examples/notebook/contrib/set_covering4.ipynb
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First, you must install ortools package in this colab.
%pip install ortools
Set partition and set covering in Google CP Solver.
Example from the Swedish book Lundgren, Roennqvist, Vaebrand 'Optimeringslaera' (translation: 'Optimization theory'), page 408.
Set partition: We want to minimize the cost of the alternatives which covers all the objects, i.e. all objects must be choosen. The requirement is than an object may be selected exactly once.
Note: This is 1-based representation
Alternative Cost Object 1 19 1,6 2 16 2,6,8 3 18 1,4,7 4 13 2,3,5 5 15 2,5 6 19 2,3 7 15 2,3,4 8 17 4,5,8 9 16 3,6,8 10 15 1,6,7
The problem has a unique solution of z = 49 where alternatives 3, 5, and 9 is selected.
Set covering: If we, however, allow that an object is selected more than one time, then the solution is z = 45 (i.e. less cost than the first problem), and the alternatives 4, 8, and 10 is selected, where object 5 is selected twice (alt. 4 and 8). It's an unique solution as well.
Compare with the following models:
This model was created by Hakan Kjellerstrand ([email protected]) Also see my other Google CP Solver models: http://www.hakank.org/google_or_tools/
from ortools.constraint_solver import pywrapcp
def main(set_partition=1):
# Create the solver.
solver = pywrapcp.Solver("Set partition and set covering")
#
# data
#
num_alternatives = 10
num_objects = 8
# costs for the alternatives
costs = [19, 16, 18, 13, 15, 19, 15, 17, 16, 15]
# the alternatives, and their objects
a = [
# 1 2 3 4 5 6 7 8 the objects
[1, 0, 0, 0, 0, 1, 0, 0], # alternative 1
[0, 1, 0, 0, 0, 1, 0, 1], # alternative 2
[1, 0, 0, 1, 0, 0, 1, 0], # alternative 3
[0, 1, 1, 0, 1, 0, 0, 0], # alternative 4
[0, 1, 0, 0, 1, 0, 0, 0], # alternative 5
[0, 1, 1, 0, 0, 0, 0, 0], # alternative 6
[0, 1, 1, 1, 0, 0, 0, 0], # alternative 7
[0, 0, 0, 1, 1, 0, 0, 1], # alternative 8
[0, 0, 1, 0, 0, 1, 0, 1], # alternative 9
[1, 0, 0, 0, 0, 1, 1, 0] # alternative 10
]
#
# declare variables
#
x = [solver.IntVar(0, 1, "x[%i]" % i) for i in range(num_alternatives)]
#
# constraints
#
# sum the cost of the choosen alternative,
# to be minimized
z = solver.ScalProd(x, costs)
#
for j in range(num_objects):
if set_partition == 1:
solver.Add(
solver.SumEquality([x[i] * a[i][j] for i in range(num_alternatives)],
1))
else:
solver.Add(
solver.SumGreaterOrEqual(
[x[i] * a[i][j] for i in range(num_alternatives)], 1))
objective = solver.Minimize(z, 1)
#
# solution and search
#
solution = solver.Assignment()
solution.Add(x)
solution.AddObjective(z)
collector = solver.LastSolutionCollector(solution)
solver.Solve(
solver.Phase([x[i] for i in range(num_alternatives)],
solver.INT_VAR_DEFAULT, solver.INT_VALUE_DEFAULT),
[collector, objective])
print("z:", collector.ObjectiveValue(0))
print(
"selected alternatives:",
[i + 1 for i in range(num_alternatives) if collector.Value(0, x[i]) == 1])
print("failures:", solver.Failures())
print("branches:", solver.Branches())
print("WallTime:", solver.WallTime())
print("Set partition:")
main(1)
print("\nSet covering:")
main(0)