eaf (version 1.9-1)

epsilon_additive: Epsilon metric

Description

Computes the epsilon metric, either additive or multiplicative.

Usage

epsilon_additive(data, reference, maximise = FALSE)
epsilon_mult(data, reference, maximise = FALSE)

Arguments

data

(matrix | data.frame) Matrix or data frame of numerical values, where each row gives the coordinates of a point.

reference

(matrix | data.frame) Reference set as a matrix or data.frame of numerical values.

maximise

(logical() | logical(1)) Whether the objectives must be maximised instead of minimised. Either a single logical value that applies to all objectives or a vector of logical values, with one value per objective.

Value

A single numerical value.

Details

Given objective vectors a and b, epsilon(a,b) is computed in the case of minimization as a/b for the multiplicative variant (respectively, a - b for the additive variant), whereas in the case of maximization it is computed as b/a for the multiplicative variant (respectively, b - a for the additive variant). This allows computing a single value for mixed optimization problems, where some objectives are to be maximized while others are to be minimized. Moreover, a lower value corresponds to a better approximation set, independently of the type of problem (minimization, maximization or mixed). However, the meaning of the value is different for each objective type. For example, imagine that f1 is to be minimized and f2 is to be maximized, and the multiplicative epsilon computed here for epsilon(A,B) = 3. This means that A needs to be multiplied by 1/3 for all f1 values and by 3 for all f2 values in order to weakly dominate B.

This also means that the computation of the multiplicative version for negative values doesn't make sense.

References

E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. Grunert da Fonseca. Performance Assessment of Multiobjective Optimizers: an Analysis and Review. IEEE Transactions on Evolutionary Computation, 7(2):117-132, 2003.

Examples

# NOT RUN {
extdata_path <- system.file(package="eaf","extdata")
path.A1 <- file.path(extdata_path, "ALG_1_dat")
path.A2 <- file.path(extdata_path, "ALG_2_dat")
A1 <- read_datasets(path.A1)[,1:2]
A2 <- read_datasets(path.A2)[,1:2]
ref <- filter_dominated(rbind(A1, A2))
epsilon_additive(A1, ref)
epsilon_additive(A2, ref)

# Multiplicative version of epsilon metric
ref <- filter_dominated(rbind(A1, A2))
epsilon_mult(A1, ref)
epsilon_mult(A2, ref)

# }