
%
\documentclass{llncs}
%



\hyphenation{op-tical net-works semi-conduc-tor}


\begin{document}
%
\title{Stable rankings in collective decision making with imprecise information}
\author{Contreras, I.$^*$, Hinojosa, M.A.$^*$, M\'armol, A.M.$^{**}$}
\institute{Pablo de Olavide University$^*$; Sevilla
University$^{**}$; Seville -Spain}
\maketitle


\begin{abstract}
The objective of the  paper is to propose procedures to construct
global ranking of a set of  alternatives in situations  in which
each member of a group is able to provide imprecise information on
his/her preferences about the relative importance of the criteria
that have to be taken into account.

We first  propose an approach based on the assumption that the
final evaluation depends on the complete group since no
possibility exists that the group might split into coalitions that
search for more favourable solutions for the the coalitions members.
To this end, the partial information on criteria weights provided
by each individual is transformed into ordinal information on
alternatives, and then the aggregation of individual preferences
is addressed within a distance-based framework.

In a second approach, the possibility of coalition formation is
considered, and the goal is to obtain rankings in which the
disagreements of all the coalitions are taken into account. These
rankings will exhibit an additional property of collective
stability in the sense that no coalition will has the incentive
to abandon the group and begin a separate evaluation process.

This last approach may be of interest in political decisions
where different sectors have to be incorporated into a joint
evaluation process with the desire to obtain a consensus across
all possible subgroups.

\bigskip%

\textit{Keywords}: Group decision making, decision analysis with multiple criteria,
imprecise information.
\end{abstract}
\section{Introduction}
In  many group decision problems a set of alternatives must be
evaluated on the basis of different and conflicting criteria which
have to be taken into account in the final decision. In general,
each  of the group members has a particular view
about the relative importance of the different criteria. The aim
of the collective decision-making process is either to identify
the best or most preferred alternative(s) from a set or to
generate a ranking of alternatives in accordance with these individual
preferences about criteria.

Recent research in the field of group decision making incorporates
the possibility of dealing with imprecise preference information
and permits the procedures to be applied in contexts where the
group members are unable or unwilling to provide a precise
representation of their preferences over alternatives. See for
instance \cite{IMA}, where a class of flexible weight indices for
ranking alternatives is proposed, and \cite{wang}, where a
preference aggregation method based on the estimation of utility
intervals is presented.



The case of imprecise information has also been addressed for
collective decisions in the multicriteria framework. A detailed
revision of group decision models with imprecise information can
be found, for instance, in \cite{Dias-Cli2005}. Recent
contributions are also \cite{Salo}, \cite{kimahn}, \cite{kimetal99}, \cite{malakooti}, \cite{SaloH}, \cite{val},
\cite{climacodias} and \cite{contreras07}.


In this paper we propose  multicriteria collective decision
procedures which consist of the construction of  compromise
solutions by using a distance function on the set of rankings when
the group wants to rank a set of alternatives. These procedures
are especially well suited for group settings where each member of
the group individually provides partial information about his/her
preferences with respect to the criteria under consideration,
whilst not having information about the preferences of the other
agents.

The final objective is the construction of a global ranking of the
alternatives that combines as accurately as possible the different
evaluations of the alternatives with respect to criteria by taking
into account the partial information sets provided by the agents.

The first approach proposed is based on the assumption that the
final evaluation depends on the complete group since no
possibility exists that the group might split into coalitions in order to seek more
favorable solutions for the coalition members.

In a second approach, this coalition possibility is considered, and the goal
is to obtain rankings in which the disagreements of all the
coalitions are taken into account.

This latter approach might be of interest in political decisions
where different sectors have to be incorporated into a joint
evaluation process with the desire to obtain a consensus across
all possible subgroups.

The rest of the paper is organized as follows. In Section 2, we
introduce the collective decision-making model to be addressed. In
Section 3, the procedure to obtain a final ranking of
alternatives is presented, and an illustrative example is provided. In Section 4
the set of group rankings obtained by the procedure
described in Section 3 is refined by taking into account the disagreement of
all the coalitions. Section 5 is devoted to conclusions.

%\hfill mds

%\hfill January 11, 2007

\section{The model}

Let us consider a multicriteria group decision problem in which $M$ alternatives, $X=\{ x_{1} ,\ldots ,x_{M} \} $, have been evaluated with respect to $N$ criteria. The evaluations of each alternative with respect to each criteria are represented by a matrix $A\in R^{N\times M} $, whose elements are denoted $a_{ij} $ with $i=1,\ldots ,N,{\kern 1pt} j=1,\ldots ,M$. These evaluations are assumed to be objective, in the sense that they do not depend on the assessment of the agents, but are measured independently. Hence, $a_{ij} $ represents the cardinal value or the score given to alternative $x_{j} $ with respect to the $i$-th criterion.

There are $K$ Decision Makers (DMs), each of whom offers some information about his/her preferences with respect to the relative importance of the criteria.

We assume that DMs' preferences can be represented by means of an additive function. Thus, the $k$-th decision makers aggregated value associated to alternative $x_{h} $ is given by,

\begin{equation} \label{GEQ107}
V^{k} (x_{h} )=\sum _{i=1}^{N} w_{i}^{k} a_{ih} ,
\end{equation}
where $w^{k} $ denotes a vector of weights that
represents the relative importance of the criteria for agent $k$.

In contrast to classic approaches which consist of the elicitation of weights, these parameters need not be  be completely determined beforehand. We allow imprecision by permitting the values of the criteria weights for each agent to vary in partial information sets, $\Phi ^{k} \subseteq R^{N} $, $k=1,\ldots ,K$.

A partial information set for an agent consists of those vectors of weights that the agent will accept as reasonable for the importance of the criteria. By convention the criteria weights are normalized to add up to one, hence,

\noindent $\Phi ^{k} \subseteq \{ w^{k} \in R^{N} ,{\kern 1pt} \sum _{i=1}^{N} w_{i}^{k} =1,w_{i}^{k} \ge 0,i=1,\ldots ,N\} $, for $k=1,\ldots ,K$. In particular, we will explore the cases where preference information is given by means of linear relations between the weighting coefficients. In this case, $\Phi ^{K} $ are polyhedral sets described by linear constraints on the criteria weights.


The process of constructing the information set for each DM can be
carried out in a sequential way (see [10]). The DMs can provide
information by stating linear relations on the weights. For
instance, they can provide partial or complete ordinal information
on the importance of the criteria. Another example of
representation of information by means of linear relations, which
is also easily interpretable by the DM,  is when the DM declares
a preference of alternative $x_{h} $ to $x_{j} $. This implies
that $x_{h} $ should not be ranked below $x_{j} $ in any ranking
of alternatives induced by the individual preferences of this
particular decision-maker. Hence, a relation $\sum _{i=1}^{N}
w_{i}^{k} a_{ih} -\sum _{i=1}^{N} w_{i}^{k} a_{ij} >0$ must be
incorporated into $\Phi ^{k} $.





\section{Rankings minimizing global disagreement}
 The main idea of the procedure proposed here is the
achievement of a final consensus or compromise solution between
DMs from the individual rankings of alternatives induced by the
partial information sets. Implicit in this problem is the
existence of a measure of global agreement or disagreement between
rankings. Therefore, the approach implies the introduction of a
distance function on the set of rankings in order to determine the
ranking that minimizes the total distance across DMs. A detailed
study of models based on distance functions can be seen in [8].

Let $R_{h} $ and $R_{k} $ be two priority vectors that represent two individual rankings, with $r_{hj} $ denoting the position assigned to alternative $x_{j} $ by DM $h$. As standard, the  first category is assigned to the most preferred alternative and when ties occur, the average of the values corresponding to the tied alternatives is assigned.

The following distance, which is based on the $L_{1} $-metric, is considered on the set of priority vectors:

\begin{equation} \label{GEQ109}
d(R_{k} ,R_{s} )=\sum _{j=1}^{M} |r_{kj} -r_{sj} |
\end{equation}

In this setting, a consensus vector, $R_{G} $, is a priority vector that satisfies

\begin{equation} \label{GEQ110}
\begin{array}{cc} {min} & {\sum _{k=1}^{K} d(R_{k} ,R_{G} )} \\ {s.t.} & {R_{G} \in {\cal R}} \end{array}
\end{equation}
where ${\cal R}$ is the set of vectors that represent ordinal rankings. Notice that neither the rankings for each DM, not the consensus vector, $R_{G} $, need to be unique.

It is important to point out that the consideration of partial information sets to represent DMs' preferences implies that, from an individual point of view, agent $k$ would accept an evaluation of alternative $x_{h} $ consisting of $V^{k} (x_{h} )=\sum _{i=1}^{N} w_{i}^{k} a_{hi} $ if $w^{k} \in \Phi ^{k} $. Hence, every ranking of alternatives that can be achieved from the dominance relations induced by any $w^{k} \in \Phi ^{k} $ is considered acceptable by the $k$-th DM. As a consequence, different rankings of alternatives (at least one, otherwise, $\Phi ^{k} =\emptyset $) can be induced by each DM's preferences. Therefore, the procedure has to include not only an objective for the group in order to determine the ranking that best agrees with the individual preferences, but also a selection criterion to choose a ranking for each DM which represents the individual preferences (those rankings that best agree with the group order).

A dominance relationship between alternatives can be derived from (\ref{GEQ107}). For a fixed vector of weights, $w^{k} \in \Phi ^{k} $, we can say that alternative $x_{j} $ strictly dominates alternative $x_{h} $ under the preference structure of the $k$-th DM, if $V^{k} (x_{j} )-V^{k} (x_{h} )>0$. The consensus ranking is induced here by the ordinal positions of the alternatives, determined by comparing the aggregated values $V^{k} (x_{i} )$ for the different values of $w^{k} \in \Phi ^{k} $.

Hence, an intermediate step consisting of the elicitation of an individual
 ranking for each DM is required.
To this end, we consider the following set of constraints, where $\delta _{ij} $ and $\gamma _{ij} $ are binary variables:

\begin{equation} \label{GEQ111}
\begin{array}{cc} {V^{k} (x_{h} )-V^{k} (x_{j} )+\delta _{hj}^{k} B\ge 0,} & {\forall {\kern 1pt} h\ne j,} \\ {V^{k} (x_{h} )-V^{k} (x_{j} )+\gamma _{hj}^{k} B\ge \varepsilon ,} & {\forall {\kern 1pt} h\ne j.} \end{array}
\end{equation}

 Here $B$ is a large number and $\varepsilon $ is a discriminating factor between alternatives, such that we say alternative $x_{h} $ strictly dominates  $x_{j} $ if $V^{k} (x_{h} )-V^{k} (x_{j} )\ge \varepsilon $.

The values of variables $\delta _{hj}^{k} $ will be equal to one each time that $x_{h} $ does not strictly dominate $x_{j} $, i.e., whenever $V^{k} (x_{j} )>V^{k} (x_{h} )$. In contrast, $\gamma _{hj}^{k} $ will be one whenever $V^{k} (x_{j} )\ge V^{k} (x_{h} )$, that is, each time $x_{h} $ is not preferred to $x_{j} $ in the $k$-th DM preferences. These variables are included in the expression in order to reflect the ties between alternatives.

The ranking position induced by the aggregate values $V^{k} (x_{h} )$ can be obtained as the sum of the binary variables divided by 2 (this fraction represents the number of alternatives that dominate $x_{h} $),  plus one. That is,

\begin{equation} \label{GEQ112}
r_{kh} =\sum _{h\ne j} \frac{\delta _{hj}^{k} +\gamma _{hj}^{k} }{2} +1,{\kern 1pt} \forall {\kern 1pt} h=1,\ldots ,M.
\end{equation}



To guarantee that the vectors $R_{k} =(r_{k1} ,\ldots ,r_{kM} )$ represent priority vectors, we incorporate the following set of constraints

\begin{equation} \label{GEQ113}
\delta _{hj}^{k} +\delta _{jh}^{k} \le 1,{\kern 1pt} \forall {\kern 1pt} h\ne j,
\end{equation}

%\begin{equation} \label{GEQ114}
%\delta _{hj}^{k} +\gamma _{hj}^{k} +\gamma _{jh}^{k} \le 2,{\kern 1pt} \forall %{\kern 1pt} h\ne j,
%\end{equation}

%\begin{equation} \label{GEQ115}
%\delta _{hj}^{k} +\gamma _{hj}^{k} +\gamma _{jh}^{k} \ge 1,{\kern 1pt} \forall %{\kern 1pt} h\ne j,
%\end{equation}

\begin{equation} \label{GEQ116}
\delta _{hj}^{k} +\gamma _{jh}^{k} =1,{\kern 1pt} \forall {\kern 1pt} h\ne j.
\end{equation}

Constraints from (\ref{GEQ113}) to (\ref{GEQ116}) are necessary in order to induce the ranking of alternatives from the values $V^{k} (x_{i} )$. These constraints guarantee that the values of the binary variables $\delta _{hj}^{k} $ and $\gamma _{kj}^{k} $ are correct, in the sense that only when $x_{j} $ strictly dominates $x_{h} $,  then $\delta _{hj}^{k} =1$ holds, and consequently $\delta _{jh}^{k} =0$. The values of $\gamma _{hj}^{k} $  also depend on the value assigned to $\delta _{hj}^{k} $ and represent ties between alternatives. It is worth noting that in each constraint we have to consider not only the dominance relation of $x_{i} $ over $x_{j} $ but also the relation of $x_{j} $ over $x_{i} $.

Finally, the following requirement is included

\begin{equation} \label{GEQ117}
\sum _{j=1}^{M} r_{kj} =\frac{M(M+1)}{2} ,{\kern 1pt} \forall {\kern 1pt} k=1,\ldots ,K.
\end{equation}

In addition to the above constraints, the condition in (\ref{GEQ117}) assures that vector $R_{k} $ is contained in set \textit{S}, that is to say, represents a priority vector.

In order to obtain the compromise solution, a set of variables
corresponding to the group has to be considered: a weighting
vector, aggregate values and a ranking of alternatives for the
group. Hence, a new agent labelled as the subindex $G$ in the set
of DMs\footnote{Subindexes  $k$,  that represent DMs, will  now
vary from 1 to K plus $G$, i.e.  $k=1{\kern 1pt} \ldots ,K,G$.} is
included in the model.

To determine the group solution we have to deal with the distance
defined in (\ref{GEQ109}). The minimization of the sum of these
nonlinear functions, as stated in (\ref{GEQ110}) can be reduced to
a linear programming model by considering a Goal Programming
formulation. By taking into account the following change of
variables proposed in [9],

\begin{equation} \label{GEQ119}
\begin{array}{c} {\alpha _{kj} =\frac{1}{2} \left[|r_{kj} -r_{Gj} |-(r_{kj} -r_{Gj} )\right]} \\ {\beta _{kj} =\frac{1}{2} \left[|r_{kj} -r_{Gj} |+(r_{kj} -r_{Gj} )\right];} \\ {} \end{array}
\end{equation}
we will include the following set of constraints in the model to measure the distance between individual priority vectors and the compromise ranking.

\begin{equation} \label{GEQ120}
r_{kj} -r_{Gj} +\alpha _{kj} -\beta _{kj} =0,{\kern 1pt} \forall {\kern 1pt} k=1,\ldots ,K;j=1,\ldots ,N.
\end{equation}

Therefore, the distance between $R_{k} $ and $R_{G} $ can be represented equivalently by the following expression

\begin{equation} \label{GEQ121}
d(R_{k} ,R_{G} )=\sum _{j=1}^{M} (\alpha _{kj} +\beta _{kj}) .
\end{equation}

All the specifications defined above yield the following model

\begin{equation}\label{GEQ122}
\begin{array}{clll}
 min & \sum_{k=1}^K\sum_{j=1}^M (\alpha_{kj}+\beta_{kj})\\
s.t. & V^k(x_h)=\sum_{i=1}^N w^k_i a_{ih}, & \forall\,h,k\\
     & V^k(x_h)-V^k(x_j)+\delta^k_{hj}B\geq 0, & \forall\, h\neq j,\,\forall\,k\\
     & V^k(x_h)-V^k(x_j)+\gamma^k_{hj}B\geq \varepsilon, & \forall\, h\neq j,\,\forall\,k\\
     & \delta^k_{hj}+\delta^k_{jh}\leq 1, & \forall\,h\neq j,\,\forall\,k\\
     & \delta^k_{hj}+\gamma^k_{hj}+\gamma^k_{jh}\leq 2, & \forall\,h\neq
j,\,\forall\,k\\
     & \delta^k_{hj}+\gamma^k_{hj}+\gamma^k_{jh}\geq 1, & \forall\,h\neq
j,\,\forall\,k\\
      & \delta^k_{hj}+\gamma^k_{jh}=1, & \forall\,h\neq j,\,\forall\,k \\
     & r_{kh}=\sum_{h\neq j} \frac{\delta^k_{hj}+\gamma^k_{hj}}{2} +1, & \forall\,h,k\\
     & \sum_{h=1}^M r_{kh}=\frac{M(M+1)}{2}& \forall\,k\\
     & r_{kh}-r_{Gh}+\alpha_{kh}-\beta_{kh}=0, & \forall\,h,k\neq G\\
     & w^k \in \Phi^k, & \forall\,k\\
     & \delta^k_{hj},\gamma^k_{hj}\in\{0,1\}, & \forall\,h,j,k \\
     & B,\varepsilon\geq 0.

\end{array}
\end{equation}

By solving (\ref{GEQ122}), the compromise ranking $r_{G} $ is obtained. The solution to the problem also provides a ranking of alternatives for each DM. This is the ranking which permits total disagreement to be minimized from among those induced by the individual preferences. Note that it does not necessarily coincide with the ranking that minimizes the DM individual disagreement. This ranking is determined by means of the aggregate values $V^{k} (x_{i} )$, hence the vector of weights which best agrees with the group solution is selected for each DM.

The aggregated values for the group, $V^{G} (x_{h} )$, are only considered for computational purposes in order to induce a ranking of alternatives for the group in the same format in which individual rankings have been constructed. They have no interpretation from a cardinal point of view since the goal of the procedure is only the determination of the group's ranking.

Some desirable properties of our procedure in the context of social choice processes which are a direct consequence of its construction are: feasibility, anonymity, neutrality and no dictatorship. That is to say, a compromise solution is always obtainable, all the group members are treated equally, all the alternatives are treated equally, and no DM exists whose individual preferences determine the consensus ranking.


\subsection{Illustrative example}

This section illustrates the proposed procedure for group decision-making problems. We have considered an example in which four decision-makers want to decide between five alternatives denoted by $\left\{x_{1} ,\ldots ,x_{5} \right\}$. The alternatives have been evaluated with respect to four different criteria. Table \ref{utilities} shows the scores of the alternatives with respect to each criterion. These evaluations have been normalized so that the sum with respect to each criterion add up to one.


\begin{table}[ht]\caption{Matrix of utilities}\label{utilities}
% Table generated by Excel2LaTeX from sheet 'Matriz'
\begin{center}
\begin{tabular}{ccccc}
\hline\hline
Alternative &    Crit. 1 &    Crit. 2 &    Crit. 3 &    Crit. 4 \\
\hline
     $x_1$ &     0.5776 &     0.1692 &     0.0481 &     0.1612 \\

     $x_2$ &     0.2054 &     0.2900 &     0.0340 &     0.2070 \\

     $x_3$ &     0.1143 &     0.1692 &     0.3107 &     0.2200 \\

     $x_4$ &     0.0717 &     0.3052 &     0.4489 &     0.1765 \\

     $x_5$ &     0.0310 &     0.0664 &     0.1583 &     0.2353 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

Four agents are to evaluate this information. They have provided partial information about the relative importance they assign to each criteria. Agent 1 considers that the importance of the criteria is ranked in order of decreasing magnitude. In addition, this agent states that the weight of criterion 5 is not less than $1\% $ of the total value. If $\Lambda ^{+} =\{ w\in R^{5} ,{\kern 1pt} \sum _{k=1}^{5} w_{k} =1,w_{k} \ge 0,k=1,\ldots ,5\} $, then the partial information set for agent 1, can be formalized as
$$\Phi^1=\{w^1\in\Lambda^+,\, w^1_1\geq w^1_2\geq w^1_3\geq
w^1_4\geq 0.01\}$$

The remaining agents preference information about the criteria are
represented in the following information sets and can be interpreted in a similar way to that above.

$\Phi^2=\{w^2\in\Lambda^+,\, w^2_2\geq w^2_1+w^2_3,\, w^2_4\geq
w^2_1+w^2_3,\,w^2_1\geq 0.01,\,w^2_3\geq 0.01\},$

$\Phi^3=\{w^3\in\Lambda^+,\, w^3_4\geq 2w^3_3\geq 4w^3_2\geq
8w^3_1\geq 0.08\},$

$\Phi^4=\{w^4\in\Lambda^+,\, w^4_3\geq w^4_1 +w^4_2+
w^4_4,\,w^4_1\geq 0.01,\,w^4_2\geq 0.01,\,w^4_3\geq 0.01\}.$

The solution to problem  (\ref{GEQ122})  yields a level of disagreement equal to 8 units, although the collective ranking is not unique. The following table (Table \ref{solutions}) summarizes the group solution that minimizes the total disagreement, represented by their respective priority vectors, and the individual disagreements each solution provides.

\begin{table}[ht]\caption{Group rankings}\label{solutions}
% Table generated by Excel2LaTeX from sheet 'Hoja1'
\begin{center}
\begin{tabular}{ccccc}
\hline
\hline
Priority vector ($R_G$) & $d(R_1,R_G)$ & $d(R_2,R_G)$ & $d(R_3,R_G)$ & $d(R_4,R_G)$ \\
\hline
(5, 4, 2, 1, 3) &          6 &          2 &          0 &          0 \\

(2, 4, 3, 1, 5) &          0 &          2 &          6 &          0 \\

(5, 3, 2, 1, 4) &          6 &          0 &          2 &          0 \\

(4.5, 3, 2, 1, 4.5) &          5 &          0 &          3 &          0 \\

(5, 3.5, 2, 1, 3.5) &          6 &          1 &          1 &          0 \\

(3.5, 3.5, 2, 1, 5) &          3 &          1 &          4 &          0 \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}

It is interesting to note that the individual disagreement of each solution can vary from one collective ranking to another. Note that we only fix the total disagreement at its minimum level which, in this case, is 8.


\section{Stable rankings across coalitions}

Unfortunately, the procedure described in Section 3 does not
always provide a unique group ranking which minimizes the level of
disagreement of the DMs. In this section we present a procedure to
refine the set of group rankings inspired by cooperative game
theory approaches.

The individual disagreement associated to the agents for  each
collective ranking can be seen as an allocation of the total
disagreement. The procedure described in Section 3 may provide
several collective rankings  which minimize the total disagreement
and, in addition, there may be different sets of individual
rankings associated to the same colletive ranking, therefore
several allocations among the DMs of the minimum level of
disagreement may exist.

Let $x=(x_1,x_2,\ldots,x_K)$  denote any of such allocations, that
is to say, for each $k=1,2,\ldots, K$, $x_k=d(R_k,R_G)$, where
$R_G$ is one of the group rankings that minimizes the total
disagreement and $R_k$ is one of the associated individual
rankings. Thus, when we choose a collective ranking, $R_G$, and a
set of associated individual rankings, $R_k$, $k=1,2,\ldots, K$,
we are assigning  a disagreement $x_S=\sum_{k\in
S}x_k$ to each subgroup (or coalition) $S\subseteq
\mathcal{K}=\{1,\ldots, K\}$. On the other hand, the procedure described in Section 3
can be applied, not only to the whole set of DMs,  but also to
each  coalition $S\subset\mathcal{K}$. In this way a minimum level
of disagreement, $D_S$, is obtained for each coalition
$S\subseteq\mathcal{K}$, by taking into account the preferences of
the DMs in $S$. We will say that $D_S$ is a reference of the
disagreement for coalition $S$.

For each coalition $S\subseteq\mathcal{K}$, we consider the
difference $x_S-D_S$, which is a measure of the dissatisfaction of
coalition $S$ with the collective ranking $R_G$ (and the
associated individual rankings) which provides allocation $x$. The
idea is to choose among all the collective rankings for which the
 the minimum level of disagreement for the whole set of DMs is achieved, those
that lexicographically minimize the maximum dissatisfaction of the
coalitions. If any vector $x=(x_1,x_2,\ldots,x_K)$, $x_k\geq 0$,
$k=1,2,\ldots,K$, $x_{\mathcal{K}}=D_{\mathcal{K}}$ could be
chosen as a allocation of the collective disagreement, then the
problem to solve would be  Problem \ref{lexmin}.
\begin{equation}\label{lexmin}
\begin{array}{clll}
lex-min & max_{S\subset \mathcal{K}} \,\, x_S-D_S\\
s.t. & x_{\mathcal{K}}=D_{\mathcal{K}}\\
\end{array}
\end{equation}
This problem has, as unique solution, the pre-nucleolus (see \cite{Schmeidler}) of the
cooperative coalitional game defined by the references of the
disagreement.

\bigskip

\noindent{\bf Example in Section 3.1 (continued)}%
{\em%
The references of the disagreement in this case are shown in Table
\ref{tabla}.

\begin{table}[ht]\caption{References of the disagreement for the coalitions}
\label{tabla}
\begin{center}
\begin{tabular}{lc|lc}
\hline\hline
 Coalition ($S$)& Disagreement ($D_S$)& Coalition ($S$)& Disagreement ($D_S$)\\
\hline
$\mathcal{K}=\{1,2,3,4\}$ & 8 & $\{2,3\}$ &          2 \\
$\{1,2,3\}$ &          8 &$\{2,4\}$ &          0 \\
$\{1,2,4\}$ &          2 &$\{3,4\}$ &          0 \\
$\{1,3,4\}$ &          6 &$\{1\}$ &          0 \\
$\{2,3,4\}$ &          2 &$\{2\}$ &          0 \\
 $\{1,2\}$ &          0 &$\{3\}$ &          0 \\
 $\{1,3\}$ &          6 &$\{4\}$ &          0 \\
 $\{1,4\}$ &          0 & & \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
In the first step to solve Problem \ref{lexmin}, we solve the
following linear problem:
\begin{equation}\label{maxmin}
\begin{array}{clll}
min  & \mu\\
s.t. & x_S-D_S\leq\mu\quad\forall\,S\subset \mathcal{K}\\
     & x_{\mathcal{K}}=D_{\mathcal{K}}.\\
\end{array}
\end{equation}
The solution of this problem is $\mu=4$ and the constraints
corresponding to coalitions $\{1,2\}$ and $\{3,4\}$ are active for
each allocation $x$ associated to the solution (it is worth noting
that for at least one of the coalitions this property is
fulfilled). The next step consists of solving Problem
\ref{maxmin2}.
\begin{equation}\label{maxmin2}
\begin{array}{clll}
min  & \mu\\
s.t. & x_1+x_2=4\\
     & x_3+x_4-2=4\\
     &x_S-D_S\leq\mu\quad\forall\,S\subset \mathcal{K}, S\neq \{1,2\},\,\{3,4\}\\
     & x_{\mathcal{K}}=D_{\mathcal{K}}.\\
\end{array}
\end{equation}
This new problem has a unique allocation
$x^*=(\frac{8}{3},\frac{4}{3},\frac{10}{3},\frac{2}{3})$,
associated to the optimal solution, $\mu=\frac{10}{3}$. $x^*$ is
called the pre-nucleolus of the cooperative coalitional game of
references of the disagreements.
}%el ejemplo

\bigskip

Unfortunately, in general,  a collective ranking, $R_G$, and a set
of associated individual rankings, $R_k$, $k=1,2,\ldots,K$, for
which the corresponding allocation
$x=(d(R_k,R_G))_{k\in\mathcal{K}}$ coincides with the
pre-nucleolus of the game, do not always exist. Nevertheless, it is
always possible to find rankings for which allocation
$x=(d(R_k,R_G))_{k\in\mathcal{K}}$ approximates $x^*$ by solving
Problem \ref{aprox}.
\begin{equation}\label{aprox}
\begin{array}{clll}
lex-min & max_{S\subset \mathcal{K}}\,\, x_S-D_S\\
s.t. & d(R_{k} ,R_{G} )=x_k,\quad \forall\, k\in \mathcal{K}\\
     & x_{\mathcal{K}}=D_{\mathcal{K}}\\
     & R_k\in{\cal R}(\phi^k)\\
     & R_G\in{\cal R}(\phi^G)\\
\end{array}
\end{equation}
Moreover, in most cases this collective ranking is unique.

\bigskip

\noindent{\bf Example in Section 3.1 (continued)}%
{\em%
The solution of Problem \ref{aprox} provides a vector of
individual disagreements $x=(3,1,4,0)$, which is as close as
possible to the pre-nucleolus, by taking into account that the
constraint in Problem \ref{aprox} only considers  distances
between rankings.

Table \ref{finalresults1} and Table \ref{finalresults2}  summarize
the individual results. In Table \ref{finalresults1} the weighting
vector selected for each individual partial information set is
included. Table \ref{finalresults2} summarizes the individual
ranking induced from these weighting vectors (note that this
ranking is constructed throughout the aggregated values
$V^k(x_i)$), and the individual distances from the group ranking
$R_G$.
\begin{table}[ht]\caption{Individual weighting vectors}\label{finalresults1}
\begin{center}
\begin{tabular}{ccccc}
\hline
\hline
           &     Crit 1 &     Crit 2 &     Crit 3 &     Crit 4 \\
\hline
     $w^1$ &      0,330 &      0,330 &      0,329 &      0,010 \\

     $w^2$ &      0,050 &      0,438 &      0,231 &      0,281 \\

     $w^3$ &      0,010 &      0,020 &      0,323 &      0,647 \\

     $w^4$ &      0,108 &      0,392 &      0,500 &      0,000 \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]\caption{Individual rankings}\label{finalresults2}
\begin{center}
\begin{tabular}{cccccc}
\hline
\hline
           &                            \multicolumn{ 5}{c}{Rank positions} \\

           &       DM 1 &       DM 2 &       DM 3 &       DM 4 &      Group \\
\hline
     $x_1$ &          2 &          4 &          5 &        3.5 &        3.5 \\

     $x_2$ &          4 &          3 &          4 &        3.5 &        3.5 \\

     $x_3$ &          3 &          2 &          2 &          2 &          2 \\

     $x_4$ &          1 &          1 &          1 &          1 &          1 \\

     $x_5$ &          5 &          5 &          3 &          5 &          5 \\
\hline
Disagreement &          3 &          1 &          4 &          0 &            \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}


Therefore, in this case, the group solutions establish that
alternative $x_4$ is ranked at the first position, followed by
$x_3$, alternatives $x_1$ and $x_2$ tie in the third position and
the least-valued alternative is $x_5$.

$$\left(
\begin{array}{c}
x_3\\
x_2\\
x_1,x_2\\
x_5
\end{array}
\right)$$

}%el jemplo


\section{Conclusions}

We have proposed a compromise method for collective decision
problems which is especially suited for situations where the
members of the group provide imprecise information about their
preferences with respect to the criteria, and there is no flow of
information between these members. In this context the group members are
not encouraged to misrepresent their true preferences in order to
manipulate the group decision. Hence, the result derived could be
considered a fair representation of the group evaluation of all
the alternatives.

Imprecise information is formalized by means of linear relations
between criteria weights. This way of providing preferential
information is one of the most easily interpreted by the DMs, and
includes interesting particular cases such as those in which they
only provide ordinal information on the weights.

The basic procedure relies on the transformation of the partial
information about criteria into ordinal information about
alternatives, expressed through rankings. A model is constructed
that provides a set of compromise rankings for the group and an
additional step enables the achievement of a unique compromise
ranking  which is stable across coalitions.

The approach presented here uses a measure of agreement based upon
the $L_{1} $-metric and, therefore, emphasizes the sum of
individual disagreements with respect to every alternative.
However, the use of other metrics could also be considered in order to analyze this class of collective decision making problems.

\bigskip

\noindent{\bf Acknowledgements:} This research has been partially
financed by the Spanish Ministry of Education and Science
project SEJ2007-62711/ECON  and by Consejería de Innovación, Ciencia y Tecnologia (Junta de Andalucía) project P06-SEJ-01801.

% (used to reserve space for the reference number labels box)
\begin{thebibliography}{1}

%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.

%\bibitem{BC86} {Bana e Costa, C.A.}
% \emph{A multicriteria decision aid methodology to deal with
%conflicting situations on the weights}. European Journal of
%Operational Research 1986; 26; 22-34


%\bibitem{borda} {Borda, J.C.}  \emph{M\'emoire sur les élections au
%scrutin}. {Histoire de l'Académie Royale des Sciences}: 1784.

\bibitem{IMA} {Contreras, I., Hinojosa, M.A., M\'armol, A.M.}
\emph{ A class of flexible weight indices for ranking
alternatives}.
  IMA Journal of Management Mathematics 2005; { 16}; 71-85.


\bibitem{wang} { Wang, Y.M., Yang, J.B., Xu, D.L.}
\emph{A preference aggregation method through the estimation of
utility interval}. {Computers and Operations Research} 2005;{ 32};
2027-2049.


\bibitem{Dias-Cli2005} {Dias, L.C., Climaco, J.N.}
 \emph{Dealing with imprecise information in group multicriteria decisions: A
 methodology and a GDSS architecture}. European Journal of Operational Research 2005; 160 (2);
  291-307.

\bibitem{Salo} {Salo, A.A.,} \emph{Interactive decision aiding for group decision
support.}. {European Journal of Operational Research} 1995; { 84},
134-149.

\bibitem{kimahn} { Kim, S.H., Ahn, B.S.} \emph{Group decision
making procedure considering preference strength under incomplete
information}. {Computers and Operations Research} 1997; { 12},
1101-1112.

\bibitem{kimetal99} {Kim, S.H., Choi, S.H., Kim, J.K.}
 \emph{An interactive procedure for multi-attribute group decision making
 with incomplete information: Range-based approach}. European Journal of Operational
 Research 1999; 118, 139-152.

\bibitem{malakooti}{Malakooti, B.}\emph{Ranking  and Screening Multiple Criteria Alternatives with Partial Information and Use of Ordinal and Cardinal Strengths of Preferences.} IEEE Transactions on Systems, Man, and Cybernetics Part A, 2000; 30 ; 355-367.

\bibitem{SaloH} {Salo, A., H$\ddot{\mbox{a}}$m$\ddot{\mbox{a}}$l$\ddot{\mbox{a}}$inen, R.P.,} \emph{Preference ratios in multiattribute
evaluation (PRIME) – elicitation and decision procedures
under incomplete information.} {IEEE Transactions on
Systems, Man and Cybernetics} 2001; {31}, 533-545.


\bibitem{val} { Valadares Tavares, L.}
\emph{A model to support the search for consensus with conflicting
rankings: Multitrident}. {International Transactions in
Operational Research} 2004;{ 11}; 107-115.

\bibitem{climacodias}{Climaco, J.N., Dias, L.C. }
\emph{Negotiation Processes with Imprecise Information on
Multicriteria Additive Models}. Group Decision and Negotiation
2006; 15; 171–184.

\bibitem{contreras07}{Contreras, I., M\'armol, A.M.}
\emph{A lexicographical compromise method for multiple criteria
group decision problems with imprecise information}. European
Journal of Operational Research 2007, 181 (3), 1530-1539.

\bibitem{ck96} {Cook, W.D., Kress, M.}
\emph{An extreme-point approach for obtaining weighted ratings in
qualitative multicriteria decision making}. Naval Research
Logistic 1996; {43}; 519-531.



%\bibitem{dyersarin} {Dyer, J.S., Sarin, R.K.}
% \emph{Group preference aggregation rules based on strength of preference}.
% Management Science 1979; 25; 822-832.

\bibitem{Pachon99}{Gonz\'alez-Pach\'on, J., Romero, C. }
\emph{Distanced based consensus methods: a goal programming
approach}. Omega 1999, 27: 341-347.




%\bibitem{keeneykirkwood} {Keeney, R.L., Kirkwood, C.W.}
% \emph{Group decision making using cardinal social welfare function}.
% Management Science 1975; 22; 430-437.

%\bibitem{k-r76}{ Keeney, R.L., Raiffa, H.} \emph{Decision with Multiple
%Objectives: Preferences and Value Trade-offs}. {John Wiley}; New
%York; 1976.

\bibitem{marmol2002}{ M\'armol, A.M., Puerto J., Fern\'andez F.R.}
\emph{Sequential incorporation of imprecise information in
multiple criteria decision processes}. European Journal of
Operational Research 2002, 137, 123-133.

\bibitem{Schmeidler}{Schmeidler, D.}
\emph{The nucleolus of a characteristic function game}. SIAM Journal of Applied Mathematics 1969, 17, 1163-1170.



%\bibitem{dea}{Charnes, A., Cooper, W.W., Rhodes, E.}\emph{Measuring the %efficiency of decision making units.} European Journal of Operational Research %1978,  2 , 429--444.


%\bibitem{salo95} {Salo, A.A.}
% \emph{Interactive decision aiding for group decision support}.
% European Journal of Operational Research 1995;  84; 134-149.







%\bibitem{weber} {Weber, M.} \emph{Decision making with incomplete
%information}. {European Journal of Operational Research} 1987;{
%28}; 44-57.








\end{thebibliography}

% biography section
%
% If you have an EPS/PDF photo (graphicx package needed) extra braces are
% needed around the contents of the optional argument to biography to prevent
% the LaTeX parser from getting confused when it sees the complicated
% \includegraphics command within an optional argument. (You could create
% your own custom macro containing the \includegraphics command to make things
% simpler here.)
%\begin{biography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{mshell}}]{Michael Shell}
% or if you just want to reserve a space for a photo:

%\begin{IEEEbiography}{Michael Shell}
%Biography text here.
%\end{IEEEbiography}

% if you will not have a photo at all:
%\begin{IEEEbiographynophoto}{John Doe}
%Biography text here.
%\end{IEEEbiographynophoto}

% insert where needed to balance the two columns on the last page with
% biographies
%\newpage

%\begin{IEEEbiographynophoto}{Jane Doe}
%Biography text here.
%\end{IEEEbiographynophoto}

% You can push biographies down or up by placing
% a \vfill before or after them. The appropriate
% use of \vfill depends on what kind of text is
% on the last page and whether or not the columns
% are being equalized.

%\vfill

% Can be used to pull up biographies so that the bottom of the last one
% is flush with the other column.
%\enlargethispage{-5in}



% that's all folks
\end{document}
