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IMAT5119 Fuzzy Logic - De Montfort University

Assignment - Theoretical

Learning outcome 1: Critically evaluate fuzzy logic approaches to solve computational problems exhibiting uncertainty and imprecision.

Learning outcome 2: Have a comprehensive understanding of the successful application of fuzzy logic to several problem domains and be capable of judging whether the fuzzy paradigm might be fruitful in a novel situation.

Question 1 - Fuzzy sets

A fuzzy set A in X (classical set of objects, called the universe, whose generic elements are denoted x) is a set of ordered pairs:

A=\{(x,μ_A (x))|x∈X,μ_A (x)∈[0,1]\}.

The α-cut (α∈[0,1]) of a fuzzy set A is the ordinary set

A_α=\{x∈X|μ_A (x)≥α\}.

The strong α-cut (α∈[0,1]) of a fuzzy set A is the ordinary set

A_(α^+ )=\{x∈X|μ_A (x)>α\}.

The set of all distinct numbers in [0, 1] that are employed as membership grades of the elements of X in A is called the level set of A, denoted by L(A).

Assume minimum and maximum operators for the intersection and union of fuzzy sets. Answer the following:

Given any two fuzzy sets A and B, prove that the following properties hold:

(A∪B)_α=A_α∪B_α and (A ∩B)_α=A_α∩B_α.

The support of A, denoted supp(A), is defined as the set of elements of X that have nonzero membership in A. The core of A, denoted core(A), is defined as the set of elements of X that have membership in A equal to 1.

How do supp(A) and core(A) relate to the α-cuts and the strong α-cuts of A?

Given the discrete fuzzy sets A=\{(0.2,x_1 ),(0.4,x_2 ),(0.6,x_3 ),(0.8,x_4 ),(1,x_5)\}, obtain L(A), and provide all the distinct α-cuts of A.

What is the relationship between A_(α_1 ) and A_(α_2 ) when α_1< α_2?

The membership function of A can be expressed in terms of the characteristic functions of its α-cuts according to the formula:

μ_A (x)=sup-(α∈[0,1] ) α⋅μ_(A_α ) (x) where μ_(A_α ) (x)={(1 iff x∈A_α@0 otherwise)¦

sup means superior (the maximum value of those obtained when multiplying the distinct values of α in the level set L(A) with 1 or 0 - the value of μ_(A_α ) (x) - depending on whether an x value belongs or not to the alpha-level set A_α).

In the case of a discrete fuzzy set we have α∈L(A). Show that this is true for the discrete fuzzy set given in c).

Question 2 - Decision making in a fuzzy environment

Fuzziness can be introduced at several points in the existing models of decision making.

Bellman and Zadeh in 1970 suggested a fuzzy model of decisions that must accommodate certain constraints C and goals G. Provide a description of this model.

Suppose we must choose one of four different jobs a, b, c, and d, the salaries of which are given by the function f such that:

f(a)=30,000,f(b)=25,000,f(c)=20,000 and f(d)=15,000.

Our goal is to choose the job that will give us a high salary given the constraints that the job is interesting and within close driving distance.

The first constraint of interest value is represented by the fuzzy set

C_1=\{(0.4,a),(0.6,b),(0.8,c),(0.6,d)\}.

The second constraint concerning the driving distance to each job is defined by the fuzzy set
C_2=\{(0.1,a),(0.9,b),(0.7,c),(1,d)\}.

The fuzzy goal G of a high salary is defined by the membership function

μ_G (x)={(0 for x<13,[email protected](x/1000-40)^2+1 for 13,000≤x≤40,000@1 for x>40,000)¦

Which is the best job when applying Bellman and Zadeh's fuzzy decision model?

Question 3 - ANFIS

Consider the IRIS data seen in the practical exercises. Explain how data such as this can be used to generate a fuzzy system using ANFIS. Illustrate your answer with examples from the lab work you did using ANFIS with the IRIS data. Include in your answer a brief discussion of when it is appropriate to use ANFIS to build a fuzzy system and identify some of the decisions that need to be made in order to generate a useful system.

Question 4 - Type 2 fuzzy logic

Write a short essay (about 2 pages not including bibliography & references) to discuss why there is a perceived need for the type-2 fuzzy logic and consider the areas of application where type-2 fuzzy sets might provide a solution. Your answer should include a brief definition of a type-2 fuzzy set (illustrate with diagram(s) and example(s)). It should also make reference to issues associated with the practical implementation of type-2 fuzzy logic and to the approaches that have been developed.

Attachment:- Fuzzy logic course.rar