The family of such functions is denoted by . A fuzzy number is said to be an LR fuzzy number if its membership function is constructed with the aid of some L and R membership functions. More precisely, a two-parameter modification of the L-type of membership function applies to all x ≤ m while the R membership function contributes to the definition of A for all x greater or m. This yields

where L, R ∈ . To explicitly articulate the parameters of the functions, we use an extended notation such as A(x; α, m, β)_LR. In this notation “m” is normally referred to as the modal value of A whereas α and β form the spreads of the LR fuzzy number.

Several selected examples of the discussed family of the membership functions are listed below,

p ≥ 0.

The membership functions of the fuzzy numbers can be derived by putting together some of the above functions. For example, one has

see also Fig. 4.1 that visualizes this fuzzy number for some parameters of the membership function.

Figure 4.1  Membership function of A for m= 2.0, α =1.0, β = 2.0 and p =1.0

4.3.2. Computing with fuzzy numbers

The calculations on fuzzy numbers rely on the extension principle (Mizumoto and Tanaka, 1976). Consider a function F transforming two fuzzy numbers A and B and producing the third number C such that

The extension principle determines the membership function of C to be

z ∈ R; f: R² → R is a real function that is pointwise consistent with F meaning that F( {x}, {y}) = f(x, y). As usual, {x} describes a single-element set.

Quite often, as implied by practice, it is enough to concentrate on a family of triangular fuzzy numbers. These are the simplest models of uncertain numerical quantities. The modal value represents an element that is the most representative to the concept. The lower and upper bound identify the region that is of interest to the discussed notion; all the elements situated beyond these boundaries are perceived as irrelevant to the linguistic concept. The analysis focusing on this class of membership functions helps reveal the most profound properties of fuzzy arithmetic. Consider two triangular numbers A = (x; a, m, b) and B = (x; c, n, d). More specifically, their membership functions are defined through the following piecewise linear relationships

Each modal value (m and n, respectively) identifies a dominant (typical) value of the corresponding quantity whereas the lower and upper bounds (a or c and b or d) reflect the spread of the concept.

Let us start with the addition operation. The sup-min composition applied to A and B yields

The resulting fuzzy number is normal; that is C(z) = 1 for z = m+n. The computations of the spreads of C are dealt with separately:

(i)  Consider first that z < m + n. In this situation the calculations involve the increasing parts of the membership functions of A and B. There exist two elements x and y such that x < m and y < n satisfying the relationship

ω ∈ [0, 1]

The linearly increasing sections of the membership functions A and B, see Fig. 4.2, embrace the interval [a + c, m + n].

Figure 4.2  Addition of triangular fuzzy numbers

Based on that we derive

along with

x, y ∈ [a+c, m+n]. Expressing x and y as functions of ω one instantaneously derives

and

Furthermore, owing to the underlying addition operation, (z = x + y), we obtain

We derive a similar relationship for the decreasing portions of the membership functions. The formulas below apply to x and y situated in [m + n, b + d]

or equivalently

Adding x and y gives

Summarizing, the membership function of A ⊕ B equals

Interestingly enough, C exhibits also a triangular membership function. To emphasize this fact, we use a concise notation

In general, by adding triangular fuzzy numbers, the result becomes a fuzzy set of the same shape. More precisely, if A_i = (x; a_i, m_i, b_i), i=1, 2, ..., n, then the sum of these numbers equals

One can observe that the spreads of the result start accumulating. For instance, let m_i = 1, a_i = 0.95, b_i =1.05 for all i = 1, 2, ... n. Iterating the addition operation, one derives a sequence of fuzzy numbers

In virtue of the fundamental representation theorem, each fuzzy number can be regarded as a family of nested α-cuts. Subsequently, these α- cuts can be utilized to reconstruct the resulting fuzzy number. For instance, we get

Generally, the use of α-cuts forms a sort of brute force method of carrying out computing with fuzzy quantities.

Let us now perform a multiplication of two triangular fuzzy numbers. As before, we look first into the increasing parts of the membership functions expressed as

Then their product becomes

If ac ≤ z ≤ mn then the membership function of C is an inverse of F₁

Similarly, consider the decreasing parts of A and B meaning that mn ≤ z ≤ bd

As before, for any z in [mn, bd] we derive

Evidently, the multiplication does not return a piecewise linear membership function. Instead, as clearly indicated by the results of the above computations, we produce a quadratic form of the resulting fuzzy number.

The linearization of the membership functions can be treated as an approximation of the previously derived membership function. We require that this linearization consists of two linear functions coinciding with C at z = mn, ac, and bd. The quality of this linearization depends very much upon the spreads of the fuzzy numbers. To quantify the resulting linearization error, let us consider that m = n = 1 and

The integral of error

is of interest here while C∗ is a piecewise linear approximation of the original fuzzy number (C).

Assuming that δ tends to zero, one can accept the linear approximation of the form