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Multiple-criteria sorting using ELECTRE TRI Assistant

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.

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Multiple-criteria sorting using ELECTRE TRI Assistant

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  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 thresholdqi is necessary • An immediate transition from indifference to preference is unrealistic, so a preference thresholdpi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. preference ab ab ab ba ba gi(b) gi(a) gi(a)-pi(b) gi(a)-qi(b) gi(a)+qi(a) gi(a)+pi(a) 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

  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 relationS 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  nonbSa  ab ab aSb  bSa  ab nonaSb  nonbSa  a?b

  5. x x x x x A x x x x x x x x x x x x x x x x x x x x x ... x x x x x x x x ELECTRE TRI: sorting problem(P) Class 1 Class 2 Class p Class p Class p-1  ...  Class 1

  6. Cl1 Cl2 Clp-1 Clp  b0 b1 b2 bp-2 bp-1 bp g1  g2  g3   a d gn 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 profilesbt, 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

  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. Ci(a,bt) preference 1 cost-type 0 gi(a) gi(bt) gi(bt)-pit gi(bt)-qit gi(bt)+qit gi(bt)+pit Ci(a,bt) preference 1 gain-type 0 gi(a) gi(bt) gi(bt)-qit gi(bt)+pit gi(bt)-pit gi(bt)+qit 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 coefficientCi(a,bt) for each criterion gi:

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

  10. Ci(a,bt) Di(a,bt) preference 1 cost-type 0 gi(a) gi(bt) gi(bt)+vit gi(bt)-pit gi(bt)-qit gi(bt)+qit gi(bt)+pit Ci(a,bt) Di(a,bt) preference 1 gain-type 0 gi(a) gi(bt) gi(bt)-vit gi(bt)-pit gi(bt)-qit gi(bt)+qit gi(bt)+pit 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 coefficientDi(a,bt) for each criterion gi:

  11. (a,bt) not yes (bt,a) (bt,a) not yes not yes a bt a ? bt bt{}a a{}bt 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 ?

  12. Cl1 Cl2 Clp-1 Clp comparison of action a to profiles bt  b0 b1 b2 bp-2 bp-1 bp g1  g2  g3   gn a ELECTRE TRI – exploitation of S and final recommendation • Assignment of actions to decision classes based on two ways of comparison of actions aAto 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

  13. Cl1 Cl2 Clp-1 Clp comparison of action a to profiles bt  b0 b1 b2 bp-2 bp-1 bp g1  g2  g3   gn a ELECTRE TRI – exploitation of S and final recommendation • Assignment of actions to decision classes based on two ways of comparison of actions aAto 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

  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 modelM 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 supposeajDMClt, Clt=[bt-1, bt] • ajDMClt ajSbt-1notajSbt 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MClt’[yaj, +xaj] • We consider set AR={a1, a2,…, ar} of reference actions, and assignment examples provided by the DM: ajDMClt, 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. n n n Inferring an ELECTRE TRI model from assignment examples • The regression problem: number of constraints& variables: r r r r 2 np n(p+1) n+n(p+1)

  23. n n n n n Inferring an ELECTRE TRI model from assignment examples Nonlinear optimization problemwith 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):

  25. gi(ai)-pit qj -pit (-pit-qit)/2 -qit ^ Ci(aj,bt) Ci(aj,bt) 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) ^

  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. qi2 pi2 qi1 pi1 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,…,ppit = 0.1 gi(bt), i=1,…,n, t=0,1,…,p  = 0.75

  29. Inferring an ELECTRE TRI model from assignment examples • Initial values of xajandyaj: • 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 xajandyaj: • 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

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