Roman Słowiński Poznań University of Technology, Poland
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Multiple-criteria ranking
using an additive utility function
constructed via ordinal regresion :
UTA method
Roman SłowińskiPoznań University of Technology, Poland
Roman Słowiński
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Problem statement
Consider a finite set A of alternatives (actions, solutions)
evaluated by n criteria from a consistent family G={g1,...,gn}
g2max
g1(x)
g2(x)
g2min
g1min g1max
A
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Problem statement
Consider a finite set A of alternatives (actions, solutions)
evaluated by n criteria from a consistent family G={g1,...,gn}
g2(x)
g1(x)
A
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Problem statement
Taking into account preferences of a Decision Maker (DM),
rank all the alternatives 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
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Basic concepts and notation
Xi – domain of criterion gi (Xi is finite or countably infinte)
– evaluation space
x,yX – profiles of alternatives in evaluation space
– weak preference (outranking) relation on X: for each x,yX
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”
n
iiXX
1
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n
iii aguaU
1
For simplicity: Xi , for all i=1,…,n
For each gi, Xi=[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 X and
where g(a) is the vector of evaluations of alternative aA on n criteria
Additive value (or utility) function on X: for each aX
where ui are non-decreasing marginal value functions, ui : Xi ,
i=1,...,n
Basic concepts and notation
n
iiiiii ,aa;,A:g
1
g
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Criteria aggregation model = preference model
To solve a multicriteria decision problem one needs
a criteria aggregation model, i.e. a preference model
Traditional aggregation paradigm:
The criteria aggregation model is first constructed and then applied
on set A to get information about the comprehensive preference
Disaggregation-aggregation (or regression) paradigm:
The comprehensive preference on a subset ARA is known a priori and
a consistent criteria aggregation model is inferred from this information
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Criteria aggregation model = preference model
The disaggregation-aggregation paradigam has been introduced
to MCDA 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 the inductive
learning used in artificial intelligence and knowledge discovery
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The comprehensive preference information is given in form of
a complete preorder on a subset of reference alternatives
ARA,
AR={a1,a2,...,am} – the reference alternatives are rearranged such
that ak ak+1 , k=1,...,m-1
Principle of the UTA method (Jacquet-Lagreze & Siskos, 1982)
A
AR
a1
a2
a5
a6a7
a3a4
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Example:
Let AR={a1, a2, a3}, G={Gain_1, Gain_2}
Evaluation of reference alternatives on criteria Gain_1, Gain_2:
Reference ranking:
Principle of the UTA method
Gain_1 Gain_2
a1 4 6
a2 5 5
a3 6 4
a1
a2
a3
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Principle of the UTA method
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Principle of the UTA method
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Let’s change the reference ranking:
One linear piece per each marginal value function u1, u2 is not enough
Principle of the UTA method
Gain_1 Gain_2
a1 4 6
a2 5 5
a3 6 4
a1
a3
a2
a1
a2
a3
u1=w1Gain_1, u2=w2Gain_2, U=u1+u2
For a1a3, w2>w1,
but for a3a2, w1>w2,
thus, marginal value functions cannot be linear
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Principle of the UTA method
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Principle of the UTA method
The inferred value of each reference alternative aAR:
where
is a calculated value function,
is a value function compatible with
the
reference ranking,
+ and - are potential errors of over- and under-estimation of the
compatible value function, respectively.
The intervals [i, i] are divided into (i–1) equal sub-intervals with
the end points (i=1,...,n)
n
iii ag'ua'U
1
g
aaaUa'U gg
n
iii aguaU
1
g
iiii
iji ,...,j,
jg
11
1
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Principle of the UTA method
The marginal value of alternative aA is approximated by a linear
interpolation: for 1 ji
jii g,gag
jii
jiij
iji
jiij
iiii gugugg
gagguagu
11
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Principle of the UTA method
Ordinal regression principle
if then one of the following holds
N.B. In practice, „0” is replaced here by a small positive number that may
influence the result
Monotonicity of preferences
Normalization
11
11
0
0
k~kkk
kkkk
aaa,a
aaa,a
iff
iff
11 kkkk a'Ua'Ua,a gg
n,...,i;,...,jgugu ijii
jii 11101
n,...,iu
u
ii
n
iii
10
11
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Principle of the UTA method
The marginal value functions (breakpoint variables) are estimated by
solving the LP problem
11
11
0
0
k~kkk
kkkk
Aa
aaa,a
aaa,a
aaFR
iff
iff
to subject
Min
n,...,i;,...,jgugu ijii
jii 11101
ji,Aa,a,a,gu
n,...,iu
u
Rjii
ii
n
iii
and
000
10
11
(C)
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Principle of the UTA method
If F*=0, then the polyhedron of feasible solutions for ui(gi) is not empty and
there exists at least one value function U[g(a)] compatible with the
complete preorder on AR
If F*>0, then there is no value function U[g(a)] compatible with the
complete preorder on AR – three possible moves:
increasing the number of linear pieces i for ui(gi)
revision of the complete preorder on AR
post optimal search for the best function with respect to Kendall’s in the area F F*+
Jacquet-Lagreze & Siskos (1982)
F F*+
polyhedron of constraints (C)
F= F*
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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
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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
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Example of UTA+
Ranking of 6 means of transportation
1
2
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Preference attitude: „economical”
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Preference attitude: „hurry”
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Preference attitude: „hurry”