An ordered semigroup \(S\) is strong Dubreil-Jacotin, if there is an ordered group \(G\) and an epimorphism \(f:S\to G\) such that the pre-image of any principal order ideal of \(G\) is a principal order ideal of \(S\). Then the pre-image of the negative cone of \(G\) has the greatest element \(\xi\), called the bimaximum element of \(S\). For any \(x\in S\), denote \(\xi:x\) the greatest element of the order ideal \(\{y\in S;xy\leq\xi\}\); the element \(x\) is called perfect, if \(x=x(\xi:x)x\); \(S\) is perfect, if any element of \(S\) is perfect. The authors give some necessary and sufficient conditions for a regular strong Dubreil-Jacotin semigroup \(S\) to be perfect; one of them is that \(S\) is naturally ordered, i.e. its order is an extension of the natural order of idempotents of \(S\) (if \(ef=fe=e\), then \(e\leq f)\). Generally, the subset \(P(S)\) of perfect elements of \(S\) is a regular strong Dubreil-Jacotin subsemigroup of \(S\); there are given necessary and sufficient conditions for \(P(S)\) to be orthodox.