A NOTE ON “EVALUATING THE RATE OF AGGREGATIVE RISK IN SOFTWARE DEVELOPMENT USING FUZZY SET THEORY”
Huey-Ming Lee1), Tsung-Yen Lee2), and Shu-Yen Lee3)
1)Chinese Culture University, ()
2)National TaiwanUniversity, ()
3)Financial Consultant China Engineering Consultants, Inc. ()
Abstract
The purpose of this article is to point out that the defuzzification by the median rule is not the same as by the classical centroid for the trapezoid or the triangular fuzzy number unless the trapezoid or the triangular is isosceles. If we defuzzify to solve the problem by the median then the computing errors will be out of our tolerance
Keywords:Median rule; Centroid; Defuzzification
- Introduction
Chen [1] defuzzified the trapezoid or triangular fuzzy numbers by the median rule to treat some problems, e.g., applied the mean rule to find the minimization of the total cost for the inventory with backorder, to evaluate the rate of aggregative risk in software development. But, we can show that the median of the trapezoid or triangular fuzzy number is not the centroid unless it's the isosceles as shown in Section 3.
- Fuzzy set theory
The fuzzy set theory was introduced by Zadeh [5] to deal with problems in which a source of vagueness is present. It has been considered as a modeling language to approximate situations in which fuzzy phenomena and criteria exist. In a universe of discourse X, a fuzzy subset A of X is a set defined by a membership function fA(x) representing a mapping which maps each element x in X to a real number in the closed interval [0, 1]. Here, the value of fA(x) for the fuzzy set A is called the membership value or the grade of the membership of x in X. The membership value represents the degree of x belonging to the fuzzy set A. The greater fA(x) the stronger the grade of membership for x in A.
The linguistic value could be used for approximate reasoning within the framework of fuzzy set theory [6] to handle effectively the ambiguity involved in the data evaluation and the vague property of linguistic expression, and normal trapezoid or triangular fuzzy numbers were used to characterize the fuzzy values of quantitative data and linguistic terms used in approximate reasoning.
Fuzzy numbers are fuzzy sets with special consideration for easy calculations. We can define operations of fuzzy numbers using the extension principle [4].
Definition 1 [5]Fuzzy number. If a fuzzy set A on the universe R of real numbers satisfies the following conditions, we call it a fuzzy number.
- A is a convex fuzzy set;
- there is only one x0 that satisfies fA(x0)=1; and
- fA(x) is continuous in an interval.
Based on the extension principle, we can derive the arithmetic of fuzzy numbers as shown in [2-4, 7].
Definition 2Trapezoid Fuzzy Number: Let , a<b<c<d, be a fuzzy set on . It is called a trapezoid fuzzy number, if its membership function is
Definition 3Triangular Fuzzy Number: Let , a<b<c, be a fuzzy set on . It is called a triangular fuzzy number, if its membership function is
Obviously, we can treat the triangular fuzzy numberas the trapezoid (a, b, b, c).
- Defuzzification
The underlying reason is that one cannot compare fuzzy numbers directly. Although many authors proposed their favorite methods, there is no universal consensus. Each method includes computing a crisp value, to be used for comparison. It means that fuzzy numbers must be first mapped to real values, which can be then compared for magnitude. This assignment of a real value to a fuzzy number is called defuzzification. It can take many forms, but the most standard defuzzification is through computing the centroid. This is defined, effectively, as the center of gravity of the curve describing a given fuzzy quantity. Because of this definition, its computation requires integration of the membership functions.
Chen [1] defuzzified the trapezoid (or triangular) fuzzy numbers by the median to evaluate the rate of risk. But, we can show that "the median is not always the centroid" as follows:
Proposition 1: For the trapezoid fuzzy number , by the bisection of area, the median only if .
Remark 1: For the triangular , we can use the formula (3) treating it as the trapezoid (a, b, b, c). We have that the median of is
Proposition 2: The centroid of the trapezoid fuzzy number is
(3)
Remark 2: By the definition of the centroid of the triangular , we obtain that the centroid of is .
Remark 3: For the triangular , we can use the formula (4) treating it as a trapezoid (a, b, b, c). We get the centroid of as
Proposition 3: The median () of the trapezoid equals to the centroid () only if the trapezoid is isosceles, i.e. d-c = b-a.
Remark 4: The median () of the triangular equals to the centroid () of the triangular only if the triangular is isosceles, i.e., c-b = b-a.
- Conclusion
Chen [1] proposed that the defuzzification of the triangular fuzzy numbers by the bisection of the trapezoid (a, b, b, c) is . But, from Proposition 3 and Remark 4, we know that if we defuzzify to solve the problem by the median then the computing results will be out of our tolerance unless the trapezoid or the triangular fuzzy number is isosceles.
REFERENCES
1)S.M. Chen, Evaluating the Rate of Aggregative Risk in Software Development Using Fuzzy Set Theory, Cybernetics and Systems: International Journal 30 (1999) 57-75.
2)A. Kaufmann and M.M. Gupta, Introduction to Fuzzy Arithmetic Theory and Applications (Van Nostrand Reinhold, New York, 1991).
3)H.-M. Lee, Applying fuzzy set theory to evaluate the rate of aggregative risk in software development, Fuzzy sets and Systems 79 (1996) 323-336.
4)Kazuo Tanaka, An Introduction to Fuzzy Logic for practical Applications (Springer-Verlag New York, Inc., 1997).
5)L. A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338-353.
6)L. A. Zadeh, The Concept of a Linguistic Variable and it's Application to Approximate Reasoning, Information Sciences 8 (1975) 199-249 (I), 301-357 (II), 9 (1976) 43-58 (III).
7)H.-J. Zimmermann, Fuzzy Set Theory and It's Applications, Second Revised Edition (Kluwer Academic Publishers, Boston/Dordrecht/London, 1991).