11institutetext: Jerzy Błaszczyński 22institutetext: Poznan Supercomputing and
Networking Center, 61-139 Poznań, Poland; 22email<EMAIL_ADDRESS>33institutetext: Salvatore Greco 44institutetext: Department of Economics and
Business, University of Catania, 95129 Catania, Italy
and
Centre for Operational Research & Logistics, Portsmouth Business School, PO1
3DE Portsmouth, UK; 44email<EMAIL_ADDRESS>55institutetext: Benedetto
Matarazzo 66institutetext: Department of Economics and Business, University of
Catania, 95129 Catania, Italy; 66email<EMAIL_ADDRESS>77institutetext:
Marcin Szeląg 88institutetext: Institute of Computing Science, Poznań
University of Technology, 60-965 Poznań, Poland; 88email:
<EMAIL_ADDRESS>
# Dominance-based Rough Set Approach, basic ideas and main trends

Jerzy Błaszczyński  Salvatore Greco  Benedetto Matarazzo  and Marcin Szeląg

###### Abstract

Dominance-based Rough Approach (DRSA) has been proposed as a machine learning
and knowledge discovery methodology to handle Multiple Criteria Decision
Aiding (MCDA). Due to its capacity of asking the decision maker (DM) for
simple preference information and supplying easily understandable and
explainable recommendations, DRSA gained much interest during the years and it
is now one of the most appreciated MCDA approaches. In fact, it has been
applied also beyond MCDA domain, as a general knowledge discovery and data
mining methodology for the analysis of monotonic (and also non-monotonic)
data. In this contribution, we recall the basic principles and the main
concepts of DRSA, with a general overview of its developments and software. We
present also a historical reconstruction of the genesis of the methodology,
with a specific focus on the contribution of Roman Słowiński.

##  1 Introduction

Among the many merits of Roman Słowiński in his so long and so rich scientific
carrier, we have to consider his pioneering approach to the use of artificial
intelligence methodologies to decision support, and, in particular, to
Multiple Criteria Decision Aiding (MCDA) (for an updated state of the art see
Greco et al (2016)). In this perspective, the proposal and the development of
the Dominance-based Rough Set Approach (DRSA) is a cornerstone in the domain.
The DRSA basic idea of a decision support procedure based on a decision model
expressed in natural language and obtained from simple preference information
in terms of exemplary decisions has attracted the interest of experts and it
is now considered one of the three main approaches to MCDA, together with the
classical Multiple Attribute Utility Theory (MAUT) Keeney and Raiffa (1976)
and the outranking approach Roy (1991). In fact, DRSA is not a mere
application to MCDA of concepts and tools already proposed and developed in
the domain of artificial intelligence, knowledge discovery, data mining and
machine learning. Indeed, consideration of preference orders typical for MCDA
problems required a reformulation of many important concepts and
methodologies, so that DRSA became a methodology viable and interesting per se
also in these domains. Consequently, after more or less 25 years from the
proposal of DRSA, we try to present a first assessment taking into
consideration the basic ideas and the main developments.

This paper is organized as follows. Next section presents some historical
notes. Section 3 recalls the basic concepts of DRSA, while section 4 describes
the main developments. Section 5 presents some available software. Section 6
collects conclusions.

##  2 Some historical notes on the Dominance-based Rough Set Approach

In the beginning of the 80’s of the last century, Roman Słowiński entered in
contact with two very relevant figures of researchers in two quite different
domains: Bernard Roy and Zdzisław Pawlak. Bernard Roy, one of the pioneers of
the Operation Research in Europe, was the founder of the European School of
Multiple Criteria Decision Aiding (MCDA). Zdzisław Pawlak, one of the founders
of the computer science and artificial intelligence, proposed the Rough Set
Theory as a mathematical tool for data analysis and knowledge discovery. Roman
Słowiński, that had already given fundamental contributions in scheduling
theory, was enthusiastically interested in both MCDA and rough set theory as
witnessed, among the others, by a translation in Polish Roy (1990) of the book
in which Bernard Roy systematically presented the basis of MCDA Roy (1985) and
by the organization of the first rough set international conference held in
1992 in Poznań (the proceedings of the conference are collected in Słowiński
(1992)). Since his first contributions in the domain, Roman Słowiński was
interested in the use of rough set theory for decision support. In particular,
he proposed the first application in real world problems related to
application of rough set theory to medical diagnosis Słowiński et al (1988).
Moreover, he realized that very interesting developments could be obtained in
rough set theory by applying concepts proposed in MCDA, as the construction of
similarity relations using indifference thresholds Słowiński (1992), inspired
by analogous concepts in outranking methods Roy and Vincke (1984). Pursuing
this research line, very soon Roman Słowiński matured the conviction that
beyond the simple application in rough set theory of some specific concepts
originating in MCDA, there was the space for a whole extension of rough set
theory that could become a relevant model for MCDA Pawlak and Słowiński
(1994a, b). In May 1994 Roman Słowiński went to Catania to give some seminars
on MCDA and rough set theory. On that occasion, discussing with Benedetto
Matarazzo and Salvatore Greco, a very interesting idea came out: what could be
obtained approximating sets of pairs of objects (binary relations) rather than
“standard” sets of single objects? The information to be processed with rough
set theory had to be the preference on single criteria and the overall
preference, in order to express the overall preference in terms of preference
on the single criteria. This idea was called rough set analysis of a pairwise
comparison table (PCT). In October of the same year Roman Słowiński, Salvatore
Greco and Benedetto Matarazzo met again at the 40th meeting of the EURO
working group on Multiple Criteria Decision Aiding, held in Paris and
Bordeaux. The program of the meeting was split in two parts, with the morning
of the 6th of October in Paris and the following day in Bordeaux, with a
transfer by train in the afternoon of the 6th of October. The travel between
Paris and Bordeaux was a very good occasion for a long discussion on the
intuition that come out in Catania. The three researchers remained in contact
continuing to work on PCT. On the first days of May 1995, Salvatore Greco
stayed one week in Poznań and during that week the first paper on PCT was
completed Greco et al (1997). In the same year the new rough set model was
presented at an international conference, more precisely at the 12th
International Conference on Multiple Criteria Decision Making held on June
19-23 in Hagen Fandel et al (1997). This was the first presentation of PCT at
an MCDA conference. The year after, Roman Słowiński, Salvatore Greco and
Benedetto Matarazzo participated in the Fourth International Workshop on Rough
Sets, Fuzzy Sets, and Machine Discovery, held on November 6-8, 1996 in Tokyo.
This was the first exposition of the PCT to a rough set conference. On that
occasion, Hirotaka Nakayama invited the three researchers to Kobe. Again
during the travels in train between Tokyo and Kobe and back there were very
rich and constructive discussions about the new rough set model. Both
researchers in MCDA and in rough set theory were interested in the idea.
However, very soon it appeared clear that the PCT was not able to represent
all the salient aspects of MCDA. Indeed, classical rough set theory is based
on indiscernibility relation, so what is taken into account is if the
considered objects have the same description or not. In MCDA, and, more in
general, in decision support, there is something more to be considered than
the equal or different description. For example, in deciding about cars, a
maximum speed of 200 km/h is greater and consequently preferred to a maximum
speed of 180 km/h. However, under the lens of indiscernibility 200 km/h is
only different from 180 km/h. How to take into account that beyond the
difference there is the preference of 200 km/h over 180 km/h? For a certain
time this was not clear and there was even a point in which Roman Słowiński,
Salvatore Greco and Benedetto Matarazzo took into consideration to abandon the
research. In the same period, Constantin Zopounidis invited Roman Słowiński to
write a contribution for a book that he was editing Zopounidis (1998). Roman
Słowiński proposed Salvatore Greco and Benedetto Matarazzo to write this
contribution by applying their approach to rough set theory to bankruptcy
evaluation, using the data that had been analyzed with classical rough set
theory in Dimitras et al (1999). Working on these data, the idea that would
permit to extend the rough set theory to MCDA was finally found. The key
concept was the substitution of the indiscernibility relation used in
classical rough set theory with the dominance relation on which MCDA is
rooted. On the basis of this intuition, definitions and results of classical
rough set theory can be reformulated in the MCDA perspective. This idea
quickly proved to be very successful in improving the results obtained using
classical rough set theory, and the contribution in the book edited by
Constantin Zopounidis Zopounidis (1998) became the first application of the
new extension of rough set theory called Dominance-based Rough Set Approach.
The new idea gave new impetus to the research, so that a systematic analysis
of the basic concepts of DRSA and of its extension was pursued. At those
times, Roman Słowiński received the invitation to write a chapter on fuzzy
sets applied to MCDA, for a book proposing a state of the art on MCDA Gal et
al (2013). To write that contribution, in the spring 1998 Salvatore Greco
stayed two months in Poznań to cooperate with Roman Słowiński, remaining
constantly in contact with Benedetto Matarazzo. The final result was a
contribution in the book in which rough set theory applied to MCDA, rather
than fuzzy set theory, was presented Greco et al (1999c). The material in that
chapter was continuously revised and improved until it became the basic paper
in which DRSA is presented Greco et al (2001b). This paper has become one of
the most read and cited of the European Journal of Operational Research. The
research on DRSA took several directions. Some years after, on December 2006,
Roman Słowiński, together with Juergen Branke, Kaylamon Deb and Kaisa
Miettinen organized a Dagstuhl seminar with the aim of opening a discussion
between researchers interested in interactive multiobjective optimization and
evolutionary multiobjective optimization (the presentations of that seminar
are collected in Branke et al (2008)). In that seminar, Roman Słowiński and
Salvatore Greco presented the result of a new research together with Benedetto
Matarazzo. The idea was the application of DRSA to guide the search of the
most preferred solution in an interactive multiobjective optimization problem
Greco et al (2008b). Again, the proposal was well accepted by the experts in
multiobjective optimization, because through the DRSA the preference
information asked from the DM is very simple and intuitive (the classification
of some solutions as good or not) and the preference model supplied by DRSA in
terms of “if …, then …” decision rules is very understandable for the DM and
easy to be managed in the optimization algorithm. Recently, Roman Słowiński
has come back to the multiobjective optimization through DRSA in a new
research conducted again with Salvatore Greco and Benedetto Matarazzo with the
addition of Salvatore Corrente. The new research Corrente et al (2021) aims at
using DRSA in an evolutionary multiobjective optimization algorithm. The use
of DRSA is justified taking into account also its good properties from the
point of view of decision psychology Lichtenstein and Slovic (2006).

##  3 Basic concepts of the Dominance-based Rough Set Approach

In this section, we want to present the basic ideas of DRSA Greco et al
(2001b) in comparison with the original Indiscernibility-based Rough Set
Approach (IRSA) proposed by Zdzisław Pawlak Pawlak (1982, 1991). Both IRSA and
DRSA consider a universe $U$ being a finite set of objects, a finite set of
attributes $Q=\\{q_{1},q_{2},\ldots,q_{m}\\}$, each one of them having a value
set $V_{q_{i}},i=1,\ldots,m$, and an information function $f:U\times Q\to V$,
with $V=\bigcup_{q\in Q}V_{q}$, such that $f(x,q)\in V_{q}$ for each $q\in Q$.

Every set of attributes $P\subseteq Q,P\neq\emptyset$, defines an
indiscernibility relation on $U$, denoted by $I_{P}$:

$I_{P}=\big{\\{}(x,y)\in U\times U:f(x,q)=f(y,q),\hbox{for all}\ q\in
P\big{\\}}.$

If $(x,y)\in I_{P}$, denoted also $xI_{P}y$, we say that the objects $x$ and
$y$ are $P$-indiscernible. The indiscernibility relation $I_{P}$ is an
equivalence relation on $U$ assigning to each object $x\in U$ its equivalence
class

$I_{P}(x)=\\{y\in U:yI_{P}x\\}.$

The family of all the equivalence classes of relation $I_{P}$ is denoted by
$U|I_{P}$. The equivalence classes of relation $I_{P}$ are called the
$P$-elementary sets or granules of knowledge encoded by $P$.

Using the indiscernibility relation $I_{P}$, to any set $X\subseteq U$ may be
associated the $P$-lower approximation

$\underline{P}(X)=\big{\\{}x\in U\colon I_{P}(x)\subseteq X\big{\\}}$

and the $P$-upper approximation

$\overline{P}(X)=\big{\\{}x\in U\colon I_{P}(x)\cap
X\neq\emptyset\big{\\}}\,.$

Intuitively, an object $x$ belongs to $\underline{P}(X)$ if it is certainly
contained in $X$, in the sense that all the objects that are indiscernible
with it also belong to $X$. Instead, an object $x$ belongs to
$\overline{P}(X)$ if it is possibly contained in $X$, in the sense that there
is at least one object indiscernible with $x$ that belongs to $X$.

Often the set of attributes $Q$ is divided into the set of condition
attributes $C\neq\emptyset$ and the set of decision attributes
$D\neq\emptyset$, such that $C\cup D=Q$ and $C\cap D=\emptyset$. The
indiscernibility relation $I_{S}$ with respect to a set of decision attributes
$S\subseteq D$, $S\neq\emptyset$, induces a partition of $U$, so that the
lower and the upper approximation of each equivalence class $I_{S}(x)$, $x\in
U$, with respect to $P\subseteq C$, $P\neq\emptyset$, can be computed, with
the aim of discovering dependencies between condition attributes from $P$ and
decision attributes from $S$. Indeed, if for $x,z\in U$, $P\subseteq C$ and
$S\subseteq D$, $x\in\underline{P}(I_{S}(z))$, then $I_{P}(x)\subseteq
I_{S}(z)$, that is, for all $y\in U$, if $f(y,q)=f(x,q)$ for all $q\in P$,
then $f(y,q)=f(z,q)$ for all $q\in S$. This can be interpreted in the sense
that the objects from $U$ suggest the following certain decision rule:

$\mathbf{\rho}_{c}$=“if $f(y,q)=f(x,q)$ for all $q\in P$, then $f(y,q)=f(z,q)$
for all $q\in S$”.

Analogously, if for $x,z\in U$, $P\subseteq C$ and $S\subseteq D$,
$x\in\overline{P}(I_{S}(z))$, then $I_{P}(x)\cap I_{S}(z)\neq\emptyset$, that
is, there is at least one $y\in U$, such that $f(y,q)=f(x,q)$ for all $q\in P$
and $f(y,q)=f(z,q)$ for all $q\in S$. This can be interpreted in the sense
that the objects from $U$ suggest the following possible decision rule:

$\mathbf{\rho}_{p}$=“if $f(y,q)=f(x,q)$ for all $q\in P$, then it is possible
that $f(y,q)=f(z,q)$ for all $q\in S$”.

The certain and possible decision rules induced from universe $U$ can also be
applied to classify objects not in universe $U$, with an easily understandable
explanation expressed in natural language. In fact, in presence of a certain
decision rule $\mathbf{\rho}_{c}$ (a possible decision rule
$\mathbf{\rho}_{p}$), if there is an object $w\notin U$ such that
$f(w,q)=f(x,q)$ for all $q\in P$, the objects from $U$ suggest that one has to
(could) expect $f(w,q)=f(z,q)$ for all $q\in S$, with the certain decision
rule $\mathbf{\rho}_{c}$ (the possible decision rule $\mathbf{\rho}_{p}$) that
can be seen as an explanation.

Very often there is a single decision attribute $d$, that is $D=\\{d\\}$. In
this case, the equivalence classes $I_{d}(x),x\in U$, can be identified with a
set of decision classes $\mathbf{Cl}$$=\\{Cl_{1},\ldots,Cl_{n}\\}$. In this
context, the lower and the upper approximation $\underline{P}(Cl_{t})$ and
$\overline{P}(Cl_{t})$ of each decision class $Cl_{t}\in\mathbf{Cl}$ with
respect to a set of condition attributes $P\subseteq C,P\neq\emptyset$, can be
obtained.

The classical IRSA has been recognized as a mathematical theory useful in
tasks considered in knowledge discovery and data mining. It has been widely
investigated from the theoretical perspective. It has also been applied in
analysis of many real world problems. However, as explained in the historical
notes of the previous section, it cannot deal with preferences and, more in
general, with data exhibiting monotonic relationships. In order to handle this
problem, Roman Słowiński with Salvatore Greco and Benedetto Matarazzo proposed
to substitute the indiscernibility relation with a dominance relation. Suppose
that to each attribute $q\in Q$ there is associated a preference relation
$\succeq_{q}$, such that, without loss of generality, for all $x,y\in U$, $x$
is at least as good as $y$ with respect to attribute $q$, denoted by
$x\succeq_{q}y$, if $f(x,q)\geq f(y,q)$. Given $x,y\in U$ and $P\subseteq Q$,
$x$ dominates $y$ with respect to $P$, denoted by $x{D_{P}}y$, if $f(x,q)\geq
f(y,q)$ for all $q\in P$. The $P$-dominance $D_{P}$ is a reflexive and
transitive binary relation, i.e., it is a preorder.

Given a set of criteria $P\subseteq Q,P\neq\emptyset$, and $x\in U$, the
granules of knowledge used for approximation in DRSA are the $P$-dominating
set $D_{P}^{+}\left(x\right)$ and the $P$-dominated set
$D_{P}^{-}\left(x\right)$ defined as follows:

$D_{P}^{+}\left(x\right)=\\{y\in U:yD_{P}x\\},\quad
D_{P}^{-}\left(x\right)=\\{y\in U:x{D_{P}}y\\}.$

Given $P\subseteq Q,P\neq\emptyset$, and $X\subseteq U$, the $P$-upward-lower
and the $P$-upward-upper approximation of $X$ are defined as follows:

$\underline{P}^{+}(X)=\left\\{x\in U:D^{+}_{P}(x)\subseteq X\right\\},$
$\overline{P}^{\;+}(X)=\left\\{x\in U:D^{-}_{P}(x)\cap
X\neq\emptyset\right\\}.$

Analogously, the $P$-downward-lower and the $P$-downward-upper approximation
of $X$ are defined as follows:

$\underline{P}^{-}(X)=\left\\{x\in U:D^{-}_{P}(x)\subseteq X\right\\},$
$\overline{P}^{\;-}(X)=\left\\{x\in U:D^{+}_{P}(x)\cap
X\neq\emptyset\right\\}.$

Also within DRSA one can consider a division of the set of attributes $Q$ into
the set of condition attributes $C\neq\emptyset$ and the set of decision
attributes $D\neq\emptyset$, so that, taken $P\subseteq C$ and $S\subseteq
D,P\neq\emptyset$ and $S\neq\emptyset$, and $x\in U$, the $P$-lower and the
$P$-upper approximation of the $S$-dominating set $D^{+}_{S}(x)$ and the
$S$-dominated set $D^{-}_{S}(x)$ can be computed, with the aim of discovering
dependencies between condition attributes from $P$ and decision attributes
from $S$. Indeed, for example, if for $x,z\in U$, $P\subseteq C$ and
$S\subseteq D$, $x\in\underline{P}^{+}(D^{+}_{S}(z))$, then
$D^{+}_{P}(x)\subseteq D^{+}_{S}(z)$, that is, for all $y\in U$, if
$f(y,q)\geq f(x,q)$ for all $q\in P$, then $f(y,q)\geq f(z,q)$ for all $q\in
S$. This can be interpreted in the sense that the objects from $U$ suggest the
following certain decision rule

$\mathbf{\rho}^{+}_{c}$=“if $f(y,q)\geq f(x,q)$ for all $q\in P$, then
$f(y,q)\geq f(z,q)$ for all $q\in S$”.

Analogously, if for $x,z\in U$, $P\subseteq C$ and $S\subseteq D$,
$P\neq\emptyset$ and $S\neq\emptyset$, $y\in\overline{P}^{+}(D^{+}_{S}(z))$,
then $D^{-}_{P}(y)\cap D^{+}_{S}(z)\neq\emptyset$, that is, there is at least
one $x\in U$, such that $f(x,q)\leq f(y,q)$ for all $q\in P$ and $f(x,q)\geq
f(z,q)$ for all $q\in S$. This can be interpreted in the sense that the
objects from $U$ suggest the following possible decision rule

$\mathbf{\rho}^{+}_{p}$=“if $f(y,q)\geq f(x,q)$ for all $q\in P$, then it is
possible that $f(y,q)\geq f(z,q)$ for all $q\in S$”.

In this context, one has to consider a semantic correlation Greco et al (2002)
between condition and decision attributes for which an improvement on a
condition attribute should not worsen a decision attribute, if the values of
the other condition attributes remain unchanged. For example, semantic
correlation implies that if the evaluation of a student on a given subject,
let us say history, improves, the overall evaluation should not decrease if
for all other subjects the evaluations remain the same. Here, history is a
condition attribute and the overall evaluation is a decision attribute.
Considering semantic correlation, taken $S\subseteq D,S\neq\emptyset$,
dominating sets $D^{+}_{S}(x)$ admit upward approximations and dominated sets
$D^{-}_{S}(x)$ admit downward approximations, and, consequently, the following
rough approximations can be considered:

$\underline{P}^{+}(D^{+}_{S}(x))=\left\\{y\in U:D^{+}_{P}(y)\subseteq
D^{+}_{S}(x)\right\\},$ $\overline{P}^{\;+}(D^{+}_{S}(x))=\left\\{y\in
U:D^{-}_{P}(y)\cap D^{+}_{S}(x)\neq\emptyset\right\\},$
$\underline{P}^{-}(D^{-}_{S}(x))=\left\\{y\in U:D^{-}_{P}(y)\subseteq
D^{-}_{S}(x)\right\\},$ $\overline{P}^{\;-}(D^{-}_{S}(x))=\left\\{y\in
U:D^{+}_{P}(y)\cap D^{-}_{S}(x)\neq\emptyset\right\\}.$

As was the case for IRSA, also for DRSA, usually, $D=\\{d\\}$, so that a
single decision attribute $d$ is considered. In this case, the decision
classes $\\{Cl_{1},\ldots,Cl_{n}\\}$ can be preferentially ordered so that for
all $x,y\in U$, with $x\in Cl_{t_{1}}$ and $y\in Cl_{t_{2}}$, if $t_{1}\geq
t_{2}$, then $x\succeq_{d}y$ and, equivalently, $f(x,d)\geq f(y,d)$.
Consequently, the dominating and the dominated sets of the decision attribute
$d$, $D^{+}_{d}(x)$ and $D^{-}_{d}(x)$, $x\in U$, can be formulated in terms
of upward and downward unions of decision classes $Cl_{t}^{\geq}$ and
$Cl_{t}^{\leq}$, $t=1,\ldots,n$, defined as

$Cl_{t}^{\geq}=\bigcup_{s\geq t}Cl_{s},\quad Cl_{t}^{\leq}=\bigcup_{s\leq
t}Cl_{s}.$

In fact, for all $x\in U$, if $x\in Cl_{t}$, then

$D^{+}_{d}(x)=\\{y\in U:f(y,d)\geq f(x,d)\\}=Cl_{t}^{\geq},$
$D^{-}_{d}(x)=\\{y\in U:f(y,d)\leq f(x,d)\\}=Cl_{t}^{\leq}.$

In this context, taking into consideration semantic correlation, the upward
lower and upper approximation $\underline{P}^{+}(Cl^{\geq}_{t})$ and
$\overline{P}^{+}(Cl^{\geq}_{t})$ of each upward union of decision classes
$Cl_{t}^{\geq}$ with respect to condition attributes $P\subseteq
C,P\neq\emptyset$, can be obtained. Analogously, the downward lower and upper
approximation $\underline{P}^{-}(Cl^{\leq}_{t})$ and
$\overline{P}^{-}(Cl^{\leq}_{t})$ of each downward union of decision classes
$Cl_{t}^{\leq}$ with respect to condition attributes $P\subseteq
C,P\neq\emptyset$, can be computed.

##  4 Developments of DRSA

In this section we discuss some of the many developments and extensions of
DRSA, related both to application of DRSA to other types of problems than just
ordinal classification, considered in the basic version of DRSA presented in
the previous section, and to adaptation of DRSA to different characteristics
of analyzed data.

###  4.1 DRSA to multicriteria choice and ranking

Choice and ranking problems are one of the main problems considered in MCDA
Roy (1985). To address them using rough sets, Greco, Matarazzo and Słowiński
proposed the concept of a pairwise comparison table (PCT) Greco et al (1995,
1997) in which binary relations, that is sets of pairs of objects, are
approximated, rather than sets of single objects as in the basic IRSA and
DRSA. Although this way it became possible to take into account the preference
over particular criteria and the overall preference between two objects, still
PCT was analyzed using IRSA, which could not take into account all types of
inconsistency observed in pairwise comparisons. This deficiency was overcome
when Greco, Matarazzo and Słowiński proposed to employ dominance relation
while processing PCT Greco et al (1996a, b).

Over the years, different ways of application of DRSA to multicriteria choice
and ranking have been proposed in the literature. In the following, we discuss
five such approaches in order of their appearance, denoting them by Greek
letters $\alpha-\epsilon$. All these approaches involve five key steps:

  1. ($s_{1}$)

elicitation of preference information in terms of pairwise comparisons of some
reference objects,

  2. ($s_{2}$)

rough approximation of comprehensive relations implied by the pairwise
comparisons, using the DRSA or a Variable Consistency DRSA (VC-DRSA, see
Section 4.7), to handle possible inconsistencies observed in the PCT,

  3. ($s_{3}$)

induction of decision rules from rough approximations of considered
comprehensive relations,

  4. ($s_{4}$)

application of induced decision rules on set $M\subseteq U$ of objects to be
ranked,

  5. ($s_{5}$)

exploitation of the resulting preference structure on $M$ to get a final
ranking of objects (total preorder).

When dealing with multicriteria ranking problem, this final ranking is the
recommendation presented to the DM. In case of multicriteria choice, the
recommendation is the object or the set of objects ranked as the best.

A common assumption of approaches $\alpha-\delta$ is that for each cardinal
criterion $q_{i}\in C$ (i.e., criterion with a cardinal scale, for which it is
meaningful to consider intensity of preference) there is given a set of
_graded preference relations_ $T_{i}=\\{P_{i}^{h},h\in H_{i}\\}$, where
$H_{i}$ is a finite set of integer numbers (“grades of intensity of
preference”) (see, e.g., Greco et al (2001b)). Relations $P_{i}^{h}$ are
binary relations over $U$, such that for $a,b\in U$:

  * •

if $aP_{i}^{h}b$ and $h>0$, then object $a$ is preferred to object $b$ by
degree $h$ w.r.t. criterion $q_{i}$,

  * •

if $aP_{i}^{h}b$ and $h<0$, then object $a$ is not preferred to object $b$ by
degree $h$ w.r.t. criterion $q_{i}$,

  * •

if $aP_{i}^{h}b$ and $h=0$, then object $a$ is similar (asymmetrically
indifferent) to object $b$ w.r.t. criterion $q_{i}$.

In approach $\epsilon$, however, it is argued that the above modeling of
binary relations $P_{i}^{h}$, involving determination of several thresholds
for each cardinal criterion, may be considered impractical. In fact, it
relates to discretization of the scale on which the strength of preference is
expressed. Therefore, in approach $\epsilon$, instead of grades of preference
intensity, one puts in a PCT for each cardinal criterion $q_{i}\in C$ just
differences of evaluations $f(a,q_{i})-f(b,q_{i})$.

The first application of DRSA to multicriteria choice and ranking, denoted by
$\alpha$, was proposed in Greco et al (1996a, b, 1998a, 1998b, 1999b) and
reminded in Greco et al (1999c); Słowiński et al (2009). It is characterized
by the following steps:

  * (s${}_{1}^{\alpha}$)

the pairwise comparisons of reference objects are expressed in terms of
outranking and non-outranking relations; given a pair of objects $(a,b)\in
U\times U$, a DM may: (i) state that object $a$ is comprehensively at least as
good as object $b$ (or, in other words, $a$ outranks $b$), denoted by $aSb$,
(ii) state that object $a$ is comprehensively not at least as good as object
$b$ (or, in other words, $a$ does not outrank $b$), denoted by $aS^{c}b$, or
(iii) abstain from any judgment;

  * (s${}_{2}^{\alpha}$)

relations $S$ and $S^{c}$ are approximated using _graded dominance relations_
(called in the following _single-graded dominance relations_) w.r.t. the set
of criteria $C$;

  * (s${}_{3}^{\alpha}$)

the approximations of $S$ and $S^{c}$ are used to generate four types of
single-graded decision rules (i.e., concerning the same grade of preference
w.r.t. each criterion present in the rule condition part), denoted by
$D_{++}$, $D_{-+}$, $D_{+-}$, $D_{--}$; if a pair $(a,b)\in U\times U$ is
covered by a rule of the first two types, it is concluded that $aSb$, while if
it is covered by a rule of the last two types, the conclusion is $aS^{c}b$;

  * (s${}_{4}^{\alpha}$)

the application of induced rules on set $M\subseteq U$ yields four outranking
relations called _true outranking relation_ , _false outranking relation_ ,
_contradictory outranking relation_ , and _unknown outranking relation_ ,
which together constitute so-called _four-valued outranking_ Tsoukias and
Vincke (1995, 1997);

  * (s${}_{5}^{\alpha}$)

the final ranking of objects from set $M\subseteq U$ is obtained using their
so-called _net flow scores_ ; the net flow score of an object $a\in M$,
denoted by $S^{NF}(a)$, is calculated as the sum of:

    * (i)

the number of objects $b\in M$ such that the induced rules suggest $aSb$, and

    * (ii)

the number of objects $b\in M$ such that the induced rules suggest $bS^{c}a$,

diminished by the sum of:

    * (iii)

the number of objects $b\in M$ such that the induced rules suggest $bSa$, and

    * (iv)

the number of objects $b\in M$ such that the induced rules suggest $aS^{c}b$.

It is worth noting that the first approach presented in Greco et al (1996a, b,
1998a, 1998b, 1999b) does not account for ordinal criteria (i.e., criteria
with ordinal scale, for which consideration of intensity of preference is not
meaningful). Moreover, the single-graded dominance relation is lacking in
precision Greco et al (1999c) as it assumes a common grade of intensity of
preference for all considered (cardinal) criteria.

The second application of DRSA to multicriteria choice and ranking, denoted by
$\beta$, was presented in Greco et al (1999c, 2000b, 2001b); Słowiński et al
(2002, 2005, 2009, 2014, 2015). It comprises of the following steps:

  * (s${}_{1}^{\beta}$)

$\equiv$ (s${}_{1}^{\alpha}$);

  * (s${}_{2}^{\beta}$)

relations $S$ and $S^{c}$ are approximated using the dominance relation that
accounts for both cardinal and ordinal criteria; w.r.t. cardinal criteria, the
_multigraded dominance relation_ is considered;

  * (s${}_{3}^{\beta}$)

the approximations of $S$ and $S^{c}$ are used to generate three types of
decision rules (that can use different grades of preference w.r.t. each
cardinal criterion present in the rule condition part), denoted by $D_{\geq}$,
$D_{\leq}$, and $D_{\geq\leq}$; if a pair of objects $(a,b)\in U\times U$ is
covered by a rule of the first type, it is concluded that $aSb$, while if it
is covered by a rule of the second type, the conclusion is $aS^{c}b$;

  * (s${}_{4}^{\beta}$)

$\equiv$ (s${}_{4}^{\alpha}$);

  * (s${}_{5}^{\beta}$)

$\equiv$ (s${}_{5}^{\alpha}$).

It is worth noting that definitions of lower approximations applied in
approaches $\alpha$ and $\beta$ appear to be too restrictive in practical
applications. In consequence, lower approximations of $S$ and $S^{c}$ are
often small or even empty, preventing a good generalization of pairwise
comparisons in terms of decision rules.

The third application of DRSA to multicriteria choice and ranking presented in
Słowiński et al (2002); Greco et al (2005), denoted by $\gamma$, is
characterized by the following steps:

  * (s${}_{1}^{\gamma}$)

$\equiv$ (s${}_{1}^{\beta}$);

  * (s${}_{2}^{\gamma}$)

relations $S$ and $S^{c}$ are approximated using the dominance relation that
accounts for both cardinal and ordinal criteria; w.r.t. cardinal criteria, the
multigraded dominance relation is considered; contrary to step
(s${}_{2}^{\beta}$), the approximations of $S$ and $S^{c}$ are calculated
using a PCT-oriented adaptation of the VC-DRSA proposed originally in Greco et
al (2000d) w.r.t. the multicriteria classification problems; as this VC-DRSA
measures consistency of decision examples using rough membership measure
$\mu$, it will be denoted by $\mu$-VC-DRSA;

  * (s${}_{3}^{\gamma}$)

the lower approximations of $S$ and $S^{c}$ are used to generate two types of
_probabilistic decision rules_ (that can use different grades of preference
w.r.t. each cardinal criterion present in the rule condition part), denoted by
$D_{\geq}$ and $D_{\leq}$; if a pair of objects $(a,b)\in U\times U$ is
covered by a rule of the first type, it is concluded that $aSb$, while if it
is covered by a rule of the second type, the conclusion is $aS^{c}b$;

  * (s${}_{4}^{\gamma}$)

$\equiv$ (s${}_{4}^{\beta}$);

  * (s${}_{5}^{\gamma}$)

$\equiv$ (s${}_{5}^{\beta}$).

The fourth application of DRSA to multicriteria choice and ranking, denoted by
$\delta$, was introduced in Fortemps et al (2008). It is characterized by the
following steps:

  * (s${}_{1}^{\delta}$)

the pairwise comparisons of reference objects are expressed in terms of
_comprehensive graded preference relations_ $\succ^{h}$, $h\in[-1,1]$; given a
pair of objects $(a,b)\in U\times U$, a DM may: (i) state that object $a$ is
comprehensively preferred to object $b$ in grade $h$, i.e., $a\succ^{h}b$ with
$h>0$, (ii) state that object $a$ is comprehensively _not_ preferred to object
$b$ in grade $h$, i.e., $a\succ^{h}b$ with $h<0$, (iii) state that object $a$
is comprehensively indifferent to object $b$, i.e., $a\succ^{0}b$, or (iv)
abstain from any judgment;

  * (s${}_{2}^{\delta}$)

_upward cumulated preference relations_ (upward unions of comprehensive graded
preference relations) $\succ^{\geq h}$ and _downward cumulated preference
relations_ (downward unions of comprehensive graded preference relations)
$\succ^{\leq h}$ are approximated using the dominance relation that accounts
for both cardinal and ordinal criteria; w.r.t. cardinal criteria, the
multigraded dominance relation is considered; analogously to step
(s${}_{2}^{\gamma}$), the approximations of $\succ^{\geq h}$ and $\succ^{\leq
h}$ are calculated using a PCT-oriented adaptation of $\mu$-VC-DRSA proposed
in Greco et al (2000d);

  * (s${}_{3}^{\delta}$)

the lower approximations of $\succ^{\geq h}$ and $\succ^{\leq h}$ are used to
generate two types of _probabilistic decision rules_ (that can use different
grades of preference w.r.t. each cardinal criterion present in the rule
condition part), denoted by $D_{\geq}$ and $D_{\leq}$; each induced rule is
additionally characterized by the attained _confidence level_ ; if a pair of
objects $(a,b)\in U\times U$ is covered by a rule of the first type, it is
concluded that $a\succ^{\geq h}b$, while if it is covered by a rule of the
second type, the conclusion is $a\succ^{\leq h}b$;

  * (s${}_{4}^{\delta}$)

the application of induced rules on set $M\subseteq U$ yields a graded fuzzy
preference relation (of level 2) over $M$; this relation is graded because of
different grades of preference, but it is also fuzzy because of different
confidence levels of rules matching pairs of objects from $M\times M$;

  * (s${}_{5}^{\delta}$)

the final ranking of objects from set $M\subseteq U$ is obtained by
exploitation of the preference structure on $M$ using either the Weighted
Fuzzy Net Flow Score (WFNFS) procedure or a Lexicographic-fuzzy Net Flow Score
procedure.

It is worth noting that elicitation of preferences in terms of comprehensive
graded preference relations $\succ^{h}$ requires, in general, a greater
cognitive effort of a DM. Moreover, it makes exploitation of the preference
structure resulting from application of induced decision rules more complex.

It is also important to note that the application of variable consistency
model of DRSA considered in approaches $\gamma$ and $\delta$, relying on rough
membership consistency measure $\mu$, leads to the situation when calculated
lower approximations of considered comprehensive relations lack several
desirable monotonicity properties Błaszczyński et al (2009).

The fifth application of DRSA to multicriteria choice and ranking, denoted by
$\epsilon$, was introduced in Szeląg et al (2013, 2014) and extended in Szeląg
(2015). In this approach, one takes into account the nature of the set of
criteria $C$ and acts accordingly, both during composition of a PCT and when
constructing the preference structure resulting from application of decision
rules. In Szeląg et al (2013), a typical MCDA assumption is adopted that set
$C$ is a _consistent family of criteria_ Roy and Bouyssou (1993), i.e., it
satisfies the properties of _completeness_ (all relevant criteria are
considered), _monotonicity_ (the better the evaluation of an object on
considered criteria, the more it is preferable to another object), and _non-
redundancy_ (there is no criterion which could be removed without violating
one of the previous two properties). Let us call this variant by
$\epsilon_{c}$. In Szeląg et al (2014), the above assumption is dropped, which
is typical for Preference Learning Fürnkranz and Hüllermeier (2010) methods.
Let us call this variant by $\epsilon_{nc}$ (for not necessarily consistent
set of criteria).

Approach $\epsilon$ consists of the following steps:

  * (s${}_{1}^{\epsilon}$)

the preference information is elicited as in step (s${}_{1}^{\gamma}$);
moreover, relation $S$ is enriched with some additional pairs of objects –
pairs $(x,y)$ such that $x$ dominates $y$ in variant $\epsilon_{c}$, and pairs
$(x,x)$ in variant $\epsilon_{nc}$;

  * (s${}_{2}^{\epsilon}$)

one proceeds as in step (s${}_{2}^{\gamma}$), but a PCT-oriented adaptation of
the VC-DRSA proposed in Błaszczyński et al (2007b, 2009) is used instead,
employing consistency measure $\epsilon$ (denoted by $\epsilon$-VC-DRSA);

  * (s${}_{3}^{\epsilon}$)

one proceeds as in step (s${}_{3}^{\gamma}$), but inducing probabilistic
decision rules using VC-DomLEM minimal cover algorithm Błaszczyński et al
(2011), employing rule consistency measure $\widehat{\epsilon}_{T}$, where
$T\in\\{S,S^{c}\\}$ Szeląg (2015);

  * (s${}_{4}^{\epsilon}$)

application of induced rules, both suggesting assignment to relation $S$ and
$S^{c}$, on set $M\subseteq U$; this way one gets a preference structure on
$M$, composed of relations $\mathbb{S}$ and $\mathbb{S}^{c}$ over $M$; both
relations can be either crisp (when it is only checked if there exists at
least one rule suggesting assignment of pair $(a,b)\in M\times M$ to relation
$S$/$S^{c}$) or valued (when it is checked what is the _strength of the
strongest rule_ suggesting assignment of $(a,b)$ to $S$/$S^{c}$), and they are
enriched with additional pairs of objects depending on the variant
$\epsilon_{c}$ or $\epsilon_{nc}$ Szeląg (2015); moreover, when constructing
valued relations, rule strength can depend on rule’s consistency only Szeląg
(2015) or also on rule’s coverage factor Szeląg et al (2014); Szeląg (2015);

  * (s${}_{5}^{\epsilon}$)

crisp/valued preference structure composed of crisp/valued relations
$\mathbb{S}$ and $\mathbb{S}^{c}$ over $M\subseteq U$ is exploited by
transforming it to valued relation $\mathcal{R}$ defined as:

$\mathcal{R}(a,b)=\frac{\mathbb{S}(a,b)+(1-\mathbb{S}^{c}(a,b))}{2},$ (1)

and by applying the well-known Net Flow Rule (NFR) Bouyssou (1992); Bouyssou
and Vincke (1997) ranking method, possessing desirable properties Szeląg
(2015), to produce the final ranking of objects from set $M$ (a weak order).

The efficiency of approach $\epsilon$ was proved experimentally in Szeląg
(2015). Moreover, it is implemented in two computer programs: ruleRank and
RUDE (see Section 5.5).

###  4.2 Case-based reasoning using dominance-based decision rules

Around the year 2005, Greco, Matarazzo and Słowiński noticed potential
application od DRSA to Case-based reasoning (CBR) (see, e.g., Kolodner
(1993)). In CBR, also called similarity-based reasoning, the more similar are
the causes, the more similar one expects the effects. This monotonic
relationship can be employed by DRSA, when applied to similarity-based
classification defined as follows. There is given a finite set of objects $U$
(called _universe of discourse_ , or _case base_) and a finite family of pre-
defined decision classes $\mathbf{Cl}$. An object $y\in U$ (a “case”) is
described in terms of features $f_{1},\ldots,f_{m}\in F$. For each feature
$f_{i}\in F$, there is given a _marginal similarity function_
$\sigma_{f_{i}}:U\times U\to[0,1]$, such that the value $\sigma_{f_{i}}(y,x)$
expresses the similarity of object $y\in U$ to object $x\in U$ w.r.t. feature
$f_{i}$; the minimal requirement that function $\sigma_{f_{i}}$ must satisfy
is the following: for all $x,y\in U$, $\sigma_{f_{i}}(y,x)=1$ if and only if
$y\textrm{ and }x\textrm{ have the same value of feature }f_{i}$. Moreover,
for each object $y\in U$ there is given an information concerning (normalized)
_credibility_ of its membership to each of the considered classes. To admit a
graded credibility, each decision class $Cl_{t}\in\mathbf{Cl}$,
$t\in\\{1,\ldots,n\\}$, is modeled as a fuzzy set in $U$ (Zadeh, 1965),
characterized by membership function $\mu_{Cl_{t}}:U\to[0,1]$. Thus, each
object $y\in U$ can belong to different decision classes with different
degrees of membership. The aim of decision aiding is to present to a DM a
recommendation concerning a new object $z$, in terms of a degree of membership
of this object to particular classes.

In similarity-based classification, the key issue is the aggregation of
marginal similarities of objects into their comprehensive similarity.
Typically, this aggregation is performed using some real-valued aggregation
function (involving operators, like weighted $L_{p}$ norm, min, etc.) (see,
e.g., (Dubois et al, 1998)) which is always arbitrary to some extent. This
motivated Greco, Matarazzo and Słowiński to propose an approach that measures
comprehensive similarity in a (more) meaningful way, avoiding the use of an
aggregation function. An approach of this type, employing an adaptation of
DRSA, was proposed for the first time in (Greco et al, 2006), and improved in
(Greco et al, 2008a, c). In the proposed approach, comprehensive similarity is
represented by decision rules concisely characterizing classification
examples. These rules are based on the general monotonic relationship
$(mr_{1})$: “the more similar is object $y$ to object $x$ w.r.t. the
considered features, the greater the membership of $y$ to a given decision
class $Cl_{t}$”, where $Cl_{t}\in\mathbf{Cl}$. This enabled to obtain a
meaningful similarity measure, which is, moreover, resistant to irrelevant (or
noisy) features because each decision rule, being a partial dominance cone in
a similarity space, may involve conditions concerning only a subset of
features. As the induced rules employ only ordinal properties of marginal
similarity functions, the considered approach is also invariant to ordinally
equivalent marginal similarity functions.

Few years later, a new monotonic relationship $(mr_{2})$ was formulated in
Szeląg et al (2011): “the more similar is object $y$ to object $x$ with
respect to the considered features, the closer is $y$ to $x$ in terms of the
membership to a given decision class $Cl_{t}$”, where $Cl_{t}\in\mathbf{Cl}$.
As observed in that paper, it is reasonable to consider $(mr_{1})$ only if the
membership of reference object $x$ to considered class $Cl_{t}\in\mathbf{Cl}$
takes a maximum value. On the other hand, $(mr_{2})$ does not require any
assumption about the membership value of reference objects and can be
considered as more general. Additionally, the authors of Szeląg et al (2011)
extended the previous approach also by proposing the way of induction of
decision rules using VC-DomLEM algorithm Błaszczyński et al (2011), and by
indicating a suitable way of application of these rules according to
Błaszczyński et al (2007a). Induced rules underline general monotonic
relationship between comprehensive closeness of objects and their marginal
similarities. An example of obtained decision rule is the following: “ _if_
similarity of flower $y$ to flower $x$ w.r.t. petal length is at least $0.7$,
_and_ similarity of flower $y$ to flower $x$ with respect to sepal width is at
least $0.8$, _then_ the membership of $y$ to class $setosa$ is between $0.7$
and $0.9$”.

The approach started in 2011 was further extended in Szeląg (2015), by further
formalizing the adaptation of DRSA, and by revising the rule classification
scheme described in Błaszczyński et al (2007a). Finally, the complete approach
was presented at a rough set conference in Szeląg et al (2016), where the
description of revised rule classification scheme was much simplified.

###  4.3 Adaptations of DRSA to handle missing attribute values

Shortly after the introduction of DRSA, Greco, Matarazzo, and Słowiński
considered two extensions of DRSA that enabled analysis of classification data
with missing attribute values Greco et al (1999a, 2000a). Also other authors
considered this problem Hu and Liu (2007); Yang et al (2008); Dembczyński et
al (2005, 2009); Liang et al (2010); Błaszczyński et al (2012); Du and Hu
(2016); Szeląg et al (2017). This research, apart from Du and Hu (2016), is
well summarized in Szeląg et al (2017), where all the approaches are given an
id, and their properties are thoroughly examined with respect to a list of 11
desirable properties.

An adaptation of DRSA to handle missing attribute values involves an adjusted
definition of dominance relation, that accounts for missing values. In some
approaches, it also involves the change of the definition of rough
approximation, to take into account the lack of some classical properties of
redefined dominance relation (e.g., lack of transitivity). In Szeląg et al
(2017), the authors denoted each adaptation identified in the literature by
DRSA-$mv_{j}$, and respective adjusted dominance relation by $D_{j}$, where
$j$ stands for the version id. In the following, we will use $j$ in the
superscript, to allow a subset of attributes $P\subseteq C$ in the subscript.
The authors of Szeląg et al (2017) pointed out, after Yang et al (2008);
Słowiński and Vanderpooten (2000), that the following generalized definitions
of $P$-lower and $P$-upper approximations of unions of decision classes
$Cl_{t}^{\geq},Cl_{t}^{\leq}$, $t=1,\ldots n$, should be employed:

$\begin{array}[]{ll}\underline{P}(Cl_{t}^{\geq})=\\{x\in
U:{d_{P}^{j+}(x)}\subseteq Cl_{t}^{\geq}\\}&\ \
\underline{P}(Cl_{t}^{\leq})=\\{x\in U:D_{P}^{j-}(x)\subseteq
Cl_{t}^{\leq}\\}\\\ \overline{P}(Cl_{t}^{\geq})=\bigcup\limits_{x\in
Cl_{t}^{\geq}}D_{P}^{j+}(x)&\ \
\overline{P}(Cl_{t}^{\leq})=\bigcup\limits_{x\in
Cl_{t}^{\leq}}{d_{P}^{j-}(x)},\end{array}$ (2)

where

${d_{P}^{j+}(x)}=\\{y\in U:x\,d_{P}^{j}\,y\\}\ \ \ \ \ \ \
{d_{P}^{j-}(x)}=\\{z\in U:z\,d_{P}^{j}\,x\\}.$ (3)

In the above definitions, $D_{P}^{j}$ denotes adapted $P$-dominance relation
($xD_{P}^{j}y$ means $x$ $P$-dominates $y$), while $d_{P}^{j}$ denotes adapted
$P$-inverse-dominance relation ($xd_{P}^{j}y$ means $x$ is $P$-dominated by
$y$). These dominance relations are defined as:

$\displaystyle yD_{P}^{j}x\Leftrightarrow{y\succeq_{q}^{j}x}\textrm{ for each
}q\in P$ (4) $\displaystyle
z\,d_{P}^{j}\,x\Leftrightarrow{z\preceq_{q}^{j}x}\textrm{ for each }q\in P,$
(5)

where $x,y,z\in U$ and versions of DRSA-$mv_{j}$ differ by definitions of
relations $\succeq_{q}^{j}$ and $\preceq_{q}^{j}$. Remark that definition (2)
can be applied even when relations $D_{P}^{j}$ and $d_{P}^{j}$ are only
assumed to be reflexive. In particular, it fits the case when $yD_{P}^{j}x$ is
not equivalent to $xd_{P}^{j}y$.

In Szeląg et al (2017), the analysis of properties of different adaptations of
DRSA was presented. It involves the following adaptations: Greco et al (1999a,
2000a) – introduces DRSA-$mv_{1}$ and DRSA-$mv_{2}$, Hu and Liu (2007) –
presents DRSA-$mv_{2.5}$, Yang et al (2008) – proposes DRSA-$mv_{1.5}$,
Dembczyński et al (2005, 2009) – introduces the concept of a lower-end
dominance relation used in DRSA-$mv_{4}$, and the concept of an upper-end
dominance relation resulting in DRSA-$mv_{5}$, Liang et al (2010) – presents
DRSA-$mv_{6}$, and Błaszczyński et al (2012) – defines DRSA-$mv_{3}$. The
analysis resulted in a conclusion that the only non-dominated approaches are
DRSA-$mv_{1.5}$, DRSA-$mv_{2}$, DRSA-$mv_{4}$, and DRSA-$mv_{5}$.

It is important to note, that taking into account the semantics of missing
values considered, e.g., in Grzymała-Busse and Hu (2001); Grzymała-Busse
(2011) (and in Stefanowski and Tsoukias (2001)), it can be said that
DRSA-$mv_{1.5}$ treats missing values as “lost” (“absent”) values, while
DRSA-$mv_{2}$ treats missing values as “do not care” values. These approaches
are defined as:

$\displaystyle{y\succeq_{q}^{1.5}x}\Leftrightarrow y\succeq_{q}x,\textrm{ or
}f(y,q)=?\ \ \ \ \ \ \ \ \ $
$\displaystyle{z\preceq_{q}^{1.5}x}\Leftrightarrow x\succeq_{q}z,\textrm{ or
}f(z,q)=?$ (6) $\displaystyle{y\succeq_{q}^{2}x}\Leftrightarrow
y\succeq_{q}x,\textrm{ or }f(y,q)=*,\textrm{ or }f(x,q)=*\ \ \ \ \ \ \ \ \ $
$\displaystyle{z\preceq_{q}^{2}x}\Leftrightarrow x\succeq_{q}z,\textrm{ or
}f(z,q)=*,$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{or
}f(x,q)=*,$ (7)

where $x,y,z\in U$, $q\in C$, $?$ denotes a “lost” missing value, and $*$
denotes a “do not care” missing value.

In Du and Hu (2016), Du and Hu proposed so-called characteristic-based
dominance relation with “do not care” values and “lost” values coexisting. It
relates to the characteristic relation considered in IRSA Grzymała-Busse
(2004, 2005). Let denote this approach by DRSA-$mv_{1.5\&2}$. Then

$\displaystyle{y\succeq_{q}^{1.5\&2}x}\ \ \Leftrightarrow\ \ f(y,q)=?,\textrm{
or }y\succeq_{q}x,\textrm{ or }f(y,q)=*,\textrm{ or }f(x,q)=*.$ (8)

Although the idea of considering “do not care” values and “lost” values
simultaneously is worth considering, the surprising part of Du and Hu (2016)
is that the authors do not employ $P$-inverse-dominance relation
$d_{P}^{1.5\&2}$, $P\subseteq C$, determined as in (5) with relation
$\preceq_{q}^{1.5\&2}$ defined as:

$\displaystyle{z\preceq_{q}^{1.5\&2}x}\ \ \Leftrightarrow\ \ f(z,q)=?,\textrm{
or }x\succeq_{q}z,\textrm{ or }f(z,q)=*,\textrm{ or }f(x,q)=*,$ (9)

where $x,y,z\in U$, $q\in P$. Instead, Du and Hu consider that
$xd_{P}^{1.5\&2}y\Leftrightarrow yD_{P}^{1.5\&2}x$. In our opinion, this is
not a proper realization of the idea outlined by Słowiński and Vanderpooten
Słowiński and Vanderpooten (2000) (even reminded in Du and Hu (2016)) who
claimed than for set $X\subseteq U$ and reflexive binary relation $R$ over
$U$, the lower approximation of set $X$ with respect to $R$ should be
calculated as:

$\underline{R}(X)=\\{x\in U:R^{-1}(x)\subseteq U\\},$ (10)

where $R^{-1}(x)=\\{y\in U:xRy\\}$.

###  4.4 Extensions of DRSA for interval evaluations on criteria

One of the ways of handling imprecision in object’s evaluation is to use
interval evaluations. Interval evaluations may also occur when replacing
missing values with a range of possible evaluations. Yet another scenario
concerns the case of hierarchical set of attributes, when a range of decisions
obtained at some lower level of attribute hierarchy is translated to an
interval of evaluations for an upper level criterion Dembczyński et al (2002).
Motivated by these observations, Dembczyński, Greco and Słowiński proposed
some extensions of the classical DRSA, that permit to analyze data with
interval evaluations on criteria Dembczyński et al (2002, 2003, 2005, 2009),
and also with interval assignments to decision classes Dembczyński et al
(2003, 2005, 2009). All these extensions involved so-called _$P$ -possible
dominance relation_ $\overline{D}_{P}$. Assuming, without loss of generality,
that each criterion is of gain type, $P$-possible dominance relation is
defined as:

$x\overline{D}_{P}y\Leftrightarrow u(x,q)\geq l(y,q)\textrm{ for all }q\in P,$
(11)

where $P\subseteq C$, and $u(x,q)$, $l(y,q)$ denote, respectively, upper limit
of the interval for object $x$ on attribute $q$, and lower limit of the
interval for object $y$ on attribute $q$.

In Dembczyński et al (2005, 2009), the authors additionally considered two
other $P$-dominance relations, $P\subseteq C$. Assuming, without loss of
generality, that each criterion is of gain type, they are defined as follows:

  * •

_$P$ -lower-end dominance relation_ $D_{P}^{l}$:

$xD_{P}^{l}y\Leftrightarrow l(x,q)\geq l(y,q)\textrm{ for all }q\in P,$ (12)

  * •

_$P$ -upper-end dominance relation_ $D_{P}^{u}$:

$xD_{P}^{u}y\Leftrightarrow u(x,q)\geq u(y,q)\textrm{ for all }q\in P.$ (13)

The above papers also took into account interval assignments to decision
classes. Suppose $d\in Q$ is a decision criterion. Then, the following rough
approximations of upward and downward unions of decision classes are
considered:

$\displaystyle\underline{Cl}_{t}^{\geq}=\\{y\in U:l(y,d)\geq t\\},$ (14)
$\displaystyle\underline{Cl}_{t}^{\leq}=\\{y\in U:u(y,d)\leq t\\},$ (15)
$\displaystyle\overline{Cl}_{t}^{\geq}=\\{y\in U:u(y,d)\geq t\\},$ (16)
$\displaystyle\overline{Cl}_{t}^{\leq}=\\{y\in U:l(y,d)\leq t\\}.$ (17)

Finally, the above rough approximations are the basis for second-order
dominance-based rough approximations Dembczyński et al (2005, 2009):

$\displaystyle\underline{P}(\underline{Cl}_{t}^{\geq})=\\{x\in
U:\overline{D}_{P}^{+}(x)\subseteq\underline{Cl}_{t}^{\geq}\\},$ (18)
$\displaystyle\underline{P}(\underline{Cl}_{t}^{\leq})=\\{x\in
U:\overline{D}_{P}^{-}(x)\subseteq\underline{Cl}_{t}^{\leq}\\},$ (19)
$\displaystyle\overline{P}(\overline{Cl}_{t}^{\geq})=\\{x\in
U:\overline{D}_{P}^{-}(x)\cap\overline{Cl}_{t}^{\geq}\neq\emptyset\\},$ (20)
$\displaystyle\overline{P}(\overline{Cl}_{t}^{\leq})=\\{x\in
U:\overline{D}_{P}^{+}(x)\cap\overline{Cl}_{t}^{\leq}\neq\emptyset\\}.$ (21)

DRSA with interval evaluations has also been considered in Yang et al (2009),
where $P$-possible dominance relation has been used for condition criteria. In
that paper, the authors also consider intervals with one or two missing
limits. Assuming that each $q\in C$ is of gain type, a missing lower limit is
then replaced with the minimal value in the value set of the respective
criterion, while missing upper limit is replaced with the maximal value in the
value set of the respective criterion.

###  4.5 Extending DRSA to address hierarchical structure of attributes

In hierarchical classification/sorting problems, where the set of attributes
has a hierarchical structure, the main difficulty is the propagation of
inconsistencies along the tree structure, i.e., taking into consideration at
each node of the tree the inconsistent information from lower level nodes. In
Dembczyński et al (2002), the inconsistencies are propagated bottom-up, in the
form of subsets of possible attribute values (for hierarchical regular
attributes), and in the form of intervals of possible values (for hierarchical
criteria). Decision rules are induced at each node of the tree. The
classification/sorting of new objects is also done from the bottom to the top
of the hierarchy, to make the final decision in the root node of the tree.

The extension of DRSA proposed in Dembczyński et al (2002) assumes that object
$y$ $P$-dominates object $x$, $P\subseteq C$, if:

  * •

for each regular attribute $q\in P$, the subset of attribute values for object
$y$ has a non-empty intersection with the subset of attribute values for
object $x$,

  * •

for each criterion $q\in P$, $u(x,q)\geq l(y,q)$, where $u(x,q)$, $l(y,q)$
denote, respectively, upper limit of the interval for object $x$ on attribute
$q$, and lower limit of the interval for object $y$ on attribute $q$.

###  4.6 Extensions of DRSA for non-ordinal data

Błaszczyński, Greco, and Słowiński proposed an approach to induction of laws
from data, which makes use of the concept of monotonic relationships between
values of condition and decision attributes, without assuming its direction a
priori and allowing local monotonicity relationships in subregions of the
evaluation space Błaszczyński et al (2012). This approach is able to discover
local and global monotonicity relationships existing in data. The
relationships are represented by monotonic decision rules. To enable the
discovery of monotonic rules, a non-invasive transformation of the input data
was proposed. The proposed transformation should be applied to all non-ordinal
attributes. Moreover, after transformation of input data, DRSA is applied to
structure data into consistent and inconsistent parts.

For the purpose of the illustration we may assume, without loss of generality,
that the value sets of both decision attribute (class labels) and condition
attributes are number-coded. As in non-ordinal classification problems the
natural complete ordering of classes $Cl_{1}$, $Cl_{2}$, $\ldots$, $Cl_{n}$
induced by number-coded class labels is not entering, in general, into some
monotonic relationships with value sets of condition attributes, we have to
consider $n$ binary ordinal classification problems with two sets of objects:
class $Cl_{t}$ and its complement $\lnot Cl_{t}$, $t=1$, $\ldots$, $n$, which
are number-coded by 1 and 0, respectively. This means that in the $t$-th
ordinal binary classification problem, set $Cl_{t}$ is interpreted by DRSA as
union $Cl_{1}^{\geq}$ and set $\lnot Cl_{t}$ as union $Cl_{0}^{\leq}$, $t=1$,
$\ldots$, $n$. Ordinal classification problems can be handled by DRSA without
altering the original number codes of the class labels.

The transformation of each non-ordinal condition attribute from $C$ is made
individually, depending on its type: numerical (number-coded) or nominal. Each
numerical (number-coded) attribute $q_{i}$ is represented, in the transformed
form, as a pair of ordinal attributes $q_{i}^{\prime}$, and
$q_{i}^{\prime\prime}$. In other words, evaluation of each object $x\in U$ by
numerical attribute $q_{i}$ is repeated twice, and the first evaluation
$f(x,q_{i})$ is renamed to $f(x,q_{i}^{\prime})$, while the second evaluation
$f(x,q_{i})$ is renamed to $f(x,q_{i}^{\prime\prime})$. Then, the first
attribute $q_{i}^{\prime}$ is set to have positive monotonic relationship with
(possibly transformed) decision attribute, while the second attribute
$q_{i}^{\prime\prime}$ is set to have negative monotonic relationship with the
decision attribute.

Each nominal attribute $q_{j}$ with value set composed of $k$ distinct values
is binarized, such that the presence or absence of the $l$-th value of this
attribute is coded by a new ordinal attribute $q_{jl}$ taking value 1 or 0,
respectively, $l=1$, $2$, $\ldots$, $k$. Then the binary attribute $q_{jl}$ is
represented, in the transformed form, by a pair of ordinal attributes
$q_{jl}^{\prime}$, and $q_{jl}^{\prime\prime}$. Again, the first ordinal
attribute in that pair has positive monotonic relationship with (possibly
transformed) decision attribute, while the second attribute in that pair has
negative monotonic relationship.

The proposed approach provides framework for analysis of heterogeneous
classification data. It has been shown experimentally in Błaszczyński et al
(2012), that the monotonic rules induced from transformed data, together with
a specific classification scheme, have at least as good predictive ability as
other well known predictors.

###  4.7 Parametric, decision theoretic, and stochastic DRSA

Greco, Matarazzo, Słowiński, and Stefanowski identified the need to relax the
definition of the lower approximation of union of classes and, in consequence,
to admit to the lower approximation some inconsistent objects (i.e., objects,
which would not be admitted to lower approximations in classical DRSA) for
which there is enough evidence for their membership to the union of classes
Greco et al (2000d). In this way, lower approximations are defined assuming an
acceptable value of a measure expressing evidence of membership to the set.
Following this idea, the evidence for this membership may be estimated by
different types of consistency or precision measures, and lower approximations
of unions of decision classes may be defined in different ways resulting in
different approaches to this kind of relaxation, called generally parametric
DRSA, including: Variable Consistency DRSA (VC-DRSA) Greco et al (2000d);
Błaszczyński et al (2006, 2007b, 2009), Greco et al (2001a, 2005), and
Variable Precision DRSA (VP-DRSA) Inuiguchi and Yoshioka (2006); Greco et al
(2008c). For example, in case of VC-DRSA, lower approximation is defined as a
subset of the approximated set. In consequence, given an upward (downward)
union of classes, objects that do not belong to this union are never included
in its lower approximation, even if they dominate (are dominated by) an object
from the considered union. On the other hand, in case of VP-DRSA, a lower
approximation is not a subset of the approximated set. All objects belonging
to dominance cones of objects from a lower approximation of a union of classes
are also included in the lower approximation of the union.

The fact that precision measure can be interpreted as a conditional
probability stimulated development of the decision-theoretic rough set model
(DTRSM) Yao (2008); Yao and Zhou (2016). DTRSM for DRSA has also been studied
in Greco et al (2007); Liu and Liang (2017). DTRSM connects definition of
approximations with conditional risk minimization in Bayesian decision theory.
In that situation, the states correspond to the decision classes, and
assignment of an object to positive, negative and boundary regions is decided
on the basis of its condition attribute values. Assuming an acceptable loss of
the classification accuracy and estimating conditional probability of the
considered decision classes, an optimal Bayes decision rule is obtained. In
this way, calculation of approximations can be seen as a classification
problem.

Another type of approach to treatment of inconsistent objects within DRSA has
also been considered in Kotłowski and Słowiński (2008); Kotłowski (2009). This
approach originates from statistical learning and statistical decision theory
and, in contrast to the previously mentioned approaches, involves relabeling
of objects (i.e., change of class to which object belongs to the more probable
one). It uses the notion of stochastic dominance and maximum likelihood
estimators of probability of object belonging to union of classes. Stochastic
lower approximations of unions of classes are composed of objects for which
values of estimators are higher than a given threshold.

It is worth mentioning that statistical interpretations of VC-DRSA and VP-DRSA
were proposed in Kusunoki et al (2021), by connecting lower approximations
with minimizers of empirical risk functions. In result, it has been
demonstrated that families of monotonic classifiers and the hinge loss
function serve as a foundation for characterization of the parametric DRSA
with consistency measures having desirable monotonicity properties.

###  4.8 Decision rules induction

In DRSA, induction of decision rules is subsequent to computation of rough
approximations. In this context, computation of approximations can be viewed
as a kind of preprocessing of data. Objects identified as sufficiently
consistent are a good basis for induction of decision rules. The purpose of
induction of decision rules is to discover strong relationships between
description of these objects and their membership to a union of classes. If
the rules are intended to be used in classification, then the goal of the
induction procedure is to find a preferably small set of rules with high
predictive accuracy. Decision rules should be short and accurate. Shorter
decision rules are easier to understand. Shorter rules also allow to avoid
_overfitting_ the training data. Overfitting occurs when the learned model
fits training data perfectly but is not performing well on new data.

In DRSA, for a given class $Cl_{t}$, $t\in\\{1,\ldots,n\\}$, we consider
decision rules of the type:

$\displaystyle\textrm{\emph{if} }f(y,q_{i_{1}})\succeq
r_{i_{1}}\land\ldots\land f(y,q_{i_{p}})\succeq r_{i_{p}}\land
f(y,q_{i_{p+1}})=r_{i_{p+1}}\land\ldots\land f(y,q_{i_{z}})=r_{i_{z}}\textrm{\
\ \ \ \ }$ $\displaystyle\textrm{\emph{then} }y\in Cl_{t}^{\geq}\textrm{,\ \ \
\ \ }$ (22) $\displaystyle\textrm{\emph{if} }f(y,q_{i_{1}})\preceq
r_{i_{1}}\land\ldots\land f(y,q_{i_{p}})\preceq r_{i_{p}}\land
f(y,q_{i_{p+1}})=r_{i_{p+1}}\land\ldots\land f(y,q_{i_{z}})=r_{i_{z}}\textrm{\
\ \ \ \ }$ $\displaystyle\textrm{\emph{then} }y\in Cl_{t}^{\leq}\textrm{,\ \ \
\ \ }$ (23)

where $q_{j},j\in\\{i_{1},i_{2},\ldots,i_{p}\\}$ denotes a criterion and
$q_{k},k\in\\{i_{p+1},i_{p+2},\ldots,i_{z}\\}$ denotes a regular attribute.
Moreover, $r_{j}\in V_{j}$,
$j=\\{i_{1},i_{2},\ldots,i_{p},i_{p+1},i_{p+2},\ldots,i_{z}\\}$, denotes a
value from the domain of criterion / regular attribute $q_{j}$. We use symbols
$\succeq$ and $\preceq$ to indicate weak preference w.r.t. single criterion
and inverse weak preference, respectively. If $q_{j}\in C$ is a gain (cost)
criterion, then elementary condition $f(y,q_{j})\succeq r_{j}$ means that the
value of covered object $y$ on criterion $q_{j}$ is not smaller (not greater)
than value $r_{j}$. Elementary conditions for regular attributes are of type
$f(y,q_{j})=r_{j}$.

The most important decision rule induction algorithms in DRSA are inspired by
LEM2 algorithm Grzymała-Busse (1992), proposed by Grzymała-Busse. The applied
heuristic strategy of rule induction in these algorithms is called sequential
covering (Han and Kamber, 2006) or separate and conquer (Michalski, 1969;
Pagallo and Haussler, 1990; Fürnkranz, 1999). It constructs a rule that covers
a subset of training objects, removes the covered objects from the training
set, and iteratively learns another rule that covers some of the remaining
objects, until no uncovered objects remain. The first of such algorithms was
proposed by Greco, Matarazzo, Słowiński, and Stefanowski and is called DomLEM
Greco et al (2000c). VC-DomLEM, proposed by Błaszczyński, Szeląg, and
Słowiński (Błaszczyński et al, 2011), is an adaptation of the same concept for
parametric DRSA. This algorithm heuristically searches for rules that satisfy
constraint with respect to chosen rule consistency measure.

##  5 Available software

In this section, we describe some computational libraries and applications
that implement DRSA and its extensions for rough set analysis of ordinal data.

###  5.1 ruleLearn

ruleLearn111https://github.com/ruleLearn/rulelearn is an open-source
computation library written in Java and hosted on GitHub (project started in
Dec 2017). Currently it implements DRSA and VC-DRSA (see Section 4.7), and
allows to handle missing attribute values according to DRSA-$mv_{1.5}$ and
DRSA-$mv_{2}$ (see Section 4.3). It also offers access to the VC-DomLEM rule
induction algorithm (see Section 4.8) and different classification strategies.
ruleLearn is utilized by other programs described in this section: RuLeStudio
and RuleVisualization.

###  5.2 RuLeStudio

RuLeStudio222https://github.com/dominieq/rule-studio is an open-source client-
server web application supporting data analysis using DRSA and VC-DRSA. It
requires Java 8+ to run, and employs ruleLearn API. This application was
finished in 2020 Dzięcioł et al (2020). It supports application of (VC-)DRSA
to analysis of ordinal data, possibly containing missing values (handled
according to DRSA-$mv_{1.5}$ or DRSA-$mv_{2}$). The application features data
editor, presentation of dominance cones (both with respect to dominance and
inverse-dominance relation), certain/possible rule generation according to VC-
DomLEM algorithm, analysis of different characteristics of induced rules,
application of rules on test objects using several classification strategies,
cross-validation, and presentation of misclassification matrix.

###  5.3 RuleVisualization

RuleVisualization333http://www.cs.put.poznan.pl/mszelag/Software/RuleVisualization/RuleVisualization.html
is an open-source client-server web application for visualization and
exploration of decision rules. It requires Java 8+ to run, and employs
ruleLearn API. This application was finished in 2019 Lewandowski (2019). This
program can read rules typical for DRSA, induced, e.g., by RuLeStudio, and
offers different visualizations of these rules (including the attributes used
by rules), their filtering, sorting, matching to some test objects, and
presentation of graphs showing co-occurence of attributes and
semantic/coverage similarity among rules.

###  5.4 jMAF

jMAF444http://www.cs.put.poznan.pl/jblaszczynski/Site/jRS.html is a rough set
data analysis software written in Java. It makes use of java Rough Set (jRS)
library. jMAF and jRS library implement methods of data analysis provided by
the DRSA and VC-DRSA, allowing to calculate lower approximations, induce
decision rules, and apply the rules to classify objects. More details with
regard to the analysis that this software package enables can be found in
Błaszczyński et al (2013).

###  5.5 RuleRank Ultimate Desktop Edition

RuleRank Ultimate Desktop Edition
(RUDE)555http://www.cs.put.poznan.pl/mszelag/Software/ruleRank/ruleRank.html
is a decision support tool for application of DRSA to multicriteria choice and
ranking. It requires Java 10+ and JavaFX to run. This application was finished
in 2018 Jówko (2018). RUDE employs approach denoted in Section 4.1 by
$\epsilon$, allowing setting different parameters characteristic for that
approach and visualization of: PCT, decision rules induced from that PCT,
preference graph resulting from application of these rules on a set of objects
to be ranked, and final ranking.

##  6 Conclusions

We presented the main ideas and developments of the Dominance-based Rough
Approach (DRSA), recalling the basic principles with a general overview of its
developments and software. At the end of this survey, we have to conclude that
DRSA continues to be appealing for experts in Multiple Criteria Decision
Aiding (MCDA) on one hand, and artificial intelligence and machine learning on
the other hand, for its most salient characteristic consisting in its capacity
of asking the decision maker for simple preference information and supplying
easily understandable and explainable recommendations. It is not difficult to
see that the DRSA properties of being simple and easily understandable have to
be acknowledged to the contribution of Roman Słowiński following the lines
traced by Bernard Roy with respect to MCDA and Zdzisław Pawlak with respect to
rough set theory.

## Acknowledgements

This research was partially supported by TAILOR, a project funded by EU
Horizon 2020 research and innovation programme under GA No 952215. Salvatore
Greco wishes to acknowledge the support of the Ministero dell’Istruzione,
dell’Universitá e della Ricerca (MIUR) - PRIN 2017, project “Multiple Criteria
Decision Analysis and Multiple Criteria Decision Theory”, grant 2017CY2NCA and
the research project “Data analytics for entrepreneurial ecosystems,
sustainable development and well being indices” of the Department of Economics
and Business of the University of Catania.

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