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The EFWA algorithm
Appendix
Definition: the input a, b, c, and d are the intervals of fuzzy membership functions, and the outputs are the
intervals of the result fuzzy membership function. Additionally, the δ s and the ζ i
s can be calculated by
i
Equations (7) and (8) respectively.
( a − a ) e + ( a − a ) e + ... + ( a a ) e
δs
=
i
e + e + ... e
1 i 1 2 i 2
n−i n
1 2
( b − b) e + ( b − b) e + ... + ( b b) e
ζ s =
i
e + e + ... e
1 i 1 2 i 2
n−i n
1 2
Description of the EFWA algorithm (Lee and Park, 1997)
n
n
(1) Sort a’s in non-decreasing order. Let (a1, a2, …, an) be the resulting sequence. Let first := 1 and last := n.
(2) Sort a’s in non-decreasing order. Let (a1, a2, …, an) be the resulting sequence. Let first := 1 and last := n.
(3) Let δ-threshold := ⎢⎣( first + last ) /2⎥⎦
. For each i = 1, 2, …, δ-threshold, let ei := di and for each i =δthreshold
+ 1, …, n, let ei := ci. For an n-tuple S = (e1, e2, …, en), evaluate and δ −
.
+
δsδ −threshold
sδ threshold
(4) If δsδ −threshold
> 0 and δsδ − threshold + 1
≤ 0 then L = fL(e1, e2, …, en) and go to Step 4; otherwise execute the
following step.
(a) If > 0 , then first :=δ-threshold + 1; otherwise last := δ-threshold, and go to Step 2.
δsδ −threshold
(5) Sort b’s in non-decreasing order. Let (b1, b2, …, bn) be the resulting sequence. Let first := 1 and last := n.
(6) Letζ-threshold := ( first + last ) /2
⎢⎣ ⎥⎦
. For each i = 1, 2, …, ζ-threshold, let ei := ci and for each i =ζthreshold
+ 1, …, n, let ei := di. For an n-tuple S = (e1, e2, …, en), evaluate and
.
ζ sζ −threshold
1
ζ s( ζ − threshold + 1)
(7) If > 0 and ζ ≤ 0 then U = fU(e1, e2, …, en) and stop; otherwise execute the following step:
− +
ζ sζ −threshold
sζ threshold
1
(b) If > 0 , then first :=ζ-threshold + 1; otherwise last := ζ-threshold, and go to step 5.
Illustrative example
ζ sζ −threshold
Assume that an instructor aims to assess five students (S1, S2, S3, S4, S5) to identify their level of understanding with
regard to five concepts (C1, C2, C3, C4, C5,). The instructor selects five test items (I1, I2, I3, I4, I5) from a test item
bank to form a test-sheet, that are relevant to concepts one to five, and each test item has its difficulty degree, D1, D2,
D3, D4, D5. In this test-sheet, each concept is possibly related to the others. The instructor then conducts the test to
assess the five students to identify the level of understanding of the individual students with regard to the five
concepts. The degree of difficulty of each test item is shown in Table 7. In addition, the relationships among the test
items and concepts are shown in Table 8, and the relationships among the concepts are shown in Table 9. After the
(7)
(8)
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