Fuzzy Logic

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FUZZY LOGIC

Ekin ERAYEmre GÖKYİĞİT

Management MathematicsAssoc. Prof. Dr. Gül Polat TATAR

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Content

• Introduction• History of Aristo Logic and Fuzzy logic• Stages of fuzzy modeling• An Academic example• Conclusion• References

ITU Graduate School of Science Engineering & Technology Construction Management - Fuzzy Logic

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• ARİSTO LOGIC

• FUZZY LOGIC


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Triangle Type Fuzzy Functions


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Trapeze Type Fuzzy Functions

a1 a4

A(x)

x

1

a2 a3


6ITU Graduate School of Science Engineering & Technology Construction Management - Fuzzy Logic

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Assingment of Membership Degree

• Intuition• Logic • Experience


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How to Create a Fuzzy Functions?

• Discuss with people who know about the subject and than make an arrangement

• Trial and error• Use the data directly and make arrangement.


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Fuzzy Set Operations

• μAUB(x) = maks {μA(x) , μB(x)}• μA∩B(x) = min {μA(x) , μB(x)}


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Stages of Fuzzy Modeling


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Rule Base

• AND A B = min (μA, μB) ∧• OR A B = maks (μA, μB) ∨


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Example

• The users of the heating system wants- Less fuel consumption- Easy to use- Inexpensive- More warranty period


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• There is;• 5 different company (A-E)• 4 different consumer needs


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1st need: FUEL

• Company A: Fuel consumption is good• Company B: Fuel concumption is high• Company C: Fuel consumption is low• Company D: Fuel consumption is normal• Company E: Fuel consumption is good


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2nd need: USAGE

• Company A: Usage is quite hard• Company B: Usage is quite easy• Company C: Usage is easy• Company D: Usage is easier• Company E: Usage is hard


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3th need: WARRANTY

• Company A: 7 years.• Company B: 8 years.• Company C: 5 years.• Company D: 6 years.• Company E: 8 years

edcbaG /8,0/6,0/5,0/8,0/7,0


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4th need: COST

• Company A: 40 M• Company B: 50 M• Company C: 60 M• Company D: 20 M• Company E: 45 M

0,8 0,8

0,6

0,55

0,4

0,5

c e

a d

Ü(x)

x

1,0

b


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• Intersections of all sets.

• We should choose the max mambership value from this set.

• Best heating system company is D.


edcbaG /8,0/6,0/5,0/8,0/7,0

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Most Common Defuzzification Process

• Maximum membership method• The center of gravity method• Weighted average method • Avarage maximum membership method.


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Mamdani Type Fuzzy Inference System

• All inputs and outputs are fuzzy functions. • can easily create • compatible with human behavior


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• Most comman defuzzification system for this model is the center of gravity method


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Sugeno Type Fuzzy Inference System

• Duductive part (THEN) of the system is a simple mathematical function of the premise part.

• It can be a constant or a linear function.• IF x=A AND y=B THEN z=f(x,y)=px+qy+r


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• ADVANTAGES• Easy to compute• Works well with the other techniques• suitable for mathematical analysis

• DISADVANTAGES• Not compatible with human behavior




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A Fuzzy Logic Implementation


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Thesis• Modeling Bid Mark-up of

International Construction Projects with Fuzzy Logic

Genç, A., 2012 Uluslararası İnşaat Projelerinde Katkı Payının Bulanık Mantık ile Modellenmesi, İTÜ Yapı İşletmesi.


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Goal of the Thesis• The goal of the survey is determining

importance levels of factors that affects amount of bid mark-up.

• The other goal is to create a fuzzy logic model to estimate amount of bid mark-up in the light of the obtained datas.


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Amount of Bid Mark-up

• Bid mark-up is a component of bidding price which is prepared by construction companies in bidding period.

Amount of bid mark-up. Genç, A., (2012)


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Factor Titles

• 61 factors which is contained with the literature survey are divided into 5 titles.

• 1-Factors associated with the employer• 2-Factors associated with the project• 3-Factors associated with the firm• 4-Factors associated with the bidding period

and the contract• 5-Factors associated with the economical milieu

and the riskITU Graduate School of Science Engineering & Technology Construction Management - Fuzzy Logic

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Survey

• 16 firms with 39 different project participated to survey.

• The questionnaire which is generally answered by bidding department managers, is provided a reliable database.


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Ranking

• The factors that affects amount of bid mark-up, ranked on an importance scale from 1 to 5 by the company representatives.

• 1: very low• 2: low• 3: medium• 4: high• 5: very high


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Fuzzy Logic Modeling

• “As complexity rises, precise statements lose meaning and meaningful statements lose precision” Lotfi Zadeh


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Fuzzy Logic Modeling

• Fuzzy logic modeling doesn’t need any acceptation contrary to statistical and stochastic processes. This is the most important advantage of fuzzy logic modeling.

• In order to create this kind of model, the

logical relations between input and output datas should be exposed.


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Method of Working

• 5 input data for the fuzzy logic model (Employer Factor, Project Factor, Firm Factor, Bidding Period and Contract Factors, Economical Milieu and Risk Factors)

• and an output data (Total estimated amount of bid mark-up which is estimated as a percentage of construction cost).


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Method of Working

• Mamdani type of fuzzy logic modeling method is used because of easily creating and its compatibility with the human behaviour and senses.

• MATLAB package program was used for creation of the model.


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Method of Working

Mamdani type of fuzzy logic model. Genç, A., (2012)


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Method of Working

• Fuzzification- First of all, fuzzification of input and output datas is required in order to create a fuzzy logic model.- The fuzzification comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets. The membership function is used to associate a grade to each linguistic term.


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Method of Working

• Forming fuzzy rule base- The second step is forming a fuzzy rule base.- The fuzzy rule base is formed by rules like IF-THAN which connects input variables to the output.


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Method of Working

• Fuzzy inference process- The relations that formed in fuzzy rule base between input and output fuzzy sets are collected in this process.- The process of fuzzy inference provides to get one result from the system- The form of the output is determined by all inferences of rules that are contained.


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Method of Working

• Defuzzification- Defuzzification is the final step of fuzzy logic modeling which converts fuzzy datas to precise results. - Defuzzification is the process of producing a quantifiable result in fuzzy logic, given fuzzy sets and corresponding membership degrees.


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Fuzzification

• Şen (2009) states that personal feelings, logic and experiences are prominent references for assigning degree of membership and membership functions. It’s quite helpful to overcome so many problems.

• Therefore, the assignments was made according to this principle. Input values assigned to membership functions and fuzzificated through expert opinions in this study.


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Fuzzification• In survey study, input datas ranked in 1-5 interval. According

to this scale: value 1 is “very low”, value 2 is “low”, value 3 is “medium”, value 4 is “high” and value 5 is “very high”.

• In model study, 5 levels are handled again for fuzzification of input datas. Input datas are fuzzificated with the same sets.

Membership function of input variable. Genç, A., (2012)


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Fuzzification

• Output variable(bid mark-up) is also fuzzificated into 5 level.• Triangle membership function is used at 4 set.• Trapeze membership function is used at 1 set.

Membership function of output variable. Genç, A., (2012)


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Fuzzification• “Very low” set: Full membership (μ=1) for 10 value. μ=0 for 17.5 value.• “Low” set: Parameters of the set are [13.5, 18.75, 24].• “Medium” set: Parameters of the set are [20.5, 25.75, 31].• “High” set: Parameters of the set are [27.5, 32.75, 38].• “Very high” set: Trapeze membership function. Starts from 35 value. μ=1 from 40 to 50

values.

Membership function of output variable. Genç, A., (2012)


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Creating Rule Base

• A systematic has developed by considering possible combinations. The rules are based on this systematic.

• 675 IF-THEN rule has written down for the model.


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Creating Rule Base

• The distribution of the rules by output fuzzy sets:

The distribution of the rules by output fuzzy sets. Genç, A., (2012)


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Creating Rule Base

• For an example, randomly selected 2 rules are shown.

• 1) IF “Employer Factors” is LOW and “Project Factors” is VERY LOW and “Firm Factors” is LOW and “Bidding Period – Contract Factors” is VERY LOW and “Economical Milieu – Risk Factors” is HIGH then “Bid Mark-up” is VERY LOW.

• 2) IF “Employer Factors” is HIGH and “Project Factors” is VERY HIGH and “Firm Factors” is VERY HIGH and “Bidding Period – Contract Factors” is LOW and “Economical Milieu – Risk Factors” is LOW then “Bid Mark-up” is HIGH.


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Fuzzy Inference Engine & Defuzzification

• In the light of the given datas fuzzy inference engine provides the calculations for finding the answers of the problems.

• Fuzzy Inference Engine applies intersection (MIN) or union (MAX) processes for input datas according to structure of rule.


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• Deffuzification is re-quantification of fuzzificated linguistic datas.

• The composite output fuzzy set is built by taking the union of all output fuzzy sets.

• Deffuzification is a kind of interpolation that requires approximate solution and smoothing a lot.

• The most commonly used defuzzification method for Mamdani type of fuzzy systems is center of gravity method.

• This method is also usable for asymmetrical membership functions.


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Overview of the model(MATLAB). Genç, A., (2012)


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Fuzzy Inference Motor & Defuzzification

Fuzzy inference system. Genç, A., (2012)

Employer=2.5 Project=2.5 Firm=2.5 İhale=2.5 Risk=2.5 Bid Mark-up=22.3


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The Results of Fuzzy Logic Model

Comparing the survey with the real system. Genç, A., (2012)


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Comparing the survey with the real system. Genç, A., (2012)


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• As a result of comparison of fuzzy logic model and estimated amount of bid mark-up, the mean absolute error is determined as 7.77%


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Comparison between model estimations and survey datas. Genç, A., (2012)


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Distribution graph of model estimations and survey datas. Genç, A., (2012)


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• R2 value which determined between data sets, is calculated as 0,916.

• This result tells us that there is a strong correlation between bid mark-up values which comes from the survey and estimated bid mark-up values which comes from fuzzy logic model.

• This point should be emphasized in terms of success of the model.


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Distribution of model and survey datas on x=y line. Genç, A., (2012)


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• x=y line shows R2=1 condition.

• In other words if the estimations of model were totally true, the blue dots would be on the line.


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Conclusion

• Fuzzy logic provides an alternative way to represent linguistic and subjective attributes of the real world in computing.

• It is compatible with human behavior and human logic.

• It doesn’t need a mathematical model for its applications.

• The software is easy and economical. • It’s easy to learn, flexible than other techniques.


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Conclusion

• Because of the complex and uncertain charachter of construction sector, fuzzy logic is a better solution for decision-making phase.


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References

• Şen, Z., 2009. Bulanık Mantık ilkeleri ve Modelleme, 3. Baskı, Su Vakfı Yayınları, İstanbul, Türkiye

• Genç,A., 2012, Uluslararası inşaat projelerinde katkı payının bulanık mantık ile modellenmesi, Yüksek Lisans Tezi, İstanbul Teknik Üniversitesi Fen Bilimleri Enstitüsü, İstanbul

• Subaşı, S., Beycicioğlu, A., Emiroğlu,M., 2008, Hafif betonlarda donatı aderansı dayanımının bulanık mantık yöntemiyle modellenmesi, Yapı Eğitimi Bölümü Teknik Eğitim Fakültesi Düzce Üniversitesi, DÜZCE

• Sarı, M., Murat, Y.Ş., Kırabalı, M., Fuzzy modeling approach and applications


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Thank You.


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Fuzzy Logic