import abc
import numpy as np
from abc import abstractmethod
from typing import List, Union
[docs]class ASFError(Exception):
"""Raised when an error related to the ASF classes is encountered."""
[docs]class ASFBase(abc.ABC):
"""A base class for representing achievement scalarizing functions.
Instances of the implementations of this class should function as
function.
"""
@abstractmethod
[docs] def __call__(self, objective_vector: np.ndarray, reference_point: np.ndarray) -> Union[float, np.ndarray]:
"""Evaluate the ASF.
Args:
objective_vectors (np.ndarray): The objective vectors to calculate
the values.
reference_point (np.ndarray): The reference point to calculate the
values.
Returns:
Union[float, np.ndarray]: Either a single ASF value or a vector of
values if objective is a 2D array.
Note:
The reference point may not always necessarily be feasible, but
it's dimensions should match that of the objective vector.
"""
pass
[docs]class SimpleASF(ASFBase):
"""Implements a simple order-representing ASF.
Args:
weights (np.ndarray): A weight vector that holds weights. It's
length should match the number of objectives in the underlying
MOO problem the achievement problem aims to solve.
Attributes:
weights (np.ndarray): A weight vector that holds weights. It's
length should match the number of objectives in the underlying
MOO problem the achievement problem aims to solve.
"""
def __init__(self, weights: np.ndarray):
self.weights = weights
[docs] def __call__(self, objective_vector: np.ndarray, reference_point: np.ndarray) -> Union[float, np.ndarray]:
"""Evaluate the simple order-representing ASF.
Args:
objective_vector (np.ndarray): A vector representing a solution in
the solution space.
reference_point (np.ndarray): A vector representing a reference
point in the solution space.
Note:
The shaped of objective_vector and reference_point must match.
"""
return np.max(
np.where(np.isnan(reference_point), -np.inf, self.weights * (objective_vector - reference_point)), axis=-1
)
[docs]class ReferencePointASF(ASFBase):
"""Uses a reference point q and preferential factors to scalarize a MOO problem.
Args:
preferential_factors (np.ndarray): The preferential factors.
nadir (np.ndarray): The nadir point of the MOO problem to be
scalarized.
utopian_point (np.ndarray): The utopian point of the MOO problem to be
scalarized.
rho (float): A small number to be used to scale the sm factor in the
ASF. Defaults to 0.1.
Attributes:
preferential_factors (np.ndarray): The preferential factors.
nadir (np.ndarray): The nadir point of the MOO problem to be
scalarized.
utopian_point (np.ndarray): The utopian point of the MOO problem to be
scalarized.
rho (float): A small number to be used to scale the sm factor in the
ASF. Defaults to 0.1.
References:
Miettinen, K.; Eskelinen, P.; Ruiz, F. & Luque, M.
NAUTILUS method: An interactive technique in multiobjective
optimization based on the nadir point
Europen Journal of Operational Research, 2010, 206, 426-434
"""
def __init__(
self, preferential_factors: np.ndarray, nadir: np.ndarray, utopian_point: np.ndarray, rho: float = 1e-6
):
self.preferential_factors = preferential_factors
self.nadir = nadir
self.utopian_point = utopian_point
self.rho = rho
[docs] def __call__(self, objective_vector: np.ndarray, reference_point: np.ndarray) -> Union[float, np.ndarray]:
mu = self.preferential_factors
f = objective_vector
q = reference_point
rho = self.rho
z_nad = self.nadir
z_uto = self.utopian_point
max_term = np.max(mu * (f - q), axis=-1)
sum_term = rho * np.sum((f - q) / (z_nad - z_uto), axis=-1)
return max_term + sum_term
[docs]class MaxOfTwoASF(ASFBase):
"""Implements the ASF used in NIMBUS, which takes the maximum of two terms.
Args:
nadir (np.ndarray): The nadir point.
ideal (np.ndarray): The ideal point.
lt_inds (List[int]): Indices of the objectives categorized to be
decreased.
lte_inds (List[int]): Indices of the objectives categorized to be
reduced until some value is reached.
rho (float): A small number to form the utopian point.
rho_sum (float): A small number to be used as a weight for the sum
term.
Attributes:
nadir (np.ndarray): The nadir point.
ideal (np.ndarray): The ideal point.
lt_inds (List[int]): Indices of the objectives categorized to be
decreased.
lte_inds (List[int]): Indices of the objectives categorized to be
reduced until some value is reached.
rho (float): A small number to form the utopian point.
rho_sum (float): A small number to be used as a weight for the sum
term.
References:
Miettinen, K. & Mäkelä, Marko M.
Synchronous approach in interactive multiobjective optimization
European Journal of Operational Research, 2006, 170, 909-922
"""
def __init__(
self,
nadir: np.ndarray,
ideal: np.ndarray,
lt_inds: List[int],
lte_inds: List[int],
rho: float = 1e-6,
rho_sum: float = 1e-6,
):
self.nadir = nadir
self.ideal = ideal
self.lt_inds = lt_inds
self.lte_inds = lte_inds
self.rho = rho
self.rho_sum = rho_sum
[docs] def __call__(self, objective_vector: np.ndarray, reference_point: np.ndarray) -> Union[float, np.ndarray]:
# assure this function works with single objective vectors
if objective_vector.ndim == 1:
f = objective_vector.reshape((1, -1))
else:
f = objective_vector
ii = self.lt_inds
jj = self.lte_inds
z = reference_point
nad = self.nadir
ide = self.ideal
uto = self.ideal - self.rho
lt_term = (f[:, ii] - ide[ii]) / (nad[ii] - uto[ii])
lte_term = (f[:, jj] - z[jj]) / (nad[jj] - uto[jj])
max_term = np.max(np.hstack((lt_term, lte_term)), axis=1)
sum_term = self.rho_sum * np.sum(f / (nad - uto), axis=1)
return max_term + sum_term
[docs]class StomASF(ASFBase):
"""Implementation of the satisfying trade-off method (STOM).
Args:
ideal (np.ndarray): The ideal point.
rho (float): A small number to form the utopian point.
rho_sum (float): A small number to be used as a weight for the sum
term.
Attributes:
ideal (np.ndarray): The ideal point.
rho (float): A small number to form the utopian point.
rho_sum (float): A small number to be used as a weight for the sum
term.
References:
Miettinen, K. & Mäkelä, Marko M.
Synchronous approach in interactive multiobjective optimization
European Journal of Operational Research, 2006, 170, 909-922
"""
def __init__(self, ideal: np.ndarray, rho: float = 1e-6, rho_sum: float = 1e-6):
self.ideal = ideal
self.rho = rho
self.rho_sum = rho_sum
[docs] def __call__(self, objective_vectors: np.ndarray, reference_point: np.ndarray) -> Union[float, np.ndarray]:
# assure this function works with single objective vectors
if objective_vectors.ndim == 1:
f = objective_vectors.reshape((1, -1))
else:
f = objective_vectors
z = reference_point
uto = self.ideal - self.rho
max_term = np.max((f - uto) / (z - uto), axis=1)
sum_term = self.rho_sum * np.sum((f) / (z - uto), axis=1)
return max_term + sum_term
[docs]class PointMethodASF(ASFBase):
"""Implementation of the reference point based ASF.
Args:
nadir (np.ndarray): The nadir point.
ideal (np.ndarray): The ideal point.
rho (float): A small number to form the utopian point.
rho_sum (float): A small number to be used as a weight for the sum
term.
References:
Miettinen, K. & Mäkelä, Marko M.
Synchronous approach in interactive multiobjective optimization
European Journal of Operational Research, 2006, 170, 909-922
"""
def __init__(self, nadir: np.ndarray, ideal: np.ndarray, rho: float = 1e-6, rho_sum: float = 1e-6):
self.nadir = nadir
self.ideal = ideal
self.rho = rho
self.rho_sum = rho_sum
[docs] def __call__(self, objective_vectors: np.ndarray, reference_point: np.ndarray):
# assure this function works with single objective vectors
if objective_vectors.ndim == 1:
f = objective_vectors.reshape((1, -1))
else:
f = objective_vectors
z = reference_point
nad = self.nadir
uto = self.ideal - self.rho
max_term = np.max((f - z) / (nad - uto), axis=1)
sum_term = self.rho_sum * np.sum((f) / (nad - uto), axis=1)
return max_term + sum_term
[docs]class AugmentedGuessASF(ASFBase):
"""Implementation of the augmented GUESS related ASF.
Args:
nadir (np.ndarray): The nadir point.
ideal (np.ndarray): The ideal point.
index_to_exclude (List[int]): The indices of the objective functions to
be excluded in calculating the first term of the ASF.
rho (float): A small number to form the utopian point.
rho_sum (float): A small number to be used as a weight for the sum
term.
References:
Miettinen, K. & Mäkelä, Marko M.
Synchronous approach in interactive multiobjective optimization
European Journal of Operational Research, 2006, 170, 909-922
"""
def __init__(
self,
nadir: np.ndarray,
ideal: np.ndarray,
index_to_exclude: List[int],
rho: float = 1e-6,
rho_sum: float = 1e-6,
):
self.nadir = nadir
self.ideal = ideal
self.index_to_exclude = index_to_exclude
self.rho = rho
self.rho_sum = rho_sum
[docs] def __call__(self, objective_vectors: np.ndarray, reference_point: np.ndarray):
# assure this function works with single objective vectors
if objective_vectors.ndim == 1:
f = objective_vectors.reshape((1, -1))
else:
f = objective_vectors
if reference_point.ndim == 1:
z = reference_point
elif reference_point.ndim == 2 and reference_point.shape[0] == 1:
z = reference_point[0]
else:
msg = "Error interpreting reference point"
raise ASFError(msg)
nad = self.nadir
uto = self.ideal - self.rho
ex_mask = np.full((f.shape[1]), True, dtype=bool)
ex_mask[self.index_to_exclude] = False
max_term = np.max((f[:, ex_mask] - nad[ex_mask]) / (nad[ex_mask] - z[ex_mask]), axis=1)
sum_term_1 = self.rho_sum * np.sum((f[:, ex_mask]) / (nad[ex_mask] - z[ex_mask]), axis=1)
# avoid division by zeros
sum_term_2 = self.rho_sum * np.sum((f[:, ~ex_mask]) / (nad[~ex_mask] - uto[~ex_mask]), axis=1)
return max_term + sum_term_1 + sum_term_2
[docs]class GuessASF(ASFBase):
"""Implementation of the naive or GUESS ASF.
Args:
nadir (np.ndarray): The nadir point of the problem being scalarized.
References:
Miettinen, K., Mäkelä, M.
On scalarizing functions in multiobjective optimization
OR Spectrum 24, 193–213 (2002)
"""
def __init__(self, nadir: np.ndarray):
self.nadir = nadir
[docs] def __call__(self, objective_vectors: np.ndarray, reference_point: np.ndarray):
nad = self.nadir
f = np.atleast_2d(objective_vectors)
z = reference_point
max_term = np.max((f - nad) / (nad - z), axis=1)
return max_term
