# Short Description

caRamel is a multiobjective evolutionary algorithm combining the MEAS algorithm and the NGSA-II algorithm.

``library(caRamel)``
``## Le chargement a nÃ©cessitÃ© le package : geometry``
``## Le chargement a nÃ©cessitÃ© le package : parallel``
``## Package 'caRamel' version 1.3``

# Test functions

## Schaffer

Schaffer test function has two objectives with one variable.

``````schaffer <- function(i) {
if (x[i,1] <= 1) {
s1 <- -x[i,1]
} else if (x[i,1] <= 3) {
s1 <- x[i,1] - 2
} else if (x[i,1] <= 4) {
s1 <- 4 - x[i,1]
} else {
s1 <- x[i,1] - 4
}
s2 <- (x[i,1] - 5) * (x[i,1] - 5)
return(c(s1, s2))
}``````

Note that :

• parameter i is mandatory for the management of parallelism.
• the variable must be named x and is a matrix of size [npopulation, nvariables].

The variable lies in the range [-5, 10]:

``````nvar <- 1 # number of variables
bounds <- matrix(data = 1, nrow = nvar, ncol = 2) # upper and lower bounds
bounds[, 1] <- -5 * bounds[, 1]
bounds[, 2] <- 10 * bounds[, 2]``````

Both functions are to be minimized:

``````nobj <- 2 # number of objectives
minmax <- c(FALSE, FALSE) # min and min``````

Before calling caRamel in order to optimize the Schafferâ€™s problem, some algorithmic parameters need to be set:

``````popsize <- 100 # size of the genetic population
archsize <- 100 # size of the archive for the Pareto front
maxrun <- 1000 # maximum number of calls
prec <- matrix(1.e-3, nrow = 1, ncol = nobj) # accuracy for the convergence phase``````

Then the minimization problem can be launched:

``````results <-
caRamel(nobj,
nvar,
minmax,
bounds,
schaffer,
popsize,
archsize,
maxrun,
prec,
carallel=FALSE) # no parallelism``````
``## Beginning of caRamel optimization <-- Wed Feb  2 07:29:28 2022``
``## Number of variables : 1``
``## Number of functions : 2``
``## Done in 6.67691111564636 secs --> Wed Feb  2 07:29:35 2022``
``## Size of the Pareto front : 76``
``## Number of calls : 1020``

Test if the convergence is successful:

``print(results\$success==TRUE)``
``## [1] TRUE``

Plot the Pareto front:

``plot(results\$objectives[,1], results\$objectives[,2], main="Schaffer Pareto front", xlab="Objective #1", ylab="Objective #2")``

``plot(results\$parameters, main="Corresponding values for X", xlab="Element of the archive", ylab="X Variable")``

## Kursawe

Kursawe test function has two objectives of three variables.

``````kursawe <- function(i) {
k1 <- -10 * exp(-0.2 * sqrt(x[i,1] ^ 2 + x[i,2] ^ 2)) - 10 * exp(-0.2 * sqrt(x[i,2] ^2 + x[i,3] ^ 2))
k2 <- abs(x[i,1]) ^ 0.8 + 5 * sin(x[i,1] ^ 3) + abs(x[i,2]) ^ 0.8 + 5 * sin(x[i,2] ^3) + abs(x[i,3]) ^ 0.8 + 5 * sin(x[i,3] ^ 3)
return(c(k1, k2))
}``````

The variables lie in the range [-5, 5]:

``````nvar <- 3 # number of variables
bounds <- matrix(data = 1, nrow = nvar, ncol = 2) # upper and lower bounds
bounds[, 1] <- -5 * bounds[, 1]
bounds[, 2] <- 5 * bounds[, 2]``````

Both functions are to be minimized:

``````nobj <- 2 # number of objectives
minmax <- c(FALSE, FALSE) # min and min``````

Set algorithmic parameters and launch caRamel:

``````popsize <- 100 # size of the genetic population
archsize <- 100 # size of the archive for the Pareto front
maxrun <- 1000 # maximum number of calls
prec <- matrix(1.e-3, nrow = 1, ncol = nobj) # accuracy for the convergence phase

results <-
caRamel(nobj,
nvar,
minmax,
bounds,
kursawe,
popsize,
archsize,
maxrun,
prec,
carallel=FALSE) # no parallelism``````
``## Beginning of caRamel optimization <-- Wed Feb  2 07:29:37 2022``
``## Number of variables : 3``
``## Number of functions : 2``
``## Done in 3.89774394035339 secs --> Wed Feb  2 07:29:41 2022``
``## Size of the Pareto front : 86``
``## Number of calls : 1011``

Test if the convergence is successful and plot the optimal front:

``print(results\$success==TRUE)``
``## [1] TRUE``
``plot(results\$objectives[,1], results\$objectives[,2], main="Kursawe Pareto front", xlab="Objective #1", ylab="Objective #2")``

Finally plot the convergences of the objective functions:

``matplot(results\$save_crit[,1],cbind(results\$save_crit[,2],results\$save_crit[,3]),type="l",col=c("blue","red"), main="Convergence", xlab="Number of calls", ylab="Objectives values")``