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\begin{document}

\mainmatter  % start of an individual contribution

\title{Building consistent pairwise comparison matrices over abelian linearly ordered groups}


\titlerunning{Building consistent pairwise comparison matrices}
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\author{Bice Cavallo
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\and Livia D'Apuzzo \and Massimo Squillante}
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\institute{University Federico II, \\ Naples, Italy\\University of Sannio,\\
Benevento, Italy\\
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%\url{http://www.unisannio.it}
}

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\toctitle{Building consistent pairwise comparison matrices}
\tocauthor{Cavallo, D'Apuzzo and Squillante} \maketitle


\begin{abstract}
In the paper,  algorithms are provided  to check the consistency of pairwise comparison matrices  and to build consistent matrices over  abelian linearly ordered groups.
 A measure of  consistency is also given; this measure improves a consistent index provided in a previous paper.

 {\bf Keywords}: Pairwise comparison matrices, consistency index, abelian linearly ordered group
\end{abstract}


\section{Introduction} \label{sec:Introduction}
Let  $X=\{x_{1},x_{2},..., x_{n}\}$ be a set of alternatives or criteria. An useful tool to determine a weighted ranking on X is a \emph{pairwise comparison matrix} (PCM)    \begin{equation}
\label{pcm}
A =\left(\begin{array}{cccc}a_{11} & a_{12} & ... & a_{1n}
 \\a_{21} & a_{22} & ... & a_{2n} \\... & ... & ... & ... \\a_{n1} & a_{n2} & ... &
 a_{nn}\end{array}\right)\
 \end{equation}
  which entry $a_{ij}$ expresses how much the alternative $x_{i}$ is preferred to alternative $x_{j}$.  A condition of \emph{reciprocity}
is assumed for the matrix $A=(a_{ij})$ in such way
 that the preference
   of $x_{i} $ over $x_{j}$ expressed by $a_{ij}$ can be exactly  read by means of the element $a_{ji}$.

  Under a   suitable condition of \emph{consistency} for $A=(a_{ij})$,
    $X$ is totally ordered and the values $a_{ij}$ can be expressed by means of the components
    $w_{i}$ and $w_{j}$ of a suitable vector \underline{$w$}, that is called \emph{consistent vector}
     for the matrix $A=(a_{ij})$; then  \underline{$w$}  provides the weights for the elements of $X$.

 The shape of the reciprocity  and consistency conditions depends on the different
meaning given to the number $a_{ij}$, as the following well known cases show.
 \begin{description}
   \item[Multiplicative case:]  $a_{ij} \in ]0, + \infty[$  is a preference ratio and the conditions of \emph{multiplicative reciprocity}  and \emph{consistency} are given respectively  by
       \begin{eqnarray*}
       % \nonumber to remove numbering (before each equation)
         a_{ji} &=& \frac{1}{a_{ij}} \quad \forall \; i,j=1, \ldots, n,\\
         a_{ik} &=&a_{ij} a_{jk} \quad \forall \; i,j, k=1, \ldots, n.
       \end{eqnarray*}
    A consistent vector is a positive vector
   $\underline{w}=(w_{1},w_{2}, ..., w_{n})$ verifying the  condition
   $\frac{w_{i}}{w_{j}}=a_{ij}$ and so   perfectly representing the preferences over
   $X$.\\

   \item[Additive case:] $a_{ij} \in ]-\infty,+\infty[$ is a preference difference and  \emph{additive reciprocity}  and \emph{consistency} are
expressed as follows
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
  a_{ji} &=& -a_{ij} \quad \forall \; i,j=1, \ldots, n, \\
  a_{ik} &=& a_{ij}+ a_{jk} \quad \forall \; i,j, k=1, \ldots, n.
\end{eqnarray*}
A  consistent vector  is a vector  $\underline{w}=(w_{1},w_{2}, ..., w_{n})$
 verifying the condition $w_{i}-w_{j}=a_{ij}$.\\

   \item[Fuzzy case:]  $a_{ij} \in [0,1]$   measures the distance from the indifference that is expressed by $0.5$;
    the conditions of \emph{fuzzy reciprocity}  and \emph{fuzzy additive consistency} are
    \begin{eqnarray*}
    % \nonumber to remove numbering (before each equation)
      a_{ji}&=&1-a_{ij} \quad \forall \; i,j=1, \ldots, n, \\
      a_{ik}&=&a_{ij}+ a_{jk}- 0.5 \quad \forall \; i,j, k=1,\ldots,n. \\
       \end{eqnarray*}  A  consistent vector  is a vector    $\underline{w}=(w_{1},w_{2}, ..., w_{n})$ verifying the
condition $w_{i}-w_{j}=a_{ij}-0.5$.
 \end{description}
The multiplicative \emph{PCMs} play a basic role in the Analytic
Hierarchy Process, a procedure developed by T.L. Saaty at the end of
the 70s \cite{Saaty77},
%\cite {Saaty78},
\cite{Saaty80}, \cite{Saaty86}.

%Saaty indicates a scale translating the
%comparisons expressed in verbal terms into the preference ratios
%$a_{ij}$. By applying this scale, $a_{ij}$ may only  take value in
% $S^{*}=\{1, 2, 3, 4, 5, 6, 7, 8, 9,$ $ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6},\frac{1}{7}, \frac{1}{8},
% \frac{1}{9} \}$.
   In \cite{BasileDApuzzo97}, \cite{BasileDApuzzo02}, \cite{BasileDApuzzo06}, \cite{BasileDApuzzo06int}, and \cite{DapuzzoMarcarelliSquillante}, properties of multiplicative PCMs are analyzed in order to determine a qualitative  ranking on the set of the alternatives and to find  vectors representing this ranking. Additive and fuzzy matrices are investigated  for instance by   \cite{Barzilai} and \cite{ViedmaHerreraChiclanaLuque}.

In the case of a  multiplicative matrix, Saaty suggests that the
comparisons expressed in verbal terms have to be translated into  preference ratios
$a_{ij}$  taking value in
 $S^{*}=\{1, 2, 3, 4, 5, 6, 7, 8, 9,$ $ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6},\frac{1}{7}, \frac{1}{8},
 \frac{1}{9} \}$.  The assumption of the Saaty scale restricts the Decision Maker's possibility to be consistent:
 indeed, if the Decision Maker (DM) expresses the following preference ratios $a_{ij}=5$ and $a_{jk}= 3$, then he will not be consistent
 because $a_{ij}a_{jk}= 15>9$. Analogously for the fuzzy case,  the assumption that  $a_{ij}\in [0,1]$, restricts the  possibility to respect the fuzzy  consistency: indeed, if  the
 DM  claims $a_{ij}=0.9$ and $a_{jk}= 0.8$, then he will not be consistent  because  $a_{ij}+a_{jk}-0.5= 1.7-0.5>1$.

In order to unify the several approaches to PCMs and  remove the
above drawbacks, in \cite{CavalloDapuzzo} the authors introduce
PCMs whose entries belong to  an  abelian linearly ordered group
(\emph{alo-group}) $\mathcal{G}= (G, \odot, \leq)$.  In this way the
reciprocity and consistency conditions
    are  expressed  in terms of the group operation
      $\odot$ and the drawbacks related to the consistency condition are removed; in fact the consistency condition is expressed by $a_{ik}= a_{ij}\odot a_{jk}$, thus, for each choice of $a_{ij}\in G$ and $a_{jk} \in G$, the result  $a_{ij}\odot a_{jk}$ is an element of $G$.
     As
a non trivial alo-group $\mathcal{G}= (G, \odot, \leq)$ has neither
the greatest element nor the least element (see \cite{CavalloDapuzzo}),
 the Saaty set S* and   the interval $[0, 1]$,  embodied with the usual order $\leq$ on $R$, can be not structured as
alo-groups.

         In reference \cite{CavalloDapuzzo}, the assumption   of $divisibility$ for
          $\mathcal{G}$ (see Section \ref{sec:Preliminaries}) allows  us
      to introduce the notion of mean $m_{\odot}(a_{1},...,a_{n})$ of $n$
      elements and  associate
     a \emph{mean vector} $\underline{w}_{m_{\odot}}$ to  a PCM.
      In order to give  a measure of consistency for  a PCM over an alo-group $\mathcal{G}$, in  \cite{CavalloDapuzzo} a notion of distance  $d_{\mathcal{G}}$, linked to $\mathcal{G}$,
        is introduced. Then,  the consistency index $I_{\mathcal{G}}(A)$ is defined as  mean of the distances $d_{\mathcal{G}}(a_{ik},a_{ij} \odot a_{jk})$, with $i<j<k$.


In this paper, we analyze the property  of consistency  for a PCM  in order to:
\begin{enumerate}
  \item provide an algorithm to check whenever or not a matrix is consistent;
  \item provide an algorithm to build a consistent matrix by means of $n-1$ comparisons;
   \item provide a new consistency index  linked to the index  $I_{\mathcal{G}}(A)$ introduced in \cite{CavalloDapuzzo} but  easier to compute.
\end{enumerate}



\section{Preliminaries on alo-groups} \label{sec:Preliminaries}
Let us recall some notions and results related to an alo-group (see \cite{CavalloDapuzzo} for details).  These results will be useful in the sequel to build a consistent PCM  over an alo-group and to define in this context a new consistency index.


 Let  $G $ be a non empty set  provided with   a total  weak order $\leq$ and a binary operation $\odot: G\times G \rightarrow G$.
$\mathcal{G}= (G, \odot, \leq)$ is called   \emph{alo-group}, if and only if
 $(G, \odot)$ is an abelian  group and the the following implication holds:
\begin{displaymath}
a\leq b\Rightarrow  a\odot c \leq b\odot c.
\end{displaymath}
The above implication is equivalent to
$$a< b\Rightarrow  a\odot c < b\odot c,$$
where $<$ is  the strict simple order associated to $\leq$.\\
If $\mathcal{G}= (G, \odot, \leq)$ is an alo-group, then $G$ is
naturally equipped with the order topology induced by $\leq$ and  $G
\times G$  is  equipped with  the related product topology, and
$\mathcal{G}$ is  a \emph{continuous} alo-group if and only if
$\odot$ is continuous.
\\ Let $\mathcal{G}= (G, \odot, \leq)$ be an
alo-group, then we assume that: $e$ denotes the \emph{identity} of
$\mathcal{G}$, $x^{(-1)}$ the \emph{symmetric}  of $x\in G$ with
respect  to $\odot$,
 $\div$ the \emph{ inverse  operation} of $ \odot$  defined by
\lq\lq$a \div b= a \odot b^{(-1)}$\rq\rq. It results $(a \div b)^{(-1)}=b \div a.$\\
Moreover we define the \emph{norm} of $a\in G$ by setting:
\begin{equation} \label{norm}  ||a|| = a\vee a^{(-1)}.
\end{equation}

\begin{Proposition}\cite{CavalloDapuzzo}
Let $\mathcal{G}= (G, \odot, \leq)$  be an alo-group.  Then, the
operation
\begin{equation}
\label{dcirc}
d_{\mathcal{G}}:(a,b)\in G^{2} \rightarrow ||a\div b|| \in G
\end{equation}
verifies the conditions:
 \begin{enumerate}
  \item $d_{\mathcal{G}}(a, b)\geq e$;
  \item $d_{\mathcal{G}}(a, b)= e  \Leftrightarrow a=b$;
%  \item  $d_{\mathcal{G}}(a, b)= e  \Leftrightarrow a=b$;
   \item  $d_{\mathcal{G}}(a, b)=d_{\mathcal{G}}(b, a)$;
  \item  $d_{\mathcal{G}}(a, b) \leq d_{\mathcal{G}}(a, c) \odot d_{\mathcal{G}}(b, c)$.
\end{enumerate}
\end{Proposition}
 \begin{Definition}
The operation $d_{\mathcal{G}}$
in \eqref{dcirc} is a $\mathcal{G}$-\emph{metric} or  $\mathcal{G}$-\emph{distance}.
\end{Definition}






  For a  positive integer $n$,  the ($n$)-\emph{power} $x^{(n)}$
of $x\in G$ is defined as follows
% $x^{(1)}= x$ and, for $ n\geq 2$
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
  x^{(1)} &=& x \\
  x^{(n)} &=&   \bigodot^{n-1}_{i=1} x_{i} \odot x_{n}= \bigodot^{n}_{i=1} x_{i},  \quad\quad x_{i }=x\quad  \; i=1,...,n, \; n\geq  2
\end{eqnarray*}
%\begin{eqnarray*}
%% \nonumber to remove numbering (before each equation)
%  x^{(1)}= x \quad and \quad
%  x^{(n)} =  \bigodot^{n-1}_{i=1} x_{i} \odot x_{n}= \bigodot^{n}_{i=1} x_{i},  \quad\quad x_{i }=x\quad  \; i=1,...,n, \; n\geq  2
%\end{eqnarray*}
 If $b^{(n)}=a$, then we say that $b$ is the ($n$)-\emph{root} of $a$
 and write $b= a^{(1/n)}$.


$\mathcal{G}$ is   \emph{divisible}  if and only if for each
positive integer $n$ and each $a\in G$ there exists the ($n$)-root of
$a$.


%
%\begin{Proposition} \label{nomaxmin}\cite{CavalloDapuzzo}
%A non trivial alo-group $\mathcal{G}= (G, \odot, \leq)$ has neither
%the greatest element nor the least element.
%\end{Proposition}
%So,
% by Proposition \ref{nomaxmin},  the interval $[0, 1]$,  embodied with the usual order $\leq$ on $R$, can be structured as an
%alo-group.


 \begin{Definition}
Let  $\mathcal{G}= (G, \odot, \leq)$ be a divisible  alo-group. Then,
the $\odot$- \emph{mean} $ m_{\odot}( a_{1},a_{2}, ..., a_{n})$ of
the  elements $a_{1},a_{2}, ..., a_{n}$  of $ G$ is  defined by
%the (n)-root of $\bigodot^{n}_{i=1}
%a_{i}$.
%So,
\begin{equation}
\label{mean} m_{\odot}( a_{1},a_{2}, ..., a_{n})=
\begin{cases}
   a_{1} & \;  n=1, \\
      (\bigodot^{n}_{i=1} a_{i})^{(1/n)}& \; n\geq 2.\nonumber
\end{cases}
%m_{\odot}( a_{1},a_{2}, ..., a_{n})=(\bigodot^{n}_{i=1} a_{i})^{(1/n)}
\end{equation}
\end{Definition}


%\paragraph{Isomorphisms between alo-groups}\label{sec:isomorphism}

\begin{Definition}\label{def:Isomorphism}
   An \emph{isomorphism } between two alo-groups $\mathcal{G}= (G,
\odot, \leq)$ and $\mathcal{G'}= (G', \circ, \leq)$ is a bijection $h:G\rightarrow G'$
 that  is   both a lattice isomorphism and a group isomorphism, that is:
\begin{eqnarray*}
\label{isoeq} x<y \Leftrightarrow h(x)< h(y)\\
h(x\odot y) = h(x) \circ h(y).
\end{eqnarray*}
\end{Definition}

  \begin{Proposition}\label{isomean}\cite{CavalloDapuzzo}
Let   $h:G\rightarrow G'$ be an
 isomorphism between the alo-groups $\mathcal{G}= (G, \odot, \leq)$ and $\mathcal{G'}= (G', \circ, \leq)$. Then,
\begin{equation*}
\label{eqisodis} d_{\mathcal{G'}}(a^{\prime}, b^{\prime}) =
h(d_{\mathcal{G}}(h^{-1}(a^{\prime}), h^{-1}(b^{\prime}))).
%\quad
%d_{\mathcal{G}}(a,b) = h^{-1} (d_{\mathcal{G'}}(h(a), h(b))).
\end{equation*}
 Moreover, $\mathcal{G}$
 is divisible if and only if $\mathcal{G'}$ is divisible and,  under the assumption of divisibility:
 \begin{eqnarray*}
 % \nonumber to remove numbering (before each equation)
m_{\circ}( y_{1}, y_{2}, ..., y_{n})= h\big(m_{\odot}(h^{-1}(y_{1}),
h^{-1}(y_{2}), ...,h^{-1}(y_{n})) \big).
 \end{eqnarray*}
%\begin{equation*}
%\label{circcircmean} m_{\circ}( y_{1}, y_{2}, ..., y_{n}) =
%h\big(m_{\odot}(h^{-1}(y_{1}), h^{-1}(y_{2}), ...,h^{-1}(y_{n}))
%\big).  \end{equation*}
\end{Proposition}


\subsection{Real alo-groups} An alo-group $\mathcal{G}= (G, \odot, \leq)$ is a \emph{real} alo-group if and only if
 $G$ is a subset of the real line R and  $\leq$ is the total order  on $G$ inherited from the  usual order on $R$.



 Let $+$ and $\cdot$ be  the usual addition  and multiplication  on R and $\otimes: ]0,1[^{2}\rightarrow ]0,1[$ the operation defined by
 $$x\otimes y=\frac{xy}{xy+(1-x)(1-y)},$$
 then, examples of real divisible and continuous alo-groups  are the following:\\

\begin{description}
 \item [\textbf{Multiplicative alo-group}] $ $\\\textbf{]0,+$\infty$[} = $(]0,+\infty[, \cdot, \leq)$; then
$e=1$, $x^{(-1)}=1/x$,  $x^{(n)}= x^{n}$  and  $x\div y
=\frac{x}{y}$. \\So   $d_{\textbf{]0,+$\infty$[}}(a,b)$=$\frac{a}{b}\vee \frac{b}{a}$ and
$ m_{\cdot}( a_{1}, ..., a_{n})$
 is the geometric mean: $\big(\prod _{i=1}^{n}
 a_{i}\big)^{\frac{1}{n}}$.\\

  \item[\textbf{Additive alo-group}]$ $\\ $ \mathcal{R}= (R, +, \leq)$;  then
   $e=0$,  $x^{(-1)}=-x$, $x^{(n)}= n x$, $x\div y =x-y$. \\So
%   the metric $d_{\mathcal{R}}$ is the usual distance over $R$:
$d_{\mathcal{R}}(a, b)= | a- b |=(a- b)\vee (b- a)$ and $ m_{+}(
a_{1},..., a_{n})$  is the arithmetic mean: $\frac{\sum
_{i}a_{i}}{n}$.\\

     \item [\textbf{Fuzzy alo-group}]$ $\\  \textbf{]0,1[} = $(]0,1[, \otimes, \leq)$;  then
%              \begin{equation}
%x\otimes y=\frac{xy}{xy+(1-x)(1-y)},
%\end{equation}
$e=0.5$, $x^{(-1)}=1-x$,  $x\div y
=\frac{x(1-y)}{x(1-y)+(1-x)y}$. \\So  $d_{\textbf{]0,1[} }(a,b)=
\frac{a(1-b)}{a(1-b)+(1-a)b}   \vee \frac{b(1-a)}{b(1-a)+(1-b)a}.$
\end{description}
%  \begin{Remark}


\begin {Remark}
Our  choice of the operation structuring the ordered interval
$]0,1[$ as an alo-group wants to obey the requests: 0,5 is the
identity element and $1-x$ is the symmetric of $x$. In this way the
condition of reciprocity for a  PCM over a fuzzy alo-group is given
again by $a_{ji}=1-a_{ij}$,  as defined in Section
\ref{sec:Introduction}. In order to obtain an operation verifying
the above requests,  we apply the following result of
\cite{CavalloDapuzzo}:
\begin{theorem} \label{aloequivalent} Let $G$ be a proper open interval of $R$ and $\leq$  the total order
    on $G$ inherited from the  usual order on $R$, then the following assertions are equivalent:
 \begin{enumerate}
  \item $\mathcal{G} = (G,\odot, \leq)$ is a  continuous  alo-group;
  \item there exists a continuous and strictly increasing function $\psi: ]0,+\infty[ \rightarrow G$ verifying  the equality
  \begin{equation*} \label{peraczel}
x\odot y =\psi(\psi^{-1}(x)\cdot\psi^{-1}(y)).
\end{equation*}
\end{enumerate}
\end{theorem}

Setting $G=]0,1[$ and
\begin{equation}\label{eq:isomorphism_v}
    \psi: t\in ]0, +\infty[ \rightarrow \frac{t}{t+1}\in ]0,1[,
\end{equation}
that is a continuous and strictly increasing function between
$]0,+\infty[$ and $]0,1[$, we get

\begin{equation*}
x\otimes y =\psi(\psi^{-1}(x)\cdot \psi^{-1}(y)).
\end{equation*}

%
%, such that $v(1)=0.5$, $v(\frac{1}{x})=1-v(x)$, with 1 and
%$\frac{1}{x}$ the identity element and the symmetric element of $x$
%respectively, in the multiplicative alo-group. The function $v$
%generates the binary operation $\otimes$, in fact, for each $x$ and
%$y$ in $]0,1[$, we have:


\end{Remark}


\begin{Remark} The operation $\otimes$ is the restriction to
$]0,1[^{2}$ of the uninorm (see \cite{FodorYagerRybalov}):
$$U(x,y)=\left\{
          \begin{array}{ll}
            0, & \hbox{($x,y) \in \{(0,1), (1,0)\}$;} \\
            \frac{xy}{xy+(1-x)(1-y)}, & \hbox{otherwise.}
          \end{array}
        \right.
$$
\end{Remark}



The multiplicative, the additive and the fuzzy alo-groups are
isomorphic; in fact the bijection
$$h: x\in ]0,+\infty[ \rightarrow \log x\in R$$ is an isomorphism
between \textbf{]0,+$\infty$[} and $\mathcal{R}$ and $\psi$ in
\eqref{eq:isomorphism_v} is an isomorphism between
\textbf{]0,+$\infty$[} and  $\textbf{]0,1[}$. So, by  Proposition
\ref{isomean}, the mean $ m_{\otimes}( a_{1}, ..., $ $a_{n})$
related to the fuzzy alo-group can be computed, by means of the
function in \eqref{eq:isomorphism_v}, as follows:
$$ m_{\otimes}( a_{1}, ...,  a_{n})=\psi\big(\big(\prod _{i=1}^{n}
\psi^{-1}(a_{i})\big)^{\frac{1}{n}}\big).$$
%\end{Remark}




\section{Consistent PCMs over a divisible alo-group}% and the property of consistency}
\label{sec:aloPCM}
In this section, $\mathcal{G}=(G, \odot, \leq)$ is a divisible alo-group and
    $A=(a_{ij})$ in \eqref{pcm} is a PCM over $\mathcal{G}$, that is $a_{ij} \in G,\; \forall i,j\in\{1,\ldots,n \}$.
\\We assume that $A$  is  \emph{reciprocal}  with respect to
$\odot$, that is :
\begin{equation}\label{def:dot_reciprocity}
    a_{ji} =a_{ij}^{(-1)} \quad \forall \; i,j=1, \ldots, n
\end{equation}
 so $a_{ii} = e$ for each
$i=1,2,...,n$ and $a_{ij} \odot a_{ji}= e$ for $i, j \in \{1,2, ...,
n\}$.



\begin{Definition}\label{def:consistency}\cite{CavalloDapuzzo},
$A=(a_{ij})$ is a \emph{consistent} matrix with respect to
$\odot$,  if and only if:
\begin{equation}\label{def:dot_consistency}
   a_{ik}= a_{ij}\odot a_{jk}\quad\forall i, j, k.
\end{equation}
Moreover, $\underline{w}=(w_{1},\dots ,w_{n} )$, with $w_{i} \in G$, is a \emph{consistent} vector  for $A=(a_{ij})$ if and only if
$w_{i}\div w_{j}= a_{ij}\; \forall$ i, j=1,2,...,n.
\end{Definition}
Let  $\underline{a}_{1}, \underline{a}_{2},\dots,\underline{a}_{n}$ be  the rows of $A$; then the  \emph{mean vector} associated to $A$ is the vector    \begin{equation}
\label{meanvec}
\underline{w}_{m_{\odot}}(A)
   =(m_{\odot}( \underline{a}_{1}),m_{\odot}( \underline{a}_{1}), \cdots, m_{\odot}( \underline{a}_{n})).
\end{equation}
If the matrix $A$ is consistent then $\underline{w}_{m_{\odot}}(A)$ is a consistent vector. Indeed in \cite{CavalloDapuzzo}, we prove the following

\begin{Proposition} The following assertions related to $A=(a_{ij})$ are equivalent:
\begin{description}
  \item[i)]   $A$ is a consistent PCM;
  \item[ii)] each column $\underline{a}^{k}$ is a consistent vector;
  \item[iii)] the mean vector $\underline{w}_{m_{\odot}}$ is a consistent vector.
\end{description}
\end{Proposition}
 \begin{Proposition}\label{Prop: dot_cons_i_less_j_less_k}\cite{CavalloDapuzzo}
$A=(a_{ij})$ is a \emph{consistent} matrix with respect to
$\odot$,  if and only if:
\begin{equation}\label{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k}
   a_{ik}= a_{ij}\odot a_{jk}\quad\forall \; i, j, k:  i< j< k.
\end{equation}
\end{Proposition}

 \begin{Proposition}\label{consid}\cite{CavalloDapuzzo}
$A=(a_{ij})$ is a \emph{consistent} matrix with respect to
$\odot$,  if and only if:
\begin{equation*}%\label{def:consistency_i_less_j_less_k}
   d_{\mathcal{G}}(a_{ik}, a_{ij}\odot a_{jk})=e \quad \forall \; i, j, k :  i< j< k
\end{equation*}
\end{Proposition}
%\subsection{Reciprocity and consistency properties}\label{sec:RecCons}
% Let us assume that
%% \begin{Definition}\label{def:reciprocity}
%$A=(a_{ij})$ is  \emph{reciprocal}  with respect to
%$\odot$ that is :
%  \begin{equation} \textbf{r}_{\odot}) \quad a_{ji} =a_{ij}^{(-1)} \quad \forall \; i,j=1, \ldots, n. \nonumber \end{equation}
%%\end{Definition}
%As a consequence,  $a_{ii} = e$  and $a_{ij} \odot a_{ji}= e$ for $i, j \in \{1,2, ...,
%n\}$.\\




\subsection{Checking the consistency}\label{sec:consistentMatrices}

%\begin{Remark}
% As $\odot$  is an group operation,  $a_{ij}\odot a_{jk} \in G$ for every choice of $a_{ij}$ and $a_{jk}$ in $G$. So the decision maker has   the possibility to be  consistent and  do not fall into the consistency drawback discussed in Section 1. \end{Remark}




%\begin{Proposition}\label{Prop:scomp_aik_dot_consistent_matrix}
%      $A$ is consistent if and only if the following equality holds:
%      \begin{equation}\label{eq:eq_in_Prop_scomp_aik_dot_consistent_matrix}
%        a_{ik}=a_{i\;i+1} \odot a_{i+1\;i+2}  \odot \ldots \odot a_{k-1\;k}\quad \forall i,k : i<k.
%      \end{equation}
%\end{Proposition}
%\begin{proof}
%By Proposition \ref{Prop: dot_cons_i_less_j_less_k}, it is sufficient to prove the equivalence between \eqref{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k} and \eqref{eq:eq_in_Prop_scomp_aik_dot_consistent_matrix}.
%\\\eqref{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k} $\Rightarrow$ \eqref{eq:eq_in_Prop_scomp_aik_dot_consistent_matrix}. Let $i<k$. By \eqref{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k}, we have that:
%       $$  a_{ik}=a_{i\;i+1} \odot a_{i+1\;k}$$
%       $$ a_{i+1\; k}=a_{i+1\;i+2} \odot a_{i+2\;k}$$
%      $$  \vdots$$
%       $$ a_{k-2\;k}=a_{k-2\;k-1} \odot a_{k-1\;k}$$
%Thus, by  associativity of $\odot$, we have \eqref{eq:eq_in_Prop_scomp_aik_dot_consistent_matrix}.
%\\ \eqref{eq:eq_in_Prop_scomp_aik_dot_consistent_matrix} $\Rightarrow$ \eqref{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k}. Let $i<j<k$. By \eqref{eq:eq_in_Prop_scomp_aik_dot_consistent_matrix} we have that:
%$$a_{ik}= a_{i\;i+1}   \odot \ldots a_{j-1\;j}  \odot a_{j\;j+1}  \ldots  \odot a_{k-1\;k},$$
%by associativity of $\odot$ and applying again \eqref{eq:eq_in_Prop_scomp_aik_dot_consistent_matrix}, we have that:
%  $$a_{ik}= (a_{i\;i+1} \odot a_{i+1\;i+2}  \odot \ldots \odot a_{j-1\;j}) \odot (a_{j\;j+1} \odot a_{j+1\;j+2}  \odot \ldots \odot a_{k-1\;k})=a_{ij}\odot a_{jk}.$$
%\end{proof}
%
%\begin{Proposition}\label{prop:Optimized_dot_consistency}
%      $A$ is consistent if and only if the following equality holds:
%      \begin{equation}\label{eq:eq_in_Prop_Optimized_dot_consistency}
%        a_{ik}=a_{i\;i+1} \odot a_{i+1\;k}\quad \forall i,k : i<k.
%      \end{equation}
%\end{Proposition}
%\begin{proof}
%By Proposition \ref{Prop: dot_cons_i_less_j_less_k}, it is sufficient to prove the equivalence between \eqref{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k} and \eqref{eq:eq_in_Prop_Optimized_dot_consistency}.
%\\\eqref{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k} $\Rightarrow$ \eqref{eq:eq_in_Prop_Optimized_dot_consistency}. It is obvious, we choose $j=i+1$.\\
%\eqref{eq:eq_in_Prop_Optimized_dot_consistency}$\Rightarrow$ \eqref{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k}.
%\end{proof}

We provide new characterizations of a consistent PCM that allows us to give an efficient algorithm to check the consistency of a PCM.

\begin{Proposition}\label{Prop:3_equivalences}The following assertions are equivalent:
\begin{enumerate}
  \item
  %  $a_{ik}= a_{ij}\odot a_{jk}\quad\forall \; i, j, k:  i< j< k.$
  $A$ is a consistent PCM with respect to $\odot$;
  \item  $   a_{ik}=a_{i\;i+1} \odot a_{i+1\;k}\quad \forall i,k :
  i<k;      $
  \item  $
        a_{ik}=a_{i\;i+1} \odot a_{i+1\;i+2}  \odot \ldots \odot a_{k-1\;k}\quad \forall i,k : i<k.
     $
\end{enumerate}

\end{Proposition}
\begin{proof}$1. \Rightarrow 2.$ It is straightforward because of Proposition \ref{Prop: dot_cons_i_less_j_less_k}.

$2. \Rightarrow 3.$  By $2.$:
       $$  a_{ik}=a_{i\;i+1} \odot a_{i+1\;k}$$
       $$ a_{i+1\; k}=a_{i+1\;i+2} \odot a_{i+2\;k}$$
      $$  \vdots$$
       $$ a_{k-2\;k}=a_{k-2\;k-1} \odot a_{k-1\;k}$$
Thus, by  associativity of $\odot$, $3.$ is achieved.

$3. \Rightarrow 1.$ By Proposition \ref{Prop: dot_cons_i_less_j_less_k}, it is enough to prove that $3. \Rightarrow$ \eqref{eq:eq_in_Prop_Prop dot_cons_i_less_j_less_k}. \\
Let $i<j<k$. By $3.$, we have that:
$$a_{ik}= a_{i\;i+1}   \odot \ldots a_{j-1\;j}  \odot a_{j\;j+1}  \ldots  \odot a_{k-1\;k};$$
so, by associativity of $\odot$ and applying again $3.$, we have that:
  $$a_{ik}= (a_{i\;i+1} \odot a_{i+1\;i+2}  \odot \ldots \odot a_{j-1\;j}) \odot (a_{j\;j+1} \odot a_{j+1\;j+2}  \odot \ldots \odot a_{k-1\;k})=a_{ij}\odot a_{jk}.$$
\end{proof}
Thus, the following corollary  follows:
 \begin{corollary}\label{consid_i_less_k}
$A=(a_{ij})$ is a \emph{consistent} matrix with respect to
$\odot$,  if and only if:
\begin{equation*}%\label{def:consistency_i_less_j_less_k}
   d_{\mathcal{G}}(a_{ik}, a_{i\;i+1}\odot a_{i+1\; k})=e \quad \forall \; i, k :  i< k
\end{equation*}
\end{corollary}


%
%By Proposition \ref{Prop: dot_cons_i_less_j_less_k} and Proposition \ref{Prop:3_equivalences}
%we have new characterizations of a consistent PCM:
%
%\begin{corollary}\label{Corollary:scomp_aik_dot_consistent_matrix}
%      $A$ is consistent if and only if the following equality holds:
%      \begin{equation}\label{eq:eq_in_Prop_scomp_aik_dot_consistent_matrix}
%        a_{ik}=a_{i\;i+1} \odot a_{i+1\;i+2}  \odot \ldots \odot a_{k-1\;k}\quad \forall i,k : i<k.
%      \end{equation}
%\end{corollary}
%
%
%
%\begin{corollary}\label{Corollary:Optimized_dot_consistency}
%      $A$ is consistent if and only if the following equality holds:
%      \begin{equation}\label{eq:eq_in_Prop_Optimized_dot_consistency}
%        a_{ik}=a_{i\;i+1} \odot a_{i+1\;k}\quad \forall i,k : i<k.
%      \end{equation}
%\end{corollary}



Finally, in order to check whenever or not a matrix is consistent, we provide Algorithm \ref{Alg:checkConsistency}, for which computational complexity order is equal to $O(n^{2})$.
In Algorithm \ref{Alg:checkConsistency}, we assume that:
\begin{itemize}
  \item $i$ is the index of the rows of $A$, it is initialized to $i=1$;
  \item $k$ is the index of the columns of $A$, it is initialized to $k=i+2$;
  \item $n$ is the order of $A$;
  \item $ConsistentMatrix$ is a boolean variable and the algorithm returns $ConsistentMatrix = true$ if and only if the matrix is
  consistent. It is initialized to $true$, but $ConsistentMatrix = false$ is immediately returned when an inconsistent triple $(a_{ik}, a_{i\;i+1}, a_{i+1\;k} )$ occurs.
\end{itemize}


\begin{algorithm}[ht]
\caption{Checking consistency}
\label{Alg:checkConsistency}
\begin{algorithmic}
\STATE  $i=1$;
\STATE  ConsistentMatrix=true;
\WHILE{$i\leq n-2$ and ConsistentMatrix}
\STATE  $k=i+2$;
\WHILE{$k\leq n$ and ConsistentMatrix}
\IF{$a_{ik} \neq a_{i\;i+1}\odot a_{i+1\;k}$}
\STATE ConsistentMatrix=false;
\ENDIF
\STATE $k=k+1$;
\ENDWHILE
\STATE  $i=i+1$;
\ENDWHILE
\RETURN ConsistentMatrix;
\end{algorithmic}
\end{algorithm}





\subsection{Building a consistent matrix}\label{sec:BuildingConsistencyMatrix}
Given $X=\{x_{1},x_{2},\ldots,x_{n}\}$ the set of alternatives, item $2.$ of Proposition \ref{Prop:3_equivalences} allow us to build a consistent PCM starting from a fixed alternative $x_{i}$ and the $n-1$ comparisons between $x_{i}$ and $x_{j}$, for $j\neq i$. These comparisons are expressed by one of the following sequences:
\begin{description}
  \item [1.] $a_{i1}, \ldots a_{i\; i-1}, a_{i\; i+1}, \ldots a_{in},$
  \item [2.]$a_{1i}, \ldots a_{i-1\; i}, a_{i+1\; i}, \ldots a_{ni}.$
\end{description}
By  item $2.$ of Proposition \ref{Prop:3_equivalences}, we can also to build a consistent PCM starting from one of the following sequences:
\begin{description}
  \item [3.]$a_{12}, a_{23}, \ldots a_{n-1\; n},$
  \item  [4.]$a_{21},  a_{32},  \ldots a_{n\; n-1}.$
\end{description}
For the fuzzy case, reference \cite{ChiclanaViedmaHerrera} builds a
consistent matrix by means of sequence \textbf{3}.  Here, we provide
Algorithm \ref{Alg:buildConsistentMatrix} to build a consistent PCM
starting from the sequence
 $a_{12}, a_{13}, \ldots, a_{1n}$; we use the equalities  $a_{i+1 \; j}= a_{i+1\; i} \odot a_{i\;
 j},$  $\forall i,j$ such that $i<j-1$, obtained from item $2$ of Proposition
 \ref{Prop:3_equivalences}.
\begin{algorithm}[ht]
\caption{Building a consistent matrix}
\label{Alg:buildConsistentMatrix}
\begin{algorithmic}
\FOR {$j=2,\ldots,n$}
\STATE $a_{j1}=a_{1j}^{(-1)}$
\ENDFOR
\FOR {$i=1,\ldots,n$}
\STATE $a_{ii}=e$
\ENDFOR
\FOR{$i=1, \ldots  n-2$}
\FOR{$j=i+2 \ldots n$}
\STATE $a_{i+1 \; j}= a_{i+1\; i} \odot a_{i\; j} $
\STATE $a_{j\; i+1}= a_{i+1 \; j}^{(-1)}$
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{Example}
Let $\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$ be a set of alternatives. We suppose that the DM prefers $x_{1}$
 to each other alternative and expresses the following preference ratios (multiplicative case):
$a_{12}=2$, $a_{13}=4$, $a_{14}=5$ and $a_{15}=6$.
By means of Algorithm \ref{Alg:buildConsistentMatrix}, we obtain:
$$a_{21}=\frac{1}{2}, a_{31}=\frac{1}{4}, a_{41}=\frac{1}{5}, a_{51}=\frac{1}{6},$$
$$a_{11}=a_{22}=a_{33}=a_{44}=a_{55}=1,$$
$$a_{23}= a_{21}a_{13}=2,\; a_{32}=\frac{1}{2},$$
$$a_{24}=a_{21}a_{14}=\frac{5}{2},\;a_{42}=\frac{2}{5},$$
$$a_{25}=a_{21}a_{15}=3,\;a_{52}=\frac{1}{3},$$
$$a_{34}=a_{32}a_{24}=\frac{5}{4},\;a_{43}=\frac{4}{5},$$
$$a_{35}=a_{32}a_{25}=\frac{3}{2},\;a_{53}=\frac{2}{3},$$
$$a_{45}=a_{43}a_{35}=\frac{6}{5},\;a_{54}=\frac{5}{6},$$
and therefore:
\begin{equation*}
A =\left(\begin{array}{cccccccccc}
1 &  & 2  & & 4 & &  5 &  & 6 &
\\
 \\\frac{1}{2} &  &  1 &  &  2 &  & \frac{5}{2} &  & 3 &
 \\
 \\\frac{1}{4}  & & \frac{1}{2}  & & 1   & & \frac{5}{4}  & & \frac{3}{2} &
 \\
  \\\frac{1}{5} & & \frac{2}{5}  & & \frac{4}{5} &  & 1 & &  \frac{6}{5} &
  \\
   \\\frac{1}{6} &  & \frac{1}{3}  & & \frac{2}{3} &  & \frac{5}{6}  & & 1   &
 \end{array}\right)\
 \end{equation*}
 \normalsize
\end{Example}



\begin{Example}
Let $\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$ be a set of alternatives. We suppose that the DM prefers $x_{1}$  to each other alternative and expresses the following preferences (fuzzy case): $a_{12}=0.6$, $a_{13}=0.7$, $a_{14}=0.8$ and $a_{15}=0.9$.
By means of Algorithm \ref{Alg:buildConsistentMatrix}, we obtain:
$$a_{21}=0.4, a_{31}=0.3, a_{41}=0.2, a_{51}=0.1,$$
$$a_{11}=a_{22}=a_{33}=a_{44}=a_{55}=0.5,$$
$$a_{23}=\frac{a_{21}a_{13}}{a_{21}a_{13}+(1-a_{21})(1-a_{13})}=0.609,$$ $$a_{32}=0.391,$$
$$a_{24}=\frac{a_{21}a_{14}}{a_{21}a_{14}+(1-a_{21})(1-a_{14})}=0.727,$$ $$a_{42}=0.273,$$
$$a_{25}=\frac{a_{21}a_{15}}{a_{21}a_{15}+(1-a_{21})(1-a_{15})}=0.857,$$ $$a_{52}=0.143,$$
$$a_{34}=\frac{a_{32}a_{24}}{a_{32}a_{24}+(1-a_{32})(1-a_{24})}=0.632,$$
$$a_{43}=0.368,$$
$$a_{35}=\frac{a_{32}a_{25}}{a_{32}a_{25}+(1-a_{32})(1-a_{25})}=0.794,$$
$$a_{53}=0.206,$$
$$a_{45}=\frac{a_{43}a_{35}}{a_{43}a_{35}+(1-a_{43})(1-a_{35})}=0.692,$$
$$a_{54}=0.308,$$
and therefore:
\begin{equation*}
A =\left(\begin{array}{cccccccccc}
0.5 &  & 0.6  & & 0.7& &  0.8   & & 0.9 &
\\
 \\0.4 &  & 0.5  &  & 0.609 & &  0.727  & & 0.857 &
 \\
 \\0.3  & & 0.391 & &  0.5  & & 0.632 & &  0.794&
 \\
  \\0.2 & & 0.273  & & 0.368 & & 0.5 &  & 0.692&
  \\
   \\0.1 &  & 0.143 & & 0.206 & & 0.308  & & 0.5&
 \end{array}\right)\
 \end{equation*}
\end{Example}


\section{A new consistency index}\label{sec:consistencyIndex}
Let $T$ be the set $\{(a_{ij}, a_{jk},a_{ik}), i<j<k\}$ and $n_{T}=|T|$.
By  Proposition \ref{consid} $A=(a_{ij})$ is inconsistent if and only if $d_{\mathcal{G}}(a_{ik},
a_{ij}\odot a_{jk}) > e$ for some triple $(a_{ij}, a_{jk},a_{ik})\in T$.
Thus, in \cite{CavalloDapuzzo} the authors have provided the following definition of consistency index and the related results:

\begin{Definition}
   The  \emph{consistency index}  of $A$ is given by:
      \begin{equation}
\label{odot_consistencyIndex} I_{\mathcal{G}}(A)=
\begin{cases}
   d_{\mathcal{G}}(a_{13},
a_{12}\odot a_{23})&   n=3, \\
     \big(\bigodot_{T}
d_{\mathcal{G}}(a_{ik}, a_{ij}\odot a_{jk})\big)^{(\frac{1}{n_{T}})}&  n>3.\nonumber
\end{cases}
\end{equation}
with $$n_{T}=\frac{n(n-2)(n-1)}{6}.$$
%  \begin{description}
%  \item[] $I_{\mathcal{G}}(A)=d_{\mathcal{G}}(a_{13},
%a_{12}\odot a_{23})$ \quad if $ n=3$;
%  \item[] $I_{\mathcal{G}}(A)=\big(\bigodot_{i<j<k}
%d_{\mathcal{G}}(a_{ik}, a_{ij}\odot a_{jk})\big)^{(1/n_{T})}$ \quad if $ n>3$.
%\end{description}
\end{Definition}




\begin{Proposition}\label{Aconsiff}
    $I_{\mathcal{G}}(A)\geq e$ and $A$ is consistent if and only if
$I_{\mathcal{G}}(A)=e$.
 \end{Proposition}

\begin{Proposition} \label{isoindexcon} Let $\mathcal{G'}= (G', \circ, \leq)$ be a divisible alo-group isomorphic
to  $\mathcal{G} $ and $A'=(h(a_{ij}))$
the
 transformed of $A=(a_{ij})$ by means of the isomorphism $h: G\rightarrow G'$. Then $I_{\mathcal{G}}(A)=h^{-1}(I_{\mathcal{G'}}(A')).$
\end{Proposition}
At the light of Corollary \ref{consid_i_less_k}, it is reasonable to define a new consistency index, considering only the distances $d_{\mathcal{G}}(a_{ik}, a_{i\;i+1}\odot a_{i+1\;k})$, with $i<k-1$; of course if $i=k-1$ then $d_{\mathcal{G}}(a_{ik}, a_{i\;i+1}\odot a_{i+1\;k})=e$. Let $T^{*}$ be the set $\{(a_{i\;i+1}, a_{i+1\;k},a_{ik}), i<k-1\}$ and $n_{T^{*}}=|T^{*}|$, then we consider the following  index:
%\begin{Definition}
%   The  \emph{consistency index}  of $A$ is given by:
\begin{equation}
I^{*}_{\mathcal{G}}(A)=
\begin{cases}
   d_{\mathcal{G}}(a_{13},
a_{12}\odot a_{23})  \quad \quad \quad \quad \quad \quad n=3,&    \\
     \big(\bigodot_{T^{*}}
d_{\mathcal{G}}(a_{ik}, a_{i\;i+1}\odot a_{i+1\;k})\big)^{(\frac{1}{n_{T^{*}}})} \;n>3.&  \nonumber
\end{cases}
\label{odot_consistencyIndex}
\end{equation}
with $$n_{T^{*}}=\frac{(n-2)(n-1)}{2}.$$

%  \begin{description}
%  \item[] $I_{\mathcal{G}}(A)=d_{\mathcal{G}}(a_{13},
%a_{12}\odot a_{23})$ \quad if $ n=3$;
%  \item[] $I_{\mathcal{G}}(A)=\big(\bigodot_{i<j<k}
%d_{\mathcal{G}}(a_{ik}, a_{ij}\odot a_{jk})\big)^{(1/n_{T})}$ \quad if $ n>3$.
%\end{description}
%\end{Definition}

By Corollary \ref{consid_i_less_k}, we have:
\begin{Proposition}\label{Aconsiff_opt}
    $I^{*}_{\mathcal{G}}(A)\geq e$ and $A$ is consistent if and only if
$I^{*}_{\mathcal{G}}(A)=e$.
 \end{Proposition}


As for $n>3$ it results $n_{T^{*}}< n_{T}$, the index $I^{*}_{\mathcal{G}}(A)$ is more easy to compute than the consistency index $I_{\mathcal{G}}(A)$, thus,  we provide the following definition:
\begin{Definition}
   A  \emph{consistency index}  of $A$ is given by $I^{*}_{\mathcal{G}}(A)$.
\end{Definition}

Moreover, proposition analogous to Proposition \ref{isoindexcon}  follows:
\begin{Proposition} \label{isoindexcon_opt} Let $\mathcal{G'}= (G', \circ, \leq)$ be a divisible alo-group isomorphic
to  $\mathcal{G} $ and $A'=(h(a_{ij}))$
the
 transformed of $A=(a_{ij})$ by means of the isomorphism $h: G\rightarrow G'$. Then $I^{*}_{\mathcal{G}}(A)=h^{-1}(I^{*}_{\mathcal{G'}}(A')).$
\end{Proposition}

\begin{Example}\label{Example:fuzzy4}
Let us consider  $$A=\left(
           \begin{array}{cccccccc}
              0.5 &  & 0.3 &  &  0.4 &  & 0.4 & \\
             \\
             0.7 & &  0.5 & &  0.1 &  & 0.2 & \\
             \\
             0.6 &  & 0.9 & &  0.5 & & 0.8 &  \\
             \\
             0.6 &  & 0.8 &  &  0.2 & &  0.5 & \\
           \end{array}
         \right)$$
         that is a  PCM over the fuzzy alo-group  \textbf{]0,1[}.
By applying the function $\psi^{-1}$, with $\psi$ in
\eqref{eq:isomorphism_v}, to the entries of $A$, we get the matrix
$$A'=\left(
           \begin{array}{cccccccc}
             1 & &  \frac{3}{7} &  &\frac{2}{3} & & \frac{2}{3}  &\\
             \\
             \frac{7}{3} & & 1 &  & \frac{1}{9} &  & \frac{1}{4} & \\
             \\
             \frac{3}{2}  & & 9 &  & 1 &  & 4  & \\
             \\
             \frac{3}{2} &  & 4 & &  \frac{1}{4}& &  1 &  \\
             \\
           \end{array}
         \right). $$
          $A'$ is a PCM over the multiplicative  alo-group \textbf{]0,+$\infty$[} and  its consistency index is
               \begin{align}I^{*}_{\emph{\textbf{]0,+}}\infty\emph{\textbf{[}}}(A')=&
           \sqrt[3]{I^{*}_{\emph{\textbf{]0,+}}\infty\emph{\textbf{[}}}(A_{123}')  \cdot I^{*}_{\emph{\textbf{]0,+}}\infty\emph{\textbf{[}}}(A_{124}')\cdot
           I^{*}_{\emph{\textbf{]0,+}}\infty\emph{\textbf{[}}}(A_{234}')} \nonumber
\\=& \sqrt[3]{14 \cdot \frac{56}{9} \cdot \frac{16}{9} }=5.37 \nonumber
\end{align} \normalsize


Applying Proposition \ref{isoindexcon_opt}, we can compute the
consistency index of $A$ by means of the isomorphism $\psi$ in
\eqref{eq:isomorphism_v}:
$$I^{*}_{\emph{\textbf{]0,1[}}}(A)=\psi(I^{*}_{\emph{\textbf{]0,+}}\infty\emph{\textbf{[}}}(A'))=\frac{5.37}{6.37}=0.84.$$
\end{Example}




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