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Roman SÅowiÅski 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

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,bA, there is gi(a)~gi(b), iI, then a~b

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

x

* *

x

*

*

*

x x

* *

x

x

x

*

*

*

x x x

x

x

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* *

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

g2max

nadir

A

ideal

g2min

g1(x)

g1min

g1max

Dominance relation- Action aA is non-dominated (Pareto-optimal) if and only if there is no other action bA such that gi(b)gi(a), iI, and on at least one criterion jI, 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

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

xy

zw

xw

yv

ut

zu

uz

DM

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x

AR

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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

A

xy

zw

xw

yv

ut

zu

uz

Preference model compatible with preference information

DM

analyst

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x

AR

t

z

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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 ?

Aggregation paradigms

- Disaggregation-aggregation (or regression) paradigm:

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

- Traditional aggregation paradigm:

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

Aggregation paradigms

- 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,yG – profiles of actions in evaluation space
- – weak preference(outranking) relation onG:for each x,yG

xy„x is at least as good as y”

xy [xyand notyx] „x is preferred to y”

x~y [xyand yx] „x is indifferent to y”

Reminder of the UTA method (Jacquet-Lagreze & Siskos, 1982)

- For simplicity: Gi, iI, 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 xG

where ui are non-decreasing marginal value functions, ui : Gi, iI

A

x2

AR

x3x4

x5

x6x7

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 actionsARA,

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

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

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=k1Gain_1, u2=k2Gain_2, U=u1+u2

For a1a3, k2>k1,

but for a3a2, k1>k2,

thus, marginal value functions cannot be linear

Principle of the UTA method (Jacquet-Lagreze & Siskos, 1982)

- The comprehensive preference information is given in form of a complete preorder on a subset of reference actionsARA,

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

- The inferred value of each reference action xAR

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 (iI)

Principle of the ordinal regression

- In the UTA method, the marginal value of action xA is approximated by linear interpolation: for

Principle of the UTA method

- Ordinal regression principle

for xkxk+1 , k=1,...,n-1

- Monotonicity of preferences
- Normalization

(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

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

xy

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uz

All instances ofpreference model compatible with preference information

A

DM

analyst

y

x

AR

t

z

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w

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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 ?

u

Includesnecessary ranking

and does not include

the complement of

necessary ranking

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w

xy

zw

yv

ut

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preference information

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possible ranking

necessary ranking

Two rankings result: necessary and possibleu

Includes necessary ranking

and does not include

the complement of

necessary ranking

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xw

additional preference information

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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 ARA
- 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)

gi

0

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

y,v,w,zAR

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

?

?

?

?

?

gi

0

i

i

yi

vi

wi

zi

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

y,v,w,zAR

Marginal values in characteristic points are unknown

gi

0

i

i

yi

vi

wi

zi

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

y,v,w,zAR

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

gi

0

i

i

yi

vi

wi

zi

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

y,v,w,zAR

The area of all compatible marginal value functions

gi

0

i

i

yi

vi

wi

zi

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

y,v,w,zAR

In the area the marginal compatible value functions must be monotone

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), iI, are then fixen in:

a

b

b

a,bAR

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,yA

xNyd(x,y) 0

where

- d(x,y) 0 means that for allcompatible value functions x is at least as good as y
- For x,yAR :

xy xNy

The UTAGMS method

- Pmeans possible(weak) preference relation
- Given a pair of actions x,yA

xPyD(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,yAR :

xy notyPx

The UTAGMS method

- Some properties:
- xNyxPy
- Nis a partial preorder (i.e. Nis reflexive and transitive)
- Pis strongly complete (i.e. for all x,yA, xPyor yPx)and negatively transitive (i.e. for all x,y,zA, notxPyand not yPznot xPz), (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,yA
- Suppose, the transitivity of N is not true, i.e. for x,y,zA

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: xNy if xNy and not yNx

indifference: xNy if xNy and yNx

incomparability: x ? y if notxNy and not yNx

- for the possible preference relation being complete

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

preference: xPy if xPy and not yPx

indifference: xPy if xPy and yPx

- 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

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

P

P

P

P

P

P

P

P

P

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

D(x,y)

„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 xNythe 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 preorderN~ with respect to credibility:

for all x,y,zA, Min{Cr(xN~y), Cr(yN~z)} Cr(xN~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 giF”

GRIP – Generalized Regression with Intensities of Preference

- DM is supposed to provide the following preference information :
- a partial preorder on AR, such that x,yAR

xy „x is at least as good as y” = not-1, = -1

- a partial preorder * on ARAR, such that x,y,w,zAR

(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 ARAR, i=1,...,m, such that x,y,w,zAR

(x,y) i* (w,z) „x is preferred to y at least as much as w is preferred to z,on criterion giF”, i* = i* noti*-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 xy
- 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 xiy, iI
- ui(x) – ui(y) ui(w) – ui(z) iff (x,y) i* (w,z), iI
- ui(x) – ui(y) > ui(w) – ui(z) iff (x,y) i* (w,z), iI
- ui(x) – ui(y) = ui(w) – ui(z) iff (x,y) i* (w,z), iI

GRIP – new contraints of the ordinal regression LP problem

- Moreover, the following normalization constraints should also be taken into account:
- ui(i)=0, iI

GRIP – new contraints of the ordinal regression LP problem

- If constraints a) – l) are consistent, then we gettwo weak preference relationsN and P , and two binary relations comparing intensity of preference*Nand *P:
- for all x,yA, a necessary weak preference relation, xNy:

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

- for all x,yA, a possible weak preference relation, xPy:

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

GRIP – fundamental properties of N, P, *N, *P

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

- For all x,yA, xNyxPy
- For all x,yAR, xyxNy
- N is a partial preorder (i.e. the relation is transitive and reflexive) andP is strongly complete and negatively transitive
- For all x,y,zA, [xNyand yPz] xPz
- For all x,y,zA, [xPyand yNz] xPz
- For all x,y,w,zA, (x,y) *N (w,z)(x,y) *P(w,z)
- For all x,y,w,zA, (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,sA, [(x,y) *N (w,z) and (w,z) *P(r,s)](x',y) *P (r,s)
- For all x,y,w,z,r,sA, [(x,y) *P (w,z) and (w,z) *N(r,s)](x,y) *P (r,s)
- For all x,x’,y,w,zA, [x’N x and (x,y) *N(w,z)](x’,y) *N (w,z)
- For all x,x’,y,w,zA, [x’N x and (x,y) *P(w,z)](x’,y) *P (w,z)
- For all x,x’,y,w,zA, [x’P x and (x,y) *N(w,z)](x’,y) *P (w,z)

GRIP – fundamental properties of N, P, *N, *P

- For all x,y,y’,w,zA, [yN y’ and (x,y) *N(w,z)](x,y’) *N (w,z)
- For all x,y,y’,w,zA, [yN y’ and (x,y) *P(w,z)](x,y’) *P (w,z)
- For all x,y,y’,w,zA, [yP y’ and (x,y) *N(w,z)](x,y’) *P (w,z)
- For all x,y,w,w’,zA, [wN w’ and (x,y) *N(w,z)](x,y) *N (w’,z)
- For all x,y,w,w’,zA, [wN w’ and (x,y) *P(w,z)](x,y) *P (w’,z)
- For all x,y,w,w’,zA, [wP 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’,yA, (x’,y) *N (x,y)x’ Nx
- For all x,x’,yA, (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

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 xPy,for anyx,yA, one should :

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

and U(x) U(y)

- Maximal *>0 xPy
- 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 xNy, 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 xNy

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,zA, 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,zA, 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: xy, x,yAR
- 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: xiy or yix, x,yA
- 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: gigj or gjgi
- 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
- comprehensive additive value
- 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 AA:
- *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 enrichN 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 impoverishP 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:

Price Speed Space Fuel_cons. Acceleration

Skoda

Opel

Ford

Citroen

Seat

VW

GRIP – illustrative example- Performance matrix

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*

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 relationP AA, 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 AA (partial preorder) N,*N
- possible w.pref. relation in A and in AA (strongly complete) P,*P
- Possible extensions:
- preference information with gradual credibility
- group decision

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