Robust Ordinal Regression and Stochastic Multiobjective Acceptability Analysis in Multiple Criteria Hierarchy Process for the Choquet integral preference model

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Robust Ordinal Regression and Stochastic Multiobjective

Acceptability Analysis in Multiple Criteria Hierarchy

Process for the Choquet integral preference model

Salvatore Corrente

Department of Economics and Business, University of Catania, Italy

Silvia Angilella, Salvatore Greco, Roman S lowi´nski

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We deal with...

The Choquet integral preference model,

SMAA and ROR to explore the whole set of parameters

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We deal with...

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We deal with...

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Plan

Problem statement Choquet integral MCHP basic concepts MCHP and Choquet integral

ROR and SMAA in MCHP applied to the Choquet integral preference model Didactic example

Conclusions

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Problem statement

Consider a ﬁnite set A of alternatives (actions, solutions, objects) evaluated on m

criteria from a consistent family G = {g1, . . . , gm}, J = {1, . . . , m} and

Ij = {gj(a), a ∈ A} for all j ∈ J .

Taking into account preferences of a Decision Maker (DM), we can deal with three main problems:

Choice Sorting Ranking

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Preference models

Utility functions(R. Keeney, and H. Raiﬀa, 1993), e.g., in additive form:

U(a) =

m

X

j=1

uj(gj(a)),

that, in the simplest case, becomes: U(a) =

m

X

j=1

wjgj(a)

Binaryoutranking relationsS (B. Roy, 1996; J.P. Brans, and P. Vincke, 1985):

aSb⇔ a is at least as good as b

Decision rules(S. Greco, B. Matarazzo, and R. S lowi´nski, 2001):

“if maximum speed of a car is at least175 km/h, and its price isat most 12,000 euros, then this car is comprehensively at leastmedium.”

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Preference models

U(a) =

m

X

j=1

uj(gj(a)),

m

X

j=1

wjgj(a)

Binaryoutranking relationsS (B. Roy, 1996; J.P. Brans, and P. Vincke,

1985):

“if maximum speed of a car is at least175 km/h, and its price isat most 12,000 euros, then this car is comprehensively at leastmedium.”

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Preference models

U(a) =

m

X

j=1

uj(gj(a)),

m

X

j=1

wjgj(a)

Binaryoutranking relationsS (B. Roy, 1996; J.P. Brans, and P. Vincke,

1985):

“if maximum speed of a car is at least175 km/h, and its price isat most

12,000 euros, then this car is comprehensively at leastmedium.”

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Interaction between Criteria and the Choquet integral

1

“Maximum speedandAccelerationso asMaximum speedandPriceare interacting criteria”,

1_{(M. Grabisch, 1996)}

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Interaction between Criteria and the Choquet integral

2

Given x ∈

R

m₊_{, and a capacity µ : 2}J _{→ [0, 1] such that:}

µ(∅) = 0, µ(J ) = 1, µ(A) ≤ µ(B), if A ⊆ B ⊆ J Ch(x, µ) = m X j=1 x(j)µ(A(j)) − A(j+1) = m X j=1 x(j)− x(j−1) µ(A(j)) where 0 = x(0)≤ x(1)≤ . . . ≤ x(m), A(j)= {i ∈ J : xi ≥ x(j)}, A(m+1)= ∅. 2_{(M. Grabisch, 1996)}

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Weightd sum vs Choquet Integral

Weighted Sum Choquet Integral

U(a) = m X j=1 wjgj(a), Ch(x, µ) = m X j=1 x(j)− x(j−1) µ(A(j))

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Shapley value and the interaction index

TheShapley value (L.S. Shapley, 1953) expressing the importance of

criterion gi ∈ G , is given by:

ϕ({i}) = X

T⊆G :i /∈T

(|G \ T | − 1)!|T |!

|G |! · [µ(T ∪ {i}) − µ(T )],

Theinteraction index (S. Murofushi and T. Soneda, 1993) expressing the

sign and the magnitude of the synergy in a couple of criteria {gi, gj} ∈ G , is

given by:

ϕ ({i, j}) = X

T⊆G :i,j /∈T

(|G \ T | − 2)!|T |!

(|G | − 1)! · τ (T , i, j),

where τ (T , i, j) = [µ(T ∪ {i, j}) − µ(T ∪ {i}) − µ(T ∪ {j}) + µ(T )].

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M¨obius representation and k-additive capacity

3 m: 2G → [0, 1] µ(R) = X T⊆R m(T ), m(R) = X T⊆R (−1)|R\T |µ(T ) 1b) m(∅) = 0, X T⊆G m(T ) = 1, 2b) ∀ i ∈ G and ∀R ⊆ G \ {i} , X T⊆R m(T ∪ {i}) ≥ 0.

A capacity is calledk-additiveif m(T ) = 0 for T ⊆ G such that |T | > k.

3_{(M. Grabisch, 1997; M. Grabisch and C. Labreuche 2005)}

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M¨obius representation of a 2-additive capacity

µ(R) =X i∈R m({i}) + X {i,j}⊆R m({i, j}) , ∀R ⊆ G . 1c) m(∅) = 0, X i∈G m({i}) + X {i,j}⊆G m({i, j}) = 1, 2c)      m({i}) ≥ 0, ∀i ∈ G , m({i}) +X j∈T

m({i, j}) ≥ 0, ∀i ∈ G and ∀ T ⊆ G \ {i} , T 6= ∅.

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Shapley value and interaction index for 2-additive capacities

Cµ(a) = X {i}⊆G m({i}) (gi(a)) + X {i,j}⊆G

m({i, j}) min{gi(a) , gj(a)}.

TheShapley value can be expressed as:

ϕ ({i}) = m ({i}) + X

j∈G \{i}

m({i, j})

2 , i ∈ G ,

Theinteraction index can be expressed as:

ϕ ({i, j}) = m ({i, j}) .

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Multiple Criteria Hierarchy Process

4

Basic concepts:

G set of criteria of all levels of the hierarchy,

IG set of indices of all criteria in the diﬀerent levels,

Gr∈ G with r = (i1, . . . , ih) ∈ IG, criterion of the level h in the hierarchy,

ELset of indices of elementary subcriteria (i.e. set of criteria in the leaves of the tree),

E(Gr) set of indices of elementary subcriteria descending from criterion Gr.

4_{S. Corrente, S. Greco, R. S lowi´}_{nski (2012)}

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Multiple Criteria Hierarchy Process

4

Basic concepts:

Gr∈ G with r = (i1, . . . , ih) ∈ IG, criterion of the level h in the hierarchy, ELset of indices of elementary subcriteria (i.e. set of criteria in the leaves of the tree),

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Multiple Criteria Hierarchy Process

4

Basic concepts:

E(Gr) set of indices of elementary subcriteria descending from criterion Gr. 4_{S. Corrente, S. Greco, R. S lowi´}_{nski (2012)}

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Multiple Criteria Hierarchy Process

4

Basic concepts:

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Multiple Criteria Hierarchy Process

4

Basic concepts:

ELset of indices of elementary subcriteria (i.e. set of criteria in the leaves of

the tree),

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Multiple Criteria Hierarchy Process

4

Basic concepts:

ELset of indices of elementary subcriteria (i.e. set of criteria in the leaves of

the tree),

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MCHP and the Choquet integral

5

Given Gr, r ∈ IG\ EL, we deﬁne:

Capacity on Gk r = n G(r,w )∈ G : (r, w ) ∈ IG∩

N

k o µk r : 2G k r → [0, 1], Given F ⊆ Gk r, µkr(F) = µ(E (F)) µ(E (Gr))

Choquet integral of a on criterion Gr, r /∈ EL,

Cµr(a) = Cµ(ar) µ(E (Gr)) where gt(ar) = ( gt(a), if t ∈ E (Gr), 0 if t /∈ E (Gr).

5_{S. Angilella, S. Corrente, S. Greco and R. S lowi´}_{nski, 2013}

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MCHP and the Choquet integral

5

Given Gr, r ∈ IG\ EL, we deﬁne:

Capacity on Gk r = n G(r,w )∈ G : (r, w ) ∈ IG∩

N

k o µk r : 2G k r → [0, 1], Given F ⊆ Gk r, µkr(F) = µ(E (F)) µ(E (Gr))

Choquet integral of a on criterion Gr, r /∈ EL,

Cµr(a) = Cµ(ar) µ(E (Gr)) where gt(ar) = ( gt(a), if t ∈ E (Gr), 0 if t /∈ E (Gr).

5_{S. Angilella, S. Corrente, S. Greco and R. S lowi´}_{nski, 2013}

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Shapley Value and the Interaction Index (1)

ϕk r G(r,w ) = X T⊆Gk r\{G(r,w )} |Gk r \ T | − 1!|T |! |Gk r|! µk r T ∪G(r,w ) − µkr (T ) , ϕk r G(r,w1), G(r,w2) = X T⊆Gk r\{G(r,w1), G(r,w2)} |Gk r \ T | − 2!|T |! (|Gk r| − 1)! . µk r (F) = X T⊆F mk_r(T )

Shapley Value and the Interaction Index (2)

ϕk r G(r,w ) = X F⊆Gk r: G(r,w )∈F mk r (F) |F| ϕk r G(r,w1), G(r,w2) = X F⊆Gk r: G(r,w1), G(r,w2)∈F mk r(F) |F| − 1

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m

_rk

in terms of the M¨obius representation m

Proposition

Letµ a capacity defined on 2EL_{, and m its M¨}_{obius representation. Let G}

r∈ G,

r∈ IG\ {EL} with µkr a capacity defined on2G

k

r and mk

r its M¨obius

representation; then for allF =G(r,w1), . . . , G(r,wα) ⊆ G

k r, mk r (F) = mkr G(r,w1), . . . , G(r,wα) = X T1⊆E(G_(r,w1)), T16=∅ ··· Tα⊆E(G(r,wα )), Tα6=∅ m({T1, . . . , Tα}) µ (E (Gr)) .

Proposition

Letµ a q-additive capacity defined on 2EL_{, then}_µk

r is a q-additive capacity defined on2Grk, for all G_r∈ G with r ∈ I_G\ {EL}.

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m

_rk

in terms of the M¨obius representation m

Proposition

Letµ a capacity defined on 2EL_{, and m its M¨}_{obius representation. Let G}

r∈ G,

r∈ IG\ {EL} with µkr a capacity defined on2G

k

r and mk

r its M¨obius

representation; then for allF =G(r,w1), . . . , G(r,wα) ⊆ G

k r, mk r (F) = mkr G(r,w1), . . . , G(r,wα) = X T1⊆E(G_(r,w1)), T16=∅ ··· Tα⊆E(G(r,wα )), Tα6=∅ m({T1, . . . , Tα}) µ (E (Gr)) .

Proposition

Letµ a q-additive capacity defined on 2EL_{, then}_µk

r is a q-additive capacity

defined on2Grk, for all G_r∈ G with r ∈ I_G\ {EL}.

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Shapley and Interaction indices in terms of m

Proposition

Letµ a 2-additive capacity defined on 2EL _{and G}

(r,w ), G(r,w1), G(r,w2)∈ G k r, with r∈ IG\ {EL}, then: 1 ϕkr G(r,w ) =         X t∈E_G(r,w) m(gt) + X t1,t2∈EG(r,w) m g_t1, g_t2 + X t1∈EG(r,w) t2∈EGk_{r \{G(r,w ) }} m(g_t1, g_t2) 2         1 µ(E (Gr)) , 2 ϕk r G(r,w1), G(r,w2) =       X t1∈E(G_(r,w1)) t2∈E(G_(r,w2)) m(gt1, gt2)       1 µ(E (Gr)) .

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Direct and indirect preference information

Direct: The Decision Maker provides all the parameters necessary to apply

the considered method (in our case, the M¨obius coeﬃcients),

Indirect: The Decision Maker provides some preference information on alternatives or criteria in order to induce parameters compatible with these preferences.

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Direct and indirect preference information

Indirect: The Decision Maker provides some preference information on alternatives or criteria in order to induce parameters compatible with these preferences.

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A first example

En Ec

Projects SoSu WaSu ExEa FiFe

a 17 14 13 18 b 14 15 18 15 c 11 21 11 20 d 15 14 15 14 m(SoSu) 0.3793 m(WaSu) 0.1724 m(ExEa) 0.0507 m(FiFe) 0.1562 m(SoSu, WaSu) −0.1724 m(SoSu, ExEa) −0.0507 m(SoSu, FiFe) −0.1562 m(WaSu, ExEa) 0.6168 m(WaSu, FiFe) 0.0039 m(ExEa, FiFe) 0

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A first example

En Ec

Projects SoSu WaSu ExEa FiFe

a 17 14 13 18 b 14 15 18 15 c 11 21 11 20 d 15 14 15 14 m(SoSu) 0.3793 m(WaSu) 0.1724 m(ExEa) 0.0507 m(FiFe) 0.1562 m(SoSu, WaSu) −0.1724 m(SoSu, ExEa) −0.0507 m(SoSu, FiFe) −0.1562 m(WaSu, ExEa) 0.6168 m(WaSu, FiFe) 0.0039 m(ExEa, FiFe) 0

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Choquet integral at intermediate level and at root level

G1(En) G2(Ec) Choquet integral values SoSu WaSu ExEa FiFe Cµ(·) a 17 14 13 18 Cµ(a) = 14.67 b 14 15 18 15 Cµ(b) = 15.15 c 11 21 11 20 Cµ(c) = 14.16 d 15 14 15 14 Cµ(d) = 14.37

G1(En) G2(Ec) Choquet integral values SoSu WaSu ExEa FiFe Cµ(·)/µ(E (Gr)) a1 17 14 0 0 Cµ1(a) = 17 a2 0 0 13 18 Cµ2(a) = 16.77 b1 14 15 0 0 Cµ1(b) = 14.45 b2 0 0 18 15 Cµ2(b) = 15.73 c1 11 21 0 0 Cµ1(c) = 15.54 c2 0 0 11 20 Cµ2(c) = 17.79 d1 15 14 0 0 Cµ1(d) = 15 d2 0 0 15 14 Cµ2(d) = 14.24

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Paradox or not?

“a is preferred to b with respect to Economic criteria”’

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Paradox or not?

“a is preferred to b with respect to Environmental criteria”’

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Paradox or not?

“b is preferred to a at root level”’

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Paradox or not?

En Ec

SoSu WaSu ExEa FiFe ϕ2 r(G(r,w )) 0.7727 0.2272 0.2450 0.7549 ϕ2 0(G(r,w )) SoSu 0.1896 WaSu 0.3965 ExEa 0.3337 FiFe 0.080

“Soil Sustainability is more important than Water Sustainability if they are considered as elementary subcriteria of Environmental criterion” Because we take into account the interaction between Soil Sustainability and

Water Sustainabilityonly.

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Paradox or not?

En Ec

SoSu WaSu ExEa FiFe ϕ2 r(G(r,w )) 0.7727 0.2272 0.2450 0.7549 ϕ2 0(G(r,w )) SoSu 0.1896 WaSu 0.3965 ExEa 0.3337 FiFe 0.080

“Water Sustainability is more important than Soil Sustainability if they are

considered as elementary subcriteria of the root criterion G0.”

Because we take into accountalsothe interactions with the elementary subcriteria

of the Economic macro-criterion.

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Direct and indirect preference information

Indirect: The Decision Maker provides some preference information on alternatives or criteria in order to induce parameters compatible with these preferences:

◮ Ordinal Regression (OR), (E. Jacquet-Lagr`eze and Y. Siskos, 1982; J.L

Marichal and M. Roubens 2000; S. Angilella, S. Greco, F. Lamantia and B. Matarazzo 2004)

◮ Robust Ordinal Regression (ROR), (S. Greco, V. Mousseau and R. S lowi´nski

2008; S. Angilella, S. Greco and B. Matarazzo 2010)

◮ Stochastic Multiobjective Acceptability Analysis (SMAA), (R. Lahdelma, J. Hokkanen and P. Salminen 1998; S. Angilella, S. Corrente and S. Greco 2015)

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Direct and indirect preference information

◮ Stochastic Multiobjective Acceptability Analysis (SMAA), (R. Lahdelma, J.

Hokkanen and P. Salminen 1998; S. Angilella, S. Corrente and S. Greco 2015)

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Direct and indirect preference information

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Direct and indirect preference information

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Robust Ordinal Regression (ROR)

6

Considering the indirect technique, there could exist more than one model compatible with the preference information provided by the DM

⇓

Robust Ordinal Regression

ais necessarily preferred tob(a %N _{b) ⇔ a is at least as good as b for all}

compatible models,

ais possibly preferred tob(a %P _{b) ⇔ a is at least as good as b for at least}

one compatible model.

6_{S. Corrente, S. Greco, M. Kadzi´}_{nski, R. S lowi´}_{nski (2013)}

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Robust Ordinal Regression (ROR)

6

⇓

compatible models,

ais possibly preferred tob(a %P _{b) ⇔ a is at least as good as b for at least} one compatible model.

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Robust Ordinal Regression (ROR)

6

⇓

compatible models,

ais possibly preferred tob(a %P _{b) ⇔ a is at least as good as b for at least}

one compatible model.

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SMAA methods

Basic assumptions:

Imprecision or lack of data (weights and evaluations)

density functions fW(w ) and fχ(ξ) over the weight space W ⊆

R

n+

and the

evaluation space χ ⊆

R

m×n_,

Computations for each alternative of:

Rank Acceptability Index: br j = Z ξ∈χ fχ(ξ) Z w∈Wr j(ξ) fW(w ) dw dξ,

Central Weight Vector: wc j = 1 b_j1 Z ξ∈χ fχ(ξ) Z w∈W1 j(ξ) fW(w )w dw dξ, where Wr j(ξ) = {w ∈ W : rank(j, ξ, w ) = r } . Pairwise Winning Index:

p(ah, ak) = Z w∈W fW(w ) Z ξ∈χ:u(ξh,w )≥u(ξk,w ) fχ(ξ)dξ dw .

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SMAA methods

Basic assumptions:

R

n+

and the

R

m×n_,

Central Weight Vector: wc j = 1 b_j1 Z ξ∈χ fχ(ξ) Z w∈W1 j(ξ) fW(w )w dw dξ, where Wr j(ξ) = {w ∈ W : rank(j, ξ, w ) = r } . Pairwise Winning Index:

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SMAA methods

Basic assumptions:

R

n+

and the

R

m×n_,

Central Weight Vector: wc j = 1 b_j1 Z ξ∈χ fχ(ξ) Z w∈W1 j(ξ) fW(w )w dw dξ, where Wr j(ξ) = {w ∈ W : rank(j, ξ, w ) = r } .

Pairwise Winning Index:

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SMAA methods

Basic assumptions:

R

n+

and the

R

m×n_,

Central Weight Vector: wc j = 1 b_j1 Z ξ∈χ fχ(ξ) Z w∈W1 j(ξ) fW(w )w dw dξ, where Wr j(ξ) = {w ∈ W : rank(j, ξ, w ) = r } .

Pairwise Winning Index: p(ah, ak) = Z w∈W fW(w ) Z ξ∈χ:u(ξh,w )≥u(ξk,w ) fχ(ξ)dξ dw .

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DM’s preference information

Comparisons between importance ofcriteria:

◮ G

(r,w )is more important than G(r,k)iﬀ ϕkr(G(r,w )) > ϕkr(G(r,k)),

◮ G

(r,w )and G(r,k)are equally important iﬀ ϕkr(G(r,w )) = ϕkr(G(r,k)), Comparisons betweenalternatives:

◮ ais preferred to b on criterion G

r iﬀ Cµr(a) > Cµr(b),

◮ aand b are indiﬀerent on criterion G

r Cµr(a) = Cµr(b),

◮ ais preferred to b more than c is preferred to d on criterion G_riﬀ

Cµr(a)− Cµr(b) > Cµr(c)− Cµr(d).

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DM’s preference information

Comparisons between importance ofcriteria:

◮ G

(r,w )is more important than G(r,k)iﬀ ϕkr(G(r,w )) > ϕkr(G(r,k)),

◮ G

(r,w )and G(r,k)are equally important iﬀ ϕkr(G(r,w )) = ϕkr(G(r,k)),

Comparisons betweenalternatives:

◮ ais preferred to b on criterion G

r iﬀ Cµr(a) > Cµr(b),

◮ aand b are indiﬀerent on criterion G

r Cµr(a) = Cµr(b),

◮ ais preferred to b more than c is preferred to d on criterion G_riﬀ

Cµr(a)− Cµr(b) > Cµr(c)− Cµr(d).

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Set of constraints translating DM’s preferences

                                                

ϕkr(G(r,w )) > ϕkr(G(r,k)), if G(r,w ) is more important than G(r,k), ϕkr(G(r,w )) = ϕkr(G(r,k)), if G(r,w ) and G(r,k)are indiﬀerent,

Cµr(a) > Cµr(b), if a is preferred to b on Gr,

Cµr(a) = Cµr(b), if a and b are indiﬀerent on Gr,

Cµr(a)− Cµr(b) > Cµr(c)− Cµr(d), if

ais preferred to b more than c is preferred to d on Gr,

m({∅}) = 0, P t∈EL m({gt}) + P {t1,t2}⊆EL m({gt1, gt2}) = 1 m({gt}) ≥ 0, ∀t ∈ EL m({gt1}) + P t2∈T m({gt1, gt2}) ≥ 0, ∀t1∈ EL and ∀ T ⊆ EL \ {t1}

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Output Information

UsingRORwe get a necessary and a possible preference relation for each

node of the hierarchy,

◮ ais necessarily preferred to b on criterion G_r

1,

◮ _bis necessarily preferred to a on criterion G

r2,

Using theSMAA methodology, we get rank acceptability indices and pairwise

winning indices for each node of the hierarchy.

◮ On criterion G

r1, alternative a is in the ﬁrst position with a frequency of the

70%,

◮ On criterion G

r2, alternative a is preferred to alternative b with a frequency of

the 40%.

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Decision problem

7

Elementary subriterion Description

Masters Graduation Rate (MGR) The percentage of new entrants that successfully completed their master programs

Masters Graduating on Time (MGOT) The percentage of graduates that graduated within the time expected (normative time) for their masters programs Number of Research Publications (NRP) The number of research publications indexed in the Web of Science database, where at least

one author is aﬃliated to the university (relative to the number of students)

Citation Rate (CR) The average number of times that the university department’s research publications (over the period 2008-2011) get cited in other research, adjusted (normalized) at the global level to take into account diﬀerences in publication years and to allow for diﬀerences

Proportion of Top Cited Publications (PTCP) The proportion of the university’s research publications that, compared to other publications in the same ﬁeld and in the same year, belong to the top 10% most frequently cited

Number of Patents Awarded (NPA) The number of patents assigned to (inventors working in) the university (over the period 2001-2010) Number of Spin-Offs (NSO) The number of spin-offs (i.e. firms established on the basis of a formal knowledge transfer arrangement

between the institution and the firm) recently created by the institution (per 1,000 fte academic staff) Research and Knowledge Transfer Revenues (RKTR) Research revenues and knowledge transfer revenues from private sources (incl. not-for profit organizations),

excluding tuition fees. Measured in e1,000s using Purchasing Power Parities. Expressed per fte academic staﬀ. 7_{S. Angilella, S. Corrente, S. Greco and R. S lowi´}_{nski (2016)}

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Performances of the universities on the considered criteria

G(0)

TL(G(1)) R(G(2)) KT(G(3)) University Country MGR(g(1,1)) MGOT(g(1,2)) NRP(g(2,1)) CR(g(2,2)) PTCP(g(2,3)) NPA(g(3,1)) NS0(g(3,2)) RKTR(g(3,3))

Bocconi University (U25) Italy 5 4 2 5 5 1 1 5

Budapest U Tech & Economics (U35) Hungary 5 3 3 3 3 2 4 2

U Cordoba (U51) Spain 3 5 3 3 3 2 3 5

Tech U Denmark (U61) Denmark 4 4 5 5 5 5 5 5

Dublin Inst. Tech (U64) Ireland 2 5 2 5 5 2 4 2

U Limerick (U108) Ireland 4 5 2 5 4 4 3 5

Lomonosow Moscow State U (U117) Russia 5 5 5 2 2 2 5 5

Mondragon U (U129) Spain 4 5 2 5 5 1 5 5

Newcastle U (U136) United Kingdom 4 5 5 5 5 5 2 5

U Salamanca (U170) Spain 5 4 4 3 3 2 2 4

U Trieste (U196) Italy 5 2 5 4 4 3 3 3

WHU School of Management (U216) Germany 5 5 2 4 4 1 5 5

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Preference information provided by the DM

R is more important than KT that, in turn, is more important than TL With respect to TL, MGOT is more important than MGR

With respect to KT, RKTR is more important than NSO that, in turn, is more important than NPA

At a comprehensive level, PTCP is more important than RKTR that, in turn, is more important than MGT

TL and R are positively interacting R and KT are positively interacting TL and KT are positively interacting

The interaction between R and KT is greater than the interaction between TL and KT

The interaction between R and TL is greater than the interaction between TL and KT

With respect to R, NRP and PTCP are positively interacting CR and PTCP are negatively interacting

NRP and RKTR are positively interacting

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MCHP and NAROR results

(a) Comprehensive level (b) Teaching and

Learning (TL)

(c) Research (R)

(d) Knowledge Transfer (KT)

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MCHP and SMAA (Rank Acceptability Indices)

Comprehensive level

University BestbBest k,0 WorstbWorst k,0 high1 bhigh1 k,0 high2 bhigh2 k,0 high3 bhigh3 k,0 U25 18 (0.11%) 103 (0.01%) 52 (3.65%) 55 (3.45%) 58 (3.39%) U35 80 (0.01%) 132 (0.05%) 106 (4.68%) 108 (4.04%) 105 (3.87%) U51 47 (0.1%) 83 (0.01%) 65 (6.18%) 60 (6%) 70 (5.71%) U61 1 (93.29%) 2 (6.71%) U64 47 (0.02%) 133 (0.05%) 90 (2.82%) 87 (2.72%) 88 (2.66%) U108 14 (0.16%) 51 (0.01%) 28 (8.55%) 34 (6.19%) 33 (5.8%) U117 23 (0.15%) 90 (0.06%) 63 (3.22%) 67 (3.15%) 64 (2.82%) U129 6 (0.2%) 68 (0.11%) 23 (4.8%) 32 (4.24%) 37 (4.12%) U136 1 (6.71%) 12 (0.12%) 2 (60.18%) 3 (14.36%) 5 (8.88%) U170 58 (0.04%) 99 (0.04%) 80 (6.04%) 82 (6.12%) 83 (7.02%) U196 40 (0.01%) 77 (0.33%) 58 (5.88%) 57 (5.64%) 55 (5.6%) U216 12 (0.04%) 81 (0.01%) 43 (4.59%) 46 (4.04%) 42 (3.65%) Knowledge Transfer

University BestbBest k,1 WorstbWorst k,1 high1 bhigh1 k,1 high2 bhigh2 k,1 high3 bhigh3 k,1 B7 30 (0.18%) 42 (0.3%) 36 (21.38%) 35 (18.18%) 34 (15.78%) B8 15 (1.28%) 23 (0.14%) 21 (24.59%) 18 (20.95%) 20 (20.65%) U25 21 (0.09%) 40 (0.09%) 30 (25.86%) 29 (11.91%) 31 (10.55%) U51 13 (1.04%) 20 (0.38%) 16 (30.46%) 17 (22.81%) 18 (21.49%) U61 1 (100.00%) 1 (100.00%) U108 7 (4.92%) 14 (0.34%) 8 (20.09%) 10 (19.71%) 11 (17.92%) U117 4 (20.81%) 14 (0.32%) 7 (22.22%) 4 (20.81%) 5 (19.14%) U136 4 (5.2%) 20 (3.17%) 7 (11.48%) 10 (10.91%) 5 (9.05%) U170 25 (0.3%) 36 (0.5%) 31 (34.4%) 32 (29.06%) 30 (9.87%) U196 26 (0.01%) 39 (0.77%) 36 (17.57%) 33 (16.3%) 35 (15.43%) Research University Best bBest k,1 Worst bWorst k,1 high1 bhigh1 k,1 high2 bhigh2 k,1 high3 bhigh3 k,1 B4 5 (6.9%) 16 (1.27%) 8 (22.16%) 10 (13.81%) 11 (13.72%) B5 15 (1.28%) 23 (0.14%) 21 (24.59%) 18 (20.95%) 20 (20.65%) B6 1 (100.00%) 1 (100.00%) U108 6 (0.32%) 19 (4.23%) 13 (16.05%) 14 (15.25%) 12 (11.63%) U117 20 (1.71%) 26 (29.18%) 25 (48.64%) 26 (29.18%) 24 (14.85%) U170 12 (3.03%) 21 (1.34%) 18 (27.28%) 17 (21.32%) 19 (16.49%) U196 4 (43.99%) 10 (0.01%) 4 (43.99%) 5 (32.95%) 6 (10.96%) U216 14 (4.66%) 21 (1.71%) 16 (23.79%) 15 (18.16%) 19 (17.19%)

Teaching and Learning

University Best bBest k,1 Worst bWorst k,1 high1 bhigh1 k,1 high2 bhigh2 k,1 high3 bhigh3 k,1 B1 3 (61.91%) 5 (18.47%) 3 (61.91%) 4 (19.62%) 5 (18.47%) B2 1 (100.00%) 1 (100.00%) B3 2 (100.00%) 2 (100.00%) U35 5 (5.93%) 9 (19.37%) 6 (32.95%) 8 (28.3%) 9 (19.37%) U51 3 (38.09%) 5 (21.64%) 4 (40.27%) 3 (38.09%) 5 (21.64%) U61 4 (21.64%) 7 (4.04%) 6 (37.63%) 5 (36.69%) 4 (21.64%) U64 4 (18.47%) 10 (0.05%) 7 (23.9%) 6 (23.52%) 4 (18.47%) U196 8 (4.48%) 13 (28.09%) 13 (28.09%) 12 (22.16%) 10 (21.72%)

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MCHP and SMAA (Pairwise Winning Indices)

Comprehensive level p0(·, ·) U25 U35 U51 U61 U64 U108 U117 U129 U136 U170 U196 U216 U25 0 99.87 77.21 0 94.75 2.69 55.06 3.36 0 94.93 54.78 19.78 U35 0.13 0 0 0 28.34 0 0 0 0 1.13 0 0 U51 22.79 100 0 0 88.08 0 32.1 0.82 0 96.44 23.16 4.65 U61 100 100 100 0 100 100 100 100 93.29 100 100 100 U64 5.25 71.66 11.92 0 0 0 9.85 0 0 26.51 2.25 2.67 U108 97.31 100 100 0 100 0 96.16 47.92 0 100 99.67 86.38 U117 44.94 100 67.9 0 90.15 3.84 0 8.82 0 96.72 45.94 21.63 U129 96.64 100 99.18 0 100 52.08 91.18 0 0.24 100 90.22 99.95 U136 100 100 100 6.71 100 100 100 99.76 0 100 100 100 U170 5.07 98.87 3.56 0 73.49 0 3.28 0 0 0 1.33 0.3 U196 45.22 100 76.84 0 97.75 0.33 54.06 9.78 0 98.67 0 18.88 U216 80.22 100 95.35 0 97.33 13.62 78.37 0.05 0 99.7 81.12 0

p1(·, ·) B1 B2 B3 U35 U51 U61 U64 U196 B1 0 0 0 100 61.91 100 81.53 100 B2 100 0 100 100 100 100 100 100 B3 100 0 0 100 100 100 100 100 U35 0 0 0 0 0 8.97 36.94 100 U51 38.09 0 0 100 0 78.36 100 100 U61 0 0 0 91.03 21.64 0 63.26 100 U64 18.47 0 0 63.06 0 36.74 0 100 U196 0 0 0 0 0 0 0 0 Research p2(·, ·) B4 B5 B6 U108 U117 U170 U196 U216 B4 0 98.73 0 100 100 94.37 14.74 100 B5 1.27 0 0 7.13 100 0 0 29.51 B6 100 100 0 100 100 100 100 100 U108 0 92.87 0 0 100 80.23 0.37 100 U117 0 0 0 0 0 0 0 1.71 U170 5.63 100 0 19.77 100 0 0 43.99 U196 85.26 100 0 99.63 100 100 0 100 U216 0 70.49 0 0 98.29 56.01 0 0 Knowledge Transfer p3(·, ·) B7 B8 U25 U51 U61 U108 U117 U136 U170 U196 B7 0 0 5.83 0 0 0 0 0 6.99 36.92 B8 100 0 100 97.27 0 52.58 0 52.26 100 100 U25 94.17 0 0 0 0 0 0 0 89.29 89.29 U51 100 2.73 100 0 0 0 0 10.75 100 100 U61 100 100 100 100 0 100 100 100 100 100 U108 100 47.42 100 100 0 0 25.01 52.24 100 100 U117 100 100 100 100 0 74.99 0 68.77 100 100 U136 100 47.74 100 89.25 0 47.76 31.23 0 100 100 U170 93.01 0 10.71 0 0 0 0 0 0 89.29 U196 63.08 0 10.71 0 0 0 0 0 10.71 0

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Rankings of the universities by the barycenter of the M¨obius representations

Comprehensive level

Position in the complete ranking University Country 1st Tech U Denmark Denmark 2nd Newcastle U United Kingdom 31th U Limerick Ireland 32th Mondragon U Spain 41th WHU School of Management Germany 53th Bocconi University Italy 54th U Trieste Italy 58th Lomonosow Moscow State U Russia 67th U Cordoba Spain 78th U Salamanca Spain 91th Dublin Inst. Tech Ireland 105th Budapest U Tech & Economics Hungary

Position in the complete ranking University Country 1st Lomonosow Moscow State U Russia WHU School of Management Germany 2nd U Limerick Ireland Mondragon U Spain Newcastle U United Kingdom 3rd Bocconi University Italy

U Salamanca Spain 4th U Cordoba Spain 5th Tech U Denmark Denmark 6th Dublin Inst. Tech Ireland 8th Budapest U Tech & Economics Hungary 12th U Trieste Italy

Research

Position in the complete ranking University Country 1st Tech U Denmark Denmark Newcastle U United Kingdom 5th U Trieste Italy 9th Bocconi University Italy Dublin Inst. Tech Ireland

Mondragon U Spain 13th U Limerick Ireland 17th WHU School of Management Germany 18th U Salamanca Spain 19th Budapest U Tech & Economics Hungary

U Cordoba Spain 25th Lomonosow Moscow State U Russia

Knowledge Transfer

Position in the complete ranking University Country 1st Tech U Denmark Denmark 6th Lomonosow Moscow State U Russia 9th Mondragon U Spain WHU School of Management Germany 10th U Limerick Ireland 11th Newcastle U United Kingdom 16th U Cordoba Spain 28th Bocconi University Italy 31th U Salamanca Spain 35th U Trieste Italy 36th Budapest U Tech & Economics Hungary

Dublin Inst. Tech Ireland

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Conclusions

We presented the Multiple Criteria Hierarchy Process (MCHP)

The added value of the MCHP is that it permits the DM expressing the preference information related to any criterion of the hierarchy.

We presented the application of the MCHP to the Choquet integral preference model,

Application of the MCHP with the Choquet integral permits the handling of importance and interactions of criteria with respect to any subcriterion of the hierarchy,

We applied the Robust Ordinal Regression (ROR) and the Stochastic Multiobjective Acceptability Analysis in order to take into account the whole set of parameters compatible with some preference information provided by the DM.

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Conclusions

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Conclusions

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Conclusions

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Conclusions

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THANKS FOR YOUR ATTENTION

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