Using MACBETH with the Choquet Integral Fundamentals to Model
Interdependencies between Elementary Concerns in the Context of
Risk Management
Diana F. Lopes
1
, Carlos A. Bana e Costa
1
, Mónica D. Oliveira
1
and Alec Morton
2
1
CEG-IST, Centre for Management Studies of Instituto Superior Técnico, Universidade de Lisboa,
Avenida Rovisco Pais, 1049-001 Lisbon, Portugal
2
Management Science Department, University of Strathclyde, 16 Richmond Street, Glasgow G1 1XQ, Scotland, U.K.
Keywords: Interactions, Value Interactions, Choquet Integral, MACBETH, Risk Evaluation.
Abstract: Effective risk management typically requires the evaluation of multiple consequences of different sources of
risk, and multicriteria value models have been used for that purpose. The value of mitigating a risk impact is
often considered by risk managers as dependent on the levels of other impacts, therefore there is a need for
procedures to identify and model these interactions within a value measurement framework. The Choquet
Integral (CI) has been used for this purpose, and several studies in the performance measurement literature
have combined the 2-additive CI operator with the MACBETH approach to model interdependencies in real
contexts. In this paper, we propose an alternative procedure to model interdependencies and determine the
CI parameters from one single MACBETH global matrix. The procedure is illustrated with the construction
of a descriptor of impacts to evaluate the risk impacts at ALSTOM Power. The paper further explains the
questioning protocol to apply the proposed procedure, as well as how decision-makers can interpret the CI
parameters.
1 INTRODUCTION
Risk management relates to all sources of risk that
can affect an organization in terms of its goals and
vision (Aven, 2008) (p. 6). Typically, risks are
characterized in terms of their probability of
occurrence and impact (or consequences) (Williams,
1996) and depending on the context, the risk impact
can be evaluated on a single concern or on multiple
concerns – also described in the literature as
dimensions or attributes (Aven, 2008) (p. 42). For
example in a project management context, the
concerns may be timely delivery, incurred costs
within the budget envelope, and conformance to
specifications of the project. Multicriteria value
models can assist in evaluating those impacts
(Linkov et al., 2006). Value models demand for the
structuring of concerns in a value tree, with key
concerns (or criteria) being independent axes for the
evaluation of impacts. Very often, key concerns are
a cluster of interdependent elementary concerns
(EC) (Bana e Costa et al., 1999) (having also been
described as interacting criteria (Clivillé et al.,
2007); (Grabisch et al., 2010)), and therefore
modelling should identify and account for possible
value interdependencies between EC. This is a major
challenge in Multiple Criteria Decision Analysis
(MCDA) literature (Grabisch et al., 2010), and
several studies have explored the use of Choquet
integral (CI) operators in modelling
interdependencies. In particular, many of them use
an extension of MACBETH (Measuring
Attractiveness by a Categorical Based Evaluation
Technique) (Bana e Costa et al., 2012) with CI
operators (Berrah et al., 2007); (Clivillé et al., 2007;
Brosig, 2012); (Merad et al., 2013). Nonetheless,
these applications entail methodological problems.
With the aim of creating a tool to be easily used
for decision aid in the context of multicriteria value
measurement and to overcome pitfalls from previous
studies, this paper exploits an alternative procedure
to use MACBETH with the CI, through the use of a
MACBETH global matrix. The use of the proposed
procedure is illustrated with a case of evaluation of
interdependent risk impacts at ALSTOM Power.
This paper is organized in four sections. Section
2 presents a literature review. Section 3 proposes the
116
Lopes D., Bana e Costa C., Oliveira M. and Morton A..
Using MACBETH with the Choquet Integral Fundamentals to Model Interdependencies between Elementary Concerns in the Context of Risk
Management.
DOI: 10.5220/0004833701160126
In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 116-126
ISBN: 978-989-758-017-8
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
MACBETH global matrix with the CI to model
interdependencies, being applied to the context of
modelling interdependencies at ASTOM Power.
Section 4 presents concluding remarks and describes
ongoing research.
2 LITERATURE REVIEW
This section starts by clarifying key concepts and the
nomenclature in use in this paper. It then reviews
previous studies that have modelled
interdependencies, emphasizing those that have used
CI operators and MACBETH.
2.1 Nomenclature
Previous studies modelling interdependencies used a
diverse terminology. We start by clarifying the
nomenclature adopted in the remaining of this paper.
Departing from the value-focused thinking
framework of Keeney (1992), we adopt “the
approach of structuring concerns” of Bana e Costa et
al. (2005), which makes clear the difference between
concerns, key concerns (KC) and elementary
concerns. A concern (also called “point of view” by
Bana e Costa et al. (1999) and “objective” by
Keeney (1992) is any aspect within a specific
decision context seen as relevant at the eyes of at
least one actor for the analysis (Bana e Costa et al.,
2005). Stated objectives, points of view, active
characteristics of the options and possible
consequences of potential options are all concerns
(Bana e Costa et al., 1998). Key concerns are the
“ends objectives [concerns]” (Keeney, 1992) that
emerge progressively from the structuring phase,
and need to respect necessary preferential
independence conditions (Bana e Costa et al., 2005),
so that they can be isolated from the other KC. Each
of them must be measurable and operational by
assigning a natural, proxy or indirect, or a
constructed descriptor of impacts (i.e., ordered set of
plausible impact levels (Bana e Costa et al., 2002;
Bana e Costa et al., 2005)). Very often, a KC is
defined by a cluster of several interrelated
elementary concerns which are usually seen as
means to achieve ends (Bana e Costa et al., 1999). It
is on the level of the EC that value dependencies (to
be modelled with the CI) occur.
Since there is no agreement in literature
regarding the meaning of concept of interaction or
judgmental dependencies, it is useful to clarify this
concept. There are two main types of preferential
independence conditions: (1) ordinal independence
and (2) cardinal independence. The first one is
verified when options or alternatives can be ranked
with respect to one KC independent of their impact
in other aspects; the second one is verified, when,
additionally, the difference in attractiveness between
the options can be measured with respect to one KC
independently of their impact in other aspects (Bana
e Costa et al., 2005). For instance, consider the
following two concerns “peak noise level during the
night” and “noise level during the whole day” (Bana
e Costa et al., 2005). Whichever the average noise
during the day, low peaks during the night are
always preferred to high peaks - this is an example
of ordinal independence. However, if the difference
of attractiveness between two night peak levels
depends of the average noise level during the whole
day, cardinal dependence occurs (Bana e Costa et
al., 2005). In this sense, interactions are related to
preferential dependence, and these can be expressed
in different ways (Bana e Costa et al., 2005), for
instance by antagonism (in which the combined
impacts are valued less than the sum of the values of
impacts) and by synergism (with the combined
impact valuing more than the sum).
Marichal (2000) suggests the existence of three
types of interactions: (1) correlation, (2)
complementarity/substitutiveness and (3)
preferential dependence. In the scope of our study
we suggest that correlations should not be
considered because they refer to environmental
(physical or statistical) and not judgmental relations
between concerns (von Winterfeldt et al., 1986).
E.g., the costs and safety of a road infrastructure are
statically correlated, in the sense that higher safety
usually requires higher investments, but they can be
taken as two (judgmentally independent) key
concerns, because the DM is able to rank levels from
one KC and can judge the difference in
attractiveness between any two impact levels from
one KC without knowing their impact on the other
KC. Hence, in contrast to what it is stated in other
studies such as Shah et al. (2013) and Marichal
(2000), correlations can be ignored unless they are
redundant (Bana e Costa et al., 2005). On the other
hand, we believe that substitutiveness and
complementarity are types of preferential
dependence, more precisely extreme situations of
antagonism and synergism, respectively. For
example, substitutiveness corresponds to a situation
when the combined impact is valued less than the
sum of individual impacts, being the value of the
individual impacts close to the value of the
combined impact (Grabisch et al., 2004). Hence,
there is no need to distinguish between preferential
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or value dependence and substitutiveness/
complementarity.
2.2 MACBETH Basics
MACBETH is a multicriteria decision analysis
approach that requires only non-numerical
judgments about differences in attractiveness
between options to help the decision maker (DM)
measure the relative attractiveness or value of
options (Bana e Costa et al., 2012). Within a
multicriteria value measurement framework,
MACBETH has been most commonly used to build
value functions and weight criteria in additive
models. For building value functions, the
MACBETH questioning protocol consists in asking
the DM to qualitatively judge the difference in
attractiveness between impact levels, two at a time,
based on seven semantic categories: “is there no
difference, or is the difference very weak, weak,
moderate, strong, very strong, or extreme?” (Bana e
Costa et al., 2012). During this questioning protocol,
a matrix with the categorical judgments of the DM is
populated. Each time a qualitative judgment is
introduced in the matrix, the M-MACBETH DSS
verifies its consistency and offers suggestions to
solve eventual inconsistencies (Bana e Costa et al.,
2008). After the consistency verification, the
software derives, by mathematical programming, an
interval numerical scale which has to be analysed
and validated by the DM (Bana e Costa et al., 2012).
A slightly different procedure than the one just
described has been used for weighting EC with CI
and MACBETH. Previous studies have shown that
MACBETH provides a simple and transparent
approach in modelling complex multidimensional
problems, and hence its wide applicability in MCDA
(examples are (Bana e Costa et al., 1999); (Bana e
Costa et al., 2002); (Carnero, 2006); (Bana e Costa
et al., 2008); (Barin et al., 2012)).
As described above, in its most common use,
MACBETH applications use an additive value
model by taking “difference independence” as a
modelling hypothesis (Bana e Costa et al., 2012). As
mentioned in Edwards et al. (1994) (p. 315)
“violations of conditional monotonicity, usually easy
to detect judgmentally, mean that additive models
should not be used”, and interdependent EC should
be combined/aggregated to form a KC (Bana e Costa
et al., 2005).
2.3 Modelling Interdependent EC
For cases of interdependent EC, there has been an
increase in studies using procedures with operators
of CI family that belong to the non-additive measure
family. These CI operators are represented by a
mathematical expression that aggregates impact
values of multiple interdependent EC into a single
global score (Mayag et al., 2011).
Several studies have applied these CI operators
in many evaluation contexts such as to: (i) evaluate
supplier’s performance (Clivillé et al., 2006);
(Berrah et al., 2007); (Feyziog
̃
lu et al., 2010),
employees’ performance (Gürbüz, 2010) and action
plans in an organization in industrial contexts
(Clivillé et al., 2007); (Merad et al., 2013); (ii)
assess academic performance of faculty members in
the education sector (Cardin et al., 2013); (iii)
evaluate classical swine fever control strategies in
the animal health literature (Brosig, 2012), (iv)
evaluate a new waste incinerator plant location in
the environment context (Bottero et al., 2013); and
(iv) more recently, in the risk management context,
to build a value- and risk-based performance
measurement and management system (Vernadat et
al., 2013).
Some of these studies lack information about the
questioning protocol applied and do not explain how
to convert impacts into value or how to determine
the CI parameters (Feyziog
̃
lu et al., 2010). Other
studies convert quantitative impacts into values
linearly and therefore presuppose that a unit of
impact has always the same value, which is a strong
assumption (in other words, problems on the
interpretation of commensurate scales - see section
3.1) (Bottero et al., 2013); (Cardin et al., 2013).
Other studies weight EC based only on the notion of
importance (Labreuche et al., 2005); (Brosig, 2012)
with no reference to impact ranges – in fact, this has
been described as the most common critical mistake
in decision analysis (Keeney, 1992). Some studies
require a direct elicitation of numerical information
about the overall value of each option from the DM
(Grabisch et al., 2000); (Grabisch et al., 2008);
(Merad et al., 2013), there being many practical and
theoretical issues regarding this type of elicitation,
as discussed by Morton et al., (2009). Several
studies make use of local judgments, instead of
global judgments, to model interdependencies
between EC – examples are (Clivillé et al., 2006);
(Clivillé et al., 2007); (Gürbüz, 2010); (Merad et al.,
2013); (Vernadat et al., 2013), with most of these
studies using the CI with MACBETH. By local
judgment we mean a judgment involving one EC
alone; by global judgment we mean a judgment
involving the interrelated EC all together.
Nonetheless, the use of local judgments is
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incoherent because under the presence of
interdependencies between several EC, one cannot
assess local judgments in an EC without considering
the levels of impact in the remaining EC. To
illustrate this problem, we recall the study by
Clivillé et al. (2007) that evaluates four actions to
improve the profit margins of a company using CI
with MACBETH. One KC concern of this study is
“service rate”, composed by three EC: “average
delay”, “order receive time” and “accuracy
shipments”. Regarding local judgments for the
“average delay” EC, the DM evaluated as “extreme”
(“6”, see the highlighted judgment in Fig. 1(a)) the
difference in attractiveness between an action “TI”
and a “neutral” action (N). However, if there exist
interdependencies between EC, that judgment may
change with the impacts in the remaining EC and
therefore these should be fixed before the
questioning (a further explanation on this is provided
in sections 2.4 and 3.3). In addition to the previous
problem, some studies such as Clivillé et al. (2006),
Clivillé et al. (2007) and Gürbüz (2010) make use of
two types of matrices – a first matrix with local
judgments (see Fig. 1(a)) and a second with global
judgments (see Fig. 1(b)) – that can be problematic.
Even if the local judgments, present in the first
matrix, were asked to the DM in the form of global
judgments, the use of two matrices could entail
incoherence and inconsistency problems. To expose
these problems, we recall once again the study by
Clivillé et al. (2007). If the “good” and “neutral”
actions present in Fig. 1(a) were taken as being
“neutral” in the remaining EC, i.e. globally
comparing the difference in attractiveness between
two actions (G
1
N
2
N
3
) and (N
1
N
2
N
3
), the same global
judgment is also present in Fig. 1(b) – see the
highlighted judgment (“weak”). However, that
judgment is inconsistent with the highlighted
judgment in Fig. 1(a). In fact, the difference in
attractiveness between an action “TI” (which is less
attractive than a “good” action in terms of “average
delay”) and a “neutral” action (N) is “extreme”,
which implies that the difference in attractiveness
between a “good” and a “neutral” action needs to be
“extreme”, being inconsistent with the previous
“weak” judgment. Taking into account the limitation
associated to the use of separated matrices, this
paper exploits the use of global judgments within
one single MACBETH global matrix. In addition, it
is worthwhile to note that some studies such as
Clivillé et al., (2006), Clivillé et al., (2007) and
Vernadat et al., (2013) apply MACBETH to build
value functions based just on the set of local
judgments between consecutive ordered levels given
by the DM (diagonal of the MACBETH matrix Fig.
1(a)). However, according to Bana e Costa et al.,
(2008), “it is recommended to ask for some
additional judgments to perform a number of
consistency checks”. If MACBETH is only applied
to the diagonal of the matrix, the procedure does not
make use of all the potential of the MACBETH
method and DSS. Thus, no consistency checks are
made and the MACBETH judgments are used to
solve an equation system by attributing a single
number to each semantic category (with ‘very
weak’=1, ‘weak’=2,..., and ‘extreme’=6) – see Fig. 1
– which is an arbitrary assumption. Therefore, it
seems also important to overcome these issues, as
other numerical values or ranges can be attributed to
each semantic category, and consistency checks are
also a key feature for a procedure to be used in real
decision aid settings.
This study exploits the use of MACBETH with
the CI to model interdependencies and overcoming
the limitations pointed above, using a single
MACBETH matrix with global judgments and all
the features of the MACBETH method and DSS.
The aim is to propose a tool that can easily be used
for decision support, as it will be described in
section 3.
2.4 Choquet Integral Operators
This section introduces basic concepts regarding CI
operators, and a brief explanation of the meaning of
CI parameters. The operators of the CI family,
introduced by Choquet (1953), belong to the non-
additive measures family and are being applied to
model interdependencies between EC. Since we are
dealing with interdependencies, the basic idea
behind these operators consists in questioning the
DM about their preferences on a set of “binary
options” to determine the value of the CI parameters
– Shapley and interaction parameters, which we
describe below. As suggested by the name, these
binary options must perform with superior and
inferior reference levels on each EC. Depending on
the context, best and worst, or good and neutral
references, or other superior and inferior references
can be used (for instance, the good corresponds to an
undoubtedly attractive level and neutral to a level
which is neither attractive nor repulsive (Mayag et
al., 2011) at the eyes of the DM; and the best
corresponds to the most attractive level and the
worst to the most repulsive level (Grabisch et al.,
2004)).
Using the best and worst references, options
which are best and/or worst on all the m EC are used
UsingMACBETHwiththeChoquetIntegralFundamentalstoModelInterdependenciesbetweenElementaryConcernsin
theContextofRiskManagement
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Figure 1: (a) Local judgments used to build the value function on the EC “average delay” and (b) the matrix that compares
interdependent EC (global judgments). Note that HR, CI, QE and TI are abbreviations of the actions that are being
evaluated, 'human resources', 'continuous improvement', 'quality and environment' and 'technical investment', respectively
(Clivillé et al., 2007).
in the CI – for instance (B
1
B
2
W
3
...W
m
), is a binary
option which is in the best (B) level on EC 1 and 2
and in the worst (W) level on the remainder EC, in a
context of m EC.
To better understand the reason why we should
ask the DM about their preferences on the set of
binary options, consider a case with three EC and
two swings: (i) a swing from an option (W
1
W
2
W
3
)
to an option (B
1
W
2
W
3
) and (ii) another swing from
(W
1
B
2
W
3
) to (B
1
B
2
W
3
). These swings just differ on
the level of EC 2 - worst and best in the first and
second swings, respectively. In the context of
independent EC, both swings are equally valued, and
there is no need to take into account of the different
baselines. Under the presence of value
interdependencies, these swings are differently
judged and one needs to consider all the swings with
different baselines.
To apply these operators, the monotonicity
condition is required. E.g., consider the following
sets of elementary concerns S, T and M, and their
corresponding complement sets S
c
, T
c
and M
c
; if S
T M, the option which is at the best level in all the
EC present in T and at the worst level in the EC
present in T
c
(B
T
,W
T
c
) dominates the option
(B
S
,W
S
c
) which is in the best level in all the EC from
S and in the worst level in the EC from S
c
, because
the former is at least as attractive as the latter, then:
)W,B(V)W,B(VMTS
cc
S
SAg
T
TAg
(1)
This condition illustrates that the EC cannot be
modelled as ordinally dependent. After having
information from the DM regarding the numerical
values of the binary options, the CI is applied,
making use of the following formula – eq. system
(2), the general formulation of CI operators.
)x(v....)x(v)x(v
0)x(v
)})]B...BB({(V
))x(v)x(v[()u(V
u)m()m(u)2()2(u)1()1(
u)0()0(
)m()1i()i(Ag
m
1i
u)1i()1i(u)i()i(Ag
(2)
Where V
Ag
represents the aggregated value (global
score) of an option u, v
(i)
corresponds to the i-th
overall value of the option u in ascending order
thatis related to a specific EC and baseline (for
instance, all worst or all best) in the other EC, x
(i)u
represents the impact level of the option u in the EC
i and V
Ag
({B
(i)
}) the aggregated value of an option
which is in the best level in the EC i and in the worst
level in the other EC. Note that this mathematical
formulation is coherent with the impact values (V
Ag
and v
(i)
) on an unipolar scale i.e. the impact values
are in the [0,1] range (Grabisch, 2005).
The exponential complexity of the general
formula of the CI has shown to be prohibitive in
many applications (Grabisch, 2005; Grabisch et al.,
2010), as 2
m
–2 variables need to be determined,
corresponding to the aggregated value of all the
binary options (2
m
) excluding the known aggregated
values of the options which are worst all over and
best all over, that for convention, value 0 and 1,
respectively (Grabisch et al., 2010). To overcome
this issue, particular CI operators have been
proposed, such as the k-additive ones, highly spread
in decision analysis literature. K-additivity fixes the
degree of interaction between EC: 1-additivity does
not allow interaction between EC (corresponding to
an additive model); 2- additivity allows interaction
up to 2 EC, etc. Note that a k-additive operator, that
allows interaction up to k EC, requires the definition
of
k
1a
a
m
parameters, which makes, in practice,
(N,G,G) (G,N,G) (G,G,N) (N,N,G) (N,G,N) (G,N,N) (N,N,N)
(N,G,G)
Null Moderate (3)
(G,N,G)
Null Extreme (6)
(G,G,N)
Null Null (0)
(N,N,G)
Null Weak (2)
(N,G,N)
Null V. Weak (1)
(G,N,N)
Null Weak (2)
(N,N,N)
Null
(a) (b)
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the 2-additivity the best compromise between
complexity and richness of the model (Grabisch et
al., 2010). In the 2-additivity case the mathematical
formulation is given by a linear part, identical to the
additive model, modified by effects coming from the
interaction represented by the second parcel (see the
first equation of the eq. system (3)). Grabisch (1997)
explains how to obtain the eq. system (3) from (2).

m
1i
i
m
1i
jujiui
ji
}j,i{
ijiuii
Ag
1s
|)x(v)x(v|I
2
1
)x(vs)u(V
(3)
where v
i
corresponds to the overall value of the
option u on the EC i considering a specific baseline
(for instance, all worst or all best) in the other EC; s
i
corresponds to the Shapley parameter of the EC i
(see section 3.3); I
ij
the interaction parameter
between the EC i and j.
The Shapley and the interactions parameters are
key concepts for the understanding and analysis of
interdependencies between EC. As remarked in
Grabisch (1997) (p. 171), the Shapley value “can be
interpreted as a kind of average value of the
contribution of element i [EC i in our study] alone in
all coalitions [considering different baselines in the
other EC]”. With regard to the interaction parameter
between 2 EC for instance, it “can be interpreted as a
kind of average value of the added value given by
putting i and j together, all coalitions being
considered” (Grabisch, 1997) (p. 171), i.e. an
average of the added value given by putting together
the 2 EC together, being all the baselines considered.
The interaction parameters I
ij
range in the interval [-
1,1] with: (a) I
ij
> 0 for a synergistic behaviour
between EC i and j, (b) I
ij
< 0 for an antagonistic
behaviour between EC i and j, and (c) I
ij
= 0 when
there is no interaction between EC i and j. In the
next section, the mathematical formulations – see
Eqs. (4) and (7) - of these CI parameters are
represented and examples are used to illustrate these
definitions to DM.
3 METHODOLOGICAL
FRAMEWORK
Departing from a real case study in risk
management, this section explains how the use of a
global MACBETH matrix with the CI can be
applicable to properly model interdependencies
between EC. Special emphasis is also given to the
protocol of questioning and to providing an intuitive
interpretation of the parameters of the CI that can be
explained to a DM.
3.1 Structuring the Evaluation
Problem
A key business area of ALSTOM Power is to
construct and commission power plants. When
building power plants, ALSTOM needs to prioritize
risks that can threaten the construction projects. In a
real case study (Figueiredo et al., 2009) developed
with a group of ALSTOM risk managers (the DM),
one KC identified was” Health and Safety” which
refers to injuries resulting from incidents that occur
inside site barriers, and takes into account two
interdependent EC: (a) number of injuries and (b)
severity of the injury.
For applying the CI, in each EC a descriptor of
impacts and two reference levels within the
descriptor need to be defined. The use of reference
levels in every EC ensures commensurateness
between EC (Grabisch et al., 2004).
“Commensurateness means that one shall be able to
compare any element of one point of view [EC in
our study] with any element of any other point of
view” (Grabisch et al., 2004) (p. 565), by
considering a baseline defined by those reference
levels since we are in the presence of
interdependencies. As described in section 2.4,
depending on the evaluation context, the reference
levels can be defined as best and worst (Labreuche
et al., 2003) or as good (G) and neutral (N) (Mayag
et al., 2011) levels or as other superior and inferior
references. It is important to note that in addition to
the two reference levels – which in the ALSTOM
example, are the “best” and the “worst” levels –
other impact levels can be defined. This is the case
of descriptors in tables 1 and 2.
Table 1: Descriptor of impacts for the EC 1 “Number of
injuries”.
Levels of impact Description
B= Best 0
I 3
W = Worst 10
Since we are dealing with interdependencies and
global preference modelling, the next step is to
define all the combinations of different impact levels
across EC, from tables 1 and 2), being then pairwise
compared in a global MACBETH matrix (see the
next section). In our example, 12 combinations of
UsingMACBETHwiththeChoquetIntegralFundamentalstoModelInterdependenciesbetweenElementaryConcernsin
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impacts are considered (from crossing impact levels
of tables 1 and 2, leading to table 3). Note in table 3
that if there are no injuries, the impact level of the
EC 2 is irrelevant, and so all levels in the
constructed descriptor which have no injuries as a
component are evaluated as being indifferent.
Table 2: Descriptor of impacts for the EC 2 “Severity of
the injury”.
Levels of impact Description
B = Best First aid incident (FAI)
M Medical treatment incident (MTI)
L Loss time incident (LTI)
W = Worst Fatality (Fatal.)
Table 3: Constructed descriptor of impacts for the KC
“Health and Safety” (B and W stand for Best and Worst,
respectively; and the indices refer to the EC 1 and EC 2).
Levels of impact Description
B
1
B
2
; B
1
M
2
; B
1
L
2
; B
1
W
2
No injury
I
1
B
2
3 FAI
W
1
B
2
10 FAI
I
1
M
2
3 MTI
W
1
M
2
10 MTI
I
1
L
2
3 LTI
W
1
L
2
10 LTI
I
1
W
2
3 fatalities
W1W
2
10 fatalities
3.2 Value Measurement with
MACBETH
Many studies, such as Berrah et al., (2007) and
Clivillé et al. (2007), use MACBETH in a first step
to build, what these authors name “elementary
expressions” (value functions) in each EC by using
local judgments (as shown in Fig. 1(a)); and in a
second step to construct a matrix to compare pairs of
EC (a matrix requiring global judgments - as shown
in Fig. 1(b)). As described in detail in section 2.3,
only a simplified version of MACBETH is used in
these studies, and the judgments used in building
‘value functions’ cannot be local judgments; and if
they are global judgments, they coincide with some
of the judgments used in the second step of ‘building
weights’ (see Fig. 1(b)), which demand for
consistency checks when filling the matrices. The
use of a single MACBETH matrix of global
judgments can avoid these judgmental
inconsistencies. Also, using such a matrix allows for
accounting for DM’s differences of opinion or
hesitation that are relevant in real decision contexts.
Accordingly, we propose using one global
MACBETH matrix with all the judgments
comparing all the levels of the constructed descriptor
from Table 3. The rationale behind the use of a
global matrix is as follows:
under the presence of interdependencies, all the
judgments required to evaluate impact levels are
global;
the use of a single global matrix allows for
depicting cases of inconsistencies between
judgments, as well as allows for a clear and
simple protocol of questioning DM; also cases of
differences of opinion or hesitation can populate
the matrix;
and the numerical scales generated by applying
the MACBETH approach to the global matrix
can be used not only with the CI, but also to test
whether other non-additive models are
appropriate (we do not pursue this research line
in this study).
Thus, we propose using a single global matrix, such
as the one depicted in Fig. 2(a) (making use of all
the levels of impact of Table 3). MACBETH is then
used with the DM to quantify the relative
attractiveness of the impact levels through a
qualitative pairwise comparison questioning mode
similar to the one described in section 2.2. For
instance, reading Fig. 2(a), the DM judged as
“moderate” the difference of attractiveness between
“no injury” and “3 FAI”. It should be emphasized
that this is a global judgment for it involves the
impact levels in the two EC, according to the
constructed descriptors described in Table 3.
Fig. 2 presents relevant information from the
single global MACBETH matrix in the M-
MACBETH DSS: Fig. 2(a) presents the matrix of
judgments filled with the DM. Note that
disagreement or hesitation between two or more
semantic categories when making judgments is
allowed – e.g. ‘mod-strg’, ‘strg-vstrg’ and ‘vstrg-
'extr’ in Fig. 2(a), corresponding to ‘moderate or
strong’, ‘strong or very strong’ and ‘very strong or
extreme’, respectively. As this matrix is consistent,
M-MACBETH DSS proposes a numerical scale that
can be adjusted until an interval scale validated by
the DM is achieved (Fig. 2(b)). Note that in Fig. 2(c)
the semantic categories are quantified by more than
one single number.
Judgments in Fig. 2(a) show an example of
judgmental dependencies between the two EC. For
instance, the DM judged the difference in
attractiveness between no FAI (B
1
B
2
) and 10 FAI
(W
1
B
2
) to be ‘moderate or strong’; and the
difference between no fatality (B
1
W
2
) and 10
fatalities (W
1
W
2
) as ‘extreme’– see the highlighted
judgments in Fig. 2(a). These judgments indicate
that the EC “number of injuries” is cardinally value
dependent with the EC “severity of the injury”; and
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other judgments also suggest that the EC “severity
of the injury” is cardinally value dependent with the
EC “number of injuries”.
Once determined the value scores of all the
combinations (see Fig. 2(b)), the next step consists
in calculating and intuitively interpreting the CI
parameters.
3.3 Interpretation of the CI Parameters
The Shapley parameters can be interpreted as an
average value of the contribution of EC i
considering different baselines in the other EC (all
worst, all best, all the combinations with worst and
best levels). Its mathematical representation is given
by eq. system (4). In the context of independency,
the Shapley parameters act like ‘weights’ in the
additive model (Grabisch et al., 2010), in the sense
that the contribution of an EC i is the same,
independently of the chosen baseline to the
remainder EC.
m
1i
i
KAgiKAg
i\NK
i
1s
)]B(V)BB(V[
!n
|!K|)!1|K|m(
s
(4)
To better explain this parameter to a DM,
consider the EC “number of injuries” of our case
study. The Shapley parameter for this EC is given by
an average of the value of two swings, e.g.,
averaging the value from improving from 10
fatalities (W
1
W
2
) to no fatality (B
1
W
2
) with the
value from improving from 10 FAI (W
1
B
2
) to no
FAI (B
1
B
2
), the two cases depicted in Fig. 3. Note
that both swings depict the case from going to the
worst to the best level in the first EC, and when
different levels of the second EC are fixed. This
means that with interdependent EC the same swing
needs to be evaluated together with different
baselines. Using the rescaled value scores (from Fig.
2(b)) in eq. system (4), the Shapley parameter for
“number of injuries” is:
30
17
45
39
45
45
2
1
45
0
45
45
2
1
)]BW(V)BB(V[
2
1
)]WW(V)WB(V[
2
1
s
21Ag21Ag21Ag21Ag1
(5)
The same logic is applied to calculate the Shapley
parameter for the EC “severity of the injury”. I.e.,
averaging the value of two swings: going from 10
fatalities (W
1
W
2
) to 10 FAI (W
1
B
2
) and going from
no fatality (B
1
W
2
) to no FAI (B
1
B
2
).
Mathematically, this parameter is given by:
30
13
45
45
45
45
2
1
45
0
45
39
2
1
)]WB(V)BB(V[
2
1
)]WW(V)BW(V[
2
1
s
21Ag21Ag21Ag21Ag2
(6)
Regarding the interaction parameter, I
A
, it can be
interpreted as an average of the added value given
Figure 2: (a) Global MACBETH matrix of judgments, (b) numerical scales (left: basic MACBETH scale; right: MACBETH
scale rescaled), (c) values or ranges of numerical values for the seven MACBETH semantic categories.
(a) (b)
(c)
UsingMACBETHwiththeChoquetIntegralFundamentalstoModelInterdependenciesbetweenElementaryConcernsin
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Figure 3: Illustrative image to help the understanding of Shapley parameter of the EC 1 “number of injuries”.
by putting all the EC in a set A together (Grabisch,
1997), when all the baselines are considered.
Mathematically, it is expressed by (7).
A,NA
),BB(V)1(
)!1|A|n(
|!K||)!A||K|n(
)A(I
LKAg
AL
|L||A|
A\NK
(7)
In case of interdependencies, the combined effect of
having two best levels is more or less than the sum
of individual effects (Bana e Costa et al., 2005).
Thus, there is a parameter to correct this situation,
that is, an interaction parameter named I
12
(in the
domain [-1,1] with positive values standing for a
synergistic behaviour between EC i and j, negative
values for an antagonistic behaviour between EC i
and j, and null values for no interaction between EC
i and j), as computed in eq. (8).
12
AgAg
AgAgAgAg
I
)2W1W(V)2B1W(V
)2W1W(V)2W1B(V)2W1W(V)2B1B(V
(8)
Rewriting eq. (8) and using the rescaled value scores
presented in Fig. 2(b), the I
12
parameter assumes the
value of -39/45. This value is negative, being in line
with expectations since there is an antagonism
between the two EC. I.e., according to the DM,
independently of the EC, a single best level has
almost the same attractiveness as two best levels,
since there is a very low difference in value between
the referred options - see Fig. 2(b). Said in another
way, no injury of any type of incident or any number
of FAI has almost the same attractiveness as no FAI.
4 CONCLUSIONS
There is scope for developing tools to assist the
modelling of interdependent EC for decision aid.
This paper has identified some problems (such as the
use of local judgments to model interdependencies)
and challenges from using CI operators, proposing a
simple procedure to use the CI operators with
MACBETH. We have shown how a single
MACBETH matrix can be populated with global
(qualitative) judgments, by applying the MACBETH
protocol of questioning and method, and how it can
be coupled with CI operators. This approach allows
for: (i) detecting inconsistent judgments and
modelling cases of differences in opinion or
hesitation; (ii) including more judgments than the
ones required in some of the previous studies
(promoting more robust results), if that is found
appropriate in the decision context; and (iii) using
the M-MACBETH DSS to assist the whole process
with the DM (for instance, detecting inconsistencies
and including judgments of hesitation or of
differences in opinion). This study has illustrated
how one single MACBETH matrix can be the
starting point to test the compatibility of judgments
with the 2-additive CI formulation; and attempted to
provide a more intuitive explanation on the values of
the Shapley and interaction parameters.
Many aspects related to the use of a global
MACBETH matrix require further research. First,
the proposed approach needs to be applied to real
cases (and we are doing so in the health and safety
context). Second, it is relevant to analyse how the
global matrix can assist in cases in which there are
infeasible combinations between impact levels of
different EC, such as with the EC of the reputation
case developed in Bana e Costa et al., (1999). In
fact, some combinations of impact levels across EC
can be infeasible, leading for instance that levels
such as (W
1
W
2
B
3
), (W
1
B
2
W
3
) and (W
1
B
2
B
3
) cannot
be observed in the decision context, and therefore it
is not possible to build “elementary expressions”
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using the same baseline. Third, it is worth exploring
the compatibility of the global MACBETH matrix
with the general formulation of the CI operators for
more than two EC and with other mathematical
formulations, such as with the multilinear one.
Finally, it seems pertinent to exploit the CI
mathematical formulation to model non-monotonic
decision problems which involve not only cardinally
but also ordinally interdependent EC.
ACKNOWLEDGEMENTS
This work was funded by National Funds from the
Portuguese Public Budget through FCT –
Fundação
para a Ciência e a Tecnologia,
within the project
PTDC/EGE-GES/119230/2010.
REFERENCES
Aven, T., 2008. Risk analysis: assessing uncertainties
beyond expected values and probabilities, Wiley.
Bana e Costa C. A., Ensslin L. and Costa A.P., 1998.
Structuring the process of choosing rice varieties at the
south of Brazil. Multi-Criteria Evaluation in Land-Use
Management, Beinat E., Nijkamp P.(eds.), Kluwer
Academic Publishers, Environment & Management. 9:
33-45.
Bana e Costa, C. A. and Beinat, E., 2005. Model-
structuring in public decision-aiding. Working Paper
LSE OR 05.79 – London School of Economics.
Bana e Costa, C. A., Carnero, M. C. and Oliveira, M. D.,
2012. A multi-criteria model for auditing a Predictive
Maintenance Programme. European Journal of
Operational Research 217: 381–393.
Bana e Costa, C. A., Corrêa, E. C., de Corte, J.-M. and
Vansnick, J.-C., 2002. Facilitating bid evaluation in
public call for tenders: a socio-technical approach. The
Internation Journal of Management Science 30: 227-
242.
Bana e Costa, C. A., de Corte, J.-M. and Vansnick, J.-C.,
2012. MACBETH. International Journal of
Information Technology & Decision Making 11(2):
359-387.
Bana e Costa, C. A., Ensslin, L., Corrêa, E. C. and
Vansnick, J.-C., 1999. Decision Support Systems in
action: Integrated application in a multicriteria
decision aid process. European Journal of Operational
Research 113: 315-335.
Bana e Costa, C. A., Lourenço, J. C., Chagas, M. P. and
Bana e Costa, J. C., 2008. Development of reusable
bid evaluation models for the Portuguese Electric
Transmission Company. Decision Analysis 5(1): 22-
42.
Barin, A., Canha, L. N., Abaide, A. R., Orling, R. B. and
Martins, L. F. G., 2012. Selection of renewable energy
technologies in landfills: The M-MACBETH decision
support. Fuelling the Future: Advances in Science and
Technologies for Energy Generation, Transmission
and Storage. A. Mendez-Vilas, Universal-Publishers:
98-102.
Berrah, L. and Clivillé, V., 2007. Towards an aggregation
performance measurement system model in a supply
chain context. Computers in Industry 58: 709-719.
Bottero, M., Ferretti, V. and Mondini, G., 2013. A
Choquet integral-based approach for assessing the
sustainability of a new waste incinerator. International
Journal of Multicriteria Decision Making 3(2/3): 157.
Brosig, J., 2012. Multi-criteria evaluation of classical
swine fever control strategies using Choquet integral
(chapter three). Alternative Classical Swine Fever
control strategies - a simulation study. Agricultural
and Food Sciences. Christian-Albrechts, University of
Kiel. PhD thesis.
Cardin, M., Corazza, M., Funari, S. and Giove, S., 2013.
Building a global performance indicator to evaluate
academic activity using fuzzy measures. Neural Nets
and Surroundings. Apolloni B., Bassis S., Esposito A.
and Morabito F. C., Springer Berlin Heidelberg. 19:
217-225.
Carnero, M. C., 2006. Predictive maintenance programme
audit with MACBETH. Safety and Reliability for
Managing Risk. C. G. Soares and E. Zio, Taylor &
Francis Ltd. 1: 109-115.
Choquet, G., 1953. Theory of capacities. Annales de
l’Institut Fourier 5: 131-295.
Clivillé, V., Berrah, L. and Mauris, G., 2007. Quantitative
expression and aggregation of performance
measurements based on the MACBETH multi-criteria
method. International Journal of Production
Economics 105: 171-189.
Clivillé, V., Mauris, G. and Berrah, L., 2006. A quantified
industrial performance measurement system based on
a Choquet fuzzy integral. IEEE International
Conference on Fuzzy Systems, Vancouver, BC,
Canada.
Edwards, W. and Barron, F.H., 1994. SMARTS and
SMARTER: Improved simple methods for
multiattribute utility measurement. Organizational
Behavior and Human Decision Processes 60: 306-325.
Feyziog
̃
lu, O. and Büyüközkan, G., 2010. Evaluation of
Green Suppliers Considering Decision Criteria
Dependencies. Multiple Criteria Decision Making for
Sustainable Energy and Transportation Systems.
Ehrgott M., Naujoks B, Stewart T.J. and W. J.,
Springer Berlin Heidelberg. 634: 145-154.
Figueiredo, M. S. M. and Oliveira, M. D., 2009.
Prioritizing risks based on multicriteria decision aid
methodology: Development of methods applied to
ALSTOM power. IEEE International Conference on
Industrial Engineering and Engineering Management,
Hong Kong.
Grabisch, M., 1997. k-order additive discrete fuzzy
measures and their representation. Fuzzy Sets and
Systems 92: 167-189.
Grabisch, M., 2005. Une approach constructive de la
UsingMACBETHwiththeChoquetIntegralFundamentalstoModelInterdependenciesbetweenElementaryConcernsin
theContextofRiskManagement
125
décision multicritère. Traitment du Signal 22(4): 321-
337.
Grabisch, M. and Labreuche, C., 2004. Fuzzy measures
and integrals in MCDA. Multiple Criteria Decision
Analysis: State of the Art Surveys. Figueira J., Greco
S. and Ehrgott M., Kluwer Academic Publishers: 563-
608.
Grabisch, M. and Labreuche, C., 2010. A decade of
application of the Choquet integral and Sugeno
integrals in multicriteria decision-aid. Annals of
Operations Reasearch 175(1): 247-290.
Grabisch M., Kojadinovic I. and Meyer P., 2008. A review
of methods for capacity identification in Choquet
integral based multi-attribute utility theory
applications of the Kappalab R package. European
Journal of Operational Research 186: 766-785.
Grabisch M. and Roubens M., 2000. Application of the
Choquet integral in multicriteria decision making.
Fuzzy Measures and Integrals - Theory and
Applications. Grabisch M., Murofushi T. and Sugeno
M., Physica Verlag: 348-374.
Gürbüz, T., 2010. Multiple criteria human performance
evaluation using Choquet integral. International
Journal of Computational Intelligence Systems 3(3):
290-300.
Keeney, R. L., 1992. Value-Focused Thinking: a path to
creative decision making. Cambridge, Harvard
University Press.
Labreuche, C. and Grabisch, M., 2003. The Choquet
integral for the aggregation of interval scales in
multicriteria decision making. Fuzzy Sets and Systems
137: 11–26.
Labreuche C. and Le Huéde F., 2005. Myriad: a tool suite
for MCDA. International Conference of the Euro
Society for Fuzzy Logic and Technology (EUSFLAT),
Barcelona, Spain.
Linkov, I., Satterstrom, F. K., Kiker, G., Batchelor, C.,
Bridges, T. and E., F., 2006. From comparative risk
assessment to multi-criteria decision analysis and
adaptive management: Recent developments and
applications. Environment International 32: 1072–
1093.
Marichal, J.-L., 2000. An axiomatic approach of the
discrete Choquet integral as a tool to aggregate
interacting criteria. IEEE Transactions on Fuzzy
Systems 8(6): 800-807.
Mayag, B., Grabisch, M. and Labreuche, C., 2011. A
characterization of the 2-additive Choquet integral
through cardinal information. Fuzzy Sets and Systems
184: 84-105.
Merad, M., Dechy, N., Serir, L., Grabisch, M. and Marcel,
F., 2013. Using a multi-criteria decision aid
methodology to implement sustainable development
principles within an organization. European Journal of
Operational Research 224: 603–613.
Morton, A. and Fasolo, B., 2009. Behavioural decision
theory for multi-criteria decision analysis: a guided
tour. Journal of the Operational Research Society 60:
268-275.
Shah, L. A., Etienne, A., Siadat, A. and Vernadat, F. B.,
2013. Value-risk graph: a decision-making tool for
supply chain and industrial system engineering. 6th
IFAC International Conference on Management and
Control of Production and Logistics. September 11-
13, Fortaleza, Ceará, Brazil.
Vernadat, F., Shah, L., Etienne, A. and Siadat, A., 2013.
VR-PMS: a new approach for performance
measurement and management of industrial systems.
International Journal of Production Research: 1-19.
von Winterfeldt D. and Edwards W., 1986. Decision
analysis and behavioral research. Cambridge,
Cambridge University Press.
Williams, T. M., 1996. The two-dimensionality of project
risk. International Journal of Project Management
14(3): 185-186.
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