#  Electre-Score: A first outranking based method for scoring actions

José Rui Figueira<EMAIL_ADDRESS>Salvatore Greco Bernard Roy
Department of Economics and Business, University of Catania, Catania, Italy
Université Paris-Dauphine, PSL Research University, CNRS (UMR 7243), LAMSADE,
Paris, France  CEG-IST, Instituto Superior Técnico, Universidade de Lisboa,
Lisbon, Portugal  Portsmouth Business School, Centre of Operations Research
and Logistics (CORL),
University of Portsmouth, Portsmouth, United Kingdom

###### Abstract

In this paper we present (to the best of our knowledge) the first outranking
method to assign a score to each alternative. It is a method of the Electre
family, and we will call it Electre-Score. Contrarily to the Multi-Attribute
Value Theory methods, Electre-Score does not construct a value function for
each criterion, and then proceeds to the aggregation into a single value. It
rather, makes use of the outranking relations to make a comparison with
reference sets of actions (to which a score is assigned) and proposes a score
range to each alternative, instead of a single value. This is a more robust
way of proceeding given the fragility of a single score. The sets of limiting
profiles are defined with the help of Electre Tri-nB and the reference scores
are assigned to them through the application of the deck of cards technique.
The fact of being able to use outranking relations, makes it also possible to
take into account the imperfect knowledge of data and avoids systematic
compensatory effects. Some fundamental theoretical results guaranteing the
consistency of the method and an illustrative example are also provided in
this paper.

###### keywords:

Multiple criteria analysis , Electre methods , Scoring methods , Outranking
relations , Decision support systems.

††journal: European Journal of Operational Research

##  1 Introduction

Multiple criteria decision aiding (MCDA) is a discipline that comprises
methods and techniques used to produce information in order to enable
decision-makers to make better and informed decisions. Over the last years
there was a tremendous growth in the development of new methods, strengthening
the maturity of the existing ones, and in increasing their application to deal
with real-world decision aiding situations of a crucial importance for
organizations. One of the most magnificent and wonderful features of these
methods is their diversity (see Greco et al., 2016, for a vast panoply od
methods).

The three major families of methods can be divided by taking into account the
form of the results provided to the decision-makers. There are scoring based
methods, which assign a score to each action or alternative (Keeney and
Raiffa, 1993), the outranking based methods that provide one or several
outranking relations between ordered pairs of alternatives and exploit such
relations (Roy and Bouyssou, 1993), and there are also rule based approaches
wich provide the decision-makers with a set of decision rules (Greco et al.,
2001).

Electre methods (Figueira et al., 2016) play a central role in the family of
outranking approaches. Since their inception in the middle of the sixties of
last century they were object of several studies, extensions, generalizations,
new developments, and many applications in real-world situations. For a
comprehensive survey of these methods see Govindan and Jepsen (2016). The most
relevant features of Electre methods that make them adequate to deal with
several situations are the following (Figueira et al., 2013): (1) they can
deal with both the quantitative and the qualitative nature of the criteria
scales; (2) the scales can be of very heterogeneous types (meters, noisy,
delay, costs, return, etc); (3) the compensatory effects are not relevant in a
systematic way (this is mainly due to the use of the concordance index and the
existence of veto thresholds that avoid some compensability); (4) they are
able to take into account the imperfect knowledge of data (uncertainty,
imprecision, and ill-determination) and the arbitrariness when building the
criteria; and, finally, (5) they are very adequate to take into account the
reasons for and the reasons against an outranking.

Of course, as all the MCDA methods, the ones from Electre family are not
perfect and also suffered from some drawbacks: (1) the intransitivity
phenomenon may occur (and even worse, it may be quite frequent); (2) the
phenomenon of the dependence with respect to irrelevant alternatives may be
present; (3) if all the criteria are of a quantitative nature and no imperfect
knowledge and arbitrariness are present, and in addition the decision-maker
allows for a systematic compensation, we can do better with other methods,
namely the ones of Multi-Attribute Value Theory (MAVT) family; and (4) if it
is necessary assigning a score to each action, these methods are note adequate
at all.

In reality the compensation between criteria is not always allowed by
decision-makers and imperfect knowledge and arbitrariness are often present
when dealing with practical decision aiding situations. Since it will be
almost impossible to avoid the two other phenomena (intransitivities and
dependence), would it be possible to build a scoring based Electre method?
This is the challenge we would like to face and the objective of this paper is
thus to propose such a method.

The new method consists of following main steps:

  1. 1.

Several reference sets of actions are built. We used the limiting profiles as
in Electre-Tri-nB (Fernández et al., 2017) as our reference sets. Such sets of
reference actions must fulfill some important separability conditions. Note
that, whichever is the procedure to define _a priori_ the set of reference
actions, it must satisfy a certain number of conditions that should be as weak
as possible in order the method to be fruitfully applied in a vast generality
of cases. To start, we consider rather restrictive conditions that must be
relaxed as much as possible in the following but only if these less
restrictive conditions would maintain the validity of the method.

  2. 2.

With a deck of cards technique, we can assign a value to each set of limiting
profiles after choosing two reference values. This is a similar technique as
the one proposed in Bottero et al. (2018) for building interval scales.

  3. 3.

The last step consists of comparing each action to the reference sets and
assigning a scoring interval to it.

It should be remarked that the nature of this scoring method is different from
those belonging to the MAVT family, since we do not consider a transformation
of the scale for each criterion into a partial value function, and then
aggregate all the partial values into a single one. The score obtained with
our procedure supplies a comprehensive evaluation without using a procedure
for converting each criterion into a value function.

The paper is organized as follows. Section 2 is devoted to some fundamental
concepts, their definitions, and corresponding notation. Section 3 presents
the new Electre-Score method, including the conditions for the construction of
the reference set, the assignment of a scoring range to each action, and the
formal definitions of the lower and upper bounds of such a range. Section 4 is
related to the conditions about the set of limiting profiles that allow the
procedure to be in accordance with the objectives. Section 5 is devoted to the
theoretical results proving desirable properties of the procedure. Section 6
presents the method by means of an illustrative example along with some
practical aspects. Finally, the last section provides the main conclusions and
some lines for future research.

##  2 Concepts, definitions, and notation

To start we need to introduce a few notation. Let
$A=\\{a_{1},\ldots,a_{i},\ldots,a_{m}\\}$ denote the set of actions to which
an interval score must be assigned to each of them,
$G=\\{g_{1},\ldots,g_{j},\ldots,g_{n}\\}$ denote the set of criteria used to
assess the performance of such actions, and $g_{j}(a_{i})$ denotes the
performances of action $a_{i}$ on criterion $g_{j}$ (with all the performances
we can build a performance table). Consider also the collection of sets of
references actions $B=\\{B_{x_{1}},\ldots,B_{x_{k}},\ldots,B_{x_{\ell}}\\}$,
for which a score is previously defined, and let
$X=\\{x_{1},\ldots,x_{k},\ldots,x_{\ell}\\}$ the set of such scores. Each set
is composed of at least one limiting profile as in Electre Tri-nB (see
Fernández et al., 2017), i.e.,
$B_{x_{k}}=\\{b_{k1},\ldots,b_{kp},\ldots,b_{kp_{k}}\\}$. As part of Electre
methods a credibility degree between all ordered pairs of actions,
$\sigma(a,b)$, must be computed. This credibility measures on a scale $[0,1]$
the degree in which action $a$ outranks action $b$. In order to pass from a
fuzzy relation to a crisp one, we need to define what is called the cutting
level $\lambda\in]0.5,1]$. A brief description of the way Electre methods
compute the degree of credibility is available in the Appendix of this paper.

The next three definitions are fundamental.

######  Definition 1 (Dominance).

Consider two actions $a$ and $b$. Action $a$ _dominates_ action $b$, whenever
$g_{j}(a)\geqslant g_{j}(b)$, for all $j=1,\ldots,n$, with at least one strict
inequality. Let $a\Delta b$ denote such a binary dominance relation.

######  Definition 2 (Fundamental outranking binary relation).

Consider two actions $a$ and $b$ from set $A$. Once fixed the cutting level
$\lambda$, we say that action $a$ outranks (or is at least as good as) action
$b$, denoted by $a\succsim^{\lambda}b$ iff $\sigma(a,b)\geqslant\lambda$. It
is easy to see that $\succsim^{\lambda}$ is a reflexive, but not necessarily
symmetric and transitive binary relation. In what follows and with some abuse
of the mathematical language, we will use simply $a\succsim b$ instead of
$a\succsim^{\lambda}b$ for denoting this $\lambda-$outranking binary relation
(the same applies for the binary relations introduced in next definition).

######  Definition 3 (Derived binary relations).

From the fundamental outranking binary relation $\succsim$, we can derive, for
two actions $a,b\in A$, the following three binary relations (which correspond
to all possible combinations of the presence and non-presence of an outranking
relation between $a$ and $b$, and $b$ and $a$, respectively).

  1. i)

$a\succ b$ (_preference_ in favor of $a$, which means that $a$ is preferred to
$b$) iff $a\succsim b$ and $\mbox{not}(b\succsim a)$;

  2. ii)

$b\succ a$ (_preference_ in favor of $b$, which means that $b$ is preferred to
$a$) iff $b\succsim a$ and $\mbox{not}(a\succsim b)$;

  3. iii)

$a\sim b$ (_indifference_ , which means that actions, $a$ and $b$, are
indifferent) iff $a\succsim b$ and $b\succsim a$;

  4. iv)

$a\parallel b$ (_incomparability_ , which means that actions, $a$ and $b$, are
incomparable) iff $\mbox{not}(a\succsim b)$ and $\mbox{not}(b\succsim a)$.

($\succ$ is irreflexive and asymmetric; $\sim$ is reflexive and symmetric;
and, $\parallel$ is irreflexive and symmetric.)

######  Remark 1.

From Definitions 2 and 3, it is easy to see that $a\succsim b$ implies, either
$a\succ b$ or $a\sim b$. Taking into account the dominance relation of
Definition 1, the following properties hold.

$\displaystyle a\Delta b\Rightarrow a\succsim b$ (.1) $\displaystyle a\succsim
b\;\mbox{and}\;b\Delta c\Rightarrow a\succsim c$ (.2) $\displaystyle a\Delta
b\;\mbox{and}\;b\succsim c\Rightarrow a\succsim c$ (.3) $\displaystyle a\succ
b\;\mbox{and}\;b\Delta c\Rightarrow a\succ c$ (.4) $\displaystyle a\Delta
b\;\mbox{and}\;b\succ c\Rightarrow a\succ c$ (.5)

##  3 Electre-Score

This section provides the basic foundations of the Electre-Score method, i.e.,
the necessary elements for the construction of a reference set, the conditions
needed for assigning a score range to each action, and the formal definition
of the lower and upper bounds of such a range. Most of the material presented
in this section is closely related to the Electre Tri-nB method (see Fernández
et al., 2017).

###  3.1 Constructing a reference set

The definition of the reference set as well as the basic assumption with
respect to such reference set are presented next. The subsection also provided
some more results in the same line as in Fernández et al. (2017).

######  Definition 4 (Set of reference actions).

Let $X=\\{x_{1},\ldots,x_{k},\ldots,x_{\ell}\\}$ denote the set of values
considered as references scores, and
$B_{x_{k}}=\\{b_{k1},\ldots,b_{kp},\ldots,b_{kp_{k}}\\}$ denote the set of
reference actions used to characterize score $x_{k}$. As a result
$B=\bigcup_{k}^{\ell}B_{x_{k}}$ denotes a set containing all the reference
actions.

######  Condition 1 (Basic assumptions).

The score $x_{k}$ is characterized by a set of reference actions,
$B_{x_{k}}=\\{b_{x_{k}1},\ldots,b_{x_{k}p},\ldots,b_{x_{k}p_{k}}\\}$, for
$k=1,\ldots,\ell$, such that:

  1. i)

For all $b_{kp},b_{kq}\in B_{x_{k}}$ there is no preference between $b_{kp}$
and $b_{kq}$ (this implies, there is only the possibility to have either
$b_{kp}\sim b_{kq}$ or $b_{kp}\parallel b_{kq}$);

  2. ii)

For all $b_{kp}\in B_{x_{k}}$ and $b_{hq}\in B_{x_{h}}$ ($x_{k}>x_{h}$), it is
not possible to have $b_{hq}\succ b_{kp}$.

######  Definition 5 (Relations between an action and a reference set).

Consider the following relations between an action $a$, and a set of reference
actions, $B_{x_{k}}$ (see Fernández et al., 2017).

  1. i)

$a\succsim B_{x_{k}}$ iff, for all $b_{kq}\in B_{x_{k}}$, either $a\parallel
b_{kq}$ or $a\succsim b_{kq}$, the latter relation being fulfilled by at least
one $b_{kq}\in B_{x_{k}}$ (note that, for all $b_{kq}\in B_{x_{k}}$, it is not
possible to have $b_{kq}\succ a$);

  2. ii)

$B_{x_{k}}\succsim a$ iff, for all $b_{kq}\in B_{x_{k}}$, either
$b_{kq}\parallel a$ or $b_{kq}\succsim a$, the latter relation being fulfilled
by at least one $b_{kq}\in B_{x_{k}}$ (note that, for all $b_{kq}\in
B_{x_{k}}$, it is not possible to have $a\succ b_{kq}$);

  3. iii)

$a\succ B_{x_{k}}$ iff, for all $b_{kq}\in B_{x_{k}}$, either $a\parallel
b_{kq}$ or $a\sim b_{kq}$, or $a\succ b_{kq}$, the latter relation being
fulfilled by at least one $b_{kq}\in B_{x_{k}}$ (note that, for all $b_{kq}\in
B_{x_{k}}$, it is not possible to have $b_{kq}\succ a$);

  4. iv)

$B_{x_{k}}\succ a$ iff, for all $b_{kq}\in B_{x_{k}}$, either $b_{kq}\parallel
a$ or $b_{kq}\sim a$, or $b_{kq}\succ a$, the latter relation being fulfilled
by at least one $b_{kq}\in B_{x_{k}}$ (note that, for all $b_{kq}\in
B_{x_{k}}$, it is not possible to have $a\succ b_{kq}$);

  5. v)

$a\sim B_{x_{k}}$ iff, for all $b_{kq}\in B_{x_{k}}$, either $a\parallel
b_{kq}$ or $a\sim b_{kq}$, the latter relation being fulfilled by at least one
$b_{kq}\in B_{x_{k}}$ (note that, since $\sim$ is symmetric, for all
$b_{kq}\in B_{x_{k}}$, it is not possible to have $b_{kq}\succ a$ or $a\succ
b_{kq}$);

  6. vi)

$a\parallel B_{x_{k}}$ iff, for all $b_{kq}\in B_{x_{k}}$, either $a\parallel
b_{kq}$ or, when $a\succ b_{kq}$, for some $b_{kq}\in B_{x_{k}}$, $b_{kp}\succ
a$, for some $b_{kp}\in B_{x_{k}}$, with $b_{kq}\neq b_{kp}$ (note that, since
$\sim$ is symmetric, it is not possible to have $B_{x_{k}}\succsim a$ or
$a\succsim B_{x_{k}}$).

######  Remark 2.

The following implications can be derived from Definitions 1 and 5 (see
Fernández et al., 2017).

  1. i)

$a\succ B_{x_{k}}$ implies $a\succsim B_{x_{k}}$;

  2. ii)

$a\succ B_{x_{k}}$ implies $\mbox{not}(B_{x_{k}}\succsim a)$, and consequently
$a\succ B_{x_{k}}$ also implies $\mbox{not}(B_{x_{k}}\succ a)$;

  3. iii)

$B_{x_{k}}\succ a$ implies $\mbox{not}(a\succsim B_{x_{k}})$, and consequently
$B_{x_{k}}\succ a$ also implies $\mbox{not}(a\succ B_{x_{k}})$;

  4. iv)

$B_{x_{k}}\succ a$ implies $B_{x_{k}}\succsim a$;

  5. v)

$B_{x_{k}}\succ a$ and $a\Delta b$ implies $B_{x_{k}}\succ b$;

  6. vi)

$a\Delta b$ and $b\succsim B_{x_{k}}$ implies $a\succsim B_{x_{k}}$;

  7. vii)

$a\Delta b$ and $b\succ B_{x_{k}}$ implies $a\succ B_{x_{k}}$.

###  3.2 Assigning a score range to each action

The aim of the ELECTRE score method we are presenting is the identification of
a range, $]s^{l}(a),s^{u}(a)[$ for the score $s(a)$ to be assigned to actions
$a\in A$. Let us first discuss some conditions that guarantee the existence of
such a range $]s^{l}(a),s^{u}(a)[$. Since we only know the relations between
$a$ and the elements of set $B$ (see Definition 5), the lower bound,
$s^{l}(a)$, and the upper bound, $s^{u}(a)$, of the range cannot be fixed _a
priori_. However, it is easy to see that we cannot have $B_{x}\succ a$, for
any score $x$ lower than or equal to the lower bound, $s^{l}(a)$. Otherwise,
the lower bound was not properly defined and we could move down (decrease) its
value, which makes no sense. Analogously, it is easy to see that we cannot
have $a\succ B_{x}$ for any score $x$ greater than or equal to the upper
bound, $s^{u}(a)$. Otherwise, the upper bound was not properly defined and we
could move up (increase) its value, which makes no sense. This reasoning led
us to establish the following two necessary conditions for the existence of
the range $]s^{l}(a),s^{u}(a)[$.

######  Condition 2 (Lower bound necessary condition).

If $x\leqslant s^{l}(a)$, then $\mbox{not}(B_{x}\succ a)$, for all $a\in A$.

######  Condition 3 (Upper bound necessary condition).

If $x\geqslant s^{u}(a)$, then $\mbox{not}(a\succ B_{x})$, for all $a\in A$.

These two conditions are necessary for the existence of the range, but they
are not sufficient, since we know nothing about the relations between $B_{x}$
and $a$, for an $x$ value strictly comprised within the range
$]s^{l}(a),s^{u}(a)[$. However, it is easy to see that we cannot have
$B_{x}\succ a$, for any score $x$ strictly greater than the lower bound,
$s^{l}(a)$. Otherwise, the lower bound was not properly defined and we could
move up (increase) its value, which makes no sense. Analogously, it is easy to
see that we cannot have $a\succ B_{x}$, for any score $x$ strictly lower than
the upper bound, $s^{u}(a)$. Otherwise, the upper bound was not properly
defined and we could move down (decrease) its value, which makes no sense.
This reasoning led us to establish the following two sufficient conditions for
the existence of the range $]s^{l}(a),s^{u}(a)[$.

######  Condition 4 (No active preference condition of reference sets in the
score range).

If $s^{l}(a)<x<s^{u}(a)$, then $\mbox{not}(B_{x}\succ a)$, for all $a\in A$.

If this condition was violated, no score $s(a)$, such that $s^{l}(a)\leqslant
s(a)<s^{u}(a)$, could be justified.

######  Condition 5 (No passive preference condition of reference sets in the
score range).

If $s^{l}(a)<x<s^{u}(a)$, then $\mbox{not}(a\succ B_{x})$, for all $a\in A$.

Analogously, if this condition was violated, no score $s(a)$, such that
$s^{l}(a)<s(a)\leqslant s^{u}(a)$, could be justified.

It is easy to see that Conditions 4 and 5 are fulfilled if and only if the
next (equivalent) condition holds.

######  Condition 6 (No preference condition of reference sets in the
interval).

For all possible scores, $x$, such that $s^{l}(a)<x<s^{u}(a)$,
$\mbox{not}(B_{x}\succ a)$ and $\mbox{not}(a\succ B_{x})$, for all $a\in A$.

This condition means that assigning to action $a$, any feasible score, i.e,
any score $x$ such that $s^{l}(a)<x<s^{u}(a)$, is only possible when, either
$a\sim B_{x}$ or $a\parallel B_{x}$.

Observe that Conditions 2 and 4 can be replaced by the following condition.

######  Condition 7 (Lower bound general condition).

If $x>s^{l}(a)$, then $\mbox{not}(a\succ B_{x})$, for all $a\in A$.

Analogously, Conditions 3 and 5 can be replaced by the following condition.

######  Condition 8 (Upper bound general condition).

If $x<s^{u}(a)$, then $\mbox{not}(B_{x}\succ a)$, for all $a\in A$.

######  Remark 3.

Conditions 2 to 5 do not imply by no means that the next condition is
automatically fulfilled.

######  Condition 9 (Indifference/incomparability).

If $a\sim B_{x}$ or $a\parallel B_{x}$, then $s^{l}(a)<x<s^{u}(a)$, for all
$a\in A$.

Figure 1 illustrates the above conditions. Notation $\mathbf{[Ck]}$ is used
instead of Condition $k$, for $k=3,4,5,6,7,8,9$.

$x$ (score)$s(a)$$\circ$$\circ$$s^{l}(a)$$s^{u}(a)$$\mbox{not}(a\succ B_{x})$
$\mathbf{[C3]}$$\circ$$\mbox{not}(B_{x}\succ a)$
$\mathbf{[C2]}$$\circ$$\mbox{not}(B_{x}\succ a)$
$\mathbf{[C4]}$,$\mbox{not}(a\succ B_{x})$
$\mathbf{[C5]}$$\mathbf{[C6][C9]}$$\mbox{not}(B_{x}\succ a)$
$\mathbf{[C8]}$$\circ$$\mbox{not}(a\succ B_{x})$ $\mathbf{[C7]}$$\circ$ Figure
1: Illustration of the necessary and no active and passive preference
conditions for the existence of a range for $s(a)$, $]s^{l}(a),s^{u}(a)[$

###  3.3 Definitions of the lower and upper bounds

The formal definitions of the lower and upper bound are as follows.

######  Definition 6 (Lower bound of the score range).

The $s^{l}(a)$ value is the highest value, such that $a\succ B_{x}$ and
$\mbox{not}(B_{x}\succ a)$, for all $x<s^{l}(a)$.

######  Definition 7 (Upper bound of the score range).

The $s^{u}(a)$ value is the lowest value, such that $B_{x}\succ a$ and
$\mbox{not}(a\succ B_{x})$, for all $x>s^{u}(a)$.

######  Remark 4.

It is obvious that Definitions 6 and 7 ensure that 2 and 3 are automatically
fulfilled.

##  4 Finding the conditions on $B$ allowing the procedure to be in accordance
with the objectives

This section presents the conditions of the set $B$ that render the procedure
or the new method coherent, i.e., in accordance with the objectives the method
have been designed for. From these conditions a set of new results needed to
be proved.

######  Condition 10 (Separability conditions).

The following are the required separability conditions on set $B$.

  1. a

_Dominance based separability conditions_.

    1. a.1

_Strong dominance_. Consider $x_{h}>x_{k}$. For any two reference actions,
$b_{hq}\in B_{x_{h}}$ and $b_{kp}\in B_{x_{k}}$, the relation $b_{hq}\Delta
b_{kp}$ holds.

    2. a.2

_Soft dominance_.

       * $a.2.p$

(_primal_): Consider $x_{h}>x_{k}$. For all $b_{kp}\in B_{x_{k}}$, there is at
least one $b_{hq}\in B_{x_{h}}$ such that $b_{hq}\Delta b_{kp}$.

       * $a.2.d$

(_dual_): Consider $x_{h}>x_{k}$. For all $b_{hq}\in B_{x_{h}}$, there is at
least one $b_{kp}\in B_{x_{k}}$ such that $b_{hq}\Delta b_{kp}$.

  2. b

_Preference based separability conditions_.

    1. b.1

_Strong preference_. For any two reference actions, $b_{hq}\in B_{x_{h}}$ and
$b_{kp}\in B_{x_{k}}$, the relation $b_{hq}\succ b_{kp}$ holds.

    2. b.2

_Soft preference_.

       * $b.2.p$

(_primal_): Consider $x_{h}>x_{k}$. For all $b_{kp}\in B_{x_{k}}$, there is at
least one $b_{hq}\in B_{x_{h}}$ such that $b_{hq}\succ b_{kp}$.

       * $b.2.d$

(_dual_): Consider $x_{h}>x_{k}$. For all $b_{hq}\in B_{x_{h}}$, there is at
least one $b_{kp}\in B_{x_{k}}$ such that $b_{hq}\succ b_{kp}$.

######  Proposition 1 (Comparisons of the actions against reference sets).

  1. i)

If the primal soft dominance separability condition holds for $B$, then, for
all $a\in A$, $a\succsim B_{x_{k}}$ implies $\mbox{not}(B_{x_{h}}\succ a)$,
for all $x_{h}<x_{k}$;

  2. ii)

If the primal soft dominance separability condition holds for $B$, then, for
all $a\in A$, $B_{x_{k}}\succ a$ implies $\mbox{not}(a\succsim B_{x_{h}})$,
for all $x_{h}>x_{k}$;

  3. iii)

If both the primal and the dual soft dominance separability condition hold for
$B$, then, for all $a\in A$, $a\succsim B_{x_{k}}$ implies $(a\succsim
B_{x_{h}})$, for all $x_{h}<x_{k}$;

  4. iv)

If both the primal and the dual soft dominance separability condition hold for
$B$, then, for all $a\in A$, $B_{x_{k}}\succ a$ implies $(B_{x_{h}}\succ a)$,
for all $x_{h}>x_{k}$;

  5. v)

If the dual soft dominance separability condition holds for $B$, then, for all
$b_{kp}\in B_{x_{k}}$, $b_{kp}\succsim B_{x_{h}}$, for all $x_{h}\leqslant
x_{k}$;

  6. vi)

If the primal soft dominance separability condition holds for $B$, then, for
all $b_{kp}\in B_{x_{k}}$, $B_{x_{h}}\succ b_{kp}$, for all $x_{h}>x_{k}$.

###### Proof.

We shall prove only point $iv)$ and $vi)$ that will be used several times in
the proofs of other propositions in the paper. The other points have similar
proofs. Analogous results have been obtained with respect to sorting
procedures in (Fernández et al., 2017).

If $B_{x_{h}}\succ a$ then there exist $b_{hp}\in B_{x_{h}}$ such that
$b_{hp}\succ a$. Since $k>h$, by Condition $a.2.p$ (primal), there exist
$b_{kq}\in B_{x_{k}}$ such that $b_{kq}\Delta b_{hp}$ and, consequently,
$b_{kq}\succ a$. By contradiction, let us suppose that there exist $b_{kr}\in
B_{x_{k}}$ such that $a\succ b_{kr}$. By Condition $a.2.d$ (dual), there
should exist $b_{hs}\in B_{x_{h}}$, such that $b_{kr}\Delta b_{hs}$. Therefore
we would get $a\succ b_{hs}$, which would imply not $(B_{x_{h}}\succ a)$
contradicting the hypothesis $B_{x_{h}}\succ a$. This concludes the proof of
point $iv)$.

If $a\succ B_{x_{h}}$ then there exist $b_{hp}\in B_{x_{h}}$ such that $a\succ
b_{hp}$. Since $h>k$, by Condition $a.2.d$ (dual), there exist $b_{kq}\in
B_{x_{k}}$ such that $b_{hp}\Delta b_{kq}$ and, consequently, $a\succ b_{kq}$.
By contradiction, let us suppose that there exist $b_{kr}\in B_{x_{k}}$ such
that $b_{kr}\succ a$. By Condition $a.2.p$ (primal), there should exist
$b_{hs}\in B_{x_{h}}$, such that $b_{hs}\Delta b_{kr}$. Therefore we would get
$b_{hs}\succ a$, which would imply not $(a\succ B_{x_{h}})$ contradicting the
hypothesis $a\succ B_{x_{h}}$. This concludes the proof of point $vi)$. ∎

######  Proposition 2 (Upper and lower bound sufficiency).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Condition 10, points $a.2.p$ and $a.2.d$), then Conditions 4
and 5 are also fulfilled.

###### Proof.

By contradiction, suppose that there exist $x_{h}\in X$ and $a\in A$ such that
$x_{h}<s^{u}(a)$ and $B_{x_{h}}\succ a$. By Definition 7, this would imply
that there should exist $x_{k}>x_{h}$ such that $a\succ B_{x_{k}}$. Thus,
there would exist $b_{kp}\in B_{x_{k}}$ such that $a\succ b_{kp}$. By
Condition $a.2.p$ (primal), there should exist $b_{hq}\in B_{x_{h}}$ such that
$b_{kp}\Delta b_{hq}$. Consequently, $a\succ b_{hq}$, which would imply
not($B_{x_{h}}\succ a$), contradicting thus the hypothesis $B_{x_{h}}\succ a$.
Therefore, Condition 4 holds.

Analogously, by contradiction, suppose that there exists $x_{h}\in X$ and
$a\in A$ such that $x_{h}>s^{l}(a)$ and $a\succ B_{x_{h}}$. By Definition 6,
this would imply that there should exist $x_{k}<x_{h}$ such that
$B_{x_{k}}\succ a$. Thus, there would exist $b_{kp}\in B_{x_{k}}$ such that
$b_{kp}\succ a$. By Condition $a.2.d$ (dual), there should exist $b_{hq}\in
B_{x_{h}}$ such that $b_{hq}\Delta b_{kp}$. Consequently, $b_{hq}\succ a$
which would imply not($a\succ B_{x_{h}}$), contradicting thus the hypothesis
$a\succ B_{x_{h}}$. Therefore, Condition 5 holds. ∎

######  Proposition 3 (Implications of soft dominance in a preference of an
action w.r.t. a set).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10), then, for any
$a\in A$, $a\succ B_{x}$ for all $x\leqslant s^{l}(a)$.

###### Proof.

By Definition 6, we have $a\succ B_{s^{l}(a)}$, and, since Points $a.2.p$ and
$a.2.d$ (primal and dual) of Condition 10 hold, by Proposition 1, we obtain
$a\succ B_{x}$, for all $x<s^{l}(a)$. ∎

######  Proposition 4 (Implications of soft dominance in a preference of a set
w.r.t. an action).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of condition 10), then, for any
$a\in A$, $B_{x}\succ a$ for all $x\geqslant s^{u}(a)$.

###### Proof.

By Definition 7, we have $B_{s^{u}(a)}\succ a$ and, since Points $a.2.p$ and
$a.2.d$ (primal and dual) of Condition 10 hold, by Proposition 1, we obtain
$B_{x}\succ a$, for all $x>s^{u}(a)$. ∎

######  Proposition 5 (Implications of soft dominance in the
indifference/incomparability region).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Condition 10, points $a.2.p$ and $a.2.d$), then Conditions 9
is fulfilled. Moreover, in such a case, the value $s^{u}(a)$ is simply the
highest value, such that $a\succ B_{x}$, for $x<s^{u}(a)$, and $s^{l}(a)$ is
the lowest value, such that $B_{x}\succ a$, for $x>s^{l}(a)$, for all $a\in A$
(i.e., the second parts of Definitions 6 and 7 can be neglected since they are
automatically fulfilled by themselves)

###### Proof.

We have already noted that Conditions 4 and 5 are equivalent to Condition 6,
in which, for all $a\in A$ and, for all values $x$ such that
$s^{l}(a)<x<s^{u}(a)$, not($B_{x}\succ a$) and not($a\succ B_{x}$). Therefore,
we have to prove only that if Points $a.2.p$ and $a.2.d$ of Condition 10 are
satisfied, then for all values $x$ such that not($B_{x}\succ a$) and
not($a\succ B_{x}$), we have $s^{l}(a)<x<s^{u}(a)$. This is true because

  * –

by Proposition 3, if $x\leqslant s^{l}(a)$, then $a\succ B_{x}$; and,

  * –

by Proposition 4, if $x\geqslant s^{u}(a)$, then $B_{x}\succ a$.

Consequently, we obtain that, under the same conditions, $s^{l}(a)$ is simply
the highest value $x$ such that $a\succ B_{x}$, and, $s^{u}(a)$ is simply the
lowest value $x$ such that $B_{x}\succ a$. ∎

##  5 Theoretical results

This section has the purpose of showing the consistency of the method with
respect to some fundamental and quite natural requirements, namely the ones of
uniqueness, independence, monotonicity, conformity, homogeneity, and
stability.

######  Definition 8 (Inserting and deleting operations).

The following two operations are considered:

  1. i)

Inserting operation:

    1. i.a)

a new set $B_{x}$ is inserted in $B$;

    2. i.b)

a new action $b$ is inserted in $B_{x}$.

  2. ii)

Deleting operation.

    1. ii.a)

a set $B_{x}$ is removed from $B$;

    2. ii.b)

an action $b$ is removed from $B_{x}$.

######  Definition 9 (Structural requirements).

The following structural requirements of the method are considered:

  1. i)

_Uniqueness_. For each action $a\in A$ there is a single value $s^{l}(a)$ and
a single value $s^{u}(a)$.

  2. ii)

_Independence_. The definition of the values $s^{l}(a)$ and $s^{u}(a)$, for
action $a\in A$, does not depend on the other actions in $A\setminus\\{a\\}$.

  3. iii)

_Monotonicity_. If $a\Delta a^{\prime}$, then $s^{l}(a)\geqslant
s^{l}(a^{\prime})$ and $s^{u}(a)\geqslant s^{u}(a^{\prime})$.

  4. iv)

_Conformity_. If $a\in B_{x_{k}}$, then $s^{l}(a)=x_{k-1}$ and
$s^{u}(a)=x_{k+1}$.

  5. v)

_Homogeneity_. If two actions, $a$ and $a^{\prime}$, compare the same way with
respect to the reference actions in $B$, then $s^{l}(a)=s^{l}(a^{\prime})$ and
$s^{u}(a)=s^{u}(a^{\prime})$.

  6. vi)

_Stability_. For $a\in A$ with $x_{r}=s^{l}(a)<s^{u}(a)=x_{s}$, let $s^{\ast
l}(a)$ and $s^{\ast u}(a)$ be the new bounds in consequence of one of the
operations of Definition 8. Then

$x_{r-1}\leqslant s^{\ast l}(a)\leqslant x_{r+1}$

and

$x_{s-1}\leqslant s^{\ast u}(a)\leqslant x_{s+1}.$

The latter requirement means that single inserting or deleting operations
according to Definition 8 imply a minimal perturbation on the score range of
each action.

Note that uniqueness, independence, monotonicity and homogeneity are clearly
satisfied. We only focus our attention on conformity and stability.

######  Theorem 1 (Conformity).

If $B$ fulfills both the primal and the dual soft dominance and preference
separability conditions (see Points $a.2.p$, $a.2.d$, $b.2.p$, and $b.2.d$ of
Condition 10,), then the conformity property of Definition 9 is fulfilled.

###### Proof.

By Points $b.2.p$ and $b.2.d$ of Condition 10, for each $a\in B_{x_{k}}$ there
exists $b_{k-1p}\in B_{x_{k-1}}$, such that

$a\succ b_{k-1p}$ (2)

and $b_{k+1q}\in B_{x_{k+1}}$, such that

$b_{k+1q}\succ a.$ (3)

By contradiction, let us suppose that there exists $b_{k-1r}\in B_{x_{k-1}}$,
such that

$b_{k-1r}\succ a.$ (4)

By Point $a.2.p$ of Condition 10, there would exist $b_{ks}\in B_{x_{k}}$ such
that $b_{ks}\Delta b_{k-1r}$ that, together with (4), would imply $b_{ks}\succ
a$, which is impossible because we cannot have a limiting profile of a class
preferred to another limiting profile of the same class. Consequently, there
cannot be any profile from $B_{x_{k-1}}$ preferred to $b_{ks}$ which, together
with (2), implies that

$a\succ B_{x_{k-1}}.$ (5)

Again, by contradiction, let us suppose that there exists $b_{k+1t}\in
B_{x_{k+1}}$, such that

$a\succ b_{k+1t}.$ (6)

By Point $b.2.d$ of Condition 10, there would exist $b_{ku}\in B_{x_{k}}$ such
that $b_{k+1t}\Delta b_{ku}$ that, together with (6) would imply $a\succ
b_{ku}$, which is impossible because we cannot have a limiting profile of a
class preferred to another limiting profile of the same class. Consequently,
$a$ cannot be preferred to any profile from $B_{x_{k+1}}$, which, together
with (3), implies that

$B_{x_{k+1}}\succ a.$ (7)

Based on the above considerations and taking any $a\in B_{k}$, we can say that

  * –

due to (1) and (5) for any $x_{h}<x_{k}$,

$a\succ B_{x_{h}},$ (8)

  * –

from Point $i)$ of Condition 1,

$not(a\succ B_{x_{k}})\mbox{ and }not(B_{x_{k}}\succ a),$ (9)

  * –

due to (1) and (7) for any $x_{h}>x_{k}$,

$B_{x_{h}}\succ a.$ (10)

Therefore,

    * $\circ$

$k-1$ is the maximum value of $h$, for which $a\succ B_{x_{h}}$ and,
consequently, by Proposition 5, $s^{l}(a)=x_{k-1}$,

    * $\circ$

$k+1$ is the minimum value of $h$, for which, $B_{x_{h}}\succ a$ and,
consequently, by Proposition 5, $s^{u}(a)=x_{k+1}$.

∎

######  Lemma 1 (Inserting a reference set: consequences w.r.t. lower bound).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10) and if a set of
limiting profiles $B_{x}$ is inserted in $B$ in such a way that these
conditions are still fulfilled, and $s^{l}(a)=x_{s}$ we have either

$x_{s}<s^{l\ast}(a)=x<x_{s+1}$

or

$s^{l\ast}(a)=s^{l}(a).$

Moreover, $s^{l\ast}(a)=x$ if and only if

  * –

$x_{s}<x<x_{s+1}$; and.

  * –

$a\succ B_{x}$.

###### Proof.

If a set $B_{x}$ is added we can have the following cases:

  1. 1.

$x<s^{l}(a)$: in this case, by Proposition 3, we obtain $a\succ B_{x}$.
Consequently, the highest value $z$ from $X\cup\\{x\\}$ such that $a\succ
B_{z}$ is again $s^{l}(a)$, which, by Proposition 5, implies
$s^{l\ast}(a)=s^{l}(a)$;

  2. 2.

$s^{l}(a)=x_{s}<x<x_{s+1}$ and not($a\succ B_{x}$): also in this case the
maximal value $z$ from $X\cup\\{x\\}$ such that $a\succ B_{z}$ is $s^{l}(a)$,
which, again by Proposition 5, implies $s^{l\ast}(a)=s^{l}(a)$;

  3. 3.

$s^{l}(a)=x_{s}<x<x_{s+1}$ and $a\succ B_{x}$: in this case the highest value
$z$ from $X\cup\\{x\\}$, such that, $a\succ B_{z}$ is $x$, which, again by
Proposition 5, implies $s^{l\ast}(a)=x$;

  4. 4.

$s^{l}(a)=x_{s}<x_{s+1}<x$: in this case we cannot have $a\succ B_{x}$ because
from Condition 7 (that is, by the joint consideration of Conditions 2 and 4)
not($a\succ B_{x_{s+1}})$, but, by Proposition 1, $a\succ B_{x}$ would imply
$a\succ B_{x_{s+1}}$.

∎

######  Lemma 2 (Inserting a reference set: consequences for the upper bound).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10), and if a set of
limiting profiles $B_{x}$ is inserted in $B$ in such a way that these
conditions are still fulfilled, and $s^{u}(a)=x_{t}$ we have either

$x_{t-1}<s^{u\ast}(a)=x<x_{t}$

or

$s^{u\ast}(a)=s^{u}(a).$

Moreover, $s^{u\ast}(a)=x$ if and only if

  * –

$x_{t-1}<x<x_{t}$; and,

  * –

$B_{x}\succ a$.

###### Proof.

If a set $B_{x}$ is added we can have the following cases:

  1. 1.

$x>s^{u}(a)$: in this case, by Proposition 3, we obtain $B_{x}\succ a$.
Consequently, the lowest value $z$ from $X\cup\\{x\\}$ such that $B_{z}\succ
a$ is again $s^{u}(a)$, which, from Proposition 5, implies
$s^{u\ast}(a)=s^{u}(a)$;

  2. 2.

$s^{u}(a)=x_{t}>x>x_{t-1}$ and not($B_{x}\succ a$): also in this case the
lowest value $z$ from $X\cup\\{x\\}$ such that $B_{z}\succ a$ is $s^{u}(a)$,
which, again by Proposition 5, implies $s^{u\ast}(a)=s^{u}(a)$;

  3. 3.

$s^{u}(a)=x_{t}>x>x_{t-1}$ and $B_{x}\succ a$: in this case the lowest value
$z$ from $X\cup\\{x\\}$, such that, $B_{z}\succ a$ is $x$, which, again by
Proposition 5, implies $s^{u\ast}(a)=x$;

  4. 4.

$s^{u}(a)=x_{t}>x_{t-1}>x$: in this case one cannot have $B_{x}\succ a$
because by Condition 8 (that is, by the joint consideration of Conditions 3
and 5) not($B_{x_{t-1}}\succ a)$, but, by Proposition 1, $B_{x}\succ a$ would
imply $B_{x_{t-1}}\succ a$.

∎

######  Remark 5 (Deleting a reference set).

If both the primal and the dual soft dominance separability conditions (see
Points $a.2.p$ and $a.2.d$ of Condition 10) hold for $B$ and a set of limiting
profiles $B_{x}$ is deleted from $B$, clearly the primal and the dual soft
dominance separability conditions continue to be fulfilled.

######  Lemma 3 (Deleting a reference set: consequences w.r.t. lower bound).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see points $a.2.p$ and $a.2.d$ of Condition 10), a set of limiting
profiles $B_{x}$ is removed from $B$, and $s^{l}(a)=x_{s}$, then

  * –

$s^{l}(a)=x_{s}=x$ and $s^{l\ast}(a)=x_{s-1}$; or

  * –

$s^{l}(a)=x_{s}\neq x$ and $s^{l\ast}(a)=s^{l}(a).$

###### Proof.

According to Remark 5, Points $a.2.p$ and $a.2.d$ of Condition 10 hold also
after $B_{x}$ is removed from $B$. Consequently, from Proposition 5,
$s^{l\ast}(a)$ continues to be the highest value $z\in X\setminus\\{x\\}$ such
that $a\succ B_{z}$. We can have two cases

  1. $s^{l}(a)=x_{s}=x$: remember from Proposition 3, that the highest value $z\in X\setminus\\{x\\}$ such that $a\succ B_{z}$ becomes $x_{s-1}$ and, consequently, $s^{l\ast}(a)=x_{s-1}$;

  2. $s^{l}(a)=x_{s}\neq x$: the highest value $z\in X\setminus\\{x\\}$ such that $a\succ B_{z}$ continues to be $x_{s}$ and, consequently, $s^{l\ast}(a)=s^{l}(a)=x_{s}$.

∎

######  Lemma 4 (Deleting a reference set: consequences w.r.t. upper bound).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10), a set of limiting
profiles $B_{x}$ is removed from $B$, and $s^{u}(a)=x_{t}$, then

  * –

$s^{u}(a)=x_{t}=x$ and $s^{u\ast}(a)=x_{t+1}$; or,

  * –

$s^{u}(a)=x_{t}\neq x$ and $s^{u\ast}(a)=s^{u}(a)$.

###### Proof.

Remember that from Remark 5, Points $a.2.p$ and $a.2.d$ of Condition 10 hold
also after $B_{x}$ is removed from $B$, and, taking into account Proposition
5, $s^{u\ast}(a)$ continues to be the lowest value $z\in X\setminus\\{x\\}$
such that $B_{z}\succ a$. We can have two cases:

  1. 1.

$s^{u}(a)=x_{t}=x$: remember that from Proposition 4, the lowest value $z\in
X\setminus\\{x\\}$ such that $B_{z}\succ a$ becomes $x_{t+1}$ and,
consequently, $s^{u\ast}(a)=x_{t+1}$;

  2. 2.

$s^{u}(a)=x_{t}\neq x$: the lowest value $z\in X\setminus\\{x\\}$ such that
$B_{z}\succ a$ continues to be $x_{t}$ and, consequently,
$s^{u\ast}(a)=s^{u}(a)=x_{t}$.

∎

######  Lemma 5 (Inserting a reference action: consequences w.r.t. lower
bound).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10), and if a limiting
profile $b_{kp}$ is added to $B_{x_{k}}$ in such a way that these conditions
are still fulfilled, and $s^{l}(a)=x_{s}$, then

  * –

$s^{l\ast}(a)=x_{s-1}$ if and only if $b_{kp}\succ a$ and
$s^{l}(a)=x_{s}=x_{k}$;

  * –

$s^{l\ast}(a)=x_{s+1}$, if and only if $a\succ b_{kp}$, $x_{k}=x_{s+1}$ and
there is no $b_{s+1q}\in B_{x_{s+1}}$ such that $b_{s+1q}\succ a$;

  * –

$s^{l\ast}(a)=s^{l}(a)$, otherwise.

###### Proof.

If $b_{kp}\succ a$ and $s^{l}(a)=x_{s}=x_{k}$, after adding $b_{kp}$,we have
not($a\succ B_{x_{s}}$). Consequently, taking into consideration Proposition
3, the highest value $z$ such that $a\succ B_{z}$ becomes $x_{s-1}$, so that
$s^{l\ast}(a)=x_{s-1}$.

If $a\succ b_{kp}$, $x_{k}=x_{s+1}$ and there is no other $b_{kq}\in
B_{x_{s+1}}$ such that $b_{kq}\succ a$, after adding $b_{kp}$, we have $a\succ
B_{x_{s+1}}$. In this case, the highest value $z$ such that $a\succ B_{z}$ is
no more $x_{s}$ and it becomes $x_{s+1}$, in such a way that, according to
Proposition 5, we obtain $s^{l\ast}(a)=x_{s+1}$.

All the other possible cases are the following:

  1. 1.

$x_{k}<s^{l}(a)$: since $a\succ B_{s^{l}(a)}$ and after adding $b_{kp}$,
Points $a.2.p$ and $a.2.d$ of Condition 10 are still satisfied, then the
highest $z$ such that $a\succ B_{z}$ remains $s^{l}(a)$, which, from
Proposition 5, leads to $s^{l\ast}(a)=s^{l}(a)$;

  2. 2.

$x_{k}=s^{l}(a)$, but non$(b_{kp}\succ a)$: in this case, after adding the
limiting profile $b_{kp}$, we continue to have $a\succ B_{x_{k}}$, in such a
way that the highest $z$ such that $a\succ B_{z}$ remains $s^{l}(a)$, which,
from Proposition 5, leads to $s^{l\ast}(a)=s^{l}(a)$;

  3. 3.

$x_{k}=x_{s+1}$, but not($a\succ b_{kp}$) or there is at least one $b_{kq}\in
B_{x_{s+1}}$, such that, $b_{kq}\succ a$: in this case, after adding the
reference action $b_{kp}$, we continue to have not($a\succ B_{x_{s+1}}$), in
such a way that the highest $z$, such that, $a\succ B_{z}$ remains $s^{l}(a)$,
which, from Proposition 5, leads to $s^{l\ast}(a)=s^{l}(a)$;

  4. 4.

$x_{k}>x_{s+1}$: since not($a\succ B_{x_{s+1}}$) and since after adding
$b_{kp}$, Points $a.2.p$ and $a.2.d$ of Condition 10 are still satisfied,
then, from Proposition 1, we obtain not($a\succ B_{x_{k}}$), in such a way
that the highets $z$ such that $a\succ B_{z}$ remains $s^{l}(a)$ which, by
Proposition 5, leads to $s^{l\ast}(a)=s^{l}(a)$.

∎

######  Lemma 6 (Inserting a reference action: consequences w.r.t. upper
bound).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10), and if a limiting
profile $b_{kp}$ is added to $B_{x_{k}}$ in such a way that these conditions
are still fulfilled, and $s^{u}(a)=x_{t}$, then

  * –

$s^{u\ast}(a)=x_{t+1}$, if $a\succ b_{kp}$ and $s^{u}(a)=x_{t}=x_{k}$;

  * –

$s^{u\ast}(a)=x_{t-1}$, if $b_{kp}\succ a$, $x_{k}=x_{t-1}$ and there is no
$b_{kq}\in B_{x_{t-1}}$ such that $a\succ b_{kq}$;

  * –

$s^{u\ast}(a)=s^{u}(a)$, otherwise.

###### Proof.

If $a\succ b_{kp}$ and $s^{u}(a)=x_{t}=x_{k}$, then, after adding $b_{kp}$, we
have not($B_{x_{k}}\succ a$). Consequently, taking into consideration
Proposition 3, the lowest value $z$ such that $B_{z}\succ a$ becomes
$x_{t+1}$, in such a way that $s^{u\ast}(a)=x_{t+1}$.

If $b_{kp}\succ a$, $x_{k}=x_{t-1}$ and there is no other $b_{kq}\in
B_{x_{t-1}}$ such that $a\succ b_{kq}$, after adding $b_{kp}$, we have
$B_{x_{t-1}}\succ a$. In this case, the lowest value $z$ such that $B_{z}\succ
a$ is no more $x_{t}$ and it becomes $x_{t-1}$, in such a way that, according
to Proposition 5, we obtain $s^{u\ast}(a)=x_{t-1}$.

All the other possible cases are the following:

  1. 1.

$x_{k}>s^{u}(a)$: since $B_{s^{u}(a)}\succ a$ and after adding $b_{kp}$,
Points $a.2.p$ and $a.2.d$ of Condition 10 are still satisfied, the lowest $z$
such that $B_{z}\succ a$ remains $s^{u}(a)$, which, by Proposition 5, leads to
$s^{u\ast}(a)=s^{u}(a)$;

  2. 2.

$x_{k}=s^{u}(a)$, but not$(a\succ b_{kp})$: in this case, after adding the
reference action $b_{kp}$, we continue to have $B_{x_{k}}\succ a$, in such a
way that the lowest $z$ such that $B_{z}\succ a$ remains $s^{u}(a)$, which, by
Proposition 5, leads to $s^{u\ast}(a)=s^{u}(a)$;

  3. 3.

$x_{k}=x_{t-1}$, but not($b_{kp}\succ a$) or there is at least one $b_{kq}\in
B_{x_{t-1}}$ such that $a\succ b_{kq}$: in this case, after adding the
reference action $b_{kp}$, we continue to have not($B_{x_{t-1}}\succ a$), in
such a way that the lowest $z$ such that $B_{z}\succ a$ remains $s^{u}(a)$,
which, by Proposition 5, leads to $s^{u\ast}(a)=s^{u}(a)$;

  4. 4.

$x_{k}<x_{t-1}$: since not($B_{x_{t-1}}\succ a$) and since after adding
$b_{kp}$ Points $a.2.p$ and $a.2.d$ of Condition 10 are still satisfied, then,
by Proposition 1, we obtain not($B_{x_{k}}\succ a$), in such a way that the
lowest $z$ such that $B_{z}\succ a$ remains $s^{u}(a)$, which, by Proposition
5, leads to $s^{u\ast}(a)=s^{u}(a)$.

∎

######  Lemma 7 (Deleting a reference action: consequences w.r.t. lower
bound).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10), and if a limiting
profile $b_{kp}$ is removed from $B_{x_{k}}$, in such a way that, these
conditions are still fulfilled, and $s^{l}(a)=x_{s}$, then

  * –

$s^{l\ast}(a)=x_{s-1}$, if $s^{l}(a)=x_{k}$, $a\succ b_{kp}$ and there is no
other $b_{kq}\in B_{x_{k}}$ such that $a\succ b_{kq}$,

  * –

$s^{l\ast}(a)=x_{s+1}$, if $b_{kp}\succ a$, $x_{k}=x_{s+1}$, there is no other
$b_{kq}\in B_{x_{s+1}}$ such that $b_{kq}\succ a$ and there exists at least
one reference action $b_{kr}\in B_{x_{s+1}}$ such that $a\succ b_{kr}$;

  * –

$s^{l\ast}(a)=s^{l}(a)$, otherwise.

###### Proof.

If $s^{l}(a)=x_{k}$, $a\succ b_{kp}$ and there is no other $b_{kq}\in
B_{x_{k}}$ such that $a\succ b_{kq}$, then after $b_{kp}$ is removed we have
not$(a\succ B_{x_{k}})$. Consequently, taking into consideration Proposition
3, the highest value $z$ such that $a\succ B_{z}$ becomes $x_{s-1}$, in such a
way that $s^{l\ast}(a)=x_{s-1}$.

If $b_{kp}\succ a$, $x_{k}=x_{s+1}$, there is no other $b_{kq}\in B_{x_{s+1}}$
such that $b_{kq}\succ a$ and there exists at least one limiting profile
$b_{kr}\in B_{x_{s+1}}$ such that $a\succ b_{kr}$, after removing $b_{kp}$ we
have $a\succ B_{x_{s+1}}$, in such a way that the highest $z$ such that
$a\succ B_{z}$ becomes $x_{s+1}$ and, consequently, from Proposition 5,
$s^{l\ast}(a)=x_{s+1}$.

All the other possible cases are the following:

  1. 1.

$x_{k}<s^{l}(a)$: since after removing $b_{kp}$ the highest $z$ such that
$a\succ B_{z}$ remains $s^{l}(a)$ and Points $a.2.p$ and $a.2.d$ of Condition
10 are still satisfied, by Proposition 5, we obtain $s^{l\ast}(a)=s^{l}(a)$;

  2. 2.

$x_{k}=s^{l}(a)$, and not($a\succ b_{kp}$) or there is at least another
$b_{kq}\in B_{x_{k}}$ such that $a\succ b_{kq}$: in this case, after removing
the limiting profile $b_{kp}$, we continue to have $a\succ B_{s^{l}(a)}$, and,
therefore, $s^{l\ast}(a)=s^{l}(a)$;

  3. 3.

$x_{k}=x_{s+1}$, but not($b_{kp}\succ a$) or there is at least another
$b_{kq}\in B_{x_{s}+1}$ such that $b_{kq}\succ a$ or there does not exist any
reference action $b_{kr}\in B_{x_{s+1}}$ such that $a\succ b_{kr}$: in this
case, after removing the limiting profile $b_{kp}$, we continue to have
not($a\succ B_{x_{s+1}}$), in such a way that the highest $z$ such that
$B_{z}\succ a$ remains $s^{l}(a)$, which gives $s^{l\ast}(a)=s^{l}(a)$;

  4. 4.

$x_{k}>x_{s+1}$: since not($a\succ B_{x_{s+1}}$) and since after removing
$b_{kp}$ Points $a.2.p$ and $a.2.d$ of Condition 10 are still satisfied, then,
by Proposition 1 we obtain not($a\succ B_{x_{k}}$), in such a way that the
highest $z$ such that $a\succ B_{z}$ remains $s^{l}(a)$, which leads to
$s^{l\ast}(a)=s^{l}(a)$.

∎

######  Lemma 8 (Deleting a reference action: consequences w.r.t. upper
bound).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10) and if a limiting
profile $b_{kp}$ is removed from $B_{x_{k}}$ in such a way that these
conditions are still fulfilled, and $s^{u}(a)=x_{t}$, then

  * –

$s^{u\ast}(a)=x_{t+1}$, if $s^{u}(a)=x_{k}$, $b_{kp}\succ a$ and there is no
other $b_{kq}\in B_{x_{k}}$ such that $b_{kq}\succ a$,

  * –

$s^{u\ast}(a)=x_{t-1}$, if $a\succ b_{kp}$, $x_{k}=x_{t-1}$, there is no
$b_{kq}\in B_{x_{t-1}}$ such that $a\succ b_{kq}$ and there is at least one
reference action $b_{kr}\in B_{x_{t-1}}$ such that $b_{kr}\succ a$;

  * –

$s^{l\ast}(a)=s^{u}(a)$, otherwise.

###### Proof.

If $s^{u}(a)=x_{k}$, $b_{kp}\succ a$ and there is no other $b_{kq}\in
B_{x_{k}}$ such that $b_{kq}\succ a$, then after $b_{kp}$ is removed one has
non$(B_{x_{k}}\succ a)$. Consequently, taking into consideration Proposition
3, the lowest value $z$ such that $B_{z}\succ a$ becomes $x_{t+1}$, in such a
way that $s^{u\ast}(a)=x_{t+1}$.

If $a\succ b_{kp}$, $x_{k}=x_{t-1}$, there is no other $b_{kq}\in B_{x_{t-1}}$
such that $a\succ b_{kq}$ and there is at least one limiting profile
$b_{kr}\in B_{x_{t-1}}$ such that $b_{kr}\succ a$, after removing $b_{kp}$ one
has $B_{x_{t-1}}\succ a$, in such a way that the lowest $z$ such that $a\succ
B_{z}$ becomes $x_{t-1}$ and, consequently, from Proposition 5,
$s^{u\ast}(a)=x_{t-1}$.

All the other possible cases are the following:

  1. 1.

$x_{k}>s^{u}(a)$: since after removing $b_{kp}$ the lowest $z$ such that
$B_{z}\succ a$ remains $s^{u}(a)$ and Points $a.2.p$ and $a.2.d$ of Condition
10 are still satisfied, by Proposition 5, we obtain $s^{u\ast}(a)=s^{u}(a)$;

  2. 2.

$x_{k}=s^{u}(a)$, and not($b_{kp}\succ a$) or there is at least another
$b_{kq}\in B_{x_{k}}$ such that $b_{kq}\succ a$: in this case, after removing
the reference action $b_{kp}$, we continue to have $B_{s^{u}(a)}\succ a$, and,
therefore, $s^{u\ast}(a)=s^{u}(a)$;

  3. 3.

$x_{k}=x_{t-1}$, but not($a\succ b_{kp}$) or there is at least another
$b_{kq}\in B_{x_{t-1}}$ such that $a\succ b_{kq}$, or there does not exist any
reference action $b_{kr}\in B_{x_{t-1}}$ such that $b_{kr}\succ a$: in this
case, after removing the reference action $b_{kp}$, we continue to have
not($B_{x_{t-1}}\succ a$), in such a way that the lowest $z$ such that
$B_{z}\succ a$ remains $s^{u}(a)$, which leads to $s^{u\ast}(a)=s^{u}(a)$;

  4. 4.

$x_{k}<x_{t-1}$: since not($B_{x_{t-1}}\succ a$) and since after removing
$b_{kp}$ Points $a.2.p$ and $a.2.d$ of Condition 10 are still satisfied, then,
by Proposition 1, we obtain not($B_{x_{k}}\succ a$), in such a way that the
lowest $z$, such, that $B_{z}\succ a$ remains $s^{u}(a)$, which leads to
$s^{u\ast}(a)=s^{u}(a)$.

∎

Putting together the results of Lemmas 1-8 we get the following general
result.

######  Theorem 2 (Stability).

If $B$ fulfills both the primal and the dual soft dominance separability
conditions (see Points $a.2.p$ and $a.2.d$ of Condition 10) then the stability
condition holds.

##  6 Practical issues and an illustrative example

This section presents some practical aspects of the method and an illustrative
example

###  6.1 Building the set of reference actions

Let us start by observing that in defining the collection of sets of
references actions

$B=\\{B_{x_{1}},\ldots,B_{x_{k}},\ldots,B_{x_{\ell}}\\}$

it is necessary to take care that all actions $a$ from $A$ the following
condition is satisfied

$B_{x_{\ell}}\succ a\succ B_{x_{1}}.$ (11)

Condition (11) permits to compare all actions $a$ from $A$ with sets $B_{x}$
of reference actions, so that a range $]s^{l}(a),s^{u}(a)[$ for the score
$s(a)$ is assigned to each $a\in A$.

There are different ways of building the set of reference actions. Next we
present two different ones.

#### Direct method

We ask the decision-maker to propose a set of reference actions that are used
to characterize some scoring levels, for example, $20$, $40$, $60$, and $80$.
Several reference actions can be proposed _a priori_ to characterize the same
scoring level. We assume that the scales are bounded from below and from above
in such a way that we can consider reference the actions $b_{0}$ and
$b_{100}$, characterizing the scores of $0$ and $100$ with actions having, on
each considered criterion, the lower bound performances and the upper bound
performances, respectively.

#### A deck of cards based technique

We can also ask the decision-maker to propose a set of actions to be taken as
reference actions and use the deck of cads method in a similar way as in
Figueira and Roy (2002). These actions must be ordered from the worst to the
best, with the possibility of some ties. The decision-maker is required to put
some blank cards between these equivalent sets of reference actions in order
to assign a score to each equivalence class, on the basis of the extreme
scores $0$ and $100$. The scores thus obtained are not necessarily equally
spaced. It should be noticed that other values different from $0$ and $100$
can be used (this is also true when using the previous method).

###  6.2 The method

The method is now presented in a very simple way. Remember that in order to
assign a score $s(a)$ to the action $a\in A$, we should firstly choose a
$\lambda$ cutting level to transform the fuzzy outranking relation into a
crispy one.

  1. 1.

Find the set of reference actions, $B_{x}$, with the highest score $x$, such
that $a\succ B_{x}$ and, for all $x^{\prime}<x$, we have either $a\succ
B_{x^{\prime}}$ or $a\parallel B_{x^{\prime}}$. It is then natural to consider
a score $s(a)$ such that $s(a)>x=s^{l}(a)$, and indeed, according Definition
6, we have $s^{l}(a)=x$. Let us remember also that by Proposition 5, if both
the primal and the dual soft dominance separability conditions (see Condition
10, points $a.2.p$ and $a.2.d$) hold, then $s^{l}(a)$ is simply the highest
value, such that $a\succ B_{x}$.

  2. 2.

Find the set of reference actions, $B_{x}$, with the lowest score $x$, such
that $B_{x}\succ a$ and, for all $x^{\prime}>x$, we have either
$B_{x^{\prime}}\succ a$ or $a\parallel B_{x^{\prime}}$. It is then natural to
consider a score $s(a)$ such that $s(a)<x=s^{u}(a)$, and indeed, according
Definition 7, we have $s^{u}(a)=x$. Let us remember also that by Proposition
5, if both the primal and the dual soft dominance separability conditions (see
Condition 10, points $a.2.p$ and $a.2.d$) hold, then $s^{u}(a)$ is simply the
lowest value, such that $B_{x}\succ a$.

###  6.3 Illustrative example

We consider the example in Figueira et al. (2009) regarding the evaluation of
some sites for the location of a new hotel. The set of criteria is as follows:

  1. 1.

Investment costs (Scale unit: $K$€; Code: ICOST; notation: $g_{1}$; preference
direction: minimization). This criterion comprises the land purchasing costs
as well as the costs for building the new hotel.

  2. 2.

Annual costs (Scale unit: $K$€; Code: ACOST; notation: $g_{2}$; preference
direction: minimization). This criterion comprises the hotel operating costs.

  3. 3.

Recruitment (Scale unit: verbal levels (seven); Code: RECRU; notation:
$g_{3}$; preference direction: maximization). This criterion models the
possibility of recruiting workers.

  4. 4.

Image (Scale unit: verbal levels (seven); Code: IMAGE; notation: $g_{4}$;
preference direction: maximization). This criterion models the perceptions of
the clients about the district where the new hotel will be located.

  5. 5.

Access (Scale unit: verbal levels (seven); Code: ACCES; notation: $g_{5}$;
preference direction: maximization). This criterion models the possibility of
recruiting workers.

The verbal scale used for the last three criteria comprises the following
levels (in between parenthesis we used a numerical code for each level): very
bad[1]; bad[2]; rather bad[3]; average[4]; rather good[5]; good[6]; very
good[7].

There are five potential sites for the location of the new hotel. The
performance table can be presented as follows (Table 1).

$a$ |  ICOST ($g_{1}$) |  ACOST ($g_{2}$) |  RECRU ($g_{3}$) |  IMAGE ($g_{4}$) |  ACCES ($g_{5}$)
---|---|---|---|---|---
$a_{1}$ | $13\;000$ | $3\;000$ | 4 | 4 | 4
$a_{2}$ | $15\;000$ | $2\;500$ | 6 | 2 | 7
$a_{3}$ | $10\;900$ | $3\;400$ | 6 | 6 | 1
$a_{4}$ | $15\;500$ | $3\;500$ | 6 | 6 | 6
$a_{5}$ | $15\;000$ | $2\;600$ | 6 | 1 | 2
Table 1: Performance table

The weights, discriminating (indifference and preference) thresholds used in
the method are the following (see Table 2). Let us consider an ordered pair of
actions $(a,b)\in A\times A$. The performance of $b$ is assumed to be worse
than the performance of $a$. This means that the variable thresholds presented
in the next table with respect to criteria $g_{1}$ and $g_{2}$ are direct
variable thresholds (see Roy et al., 2014). For the sake of simplicity, no
veto thresholds are considered in this example. For the remaining criteria the
thresholds are constant (the numbers represent the differences of levels, not
the scale levels. Thus, the indifference threshold is a difference of one
performance level, while the preference threshold corresponds to a difference
of two performance levels).

Parameters |  ICOST ($g_{1}$) |  ACOST ($g_{2}$) |  RECRU ($g_{3}$) |  IMAGE ($g_{4}$) |  ACCES ($g_{5}$)
---|---|---|---|---|---
$k_{j}$ | 5 | 4 | 3 | 3 | 3
$q_{j}(g_{j}(b))$ | $250+0.03g_{1}(b)$ | $\;\,50+0.05g_{2}(b)$ | 1 | 1 | 1
$p_{j}(g_{j}(b))$ | $500+0.05g_{1}(b)$ | $100+0.07g_{2}(b)$ | 2 | 2 | 2
Table 2: Parameters table

Our set of limiting profiles is composed of seven subsets, i.e.,
$B=\\{B_{x_{1}},B_{x_{2}},B_{x_{3}},B_{x_{4}},B_{x_{5}},B_{x_{6}},$
$B_{x_{7}}\\}$. This means that will be defined seven reference values
$X=\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}\\}$. These values were
obtained with the deck of cards method proposed in Bottero et al. (2018) to
build interval scales, by fixing two reference levels. In this work, and
without loss of generality, we will put $x_{1}=0$ and $x_{7}=100$.

The $B$ sets are characterized by at least a single limiting profile as
follows.

  * $B_{x_{1}=0}=\\{b_{11}=(18\,000,4\,000,1,1,1)\\}$;

  * $B_{x_{2}}=\\{b_{21}=(17\,000,3\,500,2,2,1),\;b_{22}=(16\,500,3\,700,1,2,1)\\}$;

  * $B_{x_{3}}=\\{b_{31}=(15\,350,3\,200,3,1,2)\\}$;

  * $B_{x_{4}}=\\{b_{41}=(14\,250,2\,850,3,4,3),\;b_{42}=(13\,750,3\,150,4,3,3)\\}$;

  * $B_{x_{5}}=\\{b_{51}=(12\,650,2\,650,4,4,5)\\}$;

  * $B_{x_{6}}=\\{b_{61}=(11\,500,2\,100,5,6,5),\;b_{62}=(11\,000,2\,500,6,5,7)\\}$;

  * $B_{x_{7}=100}=\\{b_{71}=(10\,000,2\,000,7,7,7)\\}$;

The deck of cards method applied to this problem works as follows.

  1. 1.

The subsets of reference profiles are totally ordered as follows ($\prec$
means “strictly less preferred than”):

$B_{x_{1}}\prec B_{x_{2}}\prec B_{x_{3}}\prec B_{x_{4}}\prec B_{x_{5}}\prec
B_{x_{6}}\prec B_{x_{7}}.$

  2. 2.

We then call the attention of the decision-maker to the fact that if the
difference between two consecutive sets is bigger than the difference of other
pair of consecutive subsets, she/he should add more blank cards in the first
two consecutive ones than in the second ones (for more details see Bottero et
al., 2018). We may obtain the following ranking of the subsets with the blank
cards in between consecutive ones (in brackets):

$B_{x_{1}}\;[1]\;B_{x_{2}}\;[2]\;B_{x_{3}}\;[0]\;B_{x_{4}}\;[1]\;B_{x_{5}}\;[0]\;B_{x_{6}}\;[2]\;B_{x_{7}}.$

Zero blank cards between two consecutive sets of limiting profiles does not
mean that the limiting profiles of the two consecutive sets have the same
value, but only that the difference is minimal. We know that the number of
units between $B_{x_{1}}$ and $B_{x_{7}}$ is
$\alpha=(1+1)+(2+1)+(0+1)+(1+1)+(0+1)+(2+1)=12$.

  3. 3.

In this step we should define the two reference levels. As stated before, we
considered that the value of all the limiting profiles in $B_{x_{1}}$ is $0$,
i.e., $x_{1}=0$ and that the value of all the limiting profiles in $B_{x_{7}}$
is $100$, i.e., $x_{7}=100$.

  4. 4.

Consequently, we can compute the value of the unit as follows.

$u=\frac{x_{7}-x_{1}}{h}=\frac{100-0}{12}=8.333333.$

  5. 5.

The computations of the remaining values in $X$ is easy since we know the
number of units separating two consecutive sets of limiting profiles. We have
thus,

$x_{1}=0,\;x_{2}=25,\;x_{3}=33.33333333,\;x_{4}=50,\;x_{5}=58.33333333,\;x_{6}=83.33333333,\;x_{7}=100.$

Let us consider now the comparison table of all of our five sites against the
limiting profiles (Table 3).

$b$ | $a_{1}\succ b$ | $a_{2}\succ b$ | $a_{3}\succ b$ | $a_{4}\succ b$ | $a_{5}\succ b$ | $b\succ a_{1}$ | $b\succ a_{2}$ | $b\succ a_{4}$ | $b\succ a_{4}$ | $b\succ a_{5}$
---|---|---|---|---|---|---|---|---|---|---
$b_{11}$ | $\succ$ | $\succ$ | $\succ$ | $\succ$ | $\succ$ |  |  |  |  |
$b_{21}$ | $\succ$ | $\succ$ | $\succ$ | $\succ$ | $\succ$ |  |  |  |  |
$b_{22}$ | $\succ$ | $\succ$ | $\succ$ | $\succ$ | $\succ$ |  |  |  |  |
$b_{31}$ | $\succ$ | $\succ$ | $\succ$ | $\succ$ | $\succ$ |  |  |  |  |
$b_{41}$ |  | $\succ$ |  | $\succ$ | $\succ$ |  |  |  |  |
$b_{42}$ |  | $\succ$ | $\succ$ |  |  |  |  |  |  |
$b_{51}$ |  |  |  |  |  |  |  |  | $\succ$ | $\succ$
$b_{61}$ |  |  |  |  |  | $\succ$ | $\succ$ | $\succ$ | $\succ$ | $\succ$
$b_{62}$ |  |  |  |  |  | $\succ$ | $\succ$ | $\succ$ | $\succ$ | $\succ$
$b_{71}$ |  |  |  |  |  | $\succ$ | $\succ$ | $\succ$ | $\succ$ | $\succ$
Table 3: Comparison table

Now, for the definition of the score range of each alternative we can take
advantage of this table. Let us consider action $a_{1}$ and try to identify
$s^{l}(a_{1})<s(a)<s^{u}(a_{1})$. As for the lower bound it is provided by
$x_{3}$ and the upper bound by $x_{6}$. Thus, the range for score of $a_{1}$
becomes.

$33.33333<s(a_{1})<83.33333.$

With the same procedure we can derive the range for all the five actions:

  * $33.33333<s(a_{1})<83.33333$;

  * $50.00000<s(a_{2})<83.33333$;

  * $50.00000<s(a_{3})<83.33333$;

  * $33.33333<s(a_{4})<58.33333$;

  * $33.33333<s(a_{5})<58.33333$;

##  7 Conclusions

In this paper we presented a new method for assigning a score range to each
action. In this sense it can provide a more robust conclusion about the
possible scores such an action can have. The theoretical soundness of the
method has been proved, i.e., the fundamental requirements of uniqueness,
independence, monotonicity, conformity, homogeneity, and stability with
respect to insertion and deletion operations, became properties of the method,
which provide some consistency to the method.

All the main strengths of Electre methods are present in this new method: it
deals with different types of scales without the need of converting them into
a single unit; it is able to cope with the imperfect knowledge of data and
arbitrariness when building the criteria; it takes into account the reasons
for and against an outranking; and, it avoids the compensatory phenomenon in a
systematic way. In addition, the method is able to provide a score range for
each action, which, before was considered a weak point of Electre methods.

Future research lines, start from software development and real-work
applications. They include also several extensions, as for example,

  * •

the design of a hierarchical Electre-Score method, as in Corrente et al.
(2013, 2016) and Del Vasto-Terrientes et al. (2015), allowing in our case to
give ranges of scores for different macrocriteria in the hierarchical trees of
criteria;

  * •

the application of Monte-Carlo based pseudo-robustness analysis in the same
line as in Corrente et al. (2017) to provide more accurate score ranges;

  * •

the use of robust ordinal techniques and other robustness techniques in the
same line of Greco et al. (2011) and Kadziński and Ciomek (2016);

  * •

the development evolutionary approaches for making the computations when in
presence of large sets of data as in Doumpos et al. (2009).

## Acknowledgements

José Rui Figueira acknowledges the support the Fundação para a Ciência e
Tecnologia (FCT) grant SFRH/BSAB/139892/2018 under POCH program and from the
European Regional Development Fund (ERDF) as well as FCT for supporting the
DECIdE project (LISBOA-01-0145-FEDER-024135), and for partially funding the
current research. Salvatore Greco wishes to acknowledge funding by the FIR of
the University of Catania BCAEA3 “New developments in Multiple Criteria
Decision Aiding (MCDA) and their application to territorial competitiveness”.

##  A Appendix

In this Appendix we will present the main concepts and steps that lead to the
construction of a credibility degree, $\sigma(a,b)$ for the pair of actions
$(a,b)$.

State-of-the art versions of Electre methods make use of the so-called
_pseudo-criterion_ model (Figueira et al., 2016; Roy, 1996; Roy and Bouyssou,
1993; Roy and Vincke, 1984) when comparing two actions, $a$ and $b$, on
criterion $g_{j}$, from their performances, $g_{j}(a)$ and $g_{j}(b)$,
respectively. This model associates with each criterion function,
$g_{j}(\cdot)$, two threshold functions: an _indifference threshold_ function,
denoted by $q_{j}(\cdot)$, and a _preference threshold_ function, denoted by
$p_{j}(\cdot)$. Assume that $g_{j}$ is a criterion to be maximized and that
the performance $g_{j}(a)$ is better than the performance $g_{j}(b)$. The
threshold functions or simply thresholds can be constant or may vary in a
direct way, i.e., with respect to the worst performance, $q_{j}(g_{j}(b))$ and
$p_{j}(g_{j}(b))$, or in an inverse way, i.e., with respect to the best
performance, $q_{j}(g_{j}(a))$ and $p_{j}(g_{j}(a))$. For the sake of
simplicity and without loss of generality, we consider in what follows that
the thresholds are constant and use the simple notation $q_{j}$ and $p_{j}$,
for the indifference and preference thresholds, respectively.

It is very important to note that the main purpose of these thresholds is not
to model the preferences, but rather the imperfect knowledge of data as it can
be seen in Roy et al. (2014).

The choice of a pseudo-criterion model for the comparison of two actions, $a$
and $b$, from their performances on criterion $g_{j}$, leads to the definition
of three _per_ -criterion binary relations, as follows.

  * –

A per-criterion _indifference binary relation_ , which is used to model a
situation in which $a$ is indifferent to $b$ on criterion $g_{j}$, denoted by
$a\sim_{j}b$; this occurs whenever $|(g_{j}(a)-g_{j}(b))|\leqslant q_{j}$. In
other words, a situation where no one of the two actions, $a$ and $b$, has a
significant advantage over the other on the considered criterion. Let $C(a\sim
b)$ denote the set or coalition of criteria for which $a$ is indifferent to
$b$.

  * –

A per-criterion _strict preference binary relation_ , which is used to model a
situation in which $a$ is strictly preferred to $b$ on criterion $g_{j}$,
denoted by $a\succ_{j}b$; this occurs whenever $(g_{j}(a)-g_{j}(b))>p_{j}$. In
other words, a situation where action $a$ has a significant advantage over $b$
on the considered criterion. Let $C(a\succ b)$ denote the set or coalition of
criteria for which $a$ is strictly preferred to $b$.

  * –

A per-criterion _weak preference binary relation_ , which is used to model
hesitation situations of $a$ with respect to $b$ on criterion $g_{j}$, denoted
by $a\succsim^{?}_{j}b$; this occurs whenever
$q_{j}<(g_{j}(a)-g_{j}(b))\leqslant p_{j}$. In other words, a situation where
there is an ambiguity zone between indifference and strict preference of $a$
over $b$ on the considered criterion. Let $C(a\succsim^{?}b)$ denote the set
or coalition of criteria for which $a$ is weakly preferred to $b$. Note that
the word weak has nothing to do with intensities of preferences, it models
hesitation or ambiguity (due to the imperfect knowledge of data), not
preferences.

Whenever $a$ is indifferent, weak, or strict preferred to $b$, on criterion
$g_{j}$, we say that “$a$ outranks $b$” since $a$ is at least as good as $b$
in a _stricto sensu_ on this criterion. This situation thus occurs, when
$a\sim_{j}b$, $a\succ_{j}b$, or $a\succsim^{?}_{j}b$, and can be denoted by
$a\succsim_{j}b$. In a more _lato sensu_ we can also say that “$a$ outranks
$b$” when $b\succsim^{?}_{j}a$ since there is hesitation between $b\sim_{j}a$
and $b\succ_{j}a$ on criterion $g_{j}$.

As all outranking based methods, Electre methods also make use of the _per_
-criterion outranking relations to build one or several comprehensive
outranking relations. In the method proposed in this paper only one
comprehensive outranking relation is considered, which allows to conclude
whether or not “$a$ comprehensively outranks $b$”, denoted by $a\succsim b$.
More precisely, to conclude about the assertion “$a$ outranks $b$”, the
strength of the coalition in its favor of should be powerful enough to
overcome the opposition effect of the coalition against this assertion. How
should the power of the coalition in favor (or concordant with the assertion)
and the effects of the coalition against (or discordant with the assertion) be
measured? To model the power of the concordant coalition is modeled and
measured in Electre methods through what is called in these methods a
_comprehensive concordance index_ , while the opposition effect of each
criterion is modeled and measured to what is called a _per-criterion
discordance index_. Both will be combined to devise a _credibility_
(outranking) _index_ for each ordered pair of alternatives, $(a,b)\in A\times
A$. Next, we will present the three main steps to obtain the credibility
index.

  1. 1.

Computing the _comprehensive concordance index_ $c(a,b)$. Again, for the sake
of simplicity, the formula we present next, for this index, is the classical
one as in Roy and Bouyssou (1993). A more sophisticated and recent version of
the concordance index, which takes into account the interaction between
criteria can also be used in this context (see Figueira et al., 2009) with no
additional changes in the method proposed in this paper. As said before the
concordance index is used for measuring the power of the concordant coalition,
where each criterion $g_{j}$ contributes with its relative importance
coefficient or weigh, $w_{j}$, for $j=1,\ldots,n$ (we assume w.l.o.g. that
$\sum_{j=1}^{n}w_{j}=1$). The formula for the index can thus be stated as
follows.

${\displaystyle
c(a,b)=\sum_{C(a\\{\sim,\succsim^{?},\succ\\}b)}w_{j}+\sum_{C(b\succsim^{?}a)}\varphi_{j}w_{j}},$

where

${\displaystyle\varphi=\frac{(g_{j}(a)-g_{j}(b))+p_{j}}{p_{j}-q_{j}}\in[0,1]}.$

This means that if a criterion $g_{j}$ belongs to the concordant coalition
_stricto sensu_ , i.e., $g_{j}\in C(a\\{\sim,\succsim^{?},\succ\\}b$ its
contribution to the coalition power corresponds to its total weight, $w_{j}$,
but if this criterion belongs to $C(b\succsim^{?}a)$ it only contributes with
a fraction of its weight, $\varphi_{j}w_{j}$.

  2. 2.

Computing the _per-criterion discordance indices_ $d_{j}(a,b)$,
$j=1,\ldots,n$. To model the opposition effect of each criterion against the
concordant coalition _lato sensu_ , i.e., when $g_{j}\in C(b\succ a)$, it is
necessary to introduce another concept and preference parameter, the _veto
threshold_ $v_{j}(\cdot)$. This threshold can also be constant or vary in a
direct or indirect way as in case of indifference and preference thresholds.
To render things simple and without loss of generality we will keep its value
constant, and denoted it simply by $v_{j}$. A criterion $g_{j}$ is discordant
with the assertion “$a$ outranks $b$”, when the difference of performances
$(g_{j}(b)-g_{j}(a))$ is considered significantly large to validate such an
assertion. The more or less degree of discordance of each criterion can be
measured through a _per_ -criterion discordance index of the form,

$d_{j}(a,b)=\left\\{\begin{array}[]{ccl}1&\mbox{if}&\;\;\,v_{j}>(g_{j}(a)-g_{j}(b))\\\
{\displaystyle\frac{(g_{j}(a)-g_{j}(b))+p_{j}}{p_{j}-v_{j}}}&\mbox{if}&-v_{j}\leqslant(g_{j}(a)-g_{j}(b))<-p_{j}\\\
0&\mbox{if}&\hskip 32.72049pt(g_{j}(a)-g_{j}(b))\geqslant-p_{j}\\\
\end{array}\right.$

  3. 3.

Computing the _credibility index_ $\sigma(a,b)$. This index measures the
credibility degree of the outranking relation, i.e., the degree in which $a$
outranks $b$. It can be modeled though the following formula.

${\displaystyle\sigma(a,b)=c(a,b)\prod_{j=1}^{n}T_{j}(a,b)},$

where

$T_{j}(a,b)=\left\\{\begin{array}[]{cl}{\displaystyle\frac{1-d_{j}(a,b)}{1-c(a,b)}}&\mbox{if}\;\,g_{j}(a,b)>c(a,b)\\\
1&\mbox{otherwise}\\\ \end{array}\right.$

It is thus a fuzzy measure. It can be converted into a crispy by making use of
a cutting-off level, denote by $\lambda$ as in Section 2.

For the main features, advantages, and drawbacks of Electre the reader can
refer to Figueira et al. (2013).

## References

  * Bottero et al. (2018) Bottero, M., Ferretti, V., Figueira, J.R., Greco, S., B. Roy, B., 2018.  On the Choquet multiple criteria preference aggregation model: Theoretical and practical insights from a real-world application.  European Journal of Operational Research 271, 120–140.
  * Corrente et al. (2017) Corrente, S., Figueira, J.R., Greco, S., Słowiński, R., 2017\.  A robust ranking method exending electre iii to hierarchy of interacting criteria, imprecise weights and stochastic analysis.  Omega, The International Journal of Management Science 73, 1–17.
  * Corrente et al. (2013) Corrente, S., Greco, S., Słowiński, R., 2013.  Multiple Criteria Hierarchy Process with electre and promethee.  Omega, The International Journal of Management Science 41, 820–846.
  * Corrente et al. (2016) Corrente, S., Greco, S., Słowiński, R., 2016.  Multiple Criteria Hierarchy Process for electre tri methods.  European Journal of Operational Research 252, 191–203.
  * Del Vasto-Terrientes et al. (2015) Del Vasto-Terrientes, L., Valls, A., Słowiński, R., Zielniewicz, P., 2015\.  Electre-III-H: An outranking-based decision aiding method for hierarchically structured criteria.  Expert Systems with Applications 42, 4910–4926.
  * Doumpos et al. (2009) Doumpos, M., Marinakis, Y., Marinaki, M., Zopounidis, C., 2009\.  An evolutionary approach to construction of outranking models for multicriteria classification: The case of the electre tri method.  European Journal of Operational Research 199, 496–505.
  * Fernández et al. (2017) Fernández, E., Figueira, J.R., Navarro, J., Roy, B., 2017\.  Electre Tri-nB A new multiple criteria ordinal classification method.  European Journal of Operational Research 263, 214–224.
  * Figueira and Roy (2002) Figueira, J., Roy, B., 2002\.  Determining the weights of criteria in the electre type methods with a revised Simos’ procedure.  European Journal of Operational Research 139, 317–326.
  * Figueira et al. (2009) Figueira, J.R., Greco, S., Roy, B., 2009.  Electre methods with interaction between criteria: An extension of the concordance index.  European Journal of Operational Research 199, 478–495.
  * Figueira et al. (2013) Figueira, J.R., Greco, S., Słowiński, R., Roy, B., 2013\.  An overview of electre methods and their recent extensions.  Journal of Multi-Criteria Decision Analysis 20, 61–85.
  * Figueira et al. (2016) Figueira, J.R., Mousseau, V., Roy, B., 2016.  Electre methods, in: Greco, S., Ehrgott, M., Figueira, J.R. (Eds.), Multiple Criteria Decision Analysis: State of the Art Surveys. Springer Science + Business Media Inc., New York, NY, USA, second edition. pp. 155–185.
  * Govindan and Jepsen (2016) Govindan, K., Jepsen, M.B., 2016\.  Electre: A comprehensive literature review on methodologies and applications.  European Journal of Operational Research 250, 1–29.
  * Greco et al. (2016) Greco, S., Ehrgott, M., Figueira, J.R. (Eds.), 2016.  Multiple Criteria Decision Analysis: State of the Art Surveys.  Springer Science + Business Media Inc., New York, NY, USA. second edition.
  * Greco et al. (2011) Greco, S., Kadziński, M., Mousseau, V., Słowiński, R., 2011\.  ElectreGKMS: Robust Ordinal Regression for outranking methods.  European Journal of Operational Research 214, 118–135.
  * Greco et al. (2001) Greco, S., Matarazzo, B., Słowiński, R., 2001.  Rough sets theory for multicriteria decision analysis.  European Journal of Operational Research 129, 1–47.
  * Kadziński and Ciomek (2016) Kadziński, M., Ciomek, K., 2016\.  Integrated framework for preference modeling and robustness analysis for outranking-based multiple criteria sorting with electre and promethee.  Information Sciences 352, 167–187.
  * Keeney and Raiffa (1993) Keeney, R.L., Raiffa, H., 1993\.  Decisions with Multiple Objectives: Preferences and Value Tradeoffs.  Cambridge University Press, Cambridge, U.K.
  * Roy (1996) Roy, B., 1996.  Multicriteria Methodology for Decision Aiding.  Springer Science + Business Media, Dordrecht, The Netherlands.
  * Roy and Bouyssou (1993) Roy, B., Bouyssou, D., 1993\.  Aide Multicritère à la Décision : Méthodes et Cas.  Economica, Paris, France.
  * Roy et al. (2014) Roy, B., Figueira, J.R., Almeida-Dias, J., 2014.  Discriminating thresholds as a tool to cope with imperfect knowledge in multiple criteria decision aiding: Theoretical results and practical issues.  Omega, The International Journal of Management Science 43, 9–20.
  * Roy and Vincke (1984) Roy, B., Vincke, P., 1984.  Relational systems of preference with one or more pseudo-criteria: Some new concepts and results.  Management Science 30, 1323–1335.