Sample Assessment Method Based on Fuzzy Linguistics Lily Lin , Huey-Ming Lee

ICIC International ⓒ2007 ISSN 1881-803X
ICIC Express Letters
Volume , Number , September 2010
pp. 1–6
Sample Assessment Method Based on Fuzzy Linguistics
Lily Lin 1 , Huey-Ming Lee 2 and Jin-Shieh Su 3
1
Department of International Business,
China University of Technology
56, Sec. 3, Hsing-Lung Road, Taipei (116), TAIWAN
[email protected]
2
Department of Information Management
3
Department of Applied Mathematics
Chinese Culture University, Taiwan
55, Hwa-Kung Road, Yang-Ming-San, Taipei (11114), TAIWAN
{hmlee, sjs}@faculty.pccu.edu.tw
Received March 2010; accepted September 2010
ABSTRACT. To overcome the problem of a difficulty and reflecting interviewee’s
incomplete and uncertain thought, we use fuzzy sense of sampling to express the degree of
interviewee’s feelings based on his own concept, the result will be closer to interviewee’s
real thought. In this study, we propose to analyze the sample assessment by the principle
of maximal membership grade or by the distribution of probability method. Also, we can
have the score for each fuzzy linguistic language if we assume 100 as the full score. As the
result, the proposed fuzzy assessment method on survey analysis can really reflect the
interviewee’s incomplete and uncertain thought.
Keywords: Fuzzy relation; Signed distance method
1. Introduction. Traditionally, we compute sample assessment by questionnaires according
to the thinking of binary logic. However, this kind of response may not be appropriate for
higher level thinking, performance or attitudinal outcomes, since it may lead to an
unreasonable bias since the human thinking is full with fuzzy and uncertain. There are two
different methods, multiple-item and single-item, while using linguistic variable as rating
item. We use mark or unmark to determine the choice for each item, i.e., the marked item is
represented by 1, while the other unmark item is represented by 0. Generally speaking, the
linguistic variable possesses the vague nature [1-3]. For example, one specific company
decides to conduct the satisfactory level survey on a specific product; four satisfactory
levels are used, i.e., strongly dissatisfactory; dissatisfactory; satisfactory; and strongly
satisfactory. These linguistic variables are fuzzy languages that can not be used to express
the real situation by reliability of zero or 1 to mark item. Therefore, in this paper, we apply
a value m which belongs to the interval of [0, 1] to represent the reliability or membership
grade in the fuzzy sense of marking item.
In this study, we use signed distance for analyzing sample assessment to do aggregated
assessment. The proposed fuzzy assessment method on survey analysis can really reflect
the interviewee’s incomplete and uncertain thought.
2. Preliminaries. For the proposed algorithm, all pertinent definitions of fuzzy sets are
given below [4, 5].
~
Definition 2.1. α -level set of the triangular fuzzy number A = ( p, q, r ) is
{
A(α ) = x µ A~ ( x ) ≥ α
≡ [ AL (α ),
}
AR (α )]
(1)
where
AL (α ) = p + ( q − p )α , AR (α ) = r − ( r − q)α , α ∈ [0, 1]
( 2)
~
We can represent A = ( p, q, r ) as
~
A = U [ ALα ), AR (α ); α ]
0 ≤ α ≤1
(3)
From Yao and Wu [4], we may define the signed distance from [ AL (α ), AR (α ); α ] to
~
0 as
~ 1
d ([ AL (α ), AR (α ); α ], 0 ) = [ AL (α ) + AR (α )]
2
( 4)
We have the following propositions.
~
~
Proposition 2.1. Let A1 = ( p1 , q1 , r1 ) and A2 = ( p 2 , q2 , r2 ) be two triangular fuzzy
numbers, and k>0, then, we have
~
~
( 10 ) A1 ⊕ A2 = ( p1 + p2 , q1 + q2 , r1 + r2 )
~
( 20 ) kA1 = ( kp1 , kq1 , kr1 )
~
~
Proposition 2.2. Let Aj = ( a j , b j , c j ) , B j = ( p j , q j , rj ) , j=1, 2.
If 0 ≤ AjL (α ) < AjR (α ), 0 ≤ B jL (α ) < B jR (α ), α ∈ [0, 1], j = 1, 2 then we have the following
three properties.
~ ~
( 10 ) A1 ⊗ B1 =
∪ [A
0 ≤α ≤1
1L
(α ) B1L (α ), A1R (α ) B1R (α ); α ]
~ ~
~ ~
( 20 ) ( A
1 ⊗ B1 ) ⊕ ( A2 ⊗ B2 ) =
~
~
( 30 ) k ( A1 ⊗ B1 ) =
∪ [A
0≤α ≤1
[kA
∪
α
1L
(α ) B1L (α ) + A2L (α )B2L (α ), A1R (α )B1R (α ) + A2R (α )B2R (α ); α ]
1L (α ) B1L (α ),
kA1R (α ) B1R (α ); α ], for k > 0
0≤ ≤1
~
~
Proposition 2.3. Let A1 = ( p1 , q1 , r1 ) and A2 = ( p 2 , q2 , r2 ) be two triangular fuzzy
numbers, and k ∈ R , then we have
~ ~ ~
~ ~
~ ~
( 10 ) d ( A1 ⊕ A2 , 0 ) = d ( A1 , 0 ) + d ( A2 , 0 )
~ ~
~ ~
( 20 ) d (kA1 , 0 ) = kd ( A1 , 0 )
~
~
Proposition 2.4. Let Aj = ( a j , b j , c j ) , B j = ( p j , q j , rj ) , j=1, 2. Then, we have
~ ~
~ ~ ~
~ ~ ~
~ ~ ~
d ([( A1 ⊗ B1 ) ⊕ ( A2 ⊗ B2 )], 0) = d ( A1 ⊗ B1, 0) + d ( A2 ⊗ B2 , 0) .
Definition 2.2. Fuzzy relation: Let X , Y ⊆ R be universal sets, then
~
R = {(( x, y ), µ R~ ( x, y )) ( x, y ) ⊆ X × Y }
(5)
is called a fuzzy relation on X × Y .
3. The survey based on rating items as the linguistic variables.
3.1. The statistical method. In most cases, questionnaire of survey exist many topics and questions.
For instance, one specific questionnaire regarding satisfactory level may include the survey items,
such as service attitude, communications, looks, geniality, cordial, etc. We can define them as
follows:
Items: A1, A2 , ..., An
with weights: a1 , a2 , ..., an , respectively
n
subject to: 0 ≤ a j ≤ 1 , j = 1, 2, K , n , and ∑ a j = 1
j =1
Let Lv , for v=1, 2,..., k, be the k different linguistic variables as criteria of questionnaire,
expressed in linguistic values such as very poor, poor, ordinary, good, very good, etc. We suppose
that there are m samples which are drawn from the specific population, each sample chooses Lv , for
v=1, 2,…, k respectively, the selected item denoted by 1, otherwise denoted by zero in crisp case.
The q-th interviewee assessment data is depicted as shown in Table 1.
TABLE 1. Contents of the q-th interviewee assessment form
Item
Item - weight
A1
a1
M 11q
A2
a2
M 21q
.
.
.
.
.
.
.
.
.
An
an
M n1q
L1
Linguistic variables
L2
L3
.
.
M 12 q
M 13 q
.
.
M 22 q
M 23q
.
.
.
.
.
.
.
.
M n 3q
M n 2q
.
.
.
.
.
Lk
M 1kq
M 2 kq
.
.
.
.
M nkq
In Table 1,
M ijq = 0 or 1 , and
for each i ∈{1, 2,K, n}, j ∈{1, 2,K, k}, q ∈{1, 2,K, m}.
k
∑M
j =1
ijq
=1
(6)
Since Lv is a fuzzy linguistic language and not a number, we can not join the
messages to do the analysis of survey.
3.2. The fuzzy method. In general statistics, we could not analyze the survey with the
fuzzy linguistics. Therefore, we apply the fuzzy logic to treat it.
In the fuzzy set theory, we can us the triangular fuzzy numbers to represent the fuzzy
linguistics. Let Lv be a fuzzy linguistic language, for v=1, 2, …, k, and the corresponding
~
triangular fuzzy number be Lv = ( pv , qv , rv ) .
From Table 1, we re-modify Eq. (6) as
0 ≤ M ijq ≤ 1 and
k
∑M
ijq
=1
(7)
j =1
for q ∈ {1, 2,..., m} , and for all i, j.
1 m
Let mij = ∑ M ijq , i=1, 2, …, n; j=1, 2, …, k. Then we have the following Table 2.
m q=1
TABLE 2. Contents of the average of the m interviewees’ assessment form
Item
Item - weight
A1
a1
m11
A2
a2
m 21
.
.
.
.
.
.
.
.
.
An
an
m n1
L1
Linguistic variables
L3
.
.
m13
m12
.
.
m23
m 22
.
.
.
.
.
.
.
.
mn 3
mn 2
.
.
.
.
.
L2
Lk
m1k
m2 k
.
.
.
.
mnk
In Table 2,
k
0 ≤ mij ≤ 1 and
∑m
ij
=1
(8)
j =1
, for all i = 1, 2,K , n .
n
0 ≤ a j ≤ 1 , j = 1, 2, K , n , and ∑ a j = 1
j =1
(9)
Step 1:
Let A = { A1 , A2 ,..., An } , L = {L1 , L2 ,..., Lk } , then we may represent the fuzzy set of Ai
on L as the following form.
m
m
~ m
Ai = i1 ⊕ i 2 ⊕ ⋅ ⋅ ⋅ ⊕ ik
L1
L2
Lk
(10)
for i=1, 2, …, n. Then, by the fuzzy relation on A × L with triangular fuzzy number, we
have the following fuzzy relation matrix:
~
~
~
m11L1 m12 L2 . . . m1k Lk 
 ~
~
~
m21 L1 m22 L2 . . . m2 k Lk 
.


~ 
R = .

.





 ~
~
~ 
mn1L1 mn 2 L2 . . . mnk Lk 
(11)
Step 2:
By compositional rule of inference, we have
~ ~
~
~
(b1 , b2 , ..., bk ) = ( a1 , a2 , ..., an ) o R
where
~
~
~
~
b j = ( a1 ( mij L j )) ⊕ (a2 ( m2 j L j )) ⊕ ⋅ ⋅ ⋅ ⊕ ( an ( mnj L j ))
(12)
(13)
Step 3:
Defuzzified Eq. (13) by the signed distance [4], we have
~ ~
d (b j , 0 ) =
n
∑ a m d (L , 0)
~ ~
i
ij
j
i =1
=
1
4
n
∑a m ( p
i
ij
j
+ 2q j + rj )
(14)
i =1
Step 4:
Normalize Eq. (14), we let
Dj =
~ ~
d (b j , 0 )
k
∑
~ ~
d (bq , 0 )
, j=1, 2, …, k
(15)
q =1
Then, we have 0 ≤ D j ≤ 1 and
k
∑D
j
~
= 1 . We can represent D as
j =1
D
~ D D
D = 1 + 2 + ⋅⋅⋅ + k
L1 L2
Lk
Then, we have the following proposition.
Proposition 3.1: (A) By the principle of maximal membership grade:
Let
(16)
Dt = max Dq
(17)
1≤ q≤k
t ∈ {1, 2, ..., k } . The t corresponds to the sub-index of Lt , it means that the maximal
appraisal level rating is Lt and the membership grade is Dt .
(B) By the distribution of probability method:
The probability of L j is D j , for j ∈ {1, 2, ..., k} .
(C) According to 100 as full marks, the score of each L j is as the following table:
Fuzzy linguistic
Score
L1
100 ⋅ D1
L2
100 ⋅ D2
.
.
.
.
.
.
100 ⋅ Dk
Lk
4. Conclusions. In this paper, we use the fuzzy linguistic to analyze the sample assessment
by questionnaires. We can have the analysis of the sampling survey by the principle of
maximal membership grade or by the distribution of probability method. Also, we can have
the score for each fuzzy linguistic language if we assume 100 as the full score. The
proposed fuzzy assessment method on survey analysis can really reflect the interviewee’s
incomplete and uncertain thought.
Acknowledgment. This work is partially supported by the National Science Council of
Taiwan, under grant no. NSC-99-2410-H-034-019-. The authors also gratefully
acknowledge the helpful comments and suggestions of the reviewers, which have improved
the presentation.
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