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 = 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) == MOI.OPTIMAL
    Test.@test primal_status(model) == MOI.FEASIBLE_POINT
    Test.@test objective_value(model) ≈ 14.0
    return
end

example_urban_plan()

View this file on Github.


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