\section{\label{sec:level1}First-level heading:\protect\\ The line
\section{Introduction}

The Universe today is very close to homogeneous and isotropic.
Then the most important 3-Manifolds
in connection with cosmology are $S^{3},\: H^{3}$ and $E^{3}.$
It is well known that $R\times S^{3},\: R\times H^{3}$ and $R\times E^{3}$
are solutions of Einstein's equations (EQ). It is irrelevant for the EQ, if the
spatial sections have a non trivial topology modelled on them, or not.

There are many effects that arise if space is compact. The first astrophysical
limits on the topology of the universe were obtained for a 3-torus
$T^{3}$. Accordance with the homogeneity of the cosmic microwave background
radiation CMBR puts a lower
limit on the size of the fundamental cell, about $3000$ Mpc, which
is a cube in the cases of \cite{sok} and \cite{as}. Later on, it
was shown that this result is very sensitive to the type of the
compactifications
of the spatial sections. For a universe with spatial sections $T^{2}\times R$,
the fundamental cell's size is about $1/10$ of the horizon, and is
compatible with the homogeneity of the CMBR \cite{rouk}. In compact
universes, the pair separation
histogram would present spikes for characteristic distances. At first
it was thought that this technique, known as the \textit{crystallographic
method}, was able to totally determine the topology of the universe
\cite{lll}. It turned out that the crystallographic method only applied
when the group that defines the manifold, contained at least a
Clifford translation, i.e. a translation which moves all the points by
the same distance \cite{llu} and \cite{gtrb}. Generalisations of the
crystallographic method were proposed, for example in \cite{fg}.

Also in compact universes the light front of the CMBR interacts with
itself producing circles in its sky pattern \cite{css2}.

Recent results have called our attention to the possibility that methods
based on multiple images will prove not to be efficient \cite{grt}.
The reasoning is that, according to observations, the curvature is
very small, so the fundamental regions are so big that there has not
been time enough for the formation of ghost images. The result is
that for low curvature universes such as ours, only compact universes
with the smallest volumes could be detected by pattern repetitions.

The knowledge of the spectrum of the Laplacian for compact spaces
have some motivations. One of them is the decomposition of the CMBR
in right set of modes, if space is non trivial. It has been studied
numerically for compact hyperbolic space by \cite{cornish1998,inoue1998}.
And more recently \cite{lehoucq2002} for compact spherical space.

In 1948, Casimir calculated an attraction force between
parallel non charged plates in vacuum \cite{Casimir48}. This force is
universal, in the sense that it does not depend on
the physical properties of the plates, or the electric charge.
Its origin is of topological nature, connected to the boundary
conditions. There are interesting articles, in which the
imposition of boundary conditions is addressed in a more rigorous and
realistic approach, for instance \cite{plb}.

The
Casimir effect also occurs in compact spaces. For the present era
where the size of the Universe is $\sim 3000$ Mpc the Casimir effect
is irrelevant. Anyway, soon after the cosmological singularity the
co-moving distances were on the order of the Planck scale.
It is general believed that the very primordial
Universe was dominated by strict quantum effects including Casimir
effect. Although today quantum effects are commonly disregarded,
they must have been fundamentally important in the primordial
Universe. We shall use the massless scalar field as a toy model.

The Casimir effect can be obtained by
analytic continuation of the zeta function. This procedure is known as
the zeta function regularisation, see for example \cite{bordag2001},
\cite{elizalde.et.al.}. Anyway, in a more realistic context, the
ability of the zeta function method has been questioned \cite{npb}.
For a calculation of the Casimir energy in the
spherical space, see \cite{elizalde2004}.

In this work we expose a different technique which was previously
used by us for the hyperbolic space \cite{muller2001,muller2002a,muller2002b}.

Previous calculations for the Casimir effect for the Euclidean space
were done, for instance by, DeWitt, Hart, Isham and others \cite{dWH}.
Particularly interesting is Dowker's article in which the result is
also written in terms of Epstein's zeta functions. A topological
classification of possible ``gauge fields'' over compact base manifolds, is also
given in \cite{dWH}. These and other very interesting topological
questions are not going to be addressed in this present work.

The crystallographic groups are discrete subgroups of the full isometry
group of the
Euclidean space. Among all crystallographic groups, only a few of them are of
interest in connection to manifolds. These groups must be torsion-free, for
the Euclidean and Hyperbolic spaces. In the celebrated list of problems
proposed by Hilbert, the $18th$ was answered
affirmatively by L. Bieberbach in 1910-1912 \cite{Wolf}, \cite{Ratcliffe},
see also \cite {gomero-reboucas}.
Bieberbach showed that there are only a finite number of crystallographic
groups for an Euclidean space of fixed dimension. For example, when
the dimension of the Euclidean space is $3$ there are only $10$
compact manifolds, of which $6$ are orientable and $4$ non orientable.

In this present work we shall investigate the $6$ compact orientable manifolds
and obtain the Casimir energy for each of them.
Due to the more simple
structure connected to these topological spaces compared to the hyperbolic or
spherical case, an analytical result is obtained in terms of Epstein zeta
functions. It is explicitly obtained that our result is independent of the
type of coupling with curvature $\xi$ for these particular manifolds, which
means that the calculated Casimir energy density is the same even in an
inflationary de Sitter regime.

The paper is organised as follows. Section \ref{II} is included for
completeness. In this section, the  expectation value for the energy
momentum tensor, for a massless scalar
field in closed Minkowski spaces is obtained. Section \ref{III} presents a
brief description of the compact flat manifolds. The Casimir energy is
obtained in each subsection, \ref{sec1}-\ref{sec6} and a numerical plot
is given. Natural units are
used, $c=G=\hbar=1$, and the metric $\eta_{\mu\nu}=[-1,1,1,1]$
\section{Casimir effect in closed spaces \label{II}}
The point splitting method was constructed
to obtain the renormalised (finite) expectation values for the quantum
mechanical operators. It is based on the Schwinger formalism \cite{Schwinger},
and was developed in the context of curved space by DeWitt \cite{dWitt}.
Further details are contained in the articles by Christensen \cite{chris1},
\cite{chris2}. For a review, see \cite{grib1994}.

Metric variations in the scalar action \[
S=-\frac{1}{2}\int\sqrt{-g}(\phi_{,\rho}\phi^{,\rho}+\xi R\phi^{2}+m^{2}\phi^{2})d^{4}x\,\,,
\]
give the classical energy-momentum
tensor \begin{eqnarray}
T_{\mu\nu} & = & (1-2\xi)\phi_{,\mu}\phi_{,\nu}
-\left( \frac{1}{2}-2\xi\right)\phi_{,\rho}\phi^{,\rho}g_{\mu\nu}
-2\xi\phi\phi_{;\mu\nu}\nonumber \\
 &  & +2\xi g_{\mu\nu}\phi\square\phi+\xi G_{\mu\nu}\phi^{2}-\frac{1}{2}m^{2}g_{\mu\nu}\phi^{2}\,\,,\label{tmunu}\end{eqnarray}
 where $G_{\mu\nu}$ is the Einstein tensor. As expected, for conformal
coupling, $\xi=1/6$ with massless
fields, it can be verified that the trace of the above tensor is identically
zero if $m=0.$ In the following we shall restrict to spatial Euclidean
sections and massless fields.

The renormalised energy-momentum tensor involves field products at
the same space-time point. Thus the idea is to calculate the averaged
products at separate points, $x$ and $x^{\prime}$, taking the limit
$x^{\prime}\rightarrow x$ in the end.
\begin{equation}
\langle 0|T_{\mu\nu}\left(x\right)|0\rangle=
\frac{1}{2}\lim_{x^{\prime}\rightarrow x}T(x,x^\prime)_{\mu\nu} \label{Tnl}
\end{equation}
where for the particular manifolds addressed in this article,
\begin{eqnarray}
&&T(x,x^\prime)_{\mu\nu}=\\
&&(1-2\xi)(\left\{
\partial_\mu\partial_{\nu^\prime}+\partial_{\mu^\prime}\partial_\nu) -
2\xi
(
\partial_{\beta^\prime}\partial_{\alpha^\prime}+\partial_\alpha\partial_\beta)
\right\}G^{(1)}_{\mathcal{M}}(x,x^\prime) \label{Tem_recobrimento}
\end{eqnarray}
where $G^{(1)}_{\mathcal{M}}$ is the Hadamard function, which is
the expectation value of the anti-commutator of
$\phi(x)$ and $\phi(x^{\prime})$ on the manifold $\mathcal{M}$, see below.

The causal Green function or Feynman propagator, for the infinite
covering space $R^3$, is obtained as \[
G(x,x^{\prime})=i\langle0|T\phi(x)\phi(x^{\prime})|0\rangle\,\,,\]
 where $T$ is the time-ordering operator. Taking its real and imaginary
parts, \begin{equation}
G(x,x^{\prime})=G_{s}(x,x^{\prime})+\frac{i}{2}G^{(1)}(x,x^{\prime})\,\,,
\label{dfg}\end{equation}
 we get for the Hadamard function \[
G^{(1)}(x,x^{\prime})=\langle0|\{\phi(x),\phi(x^{\prime})\}|0\rangle\
=2\mathop{\textrm{Im}}G(x,x^{\prime})\,\,.\]

For a massless scalar field
\[
G^{(1)}(x,x^\prime)=\frac{1}{2\pi^2}\frac{1}{-(t-t^\prime)^2+r^2}=
\frac{1}{2\pi^2}\frac{1}{(x-x^\prime)^2}.
\]

Green functions, as any other function defined in the spatially compact
space-time $R\times\mathcal{M}$, must have the same periodicities
of the manifold $\mathcal{M}$ itself. One way of imposing this periodicity
is by the direct summation of all the elements of the group $\Gamma$
which defines the manifold
\begin{equation}
G^{(1)}_{\mathcal{M}}(x,x^\prime)=\frac{1}{2\pi^2}\sum_{\gamma_i \in \Gamma}\frac{1}{-(t-t^\prime)^2+(\gamma_i\vec{r}-\vec{r}^\prime)^2}\,\,.
\label{f.autom}
\end{equation}
The above expression is named the Poincar\'{e} series, and when
it converges, it defines functions on the manifold
$\mathcal{M}$, in this particular case, the Hadamard function.

There are recent articles were the physical
context connected to particular boundary conditions, Dirichlet,
Neumann and periodic is addressed \cite{sugs.}. Edery, develops a
very interesting cut-off technique and obtain analytical
formulas as sums over gamma and Riemann zeta functions plus an exact
multidimensional remainder expressed as sums over modified Bessel
functions.

Anyway if the spatial sections are compact, the point $x$ and the
point $\gamma x$ where $\gamma \in \Gamma$, $x \equiv \gamma x$ for all
the elements of the fundamental group $\Gamma$. That is, the point $x$
and $\gamma x$ are the same and identical point. This means that the
only possibility is that the function (\ref{f.autom}) must be periodic
in some sense. Mathematically, the  functions defined on a closed
manifold $\mathcal{M}$ are called automorfic, see for example
\cite{balazsvoros}.

In \cite{sugs.}, for instance,
the summations are over the spectrum of the Laplace operator. It is
well known that summations over the spectrum are equivalent to
summation over the closed geodesics, also known as the method of
images, \cite{balazsvoros}. This result lies at the
hart of the
Selberg formalism for the calculations of functional traces and is
valid at least for manifolds, Lie groups included \cite{Camporesi},
see also \cite{elizalde.et.al.}

Wald's axioms for the expectation value of the energy momentum
tensor are satisfied \cite{axwald}. Albeit this does not prove that
the chosen state is the correct one, it does indeed indicates a
criterion for a candidate physical state.

The Casimir energy density is independent of $\xi$ and is given by
the $00$ component of (\ref{Tem_recobrimento}) which is the projection
to static observers
$\langle 0|T_{\mu\nu}\left(x\right)|0\rangle u^\mu u^\nu$ with $4$ velocity
$u=(1,0,0,0)$
\begin{equation}
\rho^c_{\mathcal{M}}=-\frac{1}{\pi^2}\lim_{\vec{r}\rightarrow \vec{r}^\prime}
\sum_{\gamma \in \Gamma}{}^\prime\frac{1}{(\gamma\vec{r}-\vec{r}^\prime)^4},
\label{calc_e_casimir}
\end{equation}
where the $\prime$ means that the direct path with $\gamma=1$ is to be
excluded. It can be shown that the exclusion of the direct path corresponds
to a renormalisation of the cosmological constant \cite{muller2002a}.
\section{The closed oriented Euclidean spaces \label{III}}
Differently from the hyperbolic or spherical spaces, only a finite number of
topologies can be modelled in the Euclidean space. A representation for an
arbitrary element for the rigid motion subgroup of the full
isometry group can be given
\begin{equation}
\gamma=\left(\begin{array}{cccc}
 & & &a_1\\
 &R & &a_2\\
 & & &a_3\\
0 &0 &0 &1\\
\end{array}
\right),
\label{representacao}
\end{equation}
where $R \in SO(3)$ is an orthogonal rotation matrix and $\vec{a}$ is
the translation.
The action of the group on any point $x=\{x_1,x_2,x_3\}$ of $E^3$ is linear and
multiplicative
\[
\gamma x=\left(\begin{array}{cccc}
 & & &a_1\\
 &R & &a_2\\
 & & &a_3\\
0 &0 &0 &1\\
\end{array}
\right)
\left(
\begin{array}{c}
x_1\\
x_2\\
x_3\\
1\\
\end{array}
\right)
\]

The discrete subgroups of the isometry group define the manifolds modelled
in the space of interest.
A group $\Gamma$ is said to be torsion-free if for each element
$\gamma\in \Gamma$, $\gamma^n\neq 1$ for any integer $n$. In the Euclidean
case, the torsion-free
elements must be ``screw motions", i.e. a combined rotation and translation
along the rotation
axis. The holonomy group is the rotational part of $\Gamma$, $\Psi=\Gamma/T$,
where $T$ is the translational part of the group.
Two oriented manifolds are diffeomorfic if and only if the discrete
torsion-free groups have isomorfic holonomy groups, $\Psi$. Of course,
$\Psi\subset SO(3)$ is a finite group. According to Lemma 3.5.4 of Wolf's book
\cite{Wolf}, for the $E^3$ case, the only possibilities for $\Psi$ are
$\Psi=\mathbb{Z}_1,\mathbb{Z}_2,\mathbb{Z}_4,
\mathbb{Z}_3,\mathbb{Z}_6,\mathbb{Z}_2\otimes \mathbb{Z}_2$, which
give rise to to the $6$ closed oriented manifolds \cite{Wolf}. These
manifolds and the Casimir energy are given in the subsections
\ref{sec1}-\ref{sec6}.

The very important Theorem 8.2.5 of Ratcliffe's book \cite{Ratcliffe} is
reproduced in the following \\

{\it Every compact, $n$-dimensional, Euclidean space-form is finitely
covered by an Euclidean n-torus.} \\

This theorem is in fact Bieberbach's first theorem, which states that
$T$ is a normal subgroup of finite index in $\Gamma$. This finite index
is given by the order of $\Psi$ and corresponds to the number of coverings of
the space in question by the Euclidean $n$-torus. The above theorem is
used to rewrite the summations in
the expectation values over all the elements of the group as a finite sum
of Epstein type zeta functions. The Epstein zeta function is already well
known to give the appropriate result for the Casimir energy for the $n-$
torus, see for example \cite{bv} \cite{elizalde.et.al.}. In the original
Epstein's notation let $p$ be a positive integer and
\begin{eqnarray*}
&&\vec{g}=(g_1,...,g_p), \;\; g_i \in \mbox{R}, \;\; \vec{g} \in\mbox{R}^p\\
&&\vec{h}=(h_1,...,h_p),\;\; h_i\in \mbox{R},\;\; \vec{h}\in\mbox{R}^p\\
&&\vec{m}=(m_1,...,m_p),\;\; m_i\in \mbox{Z},\;\; \vec{m}\in \mbox{Z}^p.
\end{eqnarray*}
The scalar product is defined as
\[
(\vec{g},\vec{h})=\sum_{\nu=1}^p g_\nu h_\nu,
\]
and $c_{\mu\nu}$ an invertible $p\times p$ matrix associated with a quadratic
form as follows
\[
\varphi(x)=\sum_{\mu,\nu=1}^{p}c_{\mu\nu}x_\mu x_\nu.
\]
With the above definitions Epstein's zeta function is written as
\begin{equation}
Z\left|\begin{array}{ccc} g_1 &....&g_p\\ h_1 & ....& h_p\end{array}
\right|(s)_{\varphi}=\sum_{m_1,...,m_p=-\infty}^\infty{}^\prime[\varphi(\vec{m}+\vec{g})]^
{-\frac{s}{2}}e^{2\pi i(\vec{m},\vec{h})},
\label{fze}
\end{equation}
where the $\prime$ means that $m_1=0,\; m_2=0,\;...m_p=0$ is to be excluded.
Generalisations for non homogeneous Epstein zeta functions and reflection
formulae, can be found in \cite{e.1998}, \cite{elizalde.et.al.}.
\subsection{$\Psi=\mathbb{Z}_1$ \label{sec1} }
The volume of this manifold is $V=ab_2c_3$. The Dirichlet region is the
parallelepiped given in Figure \ref{fig1}
\begin{figure}[tpb]
\includegraphics[scale=0.4]{fig4.eps}
\caption{Dirichlet Region}
\label{fig1}
\end{figure}
According to the representation (\ref{representacao}), the
matrices
\begin{eqnarray}
&&\gamma_1=\left(\begin{array}{cccc}
1 &0 &0 &a\\
0 &1 &0 &0\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),
\;\;\gamma_2=\left(\begin{array}{cccc}
1 &0 &0 &b_1\\
0 &1 &0 &b_2\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),\nonumber\\
&&\gamma_3=\left(\begin{array}{cccc}
1 &0 &0 &c_1\\
0 &1 &0 &c_2\\
0 &0 &1 &c_3\\
0 &0 &0 &1\\
\end{array} \right)
\end{eqnarray}
with the respective inverses are also the generators of the translational
part of the group $\Gamma=T=\{\gamma_1,\gamma_2,\gamma_3\}$.
The identifications are as follows
\[
\vec{r}=(x,y,z)\rightarrow (x+ma+nb_1+kc_1,y+nb_2+kc_2,z+kc_3)
\]
and according to (\ref{calc_e_casimir})
\begin{eqnarray*}
&&\rho^c_{\mathbb{Z}_1}=-\frac{1}{\pi^2}\sum_{m,n,k=-\infty}^{+\infty}{}^\prime\left\{
\left[(ma+nb_1+kc_1)^2 \right.\right. \\
&&\left. \left. +(nb_2+kc_2)^2+(kc_3)^2\right]^{-2}\right\}
\end{eqnarray*}
\begin{equation}
\rho^c_{\mathbb{Z}_1}=-\frac{1}{\pi^2}
Z\left|\begin{array}{ccc} 0 & 0 &0\\ 0 & 0 & 0 \end{array}\right|(4)_\varphi,
\end{equation}
where according to (\ref{fze}) the matrix
\[
c_{\mu\nu}=\left(
\begin{array}{ccc}
a^2 & ab_1 & ac_1\\
ab_1 &b_1^2+b_2^2 & b_1c_1+b_2c_2 \\
ac_1 & b_1c_1+b_2c_2 & c_1^2+c_2^2+c_3^2
\end{array}\right)
\]
defines the positive quadratic form $\varphi
(m)=c_{11}m^2+c_{22}n^2+c_{33}k^2
+2c_{12}mn+2c_{13}mk+2c_{23}nk$.
\subsection{$\Psi=\mathbb{Z}_2$ \label{sec2} }
The volume of this manifold is $V=ab_2c$ Figure \ref{fig2}.
Now the matrices and their inverses
\begin{eqnarray}
&&\gamma_1=\left(\begin{array}{cccc}
1 &0 &0 &a\\
0 &1 &0 &0\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),
\;\;\gamma_2=\left(\begin{array}{cccc}
1 &0 &0 &b_1\\
0 &1 &0 &b_2\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),\nonumber\\
&&\gamma_3=\left(\begin{array}{cccc}
-1 &0 &0 &0\\
0 &-1 &0 &0\\
0 &0 &1 &c\\
0 &0 &0 &1\\
\end{array} \right)
\end{eqnarray}
\begin{figure}[tpb]
\includegraphics[scale=0.4]{fig6.eps}
\caption{Dirichlet Region, DR. The opposite faces in the $z$ direction are
identified with a rotation of $\pi$.}
\label{fig2}
\end{figure}
generate the group $\Gamma$.

The generators of the translational part $T$, now, are given by
$T=\Gamma/\mathbb{Z}_2=\{\gamma_1,\gamma_2,\gamma_3^2\}$, together with the
inverses. The fundamental
domain, $FD$, for $\Gamma/\mathbb{Z}_2$ has volume $V=2ab_2c$ and is given by
two copies of the Dirichlet region of Figure \ref{fig2};
$FD=\{1.DR,\gamma_3.DR\}$.

The identifications in this manifold are provided by
\begin{eqnarray*}
&&\vec{r}=(x,y,z)\rightarrow (x+ma+nb_1,y+nb_2,z+2kc) \\
&&\rightarrow (-x+ma+nb_1,-y+nb_2,z+(2k+1)c)
\end{eqnarray*}

Using (\ref{calc_e_casimir}), the Casimir energy is given by
\begin{eqnarray*}
&&\rho^c_{\mathbb{Z}_2}=-\frac{1}{\pi^2}\sum_{m,n,k=-\infty}^{+\infty}{^\prime}\left\{
\left[(ma+nb_1)^2 +(nb_2)^2+(2kc)^2\right]^{-2}\right.\\
&&\left.+\left[(-2x+ma+nb_1)^2 +
(-2y+nb_2)^2 +((2k+1)c)^2\right]^{-2}\right\}
\end{eqnarray*}
\begin{equation}
\rho^c_{\mathbb{Z}_2}=-\frac{1}{\pi^2}\left(
Z\left|\begin{array}{ccc} 0 & 0 &0\\ g_x & g_y & g_z \end{array}\right|(4)_\varphi+
Z\left|\begin{array}{ccc} 0 & 0 &0\\ 0 & 0 & 0 \end{array}\right|(4)_\varphi \right)
\label{Ec1}
\end{equation}
where
\begin{eqnarray}
&&\vec{g}=\left(-\frac{2x}{a}+\frac{2yb_1}{ab_2},-\frac{2y}{b_2},1/2 \right) \\
&&c_{\mu\nu}=\left(
\begin{array}{ccc}
a^2 & ab_1 & 0\\
ab_1 &b_1^2+b_2^2 & 0 \\
0 & 0 & 4c^2
\end{array}\right)
\end{eqnarray}
The values of $a=1.2$, $b_1=0.7$, $b_2=1.4$ and $c=1.3$ are chosen.
The truncation in (\ref{fze}) is done such that the summation is
over a geodesic sphere of radius $R\sim 100$ of integer numbers. The
corresponding truncation error on the energy density should be less
than $1\%$. The result is shown in Figure \ref{grf1}, and is independent
of the coordinate $z$.
\begin{figure}[tpb]
\includegraphics[scale=0.4]{z2.ps}
\caption{The Casimir energy density in natural units, eq. (\ref{Ec1})
inside the DR. in Figure \ref{fig2}. The $z$ coordinate is chosen to be $z=0.6$}
\label{grf1}
\end{figure}
\subsection{$\Psi=\mathbb{Z}_4$ \label{sec3}}
The volume of this manifold is $V=a^2b$ Figure \ref{fig3}.
\begin{figure}[tpb]
\includegraphics[scale=0.4]{fig5.eps}
\caption{Dirichlet Region. The opposite faces in the $z$ direction are
identified with a rotation of $\pi/2$}
\label{fig3}
\end{figure}
The matrices
\begin{eqnarray}
&&\gamma_1=\left(\begin{array}{cccc}
1 &0 &0 &a\\
0 &1 &0 &0\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),
\;\;\gamma_2=\left(\begin{array}{cccc}
1 &0 &0 &0\\
0 &1 &0 &a\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),\nonumber\\
&&\gamma_3=\left(\begin{array}{cccc}
0 &-1 &0 &0\\
1 &0 &0 &0\\
0 &0 &1 &b\\
0 &0 &0 &1\\
\end{array} \right),
\end{eqnarray}
together with their inverses are the generators of $\Gamma$. The
translational part is generated by
$T=\Gamma/\mathbb{Z}_4=\{\gamma_1,\gamma_2,\gamma_3^{4}\}$, together
with the inverses. The fundamental
region, $FR$ for $\Gamma/\mathbb{Z}_4$ has volume $V=4a^2b$ and is given by $4$ copies of the DR in
Figure \ref{fig3}; $FR=\{DR, \gamma_3.DR,\gamma_3^{2}.DR,\gamma_3^{3}.DR\}$. The
identifications and the direct summation are given in the  \ref{ap1}.
The result for the Casimir energy according to (\ref{calc_e_casimir}) and (\ref{fze}) is the following
\begin{eqnarray}
&&\rho^c_{\mathbb{Z}_4}=-\frac{1}{\pi^2}\left(
Z\left|\begin{array}{ccc} 0 & 0 &0\\ 0 & 0 & 0 \end{array}\right|(4)_\varphi+
Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^1_x & g^1_y & g^1_z \end{array}\right|(4)_\varphi\right.\nonumber\\
&&\left.+Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^2_x & g^2_y & g^2_z \end{array}\right|(4)_\varphi+
 Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^3_x & g^3_y & g^3_z \end{array}\right|(4)_\varphi  \right)
\label{Ec2}
\end{eqnarray}
where
\begin{eqnarray}
&&\vec{g}^1=\left(-\frac{x+y}{a},\frac{x-y}{a},1/4 \right) \nonumber\\
&&\vec{g}^2=\left(\frac{-2x}{a},\frac{-2y}{a},2/4 \right) \nonumber\\
&&\vec{g}^3=\left(\frac{y-x}{a},-\frac{x+y}{a},3/4 \right) \nonumber\\
&&c_{\mu\nu}=\left(
\begin{array}{ccc}
a^2 & 0 & 0\\
0 &a^2 & 0 \\
0 & 0 & 16b^2
\end{array}\right)
\end{eqnarray}
The values of $a=1.2$, $b=1.4$ are chosen.
The truncation in (\ref{fze}) is done such that the summation is
over a geodesic sphere of radius $R\sim 100$ of integer numbers. The
corresponding truncation error on the energy density should be less
than $1\%$. The result is shown in Figure \ref{grf2}, and is independent
of the coordinate $z$
\begin{figure}[tpb]
\includegraphics[scale=0.4]{z4.ps}
\caption{The Casimir energy density in natural units, given in (\ref{Ec2})
inside the DR.  in Figure \ref{fig3}, with the coordinate $z=0.6$}
\label{grf2}
\end{figure}
\subsection{$\Psi=\mathbb{Z}_3$ \label{sec4}}
The volume of this manifold is $V=\sqrt{3}a^2b$. The Dirichlet domain is an
hexagonal prism of side $a/\sqrt{3}$ Figure \ref{fig4}
\begin{figure}[tpb]
\includegraphics[scale=0.4]{fig1.eps}
\caption{Dirichlet Region. The top and bottom faces in the $z$ direction are
identified with a rotation of $2\pi/3$}
\label{fig4}
\end{figure}
The face pairing matrices
\begin{eqnarray}
&&\gamma_1=\left(\begin{array}{cccc}
1 &0 &0 &a\\
0 &1 &0 &0\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),
\;\;\gamma_2=\left(\begin{array}{cccc}
1 &0 &0 &-a/2\\
0 &1 &0 &\sqrt{3}a/2\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),\nonumber\\
&&\gamma_3=\left(\begin{array}{cccc}
1 &0 &0 &a/2\\
0 &1 &0 &\sqrt{3}a/2\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),\nonumber\\
&&\gamma_4=\left(\begin{array}{cccc}
-1/2 &-\sqrt{3}/2 &0 &0\\
\sqrt{3}/2 &-1/2 &0 &0\\
0 &0 &1 &b\\
0 &0 &0 &1\\
\end{array} \right).
\end{eqnarray}
together with the inverses generate the group $\Gamma$.

The generators of the  translational
part are given by
$T=\Gamma/\mathbb{Z}_3=\{\gamma_1,\gamma_3,\gamma_4^3\}$, with the inverses. The fundamental
region, $FR$ for $\Gamma/\mathbb{Z}_3$ has volume $V=3\sqrt{3}a^2b$ and is given by $3$
copies of the DR in
Figure \ref{fig4}; $FR=\{DR, \gamma_4.DR,\gamma_4^{2}.DR\}$. The
identifications are given in the \ref{ap2}. The Casimir energy,
according to (\ref{calc_e_casimir}) and (\ref{fze}) is
\begin{eqnarray}
&&\rho^c_{\mathbb{Z}_3}=-\frac{1}{\pi^2}\left(
Z\left|\begin{array}{ccc} 0 & 0 &0\\ 0 & 0 & 0 \end{array}\right|(4)_\varphi+
Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^1_x & g^1_y & g^1_z \end{array}\right|(4)_\varphi\right.\nonumber\\
&&\left.+Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^2_x & g^2_y & g^2_z \end{array}\right|(4)_\varphi \right)
\label{Ec3}
\end{eqnarray}
where
\begin{eqnarray}
&&\vec{g}^1=\left(\frac{-2x}{a},\frac{x-y\sqrt{3}}{a},1/3 \right) \nonumber\\
&&\vec{g}^2=\left(-\frac{x-y\sqrt{3}}{a},-\frac{x+y\sqrt{3}}{a},2/3 \right) \nonumber\\
&&c_{\mu\nu}=\left(
\begin{array}{ccc}
a^2 & a^2/2 & 0\\
a^2/2 &a^2 & 0 \\
0 & 0 & 9b^2
\end{array}\right)
\end{eqnarray}
The values of $a=1.2$, $b=1.4$ are chosen.
The truncation in (\ref{fze}) is done such that the summation is
over a geodesic sphere of radius $R\sim 100$ of integer numbers. The
corresponding truncation error on the energy density should be less
than $1\%$. The result is shown in Figure \ref{grf3}, and is
independent of the $z$ coordinate.
\begin{figure}[tpb]
\includegraphics[scale=0.4]{z3.ps}
\caption{The Casimir energy density in natural units,
given in (\ref{Ec3}) inside the DR. in Figure \ref{fig4} with the
coordinate $z=0.6$. The region is a disc of radius $r=0.6$, $\theta$
is the angle between the $x$ and $y$ coordinates: $\theta=0$ corresponds to the
$x$ axis of Figure \ref{fig4}.}
\label{grf3}
\end{figure}
\subsection{$\Psi=\mathbb{Z}_6$ \label{sec5}}
The Dirichlet domain is a prism with hexagonal base of side $a/\sqrt{3}$,
Figure \ref{fig5}
\begin{figure}[tpb]
\includegraphics[scale=0.4]{fig2.eps}
\caption{Dirichlet Region. The top and bottom faces in the $z$ direction are
identified with a rotation of $\pi/3$}
\label{fig5}
\end{figure}
The volume of this manifold is $V=\sqrt{3}a^2b$. The face pairing matrices are
\begin{eqnarray}
&&\gamma_1=\left(\begin{array}{cccc}
1 &0 &0 &a\\
0 &1 &0 &0\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),
\;\;\gamma_2=\left(\begin{array}{cccc}
1 &0 &0 &-a/2\\
0 &1 &0 &\sqrt{3}a/2\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),\nonumber\\
&&\gamma_3=\left(\begin{array}{cccc}
1 &0 &0 &a/2\\
0 &1 &0 &\sqrt{3}a/2\\
0 &0 &1 &0\\
0 &0 &0 &1\\
\end{array} \right),\nonumber\\
&&\gamma_4=\left(\begin{array}{cccc}
1/2 &-\sqrt{3}/2 &0 &0\\
\sqrt{3}/2 &1/2 &0 &0\\
0 &0 &1 &b\\
0 &0 &0 &1\\
\end{array} \right),
\end{eqnarray}
which together with their inverses generate $\Gamma$. The translational part
is generated by
$T=\Gamma/\mathbb{Z}_6=\{\gamma_1,\gamma_3,\gamma_4^6 \}$,
together with the inverses. The fundamental
region, $FR$ for $\Gamma/\mathbb{Z}_6$ has volume $V=6\sqrt{3}a^2b$ and is
given by $6$
copies of the DR in Figure \ref{fig5}; $FR=\{DR, \gamma_4.DR,\gamma_4^{2}.DR,
\gamma_4^{3}.DR,\gamma_4^{4}.DR\,\gamma_4^{5}.DR\}$. The
identifications are given in \ref{ap3}

With a reasoning analogous to the previous sections the following result is
obtained using (\ref{calc_e_casimir}) and (\ref{fze})
\begin{eqnarray}
&&\rho^c_{\mathbb{Z}_6}=-\frac{1}{\pi^2}\left(
Z\left|\begin{array}{ccc} 0 & 0 &0\\ 0 & 0 & 0 \end{array}\right|(4)_\varphi+
Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^1_x & g^1_y & g^1_z \end{array}\right|(4)_\varphi\right.\nonumber\\
&&\left.+Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^2_x & g^2_y & g^2_z \end{array}\right|(4)_\varphi+
 Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^3_x & g^3_y & g^3_z \end{array}\right|(4)_\varphi \right.\nonumber\\
 &&\left.+Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^4_x & g^4_y & g^4_z \end{array}\right|(4)_\varphi+
 Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^5_x & g^5_y & g^5_z \end{array}\right|(4)_\varphi \right)
\label{Ec4}
\end{eqnarray}
where
\begin{eqnarray}
&&\vec{g}^1=\left(-\frac{x+y/\sqrt{3}}{a},\frac{x-y/\sqrt{3}}{a},1/6 \right) \nonumber\\
&&\vec{g}^2=\left(\frac{-2x}{a},\frac{x-y\sqrt{3}}{a},2/6 \right) \nonumber\\
&&\vec{g}^3=\left(\frac{-2x+2y/\sqrt{3}}{a},\frac{-4y/\sqrt{3}}{a},3/6 \right) \nonumber\\
&&\vec{g}^4=\left(\frac{-x+y\sqrt{3}}{a},-\frac{x+y\sqrt{3}}{a},4/6 \right) \nonumber\\
&&\vec{g}^5=\left(\frac{2y/\sqrt{3}}{a},-\frac{x+y/\sqrt{3}}{a},5/6 \right) \nonumber\\
&&c_{\mu\nu}=\left(
\begin{array}{ccc}
a^2 & a^2/2 & 0\\
a^2/2 &a^2 & 0 \\
0 & 0 & 36b^2
\end{array}\right)
\end{eqnarray}
The values of $a=1.2$, $b=1.4$ are chosen.
The truncation in (\ref{fze}) is done such that the summation is
over a geodesic sphere of radius $R\sim 100$ of integer numbers. The
corresponding truncation error on the energy density should be less
than $1\%$. The result is shown in Figure \ref{grf4}, and is independent
of the $z$ coordinate.
\begin{figure}[tpb]
\includegraphics[scale=0.4]{z6.ps}
\caption{The Casimir energy density in natural units,
given in (\ref{Ec4}) inside the DR. in Figure
\ref{fig5}, with the coordinate $z=0.6$.
The region is a disc of radius $r=0.6$, $\theta$
is the angle between the $x$ and $y$ coordinates: $\theta=0$ corresponds to the
$x$ axis of Figure \ref{fig5}.}
\label{grf4}
\end{figure}
\subsection{$\Psi=\mathbb{Z}_2\otimes\mathbb{Z}_2$ \label{sec6}}
The face pairing matrices are now
\begin{eqnarray}
&&\gamma_1=\left(\begin{array}{cccc}
1 &0 &0 &a\\
0 &-1 &0 &0\\
0 &0 &-1 &0\\
0 &0 &0 &1\\
\end{array} \right),
\;\;\gamma_2=\left(\begin{array}{cccc}
-1 &0 &0 &0\\
0 &1 &0 &b\\
0 &0 &-1 &c\\
0 &0 &0 &1\\
\end{array} \right),\nonumber\\
&&\gamma_3=\left(\begin{array}{cccc}
-1 &0 &0 &0\\
0 & 1 &0 &-b\\
0 &0 &-1 &-c\\
0 &0 &0 &1\\
\end{array} \right),
\end{eqnarray}
which together with their inverses generate $\Gamma$, and the volume of the manifold is $V=2abc$,
Figure \ref{fig6}.
\begin{figure}[tpb]
\includegraphics[scale=0.5]{fig3.eps}
\caption{Dirichlet Region. The opposite faces in the $x$ direction are
identified with a rotation of $\pi$. The adjacent faces are identified with
a rotation of $\pi$, also.}
\label{fig6}
\end{figure}
The translational part is generated by
$T=\Gamma/(\mathbb{Z}_2\otimes\mathbb{Z}_2)=\{\gamma_1^2,\gamma_2^2,
\gamma_2\gamma_3 \}$, with the inverses. The fundamental
region, $FR$ for $\Gamma/(\mathbb{Z}_2\otimes \mathbb{Z}_2)$ has volume
$V=8abc$ and is
given by $4$
copies of the DR in Figure \ref{fig6}; $FR=\{DR, \gamma_1.DR,\gamma_2.DR,
\gamma_1\gamma_3.DR\}$. The identifications are given in \ref{ap4}.
With (\ref{calc_e_casimir}) and (\ref{fze}) the Casimir energy results in
\begin{eqnarray}
&&\rho^c_{\mathbb{Z}_2\otimes \mathbb{Z}_2}=-\frac{1}{\pi^2}\left(
Z\left|\begin{array}{ccc} 0 & 0 &0\\ 0 & 0 & 0 \end{array}\right|(4)_\varphi+
Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^1_x & g^1_y & g^1_z \end{array}\right|(4)_\varphi\right.\nonumber\\
&&\left.+Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^2_x & g^2_y & g^2_z \end{array}\right|(4)_\varphi+
 Z\left|\begin{array}{ccc} 0 & 0 &0\\
      g^3_x & g^3_y & g^3_z \end{array}\right|(4)_\varphi \right)
\label{Ec5}
\end{eqnarray}
where
\begin{eqnarray}
&&\vec{g}^1=\left(1/2,\frac{-y}{b},\frac{-z}{c}\right) \nonumber\\
&&\vec{g}^2=\left(\frac{-x}{a},1/2,\frac{c-2z}{2c} \right) \nonumber\\
&&\vec{g}^3=\left(\frac{a-2x}{2a},\frac{b-2y}{2b},1/2\right) \nonumber\\
&&c_{\mu\nu}=\left(
\begin{array}{ccc}
4a^2 & 0 & 0\\
0 &4b^2 & 0 \\
0 & 0 & 4c^2
\end{array}\right).
\end{eqnarray}
The values of $a=1.2$, $b=1.4$ and $c=1.3$ are chosen.
The truncation in (\ref{fze}) is done such that the summation is
over a geodesic sphere of radius $R\sim 100$ of integer numbers. The
corresponding truncation error on the energy density should be less
than $1\%$. The result is shown in Figure \ref{grf5}. This is the only
manifold for which there is not an axis such that the energy density
expectation value is constant.
\begin{figure}[tpb]
\includegraphics[scale=0.4]{z2z2.ps}
\caption{The Casimir energy density in natural units, given in (\ref{Ec5})
inside the DR. in Figure \ref{fig6}, a cylindrical region surrounding the $x$
axis, of radius $0.762$, $\phi$ is the angle.}
\label{grf5}
\end{figure}
\section{Conclusions}
The Casimir energy for the oriented closed Euclidean space-times are
obtained in this work.

Closed manifolds are described by the discrete subgroups of the
isometry group. The crystallographic groups are discrete subgroups of
the full isometry group of
Euclidean space. Among all crystallographic groups, only a few of them are of
interest in connection to manifolds. These groups must be torsion-free, for
the Euclidean and Hyperbolic spaces. In the celebrated list of problems
proposed by Hilbert, the $18th$ was answered
affirmatively by L. Bieberbach in 1910-1912 \cite{Wolf}, \cite{Ratcliffe},
see also \cite{gomero-reboucas}.
Bieberbach showed that there are only a finite number of crystallographic
groups for an Euclidean space of fixed dimension.

When the dimension of the Euclidean space is $3$ there are only $10$ compact
manifolds, of which $6$ are orientable and $4$ non orientable.

In this present work we have investigated the $6$ compact orientable manifolds
and obtain the Casimir energy for each of them. Due to the more simple
structure connected to these topological spaces compared to the hyperbolic or
spherical case, an analytical result is obtained in terms of Epstein zeta
functions. It is explicitly obtained that our result is independent of the
type of coupling with curvature $\xi$ for these particular manifolds,
which means that the calculated Casimir energy density is the same even in
an inflationary de Sitter regime.

According to the theorem in Section \ref{II},  it is always possible to find
the normal subgroup of finite index, $T$. With this, the $n-$ torus
that covers the manifold a finite number of times is obtained.
The Casimir energy is given by a finite sum of Epstein type zeta
functions. The number of distinct zeta functions is given by the
index of $T$ in $\Gamma$, which is the number of times the manifold
is covered by this $n-$ torus.

A numerical plot inside the Dirichlet Region for each manifold is
given. The manifold in Section \ref{sec1} is the only one for which the
Casimir energy density is homogeneous. For the manifolds
Sections \ref{sec2}- \ref{sec5} the Casimir energy density depends on the
$x$ and $y$ coordinates inside the fundamental domain.
For the last manifold in Section
\ref{sec6}, the Casimir energy density
depends in all the coordinates.

The sign of Casimir energy density is
negative for all manifolds. Remind that there is a subtraction of the
divergence of the infinite $R^3$ covering space, which means that the
energy of the manifold is less than the energy of the infinite $R^3$.
The effect of choosing different representations for the fundamental group
$\Gamma$ acting on the field, like twisting, can increase the energy of the
vacuum \cite{dWH}. deWitt finds that the sign of the energy depends
on the representation of the fundamental group, the orientability of the
manifold and the spin of the field. This must have a more fundamental reason
connected to the Atiyah-Singer theorem, and is beyond our knowledge.

It is explicitly checked that the truncation error should be on the order
of $1/R$ where $R$ is the geodesic radius of the sphere of integer numbers
where the truncation of (\ref{calc_e_casimir}) was performed. It was used
$R\sim 100$ such that the truncation error should be on the order of
$1\%$. The respective values of the parameters of the manifold are
indicated in each case.
\ack
M. P. L. wishes to thank CNPq for financial support. D. M. wishes to
thank the Brazilian project {\it Nova F\'\i sica no Espa\c co} of
FAPESP.


\section{Introduction}

The facility location problem aims at finding the optimal locations for facilities from a set of candidate location nodes  in order to minimize the cost such as the fixed facility cost and the transposition cost, or to maximize the total revenue. In general, there are also some constraints to be considered, such as satisfying all customers' demands, etc. It is an NP-hard optimization problem \cite{ref5,ref6,ref7} and has attracted much attention from researchers in both the scientific community and engineering field due to its wide application in real world. The facilities could be hospitals, restaurants, post stations, bus stations, industrial plants, banks, warehouses, and distribution centers, etc. The facility location decision has high precedence in the whole logistics decisions and has a great influence on subsequent operation level decisions \cite{ref4}. Daskin et al. \cite{ref5} regards the location decisions as “the most critical and most difficult of the decisions needed to realize an efficient supply chain”.

In RFLP, the facility is not always available all the time \cite{ref5}. One or more of them may not work from time to time because of disruptions, examples include natural disasters, inclement weather, destruction of facilities by fire or flood, expiration of the contract, and any other force majeure factors. In such a situation, these are facility “failures”. The failures of the facilities will result in excessive transportation costs because the customers that were considered to be served by them must be served by other, usually more distant, facilities \cite{ref5}. Therefore, by solving RFLP, we can get a location decision which can ensure a certain level of reliability to guarantee customers can get service when facilities' failures occur.

Many models have been proposed for RFLP, in which all kinds of factors were taken into account and many of them are formulated for specific applications in real life. In addition, large-scale RFLP problems have rarely been considered. The algorithms studied in literature were mainly tested on problems of small size.

This paper focuses on two aspects: the problem formulation and the algorithm. Based on the work of~\cite{ref8,ref9}, we propose a new reliable facility location-allocation problem (RFLP) formulation, which does not fix the number of allocated facilities to each customer as a constant and is more close to reality. The resulted model is a nonlinear 0-1 integer programming model which is more complicated for traditional methods. In this paper, a hybrid evolutionary algorithm called EAMLS is proposed to solve it. EAMLS combines a memorable local search method with an evolutionary algorithm, which has a good performance on both small-scale and large-scale problems considered in this paper. It is worth mentioning that the instances used in our experiments are much larger than the ones used in previous work. Furthermore, a convergence metric l3-value is proposed for analyzing the algorithm and observing the evolutionary process.

The rest of this paper is organized as follows. Section 2 briefly reviews the related work of RFLP. In Section 3, our new RFLP formulation is introduced. We proposed a hybrid evolutionary algorithm EAMLS in Section 4. Section 5 presents computational studies, and Section 6 concludes.

\section{Related Work}

By solving a specific RFLP, decision-makers expect to get a robust location decision which is still economical when some facilities fail under various disruptions. The research can be divided into two categories according to the method used to handle facility failure or ensure reliability.

Some works \cite{ref10,ref11,ref12} use a disruptive scenarios approach to describe facility failure. In this approach, scenarios contain facility failure information, e.g., simultaneously disrupted facility sites, modified customer demands, and facility costs, etc. The disruptive scenarios approach can describe the facility failure information well, but it usually requires plenty of scenarios to cover different disruptive situations, which implies large computational cost, especially for large-scale problems.

Another approach to ensure reliability is to allocate two or more facilities to serve each customer \cite{ref8,ref9,ref16,ref17}. In this approach, the method for reliability is intuitive and easy to understand. Both a location decision (which contains how many facilities needed to build and where to build them) and an allocation decision (which shows how to allocate facilities to serve customers ) are determined before the occurrences of facilities' disruptions/failures.

Some RFLP models have been proposed, e.g., models proposed by Li et al. \cite{ref8} and Snyder and Darskin \cite{ref9}. Table \ref{tab:notations} summarizes the notations used in the models.

\begin{table}[htbp]
\centering
\caption{Description of notations.}
\label{tab:notations}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|}
\hline
Notations & Description                                             & Notations & Description                                   \\ \hline
$I$       & the set of customers, index by $i$;                     & $m$       & \# of facilities allocated for each customer; \\
$J$      & the set of candidate location sites. index by $j$;   & $p$   & the facility failure probability; \\
$NF$      & the set of candidate location sites that will not fail; & $f_i$     & the fix cost of $j$;                          \\
$F$       & the set of candidate location sites that may fail;      & $\alpha$  & weighted parameter;                           \\
$c_{ij}$ & the cost of per unit demand shipped from $j$ to $i$; & $h_i$ & the demands of customer $i$;       \\ \hline
\end{tabular}%
}
\end{table}

\noindent Besides, there are two sets of decision variables: location decision variables ($\mathbf{X}$) and allocation decision variables ($\mathbf{Y}$):
\begin{equation}
    \label{variable:X}
    X_j=\left\{\begin{array}{lc}1,&\text{if candidate location site}\;j\;\text{is selected};\\0,&\text{otherwise}.\end{array}\right.
\end{equation}
\begin{equation}
    \label{variable:Y}
    Y_{ijr}=\left\{\begin{array}{lc}1,&\text{if } j \text{ is allocated as  the level-} r \text{ facility to serve } i;\\0,&\text{otherwise}.\end{array}\right.
\end{equation}

\noindent In Eq. (\ref{variable:Y}), the "level-r" facility $j$ for customer $i$ means the facility $j$ will provide service only when the front $r$ allocated facilities (from level-0 to level-(r-1)) fail.

A classical RFLP model in \cite{ref9} is as follows.

\begin{equation}
    \label{objfun2-1}
    \operatorname{Min} \alpha w_{1}+(1-\alpha) w_{2}
\end{equation}
Subject to:
\begin{equation}
    \label{const2-2}
    w_{1}=\sum_{j \in J} f_{j} X_{j}+\sum_{i \in I} \sum_{j \in J} h_{i} c_{i j} Y_{i j 0}
\end{equation}
\begin{equation}
    \label{const2-3}
    w_{2}=\sum_{i \in I} h_{i}\left[\sum_{j \in N F} \sum_{r=0}^{m-1} c_{i j} p^{r} Y_{i j r}+\sum_{j \in F} \sum_{r=0}^{m-1} c_{i j} p^{r}(1-p) Y_{i j r}\right]
\end{equation}
\begin{equation}
    \label{const2-4}
    \sum_{j \in J} Y_{i j r}+\sum_{j \in N F} \sum_{t=0}^{r-1} Y_{i j t}=1 \quad \forall i \in I, r=0, \ldots, m-1
\end{equation}
\begin{equation}
    \label{const2-5}
    Y_{i j r} \leq X_{j} \quad \forall i \in I, j \in J, r=0, \ldots, m-1
\end{equation}
\begin{equation}
    \label{const2-6}
    \sum_{r=0}^{m-1} Y_{i j r} \leq 1 \quad \forall i \in I, \forall j \in J
\end{equation}
\begin{equation}
    \label{const2-7}
    m=|J|
\end{equation}
\begin{equation}
    \label{const2-8}
    X_{u}=1
\end{equation}
\begin{equation}
    \label{const2-9}
    X_{j} \in\{0,1\} \quad \forall j \in J
\end{equation}
\begin{equation}
    \label{const2-10}
    Y_{ijr} \in\{0,1\} \quad \forall i \in I; \forall j \in J; r=0, \ldots, m-1
\end{equation}

\noindent In this model, there are two objectives in the objective function, $w_1$ is the operating cost and $w_2$ is the expected failure cost. The objective of the model is to minimize the weighted sum of the two objectives.
Besides, there is an emergency facility $u$ which will always be selected and not fail, and all customers can get service from it.

Several shortcomings are observed in the literature:

(1) The number of facilities allocated to each customer (i.e., $m$ in Eq. (\ref{const2-7})) is fixed in models of most literature, e.g., $m = 2$ (i.e., $Y_{ij0}$ and $Y_{ij1}$) in \cite{ref8} and $m = |J|$ in \cite{ref9}. One issue of this allocation setting is the determination of an appropriate value of $m$. If $m$ is bigger than the number of selected candidate location sites, i.e., $\sum\nolimits_{j\in J}X_j$, it is not in line with the actual situation because we cannot allocate
nonexistent facilities to customers. If we set the value of $m$ smaller than $\sum\nolimits_{j\in J}X_j$, the value of $\sum\nolimits_{j\in J}X_j$ is changed during the exploration in solution space, therefore it is hard for us to set a suitable $m$ value. If we set $m = 2$ directly, which means allocate just one primary facility and one backup facility to serve each customer, the reliability is a bit weak intuitively.

(2) To our best knowledge, there is a lack of research on the large-scale problem. The largest problem instance in the related research is 150-node and the optimization solver such as CPLEX can find near-optimal or even optimal solutions for the problem.

(3) There is a lack of research on the algorithm which can solve the large-scale problems efficiently as well.

Correspondingly, this paper:

(1) constructs a new formulation in which a non-fixed allocation setting, i.e., $m=\sum_{j \in J}X_j$, is used;

(2) proposes a hybrid evolutionary algorithm EAMLS which combines a local search method with an evolutionary algorithm and performs well on both small-scale and large-scale problems;

(3) performs experimental studies on large-scale problems whose scale is much larger than any related literature;

(4) proposes a convergence metric l3-value to help observe the evolutionary process, adjust parameters and further improve the algorithm.

\section{Problem Formulation}

We propose a new RFLP formulation in which we set the number of allocated facilities to each customer as an variable instead of a fixed constant.

The mathematical formulation of our model is as follows, formulated based on \cite{ref8,ref9}. The decision variables are defined by Eqs. (\ref{variable:X}) and (\ref{variable:Y}).

\begin{equation}
    \label{objectiveFunction:3-1}
    \operatorname{Min} \sum_{j \in J} f_{j} X_{j}+\alpha \sum_{i \in I} \sum_{j \in J} \sum_{r=0}^{m-1} h_{i} c_{i j} p^{r}(1-p) Y_{i j r}
\end{equation}
Subject to:
\begin{equation}
    \label{constraint3-5}
    m=\sum_{j \in J} X_{j}
\end{equation}
\begin{equation}
    \label{constraint3-4}
    m \geq 2
\end{equation}
\begin{equation}
    \label{constraint:3-2}
    \sum_{j \in J} Y_{i j r}=1 \quad \forall i \in I ; r=0, \ldots, m-1
\end{equation}
\begin{equation}
    \label{constraint3-3}
    \sum_{r=0}^{m-1} Y_{i j r} \leq X_{j} \quad \forall i \in I, \forall j \in J
\end{equation}
\begin{equation}
    \label{constraint3-6}
    X_{j} \in\{0,1\} \quad \forall j \in J
\end{equation}
\begin{equation}
    \label{constraint3-7}
    Y_{ijr} \in\{0,1\} \quad \forall i \in I; \forall j \in J; r=0, \ldots, m-1
\end{equation}

\noindent The objective function of the model is to minimize the total cost associate with facilities construction (i.e., the term $\sum_{j \in J}f_jX_j$) and transportation between the facilities and customers (i.e., the term $\sum_{i \in I}\sum_{j \in J}\sum_{r=0}^{m-1}h_ic_{ij}p^r(1-p)Y_{ijr}$).

Constraint (\ref{constraint3-5}) makes the number of facilities allocated to each customer (i.e., $m$) a variable and its value is related to location decision variables (i.e., $\mathbf{X}$). Constraint (\ref{constraint3-4}) represents at lease two facilities are constructed to ensure reliability. Constraint (\ref{constraint:3-2}) assures only one facility can be the level-$r$ supplier of customer $i$. Constraint (\ref{constraint3-3}) means candidate location site $j$ can be allocated to customer as a supplier only when it is selected.  Constraint (\ref{constraint3-6}) and (\ref{constraint3-7}) are standard integrality constraints.

Compared with classical models shown in Section 2, the significant difference in our model is the new non-fixed facility allocation setting, i.e., constraint (\ref{constraint3-5}). In our model, the value of $m$ is not fixed but varies with decision variables $\mathbf{X}$, therefore it is more realistic, ensures reliability, but makes our model much more complex and difficult to solve by traditional methods as well.

\section{A Hybrid Evolutionary Algorithm: EAMLS}

This paper develops a new hybrid evolutionary algorithm EAMLS (Evolutionary Algorithm with Memorable Local Search) which combines a memorable local search method and an EA, and a convergence metric l3-value is proposed. In this section, the structure of EAMLS is explained first, then the design of operators of the Genetic Algorithm (GA) and EAMLS is introduced. Finally, the details of l3-value are described.

\subsection{EAMLS}

Algorithm \ref{algorithm1} is the pseudo-code of EAMLS. Compared with the GA, the main characters of EAMLS contain: (1) no crossover operation; (2) population size self-adaptation; (3) the combination of a memorable local search (MLS) and EA; and (4) the adoption of convergence metric l3-value.

In Algorithm \ref{algorithm1}, variable $allNeighborInds$ stores all non-repeating neighborhood individuals generated by MLS before current generation and is updated at the end of every generation (Algorithm 1, Line 2 and Line 13). In the evolutionary process, a new population is generated from the current population after mutation, MLS, and survival selection (Algorithm 1, Lines 5-8), and convergence metric l3-value is calculated (Algorithm 1, Line 9). If l3-value is bigger than a pre-set threshold $\beta$, population size is increased by a pre-set step size $p$ (Algorithm 1, Lines 10-12). The description of the l3-value will be shown in Section 4.3.

\begin{algorithm}[htbp]
  \caption{Evolutionary Algorithm with Memorable Local Search.}
  \label{algorithm1}
  \begin{algorithmic}[1]
    \Require
      $G$: number of generations;
      $\mu$: population size;
      $l$: individual length;
      $m$: mutation rate;
      $\beta$: threshold of l3-value;
      $p$: step size of population self-adaptation;
    \Ensure
      $bestSol$: the best individual in the final population;
    \State $initPop \leftarrow initializePop(\mu,l);$
    \State $allNeighborInds \leftarrow$ an empty set;
    \State $pop \leftarrow evaluatePop(initPop);$
    \For{$g=1 \ to\ G$}
        \State $popAfterMutation \leftarrow mutation(pop,m);$
        \State $offspring \leftarrow evaluatePop(popAfterMuation);$
        \State $offspring_{LS} \leftarrow memorableLocalSearch(pop,offspring);$
        \State $pop \leftarrow survival(pop,offspring,offspring_{LS},\mu);$
        \State l3-value $\leftarrow getl3Value(pop,allNeighborInds);$
        \If{l3-value$ > \beta$}
            \State $\mu \leftarrow \mu+p;$
        \EndIf
        \State add $ offspring_{LS}$ to $allNeighborInds;$
    \EndFor
    \State $bestSol \leftarrow selectBestIndividual(pop)$
    \RETURN{} $bestSol$
  \end{algorithmic}
\end{algorithm}

\begin{algorithm}[htbp]
  \caption{Memorable Local Search.}
  \label{algorithm2}
  \begin{algorithmic}[1]
    \Require
      $pop$: the parent population;
      $offspring$: the child population generated after mutation;
      $n$: \# of individuals which need to check whether to do local search;
      $indLSed$:the set of individuals which have already down local search before this generation;
    \Ensure
      $offspring_{LS}$: the population generated by local search;
    \State $offspring_{LS}\leftarrow$ an empty set;
    \State $parentPop \leftarrow$ combine $pop$ and $offspring$;
    \State $sortedParentPop\leftarrow$ sort $parentPop$ by fitness increasing order;
    \State $i \leftarrow 0;$
    \For{$j \leftarrow 1$ to $len(sortedParentPop)$}
        \If{$sortedParentPop[j]$ not in $indLSed$}
            \State $neighborInds \leftarrow generateNeighbor(sortedParentPop[j]);$
            \State add $neighborInds$ to $offspring_{LS};$
            \State $i \leftarrow i+1;$
            \If{$i > n$}
                \State break;
            \EndIf
        \EndIf
    \EndFor
    \RETURN $offspring_{LS}$
  \end{algorithmic}
\end{algorithm}

Algorithm \ref{algorithm2} is the pseudo-code of the memorable local search (MLS). First, we will introduce the definition of the neighborhood. The neighborhood of an individual is the set of individuals whose Hamming distance is 1 from that individual. In MLS, sort $(\mu+\lambda)$ population (variable $sortedParentPop$ in Algorithm 2) in decreasing order, i.e., good individuals are in the front. Then check individuals one by one in sorted $(\mu+\lambda)$ population whether it has been local-searched before this generation, and do local-search for those have not been local-searched (Lines 5-7 in Algorithm 2.
It looks like that the algorithm remembers all local-searched individuals and that's why we name it
Memorable Local Search). Exit the loop until the number of new individuals which have been local-searched in this generation reaches $n$ (Lines 9-12 in Algorithm 2).

\subsection{Operator Design of GA and EAMLS}

In Section 5, we use a GA for comparison. Here some operators' design for GA and EAMLS is as follows \footnote{If there is no special statement, that operator is adopted in both GA and EAMLS.}:

\textbf{Representation} This paper uses binary representation. Every bit represents a location decision variable $X_j, j \in J$.

\textbf{Population Initialization} Stochastic initialization is used in GA and EAMLS. Every gene of an individual takes 0 or 1 with equal probability.

\textbf{Fitness Function} In general, the bigger the fitness value is, the better the individual will be. Therefore, the reciprocal of the objective value of the individual is used as the fitness function.

\textbf{Selection Operator} In GA, roulette wheel selection is used to select parents to do crossover operation.

\textbf{Crossover Operator} In GA, a one-point crossover operator is used. For two parent individuals selected by the selection operator, do crossover operation according to a pre-set crossover rate.

\textbf{Mutation Operator} The bit-flipping mutation is used in GA and EAMLS. During mutation, every gene/bit of one individual mutates with a pre-set mutation rate.

\textbf{Survival Selection Strategy} We adopt $(\mu+\lambda)$ strategy to select next generation population from $(\mu+\lambda)$ population, i.e., the mixed population of the current generation population and the offspring.

\textbf{Repair Strategy} Repair strategy is working when there are individuals which do not satisfy the constraint (\ref{constraint3-4}). For an individual needed repair, check every gene in ascending order of fixed cost and change the gene with 0-value to 1 until the individual satisfies the constraint (\ref{constraint3-4}).

\textbf{How to determine $\mathbf{Y}$} For one customer, the selected candidate locations (i.e., locations whose $X_j = 1$) are allocated to it in ascending order of distance, which has been proved the optimal allocation pattern under a certain solution $\mathbf{X}$ \cite{ref9} and can satisfy the constraints (12), (13), and (15).

\subsection{Convergence Metric l3-value}

In order to observe the evolutionary process, a convergence metric l3-value is proposed.

Algorithm \ref{algorithm3} is the pseudo-code of the calculation method of l3-value. The new population generated after survival selection is checked, and the number of individuals which also belong to the set $allNeighborInds$ is counted (Lines 2-6 in Algorithm 3). Then we calculate the proportion of these individuals in the population as l3-value (Line 7 in Algorithm \ref{algorithm3}).
l3-value can be used to measure the convergence during the evolutionary process. The bigger the l3-value is, the stronger the evolution converges.

\begin{algorithm}[htbp]
  \caption{Function $getl3Value()$.}
  \label{algorithm3}
  \begin{algorithmic}[1]
    \Require
    $pop$: the new population after survival selection;
    $allNeighborInds$: the set of all individuals generated by memorable local search before this generation;
    \Ensure
     l3-value;
    \State $num\leftarrow 0;$
    \For{$ind \in pop$}
        \If{$ind \in allNeighborInds$}
            \State $num \leftarrow num+1;$
        \EndIf
    \EndFor
    \State l3-value $\leftarrow num/len(pop);$
    \State{\textbf{Return} $l3$-$value$}
  \end{algorithmic}
\end{algorithm}

\section{Computational Studies}

Because this paper proposes a new problem, and there are not any algorithms like EAMLS can be used to compare directly, we compare EAMLS with a GA and CPLEX (a commercial optimization solver of IBM) on two models: m=2 and m=${\textstyle\sum_{j\in J}}X_j$ models. The difference between the two models is the allocation setting. In the m=2 model, the number of facilities allocated to each customer, i.e. $m$, is fixed to 2, which is adopted in much literature. The m=${\textstyle\sum_{j\in J}}X_j$ model is proposed by us in this paper and $m$ varies with decision variables $\mathbf{X}$ during the search process. Section 5.1 shows the experimental design, including instances generation, parameters setting, and experimental environment. The experiments and results of the m=2 and m=${\textstyle\sum_{j\in J}}X_j$ models are presented in Sections 5.2. Analyses and discussions are given in Section 5.3.

\subsection{Experimental Design}

\textbf{Instance Generation} This paper generates problem instances uniformly at random on different scales. The parameters used to generate instances are shown in Table \ref{tab:5-1tab2}. There are eight 10-node instances, eight 50-node instances, eight 100-node instances, and four 600-node instances.

\begin{table}[]
\centering
\caption{Parameters used in instances generation}
\label{tab:5-1tab2}
\begin{tabular}{|c|c|}
\hline
Parameters                    & Ranges               \\ \hline
Candidate location coordinate & {[}0,1{]}            \\
Customer demands              & \{0,1,...,1000\}     \\
Fixed cost of facility        & \{500,501,...,1500\} \\
Facility failure probability  & 0.05                 \\ \hline
\end{tabular}
\end{table}

\noindent\textbf{Parameter Setting of Algorithms} Some parameters’ values of GA and EAMLS are shown in Table \ref{tab:5-2tab3}. Table \ref{tab:5-3tab4} presents the generation number and population size of GA and EAMLS, which associate with the scale of problem instances.
The values of parameters in Tables \ref{tab:5-2tab3} and \ref{tab:5-3tab4} are chosen arbitrarily on the basis of meeting the following conditions: (1) EAMLS converges at the end of evolution; (2) the number of fitness evaluations (FEs) of GA is not lower than EAMLS. Besides, the default parameters of CPLEX are used.

\begin{table}[]
\centering
\caption{Some parameters of GA and EAMLS}
\label{tab:5-2tab3}
\begin{tabular}{|c|c|}
\hline
Parameters                                & Value \\ \hline
Crossover rate for GA, $c$                        & 0.9   \\
Mutation rate, $m$                         & 0.1   \\
\# Local search individual, $n$            & 10    \\
l3-value threshold, $\beta$                & 0.8   \\
Step size of population self-adaption, $p$ & 100   \\ \hline
\end{tabular}
\end{table}

\begin{table}[]
\centering
\caption{Parameters associate with instance size}
\label{tab:5-3tab4}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Instance scale\\ (\# nodes)\end{tabular}} & \multicolumn{2}{c|}{GA}         & \multicolumn{2}{c|}{EAMLS}      \\ \cline{2-5}
                                                                                & \# Generation & Population size & \# Generation & Population size \\ \hline
10  & 60   & 30  & 10  & 20  \\
50  & 200  & 200 & 20  & 20  \\
100 & 400  & 200 & 50  & 100 \\
600 & 4600 & 200 & 250 & 200 \\ \hline
\end{tabular}
\end{table}

\textbf{Experimental Environment} The algorithms are implemented in Python 3.7 and run on Dell R370 server which has 2x Intel(R) Xeon(R) CPU E5-2650 v4 @ 2.20GHz CPU, 128G RAM, and CentOS 7.6 operating system.

\textbf{Statistical Test} We use the Wilcoxon sign rank test to determine whether the results between EAMLS and other methods have statistically significant differences. The Wilcoxon sign rank test is a non-parameter test which is suitable for two related or matched samples and compares data in pair, hence it is suitable to use here.

\subsection{Experiments on the m=2 and m=${\textstyle\sum_{j\in J}}X_j$ Models}

For the m=2 model, We compare EAMLS with the GA and CPLEX on small-scale (10-node), mid-scale (100-node), and large-scale (600-node) instances. There are 30 runs on small and mid-scale instances and 10 runs on large-scale instances because of time. The computational results are shown in Table \ref{tab:mIs2_10-100-600-nodetab5}.

\begin{table}[]
\centering
\caption{Computational results on m=2 model 10 (30 runs),100 (30 runs), and 600 (10 runs)-node instances. AOV is Average Objective Value. OR is the Optimal Rate and calculated by (\# runs which finding the optimal solution)/(\# all runs). Gap is calculated by (AOV(other method)-AOV(EAMLS))/AOV(EAMLS). When Gap is positive, the performance of other methods is worse than EAMLS, otherwise better. The symbol “*” in AOV represents the results between EAMLS and that method have statistically significant differences. The symbol “-” represents CPLEX cannot solve the instance or the optimal solution is unknown so no results can be given.}
\label{tab:mIs2_10-100-600-nodetab5}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Instance\\ No.\end{tabular}} & \multicolumn{4}{c|}{GA}                                 & \multicolumn{4}{c|}{CPLEX}                           & \multicolumn{4}{c|}{EAMLS}                               \\ \cline{2-13}
                                                                        & AOV              & Gap (\%) & OR            & Time      & AOV               & Gap (\%) & OR   & Time           & AOV               & Gap (\%) & OR   & Time               \\ \hline
10-1                                                                    & \textbf{2463.19} & 0.00     & \textbf{1.00} & 1.52      & \textbf{2463.19}  & 0.00     & 1.00 & \textbf{0.46}  & \textbf{2463.19}  & 0.00     & 1.00 & 6.14               \\
10-2                                                                    & \textbf{2874.03} & 0.00     & \textbf{1.00} & 1.51      & \textbf{2874.03}  & 0.00     & 1.00 & \textbf{0.41}  & \textbf{2874.03}  & 0.00     & 1.00 & 5.46               \\
10-3                                                                    & \textbf{2623.35} & 0.00     & \textbf{1.00} & 1.74      & \textbf{2623.35}  & 0.00     & 1.00 & \textbf{0.66}  & \textbf{2623.35}  & 0.00     & 1.00 & 5.41               \\
10-4                                                                    & \textbf{2323.92} & 0.00     & \textbf{1.00} & 1.93      & \textbf{2323.92}  & 0.00     & 1.00 & \textbf{0.48}  & \textbf{2323.92}  & 0.00     & 1.00 & 5.86               \\
10-5                                                                    & \textbf{2917.87} & 0.00     & \textbf{1.00} & 2.48      & \textbf{2917.87}  & 0.00     & 1.00 & \textbf{0.50}  & \textbf{2917.87}  & 0.00     & 1.00 & 5.71               \\
10-6                                                                    & \textbf{3149.31} & 0.00     & \textbf{1.00} & 2.72      & \textbf{3149.31}  & 0.00     & 1.00 & \textbf{0.41}  & \textbf{3149.31}  & 0.00     & 1.00 & 5.59               \\
10-7                                                                    & \textbf{3324.98} & 0.00     & \textbf{1.00} & 2.39      & \textbf{3324.98}  & 0.00     & 1.00 & \textbf{0.58}  & \textbf{3324.98}  & 0.00     & 1.00 & 5.64               \\
10-8                                                                    & \textbf{3165.87} & 0.00     & \textbf{1.00} & 2.10      & \textbf{3165.87}  & 0.00     & 1.00 & \textbf{0.52}  & \textbf{3165.87}  & 0.00     & 1.00 & 4.58               \\
100-1                                                                   & 13029.83*        & 22.28    & 0.00          & 2374.86   & \textbf{10645.89} & -0.10    & 1.00 & \textbf{14.30} & 10656.11          & 0.00     & 0.87 & 1431.17            \\
100-2                                                                   & 13166.44*        & 20.95    & 0.00          & 2375.71   & \textbf{10885.43} & 0.00     & 1.00 & \textbf{14.31} & \textbf{10885.43} & 0.00     & 1.00 & 1387.01            \\
100-3                                                                   & 12982.37*        & 16.90    & 0.00          & 2396.42   & \textbf{11105.21} & 0.00     & 1.00 & \textbf{14.76} & 11105.39          & 0.00     & 0.93 & 1514.12            \\
100-4                                                                   & 13379.41*        & 16.66    & 0.00          & 2388.67   & \textbf{11468.64} & 0.00     & 1.00 & \textbf{14.42} & \textbf{11468.64} & 0.00     & 1.00 & 1382.91            \\
100-5                                                                   & 14563.46*        & 16.39    & 0.00          & 2398.34   & \textbf{12505.51} & -0.05    & 1.00 & \textbf{14.80} & 12512.29          & 0.00     & 0.90 & 1415.63            \\
100-6                                                                   & 13189.74*        & 17.29    & 0.00          & 2402.44   & \textbf{11245.55} & 0.00     & 1.00 & \textbf{14.00} & \textbf{11245.55} & 0.00     & 1.00 & 1447.11            \\
100-7                                                                   & 12841.37*        & 16.11    & 0.00          & 1696.85   & \textbf{11043.70} & -0.15    & 1.00 & \textbf{15.49} & 11059.89          & 0.00     & 0.90 & 1326.41            \\
100-8                                                                   & 13886.78*        & 18.30    & 0.00          & 1242.25   & \textbf{11732.46} & -0.05    & 1.00 & \textbf{14.94} & 11738.83          & 0.00     & 0.87 & 1180.91            \\
600-1                                                                   & 144896.91*       & 281.04   & -             & 655420.67 & -                 & -        & -    & -              & \textbf{38026.65} & 0.00     & -    & \textbf{564432.00} \\
600-2                                                                   & 145508.23*       & 293.12   & -             & 656832.50 & -                 & -        & -    & -              & \textbf{37013.71} & 0.00     & -    & \textbf{572568.34} \\
600-3                                                                   & 141486.28*       & 283.41   & -             & 654632.01 & -                 & -        & -    & -              & \textbf{36902.36} & 0.00     & -    & \textbf{568824.96} \\
600-4                                                                   & 141256.35*       & 282.80   & -             & 656656.21 & -                 & -        & -    & -              & \textbf{36900.52} & 0.00     & -    & \textbf{568546.96} \\ \hline
\end{tabular}%
}
\end{table}

For the m=${\textstyle\sum_{j\in J}}X_j$ model, we compare EAMLS with the GA and CPLEX on 50 and 100-node instances, and there are 30 runs on each instance. Table \ref{tab:mIsAll_50100-node} is the computational results.

\begin{table}[]
\centering
\caption{Computational results on m=$\sum_{j \in J}X_j$ model 50 and 100-node instances, 30 runs. AOV is Average Objective Value. Gap is calculated by ((AOV(other method)-AOV(EAMLS))/AOV(EAMLS). When Gap is positive, the performance of other methods is worse than EAMLS, otherwise better. The symbol “*” in AOV represents the results between EAMLS and that method have statistically significant differences.}
\label{tab:mIsAll_50100-node}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Instance\\ No.\end{tabular}} & \multicolumn{3}{c|}{GA}        & \multicolumn{3}{c|}{CPLEX}       & \multicolumn{3}{c|}{EAMLS}                      \\ \cline{2-10}
                                                                        & AOV       & Gap (\%) & Time    & AOV        & Gap (\%) & Time     & AOV               & Gap (\%) & Time             \\ \hline
50-1                                                                    & 7053.71*  & 0.68     & 719.76  & 12589.41*  & 79.69    & 4715.56  & \textbf{7006.23}  & 0.00     & \textbf{91.52}   \\
50-2                                                                    & 7154.93   & -0.13    & 720.49  & 15734.80*  & 119.63   & 4488.26  & \textbf{7164.20}  & 0.00     & \textbf{90.75}   \\
50-3                                                                    & 6890.54*  & 0.75     & 713.25  & 12656.13*  & 85.06    & 5219.33  & \textbf{6838.95}  & 0.00     & \textbf{91.10}   \\
50-4                                                                    & 7166.63   & 0.04     & 698.45  & 12147.92*  & 69.58    & 4702.01  & \textbf{7163.42}  & 0.00     & \textbf{90.04}   \\
50-5                                                                    & 6929.29   & 0.03     & 714.86  & 11946.35*  & 72.46    & 5281.27  & \textbf{6926.95}  & 0.00     & \textbf{87.42}   \\
50-6                                                                    & 6575.09   & 0.29     & 696.80  & 13284.69*  & 102.64   & 4836.45  & \textbf{6555.87}  & 0.00     & \textbf{90.59}   \\
50-7                                                                    & 7162.83   & 0.07     & 685.04  & 12441.00*  & 73.81    & 4495.41  & \textbf{7157.76}  & 0.00     & \textbf{81.19}   \\
50-8                                                                    & 7175.89*  & 0.26     & 629.19  & 14433.41*  & 101.67   & 4522.35  & \textbf{7156.99}  & 0.00     & \textbf{70.07}   \\
100-1                                                                   & 12895.10* & 20.47    & 3976.97 & 113781.45* & 963.00   & 19451.62 & \textbf{10703.78} & 0.00     & \textbf{2266.50} \\
100-2                                                                   & 13093.80* & 19.77    & 3820.11 & 110441.89* & 910.18   & 17159.57 & \textbf{10932.89} & 0.00     & \textbf{2168.54} \\
100-3                                                                   & 13082.38* & 17.21    & 2719.68 & 114576.21* & 926.52   & 35836.91 & \textbf{11161.59} & 0.00     & \textbf{2337.35} \\
100-4                                                                   & 13484.69* & 17.04    & 2551.55 & 99484.65*  & 763.50   & 35129.74 & \textbf{11521.11} & 0.00     & \textbf{2217.00} \\
100-5                                                                   & 14484.70* & 15.22    & 2579.12 & 111338.15* & 785.68   & 17249.10 & \textbf{12570.86} & 0.00     & \textbf{2279.35} \\
100-6                                                                   & 13360.41* & 18.20    & 2626.97 & 99397.46*  & 779.39   & 19433.87 & \textbf{11302.96} & 0.00     & \textbf{2288.82} \\
100-7                                                                   & 12810.60* & 15.20    & 2553.83 & 105460.22* & 848.32   & 17271.95 & \textbf{11120.78} & 0.00     & \textbf{2036.58} \\
100-8                                                                   & 13809.12* & 17.05    & 2548.32 & 112170.76* & 850.82   & 18667.88 & \textbf{11797.25} & 0.00     & \textbf{1792.08} \\ \hline
\end{tabular}%
}
\end{table}

\subsection{Analyses and Discussions}

We compare GA, CPLEX, and EAMLS on different scale (10, 100, and 600-node) problem instances for m=2 model whose allocation setting is often used in literature, and the experimental results are shown in Table \ref{tab:mIs2_10-100-600-nodetab5}. Experimental results on 50 and 100-node instances of the new complicated m=${\textstyle\sum_{j\in J}}X_j$ model are presented in Table \ref{tab:mIsAll_50100-node}.

For m=2 model, from Table \ref{tab:mIs2_10-100-600-nodetab5}, we can see that CPLEX performs the best on both solution quality and time for small and mid-scale (10 and 100-node) instances. EAMLS can find solutions as good as CPLEX but need more time. Although CPLEX can solve small and mid-scale instances fast, it needs more RAM space as the problem scale increases. For large-scale problem (600-node) instances, EAMLS can find better solutions in less time compared with GA, while the CPLEX cannot find a solution.

The new m=${\textstyle\sum_{j\in J}}X_j$ model is more complicated to solve, especially for CPLEX. Table \ref{tab:mIsAll_50100-node} demonstrates that the performance of EAMLS is better than GA and CPLEX on both solution quality and time.

According to the observation of computational results, we can get three features of EAMLS: (1) For small- and mid-scale problems, the solutions found by EAMLS are comparable to those found by other methods; (2) For large-scale problems, EAMLS significantly outperforms other methods; (3) EAMLS especially performs well on a) the new complicated model and b) large-scale problems.
So why is EAMLS effective? Through combining MLS with EA and using l3-value to guide the population size to grow gradually, EAMLS performs a full local search while performing a global search, maintains good population diversity, as well as speeds up the convergence.

Our algorithm EAMLS performes well on large-scale problem instances of both m=2 and m=${\textstyle\sum_{j\in J}}X_j$ models, and its advantage will become more apparent as the problem scale increases. However, the larger the problem, the greater the number of FEs needed for EAMLS to converge.

\section{Conclusion}

This paper proposes a new RFLP formulation in which the number of facilities allocated to each customer (i.e., $m$) is not fixed but varies with decision variables $\mathbf{X}$. This non-fixed allocation setting makes the model more close to scenarios in real life.

A hybrid evolutionary algorithm EAMLS (which can also be viewed as a memetic algorithm) is proposed to solve the model. Combining a memorable local search method and EA, EAMLS performs well on the new complicated model and large-scale problems considered in this paper, and its advantage will become more obvious as the problem scale increases.
Besides, a convergence metric l3-value is proposed to analyze the algorithm's convergence speed and exam the evolutionary process.

Finally, we explore the large-scale problems of the two models. Under what conditions is a problem a large-scale problem? It is related to the model and whether the problem can be solved by the exact algorithm efficiently. For the m=2 model which allocates a fixed number of facilities to each customer as in the existing research, we solve large-scale problem instances (600-node) whose scale is much larger than other literature. For the new complicated $m=\sum_{j \in J}X_j$ model, 100-node instances can be treated as large-scale problems because the exact algorithm or optimization solver cannot solve them effectively. And our algorithm EAMLS has good performance on large-scale problems considered in this paper.

In the future, the model which integrates various factors should be studied, and more complicated FLPs, such as dynamic FLP and FLP under uncertain environments, should be focused. Furthermore, effective meta-heuristic algorithms for large-scale problems should be studied as well.