diff --git a/python/doc/source/reference/functions.metrics.rst b/python/doc/source/reference/functions.metrics.rst index b7e046a93..1a0f102c0 100644 --- a/python/doc/source/reference/functions.metrics.rst +++ b/python/doc/source/reference/functions.metrics.rst @@ -106,13 +106,13 @@ is defined as :cite:p:`ZitThiLauFon2003:tec` .. math:: epsilon(A,R) = \max_{r \in R} \min_{a \in A} \max_{1 \leq i \leq m} epsilon(a_i, r_i) -where :math:`a` and :math:`b` are objective vectors of length :math:`m`. +where :math:`a` and :math:`r` are objective vectors of length :math:`m`. -In the case of minimization of objective :math:`i`, :math:`epsilon(a_i,b_i)` is -computed as :math:`a_i/b_i` for the multiplicative variant (respectively, -:math:`a_i - b_i` for the additive variant), whereas in the case of -maximization of objective :math:`i`, :math:`epsilon(a_i,b_i) = b_i/a_i` for the -multiplicative variant (respectively, :math:`b_i - a_i` for the additive +In the case of minimization of objective :math:`i`, :math:`epsilon(a_i,r_i)` is +computed as :math:`a_i/r_i` for the multiplicative variant (respectively, +:math:`a_i - r_i` for the additive variant), whereas in the case of +maximization of objective :math:`i`, :math:`epsilon(a_i,r_i) = r_i/a_i` for the +multiplicative variant (respectively, :math:`r_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 diff --git a/r/R/epsilon.R b/r/R/epsilon.R index 5657326f5..4d2942150 100644 --- a/r/R/epsilon.R +++ b/r/R/epsilon.R @@ -22,11 +22,11 @@ #' #' where \eqn{a} and \eqn{b} are objective vectors of length \eqn{m}. #' -#' In the case of minimization of objective \eqn{i}, \eqn{epsilon(a_i,b_i)} is -#' computed as \eqn{a_i/b_i} for the multiplicative variant (respectively, -#' \eqn{a_i - b_i} for the additive variant), whereas in the case of -#' maximization of objective \eqn{i}, \eqn{epsilon(a_i,b_i) = b_i/a_i} for the -#' multiplicative variant (respectively, \eqn{b_i - a_i} for the additive +#' In the case of minimization of objective \eqn{i}, \eqn{epsilon(a_i,r_i)} is +#' computed as \eqn{a_i/r_i} for the multiplicative variant (respectively, +#' \eqn{a_i - r_i} for the additive variant), whereas in the case of +#' maximization of objective \eqn{i}, \eqn{epsilon(a_i,r_i) = r_i/a_i} for the +#' multiplicative variant (respectively, \eqn{b_i - r_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 diff --git a/r/man/epsilon.Rd b/r/man/epsilon.Rd index 9bff5842d..5a8309a73 100644 --- a/r/man/epsilon.Rd +++ b/r/man/epsilon.Rd @@ -33,13 +33,13 @@ reference set \eqn{R \subset \mathbb{R}^m} is defined as \deqn{epsilon(A,R) = \max_{r \in R} \min_{a \in A} \max_{1 \leq i \leq m} epsilon(a_i, r_i)} -where \eqn{a} and \eqn{b} are objective vectors of length \eqn{m}. +where \eqn{a} and \eqn{r} are objective vectors of length \eqn{m}. -In the case of minimization of objective \eqn{i}, \eqn{epsilon(a_i,b_i)} is -computed as \eqn{a_i/b_i} for the multiplicative variant (respectively, -\eqn{a_i - b_i} for the additive variant), whereas in the case of -maximization of objective \eqn{i}, \eqn{epsilon(a_i,b_i) = b_i/a_i} for the -multiplicative variant (respectively, \eqn{b_i - a_i} for the additive +In the case of minimization of objective \eqn{i}, \eqn{epsilon(a_i,r_i)} is +computed as \eqn{a_i/r_i} for the multiplicative variant (respectively, +\eqn{a_i - r_i} for the additive variant), whereas in the case of +maximization of objective \eqn{i}, \eqn{epsilon(a_i,r_i) = r_i/a_i} for the +multiplicative variant (respectively, \eqn{r_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