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3.1 Fuzzy Logic (FL)
Essentially, Fuzzy Logic (FL) is a multi-valued logic that allows middle values to be
defined between conventional evaluations like yes/no, true/false, black/white, etc.
Fuzzy Logic was introduced in 1965 by Prof. L. Zadeh at the University of California,
Berkeley [9]. The basic notion of fuzzy systems is a fuzzy set. for example, to classify
the fuzzy set of climate, which may be consisted of members like “Very cold”,
“Cold”, “Warm”, “Hot”, and “Very hot”. The theory of fuzzy sets enables us to struc-
ture and describe activities and observations, which differ from each other only
vaguely, to formulate them in models and to use these models for various purposes -
such as problem-solving and decision-making [9]. Suppose that µS(x) (or µ(S, x)) is
the degree of membership of x in set S that 0 ≤ µS(x) ≤1
µS(x) = 0 x is not at all in S,
µS(x) = 1 x is fully in S,
If µS(x) = 0 or 1, then the set S is crisp.
For example, pay attention to the diagram 1 (Fig. 1).
What is the meaning of 75 in diagram? We analyze this question as following:
A node that finished successfully 75% of its submitted jobs has simultaneously
Low, Medium, and High efficiency in various degrees. For example, it can be inter-
pret as 0.2 Low efficiency, 0.5 Medium efficiency and 0.3 High efficiency. In other
word, there is much status that needs to use of Fuzzy logic or fuzzy algorithm. For
instance, consider the following scenario:
Low
Medium
High
1.0
ȝ
60
75
90
Fig. 1. This diagram shows the Node's efficiency
Let's assume that we have to select 3 computing nodes in between 20 existing
nodes. At the first time, efficiency and availability for all nodes is evaluated. Suppose
that 70% of nodes have efficiency in range 85 to 90, and 20% have near to 95 and
10% under 75. Therefore, in this case, the range (85-90) is considered as medium
efficiency and if there are nodes with high efficiency, it is not needed to use medium
efficiency nodes.
We will not discuss fuzzy set such natural extensions here and more about fuzzy
logic can be found in [13].
3.2 Fuzzy Decision Tree Algorithm
This algorithm is a developed version of ID3 that operate on fuzzy set and it will
produce a fuzzy decision tree (FDT). Before this, other researchers [3, 12] considered

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