Reference no: EM133315909

Question 1 - Representation of 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 fuzzy set A is the ordinary set A_α={x∈X|μ_A (x)≥α}.
The strong α-cut (α∈[0,1] ) of 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).
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.
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_(Aα)(x) where μ_(Aα) (x) = {(1 if x∈A_α     0 otherwise) (*)

In the case of a discrete fuzzy set we have α∈L(A). Notice that sup in (*) means superior (for discrete fuzzy sets superior becomes 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(_Aα) (x) - depending on whether an x value belongs or not to the alpha-level set A_α).

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

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

(AUB)_α = A_αUB_α and (A∩B)_α=A_α∩B_α.

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

c) For the discrete fuzzy sets A={(x₁,0.2),(x₂,0.4),(x₃,0.6),(x₄,0.8),(x₅,1)}, obtain L(A), andprovide all the distinct α-cutsof A.

d) Show that formula (*) is true for the discrete fuzzy set of c).

Question 2 - Fuzzy sets intersection, union and complement

Consider the two discrete fuzzy sets:

A={(0,0),(1,0.2),(2,0.25),(3,(1/3)),(4,0.5),(5,0.6),(6,(2/3)),(7,1) }
¬A={(0,1),(1,(2/3)),(2,0.6),(3,0.5),(4,(1/3)),(5,0.25),(6,0.2),(7,0) }

In some sense the fuzzy sets A and A are complementary. However, the operator ¬ that has been used to make ¬A is obviously notthe negation operator of Zadeh, N(x) = 1 - x.

a) Prove whether the operator ¬ that has been used to make ¬A from Aobeys the law of "double negation", i.e: whether ¬(¬A)=A.

b) Determine the fuzzy sets ¬A∩A and ¬A∪A using the original definitions for the intersection (∩) and union (∪) operations proposed by L.A. Zadeh in 1965.

c) What should be the answers of ¬A∩A and ¬A∪A in case A and ¬A were classical sets. Explain the difference in both results.

Question 3 - 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.

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

C₁ = {(a,0.4),(b,0.6),(c,0.8),(d,0.6)}.

The second constraint concerning the driving distance to each job is defined by the fuzzy set
C₂ =  {(a,0.1),(b,0.9),(c,0.7),(d,1)}.

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

μ_G (x)={(0 for x<13,000            @-0.00125(x/1000-40)²+1 for 13,000≤x≤40,000               1 for x>40,000)

a) Provide a description of Bellman and Zadeh fuzzy model of decision making.

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

Question 4 - Type 2 defuzzification

Write a short essay (about 2 pages not including bibliography & references) discussing the issue of computational complexity in relation to the defuzzification of type-2 fuzzy sets. Your answer should include definitions of computational complexity and type-2 defuzzification and an explanation of the concept of the type-2 embedded set (illustrated diagrammatically). It should also explain the strategy of Exhaustive Defuzzification, and why embedded sets give rise to the issue of computational complexity for Exhaustive Defuzzification. Choose a type-2 defuzzification method developed at De Montfort University (other than the Exhaustive Method), and briefly present the technique, using algorithmic notation, mathematical notation, and/or a diagram as appropriate. Lastly, if you were developing a type-2 FIS, would you use the method you have chosen? Explain your reasons.