1. 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)
ab
ba
ba
ab
ab
preference

4. Four basic preference relations and an outranking relation S
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,bA:
aSb  non bSa  ab  ab
aSb  bSa  ab
non aSb  non bSa  a?b

5. ELECTRE TRI: sorting problem (P)
...
x x x
x x x
x x
x x x x
x x x A
Class 1
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 aA and t=0,1,…,p g1 g2 g3  gn b0 b1
bp-2
bp-1 bp b2
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
0qitpitvit 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), aA and t=0,1,…,p)
Concordance coefficient Ci(a,bt) for each criterion gi:
gi(bt)
preference
Ci(a,bt) 1
gain-type
gi(bt)-pit
gi(bt)-qit
gi(bt)+qit
gi(bt)+pit
gi(a)
gi(bt)
gi(a)
gi(bt)-pit
gi(bt)-qit
gi(bt)+qit
gi(bt)+pit
preference
Ci(a,bt) 1
cost-type

9. ELECTRE TRI - concordance test
Aggregation of concordance coefficients for a,bA:
C(a,bt)  [0, 1]

10. ELECTRE TRI - discordance test
Checks how strong is the coalition of criteria discordant with the hypothesis aSbt (for each couple (a,bt), aA and t=0,1,…,p)
Discordance coefficient Di(a,bt) for each criterion gi:
gi(a)
preference
Di(a,bt) 1
gain-type
Ci(a,bt)
gi(bt)-vit
gi(bt)
gi(bt)-pit
gi(bt)-qit
gi(bt)+qit
gi(bt)+pit
gi(a) 1
Ci(a,bt)
preference
cost-type
gi(bt)+vit
Di(a,bt)
gi(bt)-pit
gi(bt)-qit
gi(bt)
gi(bt)+qit
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 ?
(a,bt)
(bt,a)
a  bt
bt{}a
a ? bt
a{}bt
not
yes

12. ELECTRE TRI – exploitation of S and final recommendation
Assignment of actions to decision classes based on two ways of comparison of actions aA 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 aClt+1
Pessimistic, because for zero thresholds, aClt+1  gi(a)gi(bt) i, e.g. aCl1 g1 g2 g3  gn b0 b1
bp-2
bp-1 bp b2
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 aA 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 aClt
Optimistic, because for zero thresholds, aClt  gi(bt)>gi(a) i, e.g. aCl2 g1 g2 g3  gn b0 b1
bp-2
bp-1 bp b2
Cl1
Cl2
Clp-1
Clp a
comparison of action a to profiles bt

14. ELECTRE TRI - example
kPRICE=3
kTIME=5
kCOMFORT=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, pit, 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 xaj0 and yaj>0, such that:
(aj,bt-1)  xaj = 
(aj,bt) + yaj = 
If xaj0 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 xaj0 and yaj>0, for all j=1,…,r
the objective should be to maximize

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:
>

22. Inferring an ELECTRE TRI model from assignment examples
The regression problem: n
number of constraints & variables: r 2 np
n(p+1)
n+n(p+1)

23. Inferring an ELECTRE TRI model from assignment examples
Nonlinear optimization problem with
2r+3n(p+1)+n+2 variables 4r+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):

25. Inferring an ELECTRE TRI model from assignment examples
The approximation of Ci(aj, bt) by f(x) is such that x0=(pjt+qjt)/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-qjt) ^ qj
-pit
-qit
(-pit-qit)/2
gi(ai)-pit
Ci(aj,bt) ^

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

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
qi2
pi2
qi1
pi1
 = 0.75

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, ], 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