Theory and Applications FUZZY SETS AND FUZZY LOGIC Theory and Applications PART 4 Fuzzy Arithmetic 1. Fuzzy numbers 2. Linguistic variables 3. Operations on intervals 4. Operations on fuzzy numbers 5. Lattice of fuzzy numbers 6. Fuzzy equations

Fuzzy numbers Three properties A is a fuzzy set on R. A must be a normal fuzzy set; αA must be a closed interval for every the support of A, 0+A, must be bounded. A is a fuzzy set on R.

Fuzzy numbers

Fuzzy numbers Theorem 4.1 Let Then, A is a fuzzy number if and only if there exists a closed interval such that

Fuzzy numbers Theorem 4.1 (cont.) where is a function from that is monotonic increasing, continuous from the right, and such that ; is a function from that is monotonic decreasing, continuous from the left, and such that

Fuzzy numbers

Fuzzy numbers

Fuzzy numbers Fuzzy cardinality Given a fuzzy set A defined on a finite universal set X, its fuzzy cardinality, , is a fuzzy number defined on N by the formula for all

Linguistic variables The concept of a fuzzy number plays a fundamental role in formulating quantitative fuzzy variables. The fuzzy numbers represent linguistic concepts, such as very small, small, medium, and so on, as interpreted in a particular context, the resulting constructs are usually called linguistic variables.

Linguistic variables base variable Each linguistic variable the states of which are expressed by linguistic terms interpreted as specific fuzzy numbers is defined in terms of a base variable, the values of which are real numbers within a specific range. A base variable is a variable in the classical sense, exemplified by any physical variable (e.g., temperature, etc.) as well as any other numerical variable, (e.g., age, probability, etc.).

Linguistic variables Each linguistic variable is fully characterized by a quintuple (v, T, X, g, m). v : the name of the variable. T : the set of linguistic terms of v that refer to a base variable whose values range over a universal set X. g : a syntactic rule (a grammar) for generating linguistic terms. m : a semantic rule that assigns to each linguistic term t T.

Linguistic variables

Operations on intervals Let ＊ denote any of the four arithmetic operations on closed intervals: addition ＋, subtraction —, multiplication • , and division /. Then,

Operations on intervals Properties Let

Operations on intervals

Operations on fuzzy numbers First method Let A and B denote fuzzy numbers. ＊ denote any of the four basic arithmetic operations. for any Since is a closed interval for each and A, B are fuzzy numbers, is also a fuzzy number.

Operations on fuzzy numbers Second method

Operations on fuzzy numbers

Operations on fuzzy numbers

Operations on fuzzy numbers Theorem 4.2 Let ＊ {＋, －, •, / }, and let A, B denote continuous fuzzy numbers. Then, the fuzzy set A＊B defined by is a continuous fuzzy number.

Lattice of fuzzy numbers MIN and MAX

Lattice of fuzzy numbers

Lattice of fuzzy numbers

Lattice of fuzzy numbers Theorem 4.3 Let MIN and MAX be binary operations on R. Then, for any , the following properties hold:

Lattice of fuzzy numbers

Lattice of fuzzy numbers It also can be expressed as the pair , where is a partial ordering defined as:

Lattice of fuzzy numbers

Fuzzy equations A + X = B The difficulty of solving this fuzzy equation is caused by the fact that X = B－A is not the solution. Let A = [a1, a2] and B = [b1, b2] be two closed intervals, which may be viewed as special fuzzy numbers. B－A = [b1－ a2 , b2 －a1], then

Fuzzy equations Let X = [x1, x2].

Fuzzy equations Let αA = [αa1, αa2], αB = [αb1, αb2], and αX = [αx1, αx2] for any .

Fuzzy equations A．X = B A, B are fuzzy numbers on R+. It’s easy to show that X = B / A is not a solution of the equation.

Exercise 4 4.1 4.2 4.5 4.6 4.9