  Let $\mathcal{S}$ be a finite commutative semigroup written additively, and
let $\exp(\mathcal{S})$ be its exponent which is defined as the least common
multiple of all periods of the elements in $\mathcal{S}$. For every sequence
$T$ of elements in $\mathcal{S}$ (repetition allowed), let $\sigma(T) \in
\mathcal{S}$ denote the sum of all terms of $T$. Define the Davenport constant
$D(\mathcal{S})$ of $\mathcal{S}$ to be the least positive integer $d$ such
that every sequence $T$ over $\mathcal{S}$ of length at least $d$ contains a
proper subsequence $T'$ with $\sigma(T')=\sigma(T)$, and define the
Erd\H{o}s-Ginzburg-Ziv Theorem constant $E(\mathcal{S})$ to be the least
positive integer $\ell$ such that every sequence $T$ over $\mathcal{S}$ of
length at least $\ell$ contains a subsequence $T'$ with
$|T|-|T'|=\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil\exp(\mathcal{S})$
and $\sigma(T')=\sigma(T)$. When $\mathcal{S}$ is a finite abelian group, it is
well known that
$\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil\exp(\mathcal{S})=|\mathcal{S}|$
and $E(\mathcal{S})=D(\mathcal{S})+|\mathcal{S}|-1$. In this paper we
investigate whether $E(\mathcal{S})\leq
D(\mathcal{S})+\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil
\exp(\mathcal{S})-1$ holds true for all finite commutative semigroups
$\mathcal{S}$. We provide a positive answer to the question above for some
classes of finite commutative semigroups, including group-free semigroups,
elementary semigroups, and archimedean semigroups with certain constraints.
