  For an integral domain $R$, the {\it ring of integer-valued polynomials} over
$R$ consists of all polynomials $f(X) \in R[X]$ such that $f(R) \subseteq R$.
An interesting case to study is when $R$ is a Dedekind domain, in particular
when $R$ is the ring of integers of an algebraic number field. An algebraic
number field $K$ with ring of integers $\mathcal{O}_{K}$ is said to be a
P\'{o}lya field if the $\mathcal{O}_{K}$-module of integer-valued polynomials
on $K$ admits a regular basis. Associated to $K$ is a subgroup $Po(K)$ of the
ideal class group $Cl_{K}$, known as the {\it P\'{o}lya group of $K$}, that
measures the failure of $K$ from being a P\'{o}lya field. In this paper, we
prove the existence of three pairwise distinct totally real bi-quadratic
fields, each having P\'{o}lya group isomorphic to $\mathbb{Z}/2\mathbb{Z}$.
This extends the previously known families of number fields considered by
Heidaryan and Rajaei in \cite{rajaei-jnt} and \cite{rajaei}. Our results also
establish that under mild assumptions, the possibly infinite families of
bi-quadratic fields having a non-principal Euclidean ideal class, considered in
\cite{self-jnt}, fail to be P\'{o}lya fields.
