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Introduction to Fuzzy Logic & Intuitionistic Fuzzy Logic

Seminar 2014

Andreas Meier and Roland Schütze University of Fribourg

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Table of Contents

  What is Fuzzy Logic?   Fuzzy versus Sharp Sets   Fuzzy Classification of Online Customers   What is Intuitionistic Fuzzy Logic?   Fuzzy Logic versus Intuitionistic Fuzzy Logic   Pros and Cons   Research Center FMsquare (FMM = Fuzzy

Management Methods)

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What is Fuzziness

  Fuzziness is a concept of human thinking and speaking (linguistic)

  Fuzziness deals with subjectivity and vague concepts (all language is vague)

  Fuzzy sets and fuzzy logic express the imprecision of human thinking and behavior (by appropriate mathematical tools)

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Fuzzy Sets

  A fuzzy set is always built from a reference set called ‘universe of discourse’ (this reference set is never fuzzy)

  Suppose that X = {x1,x2,…,xn} is the universe of discourse, then a fuzzy set A in X (A ⊂ X) is defined as a set of ordered pairs {(xi, µA(xi))}, where xi∈X and µA: X→[0,1] is the membership function of A

Lotfi A. Zadeh, University of California, Berkeley, 1965

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Fuzzy versus Sharp Sets

Teenager

1

Age

µ

Age

Teenager

0

10 13 19 22

13 19

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Operations on Fuzzy Sets

  The complement of a fuzzy set A in X is

µ¬A(x) = 1 - µA(x), ∀x ∈ X

  The intersection of two fuzzy sets A and B in X is

µA∩B(x) = min(µA(x), µB(x))

  The union of two fuzzy sets A and B in X is µA∪B(x) = max(µA(x), µB(x))

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Fuzzy Logic

  In classical logic, a statement is either true or false

  Fuzzy logic consists of statements which have a degree of truth between 1 and 0

  For an element e, a fuzzy proposition ‘e is P’ is defined by a fuzzy set P

  Example: The fuzzy proposition ‘Mary is Teenager’ is defined by the fuzzy set Teenager on the domain of the variable Age

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Why Applying Fuzzy Logic?

  Fuzzy logic facilitates common sense reasoning

  Fuzzy logic deals with imprecise or vague propositions

  Fuzzy logic can serve as a basis for decision support

  Fuzzy logic can be applied for managerial analysis and control …

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Fuzzy vs. Sharp Classification

Managing customers as an asset requires measuring and treating them according to their real value (customer capital):   A sharp classification cannot asses customers

thoroughly as every customer of a class is treated the same way

  The membership degrees of a customer can determine the privileges this customer deserves

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Sharp Classification

in advance on time

1000

500

499

0

D(Payment Behavior) behind time too late

attractive payment behaviour

non-attractive payment behaviour

high turnover

low turnover

Smith:

C1: 100%

Brown:

C1: 100%

Ford:

C4: 100%

Miller:

C4: 100%

C1 C2

C4 C3

Brown

Ford

Miller

Smith

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Drawbacks for Customers

  Customer Brown has no advantage from improving his turnover or his behavior

  Brown will be surprised and disappointed if his turnover or behavior decreases slightly

  Customer Ford, potentially a good customer, may find opportunities elsewhere

  Although Smith belongs to the premium class, he is not treated according to his real value

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Fuzzy Classification

0 in

advance on time

1000

500

499

0

D(Payment Behavior)

D(Turnover)

µ high

µ low

1 µ attractive µ non-attractive

behind time too late

0.33 0.66

C1 C2

C4 C3

Brown

Ford

Miller

Smith Smith:

C1:100; C2:0; C3:0; C4:0

Brown:

C1:35; C2:17; C3:32; C4:16

Ford:

C1:16; C2:32; C3:17; C4:35

Miller:

C1:0; C2:0; C3:0; C4:100

1 0

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Advantages of Fuzzy Classes

  Similar customers can be treated similarly   The neighbors Brown and Ford receive similar

aggregated membership values (customer values)

  Although Smith and Brown belong to the top class, their memberships values are different

  Ford has interesting perspectives although he belongs to the looser class

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Mass Customization

  With a fuzzy classification, customization and personalization can be easily realized

  Personalized discount example:   Discount rates can be associated with each fuzzy class,

e.g. C1: 10%, C2: 5%, C3: 3%, C4: 0%   The individual discount of a customer can be calculated

as the aggregation of the discount of the classes he belongs to, in proportion of his membership degrees in the classes

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Personalized Discount

C1 10%

Brown

Ford

Miller

Smith Smith:

C1:100; C2:0; C3:0; C4:0

Brown:

C1:35; C2:17; C3:32; C4:16

Ford:

C1:16; C2:32; C3:17; C4:35

Miller:

C1:0; C2:0; C3:0; C4:100

  Smith: 1 * 10% + 0 * 5% + 0 * 3% + 0 * 0% = 10%   Brown: 0.35 * 10% + 0.17 * 5% + 0.32 * 3% + 0.16 * 0% = 5.3%   Ford: 0.16 * 10% + 0.32 * 5% + 0.17 * 3% + 0.35 * 0% = 3.7%   Miller: 0 * 10% + 0 * 5% + 0 * 3% + 1 * 0% = 0%

C3 3%

C2 5%

C4 0%

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Marketing Campaigns

Launching a marketing campaign can be very expensive:   How can we select the most appropriate

customers?   How can we measure the success of the

campaign?   How can we control the improvement of the

target group?

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Example

test group

100 50 49 0 1000

500

499

0

D(Turnover)

D(Loyalty)

µ high

µ low

1 0 0

1 µ positive µ negative

C1

Commit Customer

C2

Improve

Loyalty

C4

Don’t Invest

C3 Augment Turnover

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Customer’s Evolution

With a fuzzy classification, there is the possibility of monitoring the customers through the classes:

  Detect customers who are   Improving   Maintaining   Decreasing

  Avoid customer churning

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Example

1000

500

499

0

D(Turnover)

C1 C2

C4 C3 Brown

03/2006

100 50 49 0 D(Loyalty)

06/2006

09/2006

12/2006

03/2007

06/2007

µ high

µ low

1 0 0

1 µ positive µ negative

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Hierarchical Decomposition

A complex classification can be decomposed into a hierarchy of fuzzy classifications:   Keep a small number of resulting classes with

precise semantics   Derive new concepts expressing higher

semantics   Reduce the complexity of the initial problem

allowing a better definition and optimization

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Example

Profitability Loyalty

Gain

Margin Turnover

Service costs

Return rate

Payment delay

Attachment

Involvement frequency

Visiting frequency

Repurchases

Customer Lifetime Value

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Many-Valued & Fuzzy Logic

  classical logic: false   3-valued logic: ½ true   m-valued logic: 7/10 true for m=11   fuzzy logic: 0.7 true

Red ate all

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Intuitionistic Fuzzy Set

  Suppose that X = {x1,x2,…,xn} is the universe of discourse, then a intuitionistic fuzzy set A in X (A ⊂ X) is defined as a set of ordered triples {(xi, µA(xi), νA(xi))}, where xi∈X and µA: X→[0,1] is the membership function of A and νA: X→[0,1] is the non-membership function of A and 0 ≤ µA(xi) + νA(xi) ≤ 1holds.

Krassimir T. Atanassov, Bulgarian Academy of Science, Sofia, 1983

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Fuzzy Logic versus Intuitionistic Fuzzy Logic

0 1

µA(x) 1 - µA(x)

0 1

µA(x) νA(x)

πA(x)

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Example

  fuzzy logic: 0.7 true and 0.3 false

  intuitionistic fuzzy logic: 0.7 true and 0.2 false and 0.1 uncertain

Red ate all

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Conclusion

  to deal with imprecision and vague data …   to better compute with words …   to become closer to human thinking …   to work with linguistic variables and terms …   to include quantitative and qualitative

concepts …   to differentiate managerial decisions …

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Research Center for Fuzzy Management Methods (FMsquare)

www.FMsquare.org

Fuzzy Community

Building

Luis Téran

Fuzzy Reputation

Management

Edy Portmann

Fuzzy Data

Warehousing

Daniel Fasel

Fuzzy Prediction

Michael Kaufmann

Fuzzy Classification of Customers

Nicolas Werro

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Research Center for Fuzzy Management Methods (FMsquare)

www.FMsquare.org

Fuzzy Social Networks

Aleksandar Drobnjak

Fuzzy-Based Filtering of Products

Aigul Kaskina

Fuzzy-Based Service Level Management

Roland Schütze

Fuzzy Recommender

Systems

Luis Teran

Semantic Web

Monitoring

Marcel Wehrle

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Case Study @ PostFinance

Customer Data Warehouse

target group

selection

customer scoring

for product affinity

mapping customer to

advertisement message

eFinance online

individual advertisement

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Inductive Fuzzy Classification

Y=1 Y=0 Product Selling Ratio

1: Fuzzy classification 31 4939 0.63% 2: Crisp classification 15 5037 0.30% 3: Random selection 10 5016 0.20%

  An online advertisement for investment funds was shown to three different target groups

  The customers in the group defined by an inductive fuzzy classification had the highest product selling ratio

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Literature International Book Series

IGI Global, 2012

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FMsquare & Startups

•  Fasel D.: Concept and Implementation of a Fuzzy Data Warehouse. PhD Thesis, University of Fribourg, May 2012

•  Kaufmann M.: Inductive Fuzzy Classification in Marketing Analytics. PhD Thesis, University of Fribourg, May 2012

•  Portmann E.: The FORA Framework – A Fuzzy Grassroots Ontology for Online Reputation Management. PhD Thesis, University of Fribourg, February 2012

•  Teran L.: SmartParticipation – A Fuzzy-Based Recommender System for Political Community Building, January 2014

•  Werro N.: Fuzzy Classification of Online Customers. PhD Thesis University of Fribourg, May 2008

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