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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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