, and an index in 1:5. Note how JuMP
# allows indexing on arbitrary sets.
rowcol = ["R", "C"]
points = [5, 4, 3, -3, -4, -5]
@variable(model, 0 <= y[rowcol, points, 1:5] <= 1, Int)
# Objective - combine the positive and negative parts
@objective(
model,
Max,
sum(
3 * (y["R", 3, i] + y["C", 3, i]) +
1 * (y["R", 4, i] + y["C", 4, i]) +
1 * (y["R", 5, i] + y["C", 5, i]) -
3 * (y["R", -3, i] + y["C", -3, i]) -
1 * (y["R", -4, i] + y["C", -4, i]) -
1 * (y["R", -5, i] + y["C", -5, i]) for i in 1:5
)
)
# Constrain the number of residential lots
@constraint(model, sum(x) == 12)
# Add the constraints that link the auxiliary y variables to the x variables
for i in 1:5
@constraints(model, begin
# Rows
y["R", 5, i] <= 1 / 5 * sum(x[i, :]) # sum = 5
y["R", 4, i] <= 1 / 4 * sum(x[i, :]) # sum = 4
y["R", 3, i] <= 1 / 3 * sum(x[i, :]) # sum = 3
y["R", -3, i] >= 1 - 1 / 3 * sum(x[i, :]) # sum = 2
y["R", -4, i] >= 1 - 1 / 2 * sum(x[i, :]) # sum = 1
y["R", -5, i] >= 1 - 1 / 1 * sum(x[i, :]) # sum = 0
# Columns
y["C", 5, i] <= 1 / 5 * sum(x[:, i]) # sum = 5
y["C", 4, i] <= 1 / 4 * sum(x[:, i]) # sum = 4
y["C", 3, i] <= 1 / 3 * sum(x[:, i]) # sum = 3
y["C", -3, i] >= 1 - 1 / 3 * sum(x[:, i]) # sum = 2
y["C", -4, i] >= 1 - 1 / 2 * sum(x[:, i]) # sum = 1
y["C", -5, i] >= 1 - 1 / 1 * sum(x[:, i]) # sum = 0
end)
end
# Solve it
optimize!(model)
Test.@test termination_status(model) == OPTIMAL
Test.@test primal_status(model) == FEASIBLE_POINT
Test.@test objective_value(model) ≈ 14.0
return
end

example_urban_plan()

Tip