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INSTITUTE OF INFORMATION AND COMMUNICATION TECHNOLOGIES BULGARIAN ACADEMY OF SCIENCE InterCriteria Decision Making using Intuitionistic Fuzzy Sets Vassia Atanassova IICT BAS, Bulgaria 1 16-Jun-14 AComIn : Advanced Computing for

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1 http://www.iict.bas.bg/acomin 16-Jun-14

INSTITUTE OF INFORMATION AND COMMUNICATION TECHNOLOGIES BULGARIAN ACADEMY OF SCIENCE

AComIn: Advanced Computing for Innovation

InterCriteria Decision Making using Intuitionistic Fuzzy Sets

Vassia Atanassova IICT – BAS, Bulgaria

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16-Jun-14 2 http://www.iict.bas.bg

Contents

AComIn: Advanced Computing for Innovation

  • Specific problem statement
  • General problem statement
  • Proposed approach

– Intuitionistic fuzzy sets

  • Applications

– EU Competitiveness analysis

  • IF threshold analysis
  • Future directions of research
  • Publications
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16-Jun-14 3 http://www.iict.bas.bg

Specific problem formulation

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  • A problem from the petrochemical industry:

– A set of probes of mineral oil from a new shipment, tested against a set of physical and chemical criteria to determine the best way to utilize it in production. – From the set of all physical and chemical criteria, not all are equal to measure. – Any extra measurement delays the production and rises the production costs. – Can we find some correlations in our data, so that we eliminate the need of measurement along some criteria (like cetane number), while keeping the precision of the decision making process as much as possible?

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General problem formulation

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  • A set of objects is evaluated against a set of criteria

– Some measurements are cheap, quick and easy – Other measurements are expensive, time-consuming and/or difficult. We called them ‘cost-unfavourable criteria’ (CUC)

  • We need to discover dependences between the

criteria, and thus eliminate the need of making all the measurements, ideally eliminating the CUC.

  • Precision of the decision making process should be

as high as possible (higher than a predefined value).

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Proposed approach

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  • InterCriteria Decision Making, based on:

– Index matrices – Intuitionistic fuzzy sets

1 1 1,1 1, 1, ,1 , ,

... ... ... ... ... ... ... ... ... ... ... ...

s n s s m m m s m s

c c c IM

  • e

e e

  • e

e e =

( 1) pairs of objects , , 2 ( 1) i.e. termwise 2 comparisons between and

i j i j

m m

  • o

m m e e − −

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Proposed approach

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  • InterCriteria Decision Making, based on:

– Index matrices – Intuitionistic fuzzy sets

Warning

×

Yes No Cancel Want to know more about IFS?

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Intuitionistic fuzzy sets

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  • Defined in 1983 by Krassimir Atanassov
  • Titled so by him and George Gargov in relation

to Leutzen Brower’s philosophical concept of intuitionism

  • One of the most notable extensions of

Lotfi Zadeh’s fuzzy sets

  • Proven to be mathematically equivalent to many
  • ther FS extensions (like L-fuzzy sets, rough sets,

interval valued fuzzy sets)

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Intuitionistic fuzzy sets

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  • Formal definition:

Let us have a set A in the universum X and two mappings μA(x) and νA(x), so that μA(x): A → [0, 1], νA(x): A → [0, 1] 0 ≤ μA(x) + νA(x) ≤ 1 Then, the set A = {〈x, μA(x), νA(x)〉 | x ∈ X } is called intuitionistic fuzzy set.

  • Boundary conditions:

Complete non-membership (falsity) μA(x) = 0, νA(x) = 1 Complete membership (truth) μA(x) = 1, νA(x) = 0

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Intuitionistic fuzzy sets

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μ ∈ [0; 1] 1 1

0.7% MEMBERSHIP

0.15% NON-MEMBERSHIP

μ ∈ [0; 1], ν ∈ [0; 1] μ + ν ∈ [0; 1] π = 1 − μ − ν

UNCERTAINTY

Fuzzy set Intuitionistic fuzzy set

0.7% MEMBERSHIP

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Intuitionistic fuzzy sets

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  • Graphical interpretations: Linear

Standard Modified μ ν μ 1 - ν zero uncertainty 1 high uncertainty

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Intuitionistic fuzzy sets

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  • Graphical interpretations: Triangular

(0,1) (0,0) (1,0)

πA(x) νA(x) μA(x)

x

νA(x)

Elements of an IFS Elements of a FS

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Intuitionistic fuzzy sets

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  • Graphical interpretations: Triangular

(0,1) (0,0) (1,0)

IFS specific modal operators! Necessity: □A = {〈x, μA(x), 1−μA(x)〉 | x ∈ E} Possibility: ◊A = {〈x, 1−νA(x), νA(x)〉 | x ∈ E}

□x

◊x

x

πA(x) νA(x) μA(x)

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  • Operations over IFSs

Intuitionistic fuzzy sets

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Intersection A ∩ B = {〈x, min(μA(x),μB(x)), max(νA(x),νB(x))〉 | x ∈ E} Union A ∪ B = {〈x, max(μA(x),μB(x)), min(νA(x),νB(x))〉 | x ∈ E} Multiplication A . B = {〈x, μA(x).μB(x), νA(x)+νB(x)−νA(x).νB(x)〉 | x ∈ E} Addition A + B = {〈x, μA(x)+μB(x)−μA(x).μB(x), νA(x).νB(x))〉 | x ∈ E}

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Intuitionistic fuzzy sets

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  • Relations over IFSs

Inclusion A ⊂ B iff (∀ x ∈ E ) ( μA(x) ≤ μB(x) & νA(x) ≥ νB(x) ) Equality A = B iff (∀ x ∈ E ) ( μA(x) = μB(x) & νA(x) = νB(x) )

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Intuitionistic fuzzy sets

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  • Topological operators over IFSs

Closure C(A) = {〈x, sup μA(y), inf νA(y)〉 | x∈E, y∈E } Interior I(A) = {〈x, inf μA(y), sup νA(y)〉 | x∈E, y∈E }

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Intuitionistic fuzzy sets

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  • Extensions of IFSs

IFSs of Type 2, for which

0 ≤ μA(x) + νA(x) ≤ 1

changes to

0 ≤ μA(x)2 + νA(x)2 ≤ 1

and IFSs of Type n:

0 ≤ μA(x)n + νA(x) n ≤ 1

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Intuitionistic fuzzy sets

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  • More about IFS

Journals:

  • Fuzzy Sets and Systems, Elsevier
  • IEEE Transactions of Fuzzy Systems, IEEE
  • Notes on Intuitionistic Fuzzy Sets, Bulg. Acad. Sci.

Conference proceedings:

  • IFSA and EUSFLAT conferences
  • IEEE Intelligent Systems
  • International conferences and workshops on IFS

held in Bulgaria, Poland and Slovakia Website:

  • http://ifigenia.org
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Proposed approach

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  • InterCriteria Decision Making, based on

– Index matrices – Intuitionistic fuzzy sets

If R(ei,s; ej,s) is “>”, then µ++ (membership) If R(ei,s; ej,s) is “<”, then ν++ (non-membership) Otherwise, π++ (uncertainty)

, , i s j s

e e ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ = < >

, , [0;1] s s s μ ν π ∈

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Proposed approach

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  • InterCriteria Decision Making, based on

– Index matrices – Intuitionistic fuzzy sets Parameters α and β (α, β ∈ [0;1]) are used to measure the levels of consonance or dissonance between the involved criteria:

  • If (µ > α) AND (ν < β),

then (α, β)-positive consonance

  • If (µ < β) AND (ν > α),

then (α, β)-negative consonance

  • Otherwise,

dissonance

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Proposed approach

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  • InterCriteria Decision Making, based on

– Index matrices – Intuitionistic fuzzy sets We obtain back two IM-s for the positive and the negative consonances between the criteria, i.e.

1 1 1 1,1 1, 1, 1 1,1 1, 1, ,1 , , ,1 , , ,1 , ,

... ... ... ... , ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ...

i n i n i n i n i i i n i n i i i n i n n n n i n n

c c c c c c IM IM c c c c c

μ ν

μ μ μ ν ν ν μ μ μ ν ν ν μ μ μ = =

,1 , ,

.. ... ... ...

n n n i n n

c ν ν ν

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  • So far the approach is tested with:

– Petrochemical data for raw mineral oil

  • Results outline all the previously

known correlations between criteria

– Biomedical data about temperature curves of patients with multiplen myelom and carcinome

  • Results outline yet unknown relations, but the starting

data are yet scarce, work needed on problem formulation

– Data from the World Economic Forum’s Global Competitiveness Reports (2008-2014)

  • AFAWK, completely new and reliable results

which is actually good!

not yet ready to boast with

encouraging!

Applications

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  • Example with the WEF’s GCRs (2013-2014)

– 6 matrices × 28 objects × 12 criteria – Criteria (WEF Methodology):

Basic requirements

  • 1. Institutions
  • 2. Infrastructure
  • 3. Macroeconomic stability
  • 4. Health and primary education

Economy enhancers

  • 5. Higher education and training
  • 6. Goods market efficiency
  • 7. Labor market efficiency
  • 8. Financial market sophistication
  • 9. Technological readiness
  • 10. Market size

Innovation and sophistication factors

  • 11. Business sophistication
  • 12. Innovation

Application: EU competitiveness

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Application: EU competitiveness

  • Example with the WEF’s GCRs

Positive consonance 2008-2009 vs. 2013-2014

µ 1 2 3 4 5 6 7 8 9 10 11 12 1 1.000 0.844 0.685 0.757 0.788 0.833 0.603 0.828 0.823 0.497 0.794 0.802 2 0.844 1.000 0.627 0.751 0.749 0.743 0.529 0.741 0.775 0.582 0.831 0.807 3 0.685 0.627 1.000 0.616 0.638 0.664 0.653 0.648 0.693 0.434 0.651 0.667 4 0.757 0.751 0.616 1.000 0.780 0.720 0.550 0.704 0.725 0.524 0.765 0.772 5 0.788 0.749 0.638 0.780 1.000 0.746 0.622 0.728 0.757 0.558 0.767 0.796 6 0.833 0.743 0.664 0.720 0.746 1.000 0.627 0.817 0.802 0.505 0.786 0.765 7 0.603 0.529 0.653 0.550 0.622 0.627 1.000 0.664 0.611 0.389 0.563 0.590 8 0.828 0.741 0.648 0.704 0.728 0.817 0.664 1.000 0.820 0.476 0.733 0.751 9 0.823 0.775 0.693 0.725 0.757 0.802 0.611 0.820 1.000 0.548 0.817 0.815 10 0.497 0.582 0.434 0.524 0.558 0.505 0.389 0.476 0.548 1.000 0.648 0.601 11 0.794 0.831 0.651 0.765 0.767 0.786 0.563 0.733 0.817 0.648 1.000 0.860 12 0.802 0.807 0.667 0.772 0.796 0.765 0.590 0.751 0.815 0.601 0.860 1.000 µ 1 2 3 4 5 6 7 8 9 10 11 12 1 1.000 0.735 0.577 0.720 0.807 0.836 0.733 0.749 0.854 0.503 0.804 0.844 2 0.735 1.000 0.479 0.661 0.749 0.677 0.537 0.590 0.786 0.661 0.804 0.799 3 0.577 0.479 1.000 0.421 0.519 0.558 0.627 0.675 0.550 0.413 0.548 0.556 4 0.720 0.661 0.421 1.000 0.730 0.683 0.590 0.563 0.677 0.497 0.712 0.690 5 0.807 0.749 0.519 0.730 1.000 0.735 0.622 0.632 0.775 0.579 0.815 0.847 6 0.836 0.677 0.558 0.683 0.735 1.000 0.749 0.712 0.788 0.466 0.759 0.751 7 0.733 0.537 0.627 0.590 0.622 0.749 1.000 0.741 0.685 0.399 0.624 0.624 8 0.749 0.590 0.675 0.563 0.632 0.712 0.741 1.000 0.712 0.497 0.688 0.680 9 0.854 0.786 0.550 0.677 0.775 0.788 0.685 0.712 1.000 0.526 0.810 0.831 10 0.503 0.661 0.413 0.497 0.579 0.466 0.399 0.497 0.526 1.000 0.611 0.598 11 0.804 0.804 0.548 0.712 0.815 0.759 0.624 0.688 0.810 0.611 1.000 0.873 12 0.844 0.799 0.556 0.690 0.847 0.751 0.624 0.680 0.831 0.598 0.873 1.000

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Application: EU competitiveness

  • Example with the WEF’s GCRs

Negative consonance 2008-2009 vs. 2013-2014

ν 1 2 3 4 5 6 7 8 9 10 11 12 1 0.000 0.114 0.241 0.140 0.140 0.077 0.275 0.116 0.116 0.458 0.148 0.127 2 0.114 0.000 0.304 0.156 0.190 0.167 0.365 0.220 0.180 0.384 0.127 0.138 3 0.241 0.304 0.000 0.265 0.265 0.209 0.204 0.270 0.225 0.495 0.270 0.241 4 0.140 0.156 0.265 0.000 0.108 0.140 0.294 0.201 0.169 0.381 0.138 0.111 5 0.140 0.190 0.265 0.108 0.000 0.135 0.233 0.198 0.164 0.378 0.156 0.130 6 0.077 0.167 0.209 0.140 0.135 0.000 0.209 0.090 0.095 0.397 0.114 0.127 7 0.275 0.365 0.204 0.294 0.233 0.209 0.000 0.212 0.259 0.497 0.315 0.265 8 0.116 0.220 0.270 0.201 0.198 0.090 0.212 0.000 0.132 0.476 0.217 0.196 9 0.116 0.180 0.225 0.169 0.164 0.095 0.259 0.132 0.000 0.399 0.122 0.116 10 0.458 0.384 0.495 0.381 0.378 0.397 0.497 0.476 0.399 0.000 0.307 0.336 11 0.148 0.127 0.270 0.138 0.156 0.114 0.315 0.217 0.122 0.307 0.000 0.079 12 0.127 0.138 0.241 0.111 0.130 0.127 0.265 0.196 0.116 0.336 0.079 0.000 ν 1 2 3 4 5 6 7 8 9 10 11 12 1 0.000 0.220 0.386 0.188 0.132 0.077 0.185 0.172 0.090 0.452 0.138 0.111 2 0.220 0.000 0.466 0.228 0.172 0.228 0.362 0.317 0.146 0.286 0.135 0.138 3 0.386 0.466 0.000 0.476 0.405 0.344 0.286 0.251 0.394 0.537 0.394 0.389 4 0.188 0.228 0.476 0.000 0.143 0.169 0.283 0.307 0.201 0.397 0.175 0.198 5 0.132 0.172 0.405 0.143 0.000 0.153 0.272 0.259 0.135 0.341 0.098 0.079 6 0.077 0.228 0.344 0.169 0.153 0.000 0.135 0.169 0.101 0.439 0.143 0.159 7 0.185 0.362 0.286 0.283 0.272 0.135 0.000 0.146 0.209 0.505 0.267 0.275 8 0.172 0.317 0.251 0.307 0.259 0.169 0.146 0.000 0.206 0.415 0.217 0.233 9 0.090 0.146 0.394 0.201 0.135 0.101 0.209 0.206 0.000 0.405 0.119 0.101 10 0.452 0.286 0.537 0.397 0.341 0.439 0.505 0.415 0.405 0.000 0.328 0.344 11 0.138 0.135 0.394 0.175 0.098 0.143 0.267 0.217 0.119 0.328 0.000 0.071 12 0.111 0.138 0.389 0.198 0.079 0.159 0.275 0.233 0.101 0.344 0.071 0.000

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IF threshold analysis

  • How to determine the thresholds α, β?

– Initially, we used predefined pairs like (0.85; 0.15), (0.8; 0.2), (0.75; 0.25) … – Problem: when β = 1 – α, results are ‘incomparable’:

  • When (α, β) = (0.85; 0.25), α yields 2 pairs, β yields 19
  • When (α, β) = (0.80; 0.20), α yields 11 pairs, β yields 29
  • …
  • When (α, β) = (0.65; 0.35), α yields 39 pairs, β yields 51

– We need to find such α, β so that they yield similar

  • results. And if we fix α, what should β be?
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IF threshold analysis

  • How to determine the thresholds α, β?
  • Our problem needs reformulation:

If the decision maker wishes to focus

  • nly on the k most positively correlated criteria
  • ut of the totality of n criteria, what values
  • f the thresholds α and β shall he/she select?
  • In the EU Competitiveness case study, this problem

statement is in line with WEF’s address to policy- makers to “identify and strengthen the transformative forces that will drive future economic growth”

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IF threshold analysis

  • Proposed algorithm

µ 1 2 3 4 5 6 7 8 9 10 11 12 1 0.735 0.577 0.720 0.807 0.836 0.733 0.749 0.854 0.503 0.804 0.844 2 0.735 0.479 0.661 0.749 0.677 0.537 0.590 0.786 0.661 0.804 0.799 3 0.577 0.479 0.421 0.519 0.558 0.627 0.675 0.550 0.413 0.548 0.556 4 0.720 0.661 0.421 0.730 0.683 0.590 0.563 0.677 0.497 0.712 0.690 5 0.807 0.749 0.519 0.730 0.735 0.622 0.632 0.775 0.579 0.815 0.847 6 0.836 0.677 0.558 0.683 0.735 0.749 0.712 0.788 0.466 0.759 0.751 7 0.733 0.537 0.627 0.590 0.622 0.749 0.741 0.685 0.399 0.624 0.624 8 0.749 0.590 0.675 0.563 0.632 0.712 0.741 0.712 0.497 0.688 0.680 9 0.854 0.786 0.550 0.677 0.775 0.788 0.685 0.712 0.526 0.810 0.831 10 0.503 0.661 0.413 0.497 0.579 0.466 0.399 0.497 0.526 0.611 0.598 11 0.804 0.804 0.548 0.712 0.815 0.759 0.624 0.688 0.810 0.611 0.873 12 0.844 0.799 0.556 0.690 0.847 0.751 0.624 0.680 0.831 0.598 0.873

C i

1 0.854 2 0.804 3 0.675 4 0.730 5 0.847 6 0.836 7 0.749 8 0.749 9 0.854 10 0.661 11 0.873 12 0.873

,

max ( , )

i j j j i

C C μ

≠

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IF threshold analysis

  • Proposed algorithm

C i

1 0.854 2 0.804 3 0.675 4 0.730 5 0.847 6 0.836 7 0.749 8 0.749 9 0.854 10 0.661 11 0.873 12 0.873

,

max ( , )

i j j j i

C C μ

≠

Sort by value (descending for α )

C i

11 0.873 12 0.873 1 0.854 9 0.854 5 0.847 6 0.836 2 0.804 7 0.749 8 0.749 4 0.73 3 0.675 10 0.661

,

max ( , )

i j j j i

C C μ

≠

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IF threshold analysis

  • Proposed algorithm

C i

11 0.873 12 0.873 1 0.854 9 0.854 5 0.847 6 0.836 2 0.804 7 0.749 8 0.749 4 0.73 3 0.675 10 0.661

,

max ( , )

i j j j i

C C μ

≠

k = 5

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IF threshold analysis

  • Proposed algorithm

k = 5

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Future directions of research

  • Adaptive threshold analysis
  • Trend analysis and prediction
  • InterCriteria DM InterObject DM
  • Comparisons with other MCDM approaches
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Publications

  • Atanassov, K., D. Mavrov, V. Atanassova (2013). InterCriteria decision
  • making. A new approach for multicriteria decision making, based on index

matrices and intuitionistic fuzzy sets. Proc. of 12th IWIFSGN, Oct. 2013, Warsaw (in press).

  • Atanassova, V., D. Mavrov, L. Doukovska, K. Atanassov, (2014). Discussion
  • n the threshold values in the InterCriteria Decision Making Approach. Int.
  • J. Notes on Intuitionistic Fuzzy Sets, Vol. 20, No. 2, 94–99.
  • Atanassova, V., L. Doukovska, K. Atanassov, D. Mavrov (2014). InterCriteria

Decision Making Approach to EU Member States Competitiveness Analysis. 4th Int. Sym. BMSD, Luxembourg (in press).

  • Atanassova, V., L. Doukovska, K. Atanassov, D. Mavrov (2014). InterCriteria

Decision Making Approach to EU Member States Competitiveness Analysis: Temporal and Threshold Analysis. 7th Int. Conf. IEEE-IS, Warsaw, Poland,

  • Sept. 2014 (accepted).
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Thank you for your attention!

AComIn: Advanced Computing for Innovation

Vassia’s research and work visit are supported under the project AComIn "Advanced Computing for Innovation", Grant 316087, funded by the FP7 Capacity Programme