multiple-criteria sorting using electre tri assistant
DESCRIPTION
Multiple-criteria sorting using ELECTRE TRI Assistant. Roman Słowiński Poznań University of Technology, Poland. Roman Słowiński. . . . . . . . . . Weak points of preference modelling using utility (value) function. - PowerPoint PPT PresentationTRANSCRIPT
Multiple-criteria sorting
using ELECTRE TRI Assistant
Roman Słowiński
Poznań University of Technology, Poland
Roman Słowiński
2
Weak points of preference modelling using utility (value) function
Utility function distinguishes only 2 possible relations between actions:
preference relation: a b U(a) > U(b)
indifference relation: a b U(a) = U(b)
is asymmetric (antisymmetric and irreflexive) and transitive
is symmetric, reflexive and transitive
Transitivity of indifference is troublesome, e.g.
In consequence, a non-zero indifference threshold qi is necessary
An immediate transition from indifference to preference is unrealistic,
so a preference threshold pi qi and a weak preference relation
are desirable
Another realistic situation which is not modelled by U is incomparability,
so a good model should include also an incomparability relation ?
?
3
Four basic preference relations and an outranking relation S
Four basic preference relations are: {, , , ?}
Criterion with thresholds pi(a)qi(a)0 is called pseudo-criterion
The four basic situations of indifference, strict preference, weak
preference, and incomparability are sufficient for establishing
a realistic model of Decision Maker’s (DM’s) preferences
gi(b)gi(a)gi(a)-pi(b) gi(a)-qi(b) gi(a)+qi(a) gi(a)+pi(a)
ab ba baabab
preference
4
Axiom of limited comparability (Roy 1985):
Whatever the actions considered, the criteria used to compare them,
and the information available, one can develop a satisfactory model
of DM’s preferences by assigning one, or a grouping of two or three
of the four basic situations, to any pair of actions.
Outranking relation S groups three basic preference relations:
S = {, , } – reflexive and non-transitive
aSb means: „action a is at least as good as action b”
For each couple a,bA:
aSb non bSa ab ab
aSb bSa ab
non aSb non bSa a?b
Four basic preference relations and an outranking relation S
5
ELECTRE TRI: sorting problem (P)
Class 1
...
x x xx x
x x x xx
x x xx
x
x
x
x x
xx
xx x x x
x x
x x xx x x
A
Class 2
Class p
Class p Class p-1 ... Class 1
6
ELECTRE TRI
Input data: finite set of actions A={a, b, c, …, h}
consistent family of criteria G={g1, g2, …, gn}
preference-ordered decision classes Clt, t=1,…,p
Decision classes are caracterized by limit profiles bt, t=0,1,…,p
The preference model, i.e. outranking relation S is constructed for
each couple (a, bt), for every aA and t=0,1,…,p
g1
g2
g3
gn
b0 b1 bp-2 bp-1 bpb2
Cl1 Cl2 Clp-1 Clp
a d
7
ELECTRE TRI
Preferential information (i=1,…,n)
intracriteria: – indifference thresholds qit for each class limit, t=0,1,
…,p
– preference thresholds pit for each class limit, t=0,1,
…,p
intercriteria: – importance coefficients (weights) of criteria ki
– veto thresholds for each class limit vit, t=0,1,…,p
0qitpi
tvit are constant for each class limit bt, t=0,1,…,p
Concordance and discordance tests of ELECTRE TRI validate or
invalidate the assertions aSbt and btSa
8
ELECTRE TRI - concordance test
Checks how strong is the coalition of criteria concordant with
the hypothesis aSbt (for each couple (a,bt), aA and t=0,1,…,p)
Concordance coefficient Ci(a,bt) for each criterion gi:
gi(bt) gi(a)gi(bt)-pit gi(bt)-qi
t gi(bt)+qit gi(bt)+pi
t
preferenceCi(a,bt)
0
1
cost-type
gi(bt)
preferenceCi(a,bt)
0
1
gain-type
gi(bt)-pit gi(bt)-qi
t gi(bt)+qit gi(bt)+pi
t gi(a)
9
ELECTRE TRI - concordance test
Aggregation of concordance coefficients for a,bA:
C(a,bt) [0, 1]
ni i
ni tii
tk
b,aCkb,aC
1
1
10
ELECTRE TRI - discordance test
Checks how strong is the coalition of criteria discordant with
the hypothesis aSbt (for each couple (a,bt), aA and t=0,1,…,p)
Discordance coefficient Di(a,bt) for each criterion gi:
gi(a)0
1
Ci(a,bt)preference
cost-type
gi(bt)+vit
Di(a,bt)
gi(bt)-pit gi(bt)-qi
t gi(bt) gi(bt)+qit gi(bt)+pi
t
gi(a)
preferenceDi(a,bt)
0
1
gain-type
Ci(a,bt)
gi(bt)-vit gi(bt)gi(bt)-pi
t gi(bt)-qit gi(bt)+qi
t gi(bt)+pit
11
ELECTRE TRI – credibility of the outranking relation
Conclusion from concordance and discordance tests – credibility of
the outranking relation:
(a,bt)[0, 1]
What is the assignment of actions to decision classes ?
tti
Fi t
titt
b,aCb,aDiF
b,aCb,aD
b,aCb,a
: where
1
1
(a,bt)
(bt,a)(bt,a)
a btbt{}aa ? bt a{}bt
not
not not
yes
yes yes
12
ELECTRE TRI – exploitation of S and final recommendation
Assignment of actions to decision classes based on two ways of
comparison of actions aA to limit profiles bt, t=0,1,…,p
Pessimistic: compare action a successively to each profile bt,
t=p-1,…,1,0; if bt is the first profile such that aSbt, then aClt+1
Pessimistic, because for zero thresholds, aClt+1 gi(a)gi(bt) i,
e.g. aCl1
g1
g2
g3
gn
b0 b1 bp-2 bp-1 bpb2
Cl1 Cl2 Clp-1 Clp
a
comparison of action a to profiles bt
13
ELECTRE TRI – exploitation of S and final recommendation
Assignment of actions to decision classes based on two ways of
comparison of actions aA to limit profiles bt, t=0,1,…,p
Optimistic: compare action a successively to each profile bt,
t=1,…,p; if bt is the first profile such that bt{ }a, then aClt
Optimistic, because for zero thresholds, aClt gi(bt)>gi(a) i,
e.g. aCl2
g1
g2
g3
gn
b0 b1 bp-2 bp-1 bpb2
Cl1 Cl2 Clp-1 Clp
a
comparison of action a to profiles bt
14
ELECTRE TRI - example
kPRICE=3kTIME=5kCOMFORT=2
=0.75
15
ELECTRE TRI - example
??
16
ELECTRE TRI - example
=0.75
17
Inferring an ELECTRE TRI model from assignment examples
An ELECTRE TRI model M is composed of:
limit profiles bt, t=0,1,…,p
importance coefficients (weights) of criteria ki, i=1,…,n
indifference and preference thresholds qit, pi
t, i=1,…,n, t=0,1,…,p
veto thresholds vit, i=1,…,n, t=0,1,…,p
assignment procedure – either pessimistic or optimistic
We will apply a regression approach to infer an ELECTRE TRI model
compatible with a set of assignment examples
Inferring all elements of the model is computationnally hard and,
moreover, the more elements are unconstrained, the larger is the set
of all compatible models – then the choice of one model is uneasy
We assume:
no veto phenomenon occurs, i.e. vit=, i=1,…,n, t=0,1,…,p
pessimistic assignment procedure
18
Inferring an ELECTRE TRI model from assignment examples
General inference scheme of ELECTRE TRI Assistant:
AR
AR
19
Inferring an ELECTRE TRI model from assignment examples
Let us suppose aj DM Clt, Clt=[bt-1, bt]
aj DM Clt ajSbt-1 not ajSbt
i.e. (aj,bt-1) and (aj,bt)<
We define slack variables xaj0 and yaj>0, such that:
(aj,bt-1) xaj =
(aj,bt) + yaj =
If xaj0 and yaj>0, then
aj M Clt ’[yaj, +xaj]
We consider set AR={a1, a2,…, ar} of reference actions,
and assignment examples provided by the DM:
aj DM Clt , j=1,…,r
20
Inferring an ELECTRE TRI model from assignment examples
The regression problem will include the following variables:
xaj and yaj, j=1,…,r (slack variables – 2r)
(cutting level – 1)
ki, i=1,…,n (weights – n)
bit, i =1,…,n, t=0,1,…,p (limit profiles – np)
qit, i=1,…,n, t=0,1,…,p (indifference thresholds – np)
pit, i=1,…,n, t=0,1,…,p (preference thresholds – np)
The objective of the regression problem:
model M will be consistent with assignment examples iff
xaj0 and yaj>0, for all j=1,…,r
the objective should be to maximize ajaj
Aay,x
Rj min
21
Inferring an ELECTRE TRI model from assignment examples
If , then all reference actions are „correctly”
assigned by model M, for all
Objective takes into account the „worst case” only
To improve the second, third, … „worst case”, we consider objective
where is a small positive value Maximization of the new objective can be rewritten as:
aj
Aaaj
Aaxy'
Rj
Rj
min , min
R
jR
j Aaajajajaj
Aayxy,xmin
Rjaj
Rjaj
Aaajaj
Aay
Aax
y,xR
j
,
, s.t.
Max
ajajAa
y,xR
j min
0min
ajajAa
y,xR
j
>
22
Inferring an ELECTRE TRI model from assignment examples
The regression problem:
r
r
r
r
2
np
n(p+1)
n+n(p+1)
number of constraints& variables:
p,...,,tk,...,iqk
p,...,,tk,...,iqp
p,...,,tk,...,ippbgbg
,.
Aayb,a
Aaxb,a
Aay
Aax
y,x
tii
ti
ti
ti
tititi
Rjajtj
Rjajtj
Rjaj
Rjaj
Aaajaj
Rj
10 ,1 ,0 ,0
10 ,1 ,
110 ,1 ,
1 50
,
,
,
, s.t.
Max
11
1
n
n
n
23
Inferring an ELECTRE TRI model from assignment examples
p,...,,tk,...,iqk
p,...,,tk,...,iqp
p,...,,tk,...,ippbgbg
,.
Aayk
b,aCk
Aaxk
b,aCk
Aay
Aax
y,x
tii
ti
ti
ti
tititi
Rjajk
i i
ki tjii
Rjajk
i i
ki tjii
Rjaj
Rjaj
Aaajaj
Rj
10 ,1 ,0 ,0
10 ,1 ,
110 ,1 ,
1 50
,
,
,
, s.t.
Max
11
1
1
1
1 1
n
n
n
n
n
Nonlinear optimization problem with
2r+3n(p+1)+n+2 variables4r+3n(p+1)+2 constraints
24
Inferring an ELECTRE TRI model from assignment examples
Approximation of marginal concordance indices Ci(aj, bt)
Ci(aj, bt) are piecewise linear functions (non-differentiable)
We approximate Ci(aj, bt) by a differentiable sigmoidal function f(x): 01
1xxexp
xf
25
Inferring an ELECTRE TRI model from assignment examples
The approximation of Ci(aj, bt) by f(x) is such that x0=(pjt+qj
t)/2, and
(which influences the angle of inclination of Ci(aj, bt) in point (x0,0.5))
is chosen such that surface of overestimation A is equal to surface of
underestimation B – then the approximation error is minimal
= 5.55/ (pjt-qj
t)
^
qj-pit -qi
t(-pit-qi
t)/2
gi(ai)-pit
Ci(aj,bt)Ci(aj,bt)^
2555
1
1ti
ti
tijiti
ti
tjiqp
bgagqp
.exp
b,aC
26
Inferring an ELECTRE TRI model from assignment examples
Illustrative example:
There are 3 categories C1, C2, C3, separated by 2 limit profiles b1 and
b2, defined on 3 criteria g1, g2, g3 with an increasing scale [0, 100]
27
Inferring an ELECTRE TRI model from assignment examples
Optimization problem has 23 variables and 44 constraints
Additional constraints , prevent any criterion to be
a dictator
Determination of the starting point: ki = 1, i=1,…,3
where nt is the number of reference actions assigned to Ct
31 ,21 3
1,...,ikk
i ii
p,...,t,...,in
ag
n
ag
bgt
Caji
t
Caji
titjtj 1 ,31 ,
21
1
1
28
Inferring an ELECTRE TRI model from assignment examples
Initial values of indifference and preference thresholds:
qit = 0.05 gi(bt), i=1,…,n, t=0,1,…,p
pit = 0.1 gi(bt), i=1,…,n, t=0,1,…,p
= 0.75
qi2
pi2
qi1
pi1
29
Inferring an ELECTRE TRI model from assignment examples
Initial values of xaj and yaj:
Inconsistency: a4 is assigned to C1, while DM assigned a4 to C2
= xa4 = –0.083 < 0
30
Inferring an ELECTRE TRI model from assignment examples
Moreover, = 0.37, =0.629, k1=0.517, k2=1, k3=0.483
qi2
pi2
qi1
pi1
31
Inferring an ELECTRE TRI model from assignment examples
Final values of xaj and yaj:
Model M is consistent for all [0.629–0.37, 0.629+0.37],
thus for all [0.5, 1]
This proves the model is robust
32
ELECTRE TRI Assistant
ELECTRE TRI Assistant permits to infer:
weights ki, i=1,…,n
cutting level
Limit profiles, indifference and preference thresholds are fixed
Assuming , the regression problem is a Linear
Programming problem
ni ik1 1