Let RWn be the semigroup of all regular order-preserving and compressing singular transformations on a finite-chain ［n］. For an arbitrary integer r(2≤r≤n-1), the non-group rank and non-idempotent rank of the semigroup W(n,r)=｛α∈RWn:|Imα|≤r｝ were studied. The semigroup W(n,r) generated by elements of rank r is proved. Furthermore, it is shown that for 1≤l≤r, the relative rank of the semigroup W(n,r) with respect to itself each ideal W(n,l).