Description Usage Arguments Details Value Author(s) References Examples

Computes the epsilon metric, either additive or multiplicative.

1 2 3 | ```
epsilon_additive(data, reference, maximise = FALSE)
epsilon_mult(data, reference, maximise = FALSE)
``` |

`data` |
( |

`reference` |
( |

`maximise` |
( |

The epsilon metric of a set *A* with respect to a reference set *R*
is defined as

*epsilon(A,R) = \max_{r \in R} \min_{a \in A} \max_{1 ≤q i ≤q n} epsilon(a_i, r_i)*

where *a* and *b* are objective vectors and, in the case of
minimization of objective *i*, *epsilon(a_i,b_i)* is computed as
*a_i/b_i* for the multiplicative variant (respectively, *a_i - b_i*
for the additive variant), whereas in the case of maximization of objective
*i*, *epsilon(a_i,b_i) = b_i/a_i* for the multiplicative variant
(respectively, *b_i - a_i* for the additive variant). This allows
computing a single value for 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
objective 1 is to be minimized and objective 2 is to be maximized, and the
multiplicative epsilon computed here for *epsilon(A,R) = 3*. This means
that *A* needs to be multiplied by 1/3 for all *a_1* values and by 3
for all *a_2* values in order to weakly dominate *R*. The
computation of the multiplicative version for negative values doesn't make
sense.

Computation of the epsilon indicator requires *O(n \cdot |A| \cdot
|R|)*, where *n* is the number of objectives (dimension of vectors).

A single numerical value.

Manuel López-Ibáñez

ZitThiLauFon2003:teceaf

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 | ```
# Fig 6 from Zitzler et al. (2003).
A1 <- matrix(c(9,2,8,4,7,5,5,6,4,7), ncol=2, byrow=TRUE)
A2 <- matrix(c(8,4,7,5,5,6,4,7), ncol=2, byrow=TRUE)
A3 <- matrix(c(10,4,9,5,8,6,7,7,6,8), ncol=2, byrow=TRUE)
plot(A1, xlab=expression(f[1]), ylab=expression(f[2]),
panel.first=grid(nx=NULL), pch=4, cex=1.5, xlim = c(0,10), ylim=c(0,8))
points(A2, pch=0, cex=1.5)
points(A3, pch=1, cex=1.5)
legend("bottomleft", legend=c("A1", "A2", "A3"), pch=c(4,0,1),
pt.bg="gray", bg="white", bty = "n", pt.cex=1.5, cex=1.2)
epsilon_mult(A1, A3) # A1 epsilon-dominates A3 => e = 9/10 < 1
epsilon_mult(A1, A2) # A1 weakly dominates A2 => e = 1
epsilon_mult(A2, A1) # A2 is epsilon-dominated by A1 => e = 2 > 1
# A more realistic example
extdata_path <- system.file(package="eaf","extdata")
path.A1 <- file.path(extdata_path, "ALG_1_dat.xz")
path.A2 <- file.path(extdata_path, "ALG_2_dat.xz")
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)
``` |

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