2021-05-03 12:11:39 +02:00
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#!/usr/bin/env python3
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2025-01-10 11:35:44 +01:00
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# Copyright 2010-2025 Google LLC
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2018-11-18 09:04:16 -08:00
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at
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2018-11-16 16:54:37 -08:00
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#
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# http://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License.
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2023-07-01 06:06:53 +02:00
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2018-11-16 16:54:37 -08:00
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"""Creates a shift scheduling problem and solves it."""
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2020-11-18 10:50:14 +01:00
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from absl import app
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from absl import flags
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2023-11-16 19:46:56 +01:00
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from ortools.sat.python import cp_model
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2020-11-30 11:59:56 +01:00
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2022-10-07 18:21:23 +02:00
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_OUTPUT_PROTO = flags.DEFINE_string(
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"output_proto", "", "Output file to write the cp_model proto to."
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)
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_PARAMS = flags.DEFINE_string(
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"params", "max_time_in_seconds:10.0", "Sat solver parameters."
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)
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2018-11-18 09:04:16 -08:00
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2018-11-16 16:54:37 -08:00
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2023-11-16 19:46:56 +01:00
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def negated_bounded_span(
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works: list[cp_model.BoolVarT], start: int, length: int
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) -> list[cp_model.BoolVarT]:
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2025-03-07 10:33:36 +01:00
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"""Filters an isolated sub-sequence of variables assigned to True.
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2023-07-01 06:06:53 +02:00
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Extract the span of Boolean variables [start, start + length), negate them,
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and if there is variables to the left/right of this span, surround the span by
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them in non negated form.
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2023-07-01 06:06:53 +02:00
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Args:
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works: a list of variables to extract the span from.
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start: the start to the span.
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length: the length of the span.
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2023-07-01 06:06:53 +02:00
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Returns:
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a list of variables which conjunction will be false if the sub-list is
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assigned to True, and correctly bounded by variables assigned to False,
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or by the start or end of works.
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"""
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sequence = []
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# left border (start of works, or works[start - 1])
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if start > 0:
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sequence.append(works[start - 1])
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for i in range(length):
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sequence.append(~works[start + i])
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# right border (end of works or works[start + length])
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if start + length < len(works):
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sequence.append(works[start + length])
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return sequence
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2023-07-01 06:06:53 +02:00
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def add_soft_sequence_constraint(
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model: cp_model.CpModel,
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works: list[cp_model.BoolVarT],
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hard_min: int,
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soft_min: int,
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min_cost: int,
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soft_max: int,
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hard_max: int,
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max_cost: int,
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prefix: str,
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) -> tuple[list[cp_model.BoolVarT], list[int]]:
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"""Sequence constraint on true variables with soft and hard bounds.
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2023-07-01 06:06:53 +02:00
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This constraint look at every maximal contiguous sequence of variables
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assigned to true. If forbids sequence of length < hard_min or > hard_max.
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Then it creates penalty terms if the length is < soft_min or > soft_max.
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Args:
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model: the sequence constraint is built on this model.
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works: a list of Boolean variables.
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hard_min: any sequence of true variables must have a length of at least
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hard_min.
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soft_min: any sequence should have a length of at least soft_min, or a
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linear penalty on the delta will be added to the objective.
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min_cost: the coefficient of the linear penalty if the length is less than
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soft_min.
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soft_max: any sequence should have a length of at most soft_max, or a linear
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penalty on the delta will be added to the objective.
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hard_max: any sequence of true variables must have a length of at most
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hard_max.
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max_cost: the coefficient of the linear penalty if the length is more than
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soft_max.
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prefix: a base name for penalty literals.
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Returns:
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a tuple (variables_list, coefficient_list) containing the different
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penalties created by the sequence constraint.
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"""
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cost_literals = []
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cost_coefficients = []
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2018-11-18 09:04:16 -08:00
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# Forbid sequences that are too short.
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for length in range(1, hard_min):
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for start in range(len(works) - length + 1):
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model.add_bool_or(negated_bounded_span(works, start, length))
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# Penalize sequences that are below the soft limit.
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if min_cost > 0:
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for length in range(hard_min, soft_min):
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for start in range(len(works) - length + 1):
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span = negated_bounded_span(works, start, length)
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name = f": under_span(start={start}, length={length})"
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lit = model.new_bool_var(prefix + name)
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span.append(lit)
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model.add_bool_or(span)
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cost_literals.append(lit)
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# We filter exactly the sequence with a short length.
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# The penalty is proportional to the delta with soft_min.
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cost_coefficients.append(min_cost * (soft_min - length))
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# Penalize sequences that are above the soft limit.
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if max_cost > 0:
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for length in range(soft_max + 1, hard_max + 1):
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for start in range(len(works) - length + 1):
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span = negated_bounded_span(works, start, length)
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name = f": over_span(start={start}, length={length})"
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lit = model.new_bool_var(prefix + name)
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span.append(lit)
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model.add_bool_or(span)
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cost_literals.append(lit)
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# Cost paid is max_cost * excess length.
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cost_coefficients.append(max_cost * (length - soft_max))
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# Just forbid any sequence of true variables with length hard_max + 1
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2021-02-15 16:04:27 +01:00
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for start in range(len(works) - hard_max):
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model.add_bool_or([~works[i] for i in range(start, start + hard_max + 1)])
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return cost_literals, cost_coefficients
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2023-07-01 06:06:53 +02:00
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def add_soft_sum_constraint(
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model: cp_model.CpModel,
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works: list[cp_model.BoolVarT],
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hard_min: int,
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soft_min: int,
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min_cost: int,
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soft_max: int,
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hard_max: int,
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max_cost: int,
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prefix: str,
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) -> tuple[list[cp_model.IntVar], list[int]]:
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"""sum constraint with soft and hard bounds.
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2018-11-16 16:54:37 -08:00
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2023-07-01 06:06:53 +02:00
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This constraint counts the variables assigned to true from works.
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If forbids sum < hard_min or > hard_max.
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Then it creates penalty terms if the sum is < soft_min or > soft_max.
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Args:
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model: the sequence constraint is built on this model.
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works: a list of Boolean variables.
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hard_min: any sequence of true variables must have a sum of at least
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hard_min.
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soft_min: any sequence should have a sum of at least soft_min, or a linear
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penalty on the delta will be added to the objective.
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min_cost: the coefficient of the linear penalty if the sum is less than
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soft_min.
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soft_max: any sequence should have a sum of at most soft_max, or a linear
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penalty on the delta will be added to the objective.
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hard_max: any sequence of true variables must have a sum of at most
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hard_max.
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max_cost: the coefficient of the linear penalty if the sum is more than
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soft_max.
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prefix: a base name for penalty variables.
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Returns:
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a tuple (variables_list, coefficient_list) containing the different
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penalties created by the sequence constraint.
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"""
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cost_variables = []
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cost_coefficients = []
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sum_var = model.new_int_var(hard_min, hard_max, "")
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# This adds the hard constraints on the sum.
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model.add(sum_var == sum(works))
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# Penalize sums below the soft_min target.
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if soft_min > hard_min and min_cost > 0:
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delta = model.new_int_var(-len(works), len(works), "")
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model.add(delta == soft_min - sum_var)
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# TODO(user): Compare efficiency with only excess >= soft_min - sum_var.
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excess = model.new_int_var(0, 7, prefix + ": under_sum")
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model.add_max_equality(excess, [delta, 0])
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cost_variables.append(excess)
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cost_coefficients.append(min_cost)
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# Penalize sums above the soft_max target.
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if soft_max < hard_max and max_cost > 0:
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delta = model.new_int_var(-7, 7, "")
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model.add(delta == sum_var - soft_max)
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excess = model.new_int_var(0, 7, prefix + ": over_sum")
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model.add_max_equality(excess, [delta, 0])
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cost_variables.append(excess)
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cost_coefficients.append(max_cost)
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return cost_variables, cost_coefficients
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def solve_shift_scheduling(params: str, output_proto: str):
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"""Solves the shift scheduling problem."""
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# Data
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num_employees = 8
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num_weeks = 3
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shifts = ["O", "M", "A", "N"]
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# Fixed assignment: (employee, shift, day).
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# This fixes the first 2 days of the schedule.
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fixed_assignments = [
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(0, 0, 0),
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(1, 0, 0),
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(2, 1, 0),
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(3, 1, 0),
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(4, 2, 0),
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(5, 2, 0),
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(6, 2, 3),
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(7, 3, 0),
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(0, 1, 1),
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(1, 1, 1),
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(2, 2, 1),
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(3, 2, 1),
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(4, 2, 1),
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(5, 0, 1),
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(6, 0, 1),
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(7, 3, 1),
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]
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# Request: (employee, shift, day, weight)
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# A negative weight indicates that the employee desire this assignment.
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requests = [
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# Employee 3 does not want to work on the first Saturday (negative weight
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# for the Off shift).
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(3, 0, 5, -2),
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# Employee 5 wants a night shift on the second Thursday (negative weight).
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(5, 3, 10, -2),
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# Employee 2 does not want a night shift on the first Friday (positive
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# weight).
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(2, 3, 4, 4),
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]
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# Shift constraints on continuous sequence :
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# (shift, hard_min, soft_min, min_penalty,
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# soft_max, hard_max, max_penalty)
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shift_constraints = [
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# One or two consecutive days of rest, this is a hard constraint.
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(0, 1, 1, 0, 2, 2, 0),
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# between 2 and 3 consecutive days of night shifts, 1 and 4 are
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# possible but penalized.
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(3, 1, 2, 20, 3, 4, 5),
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]
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# Weekly sum constraints on shifts days:
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# (shift, hard_min, soft_min, min_penalty,
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# soft_max, hard_max, max_penalty)
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weekly_sum_constraints = [
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# Constraints on rests per week.
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(0, 1, 2, 7, 2, 3, 4),
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# At least 1 night shift per week (penalized). At most 4 (hard).
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(3, 0, 1, 3, 4, 4, 0),
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]
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# Penalized transitions:
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# (previous_shift, next_shift, penalty (0 means forbidden))
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penalized_transitions = [
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# Afternoon to night has a penalty of 4.
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(2, 3, 4),
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# Night to morning is forbidden.
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(3, 1, 0),
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]
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2025-03-07 10:33:36 +01:00
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# daily demands for work shifts (morning, afternoon, night) for each day
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2018-11-16 16:54:37 -08:00
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# of the week starting on Monday.
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weekly_cover_demands = [
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(2, 3, 1), # Monday
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(2, 3, 1), # Tuesday
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(2, 2, 2), # Wednesday
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(2, 3, 1), # Thursday
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(2, 2, 2), # Friday
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(1, 2, 3), # Saturday
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(1, 3, 1), # Sunday
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]
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# Penalty for exceeding the cover constraint per shift type.
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excess_cover_penalties = (2, 2, 5)
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num_days = num_weeks * 7
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num_shifts = len(shifts)
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model = cp_model.CpModel()
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work = {}
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for e in range(num_employees):
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for s in range(num_shifts):
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for d in range(num_days):
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2024-07-25 15:46:57 -07:00
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work[e, s, d] = model.new_bool_var(f"work{e}_{s}_{d}")
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2018-11-16 16:54:37 -08:00
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# Linear terms of the objective in a minimization context.
|
2023-11-16 19:46:56 +01:00
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obj_int_vars: list[cp_model.IntVar] = []
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obj_int_coeffs: list[int] = []
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obj_bool_vars: list[cp_model.BoolVarT] = []
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obj_bool_coeffs: list[int] = []
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2018-11-16 16:54:37 -08:00
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# Exactly one shift per day.
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for e in range(num_employees):
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for d in range(num_days):
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2023-11-16 19:46:56 +01:00
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model.add_exactly_one(work[e, s, d] for s in range(num_shifts))
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2018-11-16 16:54:37 -08:00
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# Fixed assignments.
|
2018-11-21 11:00:26 -08:00
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for e, s, d in fixed_assignments:
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2023-11-16 19:46:56 +01:00
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model.add(work[e, s, d] == 1)
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2018-11-16 16:54:37 -08:00
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# Employee requests
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2018-11-21 11:00:26 -08:00
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for e, s, d, w in requests:
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2018-11-18 09:04:16 -08:00
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obj_bool_vars.append(work[e, s, d])
|
2018-11-21 11:00:26 -08:00
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obj_bool_coeffs.append(w)
|
2018-11-16 16:54:37 -08:00
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# Shift constraints
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for ct in shift_constraints:
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shift, hard_min, soft_min, min_cost, soft_max, hard_max, max_cost = ct
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|
for e in range(num_employees):
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|
works = [work[e, shift, d] for d in range(num_days)]
|
2018-11-21 11:00:26 -08:00
|
|
|
variables, coeffs = add_soft_sequence_constraint(
|
2023-07-01 06:06:53 +02:00
|
|
|
model,
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works,
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hard_min,
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soft_min,
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min_cost,
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soft_max,
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hard_max,
|
2020-11-18 10:50:14 +01:00
|
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max_cost,
|
2024-07-25 15:46:57 -07:00
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f"shift_constraint(employee {e}, shift {shift})",
|
2023-07-01 06:06:53 +02:00
|
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|
)
|
2018-11-18 09:04:16 -08:00
|
|
|
obj_bool_vars.extend(variables)
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|
|
obj_bool_coeffs.extend(coeffs)
|
2018-11-16 16:54:37 -08:00
|
|
|
|
|
|
|
|
# Weekly sum constraints
|
|
|
|
|
for ct in weekly_sum_constraints:
|
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|
|
shift, hard_min, soft_min, min_cost, soft_max, hard_max, max_cost = ct
|
|
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|
|
for e in range(num_employees):
|
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|
|
|
for w in range(num_weeks):
|
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|
|
|
works = [work[e, shift, d + w * 7] for d in range(7)]
|
2018-11-21 11:00:26 -08:00
|
|
|
variables, coeffs = add_soft_sum_constraint(
|
2023-07-01 06:06:53 +02:00
|
|
|
model,
|
|
|
|
|
works,
|
|
|
|
|
hard_min,
|
|
|
|
|
soft_min,
|
|
|
|
|
min_cost,
|
|
|
|
|
soft_max,
|
|
|
|
|
hard_max,
|
|
|
|
|
max_cost,
|
2024-07-25 15:46:57 -07:00
|
|
|
f"weekly_sum_constraint(employee {e}, shift {shift}, week {w})",
|
2023-07-01 06:06:53 +02:00
|
|
|
)
|
2018-11-18 09:04:16 -08:00
|
|
|
obj_int_vars.extend(variables)
|
|
|
|
|
obj_int_coeffs.extend(coeffs)
|
2018-11-16 16:54:37 -08:00
|
|
|
|
|
|
|
|
# Penalized transitions
|
|
|
|
|
for previous_shift, next_shift, cost in penalized_transitions:
|
|
|
|
|
for e in range(num_employees):
|
|
|
|
|
for d in range(num_days - 1):
|
|
|
|
|
transition = [
|
2023-12-15 14:10:44 +01:00
|
|
|
~work[e, previous_shift, d],
|
|
|
|
|
~work[e, next_shift, d + 1],
|
2018-11-16 16:54:37 -08:00
|
|
|
]
|
|
|
|
|
if cost == 0:
|
2023-11-16 19:46:56 +01:00
|
|
|
model.add_bool_or(transition)
|
2018-11-16 16:54:37 -08:00
|
|
|
else:
|
2023-11-16 19:46:56 +01:00
|
|
|
trans_var = model.new_bool_var(
|
2024-07-25 15:46:57 -07:00
|
|
|
f"transition (employee={e}, day={d})"
|
2023-07-01 06:06:53 +02:00
|
|
|
)
|
2018-11-16 16:54:37 -08:00
|
|
|
transition.append(trans_var)
|
2023-11-16 19:46:56 +01:00
|
|
|
model.add_bool_or(transition)
|
2018-11-18 09:04:16 -08:00
|
|
|
obj_bool_vars.append(trans_var)
|
|
|
|
|
obj_bool_coeffs.append(cost)
|
2018-11-16 16:54:37 -08:00
|
|
|
|
|
|
|
|
# Cover constraints
|
|
|
|
|
for s in range(1, num_shifts):
|
|
|
|
|
for w in range(num_weeks):
|
|
|
|
|
for d in range(7):
|
|
|
|
|
works = [work[e, s, w * 7 + d] for e in range(num_employees)]
|
2018-11-18 09:04:16 -08:00
|
|
|
# Ignore Off shift.
|
2018-11-16 16:54:37 -08:00
|
|
|
min_demand = weekly_cover_demands[d][s - 1]
|
2023-11-16 19:46:56 +01:00
|
|
|
worked = model.new_int_var(min_demand, num_employees, "")
|
|
|
|
|
model.add(worked == sum(works))
|
2018-11-16 16:54:37 -08:00
|
|
|
over_penalty = excess_cover_penalties[s - 1]
|
|
|
|
|
if over_penalty > 0:
|
2024-07-25 15:46:57 -07:00
|
|
|
name = f"excess_demand(shift={s}, week={w}, day={d})"
|
2023-11-16 19:46:56 +01:00
|
|
|
excess = model.new_int_var(0, num_employees - min_demand, name)
|
|
|
|
|
model.add(excess == worked - min_demand)
|
2018-11-18 09:04:16 -08:00
|
|
|
obj_int_vars.append(excess)
|
|
|
|
|
obj_int_coeffs.append(over_penalty)
|
2018-11-16 16:54:37 -08:00
|
|
|
|
|
|
|
|
# Objective
|
2023-11-16 19:46:56 +01:00
|
|
|
model.minimize(
|
2023-07-01 06:06:53 +02:00
|
|
|
sum(obj_bool_vars[i] * obj_bool_coeffs[i] for i in range(len(obj_bool_vars)))
|
|
|
|
|
+ sum(obj_int_vars[i] * obj_int_coeffs[i] for i in range(len(obj_int_vars)))
|
|
|
|
|
)
|
2018-11-18 09:04:16 -08:00
|
|
|
|
2021-09-03 15:21:25 +02:00
|
|
|
if output_proto:
|
2024-07-25 15:46:57 -07:00
|
|
|
print(f"Writing proto to {output_proto}")
|
2023-07-01 06:06:53 +02:00
|
|
|
with open(output_proto, "w") as text_file:
|
2018-11-18 09:04:16 -08:00
|
|
|
text_file.write(str(model))
|
2018-11-16 16:54:37 -08:00
|
|
|
|
|
|
|
|
# Solve the model.
|
|
|
|
|
solver = cp_model.CpSolver()
|
2018-11-18 09:04:16 -08:00
|
|
|
if params:
|
2025-07-23 17:38:49 +02:00
|
|
|
solver.parameters.parse_text_format(params)
|
2018-11-21 11:00:26 -08:00
|
|
|
solution_printer = cp_model.ObjectiveSolutionPrinter()
|
2023-11-16 19:46:56 +01:00
|
|
|
status = solver.solve(model, solution_printer)
|
2018-11-16 16:54:37 -08:00
|
|
|
|
|
|
|
|
# Print solution.
|
|
|
|
|
if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE:
|
|
|
|
|
print()
|
2023-07-01 06:06:53 +02:00
|
|
|
header = " "
|
2018-11-16 16:54:37 -08:00
|
|
|
for w in range(num_weeks):
|
2023-07-01 06:06:53 +02:00
|
|
|
header += "M T W T F S S "
|
2018-11-16 16:54:37 -08:00
|
|
|
print(header)
|
|
|
|
|
for e in range(num_employees):
|
2023-07-01 06:06:53 +02:00
|
|
|
schedule = ""
|
2018-11-16 16:54:37 -08:00
|
|
|
for d in range(num_days):
|
|
|
|
|
for s in range(num_shifts):
|
2023-11-16 19:46:56 +01:00
|
|
|
if solver.boolean_value(work[e, s, d]):
|
2023-07-01 06:06:53 +02:00
|
|
|
schedule += shifts[s] + " "
|
2024-07-25 15:46:57 -07:00
|
|
|
print(f"worker {e}: {schedule}")
|
2018-11-18 09:04:16 -08:00
|
|
|
print()
|
2023-07-01 06:06:53 +02:00
|
|
|
print("Penalties:")
|
2018-11-18 09:04:16 -08:00
|
|
|
for i, var in enumerate(obj_bool_vars):
|
2023-11-16 19:46:56 +01:00
|
|
|
if solver.boolean_value(var):
|
2018-11-18 09:04:16 -08:00
|
|
|
penalty = obj_bool_coeffs[i]
|
|
|
|
|
if penalty > 0:
|
2023-11-16 19:46:56 +01:00
|
|
|
print(f" {var.name} violated, penalty={penalty}")
|
2018-11-18 09:04:16 -08:00
|
|
|
else:
|
2023-11-16 19:46:56 +01:00
|
|
|
print(f" {var.name} fulfilled, gain={-penalty}")
|
2018-11-18 09:04:16 -08:00
|
|
|
|
|
|
|
|
for i, var in enumerate(obj_int_vars):
|
2023-11-16 19:46:56 +01:00
|
|
|
if solver.value(var) > 0:
|
2023-07-01 06:06:53 +02:00
|
|
|
print(
|
2024-07-25 15:46:57 -07:00
|
|
|
f" {var.name} violated by {solver.value(var)}, linear"
|
|
|
|
|
f" penalty={obj_int_coeffs[i]}"
|
2023-07-01 06:06:53 +02:00
|
|
|
)
|
2018-11-16 16:54:37 -08:00
|
|
|
|
|
|
|
|
print()
|
2024-07-25 15:46:57 -07:00
|
|
|
print(solver.response_stats())
|
2018-11-16 16:54:37 -08:00
|
|
|
|
|
|
|
|
|
2022-06-16 07:39:30 +02:00
|
|
|
def main(_):
|
2022-10-07 18:21:23 +02:00
|
|
|
solve_shift_scheduling(_PARAMS.value, _OUTPUT_PROTO.value)
|
2018-11-18 09:04:16 -08:00
|
|
|
|
|
|
|
|
|
2023-07-01 06:06:53 +02:00
|
|
|
if __name__ == "__main__":
|
2020-11-18 10:50:14 +01:00
|
|
|
app.run(main)
|
