e m j.r. figueira e m (part i) - centralesupelec · 2 figueira, j.r., s. greco, b. roy, and r....

ELECTRE

METHODS

J.R. Figueira

1. Introduction1.1. References

1.2. Constructivism

1.3. Notation

2. Mainfeatures2.1. Preferencesituations

2.2. Preferencemodeling

2.3. Concordanceand Discordance

2.4. Illustrativeexample

2.5. Structure

ELECTRE METHODS (PART I)

José Rui FIGUEIRA ([email protected])

Technical University of Lisbon

10th MCDM Summer School, Paris, France

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Contents

1 1. Introduction1.1. References1.2. Constructivism1.3. Notation

2 2. Main features2.1. Preference situations2.2. Preference modeling2.3. Concordance and Discordance2.4. Illustrative example2.5. Structure

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2.5. Structure

Main references for this talk:

1 Figueira, J., B. Roy, and V. Mousseau (2005). ELECTRE

methods. In J. Figueira, S. Greco, and M. Ehrgott (Eds.),Multiple Criteria Decision Analysis: State of the Art Surveys,pp. 133162. New York, U.S.A.: Springer Science + BusinessMedia, Inc.

2 Figueira, J.R., S. Greco, B. Roy, and R. Słowinski, (2010).ELECTRE methods: Main features and recent developments.In C. Zopounidis and P. Pardalos (Eds.), Handbook ofMulticriteria Analysis, Chapter 4, New York, USA: Springer.

3 Greco, S., R. Słowinski, J.R. Figueira, and V. Mousseau(2010). Robust ordinal regression. In M. Ehrgott, J.R.Figueira, and S. Greco (Eds.), Trends in Multiple CriteriaDecision Analysis, pp. 273320. New York, U.S.A.: SpringerScience + Business Media, Inc.

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1. Introduction1.1. ELECTRE methods were designed according to a constructivistconception of MCDA: A decision aiding situation (Roy, 2009).

A decision aiding situation

1 Imagine that in a company or institution, a CEO isconfronted with a certain decision aiding situation andhas to make a decision.

2 The CEO needs the help of an analyst (an in-houseoperational service, a consultant, or a universityresearch team).

3 Two key elements in a decision aiding situation are:The Analyst and the Decision Maker (DM). The latter ishere represented by the CEO.

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1. Introduction1.1. ELECTRE methods were designed according to a constructivistconception of MCDA: The fundamental pillars (Roy, 2009).

The decision aiding activity is based on three fundamentalpillars:

1 The actions (formal definition of the possible actions oralternatives).

2 The consequences (aspects, attributes, characteristics,. . . of the actions that allow to compare them).

3 The modeling of a preference system (it consists of animplicit or explicit process, that for each pair of actionsenvisioned, assigns one and only one of the threepossibilities: indifference, preference, orincomparability).

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1. Introduction.1.1. ELECTRE methods were designed according to a constructivistconception of MCDA (Roy, 2009).

Based on the above three pillars:

1. The analyst should try to obtain a coherent structuredset of results in order to guide the decision aidingprocess and facilitate the communications about thedecisions.

2. The analyst must follow an approach that leads or aimsto produce knowledge from a certain number workinghypotheses defined a priori.

3. This approach should be based on models that are, atleast co-constructed interactively with the DM.

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1. Introduction1.1. ELECTRE methods were designed according to a constructivistconception of MCDA (Roy, 2009).

Based on the above three pillars:

4. During the co-construction process, that takes intoaccount the values of the DM, contradictoryjudgements or ambiguities may occur.

5. The analyst must admit that the novelty of thesequestions can bring (the DM) or the person thisquestioned to revise certain pre-existing preferencesmomentarily and locally.

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1. Introduction1.2. Notation: Basic data.

Basic data

1 A = {a1, a2, . . . , ai , . . . , am} is the set of m potentialactions. This set can be partially known a priori (it isfrequent in sorting problems).

2 F = {g1, g2, . . . , gj , . . . , gn} is a coherent family ofcriteria, with n > 3.

3 gj(ai) is the performance of action ai on criterion gj , forall ai ∈ A and gj ∈ F . A performance matrix M can thusbe built.

4 Assume w.l.g. that the higher the performance gj(a) is,the better for the DM (increasing direction ofpreference).

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2. Main features2.1. Preferences situations.

Four main comprehensive preference situations

1 I (Indifference)

2 P (strict preference)

3 Q (hesitation : weak preference)

4 R (incomparability).

(For more details see Figueira et al., 2010)

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2. Main features2.2. Preference modeling through outranking relations: The concept ofpseudo-criterion (Roy, 1996).

Pseudo-criterion

A pseudo-criterion is a function gj associated with twothreshold functions, qj(·) and pj(·), satisfying the followingcondition: for all ordered pairs of actions (a, a′) ∈ A × Asuch that gj(a) > gj(a′), gj(a) + pj(gj(a′)) andgj(a) + qj(gj(a′)) are non-decreasing monotone functions ofgj(a′), such that pj(gj(a′)) > qj(gj(a′)) > 0, for all a ∈ A.

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2. Main features2.2. Preference modeling through outranking relations: Partial binaryrelations.

Partial binary relations (1)

1 gj(a) − gj(a′) > pj(gj(a′)) ⇔ aPja′,

2 qj(gj(a′)) < gj(a) − gj(a′) 6 pj(gj(a′)) ⇔ aQja′,

3 −qj(gj(a)) 6 gj(a) − gj(a′) 6 qj(gj(a′)) ⇔ aIja′.

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2. Main features2.2. Preference modeling through outranking relations: Partial binaryrelations.

Partial binary relations (2)

1 Sj = Pj ∪ Qj ∪ Ij

2 aSja′ means that “a is at least as good as a′” oncriterion gj .

3 When aSja′ the voting power of criterion gj , denoted bywj is taken in total (assume w.l.g. thatw1 + w2 + . . . + wn = 1).

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2. Main features2.2. Preference modeling through outranking relations: Comprehensiveoutranking.

Let S = P ∪ Q ∪ I, whose meaning “is at least as good as”.

Comprehensive outranking

Consider two actions, a and a′ and the relation ≻= P ∪ Q.Four situations may occur:

1 aSa′ and not(a′Sa), i.e., a ≻ a′ (a is preferred in abroader sense to a′).

2 a′Sa and not(aSa′), i.e., a′ ≻ a (a′ is preferred in abroader sense to a).

3 aSa′ and a′Sa, i.e., aIa′ (a is indifferent to a′).

4 not(aSa′) and not(a′Sa), i.e., aRa′ (a is incomparable toa′).

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2. Main features2.3. Concordance and Discordance: Concordance.

Concordance

1 Concordance. To validate aSa′, a sufficient majority ofcriteria in favor of this assertion must occur.

2 The comprehensive concordance index c(a, a′) foreach pair of actions (a, a′) ∈ A × A, for all gj ∈ F isfundamental to all the ELECTRE methods in order tocompute a concordance matrix C.

c(a, a′) =∑

{j | gj∈C(a{P,Q,I}a′})

wj +∑

{j | gj∈C(a′Qa)}

wjϕj

where

ϕj =gj(a) − gj(a′) + pj

pj − qj

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2. Main features2.3. Concordance and Discordance: Voting power.

Voting power

This index comprises the summation of the votingpower of the criteria that clearly are in favor of theassertion aSa′, plus the summation of the fraction, ϕj ,of the voting power for those criteria included in thehesitation group.

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2. Main features.2.3. Concordance and Discordance: Graphical representation.

1

0

ϕj

gj(a′) − pj(gj(a))

gj(a′) − qj(gj(a))

gj(a′)

gj(a′) + qj(gj(a′))

gj(a′) + pj(gj(a′)) gj(a)

Figure: Variation of ϕj for a given gj(a′) and variable gj(a)

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2. Main features2.3.Concordance and Discordance: Discordance (1)

Discordance

1 Discordance. The assertion aSa′ cannot be validated ifa minority of criteria is strongly against this assertions.

2 The concept of veto threshold, vj , gives the possibilityto the criterion gj to impose its veto power. It meansthat gj(a′) is so much better than gj(a), that is notpossible to allow that aSa′

3 The computation of the partial discordance indicesleads to the construction of a discordance matrix, D.

4 The application of both types of indices is related to aspecific ELECTRE method. For example, in ELECTRE

TRI they are “combined” with c(a, a′) to define a degreeof credibility of the assertion aSa′ (fuzzy relation).

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2. Main features2.3. Concordance and Discordance: Discordance (2)

Partial discordance index

dj(a, a′) =

8

>

<

>

:

1 if gj(a) − gj (a′) < −vj (gj (a)),gj (a)−gj(a

′)+pj (gj (a))

pj (gj (a)) − vj (gj (a))if −vj(gj (a)) 6 gj(a) − gj (a′) < −pj(gj (a)),

0 if gj(a) − gj (a′) > −pj(gj (a)).

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2. Main features2.4. Reminder and additional notation

Reminder and additional notation

1 We use kj as the non-normalized weights for each criterion)

- C(aSa′) is the coalition of criteria in favor of theassertion aSa′.

- W{C(aSa′}) =∑

{j : gj∈C(aSa′)}

wj is the weight or power of

the coalition C(aSa′).

2 qj(·) is the indifference threshold of criterion gj .

3 pj(·) is the preference threshold of criterion gj .

4 vj(·) is a veto threshold of criterion gj .

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2. Main features2.4. Location of a new hotel (Figueira et al., 2009) (1)

Location of a new hotel

1 Matrix below presents the performances of the five sites - a,b, c, d , and e - according to the five criteria.

2 The performances of criterion g1 (investment costs) areexpressed in thousands of e, designated Ke.

3 The indifference and the preference thresholds assigned tothis criterion are q1(g1(x)) = 500 + 0.03g1(x) Ke andp1(g1(x)) = 1000 + 0.05g1(x) Ke, respectively, where x isthe worst of the two actions.

4 The performances of criterion g2 (annual costs) are alsoexpressed in Ke; the thresholds assigned to this criterion areq2(g1(x)) = 50 + 0.05g1(x) Ke andp2(g1(x)) = 100 + 0.07g1(x) Ke, respectively.

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2. Main features2.4. Location of a new hotel (Figueira et al., 2009) (2)

Location of a new hotel

1 The performances of criteria g3 (recruitment), g4 (image),and g5 (access) are expressed on the following seven-levelqualitative scale: very bad (1), bad (2), rather bad (3),average (5), rather good (5), good (6), and very good (7).The values between parenthesis can be used in ELECTRE

methods to code the different verbal statements.

2 Other ways of coding the verbal scale through the use ofnumerical values could be used by adjusting the thresholdsvalues (see Martel and Roy, 2006).

3 The indifference threshold for each criterion has been set atone on the seven-level scale and the preference threshold attwo levels.

4 In this example there is no veto.

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2. Main features2.4. Performances matrix (Figueira et al., 2009)

Performances matrix

1 Quantitative criteria: g1 (investment costs) and g2 (annualcosts)

2 Qualitative criteria: g3 (recruitment), g4 (image), and g5

(access)

g1[min] g2[min] g3[max] g4[max] g5[max]a 13 000 Ke 3 000 Ke Average Average Averageb 15 000 Ke 2 500 Ke Good Bad Very Goodc 10 900 Ke 3 400 Ke Good Good Very Badd 15 500 Ke 3 500 Ke Good Good Goode 15 000 Ke 2 600 Ke Good Very Bad Bad

kj 5 4 3 3 3

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2. Main features2.4. Pairwise comparison

Pairwise comparison

1 Does a outrank d , aSd? For the moment we cannotanswer this question.

2 The coalition of criteria in favor of aSd :C(aSd) = {g1, g2}

3 The power of this coalition: W{C(aSd)} = 4+518 = 0.5

(normalized)

4 What about dSa?

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2. Main features.2.5. The structure of ELECTRE methods.

Each ELECTRE method comprises two main procedures:

Two procedures

1 The first procedure is a Multiple Criteria AggregationProcedure (MCAP) that builds one or possibly severaloutranking relations aim to compare, in acomprehensive way, each ordered pair of actions.

2 The second procedure, called Exploitation Procedure(EP) is used to obtain adequate results from whichrecommendations can be derived.

3 The nature of the results depends of the specificproblematique.

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2. Main features.2.5. Example: MCAP of ELECTRE III

MCAP of ELECTRE III

It is modeled through a credibility index i.e. a fuzzy measuredenoted by σ(a, a′) ∈ [0, 1], which combines c(a, a′) anddj(a, a′):

σ(a, a′) = c(a, a′)∏

j∈J (a,a′)

1 − dj(a, a′)

1 − c(a, a′),

where j ∈ J (a, a′) if and only if dj(a, a′) > c(a, a′).

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2. Main features2.5. The nature of the results: Choosing (Mousseau, 1993; Roy, 2002).

Choosing: Selecting a restricted number as small aspossible of potential actions, which justify to eliminatingothers.

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Choosing: Selecting a restricted number as small aspossible of potential actions, which justify to eliminating allothers.

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Choosing: Selecting a restricted number as small aspossible of potential actions, which justify to eliminating allothers. Choice set

Actions rejected

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2. Main features2.5. The nature of the results: Ranking (Mousseau, 1993; Roy, 2002).

Ranking: Ranking of actions from the best to the worst, withthe of ties (ex aequo) and incomparabilities.

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Ranking: Ranking of actions from the best to the worst, withthe of ties (ex aequo) and incomparabilities.

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2. Main features2.5. The nature of the results: Sorting (Mousseau, 1993; Roy, 2002).

Ordinal classification or sorting: Assigning each potentialaction to one of the categories among those of a familypreviously defined; the categories are ordered, in general,from the worst to the best one.

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...

Category 1

Category 2

Category k

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...

Cat. 1

Cat. 2

Cat. k

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2. Main features2.5. The nature of the results: Absolute versus relative evaluation (Roy,1996).

In sorting problems there is an absolute evaluation: theassignment of an action only takes into account theintrinsic evaluation of this action on all the criteria anddoes not depend on nor influence the category to whichanother action should be assigned.

As for the remaining problematiques the actions arecompared against each other and thus there exists arelative evaluation instead of an absolute evaluation asfor the previous case.

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2. Main features.2.6. Software (Figueira et al., 2005).

Choosing: ELECTRE I, ELECTRE IV, and ELECTRE IS.

Ranking: ELECTRE II, ELECTRE III, and ELECTRE IV.

Ordinal classification or sorting: ELECTRE TRI.

New software (see later on).

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2. Mainfeatures2.6. Strong features

2.7. Weaknesses

ELECTRE METHODS (PART II)




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2.7. Weaknesses

Contents

1 2. Main features2.6. Strong features2.7. Weaknesses

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2.7. Weaknesses

2. Main featuresIntroduction

Summary

1 The qualitative nature of some criteria

2 The heterogeneity of scales

3 The non-relevance of compensatory effects

4 The imperfect knowledge and arbitrariness

5 The reasons for and reasons against and outranking

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2.7. Weaknesses

2. Main features2.6. Some strong features of ELECTRE methods

Strong features

1. They have the possibility of taking into account thequalitative nature of some criteria. They allow thus toconsider the original data.

2. They can deal with very heterogeneous scales to modelnoisy, delay, aesthetics, cost, . . . Whatever the nature ofscales, every procedure can run by preserving the originalperformances of the actions.

3. The compensatory effects are not pertinent. This is due tothe fact that the weights cannot be interpreted as substitutionrates. Contrarily to other methods there is no need inELECTRE methods to use, from the starting point of theirapplication, identical and commensurable scales.

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Strong features

Consider the following example with 4 criteria and only 2 actions(scales: [0,10]). The weighted-sum model was chosen, i.e,V (a) =

∑nj=1 wjgj(a). In the considered example, the weights, wj ,

are equal for all criteria:g1 g2 g3 g4

a1 9.5 9.5 8.1 5.4a2 8.3 8.3 7.3 8.5

V (a1) = 8.125 > V (a2) = 8.100.

This example shows, in an obvious way, the possibility that abig preference difference not favorable to a1 on one of thecriteria (g4) can be compensated by 3 differences of weakamplitude on the remaining criteria, in such a way that a1

becomes finally preferred to a2. In ELECTRE methods thiseffect does not occurs in a systematic way.

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Strong features

4. They are adequate to take the imperfect knowledge of thedata and the arbitrariness related to the construction of thecriteria. This is modeled through the indifference andpreference thresholds. Consider the same example with thefollowing (constant) discrimination thresholds:

g1 g2 g3 g4

a1 9.5 9.5 8.1 5.4a2 8.3 8.3 7.3 8.5qj 1 1 1 1pj 2 2 2 2

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Strong features

If on criterion g3 we change the performance from 7.3 to 7.1,the score moves from 8.100 to 8.050(V (a1) − V (a2) = 0.050). Consequently there is areinforcement of the preference in favor of a1.

On the other hand, with ELECTRE c(a1, a2) and c(a2, a1)remain unchanged.

Now, if we consider 7.5 instead of 7.3, then V (a2) = 8.150,and consequently a2Pa1. Again this small variation is toosmall.

When adding the discrimination thresholds and usingELECTRE methods, c(a1, a2) = 0.25 + 0.25 + 0.25 = 0.75and c(a2, a1) = 0.2 + 0.2 + 0.25 + 0.25 = 0.8. Thus, a2Pa1.

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Strong features

5. They are based in a certain sense in the reasons for and thereasons against of an outranking between two actions(concordance and discordance). Consider the sameexample and that a veto threshold should vj = 3, for allj = 1, . . . , 4.

g1 g2 g3 g4

a1 9.5 9.5 8.1 5.4a2 8.3 8.3 7.3 8.5qj 1 1 1 1pj 2 2 2 2vj 3 3 3 3

If s = 0.8 then a2Sa1 and not(a2Sa1). But, if s = 0.7, a1Ia2.Since d4(a2, a1) = 1, g4 imposes a veto, for whatever thechosen s. We get allays not(a2Sa1).

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2.7. Weaknesses

2.7. WeaknessesIntroduction

Summary

1 Scoring actions

2 The quantitative nature of family of criteria

3 The independence with respect to irrelevantalternatives

4 The intransitivities

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2.7. Weaknesses

2. Main features2.7. Some weaknesses of ELECTRE methods

Some weaknesses

1. Scoring the actions. In certain contexts it is required toassign a score to each action. When the decisionmakers require each action should appear associatedwith a score, the ELECTRE methods are not adequatefor such a purpose and the scoring based methodsshould be applied instead. The decision makersshould be, however, aware that they cannot provideinformation that leads, for example, to intransitivities orto incomparabilities between certain pairs of actions.Indeed, this score is very fragile.

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Some weaknesses

2. The quantitative nature of the family of criteria. Whenall the criteria are quantitative it is “better” to use othermethods. But, if we want to take into account acompletely or even a partial noncompensatory method,the reasons for and against, or the imperfect characterof at least one criterion, even under such conditions, wecan use the ELECTRE methods.

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Some weaknesses

3. The independence with respect to irrelevantalternatives. Except ELECTRE TRI-B, TRI-C, theremaining ELECTRE methods does not fulfill theindependence w.r.t. irrelevant alternatives (Roy, 1973).In 1973, B. Roy shows that rank reversal may occurand consequently the property of independence withrespect to irrelevant alternatives can be violated whendealing with outranking relations. Notice that rankreversal may occur only when the set of potentialactions is subject to evolve, which is quite a naturalassumption, but one that is not present in many harddecision-aiding processes where the number ofalternatives is rather small and easily identified.

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Some weaknesses

4. Intransitivities may also occur in ELECTRE methods(Roy, 1973). It is also well-known that methods usingoutranking relations (not only the ELECTRE methods)do not need to fulfill the transitivity property. This aspectrepresents only a weakness if we impose a priori thatpreferences should be transitive. There are, however,some raisons that lead us to do not impose transitivity.

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3. Somerecentdevelopments(>2000)3.1. Methodological

3.2. New approaches

3.3. Axiomatic andmeaningfulnessanalysis

3.4. Other aspects

4.Applications4.1. Someapplications areas

4.2. Real-worldapplications

5. Concludingremarks

ELECTRE METHODS (PART III)




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3.2. New approaches


3.4. Other aspects




Contents

1 3. Some recent developments (>2000)3.1. Methodological3.2. New approaches3.3. Axiomatic and meaningfulness analysis3.4. Other aspects

2 4. Applications4.1. Some applications areas4.2. Real-world applications

3 5. Concluding remarks

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3.4. Other aspects




3. Some recent developments (>2000)3.1. Methodological (1)

Recent developments

1 Pure inference based approaches after the work byMousseau and Słowinski (1998) (Software: ELECTRE

TRI-Assistant):

inferring only the weights (Mousseau et al, 2001);

inferring veto (Mousseau and Dias, 2006); and,

inferring category bounds (Ngo The and Mousseau,2002).

Some manageable disaggregation procedures forvalued outranking relations (Mouuseau and Dias,2006);

Inconsistent judgements (Mousseau et al., 2006a;Mousseau et al., 2006b) or an inadequate preferencemodel (Figueira, 2009).

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3.4. Other aspects




3. Some recent developments (>2000)3.1. Methodological (2)

Recent developments

2 The inference-robustness based approach for inferringweights and derive robust conclusions in sorting problems(Dias et al., 2002). Software: IRIS.

3 The pseudo-robustness based approach dealing withsimulation methods mainly for ranking and sorting problems(Tervonen et al., 2008, 2009). Software: SMAA-III,SMAA-TRI.

4 New robustness analysis concepts (Aissi and Roy, 2009;Roy, 2009). These papers are more general, but sometechniques can be applied to ELECTRE methods.

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3. Some recent developments (>2000)3.2. New approaches

New approaches

1 Bi-polar outranking relations implemented in RUBIS software(Bisdorff et al., 2007, 2008).

2 The weights of the interaction coefficients and themodifications in the concordance index (Figueira et al.,2009).

3 Handling with the reinforced preference and the counter-vetoeffects (Roy and Słowinski, 2009).

4 ELECTRE TRI-C, TRIN, NC (Almeida-Dias et al., 2010a,2010b).

5 The possible and the necessary approach for ELECTRE

methods (ELECTRE-GKMS) by Greco et al., (2009, 2010).

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3. Some recent developments (>2000)3.3 Axiomatic and meaningfulness

Axiomatic and meaningfulness

1 Axiomatic analysis of ELECTRE I method by using conjointmeasurement theory (Greco et al., 2001).

2 Representing preferences through conjoint measure and thedecision rule approach (Greco et al., 2002).

3 An axiomatic analysis based on a general conjoint measureframework with application to a variant of ELECTRE TRI

(Bouyssou and Marchant, 2007a,b).

4 An axiomatic analysis of the concordance-discordancerelations (Bouyssou and Pirlot, 2009).

5 Representing preferences by decision rules (Greco et al.,2002).

6 The meaningfulness of ELECTRE methods (Martel and Roy,2006).

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3. Some recent developments (>2000).3.4 Other aspects

Other aspects

1 The relative importance of criteria (Figueira and Roy,2002).

2 Concordant outranking with criteria of ordinalsignificance (Bisdorff, 2004).

3 Evolutionary approaches (Leyva-López et al., 2008;Doumpos et al., 2009).

4 The EPISSURE method for the assessment ofnon-financial performances (André and Roy, 2007;André, 2009).

5 Group decision aiding (Damart et al., 2007; Greco etal., 2009, 2010).

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4. Applications4.1. Some applications areas

Areas

1 Agriculture and Forest Management.

2 Energy.

3 Environment and Water Management.

4 Finance.

5 Medicine.

6 Military.

7 Project selection (call for tenders).

8 Transportation.

9 . . .

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4. Applications4.2. Concrete cases (1)

Areas

Sorting cropping systems (Arondel and Girardin, 2000).

Land-use suitability assessment (Joerin et al., 2001).

Greenhouse gases emission reduction (Georgopoulou,2003).

Risk zoning of an area subjected to mining-inducinghazards (Merad et al., 2004).

Participatory decision-making on the localization ofwaste-treatment plants (Norese, 2006).

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4. Applications4.2. Concrete cases (1)

Areas

Material selection of bipolar plates for polymerelectrolyte membrane fuel cell (Shanian andSavadogo).

Assisted reproductive technology (Matias, 2008).

Promotion of social and economic development(Autran-Gomes et al., 2009).

Sustainable demolition waste management strategy(Roussat et al., 2009).

Assessing the risk of nano-materials (Tervonen et al.,2009).

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5. Concluding remarks

Concluding remarks

1 ELECTRE methods have a long history of successfulreal-world applications with impact on the life of populations(see Figueira et al., 2005)).

2 When applying ELECTRE methods analysts should payattention to the characteristics of the context and also to the(theoretical) weaknesses of these methods. Note that all theMCDA methods have theoretical limitations.

3 Software implementations of high quality along with friendlyinterfaces render possible the application to a vast range ofapplications.

4 Research on ELECTRE methods is not a death field. It stillsevolving and rapidly, namely over of the first years of thisnew millennium.

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Thank You!(very much for your attention)

e m j.r. figueira e m (part i) - centralesupelec · 2 figueira, j.r., s. greco, b. roy, and r....

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