  Let $X$ be an infinite linearly ordered set and let $Y$ be a nonempty subset
of $X$. We calculate the relative rank of the semigroup $OP(X,Y)$ of all
orientation-preserving transformations on $X$ with restricted range $Y$ modulo
the semigroup $O(X,Y)$ of all order-preserving transformations on $X$ with
restricted range $Y$. For $Y = X$, we characterize the relative generating sets
of minimal size.
