Roman SÅowiÅski PoznaÅ University of Technology, Poland

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Multiple-criteria ranking using an additive value function constructed via ordinal regresion : UTA method. Roman Słowiński Poznań University of Technology, Poland.  Roman Słowiński. g 2 ( x ). g 2max. A. g 2min. g 1 ( x ). g 1min. g 1max. Problem statement.

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### Multiple-criteria ranking using an additive value function constructed via ordinal regresion :UTA method

Roman Słowiński

Poznań University of Technology, Poland

 Roman Słowiński

g2(x)

g2max

A

g2min

g1(x)

g1min

g1max

Problem statement
• Consider a finite set A of actions (actions, solutions, objects) evaluated by m criteria from a consistent family F={g1,...,gm}
• Let I={1,…,m}
What is a consistent family of criteria ?
• A family of criteria F={g1,...,gn} is consistent if it is:
• Complete – if two actions have the same evaluations on all criteria, then they have to be indifferent, i.e.

if for any a,bA, there is gi(a)~gi(b), iI, then a~b

• Monotonic – if action a is preferred to action b (ab), and there is action c, such that gi(c)gi(a), iI, then cb
• Non-redundant – elimination of any criterion from the family F should violate at least one of the above properties

A

x

* *

x

*

*

*

x x

* *

x

x

x

*

*

*

x x x

x

x

x

* *

x

x x x

Problem statement
• Taking into account preferences of a Decision Maker (DM), rank all the actions of set A from the best to the worst

x

g2(x)

g2max

A

ideal

g2min

g1(x)

g1min

g1max

Dominance relation
• Action aA is non-dominated (Pareto-optimal) if and only if there is no other action bA such that gi(b)gi(a), iI, and on at least one criterion jI, gi(b)gi(a)
Criteria aggregation model = preference model
• Dominance relation is too poor – it leaves many actions non-comparable
• One can „enrich” the dominance relation, using preference information elicited from the Decision Maker
• Preference information permits to built a preference model that aggregates the vector evaluations of elements of A
Why traditional MCDM methods may confuse their users ?
• Traditional MCDM methods require a rich and difficult preference information:
• many intracriteria and intercriteria parameters: thresholds, weights, …
• complete set of pairwise comparisons of actions on each criterion
• complete set of pairwise comparisons of criteria
• They suppose the DM understands the logic of a particular aggregation model:
• meaning of weights: substitution ratios or relative strengths
• meaning of lotteries (ASSESS)
• meaning of indifference, preference and veto thresholds (ELECTRE)
• meaning of the ratio scale of the intensity of preference (AHP)
• meaning of „neutral” and „good” levels on particular criteria (MACBETH)
Towards „easy” preference information
• Traditional methods appear to be too demanding of cognitive effort of their users
• This is why we advocate for methods requiring „easy” preference information
• „Easy” means natural and even partial
• Psychologists confirm that DMs are more confident exercising their decisions than explaining them

A

Towards „easy” preference information
• The most natural is a holistic pairwise comparison of some actions relatively well known to the DM, i.e. reference actions

holistic

preference information

A

xy

zw

xw

yv

ut

zu

uz

DM

y

x

AR

t

z

v

w

u

Towards „easy” preference information
• The most natural is a holistic pairwise comparison of some actions relatively well known to the DM, i.e. reference actions

preference information

A

xy

zw

xw

yv

ut

zu

uz

Preference model compatible with preference information

DM

analyst

y

x

AR

t

z

v

w

u

Apply the preference model on A

Towards „easy” preference information
• Question: what is the consequence of using on the whole set Athis information transformed to a compatible preference model ?

What ranking will result ?

The holistic preference on a subset ARAis known first, and then a compatible criteria aggregation model (compatible preference model)is inferred from this information to be applied on set A

The criteria aggregation model (preference model) is first constructed and then applied on set A to get information about holistic preference

• The disaggregation-aggregation paradigam has been introduced to MCDS by Jacquet-Lagreze & Siskos(1982) in the UTA method: the inferred criteria aggregation model is the additive value function with piecewise-linear marginal value functions
• The disaggregation-aggregation paradigam is consistent with the „posterior rationality” principle by March(1988) and „learning from examples” used in AI and knowledge discovery
• Other aggregation models inferred in this way:
• Fishburn (1967) – trade-off weights
• Mousseau & Słowiński(1998) – outranking relation (ELECTRE TRI)
• Greco, Matarazzo & Słowiński(1999) – decision rules or trees (DRSA – Dominance-based Rough Set Approach)
Basic concepts and notation
• Gi – domain of criterion gi (Gi is finite or countably infinte)
• – evaluation space
• x,yG – profiles of actions in evaluation space
• – weak preference(outranking) relation onG:for each x,yG

xy„x is at least as good as y”

xy [xyand notyx] „x is preferred to y”

x~y [xyand yx] „x is indifferent to y”

Reminder of the UTA method (Jacquet-Lagreze & Siskos, 1982)
• For simplicity: Gi, iI, where I={1,…,m}
• For eachgi,Gi=[i, i] is the criterion evaluation scale, i  i , where i and i, are the worst and the best (finite) evaluations, resp.
• Thus, A is a finite subset of G and
• Additive value(or utility) function on G: for each xG

where ui are non-decreasing marginal value functions, ui : Gi, iI

x1

A

x2

AR

x3x4

x5

x6x7

Principle of the ordinal regression - UTA (Jacquet-Lagreze & Siskos, 1982)
• The preference information is given in form of a complete preorder on a subset of reference actionsARA,

AR={x1,x2,...,xn} – the reference actions are rearranged such that xkxk+1 , k=1,...,n-1

a1

a2

a3

Principle of the ordinal regression
• Example:

Let AR={a1, a2, a3}, G={Gain_1, Gain_2}

Evaluation of reference actions on criteria Gain_1, Gain_2:

Reference ranking:

a1

a1

a3

a2

a2

a3

Principle of the ordinal regression
• Let’s change the reference ranking:
• One linear piece per each marginal value function u1, u2 is not enough

u1=k1Gain_1, u2=k2Gain_2, U=u1+u2

For a1a3, k2>k1,

but for a3a2, k1>k2,

thus, marginal value functions cannot be linear

• The comprehensive preference information is given in form of a complete preorder on a subset of reference actionsARA,

AR={x1,x2,...,xn} – the reference actions are rearranged such that xkxk+1 , k=1,...,n-1

• The inferred value of each reference action xAR

where

+ and - are potential errors of over- and under-estimation of the right value, resp.

• The intervals [i, i] are divided into iequal sub-intervals with the end points (iI)
Principle of the ordinal regression
• In the UTA method, the marginal value of action xA is approximated by linear interpolation: for
Principle of the UTA method
• Ordinal regression principle

for xkxk+1 , k=1,...,n-1

• Monotonicity of preferences
• Normalization

k=1,...,n-1

(C)

Principle of the UTA method
• The marginal value functions (breakpoint variables) are estimated by solving the LP problem

where  is a small positive constant

polyhedron of constraints (C)

EUTA EUTA*+

EUTA= EUTA*

Principle of the UTA method
• If EUTA*=0, then the polyhedron of feasible solutions for ui(xi) is not empty and there exists at least one value functionU[g(x)] compatible with the complete preorder on AR
• If EUTA*>0, then there is no value functionU[g(x)] compatible with the complete preorder on AR– three possible moves:
• increasing the number of linear pieces i for ui(xi)
• revision of the complete preorder on AR
• post-optimal search for the best function with respect to Kendall’s  in the area EUTA EUTA*+ 

Jacquet-Lagreze

& Siskos (1982)

Współczynnik Kendalla
• Do wyznaczania odległości między preporządkami stosuje się miarę Kendalla
• Przyjmijmy, że mamy dwie macierze kwadratowe R i R* o rozmiarze m m, gdzie m = |AR|, czyli m jest liczbą wariantów referencyjnych
• macierz R jest związana z porządkiem referencyjnym podanym przez decydenta,
• macierz R* jest związana z porządkiem dokonanym przez funkcję użyteczności wyznaczoną z zadania PL (zadania regresji porządkowej)
• Każdy element macierzy R, czyli rij (i, j=1,..,m), może przyjmować wartości:
• To samo dotyczy elementów macierzy R*
• Tak więc w każdej z tych macierzy kodujemy pozycję (w porządku) wariantu a względem wariantu b
Współczynnik Kendalla
• Następnie oblicza się współczynnik Kendalla:

gdzie dk(R,R*) jest odległością Kendallamiędzy macierzami R i R*:

• Stąd -1, 1
• Jeżeli  = -1, to oznacza to, że porządki zakodowane w macierzach R i R*są zupełnie odwrotne, np. macierz R koduje porządek a  b  c  d, a macierz R* porządek d  c  b  a
• Jeżeli  = 1, to zachodzi całkowita zgodność porządków z obydwu macierzy. W tej sytuacji błąd estymacji funkcji użyteczności F*=0
• W praktyce funkcję użyteczności akceptuje się, gdy  0.75

1

2

Example of UTA+
• Ranking of 6 means of transportation

preference information

xy

zw

yv

ut

zu

uz

All instances ofpreference model compatible with preference information

A

DM

analyst

y

x

AR

t

z

v

w

u

Apply all compatible instances on A

One should use all compatible preference models on set A
• Question: what is the consequence of using all compatible preference models on set A ?

What rankings will result ?

z

u

Includesnecessary ranking

and does not include

the complement of

necessary ranking

x

w

xy

zw

yv

ut

zu

uz

preference information

y

t

v

possible ranking

necessary ranking

Two rankings result: necessary and possible

z

u

Includes necessary ranking

and does not include

the complement of

necessary ranking

w

x

xy

zw

yv

ut

zu

uz

xw

y

t

v

necessary rankingenriched

possible rankingimpoverished

Two rankings result: necessary and possible – effect of additional preference information

„trial-and-error” interactions

• In the absence of any preference information:
• necessary ranking boils down to weak dominance relation
• possible ranking is a complete relation
• For complete pairwise comparisons (complete preorder in A):
• necessary ranking = possible ranking
The UTAGMS method(Greco, Mousseau & Słowiński 2004)
• The preference information is a partial preorder on a subset of reference actions ARA
• A value function is called compatible if it is able to restore the partial preorder reference actions from AR
• Each compatible value function induces a ranking on set A
• In result, one obtains two rankings on set A, such that for any pair of actions (x,y)A:
• x N y:x is ranked at least as good as y iff U(x)U(y) for all value functions compatible withthe preference information (necessary weak preference relation N - a partial preorder on A)
• x P y:x is ranked at least as good as y iff U(x)U(y) for at least one value function compatible with the preference information (possibleweak preference relation P - a strongly complete and negatively transitive binary relation on A)

ui(xi)

gi

0

i

i

yi

vi

wi

zi

The UTAGMS method(Greco, Mousseau & Słowiński 2004)
• The marginal value function ui(xi)

y,v,w,zAR

Characteristic points of marginal value functions are fixed on actual evaluations of actions from set A

ui(xi)

?

?

?

?

?

gi

0

i

i

yi

vi

wi

zi

The UTAGMS method(Greco, Mousseau & Słowiński 2004)
• The marginal value function ui(xi)

y,v,w,zAR

Marginal values in characteristic points are unknown

ui(xi)

gi

0

i

i

yi

vi

wi

zi

The UTAGMS method(Greco, Mousseau & Słowiński 2004)
• The marginal value function ui(xi)

y,v,w,zAR

In fact, they are intervals, because all compatible value functions are considered

ui(xi)

gi

0

i

i

yi

vi

wi

zi

The UTAGMS method(Greco, Mousseau & Słowiński 2004)
• The marginal value function ui(xi)

y,v,w,zAR

The area of all compatible marginal value functions

ui(xi)

gi

0

i

i

yi

vi

wi

zi

The UTAGMS method(Greco, Mousseau & Słowiński 2004)
• The marginal value function ui(xi)

y,v,w,zAR

In the area the marginal compatible value functions must be monotone

ui(xi)

gi

0

i

i

yi

vi

wi

zi

The UTAGMS method(Greco, Mousseau & Słowiński 2004)
• The marginal value function ui(xi)
• This means that the ordinal regression should not seek for m piecewise-linear marginal value functions, but for any compatible additive value function
The UTAGMS method
• Let ibe a permutation on the set of actions AR{x,y} that reorders them according to increasing evaluation on criterion gi:

where

• if AR{x,y}=, then =n+2
• if AR{x,y}={x} or AR{x,y}={y}, then =n+1
• if AR{x,y}={x,y}, then =n
• The characteristic points of ui(xi), iI, are then fixen in:

a

a

b

b

 a,bAR

a

a

b

b

E(x,y)

a

a

a

The UTAGMS method
• For any pair of actions (x,y)A, and for available preference information concerning AR, preference of x over y is determined by compatible value functionsU verifying set E(x,y) of constraints:

where  is a small positive constant

• For all (x,y)A, E(x,y) = E(y,x)
The UTAGMS method
• Nmeans necessary(strong) preference relation
• Given a pair of actions x,yA

xNyd(x,y)  0

where

• d(x,y)  0 means that for allcompatible value functions x is at least as good as y
• For x,yAR :

xy xNy

The UTAGMS method
• Pmeans possible(weak) preference relation
• Given a pair of actions x,yA

xPyD(x,y)  0

where

• D(x,y)  0 means that for at least onecompatible value function x is at least as good as y
• For x,yAR :

xy notyPx

The UTAGMS method
• Some properties:
• xNyxPy
• Nis a partial preorder (i.e. Nis reflexive and transitive)
• Pis strongly complete (i.e. for all x,yA, xPyor yPx)and negatively transitive (i.e. for all x,y,zA, notxPyand not yPznot xPz), (in general, Pis not transitive)
• d(x,y) = Min{U(x)–U(y)} = –Max{–[U(x)–U(y)]} = = –Max{U(y)–U(x)} = –D(y,x)
Proof of transitivity of N
• d(x,y)>0 means: Min{U(x)-U(y)}>0
• This is equivalent to the fact: for all value functions compatible with the reference preorder, U(x)>U(y)
• The set of all compatible value functions is the same for calculation of d(x,y) for any pair x,yA
• Suppose, the transitivity of N is not true, i.e. for x,y,zA

Min{U(x)-U(y)}>0, Min{U(y)-U(z)}>0, but Min{U(x)-U(z)}<0

• This means that U(x)-U(z) has achieved a minimum value d(x,z)<0 for a value function denoted by U*, such that U*(x)<U*(z),

while U*(x)>U*(y) and U*(y)>U*(z)

• In other words, U*(x)>U*(y)>U*(z)>U*(x)
• This is a contradiction, so N is transitive
The UTAGMS method
• Elaboration of the final ranking:
• for the necessary preference relation being a partial preorder

(N is supported by all compatible value functions)

preference: xNy if xNy and not yNx

indifference: xNy if xNy and yNx

incomparability: x ? y if notxNy and not yNx

• for the possible preference relation being complete

(P is supported by at least one compatible value function)

preference: xPy if xPy and not yPx

indifference: xPy if xPy and yPx

• N.B. It is impossible to infer one ranking from another because strong and weak outranking relations arenot dual

N

N

N

N

N

N

N

N

N

The UTAGMS method – final ranking (partial preorder)
• Final ranking corresponding to necessary (strong) preference :

P

P

P

P

P

P

P

P

P

P

The UTAGMS method – final ranking (complete preorder)
• Final ranking corresponding to possible (weak) preference :

1

2

Example of UTAGMS
• Ranking of 6 means of transportation

„economical”

„hurry”

Nested ranking with different credibility levels
• Reference rankings in growing sets:

with credibility levels ordered decreasingly

• The reference ranking of alternatives from ARidoes not change in ARi+1, i=1,…,p-1
• Each time we pass from ARi to ARi+1, we add to (E) new constraints concerning alternatives from {ARi+1\ ARi}
• If d(x,y)<0 in iteration iturns tod(x,y)>0 in iteration i+1, then we assign to xNythe credibility level corresponding to ranking i+1
• In this way we get a set of nested partial preorders
• In fact, we get a fuzzy partial preorderN~ with respect to credibility:

for all x,y,zA, Min{Cr(xN~y), Cr(yN~z)} Cr(xN~z), thus N~ is min-transitive and reflexive

GRIP – Generalized Regression with Intensities of Preference(Figueira, Greco, Słowiński 2006)
• GRIP extends the UTAGMS method by adopting all features of UTAGMS and by taking into account additional preference information :
• comprehensive comparisons of intensities of preference between some pairs of reference actions, e.g. „x is preferred to y at least as much as w is preferred to z”
• partial comparisons of intensities of preference between some pairs of reference actions on particular criteria,e.g. „x is preferred to y at least as much as w is preferred to z,on criterion giF”
• DM is supposed to provide the following preference information :
• a partial preorder on AR, such that x,yAR

xy  „x is at least as good as y” =  not-1,  =  -1

• a partial preorder * on ARAR, such that x,y,w,zAR

(x,y) * (w,z) „x is preferred to y at least as much as w is preferred to z” * = *  not*-1, * = *  *-1

• a partial preorder i* on ARAR, i=1,...,m, such that x,y,w,zAR

(x,y) i* (w,z) „x is preferred to y at least as much as w is preferred to z,on criterion giF”, i* = i*  noti*-1, i* = i*  i*-1

GRIP – new contraints of the ordinal regression LP problem
• A value function U : [0, 1] is called compatible if it satisfes the constraints corresponding to DM’s preference information:
• U(x)  U(y) iff xy
• U(x) > U(y) iff x y
• U(x) = U(y) iff x y
• U(x) – U(y)  U(w) – U(z) iff (x,y) * (w,z)
• U(x) – U(y) > U(w) – U(z) iff (x,y) * (w,z)
• U(x) – U(y) = U(w) – U(z) iff (x,y) * (w,z)
• ui(x)  ui(y) iff xiy, iI
• ui(x) – ui(y)  ui(w) – ui(z) iff (x,y) i* (w,z), iI
• ui(x) – ui(y) > ui(w) – ui(z) iff (x,y) i* (w,z), iI
• ui(x) – ui(y) = ui(w) – ui(z) iff (x,y) i* (w,z), iI
GRIP – new contraints of the ordinal regression LP problem
• Moreover, the following normalization constraints should also be taken into account:
• ui(i)=0, iI
GRIP – new contraints of the ordinal regression LP problem
• If constraints a) – l) are consistent, then we gettwo weak preference relationsN and P , and two binary relations comparing intensity of preference*Nand *P:
• for all x,yA, a necessary weak preference relation, xNy:

U(x) U(y) for all compatible value functions

• for all x,yA, a possible weak preference relation, xPy:

U(x) U(y) for at least one compatible value function

• for all x,y,w,z A, a necessary relation of preference intensity,(x,y) *N(w,z): [U(x) – U(y)] – [U(w) – U(z)]  0 for all compatible value functions
• for all x,y,w,z A, a possible relation of preference intensity, (x,y) *P(w,z):[U(x) – U(y)] – [U(w) – U(z)]  0 for at least one compatible value function

Theorem:If constaraints a) – l) are satisfied, then the properties hold:

• For all x,yA, xNyxPy
• For all x,yAR, xyxNy
• N is a partial preorder (i.e. the relation is transitive and reflexive) andP is strongly complete and negatively transitive
• For all x,y,zA, [xNyand yPz] xPz
• For all x,y,zA, [xPyand yNz] xPz
• For all x,y,w,zA, (x,y) *N (w,z)(x,y) *P(w,z)
• For all x,y,w,zA, (x,y) * (w,z)(x,y) *N(w,z)
• *N is a partial preorder and *P is strongly complete and negatively transitive
• For all x,y,w,z,r,sA, [(x,y) *N (w,z) and (w,z) *P(r,s)](x',y) *P (r,s)
• For all x,y,w,z,r,sA, [(x,y) *P (w,z) and (w,z) *N(r,s)](x,y) *P (r,s)
• For all x,x’,y,w,zA, [x’N x and (x,y) *N(w,z)](x’,y) *N (w,z)
• For all x,x’,y,w,zA, [x’N x and (x,y) *P(w,z)](x’,y) *P (w,z)
• For all x,x’,y,w,zA, [x’P x and (x,y) *N(w,z)](x’,y) *P (w,z)
• For all x,y,y’,w,zA, [yN y’ and (x,y) *N(w,z)](x,y’) *N (w,z)
• For all x,y,y’,w,zA, [yN y’ and (x,y) *P(w,z)](x,y’) *P (w,z)
• For all x,y,y’,w,zA, [yP y’ and (x,y) *N(w,z)](x,y’) *P (w,z)
• For all x,y,w,w’,zA, [wN w’ and (x,y) *N(w,z)](x,y) *N (w’,z)
• For all x,y,w,w’,zA, [wN w’ and (x,y) *P(w,z)](x,y) *P (w’,z)
• For all x,y,w,w’,zA, [wP w’ and (x,y) *N(w,z)](x,y) *P (w’,z)
• For all x,y,w,z,z’A, [z’N z and (x,y) *N(w,z)](x,y) *N (w,z’)
• For all x,y,w,z,z’A, [z’N z and (x,y) *P(w,z)](x,y) *P (w,z’)
• For all x,y,w,z,z’A, [z’P z and (x,y) *N(w,z)](x,y) *P (w,z’)
• For all x,x’,yA, (x’,y) *N (x,y)x’ Nx
• For all x,x’,yA, (x’,y) *P (x,y)x’ Px
• For all x,y,y’A, (x,y) *N (x,y’)y’ Ny
• For all x,y,y’A, (x,y) *P (x,y’)y’ Py 
GRIP – the linear programming problem
• In order to verify the truth or falsity of necessary and possible weak preference relations N, Pand *N, *P, one can use LP
• LP does not permit strict inequalities, such as b), e), i)They must be rewritten as:

b’)U(x)  U(y) + 

e’)U(x) – U(y)  U(w) – U(z) + 

i’)ui(x) – ui(y)  ui(w) – ui(z) + 

where >0 (small value)

• In UTA and in UTAGMS the result is dependent on the value of 
• We want to make the result of GRIP independent of 

Max

GRIP – the linear programming problem
• The following result will be useful (see e.g. Marichal & Roubens 2000):
• Proposition: x is a solution of the linear system,

if there exists >0, such that

In particular, a solution exists, iff the following LP

has optimal solution (x*,*), where *>0. Then, x* is a solution of #

#

GRIP – the linear programming problem
• According to the Proposition, if constraints b),e),i) are considered, in order to verify the truth or falsity of N and P , one should :

Max  subject to constraints a)–l), with b),e),i) written as b’),e’),i’)

• If maximal *>0, the set of compatible value functions is not empty
• Then, to verify the truth or falsity of xPy,for anyx,yA, one should :

Max  subject to constraints a)–l), with b),e),i) written as b’),e’),i’)

and U(x)  U(y)

• Maximal *>0  xPy
• This means that there exists at least one compatible value function satisfying the hypothesis U(x)  U(y)
GRIP – the linear programming problem
• In order to verify the truth or falsity of xNy, rather than to check directly that for each compatible value function U(x)  U(y), we make sure that among the compatible value functions there is no one such that U(x) < U(y) :

Max  subject to constraints a)–l), with b),e),i) written as b’),e’),i’)

and U(y)  U(x) + 

• Maximal *≤0  xNy
GRIP – the linear programming problem
• Analogously, if constraints b),e),i) are considered, in order to verify the truth or falsity of (x,y)*P(w,z)for anyx,y,w,zA, one should :

Max  subject to constraints a)–l), with b),e),i) written as b’),e’),i’)

and U(x)U(y)  U(w)U(z)

• Maximal *>0  (x,y)*P(w,z)
• In order to verify the truth or falsity of (x,y)*N(w,z)for anyx,y,w,zA, one should :

Max  subject to constraints a)–l), with b),e),i) written as b’),e’),i’)

and U(w)U(z)  U(x)U(y) + 

• Maximal *≤0  (x,y)*N(w,z)
• The value of * is not meaningful – the result does not depend on it!
Comparison of GRIP and MACBETH(Bana e Costa & Vansnick 1994)
• GRIP
• Ordinal comprehensive preference inf.
• on pairwise comparison of some
• reference actions: xy, x,yAR
• Absolute qualitative judgement of
• intensity of preference for some pairs
• of reference actions – partial and/or
• comprehensive (e.g. v_weak, weak,
• moderate,..., extreme intensity of preference for (x,y))
• OR
• Comparison of intensities of
• preference for some pairs of reference
• actions – partial and/or comprehens.:
• (x,y)i(w,z) and/or (x,y)(w,z)
• MACBETH
• Ordinal preference inf. w.r.t. each
• criterion for all not equally attractive
• pairs of actions: xiy or yix, x,yA
• Definition of „neutral” and „good” level on original scales of criteria
• Absolute qualitative judgement of
• differences of attractiveness for all not
• equally attractive pairs of actions w.r.t.
• each criterion, including „good” and
• „neutral” points (e.g. v_weak, weak,
• moderate,..., extreme int.pref. for (x,y))
• Ordinal preference inf. for all not equally
• attractive criteria: gigj or gjgi
• Absolute qualitative judgement of
• differences of attractiveness for all not
• equally attractive pairs of criteria (e.g.
• v_weak, weak, moderate,..., extreme intensity of preference for (gi,gj))

Preference information

Comparison of GRIP and MACBETHcont.
• GRIP
• Uses LP to identify a set of
• functions with interval scales,
• compatible with preference info.
• Builds necessary and possible
• weak preference relations on set A:
• N (partial preorder)
• P (strongly complete)
• Builds necessary and possible
• weak preference relations on set AA:
• *N (partial preorder)
• *P (strongly complete)
• MACBETH
• Uses LP to build a single interval scale
• for each criterion, compatible with
• preference info., and computes
• a numerical marginal value for each
• action on each criterion
• Computes a weight for each criterion
• Builds a weighted sum model on marginal
• values which is additive piecewise linear
• or discrete
• Uses the model to set up a complete
• preorder on set A

Preference model and final results

Comparison of GRIP and MACBETH(Bana e Costa & Vansnick 1994)
• Summary of crucial differences in the methodology:
• GRIP is using comprehensive and partial preference information on some pairs of actions
• MACBETH requires partial preference information on all pairs of actions
• Information about partial intensity of preference is of the same nature in GRIP and MACBETH (equivalence classes of relation i* correspond to qualitative judgements of MACBETH), but in GRIP it may not be complete
• GRIP represents „disaggregation-aggregation” approach
• MACBETH uses „aggregation” approach – needs weights to aggregate scales on particular criteria
• GRIP works with all compatible value functions, while MACBETH builds a single interval scale for each criterion, even if many such scales would be compatible with preference information
Other features of GRIP
• GRIP can be used interactively:
• In the absence of any preference information, N,*N boil down to weak dominance relation
• Each pairwise comparison  or each comparison of intensities of preference *, contributes to enrichN or*N
• In the absence of any preference information, P,*P is a complete relation
• Each pairwise comparison  or each comparison of intensities of preference *, contributes to impoverishP or*P
• For complete pairwise comparisons and comparisons of intensities:N = P and *N = *P
• GRIP permits to make preference intensity dependent on the part of criterion scale in which a difference of performances takes place, e.g. (17.000; 19.000) price(27.000; 30.000)
GRIP – illustrative example
• Car ranking problem
• Criteria: Intensity of preference:
GRIP – illustrative example
• Preference information
• Monotonicity must be respected for each criterion, e.g.

if Speed(x) Speed(y), then value[Speed(x)] value[Speed(y)]

• In the ordinal regression LP problem, monotonicity is expressed by g)
• In Tables 2-6, we skip preference labels  and that result from simple monotonicity, i.e. gi(x)=gi(y) or gi(x)gi(y), respectively
GRIP – illustrative example
• Partial information about preference intensity serves to define constraints h) and j) representing partial preorder i*

(F)

Skoda

Opel

Ford

Citroen

Seat

VW

GRIP – illustrative example
• Comprehensive information about preference intensity serves to define constraints a) and c) representing partial preorder *
GRIP – illustrative example
• Calculation of D(x,y)=Max{U(x)–U(y)}
• Since all values are  0, for all pairs (x,y) of cars there exist at least one compatible value function for which x is at least as good as y
• Therefore, possible weak preference relationP  AA, so P brings no interesting information because all weak preferences are possible
GRIP – illustrative example
• Calculation of d(x,y)=Min{U(x)–U(y)}
• As Min{U(x)–U(y)}=–Max{U(y)–U(y)}, then d(x,y)=–D(y,x)
• All values  0 correspond to pairs (x,y) of cars for which all compatible value functions are in favor of x over y
GRIP – illustrative example
• Graph of necessary weak preference relation N
Conclusions
• In GRIP,preference information is givenby the DM in terms of:
• partial preorder in the set of reference actions
• partial and comprehensive comparisons of intensities of preference between some pairs of reference actions,
• The preference information is used within regression approach to build a complete set of compatible additive value functions
• Considering all compatible value functions permits to find as result:
• necessary w.pref. relation in A and in AA (partial preorder) N,*N
• possible w.pref. relation in A and in AA (strongly complete) P,*P
• Possible extensions:
• preference information with gradual credibility
• group decision