Properties of Unimodal Membership Functions in Operations with

c Allerton Press, Inc., 2007.
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2007, Vol. 51, No. 3, pp. 73–76. c I.V. Germashev and V.E. Derbisher, 2007, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2007, No. 3, pp. 77–80.
Original Russian Text Properties of Unimodal Membership Functions
in Operations with Fuzzy Sets
I. V. Germashev* and V. E. Derbisher**
Volgograd State Technical University, pr. Lenina 28, Volgograd, 400131 Russia
Received May 10, 2006
DOI: 10.3103/S1066369X07030115
Mathematical problems with fuzzy conditions often arise in various domains of science and engineering. Examples of such problems are adduced in [1]–[2]. An important class of problems of this type is
connected with ranking or classiļ¬cation of elements of a set with a view to substantiate the decisionmaking. According to the approach described below, the main object in the solution of problems with
fuzzy conditions is a fuzzy set; we compare fuzzy sets, using the similarity indices [3]. We calculate the
indices with the help of membership functions. Let us consider unimodal membership functions.
1 is a
1 = ( (x, f (x)) a fuzzy set over X, where G = supp Q
Let X be some universe. Denote by Q
1 f (x) is a membership function (f : X → [0; 1]).
carrier of Q,
Solving problems from the mentioned class, one imposes several constraints on the class of membership functions:
f (q) = sup f (x),
f (x) ≥ h ⇔ q − δ ≤ x ≤ q + δ, δ > 0,
h = f (q − δ) = f (q + δ).
The proper choice of a membership function is a pivotal theoretical condition for the problem solution.
Let us concretize and simplify the problem, considering a more narrow class than that of functions (1).
Let f : X → [0; 1], X ⊆ R, f ∈ U (X),
∃a, h, δ ∈ R : f (a) = max f (x) = 1,
f (x) ≥ h ⇔ a − δ ≤ x ≤ a + δ,
h = f (a − δ) = f (a + δ),
then we call the function f (x) regular on the set X, and denote the class of such functions by ((X).
Here U (X) stands for the class of strictly unimodal functions ([4], p. 13). Let us study regular functions
of the form y = g(ξ(x)), where ξ(x) = b2 (x − a )2 + c; to this end, below we formulate and prove several
assertions about the properties of membership functions.
Theorem 1. Assume that fi ∈ ((X), i = 1, 2, X is compact, νi = max min (fi (x)), x1 =
x∈X i=1,2
arg max min (fi (x)), then f1 (x1 ) = f2 (x1 ).
x∈X i=1,2
Lemma 1. Let f ∈ ((X), f (x) = g(ξ(x)), where ξ(x) = b2 (x − a)2 + c, then f (a) = max f (x).
Since a coincides with the point q, where f (x) attains its maximum, further we use the notation
ξ(x) = b2 (x − q)2 + c instead of ξ(x) = b2 (x − a)2 + c.