Title:	Multi-Objective Optimization in R
Version:	0.3.2
Date:	2026-05-03
Description:	The 'rmoo' package is a framework for multi- and many-objective optimization, which allows researchers and users versatility in parameter configuration, as well as tools for analysis, replication and visualization of results. The 'rmoo' package was built as a fork of the 'GA' package by Luca Scrucca(2017) <doi:10.32614/RJ-2017-008> and implementing the Non-Dominated Sorting Genetic Algorithms proposed by K. Deb's.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding:	UTF-8
Language:	en
LazyData:	true
RoxygenNote:	7.3.3
Collate:	'AllClasses.R' 'associate.R' 'crowding_distance.R' 'data.R' 'generate_reference_points.R' 'geneticoperator.R' 'get_fixed_rowsum_integer_matrix.R' 'miscfun.R' 'AllGenerics.R' 'niching.R' 'non_dominated_fronts.R' 'parallel.R' 'utils.R' 'nsga.R' 'nsga2.R' 'nsga3.R' 'rnsga2.R' 'modified_crowding_distance.R' 'rmooControl.R' 'performance_metrics.R' 'reference_point_multi_layer.R' 'rmoo.R' 'rmoo_main.R' 'sharing.R' 'update_points.R' 'zzz.R'
Imports:	stats, utils, graphics, methods, foreach, GA, grDevices, plotly, ggplot2, BBmisc
URL:	https://github.com/Evolutionary-Optimization-Laboratory/rmoo/
BugReports:	https://github.com/Evolutionary-Optimization-Laboratory/rmoo/issues/
Suggests:	testthat, covr, rgl, ecr, emoa, cdata, dplyr, reshape2, parallel, doParallel, doRNG (≥ 1.6)
Depends:	R (≥ 2.10)
NeedsCompilation:	no
Packaged:	2026-05-03 21:27:26 UTC; Maria
Author:	Francisco Benitez [aut, cre], Diego P. Pinto-Roa [aut]
Maintainer:	Francisco Benitez <benitezfj94@gmail.com>
Repository:	CRAN
Date/Publication:	2026-05-04 00:00:02 UTC

rmoo: Multi-Objective Optimization in R

Description

The 'rmoo' package is a framework for multi- and many-objective optimization, which allows researchers and users versatility in parameter configuration, as well as tools for analysis, replication and visualization of results. The 'rmoo' package was built as a fork of the 'GA' package by Luca Scrucca(2017) doi:10.32614/RJ-2017-008 and implementing the Non-Dominated Sorting Genetic Algorithms proposed by K. Deb's.

Author(s)

Maintainer: Francisco Benitez benitezfj94@gmail.com

Authors:

• Diego P. Pinto-Roa dpinto@pol.una.py (ORCID)

See Also

Useful links:

• https://github.com/Evolutionary-Optimization-Laboratory/rmoo/

• Report bugs at https://github.com/Evolutionary-Optimization-Laboratory/rmoo/issues/

Virtual Parent Class Algorithm

Description

It will use when other algorithms are implemented. Equivalent to a Abstract class in other languages.

Association Operation in Non-Dominated Genetic Algorithms III

Description

Function that associates each member of the population with a reference point. The function calculates the perpendicular distance of each individual from each of the reference lines. This code section corresponds to Algorithm 3 of the referenced paper.

Usage

  associate_to_niches(object, utopian_epsilon = 0)
  compute_perpendicular_distance(x, y)
  compute_niche_count(n_niches, niche_of_individuals)

Arguments

Value

Returns a list with the niche count of individuals and the distances between them.

Author(s)

Francisco Benitez

References

J. Blank and K. Deb, "Pymoo: Multi-Objective Optimization in Python," in IEEE Access, vol. 8, pp. 89497-89509, 2020, doi: 10.1109/ACCESS.2020.2990567.

K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.

Calculate Normalized Preference Distance Computes the weighted normalized Euclidean distance between a set of fitness vectors and a set of reference points.

Description

Calculate Normalized Preference Distance Computes the weighted normalized Euclidean distance between a set of fitness vectors and a set of reference points.

Usage

calc_norm_pref_distance(fitness, ref_points, weight, ideal_point, nadir_point)

Arguments

Value

A matrix of distances where element (i, j) is the distance from fitness to ref_points.

Calculation of Crowding Distance

Description

A Crowded-comparison approach.

Usage

crowding_distance(object, nObj)

Arguments

Details

The crowded-comparison operator guides the selection process at the various stages of the algorithm toward a uniformly spread-out Pareto-optimal front

Value

A vector with the crowding-distance between individuals of a population.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, 'A fast and elitist multiobjective genetic algorithm: NSGA-II,' in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017.

See Also

non_dominated_fronts()

Determination of Reference Points on a Hyper-Plane

Description

A implementation of Das and Dennis's Reference Points Generation.

Usage

generate_reference_points(m, h, scaling = NULL)

Arguments

Details

The implemented Reference Point Generation is based on the Das and Dennis's systematic approach that places points on a normalized hyper-plane which is equally inclined to all objective axes and has an intercept of one on each axis.

Value

A matrix with the reference points uniformly distributed.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

K. Deb and H. Jain, 'An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints,' in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.

Das, Indraneel & Dennis, J. (2000). Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization. 8. 10.1137/S1052623496307510.

See Also

non_dominated_fronts() and get_fixed_rowsum_integer_matrix()

Accessor methods to the crowding distance for NSGA-II results

Description

Accessor methods to the crowding distance for NSGA-II results

Usage

getCrowdingDistance(obj)

## S4 method for signature 'nsga2'
getCrowdingDistance(obj)

Arguments

Value

Returns a vector with the crowding distances of class nsga2. See nsga2 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object resulting from the execution of the NSGA-II algorithm.
#
# getCrowdingDistance(out)
#

Accessor methods to the dummy fitness for NSGA-I results

Description

Accessor methods to the dummy fitness for NSGA-I results

Usage

getDummyFitness(obj)

## S4 method for signature 'nsga1'
getDummyFitness(obj)

Arguments

Value

Returns a matrix with the dummy fitness of class nsga1. See nsga1 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object resulting from the execution of the NSGA-I algorithm.
#
# getDummyFitness(out)
#

Accessor methods to the fitness for rmoo results

Description

Accessor methods to the fitness for rmoo results

Usage

getFitness(obj)

Arguments

Value

Prints the resulting fitness and when the result of the method-call is assigned to a variable, the fitness is stored as a data frame. See nsga1 nsga2, nsga3 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object resulting from the execution of the rmoo.
#
# fitness_result <- getFitness(out)
#
# fitness_result

Accessor methods to the metrics evaluated during execution

Description

Accessor methods to the metrics evaluated during execution

Usage

getMetrics(obj)

## S4 method for signature 'nsga'
getMetrics(obj)

Arguments

Value

A dataframe with performance metrics evaluated iteration by iteration.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object resulting from the execution of the rmoo.
#
# metrics_result <- getMetrics(out)
#
# metrics_result

Accessor methods to the population for rmoo results

Description

Accessor methods to the population for rmoo results

Usage

getPopulation(obj)

## S4 method for signature 'nsga'
getPopulation(obj)

## S4 method for signature 'nsga'
getFitness(obj)

Arguments

Value

Prints the resulting population and when the result of the method-call is assigned to a variable, the population is stored as a data frame. See nsga1 nsga2, nsga3 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object resulting from the execution of rmoo.
#
# population_result <- getPopulation(out)
#
# population_result

Determine the division points on the hyperplane

Description

Implementation of the recursive function in Generation of Reference points of Das and Dennis..

Usage

get_fixed_rowsum_integer_matrix(m, h)

Arguments

Details

The implemented Reference Point Generation is based on the Das and Dennis's systematic approach that places points on a normalized hyper-plane which is equally inclined to all objective axes and has an intercept of one on each axis.

Value

A matrix with the reference points uniformly distributed.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

K. Deb and H. Jain, 'An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints,' in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.

Das, Indraneel & Dennis, J.. (2000). Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization. 8. 10.1137/S1052623496307510.

See Also

non_dominated_fronts() and generate_reference_points()

KROA100

Description

A dataset containing the coord and section of 100 cities

Usage

kroA100

Format

A data frame with 100 rows and 2 variables:

COORD

City Coordinates

SECTION

City Section

References

Reinelt, G. (1991). TSPLIB—A traveling salesman problem library. ORSA journal on computing, 3(4), 376-384

KROB100

Description

A dataset containing the coord and section of 100 cities

Usage

kroB100

Format

A data frame with 100 rows and 2 variables:

COORD

City Coordinates

SECTION

City Section

References

Reinelt, G. (1991). TSPLIB—A traveling salesman problem library. ORSA journal on computing, 3(4), 376-384

KROC100

Description

A dataset containing the coord and section of 100 cities

Usage

kroC100

Format

A data frame with 100 rows and 2 variables:

COORD

City Coordinates

SECTION

City Section

References

Reinelt, G. (1991). TSPLIB—A traveling salesman problem library. ORSA journal on computing, 3(4), 376-384

Calculation of Modified Crowding Distance

Description

A Crowded-comparison approach.

Usage

modifiedCrowdingDistance(
  object,
  epsilon,
  weights = NULL,
  normalization = "front",
  extreme_points_as_ref_dirs = FALSE
)

Arguments

Details

The crowded-comparison operator maintain diversity in the Pareto front during multi-objective optimization. This version uses a reference point-based normalization and preference distance strategy.

Value

A list with:

survivors

Indices of selected individuals

indexmin

Index of individuals with minimum scalarizing value (optional)

reference_points

Updated reference points matrix

Author(s)

Francisco Benitez

References

Kalyanmoy Deb and J. Sundar (2006). GECCO '06. doi:10.1145/1143997.1144112

See Also

rnsga2()

Niche-Preservation Operation

Description

Generation of niche, by associating reference points to population members

Usage

niching(pop, n_remaining, niche_count, niche_of_individuals, dist_to_niche)

Arguments

Details

Niching procesure is a algorithms proposed by K. Deb and H. Jain in 2013.

Value

Returns the association of reference points to each individual in the population.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

K. Deb and H. Jain, 'An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints,' in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008

Felix-Antoine Fortin, Francois-Michel De Rainville, Marc-André Gardner Gardner, Marc Parizeau, and Christian Gagne. 2012. DEAP: evolutionary algorithms made easy. J. Mach. Learn. Res. 13, 1 (January 2012), 2171–2175.

See Also

associate_to_niches(), PerformScalarizing()

Calculate of Non-Dominated Front

Description

A fast approach for calculate Non-Dominated Fronts.

Usage

non_dominated_fronts(object)

Arguments

Details

Function to determine the non-dominated fronts of a population and the aptitude value.

Value

A list with 'non-dominated fronts' and 'occupied positions' on the fronts.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, 'A fast and elitist multiobjective genetic algorithm: NSGA-II,' in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017.

See Also

nsga(), nsga2() and nsga3()

Non-Dominated Sorting in Genetic Algorithms

Description

Minimization of a fitness function using Non-Dominated Genetic algorithms (NSGA). Local search using general-purpose optimisation algorithms can be applied stochastically to exploit interesting regions.

Usage

nsga(
  type = c("binary", "real-valued", "permutation"),
  fitness,
  ...,
  lower,
  upper,
  nBits,
  population = rmooControl(type)$population,
  selection = rmooControl(type)$selection,
  crossover = rmooControl(type)$crossover,
  mutation = rmooControl(type)$mutation,
  popSize = 50,
  nObj = NULL,
  dshare,
  pcrossover = 0.8,
  pmutation = 0.1,
  maxiter = 100,
  run = maxiter,
  maxFitness = Inf,
  names = NULL,
  suggestions = NULL,
  monitor = if (interactive()) rmooMonitor else FALSE,
  summary = FALSE,
  seed = NULL
)

Arguments

Details

The Non-dominated genetic algorithms is a meta-heuristic proposed by N. Srinivas and K. Deb in 1994. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (two or more).

Value

Returns an object of class nsga1-class. See nsga1 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

N. Srinivas and K. Deb, "Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms, in Evolutionary Computation, vol. 2, no. 3, pp. 221-248, Sept. 1994, doi: 10.1162/evco.1994.2.3.221.

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008

See Also

nsga2(), nsga3()

Examples

#Example
#Two Objectives - Real Valued
zdt1 <- function (x) {
 if (is.null(dim(x))) {
   x <- matrix(x, nrow = 1)
 }
 n <- ncol(x)
 g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1)
 return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g))))
}

#Not run:
## Not run: 
result <- nsga(type = "real-valued",
               fitness = zdt1,
               lower = c(0,0),
               upper = c(1,1),
               popSize = 100,
               nObj = 2,
               dshare = 1,
               monitor = FALSE,
               maxiter = 500)

## End(Not run)

Virtual Class 'nsga'

Description

The 'nsga' class is the parent superclass of the nsga1, nsga2, and nsga3 classes

Slots

call

an object of class 'call' representing the matched call.

type

a character string specifying the type of genetic algorithm used.

lower

a vector providing for each decision variable the lower bounds of the search space in case of real-valued or permutation encoded optimisations.

upper

a vector providing for each decision variable the upper bounds of the search space in case of real-valued or permutation encoded optimizations.

nBits

a value specifying the number of bits to be used in binary encoded optimizations.

names

a vector of character strings providing the names of decision variables (optional).

nvars

a

popSize

the population size.

front

Rank of individuals on the non-dominated front.

f

Front of individuals on the non-dominated front.

iter

the actual (or final) iteration of NSGA search.

run

the number of consecutive generations without any improvement in the best fitness value before the NSGA is stopped.

maxiter

the maximum number of iterations to run before the NSGA search is halted.

suggestions

a matrix of user provided solutions and included in the initial population.

population

the current (or final) population.

pcrossover

the crossover probability.

pmutation

the mutation probability.

fitness

the values of fitness function for the current (or final) population.

summary

a matrix of summary statistics for fitness values at each iteration (along the rows).

fitnessValue

the best fitness value at the final iteration.

solution

the value(s) of the decision variables giving the best fitness at the final iteration.

execution_time

a

Objects from the Class

Since it is a virtual Class, no objects may be created from it.

Examples

showClass('nsga')

Class 'nsga1'

Description

The class 'nsga1' is instantiated within the execution of rmoo and will be returned as a result of it. All data generated during execution will be stored in it.

Slots

dumFitness

a large dummy fitness value assigned to individuals from the nondominated front.

dShare

the maximun phenotypic distance allowed between any two individuals to become members of a niche.

deltaDummy

value to decrease the dummy fitness of individuals by non-dominated fronts.

Examples

showClass('nsga1')

Non-Dominated Sorting in Genetic Algorithms II

Description

Minimization of a fitness function using non-dominated sorting genetic algorithms - II (NSGA-IIs). Multiobjective evolutionary algorithms

Usage

nsga2(
  type = c("binary", "real-valued", "permutation"),
  fitness,
  ...,
  lower,
  upper,
  nBits,
  population = rmooControl(type)$population,
  selection = rmooControl(type)$selection,
  crossover = rmooControl(type)$crossover,
  mutation = rmooControl(type)$mutation,
  popSize = 50,
  nObj = NULL,
  pcrossover = 0.8,
  pmutation = 0.1,
  maxiter = 100,
  run = maxiter,
  maxFitness = Inf,
  names = NULL,
  suggestions = NULL,
  parallel = FALSE,
  monitor = if (interactive()) rmooMonitor else FALSE,
  summary = FALSE,
  seed = NULL
)

Arguments

Details

The Non-dominated genetic algorithms II is a meta-heuristic proposed by K. Deb, A. Pratap, S. Agarwal and T. Meyarivan in 2002. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (two or more).

Value

Returns an object of class nsga2-class. See nsga2 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, 'A fast and elitist multiobjective genetic algorithm: NSGA-II,' in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017.

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008

See Also

nsga(), nsga3()

Examples

#Example
#Two Objectives - Real Valued
zdt1 <- function (x) {
 if (is.null(dim(x))) {
   x <- matrix(x, nrow = 1)
 }
 n <- ncol(x)
 g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1)
 return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g))))
}

#Not run:
## Not run: 
result <- nsga2(type = "real-valued",
                fitness = zdt1,
                lower = c(0,0),
                upper = c(1,1),
                popSize = 100,
                nObj = 2,
                monitor = FALSE,
                maxiter = 500)

## End(Not run)

#Example 2
#Three Objectives - Real Valued
dtlz1 <- function (x, nobj = 3){
    if (is.null(dim(x))) {
        x <- matrix(x, 1)
    }
    n <- ncol(x)
    y <- matrix(x[, 1:(nobj - 1)], nrow(x))
    z <- matrix(x[, nobj:n], nrow(x))
    g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5))))
    tmp <- t(apply(y, 1, cumprod))
    tmp <- cbind(t(apply(tmp, 1, rev)), 1)
    tmp2 <- cbind(1, t(apply(1 - y, 1, rev)))
    f <- tmp * tmp2 * 0.5 * (1 + g)
    return(f)
}

#Not run:
## Not run: 
result <- nsga2(type = "real-valued",
                fitness = dtlz1,
                lower = c(0,0,0),
                upper = c(1,1,1),
                popSize = 92,
                nObj = 3,
                monitor = FALSE,
                maxiter = 500)

## End(Not run)

Class 'nsga2'

Description

The class 'nsga2' is instantiated within the execution of rmoo and will be returned as a result of it. All data generated during execution will be stored in it.

Slots

crowdingDistance

Crowding-comparison approach to estimate of the perimeter of the cuboid formed by using the nearest neighbors as the vertices.

Examples

showClass('nsga2')

Non-Dominated Sorting in Genetic Algorithms III

Description

Minimization of a fitness function using non-dominated sorting genetic algorithms - III (NSGA-IIIs). Multiobjective evolutionary algorithms

Usage

nsga3(
  type = c("binary", "real-valued", "permutation"),
  fitness,
  ...,
  lower,
  upper,
  nBits,
  population = rmooControl(type)$population,
  selection = rmooControl(type)$selection,
  crossover = rmooControl(type)$crossover,
  mutation = rmooControl(type)$mutation,
  popSize = 50,
  nObj = NULL,
  n_partitions = NULL,
  pcrossover = 0.8,
  pmutation = 0.1,
  reference_dirs = generate_reference_points,
  maxiter = 100,
  run = maxiter,
  maxFitness = Inf,
  names = NULL,
  suggestions = NULL,
  parallel = FALSE,
  monitor = if (interactive()) rmooMonitor else FALSE,
  summary = FALSE,
  seed = NULL
)

Arguments

Details

The Non-dominated genetic algorithms III is a meta-heuristic proposed by K. Deb and H. Jain in 2013. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (more than three).

Value

Returns an object of class nsga3-class. See nsga3 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008

See Also

nsga(), nsga2()

Examples

#Example 1
#Two Objectives - Real Valued
zdt1 <- function (x) {
 if (is.null(dim(x))) {
   x <- matrix(x, nrow = 1)
 }
 n <- ncol(x)
 g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1)
 return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g))))
}

#Not run
## Not run: 
result <- nsga3(type = "real-valued",
                fitness = zdt1,
                lower = c(0,0),
                upper = c(1,1),
                popSize = 100,
                nObj = 2,
                n_partitions = 100,
                monitor = FALSE,
                maxiter = 500)

## End(Not run)

#Example 2
#Three Objectives - Real Valued
dtlz1 <- function (x, nobj = 3, ...){
    if (is.null(dim(x))) {
        x <- matrix(x, 1)
    }
    n <- ncol(x)
    y <- matrix(x[, 1:(nobj - 1)], nrow(x))
    z <- matrix(x[, nobj:n], nrow(x))
    g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5))))
    tmp <- t(apply(y, 1, cumprod))
    tmp <- cbind(t(apply(tmp, 1, rev)), 1)
    tmp2 <- cbind(1, t(apply(1 - y, 1, rev)))
    f <- tmp * tmp2 * 0.5 * (1 + g)
    return(f)
}

#Not Run
## Not run: 
result <- nsga3(type = "real-valued",
                fitness = dtlz1,
                lower = c(0,0,0),
                upper = c(1,1,1),
                popSize = 92,
                nObj = 3,
                n_partitions = 12,
                monitor = FALSE,
                maxiter = 500)

## End(Not run)

Class 'nsga3'

Description

The class 'nsga3' is instantiated within the execution of rmoo and will be returned as a result of it. All data generated during execution will be stored in it.

Slots

ideal_point

Nadir point estimate used as lower bound in normalization.

worst_point

Worst point generated over generations.

smin

Index used to obtain the extreme points.

extreme_points

are selected using the ASF in the (PerformScalarizing()). Necessary in the nadir point generation.

worst_of_population

The worst individuals generated by objectives in the current generation.

worst_of_front

The worst individuals in the first front generated by objectives in the current generation.

nadir_point

Nadir point estimate used as upper bound in normalization.

reference_points

NSGA-III uses a predefined set of reference points to ensure diversity in obtained solutions. The chosen refenrece points can be predefined in structured manner or supplied by the user. We use the Das and Dennis procedure.

Examples

showClass('nsga3')

Virtual Class 'numberOrNAOrMatrix - Simple Class for subassigment Values'

Description

The class 'numberOrNAOrMatrix' is a simple class union (setClassUnion()) of 'numeric', 'logical', 'logical' and 'matrix'.

Objects from the Class

Since it is a virtual Class, no objects may be created from it.

Examples

showClass('numberOrNAOrMatrix')

Objective Values performance metrics

Description

Functions to evaluate the quality of the results obtained by the algorithms, evaluating their diversity and convergence, providing or not some parameters to compare.

Usage

  generational_distance(front, true_pareto_front, p, inverted, plus)

Arguments

Value

A vector with the measurement metric.

Author(s)

Francisco Benitez

References

Lamont, G., & Veldhuizen, D.V. (1999). Multiobjective evolutionary algorithms: classifications, analyses, and new innovations.

Methods for Function 'plot' in Package 'rmoo'

Description

Method used to visualize the fitness of the individuals during the execution of the algorithms.

Usage

plot(x, y, ...)

## S4 method for signature 'nsga,missing'
plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...)

## S4 method for signature 'nsga1,missing'
plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...)

## S4 method for signature 'nsga2,missing'
plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...)

## S4 method for signature 'nsga3,missing'
plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...)

## S4 method for signature 'rnsga2,missing'
plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...)

Arguments

Details

The following plots are available:

• "Scatter Plot"

• "Parallel Coordinate Plot"

• "Heat Map"

• "Polar Coordinate"

Value

A graph of the evaluated type.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object of class nsga1, nsga2, or nsga3.
# The plot method will by default plot a scatter plot.
#
# plot(out)
#
# The Parallel Coordinate Plot will be plotted if "pcp" is passed as a parameter to "type".
#
# plot(out, type="pcp")
#
# A heat map plot will be plotted if "heatmap" is passed as a parameter to "type"
# and a vector with the individuals to plot to "individual"
#
# plot(out, type = "heatmap", individual = c(1:5))
#
# A polar coordinate plot will be plotted if "polar" is passed as a parameter to "type"
# and a vector with the individuals to plot to "individual"
#
# plot(out, type = "polar", individual = c(1:5))

Methods for Function 'print' in Package 'rmoo'.

Description

Method used to print the slots and relevant values of the object.

Usage

print(x, ...)

## S4 method for signature 'nsga'
print(x, ...)

## S4 method for signature 'nsga1'
print(x, ...)

## S4 method for signature 'nsga3'
print(x, ...)

Arguments

Value

Print the slots and relevant values of the object.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object of class nsga1, nsga2, or nsga3
#
# print(out)

Methods for Function 'progress' in Package 'rmoo'

Description

Method used to save the progress of the evaluation results, similar to the summary method. Passing additional arguments to the progress method evaluates performance metrics per iteration. This method cannot be called outside of rmoo execution.

Usage

progress(object, ...)

## S4 method for signature 'nsga'
progress(object, ...)

## S4 method for signature 'nsga1'
progress(object, ...)

## S4 method for signature 'nsga2'
progress(object, ...)

## S4 method for signature 'nsga3'
progress(object, ...)

Arguments

Value

A list of length equal to the number of iterations, where the progress made during execution is saved.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object of class nsga1, nsga2, or nsga3, and callArgs are
# the additional arguments passed when calling the rmoo function, for the
# evaluation of performance metrics, reference points are expected to be passed
# as an argument to reference_dirs.
#
# progress(object, callArgs)
#

Determination of Multi-layer Reference Points

Description

A implementation of Multi-layer Reference Points Generation.

Usage

reference_point_multi_layer(...)

Arguments

Details

The Multi-layer reference point implementation is based on Blank and Deb's pymoo library, the approach generates different layers of references point at different scales, provided by the user.

Value

A matrix with the multi-layer reference points

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

J. Blank and K. Deb, "Pymoo: Multi-Objective Optimization in Python," in IEEE Access, vol. 8, pp. 89497-89509, 2020, doi: 10.1109/ACCESS.2020.2990567.

Das, Indraneel & Dennis, J. (2000). Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization. 8. 10.1137/S1052623496307510.

See Also

generate_reference_points() and get_fixed_rowsum_integer_matrix()

A function for setting or retrieving defaults non-dominated genetic operators

Description

Default settings for non-dominated genetic operators used in the 'rmoo' package.

Usage

  rmooControl(...)

Arguments

Details

If the function is called with no arguments returns the current default settings, i.e., a list with the following default components:

"binary"

• population = "rmoobin_Population"

• selection = "rmoobin_tourSelection"

• crossover = "rmoobin_spCrossover"

• mutation = "rmoobin_raMutation"

"real-valued"

• population = "rmooreal_Population"

• selection = "rmooreal_tourSelection"

• crossover = "rmooreal_sbxCrossover"

• mutation = "rmooreal_polMutation"

"permutation"

• population = "rmooperm_Population"

• selection = "rmooperm_tourSelection"

• crossover = "rmooperm_oxCrossover"

• mutation = "rmooperm_simMutation"

"discrete"

• population = "rmooint_Population"

• selection = "rmooint_tourSelection"

• crossover = "rmooint_uxCrossover"

• mutation = "rmooint_uxMutation"

"eps" =

the tolerance value used by the package functions. By default set at sqrt(.Machine$double.eps).

The function may be called with a single string specifying the name of the component. In this case the function returns the current default settings.

To change the default values, a named component must be followed by a single value (in case of "eps") or a list of component(s) specifying the name of the function for a genetic operator. See the Examples section.

Value

If the argument list is empty the function returns the current list of values. If the argument list is not empty, the returned list is invisible.

Note

The parameter values set via a call to this function will remain in effect for the rest of the session, affecting the subsequent behaviour of the functions for which the given parameters are relevant.

Author(s)

Francisco Benitez

References

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.

See Also

nsga2(), rnsga2() and nsga3()

Examples

  # get and save defaults
  defaultControl <- rmooControl()
  print(defaultControl)
  # get current defaults only for real-valued search
  rmooControl("real-valued")
  # set defaults for selection operator of real-valued search
  rmooControl("real-valued" = list(selection = "rmooreal_lrSelection"))
  rmooControl("real-valued")
  # set defaults for selection and crossover operators of real-valued search
  rmooControl("real-valued" = list(selection = "rmooreal_lrSelection",
                                   crossover = "rmooreal_spCrossover"))
  rmooControl("real-valued")
  # restore defaults
  rmooControl(defaultControl)
  rmooControl()

Monitor the execution of rmoo

Description

Functions to plotting fitness values at each iteration of a search for the 'rmoo' package.

Usage

  rmooMonitor(object, ...)

Arguments

Value

These functions plot the fitness values of the current step of the nsga3 on the console.
By default, rmooMonitor is called in interactive sessions by nsga, nsga2, or nsga3.
The function can be modified by the user to plot or print the values it considers by iteration.

Author(s)

Francisco Benitez

References

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.

See Also

nsga(), nsga2() and nsga3()

Mutation operators in non-dominated genetic algorithms

Description

Functions implementing mutation non-dominated genetic operator.

Usage

  rmoobin_raMutation(object, parent)

  rmooreal_raMutation(object, parent)
  rmooreal_polMutation(object, parent, eta = 20, indpb = 0.5)

  rmooperm_simMutation(object, parent)

Arguments

Value

Return a vector of values containing the mutated string.

Author(s)

Francisco Benitez

References

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.

Population initialization in non-dominated genetic algorithms

Description

Functions for creating a random initial population to be used in non-dominated genetic algorithms.

Usage

  rmoobin_Population(object)

  rmooreal_Population(object)

  rmooperm_Population(object)

  rmooint_Population(object)

Arguments

Details

rmoobin_Population generates a random population of object@nBits binary values;

rmooreal_Population generates a random (uniform) population of real values in the range [object@lower, object@upper];

rmooperm_Population generates a random (uniform) population of permutation values in the range [object@lower, object@upper].

rmooint_Population generates a random (uniform) population of integer values in the range [object@lower, object@upper].

Value

Return a matrix of dimension object@popSize times the number of decision variables.

Author(s)

Francisco Benitez

References

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.

See Also

nsga, nsga2 and nsga3

Half Uniform Crossover (HUX)

Description

Produces two children by swapping exactly half of the genes that differ between the two parents. Only loci where the parents disagree are eligible, making HUX more conservative than full uniform crossover.

Usage

rmoo_huxCrossover(object, parents, prob_hux = 0.5)

rmooint_huxCrossover(object, parents, prob_hux = 0.5)

rmoobin_huxCrossover(object, parents, prob_hux = 0.5)

Arguments

Value

A list with children (2 × n integer matrix) and fitness (2 × nObj NA matrix).

Linear Rank Selection

Description

Selects individuals from the population using linear rank-based probabilities. Individuals are ranked by their Pareto front, and selection probability is assigned linearly so better-ranked individuals are more likely to be chosen.

Usage

rmoo_lrSelection(object, r, q)

rmoobin_lrSelection(object, r, q)

rmooperm_lrSelection(object, r, q)

rmooreal_lrSelection(object, r, q)

Arguments

Value

A list with population and fitness of the selected individuals.

R Multi-Objective Optimization Main Function

Description

Main function of rmoo, based on the parameters it will call the different algorithms implemented in the package. Optimization algorithms will minimize a fitness function. For more details of the algorithms see nsga2(), nsga3(), rnsga2().

Usage

rmoo(
  type = c("binary", "real-valued", "permutation", "discrete"),
  algorithm = c("NSGA-II", "NSGA-III", "R-NSGA-II"),
  fitness,
  ...,
  lower,
  upper,
  nBits,
  nvars,
  population = rmooControl(type)$population,
  selection = rmooControl(type)$selection,
  crossover = rmooControl(type)$crossover,
  mutation = rmooControl(type)$mutation,
  pcrossover = 0.8,
  pmutation = 0.1,
  popSize = 50,
  maxiter = 100,
  nObj = NULL,
  names = NULL,
  suggestions = NULL,
  monitor = if (interactive()) rmooMonitor else FALSE,
  parallel = FALSE,
  summary = FALSE,
  seed = NULL,
  reference_dirs = NULL,
  epsilon = 0.001,
  normalization = NULL,
  extreme_points_as_ref_dirs = FALSE,
  weights = NULL
)

Arguments

Details

Multi- and Many-Optimization of a fitness function using Non-dominated Sorting Genetic Algorithms. The algorithms currently implemented by rmoo are: NSGA-II, NSGA-III and R-NSGA-II

The Non-dominated genetic algorithms II (NSGA-II) is a meta-heuristic proposed by K. Deb, A. Pratap, S. Agarwal and T. Meyarivan in 2002. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (two or more).

The Non-dominated genetic algorithms III (NSGA-III) is a meta-heuristic proposed by K. Deb and H. Jain in 2013. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (more than three).

The Reference point-based Non-dominated genetic algorithms II (R-NSGA-II) is a meta-heuristic proposed by K. Deb and J. Sundar in 2006. It is a modification of NSGA-II based on reference points in which the decision-maker supplies one or more preference points and a weight vector that will guide the solutions towards regions desired by the user.

Value

Returns an object of class nsga2-class, rnsga2-class or nsga3-class. See nsga2, rnsga2, nsga3 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008

Kalyanmoy Deb and J. Sundar. 2006. Reference point based multi-objective optimization using evolutionary algorithms. In Proceedings of the 8th annual conference on Genetic and evolutionary computation (GECCO '06). Association for Computing Machinery, New York, NY, USA, 635–642. doi: 10.1145/1143997.1144112

K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, 'A fast and elitist multiobjective genetic algorithm: NSGA-II,' in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017.

K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.

See Also

nsga2(), rnsga2(), nsga3()

Examples

#Example 1
#Two Objectives - Real Valued
zdt1 <- function (x) {
 if (is.null(dim(x))) {
   x <- matrix(x, nrow = 1)
 }
 n <- ncol(x)
 g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1)
 return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g))))
}

#Not run:
## Not run: 
result <- rmoo(type = "real-valued",
               fitness = zdt1,
               algorithm = "NSGA-II",
               lower = c(0,0),
               upper = c(1,1),
               popSize = 100,
               nObj = 2,
               monitor = FALSE,
               maxiter = 500)


## End(Not run)

#Example 2
#Three Objectives - Real Valued
dtlz1 <- function (x, nobj = 3){
    if (is.null(dim(x))) {
        x <- matrix(x, 1)
    }
    n <- ncol(x)
    y <- matrix(x[, 1:(nobj - 1)], nrow(x))
    z <- matrix(x[, nobj:n], nrow(x))
    g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5))))
    tmp <- t(apply(y, 1, cumprod))
    tmp <- cbind(t(apply(tmp, 1, rev)), 1)
    tmp2 <- cbind(1, t(apply(1 - y, 1, rev)))
    f <- tmp * tmp2 * 0.5 * (1 + g)
    return(f)
}

#Define uniformly distributed reference points.
ref_points <- generate_reference_points(3,12)

#Not Run
## Not run: 
result <- rmoo(type = "real-valued",
                fitness = dtlz1,
                algorithm = "NSGA-III",
                lower = c(0,0,0),
                upper = c(1,1,1),
                popSize = 92,
                nObj = 3,
                reference_dirs = ref_points,
                monitor = FALSE,
                maxiter = 500)

## End(Not run)

#Example 3
#Two Objectives - Real Valued with Preference-guided
zdt2 <- function (x)
{
  if (is.null(dim(x))) {
    x <- matrix(x, nrow = 1)
  }
  n <- ncol(x)
  g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1)
  return(cbind(x[, 1], g * (1 - (x[, 1]/g)^2)))
}

#Define uniformly distributed reference points.
ref_points <- rbind(c(1.0, 0.0), c(0.0, 1.0), c(0.5, 0.5))

#Not run
## Not run: 
result <- rmoo(type = "real-valued",
               fitness = zdt2,
               algorithm = "R-NSGA-II",
               lower = c(0,0),
               upper = c(1,1),
               reference_dirs = ref_points,
               popSize = 92,
               nObj = 2,
               monitor = FALSE,
               maxiter = 500)


## End(Not run)

Tournament Selection

Description

Binarily o por pref

Usage

rmoo_tourSelection(object, k = 2, ...)

rmooreal_tourSelection(object, k = 2, ...)

rmoobin_tourSelection(object, k = 2, ...)

rmooperm_tourSelection(object, k = 2, ...)

Arguments

Value

List with population and fitness subsets.

Examples

## Not run: 
# Creamos un "dummy" objeto mínimo
object <- list(
  population       = matrix(runif(20), nrow=5),
  fitness          = matrix(runif(10), nrow=5),
  front            = matrix(sample(1:2, 5, TRUE), ncol=1),
  crowdingDistance = runif(5),
  popSize          = 5
)
class(object) <- "nsga2"
# Llamamos al selector
sel <- rmoo_tourSelection(object, k = 2)
str(sel)

## End(Not run)

Uniform Crossover

Description

Produces two children by randomly swapping genes between two parents with equal probability at each locus. Each gene is inherited from either parent independently, giving maximum gene-level mixing.

Usage

rmoo_uxCrossover(object, parents)

rmooint_uxCrossover(object, parents)

rmoobin_uxCrossover(object, parents)

Arguments

Value

A list with children (2 × n integer matrix) and fitness (2 × nObj NA matrix).

Uniform Mutation

Description

Mutates an individual by randomly replacing each gene with a new value drawn uniformly from the integer range [lower, upper], independently at each locus with probability indpb.

Usage

rmoo_uxMutation(object, parent, indpb = 0.1)

rmooint_uxMutation(object, parent, indpb = 0.1)

rmoobin_uxMutation(object, parent, indpb = 0.1)

Arguments

Value

An integer vector of the mutated individual.

Crossover Operators in Non-Dominated Genetic Algorithms

Description

Functions implementing crossover operators for non-dominated genetic algorithms. rmoo_spCrossover (and its typed variants rmoobin_, rmooreal_, rmooint_) performs single-point crossover; rmooreal_sbxCrossover performs simulated binary crossover; rmooperm_oxCrossover performs order crossover for permutation representations.

Usage

rmooreal_sbxCrossover(object, parents, eta = 20, indpb = 0.5)

rmoo_spCrossover(object, parents)

rmoobin_spCrossover(object, parents)

rmooreal_spCrossover(object, parents)

rmooint_spCrossover(object, parents)

rmooperm_oxCrossover(object, parents)

Arguments

Value

A list with two elements:

children

A matrix of dimension 2 × nVars containing the generated offspring.

fitness

A 2 × nObj matrix of NA values, indicating that offspring fitness has not yet been evaluated.

Author(s)

Francisco Benitez

References

Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187–206, doi:10.32614/RJ-2017-008.

See Also

nsga(), nsga2(), nsga3()

Reference Point Based Non-Dominated Sorting in Genetic Algorithms II

Description

Minimization of a fitness function using reference point based non-dominated sorting genetic algorithms - II (R-NSGA-IIs). Multiobjective evolutionary algorithms

Usage

rnsga2(
  type = c("binary", "real-valued", "permutation"),
  fitness,
  ...,
  lower,
  upper,
  nBits,
  population = rmooControl(type)$population,
  selection = rmooControl(type)$selection,
  crossover = rmooControl(type)$crossover,
  mutation = rmooControl(type)$mutation,
  reference_dirs = NULL,
  epsilon = 0.001,
  normalization = c("ever", "front", "no"),
  extreme_points_as_ref_dirs = FALSE,
  weights = NULL,
  popSize = 50,
  nObj = NULL,
  pcrossover = 0.8,
  pmutation = 0.1,
  maxiter = 100,
  run = maxiter,
  maxFitness = Inf,
  names = NULL,
  suggestions = NULL,
  parallel = FALSE,
  monitor = if (interactive()) rmooMonitor else FALSE,
  summary = FALSE,
  seed = NULL
)

Arguments

Details

R-NSGA-II is a meta-heuristic proposed by K. Deb and J. Sundar in 2006. It is a modification of NSGA-II based on reference points in which the decision-maker supplies one or more preference points and a weight vector that will guide the solutions towards regions desired by the user.

Value

Returns an object of class rnsga2-class. See rnsga2 for a description of available slots information.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

Kalyanmoy Deb and J. Sundar. 2006. Reference point based multi-objective optimization using evolutionary algorithms. In Proceedings of the 8th annual conference on Genetic and evolutionary computation (GECCO '06). Association for Computing Machinery, New York, NY, USA, 635–642. doi: 10.1145/1143997.1144112

See Also

nsga(), nsga2(), nsga3()

Examples

#Example
#Two Objectives - Real Valued
zdt1 <- function (x) {
 if (is.null(dim(x))) {
   x <- matrix(x, nrow = 1)
 }
 n <- ncol(x)
 g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1)
 return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g))))
}

#Define the reference points
reference_points = rbind(c(0.2, 0.8), c(0.8, 0.2), c(0.4, 0.5))

#Not run:
## Not run: 
result <- rnsga2(type = "real-valued",
                fitness = zdt1,
                lower = c(0,0),
                upper = c(1,1),
                reference_dirs = reference_points,
                popSize = 100,
                nObj = 2,
                monitor = FALSE,
                maxiter = 500,
                seed = 45)

## End(Not run)

#Example 2
#Three Objectives - Real Valued
dtlz1 <- function (x, nobj = 3){
    if (is.null(dim(x))) {
        x <- matrix(x, 1)
    }
    n <- ncol(x)
    y <- matrix(x[, 1:(nobj - 1)], nrow(x))
    z <- matrix(x[, nobj:n], nrow(x))
    g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5))))
    tmp <- t(apply(y, 1, cumprod))
    tmp <- cbind(t(apply(tmp, 1, rev)), 1)
    tmp2 <- cbind(1, t(apply(1 - y, 1, rev)))
    f <- tmp * tmp2 * 0.5 * (1 + g)
    return(f)
}

#Define the reference points
reference_points <- rbind(c(1.0, 0.5, 0.0), c(0.0, 0.5, 1.0), c(0.5, 0.5, 0.5))

#Not run:
## Not run: 
result <- rnsga2(type = "real-valued",
                fitness = dtlz1,
                lower = c(0,0,0),
                upper = c(1,1,1),
                reference_dirs = reference_points,
                popSize = 92,
                nObj = 3,
                monitor = FALSE,
                maxiter = 500)

## End(Not run)

Class 'rnsga2'

Description

The class 'rnsga2' is instantiated within the execution of rmoo and will be returned as a result of it. All data generated during execution will be stored in it.

Slots

crowdingDistance

Crowding-comparison approach to estimate of the perimeter of the cuboid formed by using the nearest neighbors as the vertices.

reference_points

R-NSGA-II uses a set of reference points defined by the user to ensure diversity in obtained solutions.

extreme_points

are selected using the ASF in the (PerformScalarizing()). Necessary in the nadir point generation.

smin

Index used to obtain the extreme points.

Examples

showClass('rnsga2')

Scale Reference Points

Description

A implementation of Das and Dennis's Reference Points Generation.

Usage

scale_reference_directions(ref_dirs, scaling)

Arguments

Details

The implemented Reference Point Generation is based on the Das and Dennis's systematic approach that places points on a normalized hyper-plane which is equally inclined to all objective axes and has an intercept of one on each axis.

Value

A matrix with rescaled reference points uniformly distributed.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

J. Blank and K. Deb, "Pymoo: Multi-Objective Optimization in Python," in IEEE Access, vol. 8, pp. 89497-89509, 2020, doi: 10.1109/ACCESS.2020.2990567.

See Also

generate_reference_points() and get_fixed_rowsum_integer_matrix()

Calculation of Dummy Fitness

Description

Calculate of sharing distance and dummy fitness

Usage

sharing(object)

Arguments

Details

The sharing distance operator guides the selection process at the various stages of the algorithm toward a uniformly spread-out Pareto-optimal front

Value

A vector with the dummy fitness.

Author(s)

Francisco Benitez benitezfj94@gmail.com

References

N. Srinivas and K. Deb, 'Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms,' in Evolutionary Computation, vol. 2, no. 3, pp. 221-248, Sept. 1994, doi: 10.1162/evco.1994.2.3.221.

See Also

non_dominated_fronts()

Start Parallel Backend for rmoo Package

Description

This function sets up parallel computing using the parallel and doParallel packages. It supports both "snow" (PSOCK) and "multicore" backends depending on the OS.

Usage

startParallel(parallel = TRUE, ...)

Arguments

Value

An object of class logical with attributes:

• type: cluster type ("snow" or "multicore")

• cores: number of cores used

• cluster: the cluster object created or passed

Examples

## Not run: 
  cl <- startParallel(TRUE)
  stopParallel(attr(cl, "cluster"))

## End(Not run)

Stop Parallel Backend

Description

Stops the parallel backend and reverts to sequential execution.

Usage

stopParallel(cluster, ...)

Arguments

Value

invisible(NULL), used for side effects.

Examples

## Not run: 
  cl <- startParallel()
  stopParallel(attr(cl, "cluster"))

## End(Not run)

Methods for Function 'summary' in Package 'rmoo'

Description

Method used to summarize the results of the evaluations, passing additional arguments in the summary method the performance metrics is evaluated.

Usage

summary(object, ...)

## S4 method for signature 'nsga'
summary(object, ...)

## S4 method for signature 'nsga1'
summary(object, ...)

## S4 method for signature 'nsga2'
summary(object, ...)

## S4 method for signature 'nsga3'
summary(object, ...)

Arguments

Value

A summary of the values resulting from the execution of an algorithm.

Author(s)

Francisco Benitez benitezfj94@gmail.com

Examples

# Where 'out' is an object of class nsga1, nsga2, or nsga3
#
# summary(out)
#
# For the evaluation of the metrics, pass the reference point
#
# ref_points <- generate_reference_points(3,12)
# summary(out, reference_dirs = ref_points)

Adaptive normalization of population members

Description

Functions to scalarize the members of the population to locate them in a normalized hyperplane, finding the ideal point, nadir point, worst point and the extreme points.

Usage

  UpdateIdealPoint(object, nObj)
  UpdateWorstPoint(object, nObj)
  PerformScalarizing(population, fitness, smin, extreme_points, ideal_point)
  get_nadir_point(object)

Arguments

Value

Return scalarized objective values in a normalized hyperplane.

Author(s)

Francisco Benitez

References

J. Blank and K. Deb, "Pymoo: Multi-Objective Optimization in Python," in IEEE Access, vol. 8, pp. 89497-89509, 2020, doi: 10.1109/ACCESS.2020.2990567.

K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.