Roman Słowiński Poznań University of Technology, Poland

Multiple-criteria ranking

using an additive value function

constructed via ordinal regresion :

UTA method

Roman SłowińskiPoznań University of Technology, Poland

Roman Słowiński

2

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}

g2max

g1(x)

g2(x)

g2min

g1min g1max

A

3

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

4

Problem statement

Taking into account preferences of a Decision Maker (DM),

rank all the actions of set A from the best to the worst

A

**

**

x

xx

xx

x

*

*

x

x * *

x x * *

x x x * * x

x x x

x

5

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)

g2max

g1(x)

g2(x)

g2min

g1min g1max

A

ideal

nadir

6

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

7

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)

…

8

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

9


The most natural is a holistic pairwise comparison of some actions

relatively well known to the DM, i.e. reference actions

A

10


The most natural is a holistic pairwise comparison of some actions

relatively well known to the DM, i.e. reference actions

A

ARx

t z

w

v

y

u

DM

x y

z w

x w

y v

u t

z u

u z

holisticpreference information

11


Question: what is the consequence of using on the whole set A

this information transformed to a compatible preference model ?

A

ARx

t z

w

v

y

u

DM

x y

z w

x w

y v

u t

z u

u z

preference information

analyst Preference model compatible

with preference information

Apply the preference model on AWhat

ranking will

result ?

12

Aggregation paradigms

Disaggregation-aggregation (or regression) paradigm:

The holistic preference on a subset ARA is 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

13

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)

14

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 on G: for each x,yG

xy „x is at least as good as y”

xy [xy and not yx] „x is preferred to y”

x~y [xy and yx] „x is indifferent to y”

m

iiGG

1

15

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

For simplicity: Gi , iI, where I={1,…,m}

For each gi, 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

m

iii xguxU

1

Gx,...,xxg,...,xgAx,A:g mmiii 11 , and

16

The preference information is given in form of a complete

preorder on a subset of reference actions ARA,

AR={x1,x2,...,xn} – the reference actions are rearranged such that

xk xk+1 , k=1,...,n-1

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

A

AR

x1

x2

x5

x6x7

x3x4

17

Example:

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

Evaluation of reference actions on criteria Gain_1, Gain_2:

Reference ranking:

Principle of the ordinal regression

Gain_1 Gain_2

a1 4 6

a2 5 5

a3 6 4

a1

a2

a3

18


19


20

Let’s change the reference ranking:

One linear piece per each marginal value function u1, u2 is not enough


Gain_1 Gain_2

a1 4 6

a2 5 5

a3 6 4

a1

a3

a2

a1

a2

a3

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

21


22

xxxguxg'Um

iii

1

The comprehensive preference information is given in form of

a complete preorder on a subset of reference actions ARA,

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 xAR

where

+ and - are potential errors of over- and under-estimation of the

right value, resp.

The intervals [i, i] are divided into i equal sub-intervals

with the end points (iI)

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

iiii

iji ,...,j,

jx

0 11

23


In the UTA method, the marginal value of action xA is approximated

by linear interpolation: for 1 ji

jiii x,xxxg

jii

jiij

iji

jiij

iiiiii xuxuxx

xxxuxuxgu

1

1

24

UTA additive preference model

25

Ordinal regression principle

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

Monotonicity of preferences

Normalization

Principle of the UTA method

1k

1

1k1

xxxU'xU'

xxxU'xU'

~kkk

kkk

Ii,...,jxuxu ijii

jii ; 0 ,01

Iiu

u

ii

m

iii

0

11

26


The marginal value functions (breakpoint variables) are estimated by solving the LP problem

where is a small positive constant

Ii,...,jxuxu ijii

jii ; 0 01

ji,Ax,x,x,xu

Iiu

u

Rkkkjii

ii

m

iii

and 0 0 0

0

11

(C)

1

1

11

to subject

Min

k~kkk

kkkk

Ax

kkUTA

xxxU'xU'

xxxU'xU'

xxERk

k=1,...,n-1

27


If EUTA*=0, then the polyhedron of feasible solutions for ui(xi) is not empty

and there exists at least one value function U[g(x)] compatible with the complete preorder on AR

If EUTA*>0, then there is no value function U[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)

EUTA EUTA*+

polyhedron of constraints (C)

EUTA= EUTA*

28

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

gdy ,1

gdy ,50

gdy ,0

ji

i~j

ij

ij

aa

aa.

aalubji

r

29

Współczynnik Kendalla

Następnie oblicza się współczynnik Kendalla :

gdzie dk(R,R*) jest odległością Kendalla między 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

41

mm

*R,Rdk

m

i

m

jijijk *rr*R,Rd

1 121

30

Example of UTA+

Ranking of 6 means of transportation

1

2

31

Preference attitude: „economical”

32

Preference attitude: „hurry”

33

Preference attitude: „hurry”

34

One should use all compatible preference models on set A

Question: what is the consequence of using all compatible preference

models on set A ?

A

ARx

t z

w

v

y

u

DM

x y

z w

y v

u t

z u

u z

preference information

analystAll instances of

preference model compatible

with preference information

What rankings

will result ?

Apply all compatible instances on A

35

Two rankings result: necessary and possible

x

y

w

z

t

u

v

necessary ranking possible ranking

Includes

necessary ranking

and

does not include

the complement of

necessary ranking

x y

z w

y v

u t

z u

u zpre

fere

nce

info

rmati

on

36

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

possible rankingimpoverished

Includes

necessary ranking

and

does not include

the complement of

necessary ranking

x y

z w

y v

u t

z u

u z

x w

addit

ion

al pre

fere

nce

info

rmati

on

x

y

w

z

t

u

v

necessary rankingenriched

„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

37

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 with the 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 (possible weak preference relation P - a strongly complete and negatively transitive binary relation on A)

38


The marginal value function ui(xi)

ui(xi)

gi

0i yi iwi zivi

y,v,w,zAR

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

39



ui(xi)

gi

0i yi iwi zivi

??

?

??

y,v,w,zAR

Marginal values in characteristic points are unknown

40



ui(xi)

gi

0i yi iwi zivi

y,v,w,zAR

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

41



ui(xi)

gi

0i yi iwi zivi

y,v,w,zAR

The area of all compatible marginal value functions

42



ui(xi)

gi

0i yi iwi zivi

y,v,w,zAR

In the area the marginal compatible value functions must be monotone

43



This means that the ordinal regression should not seek

for m piecewise-linear marginal value functions, but for

any compatible additive value function

ui(xi)

gi

0i yi iwi zivi

44

The UTAGMS method

Let i be 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:

iiii iiii xx...xx 121

iijijiii x,..., ωjxxx

i

10 ,1for , ,

45

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 functions U verifying set E(x,y) of constraints:

where is a small positive constant

For all (x,y)A, E(x,y) = E(y,x)

R

Iiii

ii

jii

jii

~

Axxx

xu

Iixu

,...,jIixuxu

yxy'Ux'U

yxy'Ux'U

,0 ,0

1

0,

11 , ,0

1

0

1

a,bAR

E(x,y)

a

b

b

b

b

a

aaa

a

a

46

N means necessary (strong) preference relation

Given a pair of actions x,yA

xNy d(x,y) 0

where

d(x,y) 0 means that for all compatible value functions

x is at least as good as y

For x,yAR :

xy xNy

The UTAGMS method

yUxUy,xdy,xE

s.t.Min

47

P means possible (weak) preference relation

Given a pair of actions x,yA

xPy D(x,y) 0

where

D(x,y) 0 means that for at least one compatible value function

x is at least as good as y

For x,yAR :

xy not yPx

The UTAGMS method

yUxUy,xDy,xE

s.t.Max

48

Some properties:

xNy xPy

N is a partial preorder (i.e. N is reflexive and transitive)

P is strongly complete (i.e. for all x,yA, xPy or yPx) and

negatively transitive (i.e. for all x,y,zA, not xPy and not yPz

not xPz ), (in general, P is not transitive)

d(x,y) = Min{U(x)–U(y)} = –Max{–[U(x)–U(y)]} =

= –Max{U(y)–U(x)} = –D(y,x)

The UTAGMS method

49

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

50

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 not xNy 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 are not dual

51

The UTAGMS method – final ranking (partial preorder)

Final ranking corresponding to necessary (strong) preference :

N

N

N

N

N

N

N

N

N

52

Final ranking corresponding to possible (weak) preference :

The UTAGMS method – final ranking (complete preorder)

PP

P P

PP

P

P

P

P

53

Example of UTAGMS

Ranking of 6 means of transportation

1

2

54

„hurry”

D(x,y)

56

„hurry” „economical”

57

Nested ranking with different credibility levels

Reference rankings in growing sets:

with credibility levels ordered decreasingly

The reference ranking of alternatives from ARi does not change in AR

i+1,

i=1,…,p-1

Each time we pass from ARi to AR

i+1, we add to (E) new constraints

concerning alternatives from {ARi+1\ AR

i}

If d(x,y)<0 in iteration i turns to d(x,y)>0 in iteration i+1, then we assign

to xNy the 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,zA, Min{Cr(xN~y), Cr(yN~z)} Cr(xN~z),

thus N~ is min-transitive and reflexive

Rp

RR A...AA 21

58

Software demonstration

Bank

59

60

New reference action

61

New reference ranking

62

High credibility ranking embedded in low credibility ranking

Partial preorder graph (strong outranking)

63

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”

64

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

x y „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* not i*-1, i* = i* i*-1

65

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:

a) U(x) U(y) iff x y

b) U(x) > U(y) iff x y

c) U(x) = U(y) iff x y

d) U(x) – U(y) U(w) – U(z) iff (x,y) * (w,z)

e) U(x) – U(y) > U(w) – U(z) iff (x,y) * (w,z)

f) U(x) – U(y) = U(w) – U(z) iff (x,y) * (w,z)

g) ui(x) ui(y) iff x i y, iI

h) ui(x) – ui(y) ui(w) – ui(z) iff (x,y) i* (w,z), iI

i) ui(x) – ui(y) > ui(w) – ui(z) iff (x,y) i* (w,z), iI

j) ui(x) – ui(y) = ui(w) – ui(z) iff (x,y) i* (w,z), iI

66


Moreover, the following normalization constraints should also be

taken into account:

k) ui(i)=0, iI

l) 1Ii

iiu

67

If constraints a) – l) are consistent, then we get two weak preference relations N and P , and two binary relations comparing intensity of preference *N and *P :

for all x,yA, a necessary weak preference relation, x N y:

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

for all x,yA, a possible weak preference relation, x P y:

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


68

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

1. For all x,yA, x N y x P y

2. For all x,yAR, x y x N y

3. N is a partial preorder (i.e. the relation is transitive and reflexive) andP is strongly complete and negatively transitive

4. For all x,y,zA, [x N y and y P z] x P z

5. For all x,y,zA, [x P y and y N z] x P z

6. For all x,y,w,zA, (x,y) *N (w,z) (x,y) *P (w,z)

7. For all x,y,w,zA, (x,y) * (w,z) (x,y) *N (w,z)

3. *N is a partial preorder and *P is strongly complete and negatively transitive

9. For all x,y,w,z,r,sA, [(x,y) *N (w,z) and (w,z) *P (r,s)] (x',y) *P (r,s)

10. For all x,y,w,z,r,sA, [(x,y) *P (w,z) and (w,z) *N (r,s)] (x,y) *P (r,s)

11. For all x,x’,y,w,zA, [x’ N x and (x,y) *N (w,z)] (x’,y) *N (w,z)

12. For all x,x’,y,w,zA, [x’ N x and (x,y) *P (w,z)] (x’,y) *P (w,z)

13. 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

69

14. For all x,y,y’,w,zA, [y N y’ and (x,y) *N (w,z)] (x,y’) *N (w,z)

15. For all x,y,y’,w,zA, [y N y’ and (x,y) *P (w,z)] (x,y’) *P (w,z)

16. For all x,y,y’,w,zA, [y P y’ and (x,y) *N (w,z)] (x,y’) *P (w,z)

17. For all x,y,w,w’,zA, [w N w’ and (x,y) *N (w,z)] (x,y) *N (w’,z)

18. For all x,y,w,w’,zA, [w N w’ and (x,y) *P (w,z)] (x,y) *P (w’,z)

19. For all x,y,w,w’,zA, [w P w’ and (x,y) *N (w,z)] (x,y) *P (w’,z)

20. For all x,y,w,z,z’A, [z’ N z and (x,y) *N (w,z)] (x,y) *N (w,z’)

21. For all x,y,w,z,z’A, [z’ N z and (x,y) *P (w,z)] (x,y) *P (w,z’)

22. For all x,y,w,z,z’A, [z’ P z and (x,y) *N (w,z)] (x,y) *P (w,z’)

23. For all x,x’,yA, (x’,y) *N (x,y) x’ N x

24. For all x,x’,yA, (x’,y) *P (x,y) x’ P x

25. For all x,y,y’A, (x,y) *N (x,y’) y’ N y

26. For all x,y,y’A, (x,y) *P (x,y’) y’ P y

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

70

GRIP – the linear programming problem

In order to verify the truth or falsity of necessary and possible weak

preference relations N, P and *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

71


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 #

nj ijij

nj ijij

q,...,idxc

p,...,ibxa

1

1

1 ,

1 ,

nj ijij

nj ijij

q,...,idxc

p,...,ibxa

1

1

1 ,

1 ,#

nj ijij

nj ijij

q,...,idxc

p,...,ibxa

1

1

1 ,

1 ,

s.t. ,MinMax

72


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 any x,yA, one should :


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)

73


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) :


and U(y) U(x) +

Maximal *≤0 xNy

74


Analogously, if constraints b),e),i) are considered, in order to verify the truth or falsity of (x,y)*P(w,z) for any x,y,w,zA, one should :


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 any x,y,w,zA, one should :


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!

75

Comparison of GRIP and MACBETH (Bana e Costa & Vansnick 1994)

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))

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)

Pre

fere

nce

info

rmati

on

76

Comparison of GRIP and MACBETH cont.

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

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)

Pre

fere

nce

model and fi

nal re

sult

s

77

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

78

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)

79

GRIP – illustrative example

Car ranking problem

Criteria: Intensity of preference:

80


Performance matrix

Skoda

Opel

Ford

Citroen

Seat

VW

Price Speed Space Fuel_cons. Acceleration

81

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


82


Partial information about

preference intensity serves to

define constraints h) and j)

representing partial preorder i*

83


Comprehensive information about preference intensity serves to

define constraints a) and c) representing partial preorder *

(F)

Skoda

Opel

Ford

Citroen

Seat

VW

84


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 AA, so P brings

no interesting information because all weak preferences are possible

85


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

86


Graph of necessary weak preference relation N

87

In GRIP, preference information is given by 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

Conclusions

Documents

Roman Słowiński Poznań University of Technology, Poland