# The urban planning problem

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

using JuMP
import HiGHS
import Test

function example_urban_plan()
model = Model(HiGHS.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()
Presolving model
61 rows, 85 cols, 385 nonzeros
61 rows, 85 cols, 385 nonzeros
Objective function is integral with scale 1

Solving MIP model with:
61 rows
85 cols (85 binary, 0 integer, 0 implied int., 0 continuous)
385 nonzeros

( 0.0s) Starting symmetry detection
( 0.0s) Found 5 generators

Nodes      |    B&B Tree     |            Objective Bounds              |  Dynamic Constraints |       Work
Proc. InQueue |  Leaves   Expl. | BestBound       BestSol              Gap |   Cuts   InLp Confl. | LpIters     Time

0       0         0   0.00%   50              -inf                 inf        0      0      0         0     0.0s
S       0       0         0   0.00%   50              -6               933.33%        0      0      0         0     0.0s
0       0         0   0.00%   28.8            -6               580.00%        0      0      4        63     0.0s
H       0       0         0   0.00%   14              14                 0.00%     3645     99     34       642     0.2s

Solving report
Status            Optimal
Primal bound      14
Dual bound        14
Gap               0% (tolerance: 0.01%)
Solution status   feasible
14 (objective)
0 (bound viol.)
2.22044604925e-16 (int. viol.)
0 (row viol.)
Timing            0.16 (total)
0.00 (presolve)
0.00 (postsolve)
Nodes             1
LP iterations     642 (total)
0 (strong br.)
579 (separation)
0 (heuristics)

Tip