Critically evaluate fuzzy logic approaches to solve computational problems exhibiting uncertainty and imprecision.

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,?_A (x))|x?X,?_A (x)?[0,1]}.

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

A_?={x?X|?_A (x)??}.

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

A_(?^+ )={x?X|?_A (x)>?}.

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)_?=A_??B_? and (A ?B)_?=A_??B_?.

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 ?-cuts and the strong ?-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 ?-cuts of A.

What is the relationship between A_(?_1 ) and A_(?_2 ) when ?_1< ?_2?

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

?_A (x)=sup-(??[0,1] ) ???_(A_? ) (x) where ?_(A_? ) (x)={(1 iff x?A_?@0 otherwise)

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

In the case of a discrete fuzzy set we have ??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

?_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)

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