Let $n_1, n_2,\ldots, n_d$ be positive integers and $H $ be the numerical semigroup generated by $n_1,n_2, \ldots, n_d$. Let $A:=k[H]:=k[t^{n_1}, t^{n_2},\ldots, t^{n_d}]\cong k[x_1,x_2,\ldots,x_d]/I$ be the numerical semigroup ring of $H $ over $k.$ In this paper we give a condition $(*)$ that implies that the minimal number of generators of the defining ideal $I$ is bounded explicitly by its type. As a consequence for semigroups with $d=4$ satisfying the condition $(*)$ we have $\mu ({\rm in}(I))\leq 2(t(H))+1$.