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Fuzzy assessment for sampling survey defuzzification by signed distance method
Expert Systems with Applications 37 (2010) 7852–7857
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Fuzzy assessment for sampling survey defuzzification by signed distance method Lily Lin a, Huey-Ming Lee b,* a b
Department of International Business, China University of Technology, 56, Sec. 3, Hsing-Lung Road, Taipei 116, Taiwan Department of Information Management, Chinese Culture University, 55, Hwa-Kung Road, Yang-Ming-San, Taipei 11114, Taiwan
a r t i c l e
i n f o
Keywords: Sampling survey Fuzzy logic Signed distance method
a b s t r a c t Since traditional sampling survey via questionnaire is difficult in reflecting interviewee’s incomplete assessment and uncertain thought, we use fuzzy sense of sampling to express the degree of interviewee’s feelings, and find that the result is closer to interviewee’s real thought. In this study, we propose two algorithms to do aggregative assessment for sampling survey by signed distance method with the linear order character of symmetric fuzzy linguistics instead of using previous centroid method. As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify triangular fuzzy number by the signed distance is better than by the centroid method. The proposed fuzzy assessment method on sampling survey analysis is easily to assess the sampling survey and make the aggregative evaluation. Since the proposed model in this study is to measure the group evaluation, the final value is more objective than just one evaluator’s assessment. Moreover, if there is only one evaluator existing, the proposed model is also appropriate to assess. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Statistical analysis via sampling survey is a powerful market research tool to acquire the useful information. Traditionally, we compute statistics with sample data by questionnaires according to the thinking of binary logic. There usually exists two different methods, both multiple-item and single-item choices, while using linguistic variables 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 (Lin & Lee, 2008, 2009; Sun & Wu, 2006). This kind of result may lead to an unreasonable bias since the human thinking is full with fuzzy and uncertain. Fuzzy sets theory was introduced by Zadeh (1965) to deal with problem in which vagueness is present, linguistic value can be used for approximate reasoning within the framework of fuzzy set theory (Zadeh, 1975a, 1975b, 1976) to effectively handle the ambiguity involved in the data evaluation and the vague property of linguistic expression, and normal triangular fuzzy numbers are used to characterize the fuzzy values of quantitative data and linguistic terms used in approximate reasoning. With regard to fuzzy decision-making problem, Lee (1996) applied fuzzy set theory to evaluate the aggregative risk in software development under fuzzy circumstances. Lin and Lee (2008)
* Corresponding author. Tel.: +886 937 893 845; fax: +886 2 2777 4723. E-mail addresses: [email protected] (L. Lin), [email protected] (H.-M. Lee). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.04.052
presented facility site selection model using fuzzy set theory. Lin and Lee (2009) presented the fuzzy assessment on sampling survey analysis. Lin and Lee (2008, 2009) applied a value m which belongs to the closed interval [0, 1] to represent the reliability or membership grade in the fuzzy sense of marking item, and presented the method to treat the aggregated assessment of the main item Bj, and the integrated assessment. Lin and Lee (2009) presented method to treat the crisp or fuzzy multiple/single choices. As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify the triangular fuzzy number by the signed distance is better than by the centroid method. In this study, we propose a model to do assessment analysis for sampling survey with the linear order character of fuzzy linguistics by signed distance method (Yao & Wu, 2000). The proposed fuzzy assessment method on sampling survey analysis is easily to assess the sampling survey and do the aggregative evaluation. The paper is organized as follows. Section 1 is introduction. Section 2 is the preliminaries. Aggregative assessment method for sample survey with fuzzy linguistics by signed distance method is in Section 3. Section 4 is the numerical example. The conclusion of this study is given in Section 5. 2. Preliminaries For the proposed algorithm, all pertinent definitions of fuzzy sets are given below (Yao & Wu, 2000; Zadeh, 1965; Zimmermann, 1991).
L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857
Definition 2.1. Fuzzy set: If X is a collection of objects denoted e in X is a set of ordered pairs: generically by x then a fuzzy set A
e ¼ fðx; l ðxÞÞjx 2 Xg A e A
ð1Þ
leA ðxÞ is called the membership function of x in Ae which maps X to the closed interval [0, 1] that characterizes the degree of membere ship of x in A. ~ be a fuzzy set on R = (1, 1). It is Definition 2.2. Fuzzy point: Let a called a fuzzy point if its membership function is
la~ ðxÞ ¼
1 if x ¼ a 0 if x – a:
ð2Þ
a if a 6 x 6 b 0
otherwise:
ð3Þ
If a = b, we call [a, b; a] a level a fuzzy point at a.
ð4Þ
e ¼ ðq; q; qÞ. We call it the fuzzy point q ~ at q. If r = q and p = q, then A We let
Rr p
e ¼ R Cð AÞ r p
xle ðxÞdx A
ð5Þ
leA ðxÞdx
e where p and r are lower be the defuzzification by the centroid of A, and upper limits of the integral, respectively. Then, we obtain that e ¼ ðp; q; rÞ is the centroid of the normal triangular fuzzy number of A
e ¼ Cð AÞ
1 ðp þ q þ rÞ: 3
ð6Þ
e be a fuzzy set on R = (1, 1). Denote D(a) = [DL(a), DR(a)] Let D e where 0 6 a 6 1, DL(a) and DR(a) are the left and as the a-cut of D, right hand side of D(a). DL(a) and DR(a) exist and are integrable for e a 2 [0, 1]. In addition, we let F be the family of all these fuzzy sets D on R = (1, 1). e 2 F, then from the decomposition theory, D e may be repreLet D sented as
e¼ D
[ aDðaÞ ¼
06a61
[ ½DL ðaÞ; DR ðaÞ; a:
06a61
1 ½d0 ðDL ðaÞ; 0Þ þ d0 ðDR ðaÞ; 0Þ 2 1 ¼ ½DL ðaÞ þ DR ðaÞ: 2
d0 ½ðDL ðaÞ; DR ðaÞÞ; 0 ¼
ð8Þ
For each a 2 [0, 1], the crisp interval [DL(a), DR(a)] and the level a fuzzy interval [DL(a), DR(a); a] are in one to one correspondence. Therefore, we may define the signed distance from [DL(a), DR(a); a] ~ as to 0
¼
ð7Þ
As in Yao and Wu (2000), we consider the definition of the signed distance on F. Definition 2.5. The signed distance: We define d0(a, 0) = a, for a, 0 2 R. Remark 2.1. The meaning of Definition 2.5 is as the follows, if 0 < a then the distance between a and 0 is d0(a, 0) = a. If a < 0 then the distance between a and 0 is d0(a, 0) = a. Therefore, we call d0(a, 0) = a is the signed distance between a and 0.
1 ½DL ðaÞ þ DR ðaÞ: 2
ð9Þ
e 2 F, DL(a) and DR(a) exist and are integrable for Since D a 2 [0, 1], from (7), we have the following definition. ~ Let D e measured from 0: e 2 F, Definition 2.6. The signed distance of D ~ as e measured from 0 we define the signed distance of D
~ ¼1 e 0Þ dð D; 2
e ¼ ðp; q; rÞ, Definition 2.4. Normal triangular fuzzy numbers: Let A p < q < r, be a fuzzy set on R = (1, 1). It is called a normal triangular fuzzy number, if its membership function is
8 xp > < qp if p 6 x 6 q rx leA ðxÞ ¼ rq if q 6 x 6 r > : 0 otherwise:
e 2 F, from the Definition 2.5, we have that the signed disFor D tance of DL(a) and DR(a) measured from 0 are d0(DL(a), 0) = DL(a) and d0(DR(a), 0) = DR(a), respectively. Therefore, we may define the signed distance of the interval [DL(a), DR(a)], which is measured from the origin 0, by
~ ¼ d0 ð½DL ðaÞ; DR ðaÞ; 0Þ dð½DL ðaÞ; DR ðaÞ; a; 0Þ
Definition 2.3. Level a fuzzy interval: Let [a, b; a] be a fuzzy set on R = (1, 1). It is called a level a fuzzy interval, 0 6 a 6 1, a < b, if its membership function is
l½a;b;a ðxÞ ¼
7853
Z
1
½DL ðaÞ þ DR ðaÞda
ð10Þ
0
e ¼ ðp; q; rÞ, then the left endpoint and the right Remark 2.2. If C e are CL(a) = p + (q p)a and endpoint of the a-level set of C e is Cð C eÞ ¼ CR(a) = r (r q)a, respectively. The centroid of C 1 e measured from 0 ~ is ðp þ q þ rÞ, and the signed distance of C 3 e ; 0Þ ~ ¼ 1 ð2q þ p þ rÞ. dð C 4 e 2 ¼ ðp ; q ; r 2 Þ be two trie 1 ¼ ðp ; q ; r 1 Þ and A Proposition 2.1. Let A 1 1 2 2 angular fuzzy numbers, and k > 0, then, we have e1 A e 2 ¼ ðp þ p ; q þ q ; r 1 þ r 2 Þ, (10) A 1 2 1 2 e 1 ¼ ðkp ; kq ; kr 1 Þ. (20) k A 1 1 We can easily show Proposition 2.1 by extension principle. e 2 ¼ ðp ; q ; r 2 Þ be two e 1 ¼ ðp ; q ; r 1 Þ and A Proposition 2.2. Let A 1 1 2 2 triangular fuzzy numbers, and k 2 R, then we have e1 A e 2 ; 0Þ e 1 ; 0Þ e 2 ; 0Þ, ~ ¼ dð A ~ þ dð A ~ (10) dð A 0 e e ~ ~ (2 ) dðk A 1 ; 0Þ ¼ kdð A 1 ; 0Þ. e1 A e 2 ¼ ðp þ p ; q þ q ; r 1 þ r2 Þ, Proof. 10From Proposition 2.1, A 1 2 1 2 and from Definition 2.6, we have
e1 A e 2 ; 0Þ ~ ¼ 1 ½p þ p þ 2q þ 2q þ r1 þ r2 dð A 2 1 2 4 1 e 1 ; 0Þ e 2 ; 0Þ: ~ þ dð A ¼ dð A By the same way, we can prove (20) h e ¼ ðp; q; rÞ be a triangular fuzzy number. Proposition 2.3. Let A e is not an isosceles triangle, (10) If the membership function of A then, based on the maximum membership grade principle, to e by the signed distance is better than by the centroid defuzzify A method. e is (20) If the membership function of the triangular fuzzy number A e an isosceles triangle, then to defuzzify A by the signed distance is equal to by the centroid method based on the maximum membership grade principle.
7854
L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857
e is Cð AÞ e ¼ 1 ðp þ q þ rÞ, and the signed disProof. The centroid of A 3 1 e ~ tance dð A; 0Þ ¼ 4 ðp þ 2q þ rÞ. We let M ¼ 12 ðp þ rÞ be the midpoint of the closed interval [p, r]. Then, we have
e dð A; e 0Þ ~ ¼ Cð AÞ
3.1. Fuzzy linear order linguistics
1 ðp þ r 2qÞ; 12
We consider that the fuzzy linguistics L1, L2, . . . , Ln with the corresponding series of fuzzy numbers e L1 , e L2, . . . , e L n , where
e ¼ 1 ðp þ r 2qÞ; M Cð AÞ 6 e 0Þ ~ q¼ dð A;
e Lq ¼
1 ðp þ r 2qÞ: 4
Now, we consider the following three cases: e 0Þ e 2q then q < dð A; e 0ÞÞ e ~ > l ðCð AÞÞ. From Fig. 1, we have le ðdð A; eA A e < dð A; e 0Þ ~ < Case 2: If p + r < 2q, then M < Cð AÞ q. e 0ÞÞ e ~ > l ðCð AÞÞ. From Fig. 2, we have le ðdð A; e A e ¼ dð A; e A 0Þ ~ ¼ q. We have Case 3: If p + r = 2q, then M ¼ Cð AÞ e 0ÞÞ e ~ ¼ l ðCð AÞÞ. leðdð A; e A
3. Aggregative sample survey with linear order fuzzy linguistics by signed distance method
A
From the above three cases and by the maximum membership grade principle, Proposition 2.3 is proved. h Definition 2.7. Fuzzy relation: Let X, Y # R be universal sets, then
e ¼ fððx; yÞ; l ðx; yÞÞjðx; yÞ # X Yg R e
ð11Þ
R
is called a fuzzy relation on X Y.
ðq 1Þ
100 100 100 ;q ; ðq þ 1Þ nþ1 nþ1 nþ1
ð12Þ
~ < dðe ~ < for q = 1, 2, . . . , n. Then, we have dðe L 1 ; 0Þ L 2 ; 0Þ ~ ~ e e dð L 3 ; 0Þ < < dð L n ; 0Þ. Therefore, we can make a conclusion that these fuzzy linguistics L1, L2, . . . , Ln with the corresponding series of fuzzy numbers e L1 ; e L2 ; . . . ; e L k are linear order respective to the signed distance method. 3.2. Aggregative assessment for sampling survey by signed distance method In most cases, questionnaire of sampling survey exists many topics and questions, let us say, main items and sub-items. For instance, one specific questionnaire regarding satisfactory level may include main survey items such as satisfactory level for product, service and price, etc., also sub-items may exist under each main item. We can define them as follows: Main items: B1, B2, . . . , Br, with weight: b1, b2, . . . , br, respectively, P subject to: 0 6 bj 6 1,j = 1, 2, . . . , r and rj¼1 bj ¼ 1. Sub-items: Bj1 ; Bj2 ; . . . ; Bjmj under main items Bj, j = 1, 2, . . . , r; with weight: bj1 ; bj2 ; . . . ; bjmj , respectively subject to:
1
0 6 bjt 6 1;
j ¼ 1; 2; . . . ; r; t ¼ 1; 2; . . . ; mj
and
mj X
bjt ¼ 1:
t¼1
0
p
q
r
M
~~ d ( A, 0 )
~ C ( A)
Fig. 1. Case M > q.
Since the questionary of the sample survey is in fuzzy sense, we cannot evaluate the aggregative assessment by the traditional statistics. Let Lv (for v = 1, 2, . . . , m) be the m different linguistic variables as criteria of questionnaire, expressed in fuzzy language such as very low, low, medium, high, very high, etc. Suppose the evaluators assess the aggregative grades for some one main item Bj with sub-items Bj1 ; Bj2 ; . . . ; Bjmj of the company. Based on the assessment form proposed by Lin and Lee (2009), we propose the
Table 1 Contents of the proposed assessment form.
1
0
p
M
~ C ( A)
q
Main item
Itemweight
Subitem
Fuzzy linguistic
Numbers of answer
Triangular fuzzy number of the fuzzy linguistic
Bj
bj
Bj1
L1
nj11
L2
nj12
.. .
.. . Lm
.. . nj1m
e L 1 ¼ ð0; t1 ; t2 Þ e L 2 ¼ ðt1 ; t2 ; t3 Þ .. . e L m ¼ ðtm1 ; tm ; 100Þ
Bj2
L1
nj21
L2
nj22
.. . Lm
.. . nj2m
.. . L1
.. . njmj 1
L2
njmj 2
.. . e L 1 ¼ ð0; t1 ; t2 Þ e L 2 ¼ ðt1 ; t2 ; t3 Þ
.. . Lm
.. . njmj m
.. . e L m ¼ ðtm1 ; tm ; 100Þ
r .. . Bjmj
~~ d ( A, 0 )
Fig. 2. Case M < q.
e L 1 ¼ ð0; t1 ; t2 Þ e L 2 ¼ ðt1 ; t2 ; t3 Þ .. . e L m ¼ ðtm1 ; tm ; 100Þ
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L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857 Table 2 Contents of the weighted triangular fuzzy number with its signed distance for the main item Bj. Main item
Item-weight
Sub-item
Fuzzy linguistic
Numbers of answer
Weight
Weighted triangular fuzzy number (1)
Signed distance of (1)
Bj
bj
Bj1
L1
nj11
nj11/nj1
L2
nj12
nj12/nj1
ðnj11 =nj1 Þe L1 ðnj12 =nj1 Þe L2
1 4 ðnj11 =nj1 Þð0 þ 2t 1 þ t 2 Þ dj11 1 4 ðnj12 =nj1 Þðt 1 þ 2t 2 þ t 3 Þ dj12
.. . Lm
.. . nj1m
.. . nj1m/nj1
.. .
.. .
L1
nj21
nj21/nj2
ðnj1m =nj1 Þe Lm ðnj21 =nj2 Þe L1
1 4 ðnj1m =nj1 Þðt m1 þ 1 4 ðnj21 =nj2 Þð0 þ 2t 1
L2
nj22
nj22/nj2
1 4 ðnj22 =nj2 Þðt 1
.. . Lm
.. . nj2m
.. . nj2m/nj2
ðnj22 =nj2 Þe L2 .. . ðnj2m =nj2 Þe Lm
.. . L1
.. . njmj 1
.. . njmj 1 =njmj
ðnjmj 1 =njmj Þe L1
1 4 ðnjmj 1 =njmj Þð0
L2
njmj 2
njmj 2 =njmj
ðnjmj 2 =njmj Þe L2
1 4 ðnjmj 2 =njmj Þðt 1
.. . Lm
.. . njmj m
.. . njmj m =njmj
.. .
.. .
ðnjmj m =njmj Þe Lm
1 4 ðnjmj m =njmj Þðt m1
.. . Bj2
.. . Bjmj
contents of the jth (for j = 1, 2, . . . , r) main item with sub-items assessment form as shown in Table 1. From Table 1, we let
njq ¼
m X
njqt ;
j ¼ 1; 2; . . . ; r; q ¼ 1; 2; . . . ; mj :
ð13Þ
ðjÞ
P1 6 P1 ¼
2t m þ 100Þ dj1m þ t2 Þ dj21
þ 2t 2 þ t3 Þ dj22
.. . 1 4 ðnj2m =nj2 Þðt m1
þ 2t m þ 100Þ dj2m
.. .
mj X
k¼1 r X
þ 2t1 þ t2 Þ djmj 1 þ 2t 2 þ t3 Þ djmj 2
þ 2tm þ 100Þ djmj m
bjk 100 ¼ 100; ðjÞ
bj P 1 <
j¼1
r X
ð17Þ
bj 100 ¼ 100:
ð18Þ
j¼1
t¼1
Then, ðnjqt =njq Þe L t is the weighted triangular fuzzy number of e L t with respective to Bjqj . Based on Table 1, we propose the following two algorithms to tackle the assessing.
Defuzzified ðnjkq =njk Þe L q by the signed distance, we have
1 ðnjkq =njk Þ t q1 þ 2t q þ t qþ1 : 4
~ ¼ dððnjkq =njk Þe L q ; 0Þ
ð14Þ
We let
~ : djkq ¼ d ðnjkq =njk Þe Lq; 0
bjk
m X
djkq
ð19Þ
~jq ¼ ðe1q ; e2q ; e3q Þ, where We let a mj X
bjk
njkq tq1 ; njk
bjk
njkq tq ; njk
bjk
njkq tqþ1 njk
k¼1
ð15Þ
e2q ¼
mj X k¼1
(20) The aggregative assessment is r X
ðnjmj 2 =njmj Þe L2
! mjmj q nj1q e nj2q e e ~jq ¼ bj1 Lq: a L q bj2 L q bjmj nj1 nj2 njmj
e1q ¼
q¼1
mj m X njkq 1X ¼ bjk ðt q1 þ 2tq þ tqþ1 Þ: 4 k¼1 njk q¼1
P1 ¼
3 ðnj1m =nj1 Þe Lm 7 ðnj2m =nj2 Þe Lm 7 7 7 7: 7 7 5 ðnjmj m =njmj Þe Lm
where
(10) For each j 2 {1, 2, . . . , r}, the aggregative assessment of the main item Bj is
k¼1
ðnj12 =nj1 Þe L2 e ðnj22 =nj2 Þ L 2
ej; ~j1 ; a ~j2 ; . . . ; a ~jm Þ ¼ ðbj1 ; bj2 ; . . . ; bjmj Þ R ða
Proposition 3.1.
mj X
ðnj11 =nj1 Þe L1 6 6 ðn =n Þe 6 j21 j2 L 1 ej ¼ 6 R 6 6 6 4 ðnjmj 1 =njmj Þe L1
(a) Then, the first stage aggregative composition inference for the attribute Bj is as follows:
Then, we may re-modify Table 1 as Table 2. From Table 2, we have the following Proposition 3.1.
ðjÞ
Let L = {L1, L2, . . . , Ln} be the set of the n fuzzy linguistics. From e j on Bj L with the weighted Table 2, we can form a fuzzy relation R triangular fuzzy numbers as follows:
2
3.3. Algorithm I
P1 ¼
3.4. Algorithm II
e3q ¼
mj X k¼1
ðjÞ
bj P 1
for j = 1, 2, . . . , r; q = 1, 2, . . . , m. ~jq by signed distance, we have Defuzzified a
j¼1 mj r m X X njkq 1X ¼ bj bjk tq1 þ 2tq þ tqþ1 : 4 j¼1 k¼1 n jk q¼1 n
ð16Þ
Note 1. Since 0 6 njkq < 1, for all j, k, q, and 0 < tq1 < tq < jk tq+1 < 100, for all q, therefore, we have the following properties:
~ ¼ 1 ðe1q þ 2e2q þ e3q Þ ~jq ; 0Þ dða 4 mj njkq 1X bjk tq1 þ 2tq þ t qþ1 : ¼ 4 k¼1 njk
ð20Þ
7856
L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857
(b) Let B = {B1,B2, . . . , Br}, then the second stage aggregative composition inference is as follows:
2
~11 a 6a ~ 6 21 ðg~1 ; g~2 ; . . . ; g~m Þ ¼ ðb1 ; b2 ; . . . ; br Þ 6 4 ~r1 a
~12 a ~22 a ~r2 a
3 ~1m a ~2m 7 a 7 7: 5 ~rm a
where
g~q ¼
r X
~jq ¼ ðv 1q ; v 2q ; v 3q Þ; bj a
v 2q ¼ v 3q ¼
r X
bj
mj X
j¼1
k¼1
r X
mj X
bj
j¼1
k¼1
r X
mj X
j¼1
bjk
njkq t q1 ; njk
bjk
njkq tq ; njk
We use the contents of main items and sub-items shown in Lin and Lee (2008) as an example to implement the proposed algorithms. Example. Assume that we have the following main items, weights, sub-items and the answer numbers as shown in Table 3. By algorithm (I) and (II), we can have the following computing results:
j¼1
v 1q ¼
4. Example implementation
Table 3 Contents of the example.
njkq bj bjk t qþ1 njk k¼1
Main item
Weight-2 Sub-item
B1: Labor
0.3
for q = 1, 2, . . . , m. ~q by signed distance, we have Defuzzified g
~ ¼ 1 ðv 1q þ 2v 2q þ v 3q Þ dðg~q ; 0Þ 4 mj r X njkq 1X ¼ bj bjk ðtq1 þ 2t q þ t qþ1 Þ 4 j¼1 k¼1 njk
ð21Þ
for q = 1, 2, . . . , m. From (20) and (21), we have the following proposition:
(10) For each j 2 {1, 2, . . . , r}, the aggregative assessment of the main item Bj with respective to the fuzzy linguistic Lq (q = 1, 2, . . . , m), is
~ ¼ 1 ðe1q þ 2e2q þ e3q Þ ~jq ; 0Þ dða 4 mj njkq 1X bjk ðtq1 þ 2tq þ t qþ1 Þ ¼ 4 k¼1 njk
~ ¼1 ~jq ; 0Þ ¼ dða 4 q¼1
mj X
m X njkq bjk ðt q1 þ 2t q þ tqþ1 Þ njk q¼1 k¼1
(30) The aggregative assessment with respective to the fuzzy linguistic Lq (q = 1, 2, . . . , m), is
~ ¼1 dðg~q ; 0Þ 4
r X j¼1
B4: Reward
njkq bj bjk ðt q1 þ 2t q þ t qþ1 Þ: njk k¼1
0
(4 ) The integrated assessment is
P2 ¼
m X
~ ¼1 dðg~q ; 0Þ 4 q¼1
r X j¼1
bj
X k¼1
bjk
m X njkq ðtq1 þ 2tq þ tqþ1 Þ: njk q¼1
Note 2. By the same way as Note 1, we can have the following four properties:
~ < 100; ~jq ; 0Þ 0 < dða ~ 0 < dðg~q ; 0Þ < 100; ðjÞ P2
< 100; 0< 0 < P2 < 100 for all j = 1, 2, . . . , r; q = 1, 2, . . . , m.
B12: Manpower level
0.4
B21: Usage condition level of factory
0.7
B22: Nearing market level of delivery
0.3
B31: The index of industry production
0.4
B32: The index of industry modern times
0.6
B41: Reward obtain level
0.6
B42: Institution perform level
0.4
B51: Regulatory restrictions level
0.6
B52: Investment subsidy level
0.4
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
0 1 2 3 4 1 1 3 3 4
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
0 1 2 3 4 1 2 2 3 4
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
1 2 2 3 4 0 1 2 3 4
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
1 2 3 3 3 0 1 2 4 5
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
0 1 1 4 4 1 1 2 3 5
0.15
mj X
mj
0.6
B3: Economic 0.2
(20) For each j 2 {1, 2, . . . , r}, the integrated assessment of the main item Bj is m X
B11: Salary level
B2:Geograph 0.2
Proposition 3.2.
ðjÞ P2
Weight-1 Linguistic Numbers variables of answer (5) (4)
B5: Politics
0.15
ð22Þ ð23Þ ð24Þ ð25Þ
L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857
(A) By algorithm (I) (10) The integrated assessment of the main items B1, B2, B3, B4, B5 are: ð1Þ
ð2Þ
P1 ¼ 64:44444;
P1 ¼ 64:58333;
ð4Þ P1
ð5Þ P1
¼ 61:38889;
ð3Þ
P1 ¼ 63:88889;
¼ 66:55556;
respectively. (20) The integrated assessment is P1 = 64.21944 (B) By algorithm (II) (10) The aggregative assessment of the main item Bj with respective to the fuzzy linguistic Lq, for q = 1, 2, . . . , m; j = 1, 2, . . . , r, is as follows:
B1 B2 B3 B4 B5
L1
L2
L3
L4
L5
0.555556 0.416667 0.555556 0.833333 0.555556
3.111111 4 4.222222 4.444444 3.111111
11 9.5 9.333333 10.83333 6.333333
18.66667 19 18.66667 18.88889 22.66667
31.11111 31.66667 31.11111 26.38889 33.88889
(20) The integrated assessment of the main items B1, B2, B3, B4, B5 are ð1Þ
P2 ¼ 64:58333;
ð4Þ
P2 ¼ 66:55556;
P2 ¼ 64:44444; P2 ¼ 61:38889;
ð2Þ
ð3Þ
P2 ¼ 63:88889;
ð5Þ
respectively. (30) The aggregative assessment with respective to the fuzzy linguistic Lq (q = 1, 2, . . . , m), is
B
L1
L2
L3
L4
L5
0.569444
3.711111
9.641667
19.36667
30.93056
(40) The integrated assessment is P2 = 64.21944 Via implementing the proposed two algorithms, we have that the computing results are same.
7857
5. Conclusion As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify triangular fuzzy number by the signed distance is better than by the centroid method. However, if the membership function of the triangular fuzzy number is an isosceles triangle, then to defuzzify the triangular fuzzy number by the signed distance is equal to by the centroid method based on the maximum membership grade principle. In this study, we propose the linear order with symmetric fuzzy linguistics on evaluation to do aggregated assessment analysis, and present two algorithms based on signed distance method to treat the sampling survey assessment. According to the proposed algorithm (I) and (II) for above example, there are same results existing in two computational methods for group assessment to make the evaluation more unbiased. Since the proposed model described in this study to measure the group evaluation, the final value is more objective than just one evaluator’s assessment. Moreover, if there is only one evaluator existing, the proposed model is also appropriate to assess. References Lee, H.-M. (1996). Applying fuzzy set theory to evaluate the rate of aggregative risk in software development. Fuzzy Sets and Systems, 79, 323–336. Lin, L., & Lee, H.-M. (2008). A new assessment model for global facility site selection. International Journal of Innovative Computing Information and Control, 4(5), 1141–1150. Lin, L., & Lee, H.-M. (2009). Fuzzy assessment method on sampling survey analysis. Expert Systems with Applications, 36(3), 5955–5961. Sun, C.-M., & Wu, B. (2006). Statistical approach for fuzzy samples. In Proceedings of Cech–Japan seminar on data analysis and decision making under uncertainty (pp. 96–106). Kitakyushu, Japan, August 2006. Yao, J.-S., & Wu, K. (2000). Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems, 116, 275–288. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353. Zadeh, L. A. (1975a). The concept of a linguistic variable and its application to approximate reasoning, Part I. Information Sciences, 8, 199–249. Zadeh, L. A. (1975b). The concept of a linguistic variable and its application to approximate reasoning, Part II. Information Sciences, 8, 301–357. Zadeh, L. A. (1976). The concept of a linguistic variable and its application to approximate reasoning, Part III. Information Sciences, 9, 43–58. Zimmermann, H.-J. (1991). Fuzzy set theory and its applications. Boston/Dordrecht/ London: Kluwer Academic Publishers.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Fuzzy assessment for sampling survey defuzzification by signed distance method Lily Lin a, Huey-Ming Lee b,* a b
Department of International Business, China University of Technology, 56, Sec. 3, Hsing-Lung Road, Taipei 116, Taiwan Department of Information Management, Chinese Culture University, 55, Hwa-Kung Road, Yang-Ming-San, Taipei 11114, Taiwan
a r t i c l e
i n f o
Keywords: Sampling survey Fuzzy logic Signed distance method
a b s t r a c t Since traditional sampling survey via questionnaire is difficult in reflecting interviewee’s incomplete assessment and uncertain thought, we use fuzzy sense of sampling to express the degree of interviewee’s feelings, and find that the result is closer to interviewee’s real thought. In this study, we propose two algorithms to do aggregative assessment for sampling survey by signed distance method with the linear order character of symmetric fuzzy linguistics instead of using previous centroid method. As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify triangular fuzzy number by the signed distance is better than by the centroid method. The proposed fuzzy assessment method on sampling survey analysis is easily to assess the sampling survey and make the aggregative evaluation. Since the proposed model in this study is to measure the group evaluation, the final value is more objective than just one evaluator’s assessment. Moreover, if there is only one evaluator existing, the proposed model is also appropriate to assess. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Statistical analysis via sampling survey is a powerful market research tool to acquire the useful information. Traditionally, we compute statistics with sample data by questionnaires according to the thinking of binary logic. There usually exists two different methods, both multiple-item and single-item choices, while using linguistic variables 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 (Lin & Lee, 2008, 2009; Sun & Wu, 2006). This kind of result may lead to an unreasonable bias since the human thinking is full with fuzzy and uncertain. Fuzzy sets theory was introduced by Zadeh (1965) to deal with problem in which vagueness is present, linguistic value can be used for approximate reasoning within the framework of fuzzy set theory (Zadeh, 1975a, 1975b, 1976) to effectively handle the ambiguity involved in the data evaluation and the vague property of linguistic expression, and normal triangular fuzzy numbers are used to characterize the fuzzy values of quantitative data and linguistic terms used in approximate reasoning. With regard to fuzzy decision-making problem, Lee (1996) applied fuzzy set theory to evaluate the aggregative risk in software development under fuzzy circumstances. Lin and Lee (2008)
* Corresponding author. Tel.: +886 937 893 845; fax: +886 2 2777 4723. E-mail addresses: [email protected] (L. Lin), [email protected] (H.-M. Lee). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.04.052
presented facility site selection model using fuzzy set theory. Lin and Lee (2009) presented the fuzzy assessment on sampling survey analysis. Lin and Lee (2008, 2009) applied a value m which belongs to the closed interval [0, 1] to represent the reliability or membership grade in the fuzzy sense of marking item, and presented the method to treat the aggregated assessment of the main item Bj, and the integrated assessment. Lin and Lee (2009) presented method to treat the crisp or fuzzy multiple/single choices. As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify the triangular fuzzy number by the signed distance is better than by the centroid method. In this study, we propose a model to do assessment analysis for sampling survey with the linear order character of fuzzy linguistics by signed distance method (Yao & Wu, 2000). The proposed fuzzy assessment method on sampling survey analysis is easily to assess the sampling survey and do the aggregative evaluation. The paper is organized as follows. Section 1 is introduction. Section 2 is the preliminaries. Aggregative assessment method for sample survey with fuzzy linguistics by signed distance method is in Section 3. Section 4 is the numerical example. The conclusion of this study is given in Section 5. 2. Preliminaries For the proposed algorithm, all pertinent definitions of fuzzy sets are given below (Yao & Wu, 2000; Zadeh, 1965; Zimmermann, 1991).
L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857
Definition 2.1. Fuzzy set: If X is a collection of objects denoted e in X is a set of ordered pairs: generically by x then a fuzzy set A
e ¼ fðx; l ðxÞÞjx 2 Xg A e A
ð1Þ
leA ðxÞ is called the membership function of x in Ae which maps X to the closed interval [0, 1] that characterizes the degree of membere ship of x in A. ~ be a fuzzy set on R = (1, 1). It is Definition 2.2. Fuzzy point: Let a called a fuzzy point if its membership function is
la~ ðxÞ ¼
1 if x ¼ a 0 if x – a:
ð2Þ
a if a 6 x 6 b 0
otherwise:
ð3Þ
If a = b, we call [a, b; a] a level a fuzzy point at a.
ð4Þ
e ¼ ðq; q; qÞ. We call it the fuzzy point q ~ at q. If r = q and p = q, then A We let
Rr p
e ¼ R Cð AÞ r p
xle ðxÞdx A
ð5Þ
leA ðxÞdx
e where p and r are lower be the defuzzification by the centroid of A, and upper limits of the integral, respectively. Then, we obtain that e ¼ ðp; q; rÞ is the centroid of the normal triangular fuzzy number of A
e ¼ Cð AÞ
1 ðp þ q þ rÞ: 3
ð6Þ
e be a fuzzy set on R = (1, 1). Denote D(a) = [DL(a), DR(a)] Let D e where 0 6 a 6 1, DL(a) and DR(a) are the left and as the a-cut of D, right hand side of D(a). DL(a) and DR(a) exist and are integrable for e a 2 [0, 1]. In addition, we let F be the family of all these fuzzy sets D on R = (1, 1). e 2 F, then from the decomposition theory, D e may be repreLet D sented as
e¼ D
[ aDðaÞ ¼
06a61
[ ½DL ðaÞ; DR ðaÞ; a:
06a61
1 ½d0 ðDL ðaÞ; 0Þ þ d0 ðDR ðaÞ; 0Þ 2 1 ¼ ½DL ðaÞ þ DR ðaÞ: 2
d0 ½ðDL ðaÞ; DR ðaÞÞ; 0 ¼
ð8Þ
For each a 2 [0, 1], the crisp interval [DL(a), DR(a)] and the level a fuzzy interval [DL(a), DR(a); a] are in one to one correspondence. Therefore, we may define the signed distance from [DL(a), DR(a); a] ~ as to 0
¼
ð7Þ
As in Yao and Wu (2000), we consider the definition of the signed distance on F. Definition 2.5. The signed distance: We define d0(a, 0) = a, for a, 0 2 R. Remark 2.1. The meaning of Definition 2.5 is as the follows, if 0 < a then the distance between a and 0 is d0(a, 0) = a. If a < 0 then the distance between a and 0 is d0(a, 0) = a. Therefore, we call d0(a, 0) = a is the signed distance between a and 0.
1 ½DL ðaÞ þ DR ðaÞ: 2
ð9Þ
e 2 F, DL(a) and DR(a) exist and are integrable for Since D a 2 [0, 1], from (7), we have the following definition. ~ Let D e measured from 0: e 2 F, Definition 2.6. The signed distance of D ~ as e measured from 0 we define the signed distance of D
~ ¼1 e 0Þ dð D; 2
e ¼ ðp; q; rÞ, Definition 2.4. Normal triangular fuzzy numbers: Let A p < q < r, be a fuzzy set on R = (1, 1). It is called a normal triangular fuzzy number, if its membership function is
8 xp > < qp if p 6 x 6 q rx leA ðxÞ ¼ rq if q 6 x 6 r > : 0 otherwise:
e 2 F, from the Definition 2.5, we have that the signed disFor D tance of DL(a) and DR(a) measured from 0 are d0(DL(a), 0) = DL(a) and d0(DR(a), 0) = DR(a), respectively. Therefore, we may define the signed distance of the interval [DL(a), DR(a)], which is measured from the origin 0, by
~ ¼ d0 ð½DL ðaÞ; DR ðaÞ; 0Þ dð½DL ðaÞ; DR ðaÞ; a; 0Þ
Definition 2.3. Level a fuzzy interval: Let [a, b; a] be a fuzzy set on R = (1, 1). It is called a level a fuzzy interval, 0 6 a 6 1, a < b, if its membership function is
l½a;b;a ðxÞ ¼
7853
Z
1
½DL ðaÞ þ DR ðaÞda
ð10Þ
0
e ¼ ðp; q; rÞ, then the left endpoint and the right Remark 2.2. If C e are CL(a) = p + (q p)a and endpoint of the a-level set of C e is Cð C eÞ ¼ CR(a) = r (r q)a, respectively. The centroid of C 1 e measured from 0 ~ is ðp þ q þ rÞ, and the signed distance of C 3 e ; 0Þ ~ ¼ 1 ð2q þ p þ rÞ. dð C 4 e 2 ¼ ðp ; q ; r 2 Þ be two trie 1 ¼ ðp ; q ; r 1 Þ and A Proposition 2.1. Let A 1 1 2 2 angular fuzzy numbers, and k > 0, then, we have e1 A e 2 ¼ ðp þ p ; q þ q ; r 1 þ r 2 Þ, (10) A 1 2 1 2 e 1 ¼ ðkp ; kq ; kr 1 Þ. (20) k A 1 1 We can easily show Proposition 2.1 by extension principle. e 2 ¼ ðp ; q ; r 2 Þ be two e 1 ¼ ðp ; q ; r 1 Þ and A Proposition 2.2. Let A 1 1 2 2 triangular fuzzy numbers, and k 2 R, then we have e1 A e 2 ; 0Þ e 1 ; 0Þ e 2 ; 0Þ, ~ ¼ dð A ~ þ dð A ~ (10) dð A 0 e e ~ ~ (2 ) dðk A 1 ; 0Þ ¼ kdð A 1 ; 0Þ. e1 A e 2 ¼ ðp þ p ; q þ q ; r 1 þ r2 Þ, Proof. 10From Proposition 2.1, A 1 2 1 2 and from Definition 2.6, we have
e1 A e 2 ; 0Þ ~ ¼ 1 ½p þ p þ 2q þ 2q þ r1 þ r2 dð A 2 1 2 4 1 e 1 ; 0Þ e 2 ; 0Þ: ~ þ dð A ¼ dð A By the same way, we can prove (20) h e ¼ ðp; q; rÞ be a triangular fuzzy number. Proposition 2.3. Let A e is not an isosceles triangle, (10) If the membership function of A then, based on the maximum membership grade principle, to e by the signed distance is better than by the centroid defuzzify A method. e is (20) If the membership function of the triangular fuzzy number A e an isosceles triangle, then to defuzzify A by the signed distance is equal to by the centroid method based on the maximum membership grade principle.
7854
L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857
e is Cð AÞ e ¼ 1 ðp þ q þ rÞ, and the signed disProof. The centroid of A 3 1 e ~ tance dð A; 0Þ ¼ 4 ðp þ 2q þ rÞ. We let M ¼ 12 ðp þ rÞ be the midpoint of the closed interval [p, r]. Then, we have
e dð A; e 0Þ ~ ¼ Cð AÞ
3.1. Fuzzy linear order linguistics
1 ðp þ r 2qÞ; 12
We consider that the fuzzy linguistics L1, L2, . . . , Ln with the corresponding series of fuzzy numbers e L1 , e L2, . . . , e L n , where
e ¼ 1 ðp þ r 2qÞ; M Cð AÞ 6 e 0Þ ~ q¼ dð A;
e Lq ¼
1 ðp þ r 2qÞ: 4
Now, we consider the following three cases: e 0Þ e
3. Aggregative sample survey with linear order fuzzy linguistics by signed distance method
A
From the above three cases and by the maximum membership grade principle, Proposition 2.3 is proved. h Definition 2.7. Fuzzy relation: Let X, Y # R be universal sets, then
e ¼ fððx; yÞ; l ðx; yÞÞjðx; yÞ # X Yg R e
ð11Þ
R
is called a fuzzy relation on X Y.
ðq 1Þ
100 100 100 ;q ; ðq þ 1Þ nþ1 nþ1 nþ1
ð12Þ
~ < dðe ~ < for q = 1, 2, . . . , n. Then, we have dðe L 1 ; 0Þ L 2 ; 0Þ ~ ~ e e dð L 3 ; 0Þ < < dð L n ; 0Þ. Therefore, we can make a conclusion that these fuzzy linguistics L1, L2, . . . , Ln with the corresponding series of fuzzy numbers e L1 ; e L2 ; . . . ; e L k are linear order respective to the signed distance method. 3.2. Aggregative assessment for sampling survey by signed distance method In most cases, questionnaire of sampling survey exists many topics and questions, let us say, main items and sub-items. For instance, one specific questionnaire regarding satisfactory level may include main survey items such as satisfactory level for product, service and price, etc., also sub-items may exist under each main item. We can define them as follows: Main items: B1, B2, . . . , Br, with weight: b1, b2, . . . , br, respectively, P subject to: 0 6 bj 6 1,j = 1, 2, . . . , r and rj¼1 bj ¼ 1. Sub-items: Bj1 ; Bj2 ; . . . ; Bjmj under main items Bj, j = 1, 2, . . . , r; with weight: bj1 ; bj2 ; . . . ; bjmj , respectively subject to:
1
0 6 bjt 6 1;
j ¼ 1; 2; . . . ; r; t ¼ 1; 2; . . . ; mj
and
mj X
bjt ¼ 1:
t¼1
0
p
q
r
M
~~ d ( A, 0 )
~ C ( A)
Fig. 1. Case M > q.
Since the questionary of the sample survey is in fuzzy sense, we cannot evaluate the aggregative assessment by the traditional statistics. Let Lv (for v = 1, 2, . . . , m) be the m different linguistic variables as criteria of questionnaire, expressed in fuzzy language such as very low, low, medium, high, very high, etc. Suppose the evaluators assess the aggregative grades for some one main item Bj with sub-items Bj1 ; Bj2 ; . . . ; Bjmj of the company. Based on the assessment form proposed by Lin and Lee (2009), we propose the
Table 1 Contents of the proposed assessment form.
1
0
p
M
~ C ( A)
q
Main item
Itemweight
Subitem
Fuzzy linguistic
Numbers of answer
Triangular fuzzy number of the fuzzy linguistic
Bj
bj
Bj1
L1
nj11
L2
nj12
.. .
.. . Lm
.. . nj1m
e L 1 ¼ ð0; t1 ; t2 Þ e L 2 ¼ ðt1 ; t2 ; t3 Þ .. . e L m ¼ ðtm1 ; tm ; 100Þ
Bj2
L1
nj21
L2
nj22
.. . Lm
.. . nj2m
.. . L1
.. . njmj 1
L2
njmj 2
.. . e L 1 ¼ ð0; t1 ; t2 Þ e L 2 ¼ ðt1 ; t2 ; t3 Þ
.. . Lm
.. . njmj m
.. . e L m ¼ ðtm1 ; tm ; 100Þ
r .. . Bjmj
~~ d ( A, 0 )
Fig. 2. Case M < q.
e L 1 ¼ ð0; t1 ; t2 Þ e L 2 ¼ ðt1 ; t2 ; t3 Þ .. . e L m ¼ ðtm1 ; tm ; 100Þ
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L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857 Table 2 Contents of the weighted triangular fuzzy number with its signed distance for the main item Bj. Main item
Item-weight
Sub-item
Fuzzy linguistic
Numbers of answer
Weight
Weighted triangular fuzzy number (1)
Signed distance of (1)
Bj
bj
Bj1
L1
nj11
nj11/nj1
L2
nj12
nj12/nj1
ðnj11 =nj1 Þe L1 ðnj12 =nj1 Þe L2
1 4 ðnj11 =nj1 Þð0 þ 2t 1 þ t 2 Þ dj11 1 4 ðnj12 =nj1 Þðt 1 þ 2t 2 þ t 3 Þ dj12
.. . Lm
.. . nj1m
.. . nj1m/nj1
.. .
.. .
L1
nj21
nj21/nj2
ðnj1m =nj1 Þe Lm ðnj21 =nj2 Þe L1
1 4 ðnj1m =nj1 Þðt m1 þ 1 4 ðnj21 =nj2 Þð0 þ 2t 1
L2
nj22
nj22/nj2
1 4 ðnj22 =nj2 Þðt 1
.. . Lm
.. . nj2m
.. . nj2m/nj2
ðnj22 =nj2 Þe L2 .. . ðnj2m =nj2 Þe Lm
.. . L1
.. . njmj 1
.. . njmj 1 =njmj
ðnjmj 1 =njmj Þe L1
1 4 ðnjmj 1 =njmj Þð0
L2
njmj 2
njmj 2 =njmj
ðnjmj 2 =njmj Þe L2
1 4 ðnjmj 2 =njmj Þðt 1
.. . Lm
.. . njmj m
.. . njmj m =njmj
.. .
.. .
ðnjmj m =njmj Þe Lm
1 4 ðnjmj m =njmj Þðt m1
.. . Bj2
.. . Bjmj
contents of the jth (for j = 1, 2, . . . , r) main item with sub-items assessment form as shown in Table 1. From Table 1, we let
njq ¼
m X
njqt ;
j ¼ 1; 2; . . . ; r; q ¼ 1; 2; . . . ; mj :
ð13Þ
ðjÞ
P1 6 P1 ¼
2t m þ 100Þ dj1m þ t2 Þ dj21
þ 2t 2 þ t3 Þ dj22
.. . 1 4 ðnj2m =nj2 Þðt m1
þ 2t m þ 100Þ dj2m
.. .
mj X
k¼1 r X
þ 2t1 þ t2 Þ djmj 1 þ 2t 2 þ t3 Þ djmj 2
þ 2tm þ 100Þ djmj m
bjk 100 ¼ 100; ðjÞ
bj P 1 <
j¼1
r X
ð17Þ
bj 100 ¼ 100:
ð18Þ
j¼1
t¼1
Then, ðnjqt =njq Þe L t is the weighted triangular fuzzy number of e L t with respective to Bjqj . Based on Table 1, we propose the following two algorithms to tackle the assessing.
Defuzzified ðnjkq =njk Þe L q by the signed distance, we have
1 ðnjkq =njk Þ t q1 þ 2t q þ t qþ1 : 4
~ ¼ dððnjkq =njk Þe L q ; 0Þ
ð14Þ
We let
~ : djkq ¼ d ðnjkq =njk Þe Lq; 0
bjk
m X
djkq
ð19Þ
~jq ¼ ðe1q ; e2q ; e3q Þ, where We let a mj X
bjk
njkq tq1 ; njk
bjk
njkq tq ; njk
bjk
njkq tqþ1 njk
k¼1
ð15Þ
e2q ¼
mj X k¼1
(20) The aggregative assessment is r X
ðnjmj 2 =njmj Þe L2
! mjmj q nj1q e nj2q e e ~jq ¼ bj1 Lq: a L q bj2 L q bjmj nj1 nj2 njmj
e1q ¼
q¼1
mj m X njkq 1X ¼ bjk ðt q1 þ 2tq þ tqþ1 Þ: 4 k¼1 njk q¼1
P1 ¼
3 ðnj1m =nj1 Þe Lm 7 ðnj2m =nj2 Þe Lm 7 7 7 7: 7 7 5 ðnjmj m =njmj Þe Lm
where
(10) For each j 2 {1, 2, . . . , r}, the aggregative assessment of the main item Bj is
k¼1
ðnj12 =nj1 Þe L2 e ðnj22 =nj2 Þ L 2
ej; ~j1 ; a ~j2 ; . . . ; a ~jm Þ ¼ ðbj1 ; bj2 ; . . . ; bjmj Þ R ða
Proposition 3.1.
mj X
ðnj11 =nj1 Þe L1 6 6 ðn =n Þe 6 j21 j2 L 1 ej ¼ 6 R 6 6 6 4 ðnjmj 1 =njmj Þe L1
(a) Then, the first stage aggregative composition inference for the attribute Bj is as follows:
Then, we may re-modify Table 1 as Table 2. From Table 2, we have the following Proposition 3.1.
ðjÞ
Let L = {L1, L2, . . . , Ln} be the set of the n fuzzy linguistics. From e j on Bj L with the weighted Table 2, we can form a fuzzy relation R triangular fuzzy numbers as follows:
2
3.3. Algorithm I
P1 ¼
3.4. Algorithm II
e3q ¼
mj X k¼1
ðjÞ
bj P 1
for j = 1, 2, . . . , r; q = 1, 2, . . . , m. ~jq by signed distance, we have Defuzzified a
j¼1 mj r m X X njkq 1X ¼ bj bjk tq1 þ 2tq þ tqþ1 : 4 j¼1 k¼1 n jk q¼1 n
ð16Þ
Note 1. Since 0 6 njkq < 1, for all j, k, q, and 0 < tq1 < tq < jk tq+1 < 100, for all q, therefore, we have the following properties:
~ ¼ 1 ðe1q þ 2e2q þ e3q Þ ~jq ; 0Þ dða 4 mj njkq 1X bjk tq1 þ 2tq þ t qþ1 : ¼ 4 k¼1 njk
ð20Þ
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L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857
(b) Let B = {B1,B2, . . . , Br}, then the second stage aggregative composition inference is as follows:
2
~11 a 6a ~ 6 21 ðg~1 ; g~2 ; . . . ; g~m Þ ¼ ðb1 ; b2 ; . . . ; br Þ 6 4 ~r1 a
~12 a ~22 a ~r2 a
3 ~1m a ~2m 7 a 7 7: 5 ~rm a
where
g~q ¼
r X
~jq ¼ ðv 1q ; v 2q ; v 3q Þ; bj a
v 2q ¼ v 3q ¼
r X
bj
mj X
j¼1
k¼1
r X
mj X
bj
j¼1
k¼1
r X
mj X
j¼1
bjk
njkq t q1 ; njk
bjk
njkq tq ; njk
We use the contents of main items and sub-items shown in Lin and Lee (2008) as an example to implement the proposed algorithms. Example. Assume that we have the following main items, weights, sub-items and the answer numbers as shown in Table 3. By algorithm (I) and (II), we can have the following computing results:
j¼1
v 1q ¼
4. Example implementation
Table 3 Contents of the example.
njkq bj bjk t qþ1 njk k¼1
Main item
Weight-2 Sub-item
B1: Labor
0.3
for q = 1, 2, . . . , m. ~q by signed distance, we have Defuzzified g
~ ¼ 1 ðv 1q þ 2v 2q þ v 3q Þ dðg~q ; 0Þ 4 mj r X njkq 1X ¼ bj bjk ðtq1 þ 2t q þ t qþ1 Þ 4 j¼1 k¼1 njk
ð21Þ
for q = 1, 2, . . . , m. From (20) and (21), we have the following proposition:
(10) For each j 2 {1, 2, . . . , r}, the aggregative assessment of the main item Bj with respective to the fuzzy linguistic Lq (q = 1, 2, . . . , m), is
~ ¼ 1 ðe1q þ 2e2q þ e3q Þ ~jq ; 0Þ dða 4 mj njkq 1X bjk ðtq1 þ 2tq þ t qþ1 Þ ¼ 4 k¼1 njk
~ ¼1 ~jq ; 0Þ ¼ dða 4 q¼1
mj X
m X njkq bjk ðt q1 þ 2t q þ tqþ1 Þ njk q¼1 k¼1
(30) The aggregative assessment with respective to the fuzzy linguistic Lq (q = 1, 2, . . . , m), is
~ ¼1 dðg~q ; 0Þ 4
r X j¼1
B4: Reward
njkq bj bjk ðt q1 þ 2t q þ t qþ1 Þ: njk k¼1
0
(4 ) The integrated assessment is
P2 ¼
m X
~ ¼1 dðg~q ; 0Þ 4 q¼1
r X j¼1
bj
X k¼1
bjk
m X njkq ðtq1 þ 2tq þ tqþ1 Þ: njk q¼1
Note 2. By the same way as Note 1, we can have the following four properties:
~ < 100; ~jq ; 0Þ 0 < dða ~ 0 < dðg~q ; 0Þ < 100; ðjÞ P2
< 100; 0< 0 < P2 < 100 for all j = 1, 2, . . . , r; q = 1, 2, . . . , m.
B12: Manpower level
0.4
B21: Usage condition level of factory
0.7
B22: Nearing market level of delivery
0.3
B31: The index of industry production
0.4
B32: The index of industry modern times
0.6
B41: Reward obtain level
0.6
B42: Institution perform level
0.4
B51: Regulatory restrictions level
0.6
B52: Investment subsidy level
0.4
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
0 1 2 3 4 1 1 3 3 4
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
0 1 2 3 4 1 2 2 3 4
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
1 2 2 3 4 0 1 2 3 4
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
1 2 3 3 3 0 1 2 4 5
L1 L2 L3 L4 L5 L1 L2 L3 L4 L5
0 1 1 4 4 1 1 2 3 5
0.15
mj X
mj
0.6
B3: Economic 0.2
(20) For each j 2 {1, 2, . . . , r}, the integrated assessment of the main item Bj is m X
B11: Salary level
B2:Geograph 0.2
Proposition 3.2.
ðjÞ P2
Weight-1 Linguistic Numbers variables of answer (5) (4)
B5: Politics
0.15
ð22Þ ð23Þ ð24Þ ð25Þ
L. Lin, H.-M. Lee / Expert Systems with Applications 37 (2010) 7852–7857
(A) By algorithm (I) (10) The integrated assessment of the main items B1, B2, B3, B4, B5 are: ð1Þ
ð2Þ
P1 ¼ 64:44444;
P1 ¼ 64:58333;
ð4Þ P1
ð5Þ P1
¼ 61:38889;
ð3Þ
P1 ¼ 63:88889;
¼ 66:55556;
respectively. (20) The integrated assessment is P1 = 64.21944 (B) By algorithm (II) (10) The aggregative assessment of the main item Bj with respective to the fuzzy linguistic Lq, for q = 1, 2, . . . , m; j = 1, 2, . . . , r, is as follows:
B1 B2 B3 B4 B5
L1
L2
L3
L4
L5
0.555556 0.416667 0.555556 0.833333 0.555556
3.111111 4 4.222222 4.444444 3.111111
11 9.5 9.333333 10.83333 6.333333
18.66667 19 18.66667 18.88889 22.66667
31.11111 31.66667 31.11111 26.38889 33.88889
(20) The integrated assessment of the main items B1, B2, B3, B4, B5 are ð1Þ
P2 ¼ 64:58333;
ð4Þ
P2 ¼ 66:55556;
P2 ¼ 64:44444; P2 ¼ 61:38889;
ð2Þ
ð3Þ
P2 ¼ 63:88889;
ð5Þ
respectively. (30) The aggregative assessment with respective to the fuzzy linguistic Lq (q = 1, 2, . . . , m), is
B
L1
L2
L3
L4
L5
0.569444
3.711111
9.641667
19.36667
30.93056
(40) The integrated assessment is P2 = 64.21944 Via implementing the proposed two algorithms, we have that the computing results are same.
7857
5. Conclusion As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify triangular fuzzy number by the signed distance is better than by the centroid method. However, if the membership function of the triangular fuzzy number is an isosceles triangle, then to defuzzify the triangular fuzzy number by the signed distance is equal to by the centroid method based on the maximum membership grade principle. In this study, we propose the linear order with symmetric fuzzy linguistics on evaluation to do aggregated assessment analysis, and present two algorithms based on signed distance method to treat the sampling survey assessment. According to the proposed algorithm (I) and (II) for above example, there are same results existing in two computational methods for group assessment to make the evaluation more unbiased. Since the proposed model described in this study to measure the group evaluation, the final value is more objective than just one evaluator’s assessment. Moreover, if there is only one evaluator existing, the proposed model is also appropriate to assess. References Lee, H.-M. (1996). Applying fuzzy set theory to evaluate the rate of aggregative risk in software development. Fuzzy Sets and Systems, 79, 323–336. Lin, L., & Lee, H.-M. (2008). A new assessment model for global facility site selection. International Journal of Innovative Computing Information and Control, 4(5), 1141–1150. Lin, L., & Lee, H.-M. (2009). Fuzzy assessment method on sampling survey analysis. Expert Systems with Applications, 36(3), 5955–5961. Sun, C.-M., & Wu, B. (2006). Statistical approach for fuzzy samples. In Proceedings of Cech–Japan seminar on data analysis and decision making under uncertainty (pp. 96–106). Kitakyushu, Japan, August 2006. Yao, J.-S., & Wu, K. (2000). Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems, 116, 275–288. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353. Zadeh, L. A. (1975a). The concept of a linguistic variable and its application to approximate reasoning, Part I. Information Sciences, 8, 199–249. Zadeh, L. A. (1975b). The concept of a linguistic variable and its application to approximate reasoning, Part II. Information Sciences, 8, 301–357. Zadeh, L. A. (1976). The concept of a linguistic variable and its application to approximate reasoning, Part III. Information Sciences, 9, 43–58. Zimmermann, H.-J. (1991). Fuzzy set theory and its applications. Boston/Dordrecht/ London: Kluwer Academic Publishers.