InterCriteria Decision Making using Intuitionistic Fuzzy Sets - PowerPoint PPT Presentation

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 Innovation http://www.iict.bas.bg/acomin

Contents • Specific problem statement • General problem statement • Proposed approach – Intuitionistic fuzzy sets • Applications – EU Competitiveness analysis • IF threshold analysis • Future directions of research • Publications 2 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Specific problem formulation • 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? 3 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

General problem formulation • 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). 4 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Proposed approach • InterCriteria Decision Making, based on: – Index matrices – Intuitionistic fuzzy sets c ... c ... c 1 s n = IM − o e ... e ... e m m ( 1) pairs of objects 1 1,1 1, s 1, s o o , , i j 2 ... ... ... ... ... ... − m m ( 1) o e ... e ... e i.e. termwise m m ,1 m s , m s , 2 comparisons between and e e i j 5 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Proposed approach • InterCriteria Decision Making, based on: – Index matrices – Intuitionistic fuzzy sets × Warning Want to know more about IFS? Yes No Cancel 6 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • 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 other FS extensions (like L-fuzzy sets, rough sets, interval valued fuzzy sets) 7 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • Formal definition: Let us have a set A in the universum X and two mappings μ A ( x ) and ν A ( x ), so that μ A ( x ): A → [0, 1], ν A ( x ): A → [0, 1] 0 ≤ μ A ( x ) + ν A ( x ) ≤ 1 Then, the set A = { 〈 x , μ A ( x ), ν A ( x ) 〉 | x ∈ X } is called intuitionistic fuzzy set. • Boundary conditions: μ A ( x ) = 0, ν A ( x ) = 1 Complete non-membership (falsity) μ A ( x ) = 1, ν A ( x ) = 0 Complete membership (truth) 8 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets Intuitionistic fuzzy set Fuzzy set 0 1 0 1 μ ∈ [0; 1], ν ∈ [0; 1] μ ∈ [0; 1] μ + ν ∈ [0; 1] π = 1 − μ − ν 0.15% NON-MEMBERSHIP UNCERTAINTY 0.7% 0.7% MEMBERSHIP MEMBERSHIP 9 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • Graphical interpretations: Linear ν Standard μ 0 1 Modified zero uncertainty high uncertainty 1 - ν μ 0 10 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • Graphical interpretations: Triangular (0,1) Elements of a FS Elements of an IFS x ν A ( x ) (0,0) (1,0) μ A ( x ) π A ( x ) ν A ( x ) 11 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • Graphical interpretations: Triangular (0,1) IFS specific modal operators! Necessity: □ A = { 〈 x , μ A ( x ), 1 − μ A ( x ) 〉 | x ∈ E } Possibility: ◊ A = { 〈 x , 1 − ν A ( x ), ν A ( x ) 〉 | x ∈ E } □ x x ◊ x (0,0) (1,0) μ A ( x ) π A ( x ) ν A ( x ) 12 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • Operations over IFSs Intersection A ∩ B = { 〈 x , min( μ A ( x ), μ B ( x )), max( ν A ( x ), ν B ( x )) 〉 | x ∈ E } Union A ∪ B = { 〈 x , max( μ A ( x ), μ B ( x )), min( ν A ( x ), ν B ( x )) 〉 | x ∈ E } Multiplication A . B = { 〈 x , μ A ( x ). μ B ( x ), ν A ( x )+ ν B ( x ) − ν A ( x ). ν B ( x ) 〉 | x ∈ E } Addition A + B = { 〈 x , μ A ( x )+ μ B ( x ) − μ A ( x ). μ B ( x ), ν A ( x ). ν B ( x )) 〉 | x ∈ E } 13 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • Relations over IFSs Inclusion A ⊂ B iff ( ∀ x ∈ E ) ( μ A ( x ) ≤ μ B ( x ) & ν A ( x ) ≥ ν B ( x ) ) Equality ( ∀ x ∈ E ) ( μ A ( x ) = μ B ( x ) & ν A ( x ) = ν B ( x ) ) A = B iff 14 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • Topological operators over IFSs Closure C ( A ) = { 〈 x , sup μ A ( y ), inf ν A ( y ) 〉 | x ∈ E , y ∈ E } Interior I ( A ) = { 〈 x , inf μ A ( y ), sup ν A ( y ) 〉 | x ∈ E , y ∈ E } 15 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • Extensions of IFSs IFSs of Type 2 , for which 0 ≤ μ A ( x ) + ν A ( x ) ≤ 1 changes to 0 ≤ μ A ( x ) 2 + ν A ( x ) 2 ≤ 1 and IFSs of Type n : 0 ≤ μ A ( x ) n + ν A ( x ) n ≤ 1 16 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Intuitionistic fuzzy sets • 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 17 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Proposed approach • InterCriteria Decision Making, based on – Index matrices – Intuitionistic fuzzy sets > ⎧ ⎫ ⎪ ⎪ < e e ⎨ ⎬ i s , j s , ⎪ ⎪ = ⎩ ⎭ If R ( e i,s ; e j,s ) is “>”, then µ ++ (membership) μ ν π ∈ , , [0;1] If R ( e i,s ; e j,s ) is “<”, then ν ++ (non-membership) s s s Otherwise, π ++ (uncertainty) 18 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Proposed approach • 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 19 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Proposed approach • 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. c ... c ... c c ... c ... c 1 i n 1 i n = = IM , IM μ μ μ μ ν ν ν ν c ... ... c ... ... 1 1,1 1, i 1, n 1 1,1 1, i 1, n ... ... ... ... ... ... ... ... ... ... ... ... μ μ μ ν ν ν c ... ... c ... ... i i ,1 i n , i n , i i ,1 i n , i n , ... ... ... ... ... ... ... ... ... ... . .. ... μ μ μ ν ν ν c ... ... c ... ... n n ,1 n i , n n , n n ,1 n i , n n , 20 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg

Applications • So far the approach is tested with: which is – Petrochemical data for raw mineral oil actually good! • Results outline all the previously known correlations between criteria – Biomedical data about temperature curves of patients with multiplen myelom and carcinome not yet • Results outline yet unknown relations, but the starting ready to boast with data are yet scarce, work needed on problem formulation – Data from the World Economic Forum’s Global Competitiveness Reports (2008-2014) encouraging! • AFAWK, completely new and reliable results 21 16-Jun-14 AComIn : Advanced Computing for Innovation http://www.iict.bas.bg