Strategies for the incremental inference of majority-rule sorting - - PowerPoint PPT Presentation

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Strategies for the incremental inference of majority-rule sorting - - PowerPoint PPT Presentation

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Alexandru OLTEANU Lab-STICC, Universit Bretagne Sud, France 23/11/2018 DA2PL 2018 Poznan, Poland Strategies for the incremental inference of majority-rule sorting models alexandru.olteanu@univ-ubs.fr a) direct elicitation: DM gives model

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Strategies for the incremental inference of majority-rule sorting models

Alexandru OLTEANU

alexandru.olteanu@univ-ubs.fr

Lab-STICC, Université Bretagne Sud, France 23/11/2018 DA2PL 2018 Poznan, Poland

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Context

MCDA guides the DM through a decision aiding process.

?

Decision Situate Formulate Select and tune a model Solve

Model tuning a) direct elicitation: DM gives model parameters; b) indirect elicitation: parameters are inferred from holistic judgments given by the DM (assignment examples);

  • one-shot or incremental elicitation;
  • assignment examples are selected or constructed;

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 1/12

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Context

MCDA guides the DM through a decision aiding process.

?

Decision Situate Formulate Select and tune a model Solve

MR-Sort

  • ordinal classification, or sorting;
  • criteria weights wj;
  • majority threshold λ > 1

2;

  • k categories and k−1 profiles;
  • a∈ch ifg aSbh−1 and a /

Sbh Model tuning a) direct elicitation: DM gives model parameters; b) indirect elicitation: parameters are inferred from holistic judgments given by the DM (assignment examples);

  • one-shot or incremental elicitation;
  • assignment examples are selected or constructed;

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 1/12

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SLIDE 4

Context

MCDA guides the DM through a decision aiding process.

?

Decision Situate Formulate Select and tune a model Solve

Model tuning a) direct elicitation: DM gives model parameters; b) indirect elicitation: parameters are inferred from holistic judgments given by the DM (assignment examples);

  • one-shot or incremental elicitation;
  • assignment examples are selected or constructed;

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 1/12

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SLIDE 5

Context

MCDA guides the DM through a decision aiding process.

?

Decision Situate Formulate Select and tune a model Solve

Model tuning a) direct elicitation: DM gives model parameters; b) indirect elicitation: parameters are inferred from holistic judgments given by the DM (assignment examples);

  • one-shot or incremental elicitation;
  • assignment examples are selected or constructed;

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 1/12

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The protocol

Model inference Updated MR-Sort model Decision maker Generate alternatives Elicitation Assignment examples

Research question: Can we develop a strategy to reduce the amount of information required from the DM ?

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 2/12

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The protocol

Model inference Updated MR-Sort model Decision maker Generate alternatives Elicitation Assignment examples

Research question: Can we develop a strategy to reduce the amount of information required from the DM ?

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 2/12

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Proposed strategies for generating alternatives

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Strategies for generating assignment examples

Random (RND)

  • a ∶ gj(a) ∼ U(minj,maxj), ∀j ∈ J

Non-dominated same category random (NDR)

  • if

x y ch s t gj x gj a gj y then a is rejected;

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 3/12

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Strategies for generating assignment examples

Random (RND)

  • a ∶ gj(a) ∼ U(minj,maxj), ∀j ∈ J

Non-dominated same category random (NDR)

  • if ∃x,y ∈ ch s.t. gj(x) ⩾ gj(a) ⩾ gj(y) then a is rejected;

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 3/12

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Strategies for generating assignment examples

Fixing limit profiles (FLP)

  • we use the model inferred from the previous iteration;
  • we try to bound each profile from above and below;

Minimal majority coalition of criteria: J J

j J

wj and i J

j J i

wj (1) Maximal minority coalition of criteria: J J

j J

wj and i J J

j J i

wj (2)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 4/12

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Strategies for generating assignment examples

Fixing limit profiles (FLP)

  • we use the model inferred from the previous iteration;
  • we try to bound each profile from above and below;

Minimal majority coalition of criteria: J + = {J+ ⊆ J∣ ∑

j∈J+

wj ⩾ λ and ∀i ∈ J+, ∑

j∈J+−{i}

wj < λ} (1) Maximal minority coalition of criteria: J − = {J− ⊆ J∣ ∑

j∈J−

wj < λ and ∀i ∈ J − J−, ∑

j∈J−∪{i}

wj ⩾ λ} (2)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 4/12

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Strategies for generating assignment examples

Fixing limit profiles (FLP)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 5/12

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Strategies for generating assignment examples

Fixing limit profiles (FLP)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 5/12

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SLIDE 15

Strategies for generating assignment examples

Fixing limit profiles (FLP)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 5/12

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SLIDE 16

Strategies for generating assignment examples

Fixing limit profiles (FLP)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 5/12

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SLIDE 17

Strategies for generating assignment examples

Fixing limit profiles (FLP)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 5/12

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Strategies for generating assignment examples

Fixing limit profiles (FLP)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 5/12

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SLIDE 19

Strategies for generating assignment examples

Fixing limit profiles (FLP)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 5/12

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Strategies for generating assignment examples

Fixing limit profiles using a central model (FLP+)

  • identical to FLP except that the model inferred from the

previous iteration is centered within the search space;

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 6/12

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Strategies for generating assignment examples

Fixing limit profiles using a central model (FLP+)

  • identical to FLP except that the model inferred from the

previous iteration is centered within the search space;

x

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 6/12

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Strategies for generating assignment examples

Fixing limit profiles using a central model (FLP+)

  • identical to FLP except that the model inferred from the

previous iteration is centered within the search space;

x x x x

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 6/12

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Strategies for generating assignment examples

Fixing limit profiles using a central model (FLP+)

  • identical to FLP except that the model inferred from the

previous iteration is centered within the search space;

x x x

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 6/12

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Proximity of two MR-Sort models

Distance between criteria importance parameters dC = 1 2m ⋅ ∑

i∈1..m

∑

J′∈(J

i)

dJ

′

C , where

(3) dJ

′

C =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ , if ∑

j∈J′ w

′

j ⩾ λ

′ and ∑

j∈J′ w

′′

j ⩾ λ

′′

  • r ∑

j∈J′ w

′

j < λ

′ and ∑

j∈J′ w

′′

j < λ

′′

1 , otherwise (4) Distance between category profiles importance parameters dP = 1 (k − 1) ⋅ m ⋅ ∑

h∈1..k−1

∑

j∈J

∣gj(b

′

h) − gj(b

′′

h)∣

(5)

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 7/12

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Experimental validation

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Experimental framework

  • m ∈ {3,4}
  • k ∈ {2,3}
  • nin = 4
  • nst ∈ {1,4}
  • nte = 10,000
  • 50 models for each
  • config. of m, k, nst

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 8/12

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Retrieving the original model

10 60 0.85 0.90 0.95 1.00 20 24 45 28 FLP+ FLP NDR RND ∣Ai∣ Classification accuracy 3 criteria, 2 categories 10 90 0.85 0.90 0.95 1.00 46 78 85 53 FLP+ FLP NDR RND ∣Ai∣ Classification accuracy 3 criteria, 3 categories 10 100 0.85 0.90 0.95 1.00 36 45 62 41 FLP+ FLP NDR RND ∣Ai∣ Classification accuracy 4 criteria, 2 categories 140 0.85 0.90 0.95 1.00 71 80 107 FLP+ FLP NDR RND ∣Ai∣ Classification accuracy 4 criteria, 3 categories FLP+ FLP+ NDR+ RND+

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 9/12

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Distance measures

10 20 30 40 50 60 0.1 0.2 0.3 0.4 0.5 ∣Ai∣ dB 3 criteria, 2 categories 10 20 30 40 50 60 70 80 90 0.1 0.2 0.3 0.4 0.5 ∣Ai∣ dB 3 criteria, 3 categories 10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4 0.5 ∣Ai∣ dB 4 criteria, 2 categories 10 20 30 40 50 60 70 80 90 100 110 120 130 140 0.1 0.2 0.3 0.4 0.5 ∣Ai∣ dB 4 criteria, 3 categories FLP+ FLP+ NDR+ RND+

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 10/12

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Conclusions and perspectives

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Conclusions and perspectives

Conclusions:

  • Proposed strategies provide better results than the

classical approach: FLP+ ≻ {FLP,NDR} ≻ RND

  • Results currently only for small problem instances;
  • Correlation between distance measures and model

convergence; Perspectives:

  • Extend to larger instances (metaheuristics);
  • New strategies (e.g. order FLP alternatives based on

distance from targeted profile);

  • Select vs. generate alternatives? Assignment errors?

Time complexity? Vetoes? Centrality ⇐ ⇒ distance? etc.

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 11/12

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Retrieving the original model

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 0.75 0.80 0.85 0.90 0.95 1.00 ∣Ai∣ Classification accuracy

5 criteria, 3 categories FLP+ FLP+ NDR+ RND+

Alexandru OLTEANU Strategies for the incremental inference of MR-Sort 12/12